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--- Page 1 --- Robert F. Pierret EEF SL300K Semiconductor 1.0e16 9100元 5.0e16 Device Fundamentals A with Computer-Based d Exercises and Homework kProblems VG

--- Page 2 --- Electrical Engineering/Semiconduetor Devices Semiconductor Device Fundamentals Robert F Pierret Although roughly a half-century old, the field of study associated with semiconductor devices continues to be dynamic and exciting. New and improved devices are being developed at an almost frantic pace. While the number of devices in complex integrated circuits increases through the millions and the side-length of chips is measured in centimeters,the individual devices are literally being shrunk to atomic dimensions. Moreover, semiconductor properties desired for a given device structure but not available in nature are being produced artificially; in essence,the semiconductor properties themselves are now being engineered to fit device specifica- tions. This book should be viewed as a gateway to what the reader hopefully will agree is the fascinatingrealm of semiconductor devices. Highlights •Introduces and explains the basic terminology,models,properties,and concepts associated with semiconductors and semiconductor devices. Providesdetailedinsightintotheinternal workingsofthe“building-block”device structures such as the pn junction diode, Schottky diode, BJT,and MOSFET. Presents information about a wide variety of additional devices,including solar cells, LEDs, HBTs,and modern field effect devices. • Systematically develops the analyticaltools needed to solve practicaldevice problems.- Special Features •The computer-based exercises and homework problems permit the replacement of plug- and-chug boredom with more challenging and realistic problems. •The supplement-and-review mini-chapters contain an extensive review listofterms,test-like review problem sets with answers,and detailed suggestionson alternative/supplementalreadings. Providingachangeof pace withaminimumof analysis,thestrategicallyplaced read-only chapters are intendedto be fun-reading and informative. •Other special features include tables that summarize the required reading. difficulty level. and suggested w weightingofend-of-chapter problems.“crib-sheet”-likeequation summaries in the critical beginning chapters,and direct reproductions of device data and computer-generated plots. Robert F. Pierret is a Professor of Electrical and Computer Engineering at Purdue University in W.Lafayette, Indiana. He has been a member of the Purdue faculty since 1970 and administers the undergraduate Semiconductor Measurements Laboratory. Promoting an innovative total-quality approach to curriculum devel- opment,he has served in recent years as the Chairman of the ECE Curriculum Committee. Robert F. Pierret conceived and co-edits the popular Modular Series on Solid State Devices published by Addison-Wesley. He is author of four volumes 90000 in the series.He is also a consulting editor for Addison-Wesley and on the Editorial Advisory Board of the journal of Solid-State Electronic. “780201”543933 Addison-Wesley Publishing Company T-E6ES-T02-0 NS1

--- Page 3 --- SEMICONDUCTOR DEVICE FUNDAMENTALS RobertF.Pierret Schoolof Electrical andComputer Engineering Purdue University E1118775 Addison Wesley Longman Reading, Massachusetts·Menlo Park,California·New York Don Mills,Ontario·Wokingham,England·Amsterdam·Bonn

--- Page 4 --- Katherine Harutunian Associate Editor Helen Wythe Senior Production Supervisor Hugh Crawford Manufacturing Supervisor Barbara Atkinson Associate Cover Design Supervisor Peter Blaiwas Cover design Kenneth J. Wilson Text design Joyce Grandy Copyeditor Sandra Rigney Production Packaging Services G&STypesetters,Inc. Composition Publishers’ Design and Production Services,Inc. Illustrations LibraryofCongress Cataloging-in-Publication Data Pierret, Robert F. Semiconductor device fundamentals/Robert F.Pierret. p. cm. Includes index. ISBN0-201-54393-1

1. Semiconductors. I. Title. TK7871.85.P484 1996 621.3815′2-dc20 95-17387 CIP Manyofthedesignationsusedbymanufacturersandsellerstodistinguishtheirproductsareclaimed as trademarks.Wherethose designations appear inthisbook,and Addison-Wesley was awareofa trademark claim,the designations have been printed in initial capsor allcaps. MATLAB is a registeredtrademarkof The MathWorks, Inc, 24PrimeParkWay,Natick,MA01760-1500. Phone:(508)653-1415.Fax:(508)653-2997 E-mail:info@mathworks.com www:http://www.mathworks.com AccessthelatestinformationaboutAddison-WesleybooksfromourInternetgophersiteorourWorld Wide Web page: gopher aw.com http://www.aw.com /cseng/ Copyright: ©1996 by Addison-Wesley Publishing Company, Inc. Reprinted with corrections March,1996 All rightsreserved.Nopartofthispublicationmaybereproduced,stored inaretrievalsystem,or transmittedianyormorbyanymeans,lectronic,mechanical,photocopying,recording,orother- wise,without the written permission ofthe publisher. Printed in the United Statesof America 18 0807

--- Page 5 --- “The littlevoiceinsidenevergrowsanyldr.” FrankPierret (1906-1994)

--- Page 6 --- PREFACE Whyanothertextonsolidstate devices?Theauthor is awareofatleast14undergraduate texts publishedon thesubject during the past decade.Althoughseveral motivating factors couldbecited,a verysignificantfactor wasthe desireto write a bookforthe nextmillen- nium (a Book2000so to speak)that successfully incorporates computer-assisted learning. In a recent survey, members of the Undergraduate Curriculum Committee in the Schoolof ElectricalandComputer Engineering atPurdueUniversity listed integrationofthecom- puterintothelearning process as the numberone priority.Nationally,universityconsor- tiums have beenformed which emphasize computer-assisted learning.In January1992, distributionbeganoftheStudentEditionofMATLAB,essentiallya copyoftheoriginal MATLABmanualbundled with a low-cost versionofthe math-tools software.Over37,000 copiesofthebook/software weresoldinthefirst year! Textsandbookson a varietyof topics from several publishers are now availablethatmake specificuseofthe MATLAB software. The directionis clearas we proceedintothe secondmillennium:Computer- assisted learning willbecome more andmore prevalent.In dealing withsolid state devices, the computer allows one toaddress morerealisticproblems,to more readily experiment with“what-ifscenarios,and toconvenientlyobtain a graphical output.Anentire device characteristiccanoftenbecomputergenerated withless timeandeffortthan a smallsetof manuallycalculated single-point values. It shouldbeclarifiedthatthepresenttextisnotatotallynewentryinthefieldbutis derivedinpartfromVolumesI-IVoftheAddison-WesleyModular SeriesonSolidState Devices.Lest there be a misunderstanding,the latest versionsof the volumes inthe Modu- lar Series were not simply glued together. Tothecontrary,morethanhalfofthematerial coverage in the four volumes was completely rewritten.Moreover,several supplemental sectionsand twoadditional chapters were added to the VolumesI-IVoutline.The newtext also contains computer-based text exercises and end-of-chapter problems,plus a number ofother special features thatare fully described in the General Introduction. In justaboutany engineering endeavor there are tradeoffs.Device design isreplete withtradeoffs.Tradeoffsalsoenter intothe designofa book.Forexample,a few topics can becoveredindetail(depth)orlessercoveragecan begiventoseveraltopics(breadth). Similarlyone canemphasize the understanding ofconceptsoroptimize thetransmissionof factualinformation.VolumesI-IVintheModularSeriesareknownfortheirpedantic four”read-only”chapters have been specifically addedherein to broadenthecoverage and enhance thetransmissionoffactual information.Intheread-onlychapterstheemphasis is more on describing the exciting worldof modern-day devices.Compound semiconductor deviceslikewisereceive increased coverage throughoutthetext. There is alsoanatural V

--- Page 7 --- vi SEMICONDUCTORDEVICEFUNDAMENTALS tradeoffbetweentheeffortdevotedtodevelopingqualitative insightandtheimplementa- tionofaquantitativeanalysis.Carefulattentionhasbeengiventoavoidslighting thede- velopmentof“intuition”inlightofthegreatlyenhancedquantitativecapabilitiesarising fromtheintegrateduseofthecomputer.Lastly,we havenotattemptedto beallinclusive in the depthandbreadthofcoverage—many things are left for later (anothercourse,other books).Hopefully,the proper tradeoffs have beenachieved whereby the reader is reason- ablyknowledgeableaboutthesubjectmatterandacceptablyequipped toperformdevice analyses after completing the text. The present text is intended forundergraduate juniors or seniors who have had at least an introductoryexposureto electricfield theory.Chapters are groupedintothreemajor divisionsor”parts,withPartIIbeingfurthersubdividedintoIIAandIIB.Withsome deletions,thematerial in eachof thethree parts is covered during a five-week segmentof a one-semester,three-credit-hourjunior-seniorcourse inElectricalandComputer Engi- neering at PurdueUniversity.Aday-by-daycourse outline is suppliedontheInstructor’s Disk accompanying the SolutionsManual.If necessary to meet timeconstraints,read-only Chapters4,9,13,and19couldbedeletedfromthelectureschedule.（An instructormight preferablyassign thechapters as independentreadings and rewardcompliantstudents by including extra-creditexaminationquestionscoveringthe material.)StandardChapters12, 14,and15,exceptforthegeneralfield-effectintroductioninSection15.1,mayalsobe omitted with littleor noloss incontinuity. Although acomplete listing of specialfeatures is given inthe GeneralIntroduction, instructors shouldtakespecialnoteof theProblemInformation Tablesinserted priortothe end-of-chapterproblems. Thesetablesshouldproveusefulinassigning problemsandin dealing withhomeworkgraders.Whenfacedwithconstructingatestinstructorsmayalso beinterested inexamining theReviewProblem Setsfound inthemini-chapters identified bya darkened thumb tab)at theendof the three book parts.The ReviewProblem Sets are derived fromold“open-book”and“closed-book”tests.Concerning the computer-based exercises and problems,theuseofeither the studentor professional versionof MATLABis recommendedbutnotrequired.The in-textexercise solutionsandtheproblemanswers suppliedtotheinstructor,however,domakeuseofMATLAB.Althoughitwouldbehelfu, the userneed not befamiliar withthe MATLAB program atthe beginning of thebook.The MATLAB problems in successivechapters make increasingly sophisticated use of thepro- gram. Inother words,the early exercises andhomework problems provide a learning MATLABbyusingMATLABexperience.It is critical,however,thattheusercompletea large percentage of the computer-based exercises and problems in the first three chapters. Theexercises and problems found in later chapters notonlyassume areasonably competent use of MATLAB,but also build upon the programs developed in the earlier chapters. Theauthor gratefullyacknowledges theassistanceofassociates,EE305students, the respondents to an early marketing survey, the manuscript reviewers,andAddison-Wesley personnel inmaking Book200areality.Deserving of special thanks is AliKeshavarzi for arranging theauthor’s sabbaticalatIntelCorporationandforprovidingphotographsof equipmentinsidethe Albuquerquefabricationfacility.Prof.MarkLundstrom at Purdue Universitywasalsomosthelpfulin supplyingkeyinformationandfiguresforseveralbook sections.Oftheundergraduatestudentsaskedtoexaminethemanuscriptforreadability

--- Page 8 --- PREFACE vii anderrors,Eric Bragg standsoutas especiallyperceptiveandhelpful.The very conscien- tiousmanuscriptreviewers wereProf.KennethA.James,California StateUniversity,Long Beach;Prof.Peter Lanyon,Worcester PolytechnicInstitute;Prof.Gary S.May,Georgia Instituteof Technology;Prof.Dieter K.Schroder,Arizona State University;and Prof.G.W. Stillman,UniversityofIllinoisatUrbana-Champaign.Inrecognitionofafruitfulassocia- tion,a specialthanksto Don Fowley,theformereditorat Addison-Wesley whoenticedthe author into writing thebook.Last but not least,editorKatherine Harutunian is to becred- ited withsmoothlyimplementing theprojectand executiveassistantAnita Devine with cheerfully handling many oftheearly details. Prof.Robert F. Pierret School of Electrical and Computer Engineering Purdue University

--- Page 9 --- GENERALINTRODUCTION Coincidentwiththe writing ofthisbook,therehasbeenconsiderable media discussion about the”Information Superhighway.”Theenvisionedhighway itself,thephysicallink betweenpointssupportintheinformationtrafficisfiberopticcable.Relativetothetopic ofthis book,theonandoff ramps,which insertand extract the informationfrom the high- way, are semiconductor(solidstate)devices.Trafficcontrol,theinformation processing and theconversion to and from the human interface,is performed bycomputers.Thecen- tral processing unit (CPU),memory,and othermajor components inside the computerare again semiconductor devices.In the modern world,semiconductor devices are incorporated in just about every major system from automobiles to washing machines. Althoughroughlyahalf-centuryold,thefieldofstudyassociated withsemiconductor devices continues to be dynamic and exciting.New and improved devices are being devel- oped at an almostmaddening pace. Whilethe device count in complex integrated circuits increases through themillions and the side-lengthof chips is measured in centimeters,the individual devices are literally being shrunk to atomic dimensions.Moreover, semiconduc- tor propertiesdesiredfor a given device structurebutnotavailable innatureare being producedartificially;inessence,the semiconductorpropertiesthemselvesare nowbeing engineered to fit device specifications. Thisbook should be viewed as a gateway to what thereader will hopefully agree is the fascinating realmof semiconductor devices.It was written for junior-or senior-level stu- dents whohaveatleastan introductory exposureto electricfieldtheory. Thecoverage includesarepresentativesamplingofinformationabouta wide varietyofdevices.Primary emphasis,however,is placed on developing a fundamental understanding of the internal workingsof the more basic device structures.As detailedbelow,this bookcontains a num- berof uniquefeaturestoassistthereaderin learningthematerial.Alertedatanearlystage to their existence,the readercan planto take fulladvantageofthecited features. Computer-Based ExercisesandEnd-of Chapter(Homework）Problems. Themajority of chapters contain one or more MATLAB-based exercises requiring the use of a com- puter.MATLAB is amath-tools software program that has been adapted to run on most computerplatforms.Alow-cost studenteditionofMATLAB,whichcan beused torun allofthefilesassociatedwiththisbook,isavailableinbothIBM-compatibleandMac- intosh versions. The MATLAB program scripts yielding exercise answers are listed in the text and are available in electronic form as detailed below.Computer-based problems, identified byabullet)before theproblemnumber,make upapproximately2%ofthe xxi

--- Page 10 --- xxii SEMICONDUCTORDEVICEFUNDAMENTALS problem total. Althoughother math-tools programs could be employed,the useof MATLAB is recommended in answering computer-based problems. Because computer- based exercisesandproblemsintheearlychaptersarespecificallydesignedtoprogrs- sively enhance MATLAB-use proficiency,theuser need notbefamiliarwith the MATLAB programatthe beginningofthebook.Itis very important,however,tocomplete a large percentageof thecomputer-basedexercisesandproblems in the firstthreechapters.The exercisesand problemsfound inlaterchapters notonlyassume areasonablycompetent use of MATLAB,butalso builduponthe programsdeveloped in theearlierchapters. ComputerProgram Files.ProgramfilesoftheMATLAB scriptsassociated withcom- puter-basedexercises are available via the Internetftp.mathworks.com in the di- rectory pub/books/pierret)or on a floppy disk distributedfreeof charge by MathWorks,Inc.A pull-out card is provided herein forobtaining thefree program disk which is formatted for use witheitheranIBM-compatibleor Macintoshcomputer.Each floppy diskcontainstwosetsof“m-files”to beusedrespectivelywiththepre-.0stu- dent1stedition)or post-4.0(student2ndedition）versionsofMATLAB.The listings in thetext are specifically derived fromthe Macintosh post-4.0 version, but theyareiden- tical tothecorrespondingIBM-compatible versionexcept for theoccasional apearance of a Greek letter. Supplementand Review Mini-Chapters. The book is divided into three parts.At theend of each part is a Supplementand Reviewmini-chapter.The mini-chapters,identified by a darkenedthumbtab,containanalternative/supplemental reading listand information table,reference citations forthepreceding chapters,anextensive review-listof terms, andreview problem sets with answers. Thereviewproblem sets are derivedfrom “closed-book”and“open-book”examinations. Read-OnlyChapters. Chapters 4,9,13,and19have beenclassified as”read-oly Chapterswiththeread-onlydesignationcontainmostlyqualitativeinformationofasup- plemental nature. Two of the chapters survey someof the latest device structures.In- tendedtobe fun-reading,theread-onlychaptersare strategicallyplacedtoprovide a change of pace. The chapters containonly a smallnumberofequations,no exercises, andfew,if any,end-of-chapterproblems. Ina courseformat,the chapters couldbe skipped withlittle loss in continuityorpreferablyassignedasindependentreadings. Problem Information Tables.Acompacttable containing informationaboutthe end-of- chapter problems in a given chapter is inserted just before the problems.The information providedis (i)thetext sectionor subsectionafter whichtheproblemcanbecompleted (ii)theestimated problemdifficultyonascaleof1(easyorstraightforward)to（very difficultorextremelytimeconsuming),(iii)suggestedcreditorpointweighting,and (iv) a short problem description.A bullet before the problem number identifies a com- puter-based problem.An asterisk indicates computer usage for part of the problem. Equation Summaries. The verybasic carriermodeling equations in Chapter2 andthe carrieractionequations inChapter3,equationsreferenced throughoutthetext,areor- ganized andrepeated inTables2.4and3.3,respectively.These tables wouldbeidealas “crib sheets”forclosed-bookexaminations covering thematerial in PartI of thetext.

--- Page 11 --- GENERAL INTRODUCTION xxii Measurements and Data.Contraryto the impression sometimes left by the sketches and idealized plotsoftenfoundin introductory texts,devicecharacteristicsarereal,seldom perfect,and areroutinely recorded in measurementlaboratories.Herein a samplingof measurement detailsand results,derivedfrom an undergraduate EElaboratory adminis- tered bytheauthor,is included in anattempttoconvey theproper sense ofreality.For added details on the describedmeasurements,and for a descriptionofadditionalmea- surements,the reader is referred to R.F.Pierret, Semiconductor Measurements Labora- tory Operations Manual,Supplement A in the Modular Series on Solid State Devices, Addison-Wesley Publishing Company, Reading MA,© 1991. Alternative Treatment.Section2.1providestheminimumrequiredtreatmentonthe topic ofenergy quantization in atomic systems.Appendix A,which contains a more in- depthintroductiontothequantizationconceptandrlatedtopicshabeenincledo those desiring supplemental information.Section2.1may betotally replaced by Appen- dix A with no loss in continuity.

--- Page 12 --- CONTENTS General Introduction xxi PartISemiconductor Fundamentals 1 Chapter1 Semiconductors: A General Introduction 3 1.1 General Material Properties 3 1.1.1 Composition 3 1.1.2 Purity 5 1.1.3Structure 6 1.2 Crystal Structure 6 1.2.1The Unit Cell Concept 7 1.2.2 Simple 3-D Unit Cels 8 1.2.3 Semiconductor Lattices 6 1.2.4 Miller Indies 12 1.3 Crystal Growth 16 1.3.1 Obtaining Ultrapure Si 16 1.3.2Single-Crystal Formation 17 1.4 Summary 19 Problems 19 Chapter 2 Carrier Modeling 23 2.1 The Quantization Concept 23 2.2 Semiconductor Models 25 2.2.1Bonding Model 26 2.2.2Energy Band Model 26 2.2.3Carriers 29 2.2.4 Band Gap and Material Classification 31 2.3 Carrier Properties 32 2.3.1 Charge 32 2.3.2 Effective Mass 32 2.3.3Carrier Numbers in Intrinsic Material 34 ix

--- Page 13 --- X SEMICONDUCTOR DEVICEFUNDAMENTALS 2.3.4Manipulation of Carrier Numbers—Doping 35 2.3.5Carrier-Related Terminology 40 2.4 State andCarrier Distributions 40 2.4.1 Density of States 41 2.4.2 The Fermi Function 42 2.4.3 Equilibrium Distribution of Carriers 46 2.5 Equilibrium Carrier Concentrations 49 2.5.1Formulas for n and p 49 2.5.2 Alternative Expressions forn and p 52 2.5.3 n; and the np Product 53 2.5.4 Charge Neutrality Relationship 57 2.5.5 Carrier Concentration Calculations 59 2.5.6 Determination ofE 61 2.5.7 Carrier Concentration Temperature Dependence 65 2.6 Summary andConcluding Comments 67 Problems 69 Chapter 3 Carrier Action 75 3.1 Drift 75 3.1.1 Definition-Visualization 75 3.1.2 Drift Current 76 3.1.3 Mobility 79 3.1.4 Resistivity 85 3.1.5 Band Bending 89 3.2 Diffusion 94 3.2.1 Definition-Visualization 94 3.2.2 Hot-Point Probe Measurement 97 3.2.3 Diffusionand TotalCurrents 98 Diffusion Currents 98 Total Currents 99 3.2.4 Relating Diffusion Coeffi cients/Mobilities 99 Constancy of the Fermi Level 99 Current Flow Under Equilibrium Conditions 101 Einstein Relationship 101 3.3 Recombination-Generation 105 3.3.1Definition-Visualization 105 Band-to-BandRecombination 105

--- Page 14 --- CONTENTS xi R-G Center Recombination 105 Auger Recombination 107 Generation Processes 107 3.3.2 Momentum Considerations 107 3.3.3 R-G Statistics 110 Photogeneration 110 Indirect Thermal Recombination-Generation 112 3.3.4 Minority Carrier Lifetimes 116 General Information 116 A Lifetime Measurement 116 3.4 Equations of State 120 3.4.1 Continuity Equations 121 3.4.2 Minority Carrier Diffusion Equations 122 3.4.3 Simplificationsand Solutions 124 3.4.4 Problem Solving 124 Sample Problem No.1 124 Sample Problem No.2 128 3.5 Supplemental Concepts 131 3.5.1 Diffusion Lengths 131 3.5.2 Quasi-Fermi Levels 132 3.6 Summary and Concluding Comments 136 Problems 138 Chapter4 Basics of Device Fabrication 149 4.1 Fabrication Processes 149 4.1.1 Oxidation 149 4.1.2 Diffusion 152 4.1.3 Ion Implantation 155 4.1.4 Lithography 159 4.1.5 Thin-Film Deposition 162 Evaporation 162 Sputtering 162 Chemical Vapor Deposition(CVD) 164 4.1.6 Epitaxy 164 4.2 Device Fabrication Examples 165 4.2.1pn Junction Diode Fabrication 166 4.2.2 Computer CPU Process Flow 166 4.3 Summary 174

--- Page 15 --- xii SEMICONDUCTOR DEVICEFUNDAMENTALS R1 PartISupplement andReview 175 Alternative/Supplemental Reading List 175 Figure Sources/Cited References 177 Review List of Terms 178 Part I—ReviewProblem Sets andAnswers 179 Part lIA pn Junction Diodes 193 Chapter 5 pn Junction Electrostatics 195 5.1 Preliminaries 195 5.1.1Junction Terminology/IdealizedProfiles 195 5.1.2 Poisson’s Equation 197 5.1.3 Qualitative Solution 198 5.1.4 The Built-in Potential（Vbi） 203 5.1.5 The Depletion Approximation 206 5.2 Quantitative Electrostatic Relationships 209 5.2.1 Assumptions/Definitions 209 5.2.2 Step Junction with VA=0 210 Solution for ρ 210 Solution for 210 Solution forV 212 Solution for x and xp 213 5.2.3 Step Junction with VA≠0 215 5.2.4 Examination/Extrapolation of Results 219 5.2.5 Linearly Graded Junctions 223 5.3 Summary 226 Problems 227 Chapter6 pn Junction Diode: I-V Characteristics 235 6.1 The Ideal Diode Equation 235 6.1.1 Qualitative Derivation 235 6.1.2 Quantitative Solution Strategy 241 General Considerations 241 Quasineutral Region Considerations 242 Depletion Region Considerations 243 Boundary Conditions 244

--- Page 16 --- CONTENTS xiii “Game Plan” Summary 246 6.1.3 Derivation Proper 247 6.1.4 Examination of Results 249 Ideal I-V 249 The Saturation Current 250 Carrier Currents 254 Carrier Concentrations 255 6.2Deviations from the Ideal 260 6.2.1Ideal Theory Versus Experiment 260 6.2.2 Reverse-Bias Breakdown 263 Avalanching 264 Zener Process 268 6.2.3 The R-GCurrent 270 6.2.4 VA→bi High-Current Phenomena 277 Series Resistance 278 High-Level Injection 279 6.3 Special Considerations 281 6.3.1Charge Control Approach 282 6.3.2 Narrow-Base Diode 284 Current Derivation 284 Limiting Cases/Punch-Through 286 6.4 Summary and Concluding Comments 288 Problems 289 Chapter7 pn JunctionDiode: Small-SignalAdmittance 301 7.1 Introduction 301 7.2 Reverse-Bias Junction Capacitance 301 7.2.1General Information 301 7.2.2 2 C-V Relationships 305 7.2.3 Parameter Extraction/Profiling 309 7.2.4 Reverse-Bias Conductance 313 7.3 Forward-Bias Diffusion Admittance 315 7.3.1 General Information 315 7.3.2 Admittance Relationships 318 7.4 Summary 323 Problems 324

--- Page 17 --- xiv SEMICONDUCTOR DEVICEFUNDAMENTALS Chapter 8 pn Junction Diode: Transient Response 327 8.1Turn-Off Transient 327 8.1.1 Introduction 327 8.1.2 Qualitative Analysis 329 8.1.3 The Storage Delay Time 333 Quantitative Analysis 333 Measurement 334 8.1.4 General Information 338 8.2 Turn-On Transient 338 8.3 Summary 343 Problems 344 Chapter 9 Optoelectronic Diodes 347 9.1 Introduction 347 9.2 Photodiodes 349 9.2.1pn Junction Photodiodes 349 9.2.2p-i-n and Avalanche Photodiodes 352 p-i-n Photodiodes 352 Avalanche Photodiodes 355 9.3 Solar Cells 356 9.3.1Solar Cell Basics 356 9.3.2 Efficiency Considerations 357 9.3.3 Solar CellTechnology 360 9.4 LEDs 361 9.4.1General Overview 361 9.4.2Commercial LEDs 362 9.4.3LEDPackaging and Photon Extraction 366 PartlIB BJTsandOther JunctionDevices 369 Chapter 10 BJTFundamentals 371 10.1 Terminology 371 10.2 Fabrication 374 10.3 Electrostatics 378 10.4 Introductory Operational Considerations 380 10.5 Performance Parameters 382 Emitter Efficiency 382

--- Page 18 --- CONTENTS Xv Base Transport Factor 383 Common Base d.c.Current Gain 383 Common Emitter d.c.Current Gain 384 10.6 Summary 385 Problems 385 Chapter 11 BJT StaticCharacteristics 389 11.1Ideal Transistor Analysis 389 11.1.1 Solution Strategy 389 Basic Assumptions 389 Notation 390 Diffusion Equations/Boundary Conditions 390 Computational Relationships 392 11.1.2 General Solution (W Arbitrary) 393 Emitter/Collector Region Solutions 393 Base Region Solution 394 Performance Parameters/Terminal Currents 395 11.1.3 Simplified Relationships (W≤LB) 397 △B(x)in the Base 398 Performance Parameters 398 11.1.4Ebers-Moll Equations and Model 403 11.2 Deviations from the Ideal 407 11.2.1 Ideal Theory/ExperimentComparison 407 11.2.2 Base Width Modulation 410 11.2.3 Punch-Through 412 11.2.4 Avalanche Multiplicationand Breakdown 414 Common Base 414 Common Emitter 414 11.2.5 Geometrical Effects 420 Emitter Area≠Collector Area 420 Series Resistances 421 Current Crowding 421 11.2.6 Recombination-Generation Current 422 11.2.7 Graded Base 423 11.2.8 Figures of Merit 424 11.3 Modern BJT Structures 426 11.3.1 Polysilicon Emitter BJT 426 11.3.2 Heterojunction Bipolar Transistor (HBT) 429

--- Page 19 --- xvi SEMICONDUCTOR DEVICEFUNDAMENTALS 11.4 Summary 432 Problems 433 Chapter 12 BJTDynamicResponse Modeling 443 12.1 Small-Signal Equivalent Circuits 443 12.1.1Generalized Two-Port Model 443 12.1.2 Hybrid-Pi Models 446 12.2 Transient (Switching) Response 449 12.2.1 Qualitative Observations 449 12.2.2 Charge Control Relationships 452 12.2.3 Quantitative Analysis 454 Turn-on Transient 454 Turn-off Transient 456 12.2.4 Practical Considerations 457 12.3 Summary 458 Problems 459 Chapter 13 PNPN Devices 463 13.1SiliconControlled Rectifier (SCR) 463 13.2 SCR Operational Theory 465 13.3 Practical Turn-on/Turn-offConsiderations 470 13.3.1 Circuit Operation 470 13.3.2 Additional Triggering Mechanisms 471 13.3.3 Shorted-Cathode Configuration 471 13.3.4 di/dtand du/dt Effects 472 13.3.5 Triggering Time 473 13.3.6Switching Advantages/Disadvantages 473 13.4Other PNPN Devices 474 Chapter14 MSContacts andSchottky Diodes 477 14.1Ideal MSContacts 477 14.2 Schottky Diode 483 14.2.1 Electrostatics 483 Built-in Voltage 483 ρ,E,V 485 Depletion Width 486 14.2.2 I-VCharacteristics 487 14.2.3 a.c. Response 493

--- Page 20 --- CONTENTS xvii 14.2.4 Transient Response 496 14.3 Practical Contact Considerations 497 14.3.1Rectifying Contacts 497 14.3.2 Ohmic Contacts 498 14.4 Summary 500 Problems 501 R2 Part II Supplement and Review 505 Alternative/Supplemental ReadingList 505 Figure Sources/Cited References 506 Review List of Terms 507 PartII—ReviewProblem Sets and Answers 508 PartIIF FieldEffect Devices 523 Chapter 15 FieldEffect Introduction—The J-FETandMESFET 525 15.1 General Introduction 525 15.2 J-FET 530 15.2.1 Introduction 530 15.2.2 Qualitative TheoryofOperation 531 15.2.3 Quantitative ID-VD Relationships 536 15.2.4 a.c. Response 547 15.3 MESFET 550 15.3.1 General Information 550 15.3.2 Short-Channel Considerations 552 Variable Mobility Model 553 Saturated Velocity Model 554 Two-Region Model 555 15.4 Summary 557 Problems 557 Chapter 16 MOSFundamentals 563 16.1Ideal Structure Definition 563 16.2 Electrostatics—Mostly Qualitative 565 16.2.1 Visualization Aids 565 Energy Band Diagram 565 Block Charge Diagrams 566

--- Page 21 --- xvill SEMICONDUCTOR DEVICEFUNDAMENTALS 16.2.2 Effect of an Applied Bias 567 General Observations 567 Specific Biasing Regions 568 16.3 Electrostatics—Quantitative Formulation 571 16.3.1Semiconductor Electrostatics 571 Preparatory Considerations 571 Delta-Depletion Solution 576 16.3.2Gate Voltage Relationship 580 16.4Capacitance-Voltage Characteristics 584 16.4.1 Theory and Analysis 584 Qualitative Theory 584 Delta-Depletion Analysis 590 16.4.2 Computations and Observations 591 Exact Computations 591 Practical Observations 595 16.5 Summary and Concluding Comments 599 Problems 600 Chapter 17 MOSFETs-The Essentials 611 17.1 Qualitative Theory of Operation 611 17.2 Quantitative ID-VDRelationships 617 17.2.1 Preliminary Considerations 617 Threshold Voltage 617 Effective Mobility 618 17.2.2 Square-Law Theory 620 17.2.3 Bulk-Charge Thery 625 17.2.4 Charge-Sheet and Exact-Charge Theories 628 17.3 a.c. Response 630 17.3.1 Small-Signal Equivalent Circuits 630 17.3.2 Cutoff Frequency 633 17.3.3 Small-Signal Characteristics 634 17.4 Summary 637 Problems 638 Chapter 18 8 NonidealMOS 645 18.1 Metal-Semiconductor Workfunction Difference 645

--- Page 22 --- CONTENTS xix 18.2 Oxide Charges 650 18.2.1 General Information 650 18.2.2 Mobile Ions 653 18.2.3 The Fixed Charge 658 18.2.4 Interfacial Traps 662 18.2.5 Induced Charges 668 Radiation Effects 668 Negative-Bias Instability 669 18.2.6△VG Summary 670 18.3 MOSFET Threshold Considerations 674 18.3.1 V Relationships 675 18.3.2 Threshold,Terminology,and Technology 676 18.3.3 Threshold Adjustment 678 18.3.4 Back Biasing 680 18.3.5 ThresholdSummary 681 Problems 684 Chapter19 Modern FET Structures 691 19.1 Small Dimension Effects 691 19.1.1 Introduction 169 19.1.2 2ThresholdVoltage Modification 694 Short Channel 694 Narrow Width 697 19.1.3 Parasitic BJTAction 698 19.1.4 Hot-Carrier Effects 700 Oxide Charging 700 Velocity Saturation 700 VelocityOvershoot/Ballistic Transport 701 19.2 Select Structure Survey 702 19.2.1 MOSFET Structures 702 LDD Transistors 702 DMOS 703 Buried-Channel MOSFET 704 SiGe Devices 704 SOI Structures 705 19.2.2 MODFET (HEMT) 707 Problems 710

--- Page 23 --- XX SEMICONDUCTOR DEVICEFUNDAMENTALS R3PartIII Supplement andReview 713 Alternative/Supplemental Reading List 713 Figure Sources/Cited References 714 Review List of Terms 717 PartIII—ReviewProblem Sets andAnswers 718 Appendices 733 Appendix A Elements ofQuantum Mechanics 733 A.1The Quantization Concept 733 A.1.1 Blackbody Radiation 733 A.1.2The Bohr Atom 735 A.1.3 Wave-Particle Duality 737 A.2 Basic Formalism 739 A.3 Electronic States in Atoms 741 A.3.1The Hydrogen Atom 741 A.3.2 Multi-Electron Atoms 744 AppendixBMOSSemiconductor Electrostatics—ExactSolution 749 Definition of Parameters 749 Exact Solution 750 Appendix C MOSC-VSupplement 753 Appendix D MOSI-VSupplement 755 Appendix E ListofSymbols 757 Appendix M MATLAB ProgramScript 771 Exercise10.2(BJT_Eband) 771 Exercise11.7(BJT)and Exercise11.10(BJTplus) 774 Exercise16.5(MOS_CV) 778 Index 781

--- Page 24 --- PART SEMICONDUCTOR FUNDAMENTALS

--- Page 25 ---

--- Page 26 --- 1 Semiconductors: A General Introduction 1.1 GENERALMATERIALPROPERTIES The vast majority of alls solidstate devicesonthemarkettodayare fabricatedfromaclass of materials known ass semiconductors.It is therefore appropriate that we begin the discus- sion by examining the general nature of semiconducting materials. 1.1.1Composition Table1.1lists theatomiccompositionsof semiconductorsthat arelikelyto beencountered in the device literature.As noted,the semiconductor familyofmaterials includes the ele- mental semiconductors Siand Ge,compound semiconductors suchas GaAs andZnSe,and alloys likeAlxGaxAs.Due inlarge parttotheadvancedstateof itsfabricationtech- nology,Si isfarandawaythemostimportantofthe semiconductors,totallydominating the presentcommercial market.The vastmajorityofdiscrete devices and integrated circuits (ICs),includingthecentral procesing unitCPU)inmicrocomputersandtheignitionmod- ule in modern automobiles,are made from this material.GaAs,exhibiting superior electron transport properties and special optical properties,is employed in a significant number of applications ranging from laser diodes to high-speedICs.The remaining semiconductors are utilizedin“niche”applicationsthatareinvariablyofahigh-speedhigh-temperature, or optoelectronic nature.Given its present positionof dominance,we will tendtofocus our attentiononSi in the text development.Where feasible,however,GaAs will be givencom- parableconsiderationandothersemiconductors willbefeaturedasthediscussion warants. Althoughthenumberof semiconductingmaterialsisreasonably large,the listisactu- ally quite limitedconsidering thetotalnumberofelementsand possiblecombinationsof elements.Note thatreferring to theabbreviated periodicchartof theelements in Table1.2, onlyacertaingroupofelementsandelementalcombinationstypicallygiveriseto semi- conducting materials.Exceptfor theIV-VIcompounds,allof thesemiconductors listed in Table1.1are composed ofelements appearing inColumn IVof the Periodic Tableor are a combinationofelements inPeriodic Tablecolumns anequaldistancetoeithersideof +Thex（ory)inalloyformulasisafractionlyingbetween0and1.A103Ga0Aswouldindicateamaerial with 3 Al and 7 Ga atoms per every 10 As atoms. 3

--- Page 27 --- Table1.1 Semiconductor Materials. General Semiconductor Classification Symbol Name (1) Elemental Si Silicon Ge Germanium (2) Compounds (a) IV-I……SiC Silicon carbide (b) III-V … .AIP Aluminum phosphide AlAs Aluminum arsenide AlSb Aluminum antimonide GaN Gallium nitride GaP Gallium phosphide GaAs Gallium arsenide GaSb Gallium antimonide InP Indium phosphide InAs Indium arsenide InSb Indium antimonide (c) II-VI… .ZnO Zinc oxide ZnS Zinc sulfide ZnSe Zinc selenide ZnTe Zinc telluride CdS Cadmium sulfide CdSe Cadmium selenide CdTe Cadmium telluride HgS Mercury sulfide (d)IV-VI … .PbS Lead sulfide PbSe Lead selenide PbTe Lead telluride (3) Alloys (a) Binar .S_i1-Gex (b) Ternary .… …A1xGa1-xAs (or Ga{₁_xAlxAs) Alx}n{1-As (orIn₁xA1As) Cd_{1_x}M{} TGaA_s{1xP_x}GaxI{n{1-}As$(orI{n}_{1}xGaAs)$Ga_x}In{1}P$(orIn{1}_{x}GaxP)H{g}_1-xCdxTe(c)Quaternary....AlxGa1-xAsySb1-y$Gax{n{1}_A_1-y}_y 4

--- Page 28 --- SEMICONDUCTORS: AGENERAL INTRODUCTION 5 Table1.2 Abbreviated PeriodicChart of the Elements. Ⅱ III IV V VI 4 5 6 7 8 Be B C N 0 12 13 14 15 16 Mg Al Si P S 30 31 32 33 34 Zn Ga Ge As Se 48 49 50 51 52 Cd In Sn Sb Te 80 81 82 83 84 Hg T1 Pb Bi Po ColumnIV.TheColumnIIIelementGa plustheColumnVelementAs yieldstheIII-V compound semiconductor GaAs; theColumnIIelement Zn plus theColumnVIelement Se yields theII-VIcompound semiconductor ZnSe;the fractional combinationof theCol- umnIIIelementsAlandGaplus theColumnVelement As yieldsthealloysemiconductor AlxGa-xAs.This verygeneral property isrelatedtothechemical bonding in semiconduc- tors, where, on the average,there are four valence electrons per atom. 1.1.2Purity As willbe explained in Chapter 2,extremely minutetracesof impurity atoms called “dopants”canhaveadrasticeffectontheelectricalpropertiesofsemiconductors.Forthis reason,thecompositional purity of semiconductors must be very carefullycontrolledand, in fact,modern semiconductors are someof the purest solidmaterials in existence.InSi, for example,the unintentionalcontentofdopantatoms is routinely lessthanoneatom per 109Siatoms.Toassist thereader inattempting to comprehendthisincrediblelevelof purity,letus supposea forestof mapletrees was plantedfromcoast to coast,borderto border,at50-ft centersacross theUnitedStates (including Alaska).An impuritylevelof one part per10 wouldcorrespond to finding about25 crabapple trees in the maple tee forest covering theUnitedStates!Itshouldbeemphasizedthatthecitedmaterial purity referstounintnionalundeiedimpuiti.Tpicalldopanttomsat levlsangiom onepart per10tooneimpurityatomper103semiconductoratomswillbepurposelyadded to the semiconductor to control its electrical properties.

--- Page 29 --- 6 SEMICONDUCTORFUNDAMENTALS 。 。 。 (a) Amorphous (b) Polycrystalline (c) Crystalline No recognizable Completely ordered Entire solid is made up of long-range order in segments atoms in an orderly array Figure1.1Generalclassificationofsolidsbasedonthedegreeofatomicorder:(a)amorphous, (b) polycrystalline,and (c)crystalline. 1.1.3 Structure The spatial arrangementofatoms within a material plays an importantrole in determining the precise propertiesof thematerial.As shown schematically in Fig.1.1,theatomicar- rangement within a solidcauses it to beplaced into oneofthreebroadclassifications; namely, amorphous,polycrystalline,orcrystalline. Anamorphoussolid is a material in which there is no recognizablelong-range order in thepositioningofatoms withinthe material.Theatomic arrangement inany given sectionofan amorphousmaterial will look differentfromtheatomicarrangementinanyothersectionofthematerial.Crystalline solids lieat theoppositeendof the”orderspectrum;in a crystalline material theatoms are arranged in anorderly three-dimensional array.Givenany sectionof acrystallinema- terial,one canreadily reproducethe atomic arrangement inanyothersectionofthe mate- rial.Polycrystallinesolidscomprisean intermediatecase inwhichthe solidiscomposed of crystallinesubsections thatare disjointedormisoriented relativetooneanother. Upon examining themanysolidstate devices in existence,one readily finds examples of all three structural forms.An amorphous-Si thin-film transistor is used as the switching element in liquidcrystal displays (LCDs);polycrystalline Si gates areemployed in Metal- Oxide-Semiconductor Field-Effect Transistors (MOSFETs).Inthe vastmajorityof de- vices, however,the activeregion ofthe device is situated within a crystalline semi- conductor. Theoverwhelming numberof devicesfabricatedtoday employ crystalline semiconductors. 1.2 CRYSTALSTRUCTURE The discussionat theend of the preceding section leads nicely into the topic of this section. Since in-usesemiconductorsare typicallycrystallineinform,itseemsreasonabletoseek out additional informationabout the crystalline state.Ourmajor goalhere is to present a more detailedpictureoftheatomic arrangement within the principal semiconductors.To

--- Page 30 --- SEMICONDUCTORS: AGENERAL INTRODUCTION 7 achieve this goal,we first examine how one goes about describing the spatialposition- ing ofatoms withincrystals.Next,a bitof“visualizationpractice withsimplethree- dimensional lattices(atomicarrangements) is presented prior to examining semiconductor lattices themselves.The section concludes with an introduction totherelated topicofMil- ler indices.Miller indices are a convenient shorthand notationwidely employed for iden- tifying specific planes and directions within crystals. 1.2.1 The UnitCellConcept Simplystated,aunitcellisasmallportionofanygivencrystalthatcouldbeusedto reproducethecrystal.Tohelpestablishtheunitcell(orbuilding-block）concept,letus considerthetwo-dimensionallatticeshowninFig.1.2(a).Todescribethislatticeorto totallyspecifythe physicalcharacteristicsofthe lattice,one needonlyprovidetheunitcell shown in Fig.1.2(b).As indicated in Fig.1.2(c),theoriginallatticecan bereadilyrepro- duced bymerely duplicating the unitcelland stacking the duplicates next to eachother in an orderly fashion. In dealing withunitcellsthereoftenarises amisunderstanding,andhenceconfusion, relativeto two points.First,unitcellsarenotnecessarilyunique.Theunitcellshownin Fig.1.2(d)isasacceptableastheFig.1.2(b)unitcell for specifyingtheoriginallatticeof a K a D D (a) (b) 1/4 atom each corner a (c) (d) Figure1.2Introductiontotheunitcellmethodofdescribingatomicarrangementswithincrystals. (a) Sample two-dimensional lattice.(b)Unitcellcorresponding to thepart (a)latice.(c)Reproduc- tion of the original lattice.(d)An alternative unitcell.

--- Page 31 --- 8 SEMICONDUCTORFUNDAMENTALS Fig.1.2(a).Second,a unitcellneednotbeprimitive(the smallestunitcellpossible).In fact,itis usuallyadvantageous toemployaslightlylargerunitcellwithorthogonalsides insteadofa primitivecell withnonorthogonalsides.This is especiallytrue inthree dimen- sions where noncubic unit cells are quite difficult to describe and visualize. 1.2.2Simple 3-DUnitCells Semiconductor crystals are three-dimensional and are therefore described in terms of three- dimensional(3-D)unitcells.InFig.1.3(a)wehavepicturedthesimplestofallthree- dimensionalunitcells,thesimplecubicunitcell.Thesimplecubiccellis anequalsided boxorcube with anatom positioned ateachcornerof thecube.Thesimplecubiclattice associatedwiththiscell isconstructed in a manner paralleling the two-dimensionalcase. In doingso,however,itshouldbenotedthatonly1/8ofeachcorneratomisactuallyinside thecell,aspicturedinFig.1.3(b).DuplicatingtheFig.1.3(b)cellandstackingthedupli- cates likeblocks ina nurseryyieldsthesimplecubiclattice.Specifically,theprocedure generatesplanesofatomlikethapreviouslshowninFig..2).lanesofatomsl tothebaseplaneareseparatedfromoneanotherbyaunitcellsidelengthorlatticecon- D (a) Simple cubic (b)Pedantically correct simple cubic (c) bcc (d) fcc Figure1.3Simplethree-dimensional unitcells.(a)Simplecubicunitcell.(b)Pedanticallycorrect simplecubic unitcell including onlythefractional portion（1/8)ofeachcorneratom actually within thecellcube.(c)Bodycenteredcubicunitcell.(d)Facecentered cubicunitcell.

--- Page 32 --- SEMICONDUCTORS: AGENERAL INTRODUCTION 6 stant, a. When viewed normalto the base plane,atoms in a givenplane are positioned directly over atoms in a lower-lying plane. Figures1.3(c)and1.3(d)display twocommon3-Dcellsthatare somewhatmorecom- plexbutstillcloselyrelatedtothesimplecubiccell.TheunitcellofFig.1.3(c)hasanatom addedatthecenterofthecube;thisconfigurationisappropriatelycalledthebodycentered cubic(bcc)unitcell.Thefacecenteredcubic（fcc）unitcellofFig.1.3(d)containsanatom ateach faceofthecube inadditiontotheatoms ateachcorner.(Note,however,thatonly one-halfofeachfaceatomactually liesinsidethefccunitcell.)Whereasthesimplecubic cellcontainsoneatom(1/8ofanatomateachoftheeightcubecorners),thesomewhat more complex bccand fcccellscontain twoand fouratoms,respectively.Thereader should verify these factsand visualize the latticesassociated with the bccand fcccells. （C）E Exercise1.1 Thetext-associated software distributed viatheInternetor on a floppydiskcontains a directoryorfoldernamedMacMolecule.Filesstoredinsidethisdirectoryorfolder are ASCIIinputfilesgenerated by theauthorforuse withacomputerprogrambythe same name. TheMacMolecule program,copyrighted by the University of Arizona, is distributedfreeofcharge toacademicusers and isavailableatmanyacademic sites. As the name implies,the program runs only on Macintosh personal computers. The inputfiles suppliedtoIBM-compatibleusers,however,can bereadilyconverted for use on a public-access Macintosh. MacMoleculegeneratesanddisplays a“ball-and-stick”3-Dcolorrenderingof molecules,unit cells,andlattices.The input files supplied bytheauthorcanbeused to helpvisualize thesimplecubic,bcc,andfcc unit cells,plus thediamond and zinc- blende unitcells described inthe next subsection. TheAbout-MacMolecule filethat is distributed withthe MacMolecule program provides detailed informationabout the use of the program and the generation/modification of input files. Thereader is urgedto investigate and play withthe MacMoleculesoftware.The initialdisplayisalways a z-direction viewand is typicallypseudo-two-dimensional in nature.A more informative view isobtained byrotating the model.Rotation is best accomplished by”grabbing and dragging”theedges of themodel. Themore adven- turesome may wishto try theirhand at modifying the existing input filesor generat- ing new input files. 1.2.3 Semiconductor Lattices Wearefinallyinapositiontosupplydetailsrelativetothepositioningofatoms withinthe principalsemiconductors.InSi(andGe)the latticestructure is describedby theunitcell pictured inFig.1.4(a).TheFig.1.4(a）arrangementis knownasthediamond latticeunit cellbecause it also characterizes diamond,a formoftheColumnIVelementcarbon.Ex- amining thediamondlatticeunitcell,wenotethatthecell iscubic,withatoms ateach

--- Page 33 --- 10 SEMICONDUCTOR FUNDAMENTALS

D a As (a) (b) i 1 1 a/2 1 1 (c) Figure1.4(a）Diamondlatticeunitcell.(b)Zincblendelatticeunitcell(GaAsusedforillustra- tion).(c)Enlarged topcornerof thepart (a)diamond lattice emphasizing thefour-nearest-neighbor bonding within the sructure.Thecubesidelength,ais5.43Åand5.65ÅatT=300KforSiand GaAs,respectively.[(a)Adaptedfrom Shockley[].(b)From Sze[1,©1981by John Wiley &Sons, Inc. Reprinted with permission.] cornerand ateachfaceofthecube similartothefcccell.TheinterioroftheFig.1.4(a) cell,however,contains fouradditionalatoms.Oneof the interioratoms is located along a cubebody diagonalexactlyone-quarterofthe waydownthe diagonalfromthetopfront left-handcorner of the cube.Theotherthree interioratomsare displaced one-quarterofthe body diagonallengthalong thepreviouslynotedbody diagonaldirectionfromthefront, top,andleft-side faceatoms,respectively.Although itmaybe difficultto visualizefrom Fig.1.4(a),the diamond lattice canalso be described asnothing more than two interpene- trating fcc lattices.Thecornerand faceatomsofthe unitcellcanbe viewedas belonging to one fcc lattice,while theatoms totallycontained within thecellbelong to the secondfcc lattice.Thesecond lattice is displacedone-quarterofa body diagonal along a bodydiago- nal direction relative to the first fcclattice. MostoftheIII-V semiconductors,including GaAs,crystallize in the zincblende struc- ture.Thezincblendelattice,typified bytheGaAs unitcell shown in Fig.1.4(b),isessen-

--- Page 34 --- SEMICONDUCTORS:AGENERAL INTRODUCTION 11 tiallyidenticaltothediamondlattice,exceptthatlatticesitesareapportionedequallybe- tween two differentatoms. Gaoccupies sitesonone ofthetwo interpenetratingfcc sublattices; arsenic (As)populates theotherfcc sublattice. Now that the positioning of atoms within the principal semiconductors has beenestab- lished,thequestionmayariseastothe practical utilizationof such information.Although severalapplications couldbecited,geometrical-type calculations constitute a verycom- mon andreadilyexplained useoftheunitcellformalism.Forexample,in Siatroomtem- peraturetheunitcellsidelength（a)is5.43Å（where1Å=10-cm).Sincetheeare eightSiatomsperunitcellandthevolumeoftheunitcellisitfollowsthaherae 8/aoralmostexactly5×10²²atoms/cmintheSilattice.Similarcalculationscouldbe performedtodetermineatomicradii,thedistance betweenatomicplanes,andsoforth.For the purposes of the development herein,however,a major reason for the discussionof semiconductor lattices was to establishthat,as emphasized in Fig.1.4(c),atoms in the diamondand zincblendelattices have four nearest neighbors. Thechemicalbonding（or crystallineglue)withinthe major semiconductors is therefore dominated by theattraction between any givenatomand its fourclosest neighbors.This is an importantfactthat should be filed away for future reference. Exercise 1.2 P:Ifthelattice constantor unitcell side-length in Siisa=5.43×10-cm,what isthe distance (d)from thecenterofone Siatom to thecenterof itsnearestneighbor? S: As noted in the Fig.1.4 caption,theatom in theupperfrontcornerof theSiunit cellandtheatomalong thecubediagonalone-fourthofthe way down thediagonal are nearestneighbors.Sincethediagonalofacubeisequalto√timessidelengh one concludesd=(1/4)√3a=(√3/4)(5.43×10-8)= 2.35×10-8cm. (C) Exercise 1.3 P:Construct a MATLAB program that computes the numberofatoms/cm incubic crystals.Use the MATLAB input function to enter the numberof atoms/unit-cell andtheunitcellsidelength（a)foraspecificcrystal.Makea listingofyourprogram and record the program result when applied to silicon. S: MATLAB program script… %Exercise1.3 %Computation of the number of atoms/cm3 in a cubic lattice N=input(input number of atoms/unit cell,N=’); a=input(lattice constant in angstrom, a=’); atmden=N*(1.0e24)/(a^3)%numberof atoms/cm3

--- Page 35 --- 12 SEMICONDUCTORFUNDAMENTALS Program output for Si . .. input number of atoms/unit cell, N = 8 lattice constant inangstrom, a=5.43 atmden = 4.9968e+22 1.2.4 MillerIndices Singlecrystalsofsiliconused indeviceprocessing normallyassumethethin,roundform exhibited inFig.1.5.Thepictured plate-likesinglecrystals,betterknownasSi wafers,are typicalofthe starting substrates presentlyemployedbymajor manufacturers.Ofparticular interesthereisthefactthatthesurfaceofawaferiscarefullypreorientedtoliealonga specificcrystallographicplane.Moreover,a“flat”or“notch”is groundalong theperiph- eryof the waferto identifya reference direction within thesurface plane.Precise surface orientationiscritical incertaindevice processing stepsanddirectlyaffectsthecharacter- istics exhibitedby a numberof devices.Theflatornotch isroutinelyemployed,forex- ample,toorient device arrays onthewafer soastoachievehigh yieldsduring device [011] Direction “Notch”” [011] Direction “Flat” (100) Surface 200mm(8”) 150mm（6”） Figure1.5S Single-crystalsiliconwaferstypicalofthe startingsubstratespresentlyemployedby 0.625mmand0.725mmthick,respectively.Thefacing surface ispolishedandetchedyieldinga damage-free,mirror-likefinish.ThefiguredramatizestheutilityofMillerindicesexemplifiedbythe (100)planeand [011]directiondesignations.(PhotographcourtesyofIntelCorporation.)

--- Page 36 --- SEMICONDUCTORS: AGENERAL INTRODUCTION 13 Z Z 3a -2a D →y 2a D- D D x x x (a) (b) (c) Figure1.6Samplecubiccrystalplanes.a)The(632)planeusedinexplainingtheMillerindexing procedure.(b) The (001)plane.(c) The(221)plane. separation.The point we wish tomake is thatthe specificationofcrystallographicplanes and directions is of practical importance.Miller indices,exemplified by the(100) plane and[011]direction designationsusedinFig.1.5,constitutetheconventionalmeansof identifying planes and directions within acrystal. TheMiller indices for any givenplane ofatoms within acrystal areobtained by follow- ing a straightforwardfour-stepprocedure.Theprocedure isdetailedbelow,alongwthhe simultaneous sample indexing of the plane shown in Fig.1.6(a). Indexing Procedure Sample Implementation After setting up coordinate axes along the edges of the unit cell, note where the plane to be indexed intercepts the axes. Divide each intercept value by the unitcell length along therespective coordinate axis.Record the resulting normalized (pure-number) intercept set in the order x, y, Z. 1,2,3 Invert the intercept values; that is,form[1/intercept]s. $1,${2}3}$ Using anappropriatemultiplierconvertthe1/intercept settothe smallest possible set of whole numbers. 6,3,2 Enclose the whole-number set in curvilinear brackets. (632) To complete the description of the plane-indexing procedure,the user shouldalso be aware of the following special facts: axis is takento beat infinity.Thus,for example,the plane shown in Fig.1.6(b)inter- cepts thecoordinateaxes at∞∞,1,andisthereforea(01)plane.

--- Page 37 --- 14 SEMICONDUCTORFUNDAMENTALS (ii)If the plane to beindexedhasan interceptalong the negative portionofa coordinate axis,a minus sign is placedoverthecorrespondingindex number.Thus the Fig.1.6(c) plane is designated a (221) plane. (iReferrintothediamondlticeofFig..4noteththesixlanespassinou thecube faces contain identicalatomarrangements;that is,becauseof crystal sym- metry,it is impossibletodistinguishbetweenthe“equivalent”(100),(010),(001), （100),),and()planes;oritisimpossibletodistinguishbetween1}plas. A groupofequivalentplanes isconcisely referenced intheMillernotationthroughthe use of {} braces. (iv)Millerindicescannotbeestablishedforaplanepassingthroughtheoriginofcoord- nates. Theoriginofcoordinates must be moved to a lattice point noton the planeto be indexed.This procedure is acceptable becauseof the equivalent natureof parallel planes. Z Z Z →y (100) (110) (111) x x x (a) Z Z y [100] [110] [111] x x x (b) Figure1.7VisualizationandMillerindicesofcommonlyencountered(a）crystallineplanesand (b) direction vectors.

--- Page 38 --- SEMICONDUCTORS: AGENERAL INTRODUCTION 15 Table1.3 Miller Convention Summary. Convention Interpretation (hkl) Crystal plane {hkl} Equivalent planes [hkl] Crystal direction (hkl) Equivalent directions Using thesimple cubiccell as a reference,the threemostcommonlyencountered crystal- line planes are visualized in Fig.1.7(a). TheMillerindicesfordirectionsareestablishedinamanneranalogoustothewell- known procedure forfinding the componentsofa vector.First,setup a vectorofarbitrary length inthedirectionofinterest.Next,decomposethevectorintoitscomponentsby noting the projections of the vectoralong thecoordinate axes.Using an appropriate multi- plier,convertthecomponent values into the smallest possiblewhole-numberset.Thisof coursechangesthelengthoftheoriginalvectorbutnotitsdirection.Finally,enclosethe whole-number set in brackets.Square brackets,[],are used to designate specific directions within a crystal;triangularbrackets,<,designateanequivalentsetofdirections.Com- mon direction vectorsandtheircorrespondingMiller indicesare displayedinFig.1.7(b). A summaryoftheMillerbracketingconventionforplanesanddirections isgivenin Table1.3. In the foregoing discussion we presented the procedure for progressing from a given planeordirectioninacrystaltothecorrespondingMillerindices.Moreoftenthannot,one isfaced withtheinverse processvisualizingthecrystallineplaneordirectioncorrepond- ing to a given setof indices. Fortunately,one seldomencounters otherthanlow-index planesanddirectionssuchas(11),110),[001].Thusitispossibletobecomefairlyadept at the inverse process by simplymemorizingtheorientationsforplanesand directions associated withsmall-number indices.It isalsohelpfultonotethat,forcubiccrystals,a planeand the directionnormaltothe plane haveprecisely thesameindices—e.g., the [110]directionis normalto the(110)plane.Ofcourse,anyplaneor directioncanalways be deduced by reversing the indexing procedure. Exercise1.4 P:Foracubiccrystal lattice: (a) Determine the Miller indicesfortheplane and direction vector shown in Fig.E1.4(a). (b)Sketchtheplaneanddirectionvectorcharacterizedby(011)and[011],re- spectively.

--- Page 39 --- 16 SEMICONDUCTORFUNDAMENTALS S:(a) The intercepts of the plane are -1,1,and2 along the x,y, and z coordinate axes,respectively.Taking1/intercept]syields-1,1,1/2.Multiplyingby2toobtain the lowest whole-numbeeandenclosinipaenthesesives for the Millerindicesofthe plane.The direction vector has projectionsof2a,a,and0along thex,y, andz coordinate axes,respectively.TheMiller indices for the directionare then[210]]. (b) For the (011)plane,[1/intercept]s=0,1,1.The normalized x,y,z interceptsof the planeare therefore ∞,1,1;the plane intersects the y-axis ata,intersects the z-axisata,andisparalleltothe-axis.Inacubiccrystalthe[1]directionisnormal to the (011)plane. The deduced planeand direction are as sketched in Fig.E1.4(b). z 2a [011] a D- D y y D 2a x x (a) (b) Figure E1.4 1.3CRYSTALGROWTH 1.3.1Obtaining Ultrapure Si Given theobvious general availabilityand widespread use of semiconductor materials,par- ticularlySiit isreasonableto wonderabouttheoriginofthesingle-crystalSi usedin modern-day deviceproduction.Could itbe that Si isreadilyavailable in sandstone de- posits?-No.Perhaps,as a sortof by-product,Si single crystals come from South African diamond mines?-Wrong again. As suggested recently in a low-budget sciencefiction movie, maybe Si is scooped from the ocean bottom by special submarines?-Sorry, no. Although Si is the second mostabundantelement in the earth’s crustand a component in numerous compounds,chiefof which are silica (impure SiO2)and thesilicates(Si+O+ anotherelement)）,silicon neveroccursalone in natureas an element.Single-crystal Siused in device production,it turns out,is a man-made material. Given thepreceding introduction,it should beclearthat the initialstep in producing device-quality silicon must involve separating Si from its compounds and purifying the

--- Page 40 --- SEMICONDUCTORS:AGENERALINTRODUCTION 17 Very $Si{C}_{}$ Ultrapure Ultrapure Silica impure (liquid) polycrystalline silicon Si Reduced in Chlorinated Distilled, Reduced in presence of C etc. hydrogen atmosphere Figure1.8 Summaryof the process employedto produce ultrapure silicon. separated material. The ingenious separationand purification process thathas evolvedis summarizedinFig.1.8.Low-gradesiliconorferrosiliconisfirstproducedbyheatingsilica withcarboninanelectricfurnace.Thecarbonessentiallypullstheoxygen awayfromthe impure SiO2(i.e.reduces theSiO2)leavingbehindimpure Si.Next,theferrosiliconis chlorinated to yieldSiCl4orSiHCl₃,bothof whichare liquids atroom temperature.Al- thoughappearingoddatfirstglance,theliquefaction process isactuallyaclevermaneuver. Whereassolidsare verydifficultto purify,a numberof standardproceduresare available forpurifyingliquids.AnultrapureSiCl（orSiHCl）istheresultaftermultipledistillation andotherliquidpurificationprocedures.Lastly,thehigh-purityhalide is chemicallyre- duced,yieldingthedesiredultrapurelementalsilicon.Thisisaccomplished,orexample, byheating SiCl4inahydrogenatmosphere[SiCl4+2H2→4HCl+Si]. 1.3.2Single-Crystal Formation Althoughultrapure,thesiliconderivedfromtheseparationand purificationprocessas just described is nota singlecrystalbut isinsteadpolycrystalline.Additional processing is thereforerequiredtoformthelargesinglecrystals used in devicefabrication. Themost commonly employed method yielding largesingle crystals of Si isknown as theCzochral- skimethod.In thismethod theultrapure polycrystalline silicon is placed in a quartz cru- cibleandheated inaninertatmospheretoform amelt,as shown in Fig.1.9.A smallsingle crystal,or Si“seedcrystal,with thenormalto its bottom face carefully alignedalonga predetermineddirection（typicallya（1lor（100>direction),is thenclamped toametal rod and dipped into themelt.Once thermalequilibrium is established,thetemperatureof themelt inthevicinityoftheseedcrystal isreduced,andsiliconfrom themeltbeginsto freeze outontotheseedcrystal,theaddedmaterial being a structurallyperfectextension ofthe seed crystal.Subsequently,the seedcrystal is slowly rotated and withdrawn fromthe melt; this permits more and moresiliconto freeze outon the bottom of the growing crystall. A photograph showing a crystal after being pulledfrom the melt is displayed in Fig.1.9(c). Thelarge,cylindricallyshapedsinglecrystalofsilicon,alsoknownasaningotisroutinely up to 200mm(8 inches)in diameter and1to2 meters in length. The Si wafers used in device processing,suchas those previously pictured in Fig.1.5,are ultimatelyproduced by cutting the ingot into thin sections using a diamond-edged saw.

--- Page 41 --- 18 SEMICONDUCTORFUNDAMENTALS Seal Front opening Ar door Front opening chamber Seed shaft Seed holder 区 Vacuum pump Seed Valve Optical system Viewing port- Ar Silica crucible [Si] Graphite crucible Graphite heater Thermal shield Vacuum pump Crucible shaft Ar + SiO + CO (a) (b) Figure1.9 Siliconcrystal-pulling appara- tus and resulting single crystal.(a)Photo- graphof a computer-controlledCzochralski crystal puller.(b) Simplified schematic draw- ingof the puller.(c)Siliconingot.[(a)and (c) Photographs courtesy of Wacker Siltronic. (b) From Zuhlehnerand Huber(31;reprinted with permission.] (c)

--- Page 42 --- SEMICONDUCTORS: A GENERAL INTRODUCTION 19 1.4 SUMMARY This chapterprovidesbasicinformationaboutsemiconductors ingeneralandsilicon in particular.Examining theelemental compositions of semiconductors,one finds that there are,on average,four valence electrons per atom.Moreover,device-quality semiconductors are typicallysinglecrystalsofhighcompositional purity.Sicrystallizes inthediamond lattice,GaAscrystallizes in the zincblende lattice,andthelatticeatoms inbothcases have four nearest neighbors.Miller indices,introduced herein as anadjunct to the crystal struc- ture discussion,aretheacceptedmeans foridentifying planesand directions within a crystal.Finally,large singlecrystalsofdevice-grade Siarecommonlyproducedbythe Czochralski method. PROBLEMS CHAPTER1 PROBLEMINFORMATIONTABLE Complete Diffi culty Suggested Short Problem After Level Point Weighting Description 1.1 1.4 一 10(a::h-1,i-2) Short answer review 1.2 1.2.3 1 8 Unit cell for alloy 1.3

一 10 (5 each part) Ge atomic density 1.4 ” 一 8 (4 each part) Nearest-neighbor distances 1.5 1.2.4 2-3 15(a-2, b-2,c-3,d-3, e-5) Atoms/cm² on planes 1.6 ” 1 8 (2 each subpart) Determine Miller indices 1.7

1 16 (2 each part) Given (hkl), sketch plane 1.8 ” 1 16 (2 each part) Given [hkl], sketch direction 1.9

1 8 (4 each part) Perpendicular directions 1.10 ” 2 10 Direction on wafer surface 1.11 ” 2 10(a-3, b-2, c-3,d-2) Combination question 1.12 ” 2 12 (2 each part) Equivalent planes/directions 1.13 1.2.3 3 20 (5 each part) % of volume occupied 1.1Quick Quiz.Answer thefollowingquestions as concisely as possible. (a)Name(i)oneelemental semiconductorand(i)oneI-Vcompound semiconductor. (b)Whatisthedifference betweenacrystallineandapolycrystallinematerial? (c) Give a word definition of unit cell. (d)Howmanyatomsarethereina simplecubicunitcell? ina bccunitcell? inafcc unit cell? in the unit cell characterizing the diamond lattice?

--- Page 43 --- 20 SEMICONDUCTORFUNDAMENTALS (e) 1Å=?cm. (f）Intermsofthelatticeconstantawhatisthedistancebetweennearest-neighboraoms in a simple cubic lattice? (g) How manynearest-neighboratomsarethereinthediamondandthezincblende lattices? (h）Whatisbeingindicatedbythebracketsets（,[,{},and（asemployedintheMiller indexing scheme? (i) DescribetheCzochralskimethodforobtaininglargesinglecrystalsofsilicon. 1.2TheGaAs unitcellispictured inFig.1.4(b).Describe（orsketch）theunitcellfor A0.5Ga0.5As. 1.3（a）The lattice constantofGe at room temperature isa=5.65×10-8cm.Determine the number of Ge atoms/cm3. (b)Copiesof computer-required exercises employing MATLAB are includedondisk.Run the Exercise1.3programtoverifyyourresultfromart. 1.4Intermsofthelatticeconstantawhatisthedistancebetweennearest-neighbor atoms in (a) a bcc lattice? (b) an fcc lattice? 1.5 The surface of a Si wafer is a(100)plane. (a) Sketch the placement of Si atoms on the surface of the wafer. (b)Determine the number of atoms per cm² at the surfaceof the wafer. (c)/(d)Repeat parts (a)and(b),this timetaking the surfaceoftheSi wafertobea(110) plane. (e) PrimarilyforpracticeinutilizingMATLAB,constructa MATLABprogramthatcom- putes the surface densityofatomson（100)planesofcubiccrystals.The numberof atoms withcentersonthe（10)faceoftheunitcell,andthelatticeconstantinÅ,ae to bethe input variables.Confirm theresultobtained manually in part(b). 1.6Record all intermediate steps inanswering the following questions. (a)As shown inFig.P1.6(a),a crystalline plane has intercepts of1a,3a,andlaon the xyandaxe,respectively.aisthecubiccellsidelengh. (i）What is theMiller index notation for theplane? (ii)What is the Miller index notationforthe direction normalto the plane?

--- Page 44 --- SEMICONDUCTORS: AGENERAL INTRODUCTION 21 (b)AssumingthecrystalstructuretobecubicdeterminetheMillerindicesfor)thelane and (ii) the vector pictured in Fig.P1.6(b). Z Z Vector a-2 3a a y y a a x x (a) (b) Figure P1.6 1.7 Assuming a cubic crystal system,make a sketch of thefollowing planes. (a) (001), (b) (111), (c) (123), (d) (ī10), (e) (010),(f)(111), (g) (221),(h) (010). 1.8Assumingacubiccrystalsystem,useanappropriatelydirectedarrowtoidentifyeach of the following directions: [Ⅲ]（p）[I00]（)[0I]（9）[00]（e) [ε]（4[([]1[0]（) 1.9Identifytwocrystallinedirectionsinacubiccrystal whichareperpendicularto (a) the [100] direction, (b)the [111] direction. NOTE:Thecosineoftheangleθbetweentwoarbitrarydirections,[h,kl]and[h2k₂l], in a cubic crystal is h_₁h_2}+k12k₂2}+l₁l2} cos(θ)= [（h²{2}+k²+²)（h{²2+k²2+l{²)}1}$Consequently,fortwodirectionstobeperpendicular,cos(\theta )=0andonemusthave$h_₁h_2+k1k₂2+l1l₂2=0. 1.10AspicturedinFig.1.5[011]isthedirectioninthesurfaceplanenormalto themajor fatwhenthesurfaceoftheSiwaferisa(10O)plane.Toconstructa particulardevice structure,aparallel setofgroovesmust beetched inthe(100)surfaceplanealong the[10] direction.Makea sketchindicating how the grooves are to beoriented on the wafer’s sur- face. Explain how you arrived at your sketch.

--- Page 45 --- 22 SEMICONDUCTORFUNDAMENTALS 1.11AcrystallinticeischaacteiedbthecubicunitcellicturedinFigP.Te cell has a single atom positioned at the center of the cube. z ↑ y x Figure P1.11 (a)What is the name of the lattice generated by the given unit cell? (b)Determinethenumberofatomsperunitvolumeinthecrystal.Youranswersould in terms of the lattice constant, a.) (c) Suppose thecrystal has a(110)surfaceplane.Determinethe numberof atoms perunit area whose centers lie on the (110) plane. (d)AdirectionvectorisdrawnthroughthecenteroftheatomintheunitcellSpecifthe Miller indices of the direction vector. 1.12Givenacubiclatticeindicatehowmanyequivalentplanes[parts(a)to(c)]orequiva- lent directions[partsd)to]areassociatedwithacoftheollowindesignations: (a) {100}，(b){110},(c){111}, (d) (100), (e)(110),(f)(111). 1.13Treating atoms as rigid spheres withradiiequaltoone-halfthedistance between nearest neighbors,show that the ratioof the volume occupied by the atoms to the total available volume in the various crystal structures is (a) π/6 or 52% for the simple cubic lattice. (b)√3π/8or 68% for the body centeredcubic lattice. (c)√2π/6or 74% for the face centered cubic lattice. (d)√3π/16or34% for the diamond lattice.

--- Page 46 --- 2 Carrier Modeling Carriersare theentities thattransportcharge from placeto place insideamaterial and hence giverisetoelectricalcurrents.Ineverydaylifethemostcommonlyencountered typeof carrieristheelectron,thesubatomicparticleresponsibleforchargetransportinmetallic wires.Withinsemiconductors oneagainencounters thefamiliarelectron,butthere isalso a secondequallyimportanttypeofcarrier—thehole.Electronsandholesarethefocal pointofthis chapter,wherein we examine carrierrelated concepts,models,properties,and terminology. Althoughreminders willbeperiodically interjected,itshouldbeemphasizedfromthe verystartthatthedevelopmentthroughoutthischapterassumesequilibriumconditions exist within the semiconductor.”Equilibrium”is the term used todescribethe unperturbed state ofa system.Underequilibriumconditionsthereare noexternalvoltages,magnetic fields,stresses,orother perturbing forcesacting on the semiconductor.All observablesare invariantwithtime.This“rest”condition,as wewillsee,providesanexcellentframeof reference. Beingabletocharacterize the semiconductor underequilibrium conditions per- mits onetoextrapolateandascertainthesemiconductor’sconditionwhena perturbation has been applied. The readershouldbe forewarnedthat a numberof formulasandfacts willbe presented without justification in thischapter.We would liketo provideacompletediscussion,prop- erlydeveloping everyconceptandderiving everyformula.Unfortunately,this isnotpos- sible.Itisourunderlying philosophy,moreover,thatbeingabletoderive aresult is second- aryto knowing howto interpretandusearesult.Onecan,ofcourse,fill inany information gap through supplemental readings.For suggested references seethe supplementalreading listattheendofPart I. Lastly, please note that Appendix A provides a more in-depth introduction to the quantization concept and related topics considered in Section 2.1. Section 2.1 may be replaced by Appendix A with no loss in continuity. 2.1THEQUANTIZATIONCONCEPT Insteadofattemptingtodealimmediatelywithelectrons incrystallineSi,wherethereare 14electronsperatomand5×10²atoms/cmwewilltakeamorerealisticapproachand first establishcertaingroundrulesbyexamining much simpleratomic systems.Webegin 23

--- Page 47 --- 24 SEMICONDUCTORFUNDAMENTALS with thesimplestofallatomic systems,theisolatedhydrogenatom. Thisatom,asthe readermayrecall from a course inmodern physics,was under intense scrutiny at thestart ofthe twentieth century.Scientistsofthattime knewthe hydrogenatom contained a nega- tively charged particle inorbitabout amore massive positively charged nucleus.Whatthey could notexplain was the natureof the lightemittedfromthesystem whenthe hydrogen atom was heatedtoanelevated temperature.Specifically,the emitted light wasobserved at onlycertain discrete wavelengths; according to the prevailing theoryof thetime,scientists expected a continuum of wavelengths. In1913Niels Bohrproposed a solutionto the dilemma.Bohr hypothesized that theH- atom electron wasrestricted to certain well-definedorbits;orequivalently,Bohrassumed thattheorbiting electroncouldtakeononly certainvaluesofangularmomentum.This quantization.Ascanbereadilyestablishediftheelectronsangularmomentumisassumd to be nh, then m_{0^{} 13.6 EH} eV, n =1,2,3,… (2.1)

2(4πε0hn^{²n^{}$ where (also see Fig.2.1)EHis theelectronbinding energy withinthe hydrogenatom,m is the mass of a free electron,q is the magnitudeof theelectronic charge,ερis the permit- tivityoffreespace,hisPlancksconstant,h=h/2π,andnis theenergyquantumnumber or orbit identifier. The electronvolteV)isaunitofenergyequalto1.6×10-joules. Now,withthelectronlimitedtocertainenergiesinsidethehydrogenatomitfollowsfrm the Bohrmodelthatthetransitionfromahigherntoa lowernorbitwillrelease quantized energiesoflight;thisexplainstheobservationofemittedlightatonlycertain discrete wavelengths. n=1 n= 3 -13.6eV EH=-1.51eV n=2 -3.4eV Figure2.1Thehydrogenatom—idealizedrepresentationshowingthefirstthreeallowedelectron orbits and the associated energy quantization.

--- Page 48 --- CARRIERMODELING 25 $n=2 8 Electrons Six allowed levels at same energy Two allowed levels at same energy sp n=1 n= 3 2 Electrons Figure2.2 Schematic representation of an isolated Si atom. Forour purposes,themost important idea to beobtained from theBohrmodel is that the energyofelectronsinatomicsystemsisrestrictedtoa limitedsetof values.Relative to the hydrogenatom,the energy level scheme in a multi-electronatom like siliconis,as one might intuitively expect,decidedlymore complex.It is still a relativelyeasytask,how- ever, to describe thesalientenergy-relatedfeaturesofanisolatedsiliconatom.As pictured in Fig.2.2,tenof the14Si-atomelectronsoccupy very deep-lying energylevels and are tightlybound to thenucleus of theatom. Thebinding is so strong,infact,thatthese ten electronsremainessentiallyunperturbedduring chemical reactionsornormalatom-atom interactions,withthe ten-electron-plus-nucleus combinationoften being referred toas the “core”of theatom. Theremaining four Si-atomelectrons,ontheotherhand, are rather weakly boundandarecollectively calledthe valenceelectrons becauseoftheir strong participation inchemicalreactions.As emphasized inFig.2.2,the four valenceelectrons, if unperturbed,occupyfouroftheeightallowedslots （or states)having thenexthihest energyabovethedeep-lyingcorelevels.Finallyweshouldnote,forcompleteness,atth electronicconfiguration in the32-electronGe-atom(germaniumbeingtheotherelemental semiconductor)isesseniallyidenticaltotheSi-atomconfigurationexceptthattheGecore contains 28electrons. 2.2 SEMICONDUCTORMODELS Building on the information presented in previous sections, we introduceand describe in this sectiontwoveryimportantmodelsorvisualizationaidsthatareusedextensivelyinthe analysis of semiconductor devices. The inclusion of semiconductormodels in a chapter devoted tocarriermodeling mayappearoddat first glance but is actuallyquiteappropriate. We are,in effect,modeling the carrier”container”the semiconductor crystal. Actually,notonlenergybutmanyotherobservablesrlatingtoatomic-sizedparticlesarequantized.Anne field of study,Quantum Mechanics,hasbeen developed to describe the properties andactionsofatomic-szed particles and systems.

--- Page 49 --- 26 SEMICONDUCTORFUNDAMENTALS Line represents a shared valence electron Circle represents the core of a semiconductor (e.g., Si) atom Figure 2.3 The bonding model. 2.2.1Bonding Model Theisolated Siatom,or a Siatom not interacting withotheratoms,wasfound tocontain four valenceelectrons.Si atomsincorporated in thediamondlattice,on theotherhand, exhibit a bonding that involvesan attraction between eachatomand its four nearest neigh- bors (referto Fig.1.4c).Theimplication here is that,in going from isolatedatoms tothe collective crystalline state,Si atomscome to share oneoftheir valence electrons with each of thefour nearestneighbors.Thiscovalentbonding,orequalsharing of valenceelectros with nearest neighbors,and the mere fact thatatoms in the diamond lattice have four nearest neighbors,giverise to theidealized semiconductorrepresentation,the bonding model, shown inFig.2.3.Eachcircle inthe Fig.2.3bondingmodelrepresents thecoreofa semi- conductoratom,while each line represents a shared valenceelectron.(Thereare eight lines connected toeachatom because any givenatom notonlycontributesfour shared electrons butmustalsoacceptfour shared electronsfromadjacentatoms.）Thetwo-dimensional nature of the model is,ofcourse,an idealization thatfacilitates mental visualizations. Althoughconsiderable use will be madeof the bonding model insubsequent discus- sions,it is nevertheless worthwhileat this pointtoexamine sampleapplicationsofthe modeltoprovidesomeindicationof themodel’sutility.Twosampleapplications arepre- sented inFig.2.4.In Fig.2.4(a)we use the bonding modelto picture a point defect,a missing atom,in the lattice structure.In Fig.2.4(b)we visualize the breaking of anatom- to-atom bond and theassociated release or freeing ofan electron.[Bond breaking (at T> OK)and defectsoccur naturally inallsemiconductors,and hence (if we maybesomewhat overcritical)thebasicmodelofFig.2.3isstrictlyvalidforanentiresemiconductoronlyat T≈0Kwhenthe semiconductor in question contains no defects and no impurityatoms.] 2.2.2 Energy Band Model If our interests were restricted to describing thespatial aspects ofevents taking place within a semiconductor,the bonding model alone would probably be adequate.Quite often, how- ever,one is more interested in the energy-relatedaspectsof anevent.In such instances the bonding model,whichsays nothingaboutelectronenergies,isof little valueandtheenergy band model becomes the primary visualization aid.

--- Page 50 --- CARRIERMODELING 27 (a) (b) Figure2.4Sampleutilizationofthebondingmodel.（a）Visualizationofamissingatomorpoint defect.(b)Breaking of an atom-to-atom bond and freeing of an electron. Let us begin theconceptual pathleading to the energy bandmodelbyrecalling the situationinside an isolated Siatom.Reviewing theSection2.1discussion,tenof the14 electrons insidean isolatedSiatomaretightlyboundto thenucleusandare unlikelytobe significantlyperturbedbynormalatom-atom interactions.Theremaining fourelectrons are rather weakly boundand,ifunperturbed,occupyfouroftheeightallowedenergy states immediatelyabovethe last core level.Moreover,it is implicitly understood thattheelec- as the atoms are isolated,that is,far enough apart so thatthey arenoninteracting. Given the foregoing knowledgeof the isolatedatom situation,thequestion next arises as to whether we can use theknowledge to deduce informationaboutthe crystalline state. Assuredly we can omitanyfurther mentionof the core electrons because these electrons are not significantlyperturbedbynormalinteratomicforces.Theopposite,howeverisru ofthe valenceelectrons.IfNatomsarebroughtintocloseproximity(thecaseincrystalline Si),it is quitereasonableto expectamodification in theenergy statesofthe valence electrons. Themodification in the valence-electronenergystatesactuallyknown to take place is summarized inFig.2.5.Starting with N-isolated Siatoms,andconceptuallybringing the atoms closerandclosertogether,onefindsthe interatomicforcesleadtoa progressive spread in theallowed energies.The spread in energies givesrise to closely spaced setsof allowed states known as energy bands.At the interatomic distance corresponding to theSi lattice spacing,the distributionofallowed statesconsistsof twobands separatedbyan intervening energy gap. The upper bandofallowed states is calledtheconduction band; the lowerbandof allowedstates,thevalence band;andthe intervening energy gap,the forbidden gapor band gap.Infilling theallowed energy band states,electrons tendto gravitatetothelowestpossibleenergies.Noting thatelectronsarerestrictedtosingleo- cupancy inallowedstates(thePauliexclusionprinciple)andrememberingthatthe4N valence band states can just accommodate what were formerly 4N valence electrons,we typicallyfindthatthevalencebandisalmostcompletelyfilled withelectronsandthe

--- Page 51 --- 28 SEMICONDUCTORFUNDAMENTALS E (electron N Isolated E energy) Si-atoms E Etop 4N allowed states Mostly Electron energy Electron energy (conduction band) empty Ec P $n=3NostatesEGS$Ev}$ 4N allowed states Mostly 2N s-states total 6N p-states total (valence band) filled Ebottom (4N electrons total) Crystalline Si x N-Atoms ↑ Electron energy P 4N empty states S -2N+2N filled states 1 decreasing 1 1 atom spacing isolated Si lattice Si atoms spacing Figure2.5 Conceptualdevelopmentoftheenergybandmodel starting withNisolatedSiatoms onthetopleftandconcluding with a“dressed-up”versionoftheenergyband modelon thetopright. conductionbandisallbutdevoidofelectrons.Indeed,thevalencebandiscompletely filled and theconductionbandcompletelyempty at temperatures approaching =0 K. Tocompleteourplausibilizationoftheenergybandmodel,we needto introduceand utilizeoneadditionalfact:Unlike thevalenceelectrons in theisolated-atomcase,theband electronsincrystallinesiliconarenottiedtoorassociatedwithanyone particularatom. True,ontheaverage,onewilltypicallyfindfourelectronsbeing shared betweenanygiven Si atomanditsfournearest neighbors (as inthebondingmodel).However,the identityof the sharedelectronschanges as a functionoftime,withtheelectrons moving around from pointto pointinthecrystal.Inother words,theallowedelectronic statesare nolonger atomic statesbutareassociated withthecrystalas a whole;independentofthe pointex-

--- Page 52 --- CARRIER MODELING 29 Ec Ev Figure2.6 Theenergy band diagram-widelyemployed simplified versionof theenergy band model. amined in a perfectcrystal,one sees thesameallowed-stateconfiguration. Wetherefore concludethatforaperfectcrystalunderequilibriumconditions aplotoftheallowedelec- tron energies versus distancealong anypreselectedcrystalline direction (always calledthe x-direction)is asshownontheright-handsideofFig.2.5.Thecitedplot,a plotofallowed electronenergy states as a functionof position,is the basic energy band model.Eintro- duced in theFig.2.5plot is thelowest possibleconduction band energy,Eis the highest possible valence bandenergy,and EG=-Eis the band gapenergy. Finally,Fig.2.6displaystheformofthe basicenergybandmodelencounteredin practice.Inthis widelyemployed“shorthand”version,thelinetoindicatethetopnergy in theconduction band,the line to indicate thebottomenergy in the valenceband,the fil pattern drawingattentiontomostlyfilledstates inthe valence band,the labelingof they- orelectron-energy axis,andthe labelingof the x-or positionaxis areallunderstood toexist implicitly, but are not shown explicitly. 2.2.3 Carriers With the semiconductormodels having beenfirmly established,we are now in a position tointroduceandto visualizethecurrent-carryingentitieswithinsemiconductors.Referring to Fig.2.7,we notefirstfrom part (a)thatthereare nocarriersor possiblecurrentflow if the bondingmodelhasnobrokenbonds.Equivalently,intheenergy bandmodel,ifthe valence band iscompletelyfilled withelectronsandtheconductionband is devoidofelec- trons,thereare nocarriers or possiblecurrentflow. Thislackofcarriersandassociated current flow is easy to understand in termsof the bonding model where the shared electrons are viewed as being tied to theatomic cores.As accurately portrayed in the energyband model,however,the valence bandelectronsactuallymove about in thecrystal.How is it then that nocurrentcanarisefrom this groupofelectrons?As it turns out,the momentum oftheelectronsisquantizedinadditiontotheirenergy.Moreover,foreachand every possiblemomentum state in aband,there is anotherstate withanoppositely directed mo- mentumofequalmagnitude. Thus,if a band is completelyfiled withelectrons,the net momentumof theelectrons in the band is always identically zero.Itfollows that no current can arise from the electrons in a completely filled energy band. The electrons that do giverise tocharge transportare visualizedin Fig.2.7(b).When a Si-Sibond is brokenand theassociatedelectron is free to wanderaboutthelattice,the released electron is acarrier.Equivalently,in termsof the energy band model,excitation of valence band electrons into theconduction bandcreates carriers; thatis,electrons in the

--- Page 53 --- 30 SEMICONDUCTORFUNDAMENTALS Empty Ec $ECompletelyfilled,(a)NocarriersEc$Ev$(b)Theelectron$Ec$Ev$ 10 (c) The hole Figure2.7Visualizationofcarriers using thebondingmodel（(left)andtheenergybandmodel (right).(a)No-carrier situation;(b)visualizationofanelectron;(c)visualizationofahole. conduction band are carriers.Note that the energy required to break a bond in the bonding model and the band gap energy,G,are one and the same thing.Likewise,freed bonding- model electronsand conduction bandelectronsare justdifferent names for the sameelec- trons;in subsequent discussions the word”electrons,when used without a modifier,will be understood to refer to these conduction band electrons. In addition to releasing an electron,the breaking of a Si-Si bond also creates a missing bond or void in thebonding structure. Thinking intermsof the bondingmodel,one can visualize the movementof thismissing bondfrom place to place in the lattice as a result of nearbybound electrons jumping into the void (see Fig.2.7c).Alternatively,one can think interms of the energy band model where the removal of an electron from the valence band creates anemptystate inanotherwise vast seaoffilled states.Theempty state,like abubble in a liquid,movesaboutratherfreely inthe latticebecauseofthecooperativemotionofthe valence bandelectrons.What we havebeendescribing,the missing bond in thebonding

--- Page 54 --- CARRIER MODELING 31 scheme,theemptystate inthevalenceband,isthesecondtypeofcarrierfoundinsemi- conductors—the hole. Althoughperhaps notobviousfrom thepreceding introductory description,theva- lence bandholeis anentityonanequalfooting withtheconductionbandelectron.Both electrons and holes participate in the operation of most semiconductor devices.Holes are even theprimarycarrierinsomedevices.Asonesfamiliaritywithcarriermodelingrows, the comparable statusof electronsandholes becomes more and more apparent.Ultimately, it is commonplace to conceive of the hole as a type of subatomic particle. 2.2.4BandGap and MaterialClassification Weconcludethis sectionbycitingaveryimportanttiebetweenthebandgapofamaterial, the numberof carriersavailablefor transport in a material,and theoverallnatureofa material.As it turns out,although specificallyestablishedforsemiconductors,theenergy bandmodelofFig.2.6can beappliedwithonly slightmodificationtoallmaterials.The majordifferencebetweenmaterials liesnot inthe natureoftheenergybands,butrather in the magnitude of the energy gap between the bands. Insulators,as illustrated in Fig.2.8(a),arecharacterized by very wideband gaps,wih EGfortheinsulatingmaterials diamond andSiO2being ≈5eVand≈8eV,respectively.In these wide band gap materialsthethermalenergy availableatroomtemperature excites very fewelectronsfromthe valencebandintotheconduction band;thus veryfew carriers exist insidethematerial and thematerial istherefore apoorconductorofcurrent.The band Few electrons $Ec$ EG=1.42eV(GaAs) $Ec$ Wide $8eV(SiO2)EG=1.12eV(Si)E5eV(Diamond)EG=0.66eV(Ge)$E_v}$(Roomtemperature)Thermalexcitation$Ev}$moderatelyeasy(a)Insulator(b)SemiconductorVeryEc$E narrow E $E or (c) Metal Figure 2.8 Explanation ofthe distinction between(a）insulators，(b） semiconductors,and (c) metals using the energy band model.

--- Page 55 --- 32 SEMICONDUCTORFUNDAMENTALS gap in metals,by wayofcomparison,is either very small,or noband gap exists at all due to anoverlapofthe valenceandconductionbands(seeFig.2.8c).There is always an abundanceofcarriers inmetals,andhence metals are excellentconductors. Semicon- ductors present an intermediate case between insulators and metals.At room temperature (T=300K),EG=1.42eVin GaAs,EG=1.12eVin Si,andEG=0.66eVin Ge. Thermal energy,by exciting electrons from the valence band into the conductionband, creates a moderate numberofcarriers in these materials,giving rise in turnto a current- carrying capability intermediate between poor and excellent. 2.3CARRIERPROPERTIES Having formallyintroduced theelectronandthehole,we next seektolearn asmuchas possibleaboutthe natureof thesecarriers.Inthis particularsection weexamineacollage of carrier-related information,informationofa generalnatureincluding general facts, properties, and terminology. 2.3.1 Charge Both electronsand holesarechargedentities.Electrons are negativelycharged,holesare positivelychargedandthemagnitudeofthecarierchargeisthesameforthetwotys of carriers. To three-placeaccuracy inMKSunits,q=1.60×10-19 coul.Please note that,undertheconventionadoptedherein,theelectronandholechargesare -qand+q, respectively; i.e.,the sign of the charge is displayed explicitly. 2.3.2 Effective Mass Mass,like charge,is another verybasicpropertyofelectronsandholes.Unlikecharge, however,the carrier mass is nota simplepropertyandcannotbe disposedofsimplyby quoting anumber.Indeed,theapparentor“effective”massofelectrons withinacrystalis a functionof thesemiconductormaterial(Si,Ge,etc.)andis differentfromthemassof electrons within a vacuum. Seeking toobtain insight into theeffectivemassconceptletus firstconsiderthemo- tionofelectrons ina vacuum.If,asillustrated inFig.2.9(a),anelectronofrestmassm0is moving in a vacuum between two parallel plates under the influenceof an electric field, then,according to Newton’s second law,the force Fon theelectron will be du F=-q8= m0 (2.2) 1p wherev istheelectronvelocityandtis time.Nextconsiderelectrons(conductionband electrons)moving between the two parallelend faces of a semiconductorcrystal under the influenceofanappliedelectricfield,asenvisioned inFig.2.9(b).Does Eq.(2.2) alsode- scribe the overallmotionofelectrons withinthe semiconductorcrystal?The answer is no.

--- Page 56 --- CARRIERMODELING 33 8 8 (a) (b) Figure2.9 An electronmoving inresponse to anappliedelectricfield(a)withinavacuum,and (b) within a semiconductor crystal. Electrons moving inside asemiconductorcrystalwillcollide withsemiconductoratoms, therebycausing a periodic decelerationofthecarriers.However,should not Eq.(2.2)apply to the portionof the electronic motion occurring between collisions? The answer is again no.Inadditiontotheappliedelectricfield,electronsinacrystalarealsosubjecttocomplex crystalline fields not specifically included in Eq.(2.2). The foregoing discussiondelineated certain importantdifferences betweenelectrons in acrystal andelectronsina vacuum,but it leftunresolvedtheequally importantquestion as to howone properly describes themotionofcarriers in acrystal.Strictly speaking,the motionof carriers ina crystalcanbe describedonlybyusing QuantumMechanics,the formalismappropriateforatomic-sized systems.Fortunately,ifthedimensionsofthecrys- tal are large compared to atomic dimensions,the complex quantum mechanical formulation for thecarriermotionbetweencollisionssimplifiestoyieldanequationofmotionidentical to Eq.(2.2),except thatm0is replaced by an effectivecarriermass.Inother words,forthe Fig.2.9(b)electrons one can write du F=-q8=m^ * (2.3) n dt where m * is theelectroneffectivemass.Asimilarequationcanbe writtenforholes with n u pue b←b_ →m .Ineach case theinternalcrystallinefieldsandquantum- n P mechanicaleffectsareall suitablylumped intotheeffectivemassfactormultiplying du/dt. This is a very significantresult.Itallows us toconceiveofelectronsandholes as quasi- classical particles and toemploy classical particle relationships in most device analyses. Althoughtheeffectivemassformulation is a significantsimplification,itshouldbe mentionedthatthecarrieracceleration can vary with the direction of travel in acrystal;ie., the effective masses can have multiple components. Moreover, depending on how a mac- roscopicobservable isrelated to the carriermotion,a differentgrouping of mass compo- nents canleadto a different m* being utilized in a particularrelationship.Thereare,for example,cyclotron resonanceeffective masses,conductivityeffective masses,densityof

--- Page 57 --- 34 SEMICONDUCTORFUNDAMENTALS Table2.1 Density of States Effective Masses at 300 K. Material m^{* $/m0$m^{*}/m{0P Si 1.18 0.81 Ge 0.55 0.36 GaAs 0.066 0.52 stateseffectivemassesamongother.Itisalsoprobablynottoosurprisingthattheeffective masses varyomewhtwitempeature.Heeinwemakedirectuseofonlythedensiof states effectivemasses (see Subsection2.4.1).The densityofstateseffectivemassesfor electrons and holes in Si,Ge,andGaAs at 300K are listed in Table2.1. 2.3.3Carrier Numbers in Intrinsic Material The term intrinsic semiconductor in common usage refers to an extremely pure semicon- ductor sample containing an insignificantamountof impurityatoms.More precisely,an intrinsic semiconductoris a semiconductor whose propertiesare native to the material (hat is,not causedbyexternaladditives).The numberofcarriers in an intrinsic semiconductor fitsintotheschemeofthingsin thatitisanidentifiableintrinsicpropertyofthematerial. Defining,quite generally, n = number of electrons/cm 3 p = number of holes/cm³ existinginsideasemiconductor,thengivenanintrinsicsemiconductorunderequilibrium conditions, one finds n=p=n_{ (2.4) andn{}=2×10^{6}/cm^{³ in GaAs ≈1× 1010/cm^{3 in Si at room temperature ≈2×10{3/cm{3 in Ge The electronandholeconcentrations inan intrinsicsemiconductor areequalbecause carriers withina verypurematerialcanbecreatedonly inpairs.Referringto Fig.2.7,ifa semiconductorbond is broken,a free electronand a missing bondor holeare createdsi- multaneously.Likewise,theexcitationof anelectron fromthe valenceband into the con-

--- Page 58 --- CARRIER MODELING 35 ductionbandautomaticallycreates a valencebandholealong withtheconductionband electron.Alsonotethattheintrinsic carrierconcentration,althoughlarge inanabsolute sense,is relatively smallcompared withthenumberofbonds thatcouldbebroken.For example,in Sithere are5×10²² atoms/cm3andfourbondsperatommaking agrand totalof2×10²bondsorvalencebandelectronspercm.Sincen≈101/cm3onefins less thanone bond in10broken inSiatroomtemperature.Toaccuratelyrepresentthe situation inside intrinsic Si atroom temperature,we couldcoveralltheuniversity chalk- boards in the world with the bonding modeland possibly showonlyone broken bond. 2.3.4 ManipulationofCarrier Numbers—Doping Doping,in semiconductorterminology,is theadditionofcontrolled amountsof secific impurityatoms withthe expresspurposeof increasingeithertheelectronor theholecon- centration.Theaddition of dopants in controlled amounts to semiconductor materialsoc- curs routinely inthe fabricationofalmostall semiconductordevices.CommonSi dopants are listed in Table2.2.Toincrease theelectron concentration,one canaddeither phospho- rus, arsenic,or antimonyatoms to theSi crystal,with phosphorusfollowedcloselyby arsenicbeing the most commonly employed donor (electron-increasing) dopants.Toin- crease the hole concentration,oneadds either boron,gallium,indium,or aluminum atoms tothe Sicrystalwithboron beingthemostcommonlyemployedacceptorhole-ineasin) dopant. To understand how the addition of impurity atoms can lead to a manipulation of carrier numbers,it is importanttonotethatthe Table2.2donors areallfromColumnVinthe PeriodicTableofElements,whileallofthecitedacceptorsarefromColumnIIIinthe Periodic TableofElements.As visualized inFig.2.10(a)using the bonding model,when a ColumnVelement withfive valence electrons is substitutedfor a Siatom in the semicon- ductorlattice,fourofthefivevalenceelectronsfit snuglyinto thebonding structure.The fifth donorelectron,however,does notfitintothebonding structureand is weaklybound to the donor site.Atroom temperature this electron isreadilyfreed to wanderaboutthe lattice andhence becomes a carrier.Please note thatthis donation (hence thename”do- nor”)of carrierelectrons does not increase the holeconcentration. The donor ionleft be- hind when the fifthelectron isreleasedcannotmove,andthere are nobrokenatom-atom bonds associated with the release of the fifth electron. Table2.2 CommonSilicon Dopants.Arrows indicate the most widely employed dopants. Donors (Electron-increasing dopants) Acceptors (Hole-increasing dopants) B← P← ColumnV Ga Column III As← elements In elements Sb A1

--- Page 59 --- 36 SEMICONDUCTORFUNDAMENTALS p+ B (a) (b) Figure2.10Visualizationof(a）donorand(b)acceptoraction using the bonding model.In(a)the ColumnVelementPis substitutedforaSiatom;in (b)theColumnIIIelementB is substitutedfora Si atom. The explanationofacceptoractionfollowsasimilar lineofreasoning.TheColumnIII acceptors have three valenceelectrons and cannot completeoneofthe semiconductor bonds when substitutedforSi atoms inthe semiconductor lattice (see Fig.2.1Ob).The ColumnIIIatom,however,readilyaccepts （hence the name“acceptor”）an electronfrom a nearby Si-Si bond,thereby completing its own bonding scheme and in the process cre- ating ahole thatcan wander about the lattice.Hereagain there is an increase inonlyone typeofcarrier.The negativelycharged acceptorion (acceptoratomplusacceptedelectron) cannot move,and no electrons are released in the hole-creation process. The foregoing bonding-model-based explanationof dopant action is reasonably un- derstandable.There are,nevertheless,a fewloose ends.Forone, we noted that thefifth donor electron was rather weakly boundand readily freed at room temperature. How does one interpretthe relativeterm”weakly bound”?It takes ≈leV to break Si-Si bondsand very few of the Si-Sibonds are broken at room temperature. Perhaps”weakly bound” means a binding energy ≈0.1 eVor less? The question also arises as to how one visualizes dopant action in termsof theenergy band model.Bothof thecited questions,as it turns out, involve energy considerations and are actually interrelated. Letusconcentratefirstonthebindingenergyofthefifthdonorelectron.Crudely speaking,the positivelychargeddonor-core-plus-fifth-electron may be likenedtoa hydro- gen atom (see Fig.2.1).Conceptually,the donor core replaces the hydrogen-atom nucleus and the fifth donor electronreplaces the hydrogen-atom electron.In the real hydrogenatom the electron moves of course in a vacuum,can be characterized by the mass of a free electron,and,referring to Eq.(2.1),has a ground-state binding energy of-13.6eV.In the pseudo-hydrogen atom,on theother hand,theorbiting electron moves through a seaofSi atoms and is characterized by an effective mass.Hence,in the donor or pseudo-atomcase, the permittivityof free space must be replaced by the permittivity of Si and m must be replaced bym*WethereforeconcludethatthebindinenegyBofthefifhdonor electron is approximately

--- Page 60 --- CARRIER MODELING 37 P+ Figure 2.11 Pseudo-hydrogen atom model for the donor-site bond. m^{\^{} m{ 1 EB} ≈ n $EHln=1=-0.1eV(2.5)$2(4πK_sε0{)^{}m_{$whereKistheSidielectricconstant（K=11.8).Actualdonor-sitebindingenergiesinSiarelistedinTable\mathrm{2.3.}TheobservedbindingenergiesareseentobeingoodagreementwiththeEq.(2.5)estimateandconfirmingtheearlierspeculation,areroughly=1/20theSibandgapenergy.Havingestablishedthestrengthofdopant-sitebonds,wearenowinapositiontotackletheproblemofhowonevisualizesdopantactionusingtheenergybandmodel.Workingontheproblem,wenotefirstthatwhenanelectronisreleasedfromadonoritbecomesaconductionbandelectron.Iftheenergyabsorbedatthedonorispreciselyequaltotheelectronbindingenergy,thereleasedelectronwillmoreoverhavethelowestpossibleen-ergyintheconductingband—namely,.AddinganenergyEBtotheboundelectron,inotherwords,raisestheelectro{n}^{\prime }senergytoE.Hence,weareledtoconcludethattheboundelectronoccupiesanallowedelectroniclevelanenergyEBbelowtheconductionbandedge,or,asvisualizedinFig.2.12,donorsitescanbeincorporatedintotheenergybandschemebyaddingallowedelectroniclevelsatanenergyED=-EB|.Notethatthedonorenergylevelisrepresentedbyasetofdashes,insteadofacontinuousline,be-causeanelectronboundtoadonorsiteislocalizedinspace;i.e.aboundelectrondoesnotleavethegeneral∆xvicinityofthedonor.TherelativeclosenessofEDtoEcofcourereflectsthefactthatE-ED=|B|\approx （1/20)EG(Si).Table2.3Dopant-SiteBindingEnergies.Donors$|E{B$Acceptors$|E{B|$ Sb 0.039eV B 0.045eV P 0.045eV Al 0.067eV As 0.054eV Ga 0.072eV In 0.16eV

--- Page 61 --- 38 SEMICONDUCTORFUNDAMENTALS Δx 11 0.045 eV ！ Ec (P) T ED (Si)1.12 eV Donor site Ev x Figure2.12 AdditionoftheE=EDdonor levels to theenergy band diagram.Dashesof widthΔ emphasize the localized nature of the bound donor-site states. Theactualvisualizationofdopantactionusing theenergybandmodel is picturedin Fig.2.13.Examining theleft-handsideofFig.2.13(a),one findsallthedonor sitesfilled withboundelectrons attemperaturesT→0K.This is truebecause very littlethermal energy isavailabletoexciteelectronsfromthedonorsitesintotheconduction bandatthese very lowtemperatures.Thesituation changes,ofcourse,as thetemperature is increased, with more and more of the weakly bound electrons being donated to the conduction band. Atroomtemperature theionizationofthedonors isall buttotalgivingrisetothesituation pictured at the extreme rightofFig.2.13(a).Although we have concentratedon donors,the situationforacceptors iscompletelyanalogous.As visualized inFig.2.13(b),aceptors introduceallowedelectronic levels intotheforbiddengapatanenergy slightlyabove the valence band edge.At low temperatures,allof these sites will be empty—there is insuffi- E{ ED $Ev$ T→OK Increasing T Room temperature (a) $Ec$ $EA$ Ev}$ O 。 O O T→0K Increasing T Room temperature (b) Figure2.13Visualizationof(a)donorand(b)acceptoractionusingtheenergybandmodel.

--- Page 62 --- CARRIER MODELING 39 cientenergyat temperatures T→0Kfor a valence bandelectron to make the transition to an acceptor site.Increasingtemperature,implying an increased storeof thermalenerg, facilitates electrons jumping from thevalence bandontotheacceptor levels.Theremoval of electrons from the valence band ofcourse creates holes.At room temperature,essentially allof theacceptor sites are filed withanexcesselectronand an increased holeconcentra- tion is effected in the material. To complete this subsection,a few words are inorderconcerning the doping of non- elementalsemiconductorssuchas GaAs.Dopantaction in GaAsfollowsthe samegeneral principlesbut is slightlymore involved becauseof theexistenceof twodifferent lattice-site atoms.Completelyanalogous to the dopingofa Si crystal,theColumnVIelementsS, Se, and Te actas donors whenreplacing theColumnVelement As in theGaAs lattice. Similarly,theColumnIelements Be,Mg,and Znactas acceptors whenreplacingthe ColumnIIelement Ga.A new situation arises whenColumnIVelements such as Siand Ge are incorporated intoGaAs.Si typically replaces Ga in the GaAs lattice and is a popular n-type dopant.However,under certain conditions Sican be made to replace As in the GaAs lattice,thereby functioning as an acceptor.In fact,GaAs pn junctions have been fabricated where Siwasboththep-sideandn-side dopant.Animpuritythatcanactaseitheradonor or an acceptor is referred to as an amphoteric dopant. Exercise 2.1 Energy Quiz P:(a)1eV is equal to how many joules of energy? (b) kT is equal to how many eV at 300 K? (c)Theionizationenergyofaceptorsanddonors inSiisroughlyequalto? (d) EG(Si) =_? (e) EG(SiO2)=? (f) The energyrequired to ionize a hydrogenatom initially inthe n=1state is__? S:(a)1eV=1.60×10-19joules. (c)TheionitionenegofdopansisequaltoEBofthedopantsites.Asdiscussed in Subsection2.3.4,EB0.1eVfordopansin S. (d) G(Si)=1.12eV at300 K(see Subsection2.2.4). (e) EG(SiO2) =8 eV (see Subsection 2.2.4). (f) Making useof Eq.(2.1)in Section 2.1,the hydrogen atomionizationenergy is |H|n=1|= 13.6eV.

--- Page 63 --- 40 SEMICONDUCTORFUNDAMENTALS 2.3.5 Carrier-Related Terminology Since terminology is oftena stumblingblockto understanding and since this particular section isreplete withspecializedterms,it seemsappropriatetoconcludethe section with anoverviewofcarrier-relatedterminology.Approximatelyone halfofthe carrier-related terms thatfollowwere introducedand definedearlier inthissection;theremainderofthe termsarebeing listedhereforthefirsttime.Allofthetermsarewidelyemployedandther definitions should be committed to memory. Dopants—specificimpurityatoms that areaddedto semiconductors in controlled amounts fortheexpress purposeof increasingeithertheelectronortheholeconcentration. Intrinsic semiconductor-undopedsemiconducto;extremelypuresemiconductorsample containingan insignificantamountofimpurityatoms;asemiconductor whoseprop- erties are native to the material. Extrinsic semiconductor-doped semiconductor;a semiconductor whoseproperties are controlled by added impurity atoms. Donor-impurityatomthat increasestheelectronconcentration;n-type dopant. Acceptor—impurityatom that increasestheholeconcentration;p-type dopant. n-typematerial-adonor-doped material;asemiconductorcontainingmoreelectronsthan holes. p-type material-anacceptor-doped material;asemiconductorcontainingmoreholesthan electrons. Majoritycarrier—themostabundantcarrierinagivensemiconductorsample;electrons in an n-type material,holes in a p-type material. Minoritycarrier-the leastabundantcarrier inagivensemiconductorsample;holes inan n-type material,electronsinap-typematerial. 2.4 STATEANDCARRIERDISTRIBUTIONS Up to this point in themodeling process we have concentrated on carrierproperties and informationof aconceptualqualitative,or,at most,semiquantitative nature.Practically speaking,there is often a need for more detailed information.For example,most semicon- ductorsare doped,andtheprecise numericalvalueofthe carrierconcentrationsinside doped semiconductors is of routine interest.Another property of interest is the distribution of carriers as a function of energy in the respective energy bands.In this section we begin the process of developing a more detailed description of the carrier populations. The de- velopment willeventually lead torelationshipsforthecarrier distributions andconcentra- tions within semiconductors under equilibrium conditions.

--- Page 64 --- CARRIER MODELING 41 2.4.1 Density of States Whentheenergybandmodelwasfirst introduced in Section2.2weindicatedthatthetotal numberof allowed states in eachband wasfourtimesthenumberofatoms inthecrystal. Not mentioned at the time was howtheallowed states were distributed inenergy;ie,how many states wereto be found at any given energy in theconduction and valence bands.We are now interested inthis energy distributionof states,or density of states,as itis more commonlyknown,becausethestate distribution is an essential component in determining carrier distributions andconcentrations. To determinethe desired densityof states,it is necessary to perform ananalysisbased on quantum-mechanicalconsiderations.Herein we willmerely summarize theresultsof theanalysis;namely,forenergiesnottoofarremovedfromthebandedges,onefind m^{n2 E- E{) ^{(E) E≥ Ec (2.6a) π{2{3} , -E) g{v( ) E≤Ev (2.6b) $\pi 23$ where gc(E)and g(E)are the densityof states at anenergy Ein theconductionand valence bands, respectively. What exactlyshouldbeknownandrememberedaboutthecited densityofstates?For one,itisimportantto graspthegeneral densityofstatesconcept.Thedensityofstatescan belikenedto the descriptionoftheseatingina footballstadium,withthe numberofseats in the stadium a given distance from the playing field corresponding to the number of states a specifiedenergy intervalfromEor E.Second,thegeneralformof therelationships shouldbenoted.AsillustratedinFig.2.14,gc(E)is zeroatEandincreasesasthesquare rootofenergy whenone proceeds upward into theconductionband.Similarly,g(E)is precisely zeroatEand increases with thesquarerootofenergyasone proceeds downward from Eintothe valence band.Also note that differences betweengc(E)and g(E) stem from differences in the carrier effective masses. If m * were equal to m*,the seating (stats) n on both sidesof the footballfield(the band gap)wouldbe mirror imagesofeachother. Finally,considering closely spacedenergies EandE+dEin therespectivebands,one can state $gc(E)dErepresentsthenumberofconductionbandstates/cm3lyingintheenergyrangebetweenEandE+dE(ifE\ge E),$g(E)dE represents the numberof valence band states/cmlying in the energyrange between EandE+dE(ifE≤E).

--- Page 65 --- 42 SEMICONDUCTORFUNDAMENTALS E c g^{c { 1 1 E 1 g_y( {) $Ev$ Figure2.14Generalenergydependenceof gc(E)and (E)nearthe bandedges.g(E)andg(E) arethe densityof states in theconductionand valence bands,respectively. Itthereforefollowsthatgc(E)and g(E)themselves are numbers/unitvolume-unitenergy, or typically, numbers/cmeV. 2.4.2TheFermiFunction Whereasthedensityofstatestellsonehowmanystatesexistata givenenergy E,theFermi functionf(E)specifies how manyof the existing states at theenergyE willbefilled with an electron,or equivalently, f(E)specifies,underequilibriumconditions,theprobabilitythatanavailablestateat an energy E will be occupied by an electron. MathematicallpeakngtheFermiunctionissimplrobailitisiutioion. In mathematical symbols, 1 f(E) = (2.7) 1+e(E-E)/kT

--- Page 66 --- CARRIERMODELING 43 /3—=( f(E) -12 1 \fra} {2 } f(E）≧e-(E-E）/kT T 0 E} 0 Er} E E E-3kTE+3kT (a) T→0K (b) T >0 K with the energy coordinate expressed in kT units. where EF=Fermienergy or Fermilevel k=Boltzmann constant (k=8.617×10-5eV/K) T= temperature inKelvin(K) Seeking insight into the natureof the newly introducedfunction,letus beginby inves- tigatingtheFermifunction’s energy dependence.Considerfirsttemperatures where T→ OK.AsT→0K,(E-E)/kT→-∞forallenergiesE<Eand(E-EF)/kT→+∞ forallenergies E>E.Hencef(E<)→1/[1+exp(-∞)]=1and(E>)→1/ [1+exp(+∞)]=0.Thisresult is plotted inFig.2.15(a)and is simply interpreted as meaningthatallstatesatenergiesbelowEFwillbefilledandallstatesatenergiesabove EF willbeemptyfortemperaturesT→0K.Inother words,there is a sharpcutoff in the filling ofallowed energy states attheFermienergyEF whenthesystem temperatureap- proaches absolute zero. Let us next consider temperatures T>0K.Examining the Fermi function, we make the following pertinent observations. (i) IfE=EF,f(EF)=1/2. (ii) IfE≥EF+3kT,exp[(E-)/kT]≥1andf(E)≈exp[-(E-E)/kT].Conse- quently,above EF3kTthe Fermi functionor filled-state probability decaysexpo- nentiallyto zero withincreasingenergy.Moreover,most states atenergies3kTor more above EF will beempty.

--- Page 67 --- 44 SEMICONDUCTORFUNDAMENTALS (iii)IfE≤EF-3kT,exp[(E-EF)/kT]≤1andf(E)≈1-exp[(E-E)/kT].Below EF-3kT,therefore,[1-f(E)],theprobabilitythata givenstate willbeempty, decays exponentially to zero withdecreasing energy.Moststates atenergies 3kTor more below EF will be filled. Comparedto the Si band gap,the3kTenergy intervalthat appears prominently in the T>0Kformalism is typically quite small. Theproperties justcited arereflectedand summarized intheT>0KFermifunctionplot displayed in Fig. 2.15(b). Before concluding the discussionhere,it is perhaps worthwhile to reemphasize that the Fermifunctionappliesonly underequilibriumconditions. The Fermifunction,how- ever,is universalinthesensethatitapplies withequalvaliditytoallmaterialsinsulators, semiconductors,andmetals. Although introduced inrelationship to semiconductors, the Fermi function is not dependent inany wayon the special natureof semiconductors but is simplyastatisticalfunctionassociatedwithelectronsingeneral.Finally,therelativeposi- tioningoftheFermienergyEFcomparedtoE（orE),an itemofobviousconcern,is treated in subsequent subsections. Exercise 2.2 P:Theprobabilitythatastateisfilledattheconductionbandedge（E)isprecisely equaltotheprobabilitythatastateisemptyatthe valencebandedge(E).Where is the Fermi level located? S:The Fermi function,f(E),specifies theprobabilityofelectronsoccupying states Eis equal to 1-f(E).Here we are told f(Ec)=1-f(E) Since 1 f(E{)= 1+e(Ec-EF)/kT and 1 1 1-f(Ey)=1- 1+e(Ev-EF)/kT 1+e(EF−Ev)/kT

--- Page 68 --- CARRIERMODELING 45 we conclude E{c- $EF$E$-$E_ kT kT or $Ec+E$

2 The Fermi level is positioned at midgap. (C) Exercise 2.3 In this exercise we wish to examine in more detailhow the Fermi function varies with temperature. P:Successively setting T=100,200,300,andthen400K,computeandplotf(E) versus △E=E-EFfor-0.2eV≤ΔE≤0.2eV.Allf(E)versus △Ecurves should be superimposed on a single set of coordinates. S: MATLAB program script… %FermiFunctionCalculation,f(∆E,T) % Constant k=8.617e-5; %Computation proper for ii=1:4; T=100i; kT=kT; dE(ii,1)=-5kT; for jj=1:101 f(ii,jj)=1/(1+exp(dE(ii,jj)/kT)); dE(ii,jj+1)=dE(ii,jj)+0.1kT; end end dE=dE(:,1:jj); %This step strips the extra dE value

--- Page 69 --- 46 SEMICONDUCTORFUNDAMENTALS %Plotting result close plot(dE’,f); grid;%Note thetranspose()toform datacolumns xlabel(‘E-EF(eV)’);ylabel(‘f(E)’); text(.05,.2,‘T=400K’);text(-.03,.1,T=100K’); Program output … 1 0.8 0.6 E 0.4 T=400K 0.2 T=100K … 0 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 E-EF(eV) 2.4.3 Equilibrium Distribution of Carriers Havingestablishedthedistributionofavailablebandstatesandtheprobabilityoffilling those states underequilibrium conditions,we can noweasily deduce the distributionof carriers in the respective energy bands.To be specific,the desired distribution is obtained by simply multiplying the appropriate densityof states by the appropriate occupancy factor—gc(E)f(E) yields the distribution of electrons in the conduction band and gy(E)[1-f(E)]yieldsthe distributionofholes(unfilled states)in thevalence band. Sample carrier distributions forthreedifferentassumed positionsof the Fermi energy (along withassociated energy band diagram,Fermifunction,and densityof states plots) are pictured in Fig. 2.16. Examining Fig.2.16,we note in general thatallcarrier distributionsare zero atthe

--- Page 70 --- CARRIERMODELING 47 Energy band Density Occupancy Carrier diagram of states factors distributions E 1 1-f(E) 1 1 Ec $gc(E)$gc(E)f(E) →$8(E)gv(E)[1-f(E)]Ev$ f(E) $Ev$ (a) EF above midgap E Electrons Ec Ec $gc(E)1$EF$E$Ev}$Holes(b)EFnearmidgapEEc$Ecgc(E) 1 —EF g_) Ev 1 Ev (c)EF belowmidgap Figure2.16Carrierdistributions(notdrawntoscale)intherespectivebandswhentheFermilevel is positioned(a) abovemidgap,(b)near midgap,and(c) below midgap.Also shown ineachcase are coordinated sketchesof the energy band diagram,densityofstates,and theoccupancy factors (he Fermi functionand one minus the Fermifunction).

--- Page 71 --- 48 SEMICONDUCTORFUNDAMENTALS band edges,reach a peak value veryclose to Eor E,andthen decay very rapidlytoward zero as one moves upward into the conduction band or downward into the valence band.In other words,mostofthecarriersaregroupedenergetically in the near vicinityof theband edges.AnothergeneralpointtonoteistheeffectoftheFermilevelpositioning onthe relative magnitudeof thecarrierdistributions.WhenEFis positioned inthe upperhalfof theband gap (or higher),theelectron distribution greatlyoutweighsthehole distribution. Althoughboththefiled-stateoccupancyfactor f(E)]andtheempty-stateoccupancy factor [1-f(E)]falloffexponentiallyasoneproceedsfromthebandedgesdeeperintothe conductionandvalence bands,respectively,[1-f(E)] is clearly muchsmaller thanf(E) atcorrespondingenergiesifEFlies intheupperhalfoftheband gap.Lowering EFeffec- tivelyslidestheoccupancyplotsdownwardgivingrisetonearlyequalnumberofcariers when EFis at themiddleof the gap.Likewise,a predominanceofholesresults whenE lies below themiddleof the gap. The argument here assumes,of course,that g(E)and gv(E)areofthesameorderofmagnitudeatcorrespondingenergies(asis thecase for Si and Ge).Also,referring backto theprevious subsection,the statements concerningthe occupancyfactorsfallingoffexponentiallyintherespectivebandsarevalidprovidd E-3kT≥EF≥E+3kT. Theinformation justpresentedconcerning thecarrier distributionsandtherelative magnitudesofthecarrier numbers finds widespreadusage.Theinformation,howeveris often conveyed in anabbreviated or shorthandfashion.Figure2.17,for example,showsa common wayofrepresentingthe carrier energy distributions. Thegreatestnumberof circlesordotsaredrawnclosetoEandE,reflectingthepeakinthecarrierconcentrations neartheband edges. The smallernumberof dots as one progresses upward into thecon- ductionbandcrudelymodelstherapidfallofinthelctrondensitywithincreasingerg An extensively utilized meansofconveying the relativemagnitudeof thecarriernumbers is displayed inFig.2.18.Torepresentanintrinsicmaterial,adashedlineis drawninap- proximatelythemiddleofthebandgapandlabeledEThenear-midgappositioningofE, theintrinsicFermilevel,isofcourseconsistentwiththepreviouslycitedfactthattheelec- tron and hole numbers areaboutequal whenEis nearthecenterofthe bandgap.Similarly, a solidline labeledEF appearingabovemidgaptellsoneata glancethat the semiconductor in question isn-type;a solidlinelabeledEFappearingbelowmidgapsignifiesthatthe semiconductor is p-type.Note finallythatthe dashedElinealso typicallyappearsonthe -Ec Ev 。 Figure2.17 Schematic representationof carrier energy distributions.

--- Page 72 --- CARRIER MODELING 49 Ec$Ec$ -Ec -E$E；-Ei$ E; -Ev}Ev$Ev$ Intrinsic n-type p-type Figure2.18 8”Ataglance”representationofintrinsic(left),n-type(middle),andptype(right) semiconductor materials using the energy band diagram. energybanddiagramscharacterizingextrinsicsemiconductors.TheEline insuchcases representstheexpected positioningof theFermilevelif thematerial were intrinsic,and it serves as a referenceenergy level dividing the upperand lower halvesofthe band gap. 2.5 EQUILIBRIUMCARRIERCONCENTRATIONS Wehavearrivedatan important point in thecarriermodelingprocess.Forthemostpart, this section simplyembodies theculminationofourmodelingefforts,with working rela- tionshipsfortheequilibriumcarrierconcentrationsbeingestablished tocomplementthe qualitativecarrier informationpresented in previoussections.Unfortunately,the emphasis on the developmentofmathematicalrelationshipsmakesthefinalassaulton the carrier modelingsummitalittlemorearduousand perhapsjustbitboring).Hopefullythereader can stay focused.A comment is also inorderconcerning the presentation herein of alter- native formsfor thecarrierrelationships.Thealternativeformscanbe likened to thediffer- ent kindsof wrenchesused,forexample,inhomeand automobilerepairs. Theoen-end wrench,thebox wrench,and theratchet wrenchall serve thesame general purpose.In some applicationsonecan use anyofthe wrenches.Inotherapplications,however,aspe- cial situationrestricts the typeof wrenchemployedorfavors the useofone wrenchover another.Thesameistrueofthealternativecarrierrelationships.Finally,boxesaredrawn around expressions that find widespread usage.A single-walled box signifies amoderately important result; a double-walled box, a very important result. 2.5.1Formulas for nand p Since gc(E)dErepresentsthe numberofconductionband states/cm3lyingin theEtoE+ dEenergyrange,andfE)specifies theprobabilitythatanavailablestateatanenergyE willbeoccupiedby an electron,it thenfollows thatgc(E)f(E)dEgives the numberof conductionbandelectrons/cm3lying in theEtoE+dEenergy range,and g(E)f(E)dE integratedoverallconduction band energiesmust yieldthetotal numberofelectrons inthe conductionband.Inotherwords,integrationovertheequilibriumdistributionofelectrons in theconductionband yields theequilibriumelectronconcentration.Asimilar statement can be made relative to the hole concentration. We therefore conclude

--- Page 73 --- 50 SEMICONDUCTORFUNDAMENTALS Etop n 二 gc(E)f(E)dE (2.8a) JEc $v$ g(E)[1−f(E)]dE (2.8b) p

Ebottom Seeking explicitexpressions forthecarrierconcentrations,letus focusoureffortson the n-integral.(Theanalogous p-integral manipulations are left to the reader as an exer- cise.) Substitutingthe Eq.(2.6a)expression for gc(E)and the Eq.(2.7)expression forf(E) into Eq.(2.8a),one obtains m^{*\2 Etop √E-Ec dE (2.9) n =二 π{2{3} JE{1+ e(E-EF)/kT Now letting (E-Ec) (2.10a) n

kT (EF-E{) η_c (2.10b) kT E_{op→ (2.10c) yields m* \2m{ (kT )^3/2∞ $\eta d$ n n (2.11) n $\pi 23$ Jo 1+en-n_c The(2.1Oc)simplificationontheupperintegration limitmakes useofthefactthatthe integrand inquestionfallsoff rapidly withincreasing energyand isessentially zerofor energies only a fewkT above E.Hence,extending theupper limit to ∞has a totallyneg- ligible effect on the valueof the integral. Even withthecited simplification,the Eq.(2.11)integral cannotbeexpressed in a closedformcontaining simplefunctions.The integral itself is in facta tabulatedfunction that can be found in a variety of mathematical references.Identifying ∞ $\eta d$ F{2{c= the Fermi-Dirac integralof order1/2 (2.12) Jo 1+en-nc ,

--- Page 74 --- CARRIER MODELING 51 and also defining m{kT 3/2 $Nc=2the“effectiv{e}^{\prime }densityofconductionbandstates(2.13a)$2πh^{} , m^{*kT 3/2 N_v}2 the“effective’density of valence band states (2.13b) $2\pi h^{}$ one obtains 2 $n=Nc$ \or}{} F_{12{ (2.14a) and, by analogy, 2 p= N_vF{1/2() (2.14b) \r} whereη=(E-EF)/kT. The Eq.(2.14)relationships are a very general result,validfor anyconceivable posi- tioning of the Fermi level. The constants Nc and N are readily calculated; at 300 K, NC.V=(2.510×10^{/cm{³)(m n,p’ from availabletables,from plots,or by drectcomputation.The general-form relationships, nonetheless,are admittedly cumbersome and inconvenient to use in routine analyses. For- tunately,simplifiedclosed-form expressions doexist thatcan beemployed inthe vast ma- jorityofpracticalproblems.Tobespecific,ifEFisrestrictedto values EF≤E-3k pue0=)=[°u-l）-x≈[u-）dx+u Vπ{12}n{c=$e(EF-c)/kT(2.15a)2Likewise,ifEF\ge E+3kT,then$}F_{12}{=$ e(Ev-EF)/kT (2.15b) 2

--- Page 75 --- 52 SEMICONDUCTORFUNDAMENTALS 3kT EF here …. Degeerate $Ec$ semiconductor EF here … Nondegenerate semiconductor Ev} EF here … Degenerate 3kT semiconductor Figure2.19 Definition of degenerate/nondegenerate semiconductors. Itthereforefollows that,ifE+3kT≤EF≤Ec-3kT, n= NCe(F−EckT (2.16a) p= Nve(Ev-EF)/kT (2.16b) The mathematical simplification leading to Eqs.(2.16) is equivalentto approximating the occupancy factors,f(E) and1-f(E),by simple exponential functions—an approxi- mation earlier shown to be valid providedEF was somewhere in the band gap no closer than 3kTto either bandedge.WheneverEFis confined,as noted,to E+3kT≤E≤ E-3kT,insteadofcontinually repeatingthe EFrestriction,the semiconductor is simply said to be nondegenerate.Whenever EF lies in the band gap closer than 3kTto either band edge or actually penetrates oneof thebands,the semiconductor is said to be degenerate. These very important terms are also defined pictorially in Fig.2.19. 2.5.2 Alternative Expressions for nand p Although inclosedform,the Eq.(2.16)relationshipsare not in the simplestform possible, and,moreoftenthannot,it is thesimpleralternative formof theserelationships thatone encounters in device analyses. Thealternative-formrelationships can beobtained by re- calling thatE,theFermi levelfor an intrinsic semiconductor,liesclosetomidgap,and hence Eqs.(2.16)most assuredly apply to an intrinsic semiconductor.If this be the case, then specializingEqs.（2.16)toanintrinsicsemiconductor,ie.settingn=p=nandE= $E, one obtains n₁=NCe(Ei-c)/kT (2.17a)

--- Page 76 --- CARRIER MODELING 53 and n₁=Nve(Ev-E₁)/kT (2.17b) Solving Eqs.(2.17)forNandNyyields NC=n₁e(Ec-E₁)/kT (2.18a) NV=n;e(E₁-Ev)/kT (2.18b) Finally,eliminatingNandNintheoriginalEq.(2.16)relationshipsusingEqs.(2.18) gives n=n;e(EF−E₁)/kT (2.19a) p=n;e(Ei-EF)/kT (2.19b) Like Eqs.(2.16),the Eq.(2.19)expressions are valid for any semiconductor in equilib- rium whose doping is such as to give rise to a nondegenerate positioningof the Fermi level. Whereas two constantsand three energy levels appear intheoriginalrelationships,how- ever,onlyoneconstantand twoenergy levels appear in thealternativerelationships.Be- cause of their symmetrical nature,the alternative expressions are also easier to remember, requiring merely an interchange of EF and Ein going from the n-formulato the p-formula. 2.5.3n,and thenp Product As canbeinferredfrom itsappearance in Eqs.(2.19),theintrinsic carrierconcentration canfigure prominently inthequantitativecalculationofthecarrierconcentrations.Con- tinuing toestablish pertinentcarrierconcentrationrelationships,we next interjectconsid- erations specifically involving this important material parameter. First,ifthecorrespondingsidesofEqs.(2.17a)and(2.17b)aremultipliedtogether, one obtains {2=NCNVe-(E−Ev)/kT=NCNVe−EG/kT (2.20) or n₁=√NCNve-EG/2kT (2.21)

--- Page 77 --- 54 SEMICONDUCTORFUNDAMENTALS $10^{1}11÷14÷$10^{5 Ge Si T(℃) $n_{;}(cm-3)$ 0 8.86× 108 $10Si51.44\times 109102.30\times 109153.62\times 109$10{}$205.62\times 109258.60\times 109301.30\times 1010Intrinsiccarrierconcentration(cm3)GaAs$10^{{2}$351.93\times 1010402.85\times 1010454.15\times 1010505.97\times 1010$10$1300K1.00\times 1010$1010 GaAs T(℃) $n_{;}(cm-3)$ $10$ : 0 1.02× 105 5 1.89× 105 10 3.45× 105 $10^8$ 15 6.15 5× 105 20 1.08 × 106 25 1.85 × 106 $10^{}$ . 30 3.13 ×106 2: 35 5.20 ×106 ÷ -÷ 40 8.51 ×106 45 1.37 ×10 $10^{6}502.18\times 101300K2.25\times 106$10^{5 200 300 400 500 600 700 T(K) Figure 2.20 IntrinsiccarrierconcentrationsinGe,Si,andGaAsas afunctionoftemperatur.

--- Page 78 --- CARRIER MODELING 55 Equation（2.2)expressesnasafunctionofknownquantitiesandcanbeusedtocomue intrinsiccarrierconcentrationinSiandGeatroomtemperaturewerecitedpreviously;the best available plotsofnas functionsof temperature in Si,Ge,and GaAs are displaedin Fig. 2.20. Asecondveryimportant-basedrelationshipfollowsdirectlyfromEqs.(2.)Mu- tiplying thecorresponding sides of Eqs.(2.19a)and(2.19b)together yields np =n{} (2.22) Althoughappearing trivial,thenpproductrelationship(Eq.2.22)oftenprovestobe extremelyuseful inpractical computations.Ifoneofthecarrierconcentrations isknown, the remaining carrier concentration is readily determined using Eq.(2.22)—providedof course that the semiconductor is in equilibrium and nondegenerate. (C) Exercise 2.4 Substituting the Eq.(2.13）definitionsofNCandNinto Eq.(2.21),introducing normalizationfactors,and replacingknown constants by theirnumerical values,one obtains 3/4 3/2 T m* n e-E/2kT m0 m{0 300 EG and theeffective masses exhibit a weak but non-negligible temperature depen- dence. The EG versus T variationcan be modeledto four-placeaccuracy by thefit relationship noted inProblem2.1(a).As deduced from the analysis by Barber[Solid- StateElectronics,10,1039(1967)],thetemperature dependenceoftheeffective masses over the range 200K≤T≤700K can be approximated by $m\ast$ n 1.028+(6.11×10-4)T-(3.09 ×10-7)T² m{o m*$

0.610+(7.83×10-4)T-(4.46×10-7)2 mo P:(a) Confirm that the n versus Tcurve for Si graphed in Fig.2.20 is generated employing therelationships justcited,provided EG in the nexpression is replaced byEG- Eex，where Eex=0.0074eV.(In the previously cited article,Barber sug- gested usinganexcitoncorrectionfactorofEex=0.007eV.The slightly larger value employed in the Fig.2.20computation was specifically chosen to give n=100/ cm^{3}at 300 K.)

--- Page 79 --- 56 SEMICONDUCTORFUNDAMENTALS (b) The accepted valueof nin Si at 300K has been revisedrecently to be in agree- ment withtheexperimental n versusTdataacquired bySproulandGreen[Journal of AppliedPhysics,70,846(July1991)].Theauthorsconcludedn=1.00±0.03 ×1010/cm3in Si at 300K.Over the probed temperature range of 275K≤T≤ 375K,their experimental data could be fitted by the relationship 2 e-0.5928/kT 300 Comparethecomputedromthisexprimentalfitrlationhiptothepart)outu over the temperature range of mutual validity. S: (a) MATLAB program script . .. %ni vs.Tcalculation for Si(200K-700K)used in Fig.2.20 %Initialization format short e %Constants and T-range k=8.617e-5; A=2.510e19; Eex=0.0074; %Value was adjusted to match S&G ni(300K)value T=200:25:700; %Band Gap vs.T EGO=1.17; a=4.730e-4; b=636; EG=EGO-a.(T.^2)./(T+b); %Effective mass ratio (mnr=mn/m0,mpr=mp*/m0) mnr=1.028+(6.11e-4).*T-(3.09e-7).T.^2; mpr=0.610+(7.83e-4).T-(4.46e-7).T.^2; %Computation of ni （（**/x-）-）x（)（))）（/）= %Display output on screen j=length(T); fprintf(‘\n \n T ni\n’); %There are ten spaces between Tand ni. for ii=1:j, fprintf(’%-10.f%-10.3e\nT(i),ni(ii)); end

--- Page 80 --- CARRIER MODELING 57 (b) MATLAB program script . .. %ExperimentalfitofSproul-Green ni data(275K-375K) %ni calculation T=275:25:375; k=8.617e-5; ni=(9.15e19).*(T./300).^2.exp(-0.5928./(kT)); %Display result on screen j=length(T); fprintf(‘\n\n T ni\n’);%There are ten spaces between T and ni. for ii=1:j, fprintf(’%-10.f%-10.3e\n’,T(ii),ni(ii)); end Output from thetwo programs isreproduced below.Note the excellentagreement of the computationalresults(to within2%) over the temperature range of mutual validity. T(K) (a)n;(cm-{3) (b)n;(cm- {3) 200 5.246×104 275 1.059×109 1.051×109 300 1.000×1010 1.006×1010 325 6.798× 1010 6.887×1010 350 3.565×1011 3.623×1011 375 1.518×1012 1.542×1012 400 5.449×1012 500 2.716×1014 600 3.988× 1015 700 2.865×1016 2.5.4Charge NeutralityRelationship Therelationshipsestablishedtothispointare devoidofanexplicitdependenceonthe dopantconcentrations introduced into asemiconductor.It is thecharge neutralityrelation- ship that provides thegeneral tie between thecarrierand dopantconcentrations. Toestablish the charge neutrality relationship, let us consider a uniformly doped semi- conductor,a semiconductor where the number of dopant atoms/cm3is the same every- where.Systematicallyexamining littlesectionsof the semiconductor far fromany surfaces, and assuming equilibriumconditions prevail,one must invariably findthateachandevery sectionischarge-neutraliecontainsnonetcharge.Ifthiswerenotthecaseelectricfields

--- Page 81 --- 58 SEMICONDUCTORFUNDAMENTALS wouldexist insidethesemiconductor. Theelectric fields in turn would giverise to carrier motion andassociated currents—a situation totally inconsistent withtheassumed equilib- rium conditions. Thereare, however,charged entities inside all semiconductors.Electrons, holes,ionized donors(donoratoms thathave become positively charged by donating an electronto theconductionband)and negatively-charged ionizedacceptors canall exist simultaneously inside any given semiconductor. For the uniformly doped material tobe everywhere charge-neutral clearly requires charge

qp-qn+q-qN=0 (2.23) $c{m}^3$ or p-n+N-NA=0$(2.24)where,bydefinition,N=numberofionized（positivelycharged)donors/cm,N=numberofionized(negativelycharged）acceptors/cm.Aspreviouslydiscussedthereissufficientthermalenergyavailableinasemiconduc-toratroomtemperaturetoionizealmostalloftheshallow-leveldonorandacceptorsites.DefiningND=totalnumberofdonors/cm3NA=totalnumberofacceptors/cm3,andsetting$N{}DN{{} NA}$ one then obtains assumes total ionization p-n+ND-NA=0 (2.25) of dopant atoms Equation (2.25) is the standard form of the charge neutrality relationship.

--- Page 82 --- CARRIER MODELING 59 2.5.5CarrierConcentration Calculations Wearefinally ina positiontocalculatethecarrierconcentrations in a uniformlydoped semiconductor under equilibrium conditions.In the computations to be presented we spe- cificallymake theassumptionsof NONDEGENERACY(allowing us to use the np product relationship)and TOTALIONIZATIONof the dopantatoms.Note that ,which appears in the np productexpression,has beencalculatedand plotted and mustbeconsidered a known quantity.Likewise,NAand ND,whichappear in thechargeneutralityrelationship, are typicallycontrolledanddeterminedexperimentallyandshouldalsobeconsidered known quantities.Theonlyother symbols used in thetwo equations are n and p.Thus, under thecited assumptionsof nondegeneracy and total ionizationofdopantatoms,we have two equations and two unknowns from which n and p can be deduced. Starting with the np product expression,one can write n^{{}{2} (2.26) p

n Eliminatingpin Eq.(2.25)using Eq.(2.2)gives n^{{}{2} -n +ND -NA= 0 (2.27) n or n^{2-nND-A)-n{2 (2.28) Solving the quadratic equation for n then yields N{D-$NA2(2.29a)n=2and[(Na-Nn)+$n^{a{{2}N_{ANDD$ 2 1/2 P

(2.29b) n 2 Onlythe plusroots have beenretained in Eqs.(2.29)because physically thecarrierconcen- trations must be greater than or equal to zero.

--- Page 83 --- 60 SEMICONDUCTORFUNDAMENTALS Equations (2.29)are general-case solutions.In the vastmajorityofpractical compu- tations it is possible to simplifythese equationspriorto substituting in numerical values for ND,NA,and n.Specialcasesof specificinterest areconsiderednext. (1）IntrinsicSemiconductor（NA=0,ND=O).WithNA=0andND=0,Eqs.(2.29) simplify to n=n;and p=n.n=p=nis of course the expected result for the equilibrium carrier concentrations in an intrinsic semiconductor. (2) DopedSemiconductor where either ND-NA=ND≥n₁orNA-ND=NA>n₁. Thisisthespecialcaseofgreatestpracticalinterest.Theunintentionaldopinglevlsin Si are such that thecontrolledadditionofdopants routinely yields ND≥NAor NA> ND.Moreover,the intrinsic carrierconcentration in Si atroom temperature is about 1010/cm3,whilethedominantdopingconcentration（NAor ND)isseldomlessthan 1014/cm3.Thusthespecialcaseconsidered hereis the usualcaseencountered inprac- tice.IfND-NA≈ND≥n,thesquareroot in Eq.(2.29a)reducesto ND/2and n =N (2.30a) ND>NA，ND>n₁ $p\approx$n{2D (nondegenerate, total ionization) (2.30b) Similarly p =N{ (2.31a) NA>ND，NA>n₁ n =n{2A (nondegenerate, total ionization) (2.31b) As a numerical example,suppose a Si sample maintained at room temperature is uni- formlydoped withND=101/cm3donors.Using Eqs.(2.30),onerapidlyconcludes n≈$1015/cm3andp=10³/cm^3 (3) Doped Semiconductor where n;≥|ND-NA|.Systematically increasing theami- ent temperature causes a monotonicrise in theintrinsic carrierconcentration(see Fig.2.20).At sufficientlyelevated temperatures,nwill eventuallyequal and thenex- ceed the net doping concentration.Ifn≥|ND-NA|,the square roots in Eqs.(2.29) reduce to nandn=p≈n.Inother words,allsemiconductorsbecome intrinsic at sufficiently high temperatures where n;≥|ND-NA|. (4) Compensated Semiconductor. As is evidentfrom Eqs.(2.29),donors and acceptors tend to negateeachother.Indeed,it is possible to produce intrinsic-like material by making ND-NA=0.In some materials,such as GaAs,NA may be comparable to ND in the as-grown crystal.When NAand NDare comparable and nonzero,the material is said to be compensated.Ifthe semiconductor is compensated,both NA and NDmust be retained in all carrier concentrationexpressions.

--- Page 84 --- CARRIER MODELING 61 In summary,Eqs.(2.29)can always be used to compute the carrier concentrationsif the semiconductor is nondegenerate and thedopantatoms are totally ionized.In thevast majorityofpracticalsituations,howeveritispossibletosimplifytheseequations priorto performing numericalcomputations.Equations(2.29)must be usedtocompute the carier concentrations only in those rare instances when |Np-NA|~n.The simplified relation- ships ofgreatest practical utilityare Eqs.(2.30) and(2.31). Exercise 2.5 P:A Si sample is doped with 1014 boron atoms per cm3. (a) Whatare thecarrierconcentrations in the Si sample at300K? (b) What are the carrier concentrations at 470K? S:(a)Boron in Siis anacceptor(see Table2.2).Thus NA=1014/cm3.At300K, n=1.00×10/cmandthegiven NAisclearlymuchgreaterthannMoreover, since the ND doping was omitted from the problem statement, we infer ND≤NA. WithNA≥nand NA≥ND,Eqs.(2.31)maybeusedtocalculatethecarrierconcen- trations:p=NA=10/cm3；n=n²/NA=106/cm3. (b)As deducedfrom Fig.2.20,n≈104/cmat 470 K.Because n;is comparable to NA,Eqs.(2.29)must be used to calculate at leastoneof the carrier concentra- tions.(Once oneof the carrier concentrations is known,the second is more readily computed using thenpproductexpression.)Performing the indicatedcalculations givesp=NA12+[(NA/2）²+n²]1²=1.62×104/cm³；n=n²/p=6.18× $10$1{3}/cm^{3 2.5.6DeterminationofEF Knowledge concerning theexact positionof the Fermi level on the energy band diagram is often of interest. For example, when discussing the intrinsic Fermi level, we indicated that Eiwas located somewhere near the middleof the band gap.It wouldbeuseful toknow the precise positioning ofEin theband gap.Moreover, we have developed computational formulas forn and p appropriatefor nondegenerate semiconductors. Whether a doped semiconductor is nondegenerate or degenerate depends,of course,on the value or position- ing of EF. Beforerunning thoughthemechanics of finding the Fermi level in selected casesof interest,it is useful tomake a generalobservation; namely,Eqs.(2.19)or (2.16)[oreven more generally,Eqs.(2.14)] provide aone-to-one correspondence between the Fermien- ergy and the carrierconcentrations. Thus,having computed any one of the three var- ables—n,p,or EF—one can always determine the remaining two variables under equili- rium conditions.

--- Page 85 --- 62 SEMICONDUCTORFUNDAMENTALS (1)Exact positioningofE.Inanintrinsicmaterial $n=p(2.32)SubstitutingfornandinEq.(2.32)usingEqs.(2.16)andsettingEF=dsNCe(E-Ec)kT=Nve(Ev-E)/kT(2.33)SolvingforE,oneobtains(N)$E{c+E$kT$ $Ei==$ + ln (2.34) 2 2 But 3/2 N_v} m{ (2.35) $N$ 二 m{\} Consequently, E_{ }+E} 3 $m{

kTln (2.36) 2 4 $m\}$ According to Eq.(2.36),Elies precisely at midgaponly if m* =m* orifT= P n 0K.Forthe morepracticalcaseofsilicon atroomtemperature,Table2.1gives belowmidgap.Although potentiallysignificantin certainproblems,this small devia- tion from midgap is typically neglected in drawing energy band diagrams,etc. (2) Doped semiconductors(nondegenerate,dopantstotallyionized).Thegeneral position- ing of the Fermi level in donor- and acceptor-doped semiconductors assumed tobe nondegenerate,inequilibrium,and maintained at temperatures where the dopantsare fully ionized,is readily deducedfrom Eqs.(2.19).Specifically,solving Eqs.(2.19)for EF-Ej,one obtains EF-E=kTln(n/n)=-kTln(p/n) (2.37) Depending on thesimplifications inherent in a particularproblem,theappropriatecar- rierconcentrationsolution[Eqs.(2.29),(2.30),(2.31),etc.]is thensubstitutedinto Eq.(2.37)to determine the positioning of F.For example,per Eqs.(2.30a)and (2.31a),n≈NDin typical donor-doped semiconductors andp≈NAin typical accep-

--- Page 86 --- CARRIER MODELING 63 tor-doped semiconductors maintained ator nearroomtemperature.Substituting into Eq.(2.37), we therefore conclude EF-E₁=kTln(ND/n) ..ND≥NA，ND >n{ (2.38a) $E-EF=kTln(NA/n_1)...NA>ND，NA>n(2.38b)FromEqs.(2.38)itisobviousthattheFermilevelmovessystematicallyupwardinenergyfromEwithincreasingdonordopingandsystematicallydownwardinenergyfromE;withincreasingacceptordoping.TheexactFermilevelpositioninginSiatroomtem-perature,nicelyreinforcingtheforegoingstatemenisdisplayedinFig.2.2.Alonotethatforagivensemiconductormaterialandambienttemperature,thereexistmaximumnonde-generatedonorandacceptorconcentrations,dopingconcentrationsabovewhichthematerialbecomesdegenerate.InSiatroomtemperaturethemaximumnondegeneratedopingcon-centrationsareND\approx 1.6\times 1018/cm3andNA\approx 9.1\times 1017/cm.ThelargeSidopingvaluesrequiredfordegeneracy,weshouldinterject,haveledtothecommonusageof"highlydoped"（orn+-material/p+-material)and"degenerate”asinterchangeableterms.Finally,thequestionmayarise:WhatprocedureshouldbefollowedincomputingEwhenoneisnotsurewhetheramaterialisnondegenerateordegenerate?Unlessamaterialisknowntobedegenerate,alwaysassumenondegeneracyandcomputeEFemployingtheappropriatenondegeneraterelationship.IfEFderivedfromthenondegenerateformulaliesinthedegeneratezone,onemustthen,ofcourse,recomputeEFusingthemorecomplexformalismvalidfordegeneratematerials.EcEF-Donor-doped3kTE$E$EF-Acceptor-doped3kTEv1$10^{3}10{f}10^{ }10^{10{ 10{8 10^{1 $10^{ NA Oor ND(cm^-^³3) Figure2.21 Fermi level positioninginSiat300Kasa functionofthedopingconcentration.The solidElines wereestablishedusingEq.(2.38a)fordonor-dopedmatrialandE.2.38b)foraccep- tor-doped material (kT=0.0259eV,andn=1010/cm³).

--- Page 87 --- 64 SEMICONDUCTORFUNDAMENTALS Exercise 2.6 P:Foreachoftheconditions specified in Exercise2.5,determine thepositionofE, computeEF-E,and drawacarefully dimensionedenergy band diagram fortheSi sample.Note that EG(Si)=1.08eVand m*/m* =0.71at470K. p n S:（a)In part（a)ofExercise2.5,the NA=1014/cm3Si sampleis mainted at300K. Using Eq.(2.36),weconclude Eislocated0.0073eVbelowmidgap.(Theposition- ing of Ein Si at300Kis also noted inthe textfollowing Eq.2.36).Nextapplying Eq.(2.38b), we find $E-EF=kTln(NA/n_1)=0.0259ln(1014/1010)=0.239eVTheenergybanddiagramconstructedfromthededucedpositioningofEandEFisshowninFig.E2.6(a).(b)Inpart(b)ofExercise2.5theSisampleisheatedto470K.Withm\ast /m\ast =0.71andkT=0.0405eVat470K,(3/4)kTln(m\ast /m\ast ）=-0.0104eVandEisdeducedtobelocated0.0104eVbelowmidgap.BecauseNAiscomparabletonat470K,Eq.(2.37)mustbeusedtocomputethepositioningofEF.Specifically,with=1014/cm3andp=1.62\times 1014/c{m}^{3}3,E-EF=kTln(p/n_1)=0.0405ln(1.62\times 1014/1014)=0.0195eVHereEFisonlyslightlyremovedfromEaspicturedinFig.E2.6(b).-cEc$ _-Ei $E_i$ EF EF Ev $E_v$ (a) (b) Figure E2.6

--- Page 88 --- CARRIER MODELING 65 2.5.7 Carrier Concentration Temperature Dependence Anumberofisolatedfactsaboutthecarrierconcentrationtemperature dependence have alreadybeenpresentedatvariouspoints inthischapter.Thediscussion inSection2.3con- cerned withdopantaction,for example,describedthe increased ionizationof dopantsites and theassociated increase inthe majority carrier concentration when the temperatureofa semiconductor is raisedfrom nearT=0Ktoward room temperature.More recentsub- sectionshaveincluded a plotoftheintrinsic carrierconcentration versus temperature (Fig.2.20)andacalculationindicatingthatallsemiconductors becomeintrinsic(n→n, p→natsufficientlyhightemperatures.Inthissubsectionwhichconcludesthecarrier concentration discussion,temperature-related facts are combined and embellished topro- vide a broader,more complete descriptionof how the carrier concentrations vary with temperature. Figure2.22(a),a typicalmajority-carrierconcentration-versus-temperatureplotcon- structedassuming a phosphorus-dopedND=10/cm3Sisample,nicelyillustratesthe generalfeaturesofthe concentration-versus-temperature dependence. ExaminingFig. 2.22(a),we find that n is fixed atapproximately NDover abroadtemperature rangeextend- ing roughlyfrom150 K to 450K for the given Si sample. This n=NDor”extrinsic temperature region’constitutes thenormaloperating range formost solid-statedevices. Below100Kor so,inthe“freeze-outtemperature region,”n drops significantlybelow NDand approaches zero asT→0K.Inthe”intrinsictemperature regionatthe oposite endofthetemperaturescale,nrisesabove ND,asymptoticallyapproachingwithincras- ingT. To qualitatively explainthe just-describedconcentration-versus-temperature depen- dence,itisimportanttorecallthattheequilibriumnumberofcarriers withinamaterialis affected by two separate mechanisms.Electrons donatedtotheconduction band from do- nor atoms and valence bandelectronsexcited across the band gap into theconduction band (brokenSi-Sibonds）bothcontributeto themajority-carrierelectron concentrationina donor-doped material.At temperaturesT→0Kthe thermal energy available in the system is insufficienttoreleasetheweaklyboundfifthelectronondonorsitesandtotallyinsuffi- cient to excite electrons across the band gap.Hence n=0atT=0K,as visualizedon the left-hand sideofFig.2.22(b).Slightly increasing thematerialtemperatureaboveT=0K “defrosts”or frees some ofthe electrons weakly bound to donor sites.Band-to-band exci- tation, however,remains extremely unlikely, and therefore thenumberof observed elec- trons in thefreeze-outtemperature regionequals the numberof ionized donorsn=N. Continuing to increase the system temperature eventually frees almostallof the weakly bound electronson donor sites,n approaches ND,andoneenters theextrinsic temperature region.Inprogressing through the extrinsic temperature region,moreand more electrons are excited across the bandgap,butthenumberofelectrons supplied in this fashion stays comfortably below ND.Ultimately,ofcourse,electronsexcited across the band gap equal, then exceed,and,as picturedon theright-handsideofFig.2.22(b),finally swamp thefixed number of electrons derived from the donors. As a practical note,it shouldbe pointedout that the wider the band gap,the greater the energy required toexciteelectronsfrom the valence band into theconduction band, and

--- Page 89 --- 66 SEMICONDUCTORFUNDAMENTALS 2.0 1 Freeze 1 1.5 out 1 -Extrinsic T-region n 1.0 Intrinsic T-region 0.5 $n;/N$ 1 1 0 100 200 300 400 500 600 T(K) (a) n= 0 n = N} $n=N$ n =n_} 44 44 Ec E{e -Negligible -Negligible -Dominant E_v} OK LowT Moderate T HighT (b) Figure2.22 (a) Typical temperature dependence of the majority-carrierconcentration in a doped semiconductor.Theplot wasconstructedassuming aphosphorus-doped ND=101/cmSisamle. n/NDversusT（dashed line)hasbeenincludedforcomparison purposes.(b)Qualitativeexplanaion of the concentration-versus-temperature dependence displayed in part (a). the higher the temperature at the onsetof the intrinsic temperature region.Since thetem- perature attheonsetoftheintrinsictemperatureregioncorresponds totheupperendofthe normal operating range for most solid-state devices,GaAs devices can inherentlyoperate at a higher maximum temperature than similarly doped Si devices, which in turn can op- erate at a higher maximum temperature than similarly doped Ge devices.If we assume,for examplethecriticaldopingconcentrationisND=101/cmandtheonsetoftheintrinsic temperature region is approximated as the temperature where n=ND,then from Fig.2.20

--- Page 90 --- CARRIERMODELING 67 themaximumoperating temperatures are deducedto be385K,540K,and>700Kfor Ge, Si,andGaAs,respectively.Indeed,GaAs and SiC(G>2eV)devicescontinuetobe under development for use in high temperature environments. 2.6 SUMMARYANDCONCLUDINGCOMMENTS Under the general heading of carrier modeling we have described,examined, and charac- terized the carriers within a semiconductor under”rest”orequilibriumconditions.The many importanttopicsaddressed in this chapter included the introductionoftwo“visual- ization”models: thebondingmodeland theenergybandmodel. The extremely useful energy bandmodel is actuallymore than just a model—it is a sophisticated sign language providing a concise means of communicating on a nonverbal level.Relative to the carriers themselves,thereader bynow has beensuccessfullyprodded intothinking ofelectrons and holes asclassical ball-like”particles,wherethecharge onanelectron is-q,thecharge on a hole is +q, and the effective masses of the particles are m * andm * ,respectively. The n reader shouldalsoknowthatthecarriernumbers inanintrinsicmaterialare equaland relativelysmall;thecarrierconcentrationshowevercanbeselectivelyincreasedbydding special impurity atoms or dopants to the semiconductor. In addressing the problem of determining the carrier concentrations in doped semi- conductors, we developed or derived a numberof usefulmathematical relationships.The densityof states functions (Eqs.2.6),the Fermifunction (Eq.2.7),the symmetrical non- degenerate relationships fornandp(Eqs.2.19),thenpproduct(Eq.2.22),thecharge neutralityrelationship(Eq.2.25)andthesimplifiednandpexpressionsapproprateora typical semiconductormaintainedatroomtemperature(Eqs.2.30and2.31)deservespecial mention.Thecitedequationsand a fewothersarecollected in Table2.4.Withregardto the use of these relationships,thereader shouldbe cautioned against”no-thinkplug and chug.” Because semiconductor problems are replete withexceptions,special cases, and nonidealsituations,it is imperativethatthe formulauser beawareofderivationalassump- tions andthe validity limitsofanyandallexpressionsusedinananalysisorcomputation. In addition to the quantitative carrier relationships,the reader shouldalso have a qualitative “feel”for thecarrier distributions in the respective energy bands,the temperature depen- dence of the intrinsicconcentration,and the typical temperature dependence of the majority carrier concentration in a doped semiconductor. Finally,some attention should begiven to the many technical terms and the key para- metricvalueresntedinthichaptr.Thetrmsxrisicemconductodonorc nondegenerate semiconductor,Fermi level,andsoon,will beencounteredagainandagain in the discussion of semiconductor devices. Likewise,a knowledge of typical numerical values forkey parameters,suchas EG=1.12eVandn=1010/cm3in Siat300K,will be invaluable in subsequent work when performing both “back-of-the-envelope”and computer-assistedcomputations.Key parametric values also serve asyardsticksforgaug- ing whethernewlyencountered quantitiesare relativelysmallorrelatively large.

--- Page 91 --- 68 SEMICONDUCTOR FUNDAMENTALS Table2.4 Carrier Modeling Equation Summary. Density of States and Fermi Function m^{n2CE

E{ ^{( ) == E ≥c $\pi 23$ , 1 f(E)= $2m\ast (EE)1+e(E-EF)/kT$g{v(E)=$p$E≤E π^{2}{}3} Carrier Concentration Relationships n=NCe(EF-Ec)/kT 2 m^{*2k 3/2 F_{12{ n p=Nve(E-E)/kT $2\pi h^{}$ 2 m{2k 7 3/2 $p=N_v$ \rat} F_{2}{) N_v}=2 n=n;e(EF-E₁)/kT $2\pi h^2$ p=n;e(Ei-EF)/kT n,np-Product, and Charge Neutrality n₁=NCNve-EG2 np = n{^} P-n+ND-NA=0 n,p, and Fermi Level Computational Relationships 2 2+n^^$1/23$m$ n= 2 2

2 4

kTln $m$ $n=D$ ND≥NA，ND≥n₁ EF-E₁=kTln（n/n₁）=-kTln(p/n;) p≈n{2D p = N_{}$EF-E_1=kTln(ND/n）ND\ge NA，NnNA\ge ND，NA\ge n_1$n n{²}{A}$ E-EF=kTln(NA/n）NA≥ND，NA≥n

--- Page 92 --- CARRIER MODELING 69 PROBLEMS CHAPTER2 PROBLEM INFORMATIONTABLE Complete Difficulty Suggested Short Problem After Level Point Weighting Description 2.1 2.2.4 2 10 (5 each part) EG vs.T computation 2.2 2.3.4 1 10 (2 each part) Bonding model applications 2.3.4 (a-d) 2.3 1 24 (2 each part) E-band model applications 2.5.7 all 2.4 2.3.4 2 10(2 each part) Si-doped GaAs 2.5 2.4.1 2 5 States/cm3in△E 2.6 2.4.2 2 8(a-2,b-3,c-3) Fermi function questions 2.7 2.5.1 2-3 10 Find distribution peak 2.8 ” 2 5 Population at E+△E 2.9 ” 2 15 Plot distributions in bands 2.10 3 20(a-3,b-17..discuss-2) Tvariation of distribution 2.11 ” 2 10 Derive Eqs.(2.14b),(2.16b) 2.12 ” 3 15(a-3,b-12) Hypothetical gc= constant 2.13 ” 2 10(a-7,b-3) Compute NC,N_ 2.14 2.5.3 1 8(4 each part) n; comparison 2.15 ” 1 5 Plot n; vs.Tfor Ge 2.16 2.5.5 2 12(a::d-2, e-4) Tricky conc. questions 2.17 ” 1-2 10 (2 each part) Compute n and p 2.18 2.5.6 2 15 (3 each part) Determine E,EF-E,etc 2.19 ” 2 15(a-12,b-3) Computer check P2.17/2.18 2.20 ” 2 8 Verify nondegenerate limits 2.21 ” 2 10 PlotEF-E₁vs.NA，ND 2.22 ” 2-3 12(a-3,b-3,c-2,d-4) GaAs considerations 2.23 2.5.7 4 25(a-8,b-10,c-2,d-3,e-2) EF variation with T 2.1EG versus TComputation With increasingtemperatureanexpansionofthecrystal lattice usually leadstoaweakening of the interatomicbonds and an associated decrease in the band gap energy. For many semiconductors the cited variation of the band gap energy withtemperature can be modeled by the empirical relationship $\alpha T$ EG(T)=(O) )一 (T+β)

--- Page 93 --- 70 SEMICONDUCTORFUNDAMENTALS whereandareconstantschosentoobtainthebestfittoexperimentaldataand)is the limiting valueoftheband gapatOK.As far as Si is concerned,a fitaccurate to four places is obtained by employing EG(0)=1.170eV α=4.730×10-4eV/K β= 636 K Tin Kelvin (a) Make aplotofEG versusTfor Si spanning thetemperature rangefromT=0Kto T=600K.Specifically note the value of EG at 300K. (b) ForT>300K,thetemperature variation isnearlylinear.Notingthisfact,someau- thors have employed EG(T)=1.205-2.8×10-4T …T>300K Howdoesthissimplifiedrelationshipcompare withthemorepreciserelationshipoverthe temperature region of mutual validity? 2.2Usingthebondingmodelforasemiconductorindicatehowone visualizes(a)miss- ing atom,(b) an electron,(c)a hole,(d) a donor,(e) an acceptor. 2.3Using theenergy band model for a semiconductor,indicate how one visualizes(a)an electron,(b)a hole,(c)donor sites,(d)acceptor sites,(e)freeze-outof majority carrier electrons at donorsitesas thetemperatureis loweredtoward0K,(f)freeze-outofmajority carrier holes at acceptor sitesas thetemperature is lowered toward0K,(g)theenergy distributionofcarriersintherespectivebands(h)anintrinsic semiconductor,(i)ann-type semiconductor,(j) a p-typesemiconductor,(k)a nondegenerate semiconductor,(l) ade- generate semiconductor. 2.4 The bonding modelfor GaAs is pictured in Fig.P2.4. (a)Draw the bonding modelfor GaAs depicting the removalof the shaded Gaand As atomsinFig.P2.4.HINT:GaandAstaketheir bondingelectrons withthem whentey are removedfromthe lattice.Alsosee Fig.2.4(a)showing theresultsofremoving an atom from the Si lattice. (b)RedrawthebondingmodelforGaAsshowingtheinsertionofSiatomsintothemissing Ga and As atom sites. (c) Is the GaAs doped p-or n-type when Si atoms replace Ga atoms? Explain. (d) Isthe GaAsdoped p-or n-type when Siatomsreplace Asatoms?Explain. (e)Draw the energy band diagram for GaAs when theGaAs is doped with Sion(iGa sites and (ii) on As sites.

--- Page 94 --- CARRIER MODELING 71 Ga As Ga As Ga As Ga As Ga As Ga As Ga As Ga As Ga As Ga As FigureP2.4 2.5Developanexpressionfor thetotal numberof availableSTATES/cm3in theconduc- tion band betweenenergies Eand E+kT,whereγis an arbitraryconstant. 2.6（a)UnderequilibriumconditionsandT>0K,whatis theprobabilityofanelectron state being occupied if it is located at the Fermi level? (b)IfEF is positioned at E,determine (numericalanswerrequired)theprobabilityof finding electrons in states at E+kT. (c）TheprobabilitastateisfilledatEkTisequaltotheprobabilityastateisemptyat E+kT.Where is the Fermi level located? 2.7Thecarrierdistributionsornumbersofcarriersasafunctionofenergy intheconduc- tion and valence bands were noted to peak at an energy very close to the band edges.(See the carrier distribution sketches inFig.2.16.) Taking the semiconductor to be nondegener- ate, show that the energy at whichthecarrier distributions peak is E+ kT/2 and E-kT/2for theconductionand valence bands,respectively. 2.8Foranondegeneatesemiconductorthepeakinthelectrondistributionversusenerg insidetheconductionbandnotedinFig.2.16occursatE+kT/2.Expressed asafraction ofthe electronpopulation atthepeakenergy,whatis theelectronpopulationinanonde- generate semiconductor atE=E+5kT? 2.9TheFermi level in a Si sample maintained atT=300Kis located atE-E/4. Computeandplottheelectronandholedistributionsnumbers/cmeV)asafunctionof energy in theconduction and valence bands,respectively. ●2.10Letus investigatehowtheelectronenergy distributionin theconductionband varies as a function of temperature.

--- Page 95 --- 72 SEMICONDUCTORFUNDAMENTALS (a)Assuming the semiconductor to be nondegenerate and employing the Eq.(2.16a) ex- pressionfornconfirmthatthelectrondistributionintheconductionbandnormalized to the total electron concentration is given by $2\text{E}-Ec$ gc(E)f(E) e-(E-Ec)/kT $\text{π}(kT)32$ n (b)Computeand plotthe normalizedelectron distribution in theconductionbandversus E-Ecfor temperaturesT=300K,600K,and1200K.Plot the distribution values alongthex-axis(0≤gc（E)f(E)/n≤20eV-1)andE-Ec(0≤E-Ec≤0.4eV) along the y-axis on a single set of coordinates.Discuss yourresults. 2.11Starting with Eq.(2.8b)andfollowing a procedure analogous to thatoutlined in the text,presenttheintermediate stepsandargumentsleadingto Eqs.(2.14b)and (2.16b). 2.12Thedensityofstatesintheconductionbandofahypotheticalsemiconductoris $gc(E)=constant=NC/kT T …E≥c (a) Assuming EF<-3kT,sketchtheelectrondistributionintheconductionbandof the hypothetical semiconductor. (b)Followintheprocedureoutlinedinthetexestablishrelationshipsfortheectron 2.13(a）Verify thestatement in Subsection2.5.1that,at300K, NCv=(2.510×101/cm3)（m*/m0)3/2 whereonesets m*=min computing Nandm=min computingNv.m0= 9.109×10-31kg;h=6.625×10-34joule-sec;andq=1.602×10-19coul. (b)Usingtheeffectivemass valuesrecordedin Table2.1,constructatablethatliststhe numerical valuesof NC and Nyfor Si,Ge,and GaAs at 300K. 2.14(a) Determine thetemperature at whichthe intrinsic carrierconcentration in(i)Si and (ii)GaAsareequaltotheroomtemperature(00)intinsiccarrierconcenration of Ge. (b) Semiconductor A has a band gap of1eV,while semiconductor B has a band gap of2 eV.What is theratioof the intrinsic carrier concentrations in the two materials (niA/nB) at300K.Assume anydifferences in thecarrier effectivemasses may be neglected.

--- Page 96 --- CARRIER MODELING 73 ●2.15Confirmthatthe n; versus Tcurve for Ge graphed inFig.2.20canbegenerated employing the empirical-fit relationship n(Ge)=(1.76×1016)T31²e-0.392/kT 2.16 Concentration questions with a twist. (a)Asilicon waferis uniformlydoped p-type with NA=101/cm3.AtT≈0K,whatare the equilibrium hole and electron concentrations? (b) Asemiconductoris doped withanimpurityconcentrationNsuchthatN>andall theimpuritiesareionized.Also,n=Nand=n/N.Istheimpurityadonororan acceptor? Explain. (c) TheelectronconcentrationinapieceofSimaintainedat300Kunderequilibrium conditions is 10/cm3.What is the hole concentration? (d) Forasilicon samplemaintained atT=300K,the Fermi level islocated0.259eV abovethe intrinsic Fermi level.Whatare thehole andelectronconcentrations? (e) In anondegenerategermaniumsamplemaintainedunderequilibiumconditionsnear roomtemperatureitisknownthatn=103/cm,n=2p,andNA=0.Determinen and ND. 2.17Determinetheequilibiumlctonandholeconcnrtionsinideuniormlyoped sample of Si under the following conditions: (a)T=300K,NA≤ND，ND=$1015/cm^3. (b）T=300K,NA=1016/cm3,ND≤NA. (c)T=300K,NA=9×1015/cm^3,ND=1016/cm3. (d)T=450K,NA=0,ND=1014/cm³3. (e)T=650K,NA=0,ND=1014/cm3. 2.18(a toe）Foreachof theconditions specified inProblem2.17,determinetheposition ofE,compute EF-E,and drawa carefully dimensionedenergy band diagram fortheSi sample.NOTE:EG(Si)=1.08eVat450Kand1.015eVat650 K. 2.19(a)Assuming a nondegenerate Si sampleandtotal impurity atom ionization,con- structa MATLAB program that computes n,p,andEF-Egiven acceptable input valuesofT(temperature in Kelvin),ND(cm-3),andNA（cm-3).Incorporatethe program presented in part(a)of Exercise2.4tocompute n at a specifiedT.Make use ofthe MATLAB input function to enter the input variables from the command window. (b)Use your program to check the relevant answers to Problems2.17and2.18.

--- Page 97 --- 74 SEMICONDUCTORFUNDAMENTALS 2.20 According to the text,the maximum nondegenerate donorand acceptor doping con- centrationsin Siatroomtemperatureare ND≈1.6×10/cmandNA≈9.1×107/ cm3,respectively.Verify the text statement. 2.21Constructacomputer programtoproduce aplotofF-Eversus NAor ND similar to Fig.2.21.Use theprogram to verify theaccuracyofthe cited figure.(You mayfind it convenient to employ the MATLAB logspace function in constructing your program.) 2.22 GaAs considerations. (a)MakeasketchsimilartoFig.2.14thatisspecificallyappropriateforGaAs.Besureto take into account the fact thatm *□ m * in GaAs. (b) Basedonyouranswertoparta),wouldyouexpectEinGaAstolieaboveorbelow midgap? Explain. (c) DetermnethecieoitonoftheiniiFrmivliGAtoomtmpature (300 K). (d)Determinethemaximumnondegeneratedonorandacceptordopingconcentrationsin GaAs at room temperature. 2.23 Given an NA=104/cm3doped Sisample: (a)Presentaqualitativeargumentthatleadstotheapproximatepositioningofthe Fermi level in the material asT→0K. (b)ConstructaMATLABcomputer)programtocalculateandplotEinthematerial as a function of temperature for 200 K≤T≤500K. (c)WhatdoyouconcluderelativetothegeneralbehavioroftheFermilevelpositioning as a function of temperature? (d)Run your part（b）programto determine whathappens ifNAis progressivelyin- creasedin decadesteps fromNA=1014/cm³toNA=1018/cm3.Summarizeyour observations. (e) How wouldyourforegoing answers bemodified if theSi sample was doped with donors instead of acceptors?

--- Page 98 --- 3 Carrier Action Carriermodeling under“rest”or equilibrium conditions,considered in Chapter 2,is im- portantbecause itestablishes the properframeof reference.From a device standpoint, however,the zerocurrentobserved underequilibriumconditions isratheruninteresting. Only whenasemiconductorsystemisperturbed,givingrisetocarrieractionoranetcarrier response,cancurrentsflowwithinandexternaltothe semiconductorsystem.Action,a- rier action,is the general concern of this chapter. Under normal operating conditions thethree primary typesof carrieraction occurring inside semiconductors are drift,difusion,and recombination-generation.In this chapter we first describeeachprimary typeofcarrieractionqualitatively and thenquantitatively relate theactionto thecurrentflowing withinthe semiconductor. Specialemphasisis placedoncharacterizingtheconstantsofthemotionassociatedwitheachtypeofaction and wherever appropriate the discussion is extended to subsidiary topics of a relevant na- to occur simultaneously inside any given semiconductor.Mathematicallycombining the variouscarrieractivitiesnext leads totheculminationofourcarrier-actioneforts;weob- tainthe basicsetof startingequationsemployedin solvingdeviceproblemsofanelectrical nature.Finallysimpleexampleproblemsareconsideredtoillustratesolutionapproaches and to introduce key supplemental concepts. 3.1DRIFT 3.1.1Definition-Visualization Drift, by definition,is charged-particle motion in response to anappliedelectric field. Within semiconductors the driftingmotionof thecarriers on a microscopicscalecanbe described asfollows:Whenanelectricfield()is appliedacross asemiconductor as visu- alized inFig.3.1(a),theresulting forceon thecarriers tends toacceleratethe +q charged holes in the directionof theelectric fieldandthe-qchargedelectrons in the direction oppositetotheelectricfield.Becauseofcollisions withionizedimpurityatomsandther- mallyagitatedlatticeatomshoweverthecarrieraccelerationisfrequentlyinterrupdh carriersaresaidtobescattered).ThenetresulpicturedinFig.3.b)iscarriermotion generally alongthe directionoftheelectricfieldbutinadisjointedfashioninvolving repeated periods of acceleration and subsequent decelerating collisions. The microscopic drifting motionofa single carrier isobviouslycomplexandquite tedious to analyze in any detail. Fortunately,measurable quantities are macroscopic 75

--- Page 99 --- 76 SEMICONDUCTORFUNDAMENTALS 8 + vd + (a) (b) (c) Figure3.1Visualizationofcarrierdrift:（a）motionofcarriers withinabiasedsemiconductorbar; (b)driftingholeonamicroscopicoratomic scale;(c)carrierdrifonamacroscopicscale. observablesthatrefecttheaverageoroverallmotionofthecarriers.Averaging overall electronsor holes atany giventime,wefindthattheresultantmotionofeach carriertype can be described in terms ofaconstantdrift velocity,v.In other words,on a macroscopic scale,driftmay be visualized (see Fig.3.lc) as nothing more thanallcarriers of a given typemovingalonatconstantvelocityindirectionparallelorantiparalleltotheapplied electric field. By wayofclarification,it is importantto pointoutthatthe drifting motionofthe carriers arising inresponse to anappliedelectricfield isactually superimposed uponthe always-presentthermal motionof thecarriers.Electrons in the conductionbandandholes in the valence band gainand loseenergy via collisions with the semiconductor latticeand are nowhere near stationary even underequilibrium conditions.In fact,underequilibrium conditionsthethermallyrelatedcarriervelocitiesaverage~1/1000the speedoflightat room temperature! As pictured in Fig.3.2,however,the thermal motionof the carriers is completely random.Thermal motion thereforeaverages out tozeroon amacroscopic scale, does not contribute to current transport,and can be conceptually neglected. 3.1.2 DriftCurrent Let us nextturntothetaskofdevelopingananalyticalexpressionforthecurrentflowin within a semiconductor as a resultof carrier drift.By definition Figure 3.2 Thermal motion of a carrier.

--- Page 100 --- CARRIERACTION 77 8 I A Figure3.3 Expanded viewofa biased p-type semiconductor bar of cross-sectional area A. Icurrent)=thechargeperunittimecrossinganarbitrarilychosenplaneofobservaion oriented normal to the direction of current flow. Consideringthep-typesemiconductorbarofcrosssectionalareaAshowninFig.3.3,and specificallynoting thearbitrarilychosenv-normalplanelyingwithinthebar,wecan argue: v_{ Allholesthisdistance backfrom thev-normalplane willcrossthe plane in a time t. _{ { All holes in this volume willcross the plane in a time t. p{ Holes crossing the plane in a time t. qp A$Chargecrossingtheplaneinatimet.$qpA$ Charge crossing the plane per unit time. The word definitionofthelastquantityisclearlyidenticaltotheformaldefinitionofcur- rent. Thus IPldrift=qpAhole drif current (3.1) As apracticalmatter,the cross-sectional area Aappearing in Eq.(3.1)andother cur- rentformulasisoftenexcessbaggage.Current,moreoverisgenerallythoughtofas ascalar quantity,while inrealityitisobviously a vector.Thesedeficienciesareovercome byintro- ducing a related parameter known as the current density,J.J has the same orientationas the directionofcurrentflowand is equal in magnitudeto thecurrent per unitarea（or J=I/A).Byinspection,thecurrentdensityassociated with holedriftis simply JPldrift = qpd (3.2) Sincethedriftcurrentarises inresponsetoanappliedelectricfield,it isreasonableto proceedonestepfurtherandseekaformofthecurrentrelationshipthatexplicitlyrelates JPldrifttotheapliedlctricfild.Tothisend,wemakererencetothepeenttivef velocity versuselectricfielddatapresentedinFig.3.4.Notethatis proportionaltoat

--- Page 101 --- 78 SEMICONDUCTORFUNDAMENTALS 107 vd=saElectrons (cm/sec) Holes 106 105 10² 103 104 105 106 (V/cm) Figure3.4Measureddriftvelocityofthecarriersinultrapuresiliconmaintainedatroomtempera- tureasfunctionoftheappliedlectricfieldConstructedromthedatafitsandthedatarespectvl inJacoboniet al.[4and Smithetal.[5] low electricfields,whileathighelectricfieldsvsaturatesandbecomesindependentof&. To be more precise, μ{0\μ$...8\to 0$}μ_

U_sat …8→∞ (3.3) 1+ _sat$ whereβ≈1for holesandβ≈2 for electrons in silicon,is the constantof proportion- alitybetween vandEatlowtomoderateelectric fields,andva is the limiting or satura- tion velocityapproachedat veryhighelectricfields.Obviously,inthe high-field limitvin Eq..2)issimplreplcebndeotexiitlddenceI the low-fieldlimit,which isof greatest practical interest and assumed herein unles speci- fiedotherwise,=μp(0→pfor holes)and substitution into Eq.(3.2)yields JPldrift = qμpp8 (3.4a) Similarly,for electrons one obtains JNldrift = 9μ{n8 (3.4b)

--- Page 102 --- CARRIER ACTION 79 Respectively knownas theelectronmobility andholemobility,μandμarealwaystaken tobepositivequantities.Notethatalthoughelectronsdriftinthedirectionoppositeohe appliedelectricfield(=-μn),thecurrenttransported bynegativelychargedparticles isinturncountertothedirectionofdri(Jrinv).Thentresulasinicatedin Eq.(3.4b),isanelectroncurrent inthedirectionoftheappliedelectricfield. 3.1.3 Mobility Mobility isobviously acentral parameter incharacterizingelectronandhole transportdue to drift.As furtherreadings willreveal,thecarriermobilitiesalso play a keyrole in char- acterizing the performanceof many devices.It is reasonable therefore to examine μ and core of useful information for future reference. StandardUnits:cm²/V-sec. Sample Numerical Values:μ≈1360cm²/V-sec andμp≈460cm²/V-sec at 300Kin ND=10/cm3andNA=104/cmdopedSi,respectively.Inuncompensatedhigh-purity (NDorNA≤10/cm)GaAs,theroom-temperaturemobilitiesareμ≈8000cm²/V-sec andμ≤400cm²/V-sec.The quoted values are usefulforcomparison purposes and when performingorder-of-magnitudecomputations. Also note that μn>μpforbothSiand GaAs.Inthe major semiconductors,μis consistentlygreaterthanμfora given doping and system temperature. Relationship to Scattering: The wordmobility in everyday usage refers to a general free- dom of movement.Analogously,in semiconductor work the mobility parameter is amea- sure of theeaseofcarriermotion in a crystal.Increasing themotion-impeding collisions withinacrystaldecreasesthemobilityofthecarriers.Inotherwords,thecarriermobility varies inversely withtheamountofscattering taking place within the semiconductor.As visualized inFig.3.1(b),the dominant scattering mechanisms innondegenerately doped materialsofdevicequalityaretypicallyi)latticescatteringinvolvingcollisions withther- mallyagitatedlatticeatoms,and(ii)ionizedimpurity(i.e.donor-siteand/oracceptor-site) scattering.Relativeto latticescatteringitshouldbeemphasizedthat itisthethermal vibra- tion,thedisplacementoflatticeatomsfromtheirlatticepositions,thatleadsto carrier scattering.Theinternalfieldassociated withthestationaryarrayofatomsina crystalis already taken into account in the effective mass formulation. Althoughquantitativerlationshipsconnectingthemobilityandscatteringcan become quiteinvolveditisreadilestablishedthat=τ/m*,where（isthemeanfrime betweencollisionsand m*istheconductivityeffectivemass.Sinceincreasingthe number of motion-impedingcollisions decreases themeanfreetime betweencollisions,weaain concludeμ varies inversely with the amount of scattering.However,μis also noted to vary in GaAs is significantly smaller than them * in Si, thereby explaining the higher mobility of n the GaAs electrons.

--- Page 103 --- 80 SEMICONDUCTORFUNDAMENTALS Silicon T=300K Electrons 1000 n op (cm²2/V-sec) Holes NA Or ND(cm-3) μn μ_p (cm^{}$ V-sec) 1×1014. 1358 461 2 1357 460 100 5 1352 459 1×1015 1345 458 2 1332 455 5 1298 448 1×1016. 1248 437 2 1165 419 5 986 378 1×1017 801 331 10 .. 1014 1015 1016 1017 1018 NA Or ND(cm-3) (a) 10000 GaAs-electrons T=300K Ge-electrons n orp(cm²/V-sec) Ge-holes 1000 GaAs-holes 100 1014 1015 1016 1017 1018 NA or ND(cm-3) (b) Figure3.5Roomtemperaturecarriermobilitiesasafunctionofthedopantconcentrationin（a)Si and(b) Ge and GaAs.μn is the electron mobility; pis the hole mobility.

--- Page 104 --- CARRIER ACTION 81 Doping Dependence:Figure 3.5 exhibits the observed doping dependence of the electron and hole mobilities in Si,Ge,and GaAs. Allsemiconductors exhibitthe same general dependence.Atlowdopingconcentrationsbelowapproximately10/cminSithecarrier mobilitiesare essentially independentof the doping concentration.For dopings inexcess of101/cm,the mobilities monotonically decrease with increasing NAorN. With theaidof Fig.3.6 theexplanationof theobserved doping dependence isrela- tively straightforward.Invoking an electricalanalogy,one canassociate a resistanceto motion with each of the scattering mechanisms. These resistances are in series. At suffi- cientlylowdoping levels,ionizedimpurityscatteringcanbeneglectedcomparedto lattice scattering,orinthe analogy,RToTAL=RL+RIRL.When latticescattering,whichis not a function of NA or ND,becomes the dominant scattering mechanism,it automatically follows thatthecarriermobilities willbe likewiseindependentofNAorND.Fordopings in excess of10/cm3in Si,ionized impurity scattering and the associated resistance to motion can no longer be neglected.Increasing the numberof scattering centers byadding more and more acceptors ordonors progressively increases the amount of ionized impurity scattering and systematically decreases the carrier mobilities. Temperature Dependence: The temperature dependenceof theelectronand holemobili- ties in Si with doping as a parameter is displayed in Fig.3.7.For dopings of NAor ND 1014/cm,the datamergeintoasinglecurveandthere is anearpower-lawincrease in the carrier mobilities as the temperature is decreased. Roughly,T-2.3±0.1 and μp T-2.20.1.Forprogressivelyhigherdopings,thecarriermobilitiesstillincreasewithde- creasingtmeraturebuttsystmaticallydecreasinate.InacomeN/cm experimental data (not shown)exhibit a decrease inμ as Tis decreased below 200K. The general temperature dependenceof thecarriermobilities is relativelyeasy toex- plain in the lowdoping limit.As noted in the doping dependence discussionlatticescat- tering is thedominantscateringmechanismRToTALRL)inlightlydopedsamples. Decreasingthe systemtemperaturecauses anever-decreasingthermalagitationofthe semi- conductoratomswhichinturn decreasesthe latticescattering.Thedecreased scatring RL R{I Impedance to motion due to Impedance to motion due to lattice scattering: ionized impurity scattering: —No doping dependence. —Increases with NA or ND -Decreases with decreasing T. -Increases with decreasing T. Figure3.6Electricalanalogyforscatteringinasemiconductor.RLandRrepresenttheimpedance to motion due to lattice scattering and ionized impurity scattering,respectively.

--- Page 105 --- 82 SEMICONDUCTOR FUNDAMENTALS $10^{}$ D51014 Silicon 4 D.=1016 -2.3 μn(cm2}/.) N =10{7 10^{3}$D$N{ }=10{}10^{ 100 200 500 1000 T(K) (a) $10^{}$ Silicon NA (cm^{2}N}-s) -2.2 10^{3 10+6 A A1017 N{A}=101}10^{}$ 100 200 500 1000 T(K) (b) Figure3.7Temperature dependenceof (a)electronand (b)holemobilities in silicon for dopings rangingfrom10/cm3to10/cm3.Thecurveswereconstructedusingtheempiricalfitrelation- shipsand parameters presented in Exercise3.1.The dashed lineportionofthecurvescorrespondto aslightextensionofthefitbeyondtheverified200K≤T≤500Krangeof validity.

--- Page 106 --- CARRIERACTION 83 roughlyfollows a simplepower-law dependence. Being inversely proportional to the amountof lattice scattering,themobilityof lightlydopedsamples isthereforeexpectedto increase with decreasing temperature,varying roughly as the temperature to a negative power. The more complex dependence of the higher doped samples reflects the added effect ofionizedimpurity scatering.In theelectrical analogy,RI can no longer be neglected. Moreover,whereas lattice scattering (RL)decreases with decreasing T,ionized impurity scattering(R)increases with decreasingT.Ionized impurities become moreandmore effective in deflecting thecharged carriers as the temperatureand hencethe speedofthe carriers decreases.Thus ionized impurity scattering becomes a largerand larger percentage oftheoverallscattering as thetemperatureis decreased(TOTAL→R).Clearly,thisex- plainsthe decreased slopeofthemobility versus temperature dependenceexhibitedbythe higher doped samples. (C) Exercise 3.1 There exist surprisinglyaccurate“empirical-fit”relationshipsthatare widelyem- ployedtocomputethecarriermobilitieataivendopinandtemperatureFis 3.5and 3.7 wereconstructedusingsuchrelationships.Theformofacomputational relationship is typicallyestablishedonanempirical basis by notingthegeneral func- tional dependenciespredictedtheoreticallyandobservedexperimentally.Parameters intherelationshiparethenadjusteduntilanacceptablematchisobtainedtothebest available experimental data. Themajority carrier mobility versus doping at room temperature is popularly computed from μo μ= min + 1+(N/Nref)α whereisthecarriermobility(nor）,Nisthedopingconcentration（NAor ND), and allotherquantitiesarefit parameters.Tomodelthetemperature dependence,one additionally employs （I A = A3000$ Aintheaboveequationrepresentsmin,Nef,orα;A0isthe300Kvalueofthe parameter,Tis temperature in Kelvin,andηis the temperature exponent.Theft parameters appropriate for Si are listed in the following table:

--- Page 107 --- 84 SEMICONDUCTORFUNDAMENTALS Value at 300K Temperature Parameter Electrons Holes Exponent (n) ref(cm- 3}) 1.3× 1017 2.35×1017 2.4 μmin(cm^²/V-sec) 92 54.3 -0.57 $μ0(cm²/V-sec) 1268 406.9 -2.33 electrons -2.23 holes α 0.91 0.88 -0.146 P:(a)Constructalog-logplotofandμpversus NAorNDfor101/cm≤NAor ND≤101/cmusing thequotedfitrelationshipandthe listedSifitparameters. Compare your result with Fig.3.5(a). (b) Construct log-log plotsofn versus Tandp versusTfor200K≤T≤500K andNDorNAsteppedindecade valuesfrom104/cm3to1018/cm3Compareyour results with Figs.3.7(a) and3.7(b),respectively. S: (a) MATLAB program script… %Mobility versus DopantConcentration(Si30K) %Fit Parameters NDref=1.3e17;NAref=2.35e17; μnmin=92;μpmin=54.3; μn0=1268;μp0=406.9; an=0.91;ap=0.88 %MobilityCalculation N=logspace(14,19); μn=μnmin+μn0.(1+(N/NDref).^an; p=pmin+pO./(1+(N/NAref).^ap); %Plotting results close loglog(N,μn,N,μp); grid; axis([1.0e141.0e191.0e11.0e4]); xlabel(‘NAor ND (cm-3)’); ylabel(‘Mobillity (cm2/V-sec)); text(1.0e15,1500,‘Electrons’); text(1.0e15,500,‘Holes’); text(1.0e18,2000,‘Si,300K’);

--- Page 108 --- CARRIER ACTION 85 The results obtained byrunning thepreceding program shouldbe numerically identical to Fig. 3.5(a). (b)Part (b) is left for the reader to complete. 3.1.4 Resistivity Resistivityisanimportantmaterialparameterthatiscloselyrelatedtocarrierdrift.Quali- tatively,resistivity is ameasureof amaterial’sinherentresistance tocurrentflow—a”nor- malized”resistance that does not depend onthe physical dimensionsof the material.Quan- titatively,resistivity(isdefinedastheproportionalityconstantbetweenthelectricfeld impressed across a homogeneousmaterial and the total particle current per unit area flow- ing in the material; that is, 8=ρJ (3.5a) or 1 J=o8=-8 (3.5b) ρ whereσ=l/pisthematerialconductivity.Inahomogeneousmaterial,J=Jdrifand,as established with the aid of Eqs.(3.4), Jdrift = JNldrift + JPldrift=q(（μnn +pp)8 (3.6) It therefore follows that 1 (3.7) ρ = q(μnn +pp) In a nondegenerate donor-doped semiconductor maintained in the extrinsic tempera- tureregion where ND≥n,n=NDandp≈n/ND≤n.This result was established in Subsection 2.5.5.Thus,for typical dopings and mobilities,n+μpp≈nNDin an n-type semiconductor.Similar arguments yieldnn+μpp≈pNAina p-type semicon- ductor.Consequently, underconditions normallyencountered in Si samplesmaintainedat or near room temperature, Eq.(3.7)simplifies to 1 (3.8a) ρ

$qμ_{nD . .. n-type semiconductor

--- Page 109 --- 86 SEMICONDUCTORFUNDAMENTALS $103$ 10^{ Silicon 300K p-type 10 n-type ρ(ohm-cm) 1 10-1 $10-2$10-3 10-4 1013 10{4 1015 1016 1017 1018 10{9 $10{v NA or ND(cm-3) (a) Figure3.8Resistivityversusimpurityconcentrationat300Kin(a)Siand(b)othersemiconduc- tors.[(b) From Szel2,©1981by John Wiley & Sons,Inc. Reprinted with permission.] and 1 (3.8b) ρ

qp{A … p-type semiconductor When combined withmobility-versus-doping data,Eqs.(3.8)provide a one-to-one corre- spondence betweentheresistivity,a directlymeasurablequantity,andthe doping inside a semiconductor.In conjunction withplots ofρ versus doping (see Fig.3.8),the measured resistivity is in fact routinely used to determine NA or ND. The measured resistivity required in determining the doping can be obtained in a num- ber of different ways.A seemingly straightforward approach would be to form the semi- conductor into a bar,apply a bias V across contacts attachedto theends ofthe baras in

--- Page 110 --- CARRIER ACTION 87 $10$ T=300K n-type —p-type 10{ p-GaP n-GaP 1 ρ(ohm-cm) p-GaAs $10-1n-GaAs$10^-2 p-Ge n-Ge $10-310-41014$101}$10161017$1018 1019 $1020 10^21 NA or ND(cm-3) Figure 3.8 (b) Fig.3.1(a),measure the current I flowing in the circuit,and then deduceρ from the mea- sured resistance.[R(resistance)=V/I=pl/A,wherelis thebar length and Aisthecross- sectional area.]Unfortunately,the straightforward approach is deceptively dificult,is de- structive (wastes semiconductor material),and is notreadilyadaptable to the wafersused in device processing. A measurement method finding widespread usage in practice is the four-point probe technique.Inthestandardfour-pointprobetechnique,fourcollinear,evenlyspacedproes, as shown in Fig.3.9(a),are brought into contact with the surface of the semiconductor.A known currentI is passedthroughtheoutertwo probes,and the potentialVthereby devel- oped is measured across the inner two probes. The semiconductor resistivity is then com- puted from V (3.9) ρ=2s-Γ I where s is the probe-to-probe spacing and Γis a well-documented“correction”factor.The correction factor typically depends on the thicknessof thesampleand on whetherthe bot- tom of the semiconductor is touching an insulator or a metal.Commercial instruments are

--- Page 111 --- 88 SEMICONDUCTORFUNDAMENTALS 1 I S S↑ S Semiconductor sample (a) Sound generator Sound receiver Wafer RFcoil Sound Ferritecore receiver Sound generator (b) Figure3.9 Resistivitymeasurementtechniques.(a)Schematicdrawingoftheprobearrangement, placement,andbiasing in the four-point probe measurement.(b)Schematicof a commercialeddy- currentapparatusshowingtheRFcoilsandsoniccomponents.(b)From Schroder[6,©1990by John Wiley & Sons,Inc. Reprinted with permission.]

--- Page 112 --- CARRIERACTION 89 availablethatautomaticallycomputetheappropriatecorrectionfactorbasedonthesample thickness entered by the operator. Unlike the semiconductor-bar measurement, the four- pointprobetechniqueisobviouslyeasytoimplement,causesonlyslight surface damage in the vicinityoftheprobecontacts,and is ideally suitedforworking with wafers.The surface damage,althoughslight, does excludethetechniquefrombeing used on wafers intended for producing high-yield devices and ICs. A second technique worthyof note utilizes a noncontacting eddy-currentapproach. Theschematicofacommercial eddy-currentapparatus is pictured in Fig.3.9(b).Thereare fringing fieldsnear a ferritecoreexcitedwithanRFcoil.Ifaconductingmaterialsuchas a semiconductor wafer is placed neartheferritecore,thefringing fieldscause localized (eddy)currentstoflow in theconducting material.Thecurrentflow inturnabsorbs some of the RFpower.The sheetresistivityoftheconducting material is deduced bymonitoring the power consumption in a calibrated system. The ultrasound generators andreceivers built into the commercial apparatus are for the in-situ determination of the wafer thickness. Thechange in phase of ultrasoundbouncedoffthe topand bottom wafer surfaces permits theinstrument to compute the intercore spacing occupied by the wafer.The sheetresistivity multiplied by the waferthickness yieldsthe waferresistivity. 3.1.5BandBending In previousencounters withtheenergy band diagram,we have consistently drawnEcand Etobeenergiesindependentofthepositioncoordinatex.Whenanelectricfield()exists inside a material,the bandenergies becomea functionof position.The resulting variation of Ec and Ewithpositionontheenergyband diagram,exemplifiedby Fig.3.10(a),is popularly referred to as “band bending.” Seeking toestablishtheprecise relationshipbetween theelectricfield within a semi- conductorand the induced band bending, let us carefully re-examine theenergy band dia- gram.The diagram itself,asemphasized in Fig.3.10(a),is aplotoftheallowedelectron energies within the semiconductor as a function of position, where E increasing upward is understood to be the total energy of the electrons.Furthermore,we know from previous discussions that if anenergyofpreciselyEG is addedto breakanatom-atombond,the createdelectronandholeenergies would be Eand E,respectively,andthecreated carriers would be effectively motionless.Absorbing an energy in excess of EG,on theother hand, wouldinallprobabilitygiverisetoanelectronenergygreaterthanEandaholeenergy less than E,withboth carriers moving around rapidly within the lattice. We are led,there- fore,tointerpretE-Eto bethekineticenergy(K.E.)oftheelectrons andE-Etobe thekineticenergyof theholes [seeFig.3.10(b)].Moreover,since the total energy equals the sumofthekineticenergyand the potentialenergy (P.E.),Eminustheenergyreference level(Eref)mustequaltheelectronpotentialenergy,asilustratedinFig.3.10(c).(Potentil energy,it shouldbe noted,is arbitraryto within a constant,and the position-independent reference energy,Eref,may be chosen to be any convenient value.) Thepotential energy is thekey in relating theelectric field within a semiconductorto positional variations in theenergy bands.Specifically,assuming normaloperational con ditions,wheremagneticfield,temperaturegradientand stress-inducedeffectsarengli- gible,onlytheforceassociated withanexistingelectricfieldcangiverisetochangesinthe

--- Page 113 --- 06 SEMICONDUCTORFUNDAMENTALS E (Total $Ec$ electron energy) Ev \o}$(a)K.E.$Ec$EvK.E.(b)K.E.EcEP.E.$E_v}$ Eref (c) V → x (d) 8 x (e) Figure3.10 Relationship between band bending and theelectrostatic variables inside a semicon- ductor:(a)sampleenergyband diagram exhibiting band bending;(b)identificationof thecarrier kinetic energies;(c)specificationof the electronpotentialenergy;(d)electrostatic potentialand (e) electricfieldversuspositiondependence deducedfromandassociated withthepart(a)energy band diagram.

--- Page 114 --- CARRIER ACTION 91 potentialenergyofthecarriers.Elementaryphysics,infacttellsusthatthepotentialen- ergyofa-qcharged particleundersuchconditionsis simplyrelated totheelectrostatic potentialV at a given point by P.E.=-qV (3.10) Having previously concluded that P.E.= Ec-ref (3.11) we can state V=—(Ec-ef) 1 (3.12) q By definition, moreover, A△-=8 (3.13) or, in one dimension, dV 8= (3.14) dx Consequently, 1 dEc 1dE_ 1dE{ 8= (3.15) xp b$ xp b q dx ThelatterformsofEq.(3.15)followfromthefactthatE,E,andEdifferbyonlyan additive constant. The preceding formulation provides a means ofreadilydeducing the general formof the electrostatic variablesassociated withthe“bandbending”in Fig.3.10(a)andother energyband diagrams.Making useofEq.(3.12),or simply invertingE(x)inFig.3.10(a), one obtains the V versus x dependence presented inFig.3.10(d).[Please note thatthe electrostaticpotential,likethepotentialenergy,is arbitrary to withina constant—the Fig.3.10(d)plotcan betranslated upward or downwardalong thevoltage axis without modifyingthephysicalsituation insidethe semiconductor.] Finally,recording theslopeof E versus position,as dictated by Eq.(3.15),produces the versus x plot shown in Fig.3.10(e). In summary, thereadershouldbeawareofthefactthattheenergyband diagram containsinformationrelatingtotheelectrostaticpotential andelectricfield withinthe semi- conductor. Moreover,the generalformof the V and & dependencies within the semi- conductorcanbeobtainedalmostbyinspection.TodeducetheformoftheVversusx

--- Page 115 --- 92 SEMICONDUCTORFUNDAMENTALS relationship,merely sketch the“upside down”of E（or Eor E)versus x; to determine the general versus x dependence,simply note the slopeofE(or Eor E)as a function of position. Exercise3.2 P:Considerthe following energy band diagram.Take the semiconductor represented to be Simaintained at 300KwithE-EF=E/4atx=±LandEF-E= EG/4atx=0.NotethechoiceofEFastheenergy referenceleveland theidentiffi- cation of carriers at various points on the diagram. →E (electrons) 2 3 $Ec$ 1 $Ei$EF$0+Ev$ 1 E (holes) 2 3 I 1 →x -L 0 L (a) Sketchtheelectrostaticpotential)insidethesemiconductorasafunctionof. (b) Sketchthe electric field()inside the semiconductoras a functionof. (c) AscertaintheK.E.andP.E.oftheelectronsandholespicturedonthediagam. (d) Determinethe resistivityof thex>Lportionof the semiconductor. S:(a)The Vversusxrelationshipmusthavethe samefunctionalformasthe“upside downof（orEorE).IfthearbitraryvoltagereferencepointistakentobeV= Oatx=L,thenoneconcludesthe Vversusdependence is as sketchedhere: V →x 7- 0 L

--- Page 116 --- CARRIER ACTION 93 (b) The electric field,E,isproportional tothe slope ofthe bands: 8 -L 0 →x L (c) Fortheelectrons,K.E.=E-Eand P.E.=-ef=E-.Sincethe total energyof the holes increases downward on the diagram,K.E.=E-Eand P.E.=ref-E=F-Efor the holes. The energy differences are of course evaluated at the same x.Thededuced K.E.andP.E.valuesaresummarized inthe following table: Carrier K.E.(eV ) P.E.(eV ) Electron 1 0 0.28 Electron 2 0.56 0.28 Electron 3 0 0.84 Hole1 0 0.28 Hole 2 0.56 0.28 Hole 3 0 0.84 (d)Inthe>Lregion,E-F=E/4=0.28eVand NA=p=ne(E-EF)/kT=1010e0.28/0.0259=4.96×1014/cm3 From Fig.3.5(a)one deduces μp=459cm²/V-sec,and therefore, 1 1 ρ = 27.5 ohm-cm q{p{} (1.6×10-19)(459)(4.96×10{4) Alternativelymaking useof theFig.3.8(a) Siresistivityplotone similarlyconcludes ρ≡25 ohm-cm.

--- Page 117 --- 94 SEMICONDUCTORFUNDAMENTALS 3.2 DIFFUSION 3.2.1Definition-Visualization Diffusionisaprocess wherebyparticlestendtospreadoutorredistributeasaresultoftheir random thermal motion,migrating on a macroscopic scale from regions of high particle concentrationintoregionsof low particleconcentration.Ifallowed to progress unabated, the diffusionprocess operates so as to produce a uniform distributionof particles. The diffusing entity,it shouldbe noted,need notbecharged;thermalmotion,not interparticle repulsion,is the enabling action behind the diffusion process. To cite aneveryday exampleof diffusion,suppose anopen bottleof perfume is placed in onecorner of a room. Even in the absence of air currents,random thermal motion will spread the perfumemoleculesthroughouttheroom in arelatively shortperiodoftime,with intermolecularcollisions helping to uniformlyredistribute theperfumeto every nookand cranny within the room. Seeking to obtain a more detailedunderstanding of the diffusion process,letus next conceptually“monitor”the processon amicroscopic scaleemploying a simple hypotheti- cal system. The system we propose to monitor is a one-dimensional box containing four compartmentsand1024mobileparticles(seeFig.3.11).Theparticleswithintheboxobey certain stringentrules.Specifically,thermal motioncausesall particles in a givencompart- ment to “jump”intoan adjacentcompartmenteveryτseconds.Inkeeping withtherandom nature of the motion,each andevery particle has an equal probabilityof jumping to the left reflects the particlebackto its pre-jumpposition.Finally,attimet=Oitis assumedthat allof the particles are confined in the left-most compartment. Figure3.11records theevolutionofour1024 particlesystem as a functionoftime.At timet=,512ofthe1024particlesoriginally incompartment1 jumpto therightand come to rest in compartment2. The remaining512 particles jumpto the left andare re- flectedbackintotheleft-mostcompartment.Theendresultis512particleseachincom- partments1and2afterτseconds.Attimet=2,256ofthe particles incompartment2 jump into compartment 3 and the remainder jump back into compartment1.In the mean- time,256ofthe particles fromcompartment1jumpintocompartment2and256undergo areflectionatthe left-handwall.Thenetresultafter2τseconds is 512particles incom- partment1and256particleseach incompartments2and3.Thestateofthe systemafter 3τ and 6τ seconds,also shown inFig.3.11,can be deduced in asimilar manner.Byt=6, the particles,once confined to the left-most compartment, have becomealmost uniformly distributed throughoutthe box,and it is unnecessary to consider later states.Indeed,the fundamental natureofthe diffusion process is clearly self-evidentfrom anexaminationof the existing states. In semiconductors the diffusion process on a microscopic scale is similarto thatoc- curring in the hypothetical system except,of course,the randommotion of the diffusing particles is three-dimensional and not“compartmentalized.”On a macroscopic scalethe net effectof diffusion is precisely the same within both the hypothetical system and semi-

--- Page 118 --- CARRIERACTION 95 1024 t=0 512 512 1=T 512 256 256 t=2τ 384 384 128 128 t=3 ● ● 320 256 256 192 t=6 Figure3.11Diffusiononamicroscopicscaleinahypotheticalone-dimensional system.Thenum- bers overthearrows indicatethe numberof particles inagivencompartment;observationtimesare listed to the extreme right. conductors; there is anoverallmigrationof particlesfromregionsofhigh particlecon- centration to regionsof low particleconcentration.Within semiconductors the mobile par- ticles—the electrons and holes-are charged,and diffusion-related carrier transport therefore gives rise to particlecurrents as pictured in Fig.3.12. J_|diff N ldiff Diffuse + Diffuse + + + + x x Figure3.12 2Visualizationofelectron and hole diffusion on amacroscopic scale.

--- Page 119 --- 96 SEMICONDUCTORFUNDAMENTALS (C) Exercise 3.3 The”DiffDemo”MATLABfile listedbelow was writtentohelp the user visualizethe diffusion process. The program provides a pseudo-animation of a one-dimensional particle systemsimilartothatdescribedinthetext.The y=[]statementinthe program controlstheinitialcondition,N specifiesthemaximumnumberofmoni- tored“jumpsandthenumber inthe pause statementscontrolsthetimebetween jumps. %Simulationof Diffusion (DiffDemo) %One-dimensional system,rightandleft jumpsequallyprobable %Initialization close x=[0.51.52.53.54.5]; y=[1.0e60000]; %NOTE:initial positioncan bechanged [xp,yp]=bar(x,y); plot(xp,yp);text(0.5,1.1e6,t=0); axis([0,5,0,1.2e6]); pause (0.5) N=15; %NOTE: increase N for extended run %Computations and Plotting for ii=1:N, %Diffusion step calculation bin(1)=round(y(1)/2+y(2)/2); bin(2)=round(y(1)/2+y(3）/2); bin(3)=round(y(2)/2+y(4)/2); bin(4)=round(y(3)/2+y(5)/2); bin(5)=round(y(4)/2+y(5)/2); y=bin; %Plotting the result [xp,yp]=bar(x,y); axis(axis); plot(xp,yp);text(0.5,1.1e6,[‘t=‘,num2str(ii)]); axis([0,5,0,1.2e6]); pause (0.5) end P:(a) Enter the program into your computer or locate the copy supplied on disk.Run the program.(The command-period on a Macintosh computer or break on an IBM- compatible computer can be used to prematurely terminate the program.) (b) Rerun theprogram after changing the initial conditions so thatallofthecarriers are in the middle box att=0.

--- Page 120 --- CARRIER ACTION 97 (c) Experiment with initiallyplacingparticles in morethanone boxand changing the number in the pause statements.You mightalso try increasing the number of par- ticle boxes. S: Sample output is reproduced in Fig. E3.3. ×105 ×105 12 12 t=0 t=15 10 10 8 8 6 6 4 4 2 2 0 1 0 0 1 2 3 4 5 0 1 2 3 4 5 Figure E3.3 3.2.2 Hot-Point Probe Measurement Before establishing the currentassociated with diffusion,letus digress somewhatand briefly consider the hot-point probe measurement. Thehot-point probe measurement is a common technique for rapidly determining whether a semiconductor is n-or p-type. From apractical standpoint,knowledgeofthesemiconductortypeisessential indevice process- ing and must be known even before determining the doping concentration from resistivity measurements(referbackto Fig.3.8).The hot-pointprobe“typingexperimentisconsid- ered here because it simultaneously provides an informative example of the diffusion process. Examining Fig.3.13(a),we findtheonlyequipmentrequiredforperforming the ho- point probemeasurement is a hot-point probe,a cold-point probe,and a center-zeromil- liammeter. The hot-point probe is sometimes a refugee from a wood burning set or simply a solderingiron;thecold-pointprobetypicallyassumestheformofanelectricalprobelike that used with hand-held multimeters.No special requirements are imposed on the center- zero milliammeter connected between the probes. The measurement procedure itself is also extremely simple: After allowing the hot probe to heat up,one brings the two probes into contactwiththe semiconductor sample,andtheammeter deflectstotherightorlef, thereby indicating the semiconductor type.Observe that the spacing between the probes is set arbitrarily and can be reduced to enhance the ammeter deflection. A simplifiedexplanationof howthe hot-point probemeasurement works is presented in Fig.3.13(b).In the vicinityof theprobecontact the heated probecreates an increased

--- Page 121 --- 98 SEMICONDUCTORFUNDAMENTALS Hot Cold probe probe A 10H Hot (一) (+) p-type n-type Energetic holes Energetic electrons diffuse away diffuse away (a) (b) Figure3.13Thehot-pointprobemeasurement:(a)requiredequipmen;(b)simplifiedexplanation of how the measurement works. numberofhigher-energycarriers. Theseenergeticcarriers willbe predominantly holesin carriers near the heated probethanelsewhere,diffusion acts so as to spread thehigher- energy carriers throughoutthe semiconductor wafer.Theneteffect isa deficitof holes or a net negative charge in the vicinityof the hot-point probefor a p-type material,and a posi- tive charge buildupnearthe heatedprobe within an n-type material. Accordingly,the center-zero meter deflects in a different direction for p-and n-type materials. 3.2.3Diffusionand TotalCurrents DiffusionCurrents In defining thediffusionprocessand incitingdiffusionexamples,we have soughttoem- phasizethedirectcorrelationbetweendiffusionandaspatial variationinparticlenumber. For diffusion tooccur,moreof the diffusing particles must exist atone point than atother points or,in mathematical terms,there must be a nonzero concentration gradient (∇p≠0 forholes,n≠0for electrons).Logically,moreover,the greatertheconcentrationgradient, the largerthe expectedfluxof particles.Quantitativeanalysisofthe diffusion process in- deed confirms the foregoing and leads to what is known as Fick’s law: F=-D∇η (3.16) where Fis thefluxor particles/cm-seccrossing a plane perpendicularto the particle low, ηis theparticleconcentrtionandDisthediusioncoefficienpositiveproportioali factor.The diffusioncurrent density due toelectronsand holes isobtainedby simplymul- tiplying the carrier flux by the carrier charge: Jpldiff= - Dp∇p (3.17a) JNldiff= qDN∇n (3.17b)

--- Page 122 --- CARRIER ACTION 66 TheconstantsofproportionalityDpand DN,haveunitsofcm²/secandarereferredtoas the holeandelectrondiffusioncoefficients,respectively. Upon examining Eqs.(3.17),note thatthecurrent directions deducedfrom theequa- tions areconsistentwiththeFig.3.12visualizationofthemacroscopicdiffusioncurrents. Forthepositiveconcentration gradient shown in Fig.3.12(dpldx>0anddnldx>0for thepicturedone-dimensionalsituation),bothholesandelectrons willdiffuse inthe-x directionldiwillthereforebenegativeorirectedinthedirectionwillifwill beoriented in the +x direction. TotalCurrents Thetotal or net carriercurrents in asemiconductorariseas thecombined resultof drift and diffusion.Summing the respectiven andp segmentsof Eqs.(3.4) and Eqs.(3.17),one obtains JP = JPldrift + JPlif= qpp8-q∇p (3.18a) diffusion JN=JNldrift+JNldi=qnn8+qDNn (3.18b) Thetotal particle current flowing in a semiconductor is in turn computed from J =JN +J_P (3.19) Thedoubleboxesemphasizetheimportanceofthetotal-currentrlationships;theyare used directlyorindirectlyinthe vastmajorityofdeviceanalyses. 3.2.4 Relating Diffusion Coefficients/Mobilities Thediffusion coefficients,theconstantsof themotionassociated withdiffusion,areobvi- ouslycentral parameters incharacterizingcarrier transport dueto diffusion.Giventhe im- portanceof the diffusioncoefficients,one mightanticipate an extended examinationof relevantpropertiesparallelingthemobilitypresentation inSubsection3.1.3.Fortunately, an extended examination is unnecessary because the D’s and the ‘s areall interrelated.It is only necessary toestablishthe connecting formulaknown as the Einsteinrelationship. In deriving the Einstein relationship,we consider a nonuniformly doped semiconduc- tor maintainedunderequilibriumconditions.Specialfactsrelatedtothenonuniform dop- ing and theequilibrium state are invoked in thederivation.These facts,important in them- selves, are reviewed priorto presenting the derivation proper. Constancy of the Fermi Level Consider a nondegenerate, nonuniformly doped n-type semiconductor sample, a sample where the doping concentration varies as a functionof position.A concrete exampleof what we have inmind is shown in Fig.3.14(a).Assuming equilibrium conditions prevail,

--- Page 123 --- 100 SEMICONDUCTORFUNDAMENTALS the energy band diagram characterizing the sample willhave theform sketchedin Fig.3.14(b).In Chapter 2the Fermi level in uniformly doped n-type semiconductors was found to move closer and closer to Ewhen the donor doping was systematically increased （see Fig.2.21).Consistent withthis fact,Ewas drawnclosertoEin going fromx=0to x=Lon theFig.3.14(b)diagram.Constructionof the diagram,however,couldnothave been completed withoutmaking useof a critical new piece of information.Specifically, underequilibriumconditions,dE/dx=dE/dy=dE/dz=0;i.e.,theFermi level inside amaterial oragroupofmaterials inintimatecontact is invariant asafunctionofposition. TheconstancyoftheFermilevelmeans thatEFis toappearasahorizontallineon equilib- rium energy band diagrams. ThepositionindependenceoftheFermienergy isestablishedbyexaminintherans- ND ↑ +→x 0 L (a) $EEFFEiEv$ x 0 L (b) Figure3.14 Nonuniformlydoped semiconductor:（a）assumeddoping variation withposition; (b) corresponding equilibrium energy band diagram.

--- Page 124 --- CARRIER ACTION 101 fer ofcarriers betweenallowed states withthesameenergy butat adjacent positions inan energyband.Itisconcludedtheprobabilityoffillingthestatesatagivenenergy,f(E),must be the same everywhere in the sample underequilibrium conditions.If this were not the case,carriers wouldpreferentiallytransferbetween statesand thereby give rise to a net current. The existenceof a netcurrent is inconsistent withthe specifiedequilibriumcon- ditions.Referring to the Eq.(2.7)expression for f(E),we findthattheconstancyofthe Fermi function in turn requires the Fermi level to be independent of position. Current Flow Under EquilibriumConditions Underequilibrium conditionsthetotalcurrent isofcourseidentically zero.Becauseelec- tronand holeactivityis totallydecoupledunderequilibriumconditions,theelectronand Eq.(3.18),however, leads to theconclusion thatthe driftand diffusioncurrents associated with a given carrierare merelyrequired to beofequal magnitude andopposite polarity.In fact,thedriftanddiffusioncomponents willvanishunderequilibriumconditionsonlyif 0=dΔ=u△pu0=8 To provide a concrete illustrationof non-vanishing equilibrium current components, let us return to an examination of the nonuniformly doped semiconductor samplecharac- terized byFig.3.14.Thedopant variationpictured inFig.3.14(a）clearlygives rise toa significantelectronconcentrationgradient.Anelectrondiffusioncurrentflowinginthe+x direction must exist inside the sample.Thebandbending shown in the band diagramof Fig.3.14(b),moreover,implies theexistenceof a”built-in”electric fieldoriented in the -x direction. This gives rise to a drift current also in the-x direction.Notethat the drift and diffusion components areofoppositepolarity. Since equilibriumconditions are as- sumed to prevail,thecomponents must also beofequal magnitude.In pointof fact,the electricfieldarises insidethe semiconductor to counteract thediffusive tendencies of the carriers caused bythenonuniform doping.The major point here is that,evenunderequili- riumcondition,nonuniformdoping willgiverisetocarrierconcentrationgradients,uil- in electric field,and non-zero current components. EinsteinRelationship Having laid the proper foundation,we can now proceed to derive the connecting formula between the D’s and the μ’s known as the Einstein relationship. To simplify the develop- ment,we work inonlyone dimension. The sample under analysis is taken to be a nonde- generate, nonuniformly doped semiconductor maintained under equilibrium conditions. Citingthefactthatthenetcarriercurrentsmustbeidenticallyzerounderequilibriumcon- ditions,and focusing on the electrons,we can state dn JNldrift +JNlif= qnn+DN =0 (3.20) xp

--- Page 125 --- 102 SEMICONDUCTOR FUNDAMENTALS However, $1dE$ 8= (3.21) xp b$(sameas3.15)andn=n;e(E-Ei)/k(3.22)Moreover,withdE/dx=0(duetothepositionalinvarianceoftheFermilevelunderequilibriumconditions),dn$n_}dE}$9e(EF-E)/kTno(3.23)dx$kT dx kT SubstitutingdndxfromEq.(3.23)intoEq.(3.20),andrearrangingtheresultslightly,one obtains (qnE)μn-(qn) \a} DN=0 (3.24) kT Since≠0(aconsequenceof the nonuniform doping),it follows from Eq.(3.24)that DN$kTEinsteinrelationshipforelectrons(3.25a)$μ_{} q A similar argument for holes yields D{ kT Einsteinrelationshipforholes (3.25b) μ{ q Although it wasestablished whileassuming equilibrium conditions,we can present more elaboratearguments that showthe Einsteinrelationship to be valideven undernon- equilibrium conditions. Thenondegeneraterestriction,however,stillapplies;slightly modifiedformsof Eqs.(3.25)result whenthe argument is extendedto degenerate mate-

--- Page 126 --- CARRIER ACTION 103 rials.We shouldemphasize againthatthepreliminaryresults,suchasthe positionalinvari- ance of the equilibrium Fermi level and the situation inside a nonuniformly doped semi- conductorunder equilibrium conditions,are important in themselves.Relative to numerical values,note thatkT/qisa voltage,and at room temperature(300K)is equal to0.0259V. Hence,for an ND=104/cm3 Si samplemaintained at roomtemperature, DN=(kT/q)μn =(0.0259)(1358cm²/V-sec)=35.2cm²/sec.Finally,theEinstein relationship is oneof the easiest equations to remember because it rhymes internally—D overμ equals kT over q. The rhyme holds even if the equation is inverted—over D equals q over kT. Exercise 3.4 P:In this exercise we continue the examination of the semiconductor sample repre- sented by the energy band diagram presented in Exercise 3.2. (a) The semiconductor is inequilibrium.Howdoesone deducethis factfromthe given energy band diagram? (b)Whatistheelectroncurrentdensity（J）andholecurrentdensity（Jp)atx= ±L/2? (c) Roughly sketch n and p versus x inside the sample. (d)Is thereanelectrondiffusioncurrentatx=±L/2?Ifthereisadiffusioncurrent at a given point,indicate the direction of current flow. (e) Noting thesolutionfortheelectricfieldpresentedin Exercise3.2,is therean electrondrifcurrnttL/2Ifthereidifcurrentatagivenpoininc the direction of current flow. (f) What is thehole diffusioncoefficient(Dp)in thex>Lregionofthesemi- conductor? S:(a)The semiconductor isconcluded to be in equilibriumbecausetheFermilevel is invariant as a function of position. (b) Jn}=0 and Jp=0atbothx=-L/2and=+L/2.Thenetelectronand holecurrentsarealwaysidenticallyzeroeverywhereunderequilibriumconditions. AtroomtemperaturekT=0.0259eV=(1.6×10-19)(0.0259)joule.ThuskT/q=(1.6×10-19)(0.0259)/ (1.6×10-19)joule/coul=0.0259V.

--- Page 127 --- 104 SEMICONDUCTORFUNDAMENTALS (c) Sincen=n;exp[(EF−E)/kT] andp=n;exp[(E-F)/kT],weconclude (Log scale) p $ni$ n x -L L (d) There is anelectron diffusioncurrentat bothx=-L/2andx=+L/2.Fromthe answer to the preceding question we note that dnldx≠O at the cited points. With dn/dx>0,JNldiat=-L/2flowsin the+xdirection.Conversely,dnldx<0at x=L/2and JNldif atthis point flows in the-xdirection. (e) There is an electron drift current at bothx=-L/2andx=+L/2.Since bothn and are non-zero atx=±L/2itfollows thatJNldrift=qμn≠0.The drift componentoftecralyshastheamedrctionastectricfilJdriis in the-x direction atx=-L/2 and in the+x directionatx=+L/2.The drift componentofthecurrentmustofcoursecancelthe diffusioncomponentof thecur- related answers to parts (d)and(e),summarized next,are consistent withthis requirement: x=-L/2 x=+L/2 JNdff → ← JNldrift ← → (f) In working part (d)of Exercise 3.2,we concluded the x>Lregion of the semi- conductor (Si,300K)had a dopingof NA≈5×1014/cmand a corresponding hole mobilityofμ=459cm²/V-sec.Thus,employing the Einstein relationship, =(kT/q)p=(0.0259)(459)=11.9cm²/sec

--- Page 128 --- CARRIER ACTION 105 3.3 RECOMBINATION-GENERATION 3.3.1 Definition-Visualization When a semiconductor is perturbedfrom theequilibrium state,an excess or deficit inthe carrierconcentrationsrelativetotheirequilibriumvaluesis invariablycreatedinsidethe semiconductor.Recombination-generation (R-G) is nature’s order-restoring mechanism, the means wherebythe carrierexcess or deficit inside thesemiconductor is stabilized(if the perturbation is maintained)or eliminated (if the perturbation is removed).Since non- equilibriumconditionsprevailduring deviceoperation,recombination-generationmore often than not plays a majorrole in shaping thecharacteristics exhibited by a device.For- mally,recombinationand generation can be defined as follows: Recombination-a process whereby electrons and holes(carriers) are annihilated or destroyed. Generation—a process whereby electrons and holes are created. Unlikedriftanddifusion,thetermsrecombinationandgenerationdonotrefertoasingle process.Rathertheyarecollectivenamesforgroupofsimilarprocessescarrierscane created and destroyed within a semiconductor in a numberof ways. The most common R-Gprocesses are visualized inFig.3.15.The individual processes are described and discussed next. Band-to-Band Recombination Band-to-bandrecombinationisconceptuallythesimplestofallrecombinationprocesses. AspicturedinFig.3.15(a),itmerelyinvolves the directannihilationofaconductionband electron and a valence band hole.An electron and hole moving in the semiconductor lattice stray into the same spatial vicinityand zap!—theelectronand holeannihilate eachother. Theexcessenergyreleased during theprocesstypicallygoesintotheproductionofapho- ton (light). R-GCenterRecombination Theprocess pictured inFig.3.15(b)involvesa“thirdparty”or intermediary,andtakes placeonlyat speciallocations withinthesemiconductorknownasR-Gcenters.Physically, R-Gcenters are lattice defects or special impurityatoms such as gold in Si.Lattice defects and the special atoms in the formof unintentional impurities are presenteven in semicon- ductors ofthe highestavailable purity.The R-Gcenterconcentration,however,is normally very low compared to the acceptor and donorconcentrations in device-quality materials. The most important propertyof the R-G centers is the introductionof allowed electronic levels nearthecenterof the band gap.These levels are identified by the ETenergy label in Fig.3.15(b).The nearmidgap positioning of thelevels is all-important because it distin- guishes R-G centers from donors and acceptors.Themidgap levels introduced by select R-G center impurities in Si are summarized in Fig.3.16.

--- Page 129 --- 106 SEMICONDUCTORFUNDAMENTALS $Ec$ Ec Thermal W Photon energy (Light) 1O W Light Ev} Ev} X←0 (a) Band-to-band recombination (d) Band-to-band generation E{cEc or —ET Thermal Thermal ET energy energy E_v} $Ev$ (b) R-G center recombination (e) R-G center generation Ec Ec EvEv}$ (c) Auger recombination (f) Carrier generation via impact ionization Figure3.15 Energy band visualizationof recombination and generation processes. As noted in Fig.3.15(b),recombination at an R-Gcenter is a two-step process.First, onetypeofcarrier,sayanlectronstraysintothevicinityofanRGcenteriscaughtb the potential well associated withthe center,loses energy,and is trapped.Subsequently,a hole comes along,is attracted to the trapped electron, loses energy,andannihilatesthe electronwithinthecenter.Alternatively,as also showninFig.3.15(b),onecanthinkofthe electron as losing energy a secondtimeandannihilatingthe hole in the valence band.R-G centerrecombinationalsocalledindirectrecombination,typically releasesthermalenergy (heat) during the process or,equivalently,produces lattice vibrations.

--- Page 130 --- CARRIER ACTION 107 -$Ec$ =---Midgap Ev Au Cu Mn Cr Fe Figure3.16 Near-midgapenergy levels introduced by some common impurities in Si.Whenan impurityintroduces multiple levels,oneofthe levels tends to dominate in a given semiconductor sample. AugerRecombination In the Auger (pronouncedOh-jay)process pictured in Fig.3.15(c),band-to-band recombi- nation occurs simultaneously withthecollisionbetween twolike carriers.Theenergyre- leased by the recombination is transferred during thecollisionto the surviving carrier.This highly energetic carriersubsequently“thermalizes”—losesenergy in small steps through heat-producing collisions with thesemiconductorlattice.The”staircase”inFig.3.15(c) represents the envisioned stepwise loss of energy. GenerationProcesses Anyof the foregoing recombination processes can bereversed to generatecarriersBand- to-bandgeneration,whereanelectron is excited directlyfrom the valence band intothe conductionband,is picturedinFig.3.15(d).Notethateitherthermalenergyorlightcan providetheenergyrequiredfortheband-to-bandtransition.Ifthermalenergy isabsorbed, the processisalternativelyreferredtoasdirectthermalgeneration;ifexternallyintroduced light is absorbed,theprocess is called photogeneration.Thethermallyassisted generation ofcarriers with R-Gcenters acting as intermediaries is envisioned inFig.3.15(e).Finally, impact ionization,the inverseofAugerrecombination,is visualized inFig.3.15f).In this process anelectron-hole pair is produced as a resultofenergyreleased when a highly energeticcarriercollideswiththecrystalltice.Thenerationofcairsthrouhimact ionizationroutinelyoccursinthehigh-fieldregionsofdevices.Wewillhavemoretosay about this process when treating breakdown in pn junctions. 3.3.2MomentumConsiderations Allof the variousrecombination-generation processesoccur atalltimes inall semicon- ductors—evenunderequilibriumconditions.Thecriticalissueisnotwhetherthercess are occurring,butrather,therates at which the various processesareoccurring.Typically, one need beconcernedonly with the dominant process,the process proceeding atthefastest rate.Anumberofthe processes areimportantonly under special conditionsor ae highly improbable,occurring at a much slower rate than competing processes.Auger recombia-

--- Page 131 --- 108 SEMICONDUCTORFUNDAMENTALS tion provides a case inpoint. Becausethe numberofcarrier-carriercollisions increases with increased carrier concentration,thefrequencyofAugerrecombinationlikewisein- creases with carrier concentration.Auger recombination must beconsidered,for example, in treating highlydoped regionsof a devicestructure,but it is usuallynegligibleotherwise. Similarly,significantimpactionizationoccursonlyinthehigh-feldregionsofadevic. In a somewhat different vein,photogeneration at room temperature becomes a significant processonlywhenthesemiconductor isexposed to external illumination.Whatthen is the R-Gprocesspairdeservingofspecialattention,theprocesspairthattypicallydominates in thelow E-field regions of a nondegenerately doped semiconductormaintained atroom temperature?Fromthe information provided, the choice is obviously between thermal band-to-band R-Gand recombination-generation via R-G centers. From theenergybandexplanationaloneonemightspeculatethatband-to-band recombination-generation woulddominateunderthe cited“standard”conditions.How- ever, visualizationof R-G processes using the energy band diagram can be misleading. The E-x plot examines only changes in energy,whereas crystalmomentum inadditionto energy must beconserved in any R-G process.Momentumconservationrequirements,as it turns out,play an important role in setting the process rate. As firstnoted inSubsection2.2.3,themomentumofanelectron inan energyband, like the electron energy,can assumeonly certain quantized values.Momentum-related as- pectsofR-Gprocessesareconvenientlydiscussed withtheaidofplots wheretheallowed electron energies in theconductionand valence bands are plotted versustheallowedmo- mentum.These are referredto asE-k plots,wherek is a parameter proportionaltothe electroncrystalmomentum. TheE-k plotsassociated withsemiconductors fall intotwo generalcategories.InthefirstsketchedinFig.3.17a)boththeminimumconductonnd energy and maximum valence band energy occur atk=0.In the second,pictured in Fig.3.17(b),the conduction band minimum is displaced to ak≠0.Semiconductors exhib- iting the former typeof plotare referred to as direct semiconductors,and semiconductors exhibiting the latter typeof plot,as indirect semiconductors.GaAs is a notable memberof the direct semiconductor family;Ge and Si are both indirect semiconductors. E E 4 4 E $Ec$ E} E}$ k (a) Direct semiconductor (b) Indirect semiconductor Figure3.17 GeneralformsofE-k plots for direct and indirect semiconductors.

--- Page 132 --- CARRIER ACTION 109 E E Phonon(s) WPhoton W k k (a) Direct semiconductor (b) Indirect semiconductor Figure3.18 E-kplot visualizationsofrecombination indirectandindirectsemiconductors. To make use of the E-k plots in visualizing an R-G process,one needs to know the natureof transitionsassociated withtheabsorptionoremissionof photons(light)and lat- tice vibration quanta called phonons. Photons,being massless entities,carry very little momentum,and a photon-assisted transition is essentially verticalonthe E-kplot.Con- versely,thethermalenergyassociated with lattice vibrations (phonons)is very small(in the10-50meVrange), whereas the phonon momentum is comparativelylarge. Thuson anE-kplota phonon-assistedtransition is essentiallyhorizontal.It is alsoimportantto pointoutthat,aspicturedinFig.3.18,electronsandholesnormallyonlyoccupysates very close to the Ecminimumand Emaximum,respectively.This isof course consistent with our earlier discussion of the distribution of carriers in the bands. Let us now specifically re-examine the band-to-band recombination process.In a direct semiconductor wherethek-valuesofelectrons andholes areallbunched neark=0,little change inmomentum is required for the recombination process to proceed. The conserva- tion of both energy and momentum is readily met simply by theemission of a photon (see Fig.3.18a).In an indirect semiconductor,on theother hand,there is a large change in momentum associated with the recombination process. The emission of a photon will con- serve energy but cannot simultaneously conserve momentum. Thus for band-to-band re- combination to proceed in an indirectsemiconductor, theemissionof a photon must be accompanied by the emission or absorption of a phonon (see Fig.3.18b). Therather involved natureoftheband-to-band process inindirect semiconductors understandably leads to a vastly diminished recombination rate. Band-to-band recombina- tion is infacttotallynegligiblecomparedtoR-GcenterrecombinationinSiandother indirect semiconductors.Band-to-band recombination does proceed at a muchfasterrate in direct semiconductorsand is the process producing the lightobserved fromLEDsand junction lasers.Even in directmaterials,however,the R-G center mechanism is oftenthe dominant process.Given theuniversal importanceof theR-Gcenterprocess,and thefact thatmost devices are made from Si,which is an indirectmaterial,the quantitativeconsid- erations in the next subsectionaredevoted primarilyto theR-Gcentermechanism.Other processes are considered as the need arises.

--- Page 133 --- 110 SEMICONDUCTOR FUNDAMENTALS 3.3.3 R-G Statistics R-G statistics is thetechnicalname given to themathematicalcharacterization ofrecom- bination-generation processes.“Mathematical characterization” inthe caseofrecombi- nation-generation does notmeanthe developmentof a current-densityrelationship.An R-Geventoccurs at localized positions in the crystal,at a given x-value,and therefore the R-Gaction does notlead per se to charge transport. Rather,recombination-generation acts tochangethecarrierconcentrations,therebyindirectlyaffectingthecurrentflow.Itis thetimerateofchangeinthecarrierconcentrations（∂n/p/t)thatmustbespecfiedto achievethedesiredmathematicalcharacteriaion.ecauseitscharacterizationisrltil simpleand its involvement in deviceoperationquitecommon,we beginby considering photogeneration. Themajor portionof the subsection is ofcourse devoted to recombina- tion-generation via R-Gcenters,alternativelyreferred to as indirect thermal recombina- tion-generation. Photogeneration Light striking thesurfaceofa semiconductoras pictured inFig.3.19willbepartially reflectedand partiallytransmittedintothematerial.Assumethe lightto bemonochromatic with wavelengthλandfrequency ν.Ifthe photonenergy (hν)is greaterthan the bandgap energy,thenthelightwillbeabsorbedandelectron-holepairs willbecreatedasthe light passesthroughthe semiconductor.Thedecrease inintensityasmonochromatic lightpasses through a material is given by I=$Ioe-ax$ (3.26) whereI0is the lightintensity justinsidethematerialat=0+andαis theabsorption coefficient.NotefromFig.3.20thatα is material dependentand a strong functionofλ. Since there is a one-to-one correspondence betweenthe absorptionof photons andthe creationofelectron-hole pairs,the carrier creationrate shouldlikewise exhibit an exp(-αx) dependence. Moreover,referring to Fig.3.15(d),the photogeneration process always acts so as to create anequal numberofelectrons andholes. Wethereforeconclude ∂n ∂p =G{L(x,λ) (3.27) ∂t 1e light light where, for the situation analyzed, GL(x,λ)= GLoe-ax (3.28) GListhesimplifiedsymbolforthephotogenerationrate(anumber/cmsec,whileG0is essary because,as we have seen,n and p can be affected by a number of processes. In working problems involving the illuminationof a sample or device,we will often invokeone of two simplifying assumptions:(i) The illumination is assumed tobe uniform everywhere inside thesampleand (ii)the light is assumedtobeabsorbedinan

--- Page 134 --- CARRIERACTION 111 Light (λ,v) +0 Semiconductor L x Figure3.19Semiconductorlightpropagationandcoordinateorientationinthephotogertion analysis.The semiconductor is assumed to be sufficiently thick(L>l/α)so that bottom surface reflections may be neglected. 105 104 103 $α(cm-1) 102 GaAs- Ge Si 10 InP GaP 1 1 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 λ(μm) Figure3.20 Theabsorptioncoefficientasafunctionof wavelengthinSiandotherselectsemicon- ductors.(From Schroder[6l,©1990by John Wiley & Sons,Inc. Reprinted with permission)

--- Page 135 --- 112 SEMICONDUCTORFUNDAMENTALS infinitesimallythinlayer nearthe semiconductor surface.Seekingtoidentify whenthe assumed conditionsare approached in practice,we note first thatl/α,which has the dimen- sions of a length,represents the average depthof penetration of the light into a material.At wavelengthswherehν≈EG,theabsorptioncoefficientis verysmalland1/isvery large, about1 cm inSi.This makesexp(-x)≤1forallpractical depths. To a very good approximation,GL= constantandthe lightis being uniformly absorbed.Ontheother hand,ifλ=0.4μm,an (Si)10³cm-is deducedfrom Fig.3.20.Atthis wavelength the lightpenetration depth isonly a few thousandangstromsand theassumptionof surface absorption is closely approximated. Indirect ThermalRecombination-Generation Althoughthe general case developmentofR-Gcenterstatistics is beyond the scopeofthis textitispossibetopresntrgumentsleaintothespcialcaseexpressionemploin many practicalproblems.Letus begin bycarefullydefining the variouscarrierandcenter concentrations involved in the analysis. n_{，p carrier concentrations inthematerialunderanalysis whenequilib- rium conditions prevail. n,p carrier concentrations in the material underarbitrary conditions. Δn=n-n_0 . . deviations inthecarrierconcentrationsfromtheirequilibrium values. Δp=p-Po Δn and Δp can be both positive and negative,where a positive devia- tion corresponds to a carrier excess and a negative deviation corre- sponds to a carrier deficit. No} number of R-G centers/cm3. In the special-case development,perturbationof the carrierconcentrations from their equilibriumvaluesmustcorrespondtolowlevelinjection.Thisrequirementisjustanal- ternative wayofsayingtheperturbationmustberelatively small.Tobemore precise, low-level injection implies ∆p≤n{, n ≈n{ in an n-type material Δn≤p0，p≈P0 in a p-type material Consider aspecificexampleofNp=104/cm3dopedSiatroomtemperaturesubject toa perturbationwhere△p=∆n=10/cm3.Forthegivenmaterial,n0=N=1014/cm3and p0≈n²/ND=10/cm3.Consequently，n=n0+Δn=n0andΔ=10/cm≤n0=14 cm.Thesituation isclearlyoneoflow-levelinjection.Observe,however,thatp P0.Although themajoritycarrierconcentrationremainsessentiallyunperturbedunder low-levelinjection,theminoritycarrierconcentrationcan,androutinelydoes,increaseby many orders of magnitude.

--- Page 136 --- CARRIER ACTION 113 Perturbation △P_0 △p<△P0 △p=0 t=0 1>0 1→∞ n
no E E} c $E$ E}$Po$ -△p<n_{0 Figure3.21 Situationinsideann-typesemiconductorafter a perturbationcausing thelow-level injection of excess holes. We are nowready to analyze the specific situationenvisioned in Fig.3.21.A pertur- bation hascaused aΔpn0excessofholes in an n-typesemiconductor sample.Assuming the system is in themidstofrelaxingbackto theequilibrium statefromthe pictured per- turbed state via the R-Gcenter interaction,whatfactors wouldyou expectto havethe greatesteffecton∂p/∂t|R,therateofholerecombination?Well,to eliminate a hole,the holemustmakethetransitionfromthevalencebandtoanelectron-filledR-Gcenter. Logically,thegreaterthe numberoffilled R-Gcenters,thegreatertheprobabilityofahole annihilating transitionandthefaster therateofrecombination.Underequilibriumcondi- tions,essentiallyalloftheR-Gcentersarefilledwithelectronsbecausetheequilibrium Fermi level is comfortably aboveET.With ∆p≤n0,electrons always vastly outnumber holesandrapidlyfill R-Glevelsthat become vacant.Thiskeepsthenumberof filledR-G centers during the relaxation process≈N.Thus,we expect∂p/tto beapproximately proportionaltoNT.Moreover,italsoseemslogicalthatthenumberofhole-annihilating transitionsshouldinceasealmostlnearlwiththenumberofholes.Allotheractorsbeing equal,the more holes available for annihilation,the greater the numberof holes removed from the valence band per second.Consequently,we also expect∂p/∂tto be approxi- matelyproportionalto p.Consideringadditionalfactorsleadstonoadditionaldependen- cies.HeceinroducinoitivroortionlitotnnrlinRis negative because p is decreasing, we conclude ∂p$ =- (3.29) 1e R

--- Page 137 --- 114 SEMICONDUCTORFUNDAMENTALS Ifthe holegeneration process is analyzed in a similarmanner,∂p/∂t|G is deduced to dependonlyon the numberofempty R-G centers. This number,which is smallforthe situationunderconsideration,isheldapproximatelyconstantatitsequilibrium valueover the rangeof perturbations whereholegenerationis significant.This meansthat∂p mustpreciselybalanceunderequilibriumconditionor∂ppGequilibrium

lumukuso ∂p$=p0$ (3.30) 1e G Finally,thenetrateofchange intheholeconcentrationdue tothe R-Gcenterinter- action is given by ∂p$\partial p$ ∂p + =-p(p-0) (3.31) ∂t i-thermal 1e 1e R G R-G or ∂p$forholesinann-typematerial(3.32a)\partial ti-thermalR-GAnanalogoussetofargumentsyields\partial n-nN$ for electrons in a p-type material (3.32b) 1e i-thermal R-G cnlikepisositiveproportionalitconstant.nandarerefredtoasthelectronand hole capture coefficients,respectively. Althoughcompletelyfunctionalas is,it is desirabletomanipulate Eqs.(3.32)inoa slightlymore compactandmeaningfulform. Anexaminationoftheleft-handsidesof Eqs.(3.32)revealsthedimensionalunitsarethoseofaconcentrationdividedbytime.Since Δp and Δnon theright-hand sidesofthe sameequations arealsoconcentrations,the con- stantscpNandnNTmusthaveunitsof1/time.Itisthereforereasonabletointroducethe time constants Underequilibriumconditionseachfundamentalprocessand its inverse mustself-balance independentofany otherprocess thatmaybeoccurring insidethe material.Thisisknownas thePrincipleof Detailed Balance.

--- Page 138 --- CARRIER ACTION 115 1 二 $cT(3.33a)1$N{}$(3.33b)which,whensubstitutedintoEqs.(3.32),yield$\p$\Delta p$ for holes in an n-type material (3.34a) 1e i-thermal R-G ∂n Δn forelectrons in a p-type material (3.34b) 1e i-thermal Tn$R-GEquations(3.34)arethedesiredendresult,thespecial-casecharacterizationofR-Gcenter(indirectthermal)recombination-generation.AlthoughsteadystateorslowlyvaryingconditionsareimplicitlyassumedinthedevelopmentofEqs.(3.34),therelation-shipscanbeappliedwithlittleerrortomosttransientproblemsofinterest.Itshouldberestatedthata\Delta <0ispossibleandwillgiverisetoa\partial p/thermalG>0.Apositive\partial p/\partial t|i-thermalR-GsimplyindicatesthatacarrierdeficitexistsinsidethesemiconductorandgenerationisoccurrinatamorerapidratethnrecombinationplithermlRGandbinationandthermalgenerationprocesses.Asdulyemphasized,theEq.(3.34)relationshipsapplyonlytominoritycarriersandtosituationsmeetingthelow-levelinjectionrequirement.Themoregeneralsteadystateresult[7],validforarbitraryinjectionlevelsandbothcarriertypesinanondegeneratesemi-conductor,isnotedforcompletenessandfuturereferencetobe\partial p$ ∂n $n^{2 }p (3.35) 1e i-thermal ∂t i-thermal p(n+n1)+τn(p+p1 R-G R-G where (3.36a) n₁=n₁ e(EE)kT P₁≡n;e(E-ET)/kT (3.36b)

--- Page 139 --- 116 SEMICONDUCTOR FUNDAMENTALS It is leftasanexercisetoshowthatthegeneral-caserelationshipreducestoEqs.(3.34) under the assumed special-case conditions. 3.3.4MinorityCarrier Lifetimes General Information The timeconstants τnand,which wereintroduced withoutcommentin writing down Eqs.(3.34),areobviously the“constantsof theactionassociated withrecombination- generation.Seeking to provide insightrelative to the standard interpretationand namingof theT’s,letusconsideronceagainthesituation picturedinFig.3.21.Examiningthechange in the holeconcentration with time,it goes almost without saying that the excess holesdo not alldisappearatthe sametime.Rather,the holeexcess presentatt=Ois systematically eliminated,withsomeof theholesexisting foronly ashort periodandothers”living”for comparativelylongperiodsoftime.Ifthermalrecombination-generation isthesolepro- cess acting to relax the semiconductor,the average excess hole lifetime,(t),can be com- putedinarelativelystraightforward manner.Withoutgoingintodetails,thecomputation yields<t）=τn（orτp).Physically,therefore，τnandτhavecometobeinterpretedasthe average time an excess minority carrier will live in a seaof majority carriers.For identi- fication purposes,τandτare simplyreferredtoas theminority carrier lifetimes. Like theμ’s and the Ds,the s are importantmaterial parameters invariablyrequired in the modelingof devices.Unlike the μs and the D’s,thereare few plots indicating the -values to be employed in a given device structure.In fact,subsidiary experimental mea- surements(see thefollowing subsection）must be performedto determine theminority carrierlifetime in a given semiconductor sample.Thereason forthe lackofcatalogued informationcan betracedtotheextreme variabilityoftheτnand τparameters.Referring to Eqs.(3.33),we seethatthecarrier lifetimes dependontheoften poorlycontrolledR-G centerconcentration(N),notonthecarefullycontrolleddopingparameters(NAandN). Moreover,the physical natureof thedominant R-G center concentration changes even withina given sampleduring devicefabrication.Afabricationprocedurecalledgettering can reduce the R-Gcenterconcentration to a very lowlevel and give rise to aτn(）~ 1msec inSi.The intentional introductionof goldintoSi,on theotherhand,cancontrol- lablyincrease the R-Gcenterconcentrationand giverise toaτn(p)~1nsec.Minority carrierlifetimesin completedSidevicestendtolieaboutmidway betweenthecited extremes. ALifetimeMeasurement The carrier lifetimes are measured using a varietyof differentmethods. Themajorityof measurementsemploy a device structure of some sort and deduce τn and/orτby matching the theoreticaland experimentallyobserved device characteristics. The measurement to be described employs a bar-like pieceof semiconductor withohmic contacts at its two ends. In a commercial implementation this structure is called a photoconductor,a typeof photo- detector.Asituation very similar tothat envisioned inFig.3.21is created inside thesemi-

--- Page 140 --- CARRIER ACTION 117 Light pulses Rs + VA ↑！ R{ U_L Scope Figure3.22 Schematic illustration of the photoconductivity decay measurement. conductor during themeasurement. Theperturbation is in the formofa lightpulse that gives rise to a carrierexcess inside the semiconductor and a detectable change in conduc- tivity.Once the light isremoved the excesscarriersareeliminated via recombinationand theconductivitydecaysbacktoits valueinthedark.Thecarrierlifetimeisdeterminedy monitoring the conductivity decay. Theexperiment is schematically illustrated inFig.3.22.Ris the sampleresistance, RLaloadresistor,VAtheappliedd.c.voltage,andvthemeasured loadoroutput voltage. Obviously,Randtherefore varefunctionsoftime,withvrefectingthechanesin conductivity inside the semiconductor sample. Weare specificallyinterestedinhowvchangeswithtimeafterthe light isturnedoff. vLisrelatedtotheconductivitywhichinturnisrelatedtotheexcesscarrierconcenrtions. Given an n-type sample and assuming uniform photogenerationthroughout the semicon- ductor,wewillverifylaterinthechapterSection3.5)thatthedecayoftheexcessmority carrier hole concentration (∆p) is described by ∆p(t)= ∆p0e−/p (3.37) ∆p0istheexcessholeconcentrationatt=0whenthelightisturnedoffNextconsidering the conductivity, we note σ = 1/ρ =q(μn +pp) (3.38a) =q[n（n0+∆n)+（0+∆p)] (3.38b) =qμnND+q(n+)Δ (3.39) ∆σ(t) In establishing Eq.(3.39),we made use of thefact that0≈ND≥p0 in an n-type semi- conductor.Also Δn was assumedequalto Δp.darkis theunperturbedsampleconductivity in the dark and ∆(t)α ∆prepresents the time decaying componentof theconductivity.

--- Page 141 --- 118 SEMICONDUCTORFUNDAMENTALS Thelaststepistorelateνand.Inspecting Fig.3.22,oneconcluds U_ = iRL= VAR2+_ $R_L$ (3.40) but R_{s

ρl l l

sd$dark(3.41)A\sigma A(dark+∆\sigma )Adark+\Delta \sigma wherelisthelength,Athecross-sectionalarea,andRd=ldarkAthedarkresistanceofthesample,respectively.SubstitutingEq.(3.41)intoEq.(3.40)andrearrangingyields$VA}$(3.42)$R{_{}_L}$1+$1+∆/σdarkk or,ifRLismatchedtoRdandthelightintensityrestrictedsoΔ/dark1(equivalentto specifying low-level injection conditions), we obtain $VA$ △σ $UL 1+

V+v0 e-1/ (3.43) 2 dark □ d.c. decay transient An exponential decay is predicted with τp being the decay constant. An actual implementation of the measurement is shown in Fig.3.23.The required VA do),while the Tektronix11401DigitizingOscilloscope is usedto capture the photocon- ductive decay.The bar-shaped Si test structures are housed in a special measurements box along withthe RL=Rsdloadresistor.Theboxpermitsconvenienthandlingofthesamples andfacilitates switching between samples. The light pulsesrequired in themeasurement are derivedfromthe GeneralRadio1531-ABStroboscope. TheGRStroboscope putsout high-intensity light flashesof~1μsec duration spaced at controllable time intervals rang- ing from2.5msec to 0.5 sec.Fairly uniform photogenerationcan beachieved in a sample Thisbeingthefirst textreference,itis perhaps worthwhilereiterating the GeneralIntroduction statementthat measurement particulars,included at various points in the text,are derived from experiments performed in an undergraduateEElaboratory administered by theauthor.A somewhat expanded discussionof backgroundtheory, the measurement system, and measurement procedures can be found in R.F.Pierret, Semiconductor Measure- ments Laboratory Operations Manual,Supplement A,in the Modular Series on SolidState Devices,Addison- Wesley Publishing Co., Reading MA, © 1991.

--- Page 142 --- CARRIER ACTION 119 GPIB to Cable Computer TEK 11401 Menu 11A32 PS5004 FG5010 DM5010 touch □ □2 。 。 ⅢⅢ O Stroboscope Sample1 Sample 2 Sample3 Sample4 Figure 3.23 Photoconductive decay measurement system. by first passing the stroboscope lightthrough a small-diameter Si wafer. TheSi wafer acts as a filterallowingonlynear-bandgapradiationto reachthe sampleundertest.Although the11401Oscilloscope is capableof in situdataanalysis,a computer isconnectedtothe oscilloscopetoadd data manipulationand display flexibility. An example of the photoconductive decay as displayed on the computer screen is re- produced in Fig.3.24.The transientcomponentofvinmillivolts is plottedon the y-axis versus time inmilliseconds on the x-axis.The part (a)linear-scale plot nicely exhibitsthe generally expected dependence.The part (b) semilogplot verifies the simpleexponential nature of the decay (at least for times 0.1 msec ≤t≤0.6msec)and is more useful for

--- Page 143 --- 120 SEMICONDUCTORFUNDAMENTALS x1E-3 1E-1- 150 VOLTS 100- VOLTS t∞ 1/slope 1E-2- 50- 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x1E-3 x1E-3 SECONDS SECONDS (a) (b) Figure3.24Photoconductivitytransientresponse.Thetime-varyingcomponentofinmillivolts is plottedon the y-axis versus time in millisecondsonthe x-axis.(a)is a linearand (b)is a semilog plot. extractingtheminority carrierlifetime.Specifically,fromtheslopeofthestraight-line regionin thelatterplottheminoritycarrier lifetime intheprobedsampleis readilyd- duced to be p ≤ 150 μsec. 3.4 EQUATIONSOFSTATE In the first three sections of this chapter we examined and separately modeled the primary types of carrier action taking place inside semiconductors. Withinactual semiconductors all the various typesofcarrier actionoccuratthesametime,andthestateof any semicon- ductor can be determined only by taking intoaccount the combined effect of the individual typesofcarrieraction.Putingitalltogethersotospeakleadstothebasicsetofaring equationsemployed in solvingdevice problems,herein referredto astheequationsofstate. The first portionof thissection is devotedtodevelopingtheequationsofstate;thelatr portionincludes a summaryofcommonsimplifications,a listing of special-case solutions, and problem-solving examples.

--- Page 144 --- CARRIERACTION 121 3.4.1 Continuity Equations Each and every typeofcarrieraction—whether itbe drift,diffusion,indirector direct thermal recombination,indirector direct generation,or someother typeofcarrieraction— gives rise to a change in thecarrier concentrations withtime.Thecombinedeffectofall types ofcarrieractioncanthereforebetaken into accountbyequating theoverallchange in the carrier concentrations per unit time(∂n/∂tor ∂p/∂t)to the sumof the ∂n/∂t’sor ∂pl∂t’s due to the individual processes; that is, ue ∂n ∂n ∂n ∂n 1e 1e + 1e + 1e thermal + 1e other processes (3.44a) uup JJIp R-G (light, etc.) ∂p de ∂p ∂p ∂p$ ∂t 1e + 1e + 1e thermal + ∂t other processes (3.44b) uup JJ!P R-G (light, etc.) Theoveralleffectof theindividualprocesses is established,inessence, by invokingthe requirementof conservationof carriers.Electrons and holes cannot mysteriously appear and disappear at a given point, but must be transportedto or created at the given point via some typeofongoing carrieraction. There mustbe a spatialandtimecontinuity inthe carrier concentrations.For this reason,Eqs.(3.44)areknownas thecontinuity equations. The continuityequations can be written in a somewhat more compactform by noting ∂n 1e uup + ∂n 1e JJ!P

1 $a{J}_x+xeaJNay+aJNz$ ∂z

q 1 ∇.JN (3.45a) ∂p ∂p$1$aJ_x a_} aJ_z}$ —∇.J_ 1 1e + 1e 二 q e + ay + ∂z

q (3.45b) !p JJ!P Equations (3.45),which can beestablished by a straightforward mathematicalmanipula- tion,merelystatethattherewillbeachangeinthecarrierconcentrationswithnaien smallregionof the semiconductor if an imbalance existsbetween the total carrier currents into and out of the region. Utilizing Eqs.(3.45),we obtain ∂n 1 ∇.JN ∂n ∂n 1e q + 1e R-G thermal + 1e other processes (3.46a) ∂p ∇.J_$ 1 ∂p ∂p 1e q + 1e R-G thermal + 1e other processes (3.46b) The (3.46)continuity equations are completely general and directlyor indirectly con- stitute the starting point in most deviceanalyses.In computersimulations thecontinuity

--- Page 145 --- 122 SEMICONDUCTORFUNDAMENTALS equationareoteemloyeddirctlThearoriaterelationipsornerlG ∂p/tthermal R-Gwhich mayor may not be the special-case relationships given by Eqs.(3.34)],along withtheconcentrationchangesdueto”otherprocesses,are substituted into Eqs.(3.46),and numerical solutions are sought forn(x,y,z,t) and p(x,y,,t).In problems where a closed-form typeof solution is desired,the continuity equations are typicallyutilizedonlyinanindirectfashion.Theactualstartingpointinsuchanalysesisa simplified versionof the continuity equationsto be established in the next subsection. 3.4.2 MinorityCarrier Diffusion Equations The“workhorse”minoritycarrierdiffusionequationsarederivedfromthecontinuity equations by invoking the following set of simplifying assumptions: (1)Theparticularsystemunderanalysisisone-dimensional;ieall variablesareatmost a functionofjustonecoordinate (say the -coordinate. (2) Theanalysisislimitedorrestrictedtominoritycarirs. (3)≈0inthe semiconductororregionsofthesemiconductor subjecttoanalysis. (4）Theequilibriumminoritycarrierconcentrationsarenotafunctionofposition.Inother words,n0≠n0（x）,p0≠p0(x). (5) Low-level injection conditions prevail. (6)Indirect thermal recombination-generation is thedominant thermal R-Gmechanism. (7) Thereare no”other processes,“exceptpossiblyphotogeneration,takingplace within the system. Working on the continuity equation for electrons, we note that 1∂JN$1\nabla \cdot JN\to (3.47)qxeb$ if the system is one-dimensional. Moreover, ∂n =qDN ∂n JN=q{nn+DN (3.48) xe ax when≈0 andone is concernedonly withminoritycarriers[simplifications(2)and()]. By way ofexplanation,the drift componentcan be neglected in the current density expres- sion becauseis small by assumption and minority carrierconcentrations are also small, making the n product extremely small.(Note that the same argumentcannot be applied to majoritycarriers.)Since by assumption n0≠n0(x,andby definitionn=n0+∆n,we can also write

--- Page 146 --- CARRIER ACTION 123 ∂n $a{n}_o$ ∂Δn ∂△n + (3.49) xe ∂x ∂x$xeCombiningEqs.(3.47)through(3.49)yields1$a {2}Δn$\nabla \cdot JN\to D(3.50)$ax^{2}$ q Turning to the remaining terms in the continuity equation for electrons,theassumed dominance of recombination-generation via R-G centers,combined with the low-level injectionandminoritycarrierrestrictions,allows us to replace thethermal R-Gtermwith the Eq. (3.34) special-case expression. ∂n Δn (3.51) 1e thermal

Tn$ R-G In addition,applying simplification (7)yields ∂n 1e other

GL (3.52) processes where it is understood thatGL=0 if thesemiconductor is not subject to ilumination. Finallytheequilibriumelctronconcentationisneverfunctionoftime≠n(,and we can therefore write ∂n ∂no$\partial \Delta n\partial △n+(3.53)1e1e1e1eSubstitutingEqs.(3.50)through(3.53)intothe(3.46a)continuityequation,andsimulta-neouslyrecordingtheanalogousresultforholes,oneobtains$aΔn{}a^{2Δ{ Δ_n{p

DN +GL (3.54a) 1e $\partial x^2$ n Minority carrier a△p{n a^{2△_{ ΔP_{$

DP 7+ (3.54b) ∂t $a{x}^2$ Tp Subscripts are addedto thecarrierconcentrations in Eqs.(3.54)to remindtheuserthatthe equationsare validonlyforminoritycarriers,applyingtoelectronsinp-typematerialsand to holes in n-type materials.

--- Page 147 --- 124 SEMICONDUCTOR FUNDAMENTALS Table 3.1 Common Diffusion Equation Simplifications. Simplification Effect 0} Steady state $a{\Delta }_n$ →0 1e No concentration gradient or no a²2Δ{ $a_{}^{2△}$ →0 (D}D{N p →0 diffusion currentax^{2}∂x^{2}$Nodriftcurrentor=0Nofurthersimplification.(\approx 0isassumedinthederivation.)$\qr{0}Δ_n{_}$NothermalR-G\to 0\tau n$ No light 0←9 3.4.3 Simplifications and Solutions In the course of performing device analyses,the conditions of a problemoften permitad- ditionalsimplificationsthatdrasticallyreducethecomplexityoftheminoritycarrierdiffu- sion equations.Common simplifications and theireffecton theminority carrier diffusion equationsare summarized in Table 3.1.Extensivelyutilized solutions to simplified forms of theminoritycarrierdiffusionequations are collected forfuturereference in Table3.. 3.4.4 Problem Solving Although we havepresented allnecessary information,itmay notbe completelyobvious how one works with the diffusion equations toobtain thecarrierconcentration solutions. Device analyses of course provide examplesof”problem”solutions,and a numberof device analyses are presented later in thetext.Nevertheless,it is worthwhile to examine a few well-definedsimple problemsto illustrate problem-solvingprocedures. Thechosen sample problemsalsoprovide a basis forthetwo supplemental concepts introduced inthe next section. Sample Problem No.1 P:A uniformly donor-dopedsilicon wafermaintained atroom temperature is suddenly illuminatedwithlightattimet=0.AssumingN=101/cm3,τ=10-6sec,andalight- inducedcreationof107electronsandholespercm-sec throughoutthesemiconductor, determine ∆n()for t>0. S: Step1—Review precisely what information is givenor implied in the statementof the problem.

--- Page 148 --- CARRIER ACTION 125 Table 3.2 Common Special-Case Diffusion Equation Solutions. Solution no. 1 GIVEN:Steady state,no light. SIMPLIFIED d^{2{n{} Δ_n_{}$DIFF.EQN:O=DN$dx^{}$\tau n$ SOLUTION: ∆np（x）=Ae-x/LN+BeLN where LN=√Dn$andA,Baresolutionconstants.Solutionno.2GIVEN:Noconcentrationgradient,nolight.SIMPLIFIED$dΔ{n{}Δ_n{}$DIFF.EQN:二1pnSOLUTION:∆np(t)=∆n(0)e-t/nSolutionno.3GIVEN:Steadystate,noconcentrationgradient.SIMPLIFIED$Δ_n{}$

• GL DIFF.EQN:

• = 0 Tn$SOLUTION:$∆np=GLn$Solutionno.4GIVEN:Steadystate,noR-G,nolight.SIMPLIFIED$^{2}{}d^{2}{}$DIFF.EQN:O=DNor0=$d_x^{}d_x^{2}$SOLUTION:$∆n{(x)=A+B The semiconductor is silicon,T=300K,the donor doping is the same everywhere withND=10/cm,andGL=1017/cm-secatallpointsinsidethesemiconducorAlo the statement of the problem implies equilibrium conditions exist for t<0. Step 2—Characterize the system under equilibrium conditions. In Siatroomtemperature n₁=1010/cm3.Since ND>n，0=ND=101/cm³3and 0=n²/ND=10/cm3.Withthedopingbeinguniformtheequilibriumnandp0valus are the same everywhere throughout the semiconductor. Step 3—Analyze the problem qualitatively. Priortot=0,equilibriumconditionsprevailand ∆pn=0.Startingatt=0thelight creates addedelectrons andholesand p willbegin to increase. The growing excess

--- Page 149 --- 126 SEMICONDUCTOR FUNDAMENTALS carrier numbers,however,in turn lead to an increased indirect thermalrecombination rate which is proportional to Δp.Consequently,as Δpn grows as a result of photogeneration, more and moreof the excess holes are eliminated per second by recombination through R-Gcenters.Eventually,a point is reached where the carriers annihilated per secondby indirectthermal recombinationbalance the carriers created per second by the light,and a steady state condition is attained. Summarizing,we expect Δpn（t)to start from zero att=0,to build up at a decreasing rate,andto ultimatelybecome constant.Sincethelight generationand thermalrecom- binationrates must balance under steady state conditions,we can even state GL = ∆n(t→∞)/or△(t→∞)=∆|max=GLpiflow-levelinjectionprevails. Step 4—Perform a quantitative analysis. Theminoritycarrierdiffusionequation is the startingpoint formost first-orderquan- titative analyses.Afterexaminingtheproblemforobviousconditionsthat wouldinvalidate the useof thediffusionequation,theappropriateminority carrierdiffusionequation is written down,the equation is simplified,and a solution is sought subject to boundarycon- ditions stated or implied in the problem. Fortheproblemunderconsideration acursoryinspectionreveals thatall simplifying assumptionsinvolvedindivinthedifuioqutionsreadilisfiedpfill onlytheminoritycarrierconcentrationisofinteres;theequilibriumcarrierconcentrations are not a function of position,indirect thermal R-G is dominant in Si,andthere are no “otherprocesses”exceptforphotogeneration.Becausethephotogeneration is uniform throughout the semiconductor, the perturbed carrier concentrations are also position- independentand theelectricfieldmustclearlybe zero inthe perturbed system.Finally prevailing at all times. With noobstaclestoutilizingthediffusionequation,thedesiredquantitativesolution can now be obtained by solving a△p_{n a^{2p{ $ΔP_{+ GL

D{}$(3.55)1e$∂x^{2}$psubjecttotheboundarycondition∆n(t)|t=0=0(3.56)Since\Delta pnisnotfunctionofpositionthedifusionequationbecomesanordinardifr-entialequationandsimplifiesto$dΔp{n${\Delta }_n=G_L$ + (3.57) 1p Tp

--- Page 150 --- CARRIER ACTION 127 The general solution of Eq.(3.57) is ∆n(t)=Gp+Ae−-/ (3.58) Applying the boundary condition yields A=-G p$(3.59)and∆n(t)=GL\tau p(1-e-)solution(3.60)Step5—Examinethesolution.Failingtoexaminethemathematicalsolutiontoaproblemislikegrowingvegetablesandthenfailingtoeattheproduce.RelativetotheEq.(3.60)result,GL\tau hasthedimen-sionsofaconcentration(number/cm)andthesolutionisatleastdimensionallycorrect.AplotoftheEq.(3.60)resultisshowninFig.3.25.Notethat,inagreementwithqualitativeprediction,psrromroa0ndntullyutatGtwpEpilogue—Wewouldberemissifwedidnotpointouttheconnectionbetweenthehypo-theticalproblemjustcompletedandthephotoconductivedecaymeasurementdescribedinSubsection\mathrm{3.3.4.}LightoutputfromthestroboscopeusedinthemeasurementcanbemodeledtofirstorderbythepulsetrainpicturedinFig.3.26(a).TheEq.(3.60)solutionthereforeapproximatelydescribesthecarrierbuild-upduringalightpulse.Itshouldbenoted,however,thatthestroboscopelightpulseshaveadurationofon\approx 1\mu seccomparedtoaminoritycarrierlifetimeof\tau p\le 150\mu sec.Withon/\tau p\le 1,oneseesonlythevery$G_{{P$11110123456$1/_p$ Figure3.25 Solutionto SampleProblemNo.1.Photogeneration-inducedincrease in theexcess hole concentration as a function of time.

--- Page 151 --- 128 SEMICONDUCTOR FUNDAMENTALS GL(t) GLO →t 1 1 (a) 1 : 1 1 1 1 1 1 $∆pn() (b) Figure3.26(a)Approximatemodel for thelightoutput from the stroboscope in thephotoconduc- tivity decaymeasurement.(b)Sketchofthecombinedlight-on/light-offsolutionfortheexcess minority-carrier concentration. first portionoftheFig.3.25transientbefore the light is turnedoffandthesemiconductor is allowed to decay back to equilibrium. We arealso interested in verifying thecorrectnessof the light-off Δp(t)expression used inthephotoconductivitydecayanalysis.Derivationofthisexpressioncloselyparallels the light-on sample problem solution.Exceptions are Δpn at the beginning of the light-off transientequals ∆pn at theendofthe light-ontransientandGis setequalto zeroin Eqs.(3.55)and(3.57).Thesolutionto Eq.(3.57)withG=0is the same as SolutionNo.2 in Table 3.2; i.e., ∆n（t）=∆pn(0）e-t/p (3.61) wheret=0has been resetto correspond to the beginningofthe light-off transient.Equa- tion(3.37)in the measurementanalysis and Eq.(3.61)areof course identical.Forcom- pleteness,thecombinedlight-onand light-off Δpn(t)solution is sketched inFig.3.6(b). Sample Problem No.2 P:As pictured in Fig.3.27(a),thex=0endofauniformlydoped semi-infinitebarof siliconwithND=101/cm3isilluminatedsoastocreate∆0=1010/cm3excessholesat

--- Page 152 --- CARRIER ACTION 129 △p_n} Light absorbed at x=0 edge of bar Light $\Delta pn(xSibarx0x0123456$\r{}$ (a) (b) Figure3.27 (a）Pictorialdefinitionof Sample Problem No.2.(b)Solutionto SampleProblem No.2showingtheexcess holeconcentrationinside the Si bar as a functionof position. x=0.Thewavelengthoftheilluminationissuchthatnolightpenetratesintotheinterior (x>0)of the bar. Determine ∆pn(x). S:Thesemiconductorisagainsiliconuniformlydoped withan ND=10/cm3.Steady state conditions are inferredfromthestatementoftheproblem,since we areaskedfor ∆pn（x）andnot∆pn（x，t）.Morever,at=0,∆pn（0）=△pn0=100/cm,and∆pn→0 asx→∞.Thelatter boundaryconditionfollowsfrom thesemi-infinite natureofthe bar. The perturbation in Δpndue to the nonpenetrating light can’t possibly extend out tox=∞. Thenonpenetratingnatureof the lightallows ustosetG=0forx>0.Finally,notethat the problem statementfails to mention the temperatureofoperation. Whenthis happens,it is reasonable to assume an intended T=300K. Ifthe lightwereremoved,the siliconbar in SampleProblem2maintained at300K wouldrevertto anequilibriumconditionidenticalto that described in SampleProblem1. Underequilibriumconditions,then0=101/cm,0=10/cmandthecarrierconcen- trations are uniform throughout the semiconductor bar. Qualitatively it is a simple matter to predict the expected effectof the nonpenetrating lighton thesilicon bar. The lightfirstcreatesexcess carriers rightatx=0.Withmore carriers atone point thanelsewhere in the bar,the diffusion process nextcomes intoplay and the carrier excess spreads into the semiconductor proper. At the same time,however, the appearance of an excess hole concentration inside the barenhances the thermal recom- binationrate. Thus,as thediffusing holesmove into the bar their numbers are reduced by recombination.Inaddition,sincetheminoritycarrierholesliveforonlya limitedperiod, a timeτpontheaverage,fewerandfewerexcess holeswill survive as thedepthof penetra- tion into the bar becomes largerand larger.Understeady stateconditions it is therefore reasonabletoexpect anexcess distributionof holesnear=0,with △pn(x)monotonically decreasing from ∆pn0atχ=0to∆p=0asχ→∞.

--- Page 153 --- 130 SEMICONDUCTORFUNDAMENTALS In preparation forobtaining a quantitative solution, we observe that the system under consideration isone-dimensional,theanalysis is restricted totheminority carrierholes,the equilibrium carrierconcentrations are position-independent,indirect thermal R-G domi- nates,there are no”other processes”forx>0,andlow-level injectionconditions clearly raisedconcerning the useof the diffusionequationas the starting pointfor thequantitative analysis is whether&≈0.With the light on,a nonuniform distributionof holes and asso- ciated distributionof positive charge willappear nearthe x=0 surface.Theexcess hole pile-up,however,isverysmallmaxn)andtheassociatedelectricfieldistherefore expected to becorrespondingly small.Moreover,inproblemsofthis type it is foundthat themajoritycarriers,negativelycharged electronsinthegivenproblem,redistributein such a way as to parly cancel the minority carrier charge. Thus experience indicates the=0 assumptionto be reasonable,anduseof the minority carrier diffusion equationtobe justified. Under steadystate conditions withGL=0forx>0theholediffusionequation reduces to the form d^{}Δp{ $\Delta p_{}$ D{P}$=0forx>0(3.62)$dx^{}$whichistobesolvedsubjecttotheboundaryconditions∆n|x=0+=∆pn|=0=\Delta pn0(3.63)and$∆n|x→∞=0 (3.64) Equation(3.62)shouldberecognizedasoneofthesimplifieddiffusionequationscited in Table 3.2, with the general solution ∆pn（x)=Ae-/P+BexLP (3.65) where = √DP (3.66) Because exp(x/Lp)→∞ as x→∞,theonly way that the Eq.(3.64)boundary condition can be satisfied is forBto beidentically zero.WithB=0,applicationof the Eq.(3.63) boundary condition yields A=∆pn0$ (3.67)

--- Page 154 --- CARRIER ACTION 131 and ∆n(x)=∆pn0e-/p solution (3.68) The Eq.(3.68)result is plottedin Fig.3.27(b).Inagreement withqualitativeargu- ments,the nonpenetrating lightmerelygives rise toamonotonicallydecreasing Δ(x) starting from ∆pn0 atx=0 and decreasing to ∆pn=0asx→∞.Note that the precise functionalformofthefalloff intheexcesscarrierconcentrationisexponential with achar- acteristic decay length equal to Lp. 3.5 SUPPLEMENTALCONCEPTS 3.5.1 Diffusion Lengths The situation just encountered in SampleProblem No.2—thecreation (orappearance)of an excessof minoritycarriers along a given plane in a semiconductor,the subsequent dif- fusionof theexcessfromthe pointofinjection,and theexponentialfalloff intheexcess carrier concentration characterizedby a decaylength(Lp)—occurs often enough in semiconductor analyses that the characteristic length has been given a specialname. Specifically, L≡√DP associated with the minority carrier (3.69a) holes in an n-type material and associated with the minority carrier LN=√D$ (3.69b) electrons in a p-type material are referred to as minority carrier diffusion lengths. Physically,LpandLNrepresenttheaverage distanceminoritycarrierscandiffuseinto a seaofmajoritycarriers before beingannihilated.This interpretationisclearlyconsistent with Sample Problem No.2,where the average positionof the excess minority carriers inside the semiconductor bar is ∞ 8 <x）

xΔpn(x)dx/ ∆pn(x)dx=Lp (3.70) 0 0 Formemory purposes,minority carrier diffusion into a sea ofmajority carriers mightbe likened to asmall groupofanimalsattemptingtocrossa piranha-infestedstretchofthe Amazon River.In the analogy,Lp and LN correspondto theaverage distance theanimals advance into the river before being eaten.

--- Page 155 --- 132 SEMICONDUCTORFUNDAMENTALS Seeking an idea as to the sizeof diffusionlengths,letus assumeT=30oK,the semiconductorisND=10/cm3dopedSi,andp=10-6sec.Inthissample LP=√Dp=√(kT/q)pτ=[(0.0259)(458)(10-6)]1/2 =3.44 × 10-3 cm Althoughthecomputedvalueisfairlyrepresentativeitshouldbeunderstoodthatdiffusion lengthscanvaryoverseveralordersofmagnitudebecauseofwidevariationsinthecarrier lifetimes. 3.5.2Quasi-Fermi Levels Quasi-Fermi levelsareenergylevelsusedto specifythecarrierconcentrations insidea semiconductor under nonequilibrium conditions. To understand the need for introducing quasi-Fermi levels,let us first refer to Sample Problem No.1.Inthis problem,equilibriumconditionsprevailedpriortot=0,withn0= ND=101/cmandp0=10/cm.Theenergybanddiagramdescribingtheequilibrium situation is shown in Fig.3.28(a).A simple inspectionof the energy band diagram and the Fermi level positioning also conveys theequilibrium carrier concentrations,since n0=n;e(EF-E₁)/kT (3.71a) P0 =n;e(E₁-EF)/kT (3.71b) The point we wishto emphasize is that underequilibrium conditions there is a one-to-one correspondencebetweentheFermi levelandthecarrierconcentrations.Knowledgeof completely specifies n and po and vice versa. Let us turn next to the nonequilibrium (steadystate）situation inside theProblem1 semiconductorattimest≥.Fortimest≥τ，Δn=GL=101/cm,p=0+∆p 101/cm3,andn≈n0=10/cm3.Althoughnremainsessentiallyunperturbed,phas E{ E{ E} F{ E

E} P Ev $Ev$ (a) (b) Figure3.28 Sample use ofquasi-Fermi levels.Energy band descriptionof the situation insidethe semiconductorof Sample Problem No.1under (a)equilibriumconditionsand(b)nonequilibrium conditions (t≥τ).

--- Page 156 --- CARRIER ACTION 133 increased bymanyordersof magnitude;and it is clear thatthe Fig.3.28(a)diagram no longer describesthestateofthesystem.InfacttheFermilevelisdefinedonlyforasystem underequilibriumconditionsandcannotbeusedtodeducethecarrierconcentrations inside a system in a nonequilibrium state. The convenience ofbeing able to deduce the carrier concentrations by inspection from theenergybanddiagram isextendedtononequilibriumconditionsthroughtheuseofqus- Fermi levels.Thisisaccomplishedbyintroducingtwoenergies,N(thequasi-Fermilevel forelectrons)andFp(thequasi-Fermi levelforholes),whichareby definitionrelatedto the nonequilibrium carrierconcentrations in the same wayEFisrelatedto theequilibrium carrierconcentrations.To bespecific,undernonequilibriumconditionsand assumingthe semiconductor to be nondegenerate, n =n;e(FN−Ei)/kT or (3.72a) and n(n) p ≡ n;e(Ei-Fp)/kT or =E₁-kTln (3.72b) Notethat FNandFpareconceptualconstructs,withthevaluesofFNandFpbeing totally determinedfrom a priorknowledgeofnand p.Moreover,thequasi-Fermileveldefinitions have beencarefully chosenso that when a perturbed systemrelaxes backtowardequili- rium,N→EF,p→,and Eqs.(3.72)→Eqs.(3.71). To providea straightforwardapplicationof thequasi-Fermilevelformalism,letus again consider thet>τstateof the semiconductor in Sample Problem No.1.First,since n=n0,N≈F.Next,substitutionofp=10/cm³（n=100/cm3andkT=0.259eV） into Eq.(3.72b)yields p=E-0.06eV.Byeliminating EFfrom theenergy band dia- gramanddrawing linesattheappropriateenergiestorepresent FandFp,oneobtainsthe diagramdisplayed inFig.3.28(b).Figure3.28(b)clearlyconveys toanyobserverthatthe system underanalysis is in anonequilibrium state.Whenreferencedtopart(a)part(b)of Fig.3.28furtherindicatesata glancethatlow-levelinjectionistakingplaceinsidethe semiconductor,creating aconcentrationofminoritycarrier holes inexcess of n.A second example use of the quasi-Fermi level formalism,a use based on Sample Problem No.2 and involving positiondependentquasi-Fermi levels,is presented in Exercise3.5at theendof the section. As a finalpoint,it shouldbementioned thatthequasi-Fermi levelformalism canbe used to recast someof the carrier-actionrelationships in a more compactform.Forex- ample,the standard formof the equation for the total hole current reads Jp = qp-p∇p (3.73) (same as 3.18a)

--- Page 157 --- 134 SEMICONDUCTORFUNDAMENTALS DifferentiatingbothsidesofEq.3.72b)withrespecttoposition,oneobtains n_} ∇p

e(Ei-Fp)/kT(∇E-∇) (3.74a) kT $qpp$∇$ 二 kT 8

kT (3.74b) Theidentity =∇E/q (thethree-dimensionalversionof Eq.3.15)isemployedin progressing fromEq.(3.74a)to Eq.(3.74b).Next,eliminating ∇pin Eq.(3.73)using Eq.(3.74b) gives qDp$(cid:P$d:P Jp = q(_p{8+$ (3.75) kT kT FromtheEinsteinrelationship,however,p/kT=.Wethereforeconclude p =p∇P (3.76a) Similarly, JN =μn∇ (3.76b) Since Jp∇Fp and Jα∇in Eqs.(3.76),one is ledtoa very interesting general interpretationof energy band diagrams containing quasi-Fermilevels.Namely, a quasi- Fermilevelthat varies withposition(dFpldx≠0ordF/dx≠0)indicates at a glancethat current is flowing inside the semiconductor. Exercise3.5 P:In Sample Problem No. 2,nonpenetrating illuminationof a semiconductor bar was foundto cause a steady state,excess-hole concentrationof ∆p(x)= ∆p0exp(-x/Lp).Giventheprevailing low-level injectionconditions,and noting thatp=p0 +∆p,we can therefore state n ≈n p=0+∆pn0e-/p for the illuminated sample. (a) Making useof Eqs.(3.72),establishrelationships for FN andFpin theilumi- nated bar.

--- Page 158 --- CARRIERACTION 135 (c)Usingthe resultsof parts (a)and (b),sketchtheenergy banddiagrams describing the semiconductor barof SampleProblem No.2 underequilibrium and illuminated steady state conditions.(Assume =0in the illuminated bar.) (d)Is there a holecurrentin theilluminatedbar under steady state conditions? Explain. (e) Is thereanelectroncurrent inthe iluminated bar understeady stateconditions? Explain. S:(a)Sincenn0,itfollowsfromEq.(3.72a）thatFNEF.Likewise,substituting the preceding p-expression into Eq.(3.72b), we conclude F=E₁-kTln(p/n₁）=E₁-kTln[p/n+（∆pn0/n₁）e-x/LP] (b)If∆pn（x)≥p0，then（∆pn0/)exp（-/Lp)≥P0/n;and ≈E₁-kTln[(∆n0/n₁）e-xLP] or P=E₁-kTln(∆n0/n₁）+kT(x/Lp) (c) WeknowfromSampleProblemNo.2that△n0=1010/cm,n=1010/cm3,and P{=n²ND=10/cm3.Th (i) Near=0,∆pn（x）≥p0andFpisalinearfunctionof. (ii)At=0,∆pn0=and wededucefromthepart(b)resultthat =E. (iii)For large x,Fpeventually approaches =F. (iv)N-E≈EF-E=kTln(ND/n）=0.30eV. Utilizing the preceding information,one concludes EcEcFNF_}$E●●$Ev}$ E →x x 0 0 Equilibrium Steady state illuminated (d) Assuming p≠0,it follows from Eq.(3.76a)that there willbe a hole current wheneverdFpldx≠0.Thereisobviouslyaholecurrent in theilluminated barnar x= 0.

--- Page 159 --- 136 SEMICONDUCTORFUNDAMENTALS (e) Appearances can sometimes be deceiving;JN≠0 nearx=0!One might conclude from the part (c)resultthatdF/dx=0andthereforeJN=0.Under steadystate points in the bar.Since no current leaves the bar atx=0,Jmoreover must vanish at x=0.Thus JN(x）=-Jp(x)and we know Jp≠0near=0.The apparent discrep- ancy here stems from the fact that JNis proportionalto both n and ∇FN.Becausethe majoritycarrier electronconcentration ismuchlarger thantheminoritycarrierhole concentration,dFN/dxmustbecorrespondingly smallerthandFp/dx.Theslopein FN simply cannot be detected by inspecting the energy band diagram. 3.6 SUMMARYANDCONCLUDINGCOMMENTS Mostofthechapter was devoted to examining the three primary types of carrieraction occurring inside ofsemiconductors: drift, diffusion,andrecombination-generation.In each casethe carrieraction was first defined and visualized.Drift is charged particle motion concentration to regions of low particle concentration due to the random thermalmotion of the carriers.Recombination and generation are respectively the annihilation and creation of carriers.Nextthequantitativeeffectof each typeofcarrieraction was analyzed.Drift anddiffusiongiveriseto particlecurrents(Eqs.3.4and3.17-3.19);recombination- generationacts tochangethecarrierconcentrations asa functionof time(Eqs.3.34).As- sociated withthequantitative analysis ofeachcarrieractionthere arises a”constantofthe motion,“animportantmaterial-basedparameterthatspecifiesthemagnitudeofthecarrier action in a givensemiconductor sample. Thecarriermobilities,the diffusion coefficients, and theminoritycarrier lifetimes are thematerial parameters associated with drift,diffu- sion,and recombination-generation,respectively.In the major semiconductors the carrier mobilities are very well characterized as a function of doping and temperature.Selectmo- bility data are graphed in Figs.(3.5)and(3.7);anempirical fitrelationship forthecarrier mobilities inSi is presentedin Exercise3.1.Thediffusioncoefficients innondegenerate semiconductors are computedfrom the carriermobilities using theEinsteinrelationship (Eqs.3.25).Conversely,an experimental measurement is normally performed to determine the carrier lifetimes in a given semiconductor sample. Althoughintroducedandexaminedindividually,the various typesof carrieraction occur simultaneously inside semiconductors.Mathematicallycombining theoverall effect of the variouscarrieractivities leadstothecontinuityequations(Eqs.3.46).Thecontinuity equations inturnsimplifytotheminoritycarrierdiffusionequations(Eqs.3.54)under conditions encountered in many practical problems.Additional simplifications and their effectontheminoritycarrierdiffusionequationsarenotedinTable3.;extensivelyutilized solutionstosimplifiedformsoftheequationsarecollectedin Table3.2.Thecontinuityor minoritycarrier diffusionequationscombined withtheotherrelationshipsestablishedin the chapterallowone to model the stateof a semiconductor subjectto an external pertur- bation.The more important workingrelationshipsareorganized andrepeated in Table3.3.

--- Page 160 --- CARRIERACTION 137 Table3.3 Carrier Action Equation Summary. Equations of State aΔn{_} a^{}Δ{n{} Δn_} ∂n 1 ∇·JN ue ue 1+ 1e q + 1e thermal + 1e other 1e

DN $a{x}^2$ τn$R-Gprocesses\partial p1e二1q.JP+\partial p\partial tR-Gthermal+\partial p1eprocessesother$a△p_{$ 1e

D{}a^{△_{nax^{}Δp{}$ Tp +GL Current and R-G Relationships JN=JNldrift+JNldf=9μnn+qDN∇n ∂n Δn

drift diffusion 1e i-thermal τn$R-GJp=JPldrift+Jldf=q\mu pp8-q\nabla p\partial p$Δp$1ei-thermalJ=JN+JPR-GKeyParametricRelationshipsDN$ kT 1 $LN\equiv \surd DN$ τnN{}μ_{}$qDP$ kT 1 Tp 二 c{T $\mu$ q Resistivity and Electrostatic Relationships 1 ρ … n-type semiconductor 1 q_{nD ρ

q(μ{nn +pp) 1 ρ= . .. p-type semiconductor q_p} $1Ec$ $1dEy$ 1dE} 1 8= V=— (Ec− e) q dx q dx xp b q Quasi-Fermi Level Relationships JN =μ{ n∇F{$ J= ppP

--- Page 161 --- 138 SEMICONDUCTORFUNDAMENTALS Anumberof topics related to carrier action were also addressed in the chapter,includ- ing resistivityandresistivitymeasurements,the hot-pointprobemeasurement,constany of theequilibrium Fermi level,nonuniform dopingandtheassociatedbuilt-inelectricfield, E-k diagrams,measurementof theminority carrier lifetimes,and diffusion lengths.More- over, although notidentified as such,theenergy band diagram was subjecttofurther de- velopment.Specifically,it was pointedoutthat theexistenceof an electric fieldinside the semiconductor causes band bending or a variationof the energy bands with position.By simplyinspectinganenergybanddiagramitispossibletoascertainthegeneralfunctional dependenceoftheelectrostaticpotentialandelectricfieldpresent in thematerial.Inthe discussionof recombination-generation,another level was added nearmidgap. This level, arising from R-G centers,plays a dominantrole in thethermalcommunication between the energy bands.Lastly,quasi-Fermi levels were introduced to describe nonequilibrium conditions. PROBLEMS CHAPTER 3 PROBLEMINFORMATIONTABLE Complete Dificulty Suggested Short Problem After Level Point Weighting Description 3.1 3.3.1 1 16 (2 each part) Energy band visualization 3.1.4 (a-d) 3.2 1-2 16 (2 each part) Short answer 3.3.4 (e-h) 3.3 3.1.3 3 25 μvs.T plots 3.4 ” 3 18(b-15, c-3) μn vs. T student data 3.5 3.1.4 a-1,b-3 12(a-6, b-6) Intrinsic/maximum ρ 3.6 ” 2 15 (3 each part) Resistivity questions 3.7 ” 2 15 ρVS.NA,ND plot 3.8 ” 3-4 20 (10 each part) ND(x)variation in resistor 3.9 ” 3 18(a-12,b-6) ρ vs.T plots 3.10 ” 4 20 (5 each part) Temperature sensor 3.11 3.1.5 2 5 Compute thermal speed 3.1.5 (a-d) 16/diagram (a::c-2, 3.12 2, e-3 Interpret E-band diagrams 3.2.4 (e,f) d-4, e-2, f-4) 3.13 3.2.4 3 15(a-10,b-3, c-2) Built-in electric field 3.14 ” 2 15(a-3,b-10, c-2) Dvs.NA,ND computation 3.15 3.3.3 3 10 Eq.(3.35)→Eqs.(3.34) 3.16 3.4.2 1 6 (2 each part) Simple Diff.Eq. questions

--- Page 162 --- CARRIER ACTION 139 3.17 3.4.4 一 6 Diff. Eq., no R-Gregion 3.18 ” 2 12(a::c-2,d-6) Diff. Eq., mysterious ray 3.19

2 8 Diff.Eq.,G→G/2 3.20 ” 2-3 10 Diff. Eq., light + edge R 3.21 ” 3 15 (a::d-2, e-7) Dff . Eq., double-ended 3.22 ” 3-4 15(a::c-2,d-9) Diff.Eq., half-lluminated 3.23 ” 2 8(a-2,b-4, c-2) CdS Photoconductor 3.24 3.5.2 2 12 (a-2,b-4, c::e-2) Quasi-Fermi levels 3.25 ” 3-4 15 Quasi-Fermi levels 3.26 ” 3-4 15(a-13,b-2) Plot quasi-Fermi levels 3.1Using the energy band diagram,indicate howone visualizes (a) The existence of an electric field inside a semiconductor. (b)An electron with a K.E.= 0. (c) A hole with a K.E.= EG/4. (d) Photogeneration. (e) Direct thermal generation. (f) Band-to-band recombination. (g) Recombination via R-Gcenters. (h) Generation via R-G centers. 3.2 Short Answer 1-cm-long semiconductor bar.What is the hole mobility insidethe bar? (b)Namethetwo dominantcarrier scatteringmechanisms innondegenerately doped semiconductors of device quality. (c)Foragivensemiconductorthecarriermobilitiesinintrinsicmaterialarechooseo: higherthan,lowerthan,thesameas)thoseinheavilydopedmaterial.Brieflyexplain whythemobilities inintrinsicmaterialare(chosenanswer）thoseinheavilydoped material. (d) Two GaAs wafers,one n-type and one p-type,are uniformly doped such that ND(wafer1)=NA(wafer2)>n.Which wafer will exhibitthe largerresistivity? Explain.

--- Page 163 --- 140 SEMICONDUCTORFUNDAMENTALS (e)Theelectronmobilityinasiliconsampleisdeterminedtobe1300cm/V-secatroom temperature.What is the electrondiffusion coefficient? (f)What isthealgebraicstatementof low-level injection? (g)Lightis usedtocreateexcess carriers insilicon.Theseexcess carriers willpre- dominantlyrecombine via(choose one: band-to-band,R-G center,or photo)re- combination. (h）Priortoprocessing,asiliconsamplecontainsN=1014/cmdonorsandN=101/ cm³R-G centers.Afterprocessing (say in the fabricationof a device,thesample containsND=1017/cm3donorsandN=1010/cm3R-Gcenters.Didprocessingin- crease or decrease theminoritycarrier lifetime?Explain. 3.3Complete part (b)of Exercise 3.1. ●3.4(a)(Optional）ReadtheExperiment No.7IntroductionandMeasurementSystem description in R. F. Pierret,Semiconductor MeasurementLaboratory Operations Manual,Supplement A,in the Modular Serieson Solid State Devices,Addison-Wesley Publishing Co., Reading, MA, © 1991. (b)The measurementdescribed in thepart (a)reference is used to determinethe mobility in an ND<10/cm3Sisample asafunctionoftemperaturefromroughlyroomtem- peraturetoT=150K.Representative versusT(K)data derived by undergraduate students is tabulated in thefollowing table.Assuming μnT-b[orln(n)=con- stant -bln(T)] can be fited to the data,use theMATLAB polyfit function oran equivalentleastsquaresfittodeterminethebestfitvalueofthepowerfactorb.Draw the fit line and note the b-fit valueon a log-log plotof the data. (c)Compareyourpart(b)resultwithFig.3.7(a).Isitsignificanthatheexperimentalata was derivedfrom a lowlydoped sample?Istheexperimentallyderived b-valueconsis- tent withthe b-value noted on the text plot?Are the measured mobility valuesofthe proper magnitude? Given N data points,（x,y)withi=1to N,thestraight liney=a+bx,whichis the“bestfitto thedata, will have a y-axis intercept and slope respectively equal to ∑yΣx²-Σ∑x₁y NΣx₁y-∑x}y and b=$D$N∑x{2-(Σx{^ $N∑x{2-(Σx{^ All summations are fromi=1toi=N.The“bestfit”criterionleading to the expressions fora and bis that the square of the difference between the data points and the fited line is minimized; hence the name “least squares fit.”

--- Page 164 --- CARRIERACTION 141 T(K) $μ{n(cm^{2}/V-sec) 290 1501 280 1646 270 1805 260 1985 250 2185 240 2415 230 2675 220 2978 210 3306 200 3743 190 4209 180 4619 170 5216 160 5910 150 6757 3.5Intrinsic and Maximum Resistivity (a) Determine the resistivityof intrinsic Ge,Si,andGaAs at300K. (b)Determine the maximum possible resistivityofGe,Si,andGaAs at 300 K. 3.6 More Resistivity Questions (a)Asilionamplemntanedtroomtemrtureisuormlydopedwith0/ cm3donors.Calculate the resistivityof the sample using Eq.(3.8a).Compare your calculated result with the ρ deduced from Fig.3.8(a). (b) The siliconsampleof part（a)iscompensated”byadding NA=1016/cmacceptors. Calculatetheresistivityofthecompensated sample.(Exercisecautioninchoosingthe mobility values to be employed in this part of the problem.) (c) Computetheresistivityofintrinsic（NA=0,ND=0)siliconatroomtemperature. How does your result here compare with that for part (b)?

--- Page 165 --- 142 SEMICONDUCTORFUNDAMENTALS (d)A500-ohmresistor is to be made from a bar-shaped pieceof n-typeSi.The bar has a cross-sectional areaof10-²cm² and acurrent-carrying lengthof1cm.Determinethe doping required. 100C.N>nat bothroomtemperatureand100C.Istheresistivityofthe sample expected to increase or decrease? Explain. 3.7Making useof themobilityfitrelationships and parameters quoted in Exercise3.1, constructaplotofthesiliconresistivityversusimpurityconcentrationatT=3K.In- cludecurvesforbothn-andp-typesiliconovertherange101/cm≤NAorND≤10²0/ cm3.Compare your result with Fig.3.8(a). 3.8Resistors inICs are sometimes thin semiconductor layers near thesurface of a wafer. However,formationof the layer by diffusionor ion implantation (discussed inChapter4) is likely togiverisetoadopingconcentrationthatvaries with depthintothe layer.Letus examine how the resistance is computed when the doping varies with depth. (a)Given a bar-shaped layerof width W, length L,anddepthd,and assuming an arbitrary ND(x)variation withthedepthxfromthe wafer surface,showthattheresistanceofthe layer is to be computed from L 1 R

W pJ q $μn（x)ND(x)dx Jo (b）Taking ND(x)=ND0exp(-ax）+NDB，computeandplotRversus ND0for1014/cm³ ≤ND0≤1018/cm3whenL=W,NDB=1014/cm,d=5μm,and1/a=1μm. ●3.9(a)Modify yourμ versus Tprogram fromProblem3.3tocompute and construct semi- log plots ofρ(log scale)versus Tfor 200K≤T≤500Kand NDor NAstepped in decadevaluesfrom104/cmto 101/cm.[NOTE:Thelowerdoping curvesarein error at the higher temperatures if Eqs.(3.8)are used to compute ρ.Can you explain why? Assuming Eqs.(3.8) were used in the computation, how must theρ versus T program be modified to correct the error?] (b)Compareyourn-typeplotwithFig.7in LiandThurber,Solid-StateElectronics,20, 609(1977).Compare your p-type plot with Fig.7in Li,Solid-State Electronics,21, 1109 (1978).Comment on the comparisons. 3.10Yourboss asks youtoconstructatemperature sensor formeasuring theambientout- side temperature in W.LafayetteIN（-30C≤T≤40°C).You decide to base operation of the temperaturesensoron thechange inresistance associated withabar-shaped piece of Si. (a)Restricting yourself to nondegenerate dopings andreasonable device dimensions,and alsorequiring the resistance to bereadilymeasured witha hand-heldmultimeter(say 1Ω≤R≤1000 Ω),indicate the doping and dimensions of your sensor.

--- Page 166 --- CARRIER ACTION 143 (b)Derivean expressionforthe sensitivity(dR/dTinΩ/C)of your sensorover theoper- atingtemperaturerange.Relativetosensitivityisitpreferabletouseloworhighdop- ings? Explain. (c) What are the upperand lower temperature limitsofoperationofyour sensor(approxi- mately)? Explain. (d) Plot R versus Tof your sensor over the required temperature range ofoperation. 3.11Thermalenergyalone was notedto contributetorelatively highcarrier velocities inside of a semiconductor.Considering a nondegenerate semiconductor maintained at room ing to the peak of the electron distribution in theconduction band.Set m*=min perform- ing your calculation.(This problem assumes theprior workingof either Problem2.7 or Problem 2.8.) 3.12InterpretationofEnergy Band Diagrams Sixdifferentsiliconsamplesmaintainedat300Karecharacterizedbytheenergyband diagramsinFig.P3.12.Answerthequestionsthatfollowafterchoosingaspecificdiagram foranalysis.Possibly repeat using other energy band diagrams.(Excessive repetitions have been known to lead to the onset of insanity.) (a) Doequilibriumconditionsprevail?Howdo youknow? (b) Sketch theelectrostatic potential(V)insidethe semiconductor as a functionof (c) Sketch theelectric field()insidethe semiconductorasafunctionof. (d) The carrier pictured on the diagram moves back and forthbetweenx=0andx= L withoutchanging itstotalenergy.SketchtheK.E.andPE.ofthecarrierasafunction of position inside thesemiconductor.Let EF be the energyreference level. (e) Roughly sketch n and p versus x. (f) Onthe same setof coordinates,make a roughsketchoftheelectron drift-current den- sity(JNldrif）andtheelectrondiffusion-currentdensityJNldif）insidetheSisampleas a functionofposition.Be suretographtheproperpolarityofthecurrent densitiesat allpointsandclearlyidentifyyourtwocurrentcomponents.Alsobrieflyexplainhow you arrived at your sketch. 3.13Thenonuniformdoping inthecentralregionofbipolar junctiontransistors (BJTs) creates abuilt-infieldthatassistsminoritycarriersacross theregionandincreases the maximum operating speedof thedevice.Suppose the BJT is a Si device maintained under equilibriumconditions atroomtemperature withacentralregionof length L.Moreover, the nonuniform acceptor doping is such that p(x)≡NA(x)=n; e(a-x)bb 7≥x≥0 where a=1.8 m,b=0.1 m, and L=0.8m. (a) Drawtheenergy banddiagramforthe0≤x≤Lregion specifically showing ,, E,and Eon your diagram.Explain how you arrived at your diagram.

--- Page 167 --- 144 SEMICONDUCTORFUNDAMENTALS $Ec$ Electron EG/4 $Ec$EF EF E_v}E{ 1 1 1 1Ev}$Hole111\to x+\to x0L/2L0L/4L/23L/4L(a)(b)$Ec$Ei$Ec EF EF Ev} Ei Ev} →x →x 0 L/2 L 0 L/2 L (c) (d) $Ec$ 1 1 1 1 Ec 1 $Ei$ 1 1 1 Ei $Ev$ 1 Ev$ 1 1 1 1 1 1 1 ！ + →x →x 0 L/2 L 0 L/4 L/2 3L/4 L (e) (f) Figure P3.12

--- Page 168 --- CARRIERACTION 145 (b)Make a sketchofthe-fieldinsidetheregionas afunctionofposition,andcompute the value of atx= L/2. (c)Isthebuilt-inelectricfieldsuchastoaidthemotionofminoritycarrierelecronsin going fromx=0 tox= L?Explain. 3.14(a)Basedoninformationfoundinthetext proper,roughly sketchtheexpectedvaria- tionofDNandDpversusdopingappropriatefor1014/cm≤NAorND≤108/cm³ doped SimaintainedatT=300K.Explain howyouarrivedattheformofyoursketch. （b)MakinguseofthefitrelationshipsandparametersquotedinExercise3.1construta plotof DNand DpversusNAorNDfor1014/cm3≤NAorND≤108/cm³doedSi maintained at T= 300K. (c)Why wasthe upperdoping limittakentobe10/cmin thepart(b)computation? -do sn pu=uu=d=uos=u rabletop,showthatthe general-case R-Grelationshipof Eq.(3.35)reduces tothe special-case relationshipsof Eqs.(3.34)whenone haslow-level injection in asecific material type. 3.16∂∆n/t=DN²Δn/∂x²-△n/+GLisknownastheminoritycarrierdiffusion equation for electrons. (a)Why is it called a diffusion equation? (b)Why is it referred to as a minority carrier equation? (c) Theequation is validonly underlow-level injectionconditions.Why? 3.17 Show that,under steady state conditions ∆pn（x)=∆pn0(1-x/L） …0≤x≤L is the special-case solutionoftheminoritycarrierdiffusionequationthatwillresultif (1)one assumesall recombination-generation processes are negligible within an n-type semiconductoroflength L,and(2)oneemploystheboundaryconditions ∆p(0)=Δ0 and ∆pn(L)=0.(Neglecting recombination-generation is anexcellentapproximation when Lis much less thanaminority carrier diffusionlength.A Δp(x)solutionof theabove type is frequently encountered in practical problems.) 3.18Theearth ishitby amysteriousraythatmomentarily wipesoutallminoritycarriers. Majoritycarriers are unaffected.Initially inequilibriumand notaffectedbyroom light,a uniformly dopedsilicon wafer sitting on your desk is struck by the ray at timet=0.The wafer dopingis NA=1016/cm3,τ=10-6sec,andT=300K. (b) Does generation orrecombination dominate att=0+? Explain. (c) Dolow-levelinjectionconditionsexistinsidethe waferatt=0?Explain. (d) Startingfromtheappropriatediferentialequation,derive∆np()for>0.

--- Page 169 --- |46 SEMICONDUCTORFUNDAMENTALS 3.19Asilicon wafer（NA=104/cm,τn=1μsec，T=roomtemperature)isfirst illumi- natedforatimet>τnwithlight whichgenerates GL0=1016electron-holepairspercm- sec uniformlythroughoutthevolumeofthesilicon.Attimet=0the lightintensityis reduced,making GL=GL0/2fort≥0.Determine ∆np()fort≥0. 3.20Asemi-infinite p-typebar(seeFig.P3.20)is iluminated with light whichgeneraes GLelectron-hole pairs percm-sec uniformly throughoutthe volumeof the semiconductor. Simultaneously,carriers are extracted atx=0making Δnp=0atx=0.Assuming a steady stateconditionhasbeenestablishedand∆p(x)≤p0forall,determine ∆np(x). Light Δ_n{ 0 p-type x 0 Figure P3.20 3.21Thetwoends of a uniformly dopedn-type Sibaroflength L aresimultaneously illuminated so as tocreateγNDexcess holes at both=0andx=L(see Fig.P3.2).he wavelengthandintensityoftheilluminationaresuchthatno lightpenetrates into theinte- rior（0<x<L)ofthebarand=10-3.Also,steadystateconditionsprevail,T=300K, $and ND \ n{. △pn（O）=△pn(L）=γND M Sin-type W (7>x>0[ou) — →x 0 L Figure P3.21 (a) Basedon qualitative reasoning,sketchthe expected generalformof the ∆n(x) solution. (b)Dolow-levelinjectionconditionsprevailinsidetheilluminatedbar?Explain. (c) Writedownthediferentialequation(simplestformpossible)youmustsolvetodeter- mine ∆pn(x) inside the bar. (d) Writedownthegeneralformofthe△pn()solutionandtheboundarycondition(s) appropriate for this particular problem.

--- Page 170 --- CARRIERACTION 147 (e) Establishanexpressionfortheholecurrent(p)flowingintheilluminatedbarat=. [Youranswermaybeleftintermsofthearbitraryconstant(s)appearinginthegeneral form of the ∆pn(x)solution.] 3.22Aspictured in Fig.P3.22,thex>0portionof an infinite semiconductor is illumi- nated withlight.ThelightgeneratesGL=105electron-holepairs/cm-secuniformly throughoutthex>0regionofthe bar.GL=0forx<0,steadystate conditionsprevail, thesemiconductorissilicon,theentirebarisuniformlydoped withND=10/cm3,p= 10-6sec,and T=300 K. + →x 0 Figure P3.22 (a)What is the hole concentration atx=-∞? Explain. (b) What is the hole concentration atx=+∞? Explain. (c) Dolow-level injection conditions prevail?Explain. (d) Determine ∆pn（x)for all x. NOTE:(1）Separate ∆pn(x)expressions apply for>0and<0. (2) Both ∆pand dΔpn/dx must be continousat=0. 3.23CdSis themost widelyused materialforconstructing commercial photoconductors operating in the visible portionof the spectrum.The CdS photoconductor has a high sen- sitivityand its spectral response closelymatchesthatofthe human eye.AmodelVT33 CdS Photoconductor is pictured in Fig. P3.23. Figure P3.23 (a)Speculate why the conducting film has a snake-like pattern. (b) The VT333resistor pattern is approximately0.3mm wideand3cmlong.Estimating the deposited CdSfilmto be5μm thick，ND=1013/cm³≥n,andμn=100cm²/ V-sec,compute the dark resistance of the device. (c) TheVT333exhibiteda250Ωresistance whenilluminated with amicroscope light. Can the usualrelationships beusedto determinethe GLrequired to produce thecited resistance? Explain.

--- Page 171 --- 148 SEMICONDUCTORFUNDAMENTALS 3.24Theequilibriumandsteadystateconditionsbeforeandafteriluminationofa semi- conductorarecharacterized by theenergy band diagrams shown in Fig.P3.24.T=300K, n=1010/cm³3,=1345cm²/V-sec，andp =458cm²/V-sec.From the information provided, determine (a) n0 and p0,the equilibrium carrier concentrations. (b)n and p under steady state conditions. (c ) ND. (d)Do we have“low-level injection”when the semiconductor is illuminated?Explain. (e)Whatistheresistivityofthe semiconductorbeforeand after illumination? -Ec $Ec$FN$ 0.3eV 0.318eV

E{ $Ei$ 0.3 eV F_{ E E_ (a) Before (b) After Figure P3.24 3.25Aportion（0≤x≤L)ofaSi sample,uniformlydoped with ND=10/cm³donors and maintained at room temperature,is subject to a steady state perturbation such that $n=NDD$ 7≥x≥0 p=n₁（1-xL）+n²/ND Since n≈ND,it is reasonable to assume≈0in the0≤x≤Lregion.Given≈0, sketchtheenergybanddiagramfortheperturbedregionspecificallyincludingEE,E, FN,and Fpon your diagram. 3.26（a)WriteaMATLAB(computer)programthatautomaticallyplots FNandFpversus forthe experimental situationfirstconsidered inSampleProblemNo.2 andfurther analyzed in Exercise 3.5.Assume a Si sample maintained at 300K with τ=10-6 p sec.In writing the program,take ND and Δpn0 to be input variables. Scale the x-axis inLpunitswith（x/Lp)max=ln[100△pnp0].Scalethey-axis inkTunitsplotting (FN-E)/kTand（-E)/kT.Setymin=-5andymax=45.Identifythepositions of Eand E on your plot. (b) Runyourprogram with ND=101/cm3and Δpn0=10/cm3.Compare theresultant plot with the corressponding diagram sketched in Exercise 3.5. (c) Rememberingtheassumedlow-levelinjectionrequires Δpn0ND,tryrunningyour program using different values of ND and Δp0

--- Page 172 --- 4 Basics of Device Read Only Fabrication This chapter provides a very briefoverviewofsilicondevice fabrication. Thefabrication overview constitutesthefinalpreparatory stepbeforetheconsiderationandanalysisof specificdevices.Thegoalis to developa generalfeelforthephysical natureofdevice structures.Moreover,the way a device is made affects key device parameters,which in turn affect thesimplifications,assumptions,and soforthpossibleinthe deviceanalysis.Even in purelytheoreticalendeavors,some familiaritywithfabrication processesand procedures is obviously necessary. In what follows we first examine common“building block”processes.The proesses are used in combination and areroutinely repeated several times in producing a modern device or IC. Subsequently we describe the sequencing of processes to produce a pn junc- tion diode,thefirstdevicetobeanalyzedinPartIIofthetext.TobetterillustrateIC fabrication,thechapterconcludes witha subsectiononthemakingofaCPU(central processing unit).Foradditional information thereader isreferred tooneof theexcellent device-fabricationbooks cited in the Part Ireading list. Finally,itshouldbenotedthatthisisthefirst“readonly”chapter.Asexplainedinthe GeneralIntroduction,chapterswiththeread-onlydesignationareprimarily intendedto conveyqualitativeinformationofgeneralinterest.Theycontainasmallnumberofequa- tions, no exercises,and few,if any,end-of-chapter problems.Also,there is no reference to thecontentsofread-onlychapters inthetest-like ReviewProblemSets thatare includedat the endofthethree textsegments.Ifthematerial inthischapter werebeing transmitted verbally,you wouldbeadvisedto put down your note-taking pencils and to justsitback and listen. 4.1 FABRICATIONPROCESSES 4.1.1Oxidation Theexistenceofareadilyproducedhigh-qualityoxidehelpedestablishthedominanceof Siintheproductionofcommercialdevices.SiOfunctionsasbothaninsulator in anumber of device structures and as a barrier to diffusion (seeSubsection 4.1.2)during device fab- rication.Siliconishighlyreactiveandathinnativeoxidelayerisrapidlyproducedby simplyexposing virginSi to theatmosphere atroom temperature.SiO₂layersofcontrol- lablethickness are produced during devicefabricationbyreacting Si witheitheroxygen gas or water vaporat anelevatedtemperature.Ineithercasetheoxidizing species diffuses 149

--- Page 173 --- 150 SEMICONDUCTORFUNDAMENTALS A Figure4.1Production-line(IntelFab-9fabricationfacility)bankoffurnacesof thetypeusedfor oxidationanddiffusion.Theendsof threehorizontalfurnace tubesareclearlyvisible inthe center- right of the photograph.Anoperator in standard cleanroom garb is seated next to the furnace controls. (Photograph courtesy of IntelCorporation.) throughtheexisting oxideandreactsattheSi-SiO2interfacetoformmoreSiO2.The relevant overall reactions are Si +O_2}→SiO_2 (4.1a) Si+2H_₂2O→SiO_₂2+2H₂ (4.1b) GrowthofSiO2using O2 and water vaporare referred to as dry and wetoxidation,respec- tively.Dryoxidationisusedtoformthemostcriticalinsulatorregionsinadevicestructure such asthe gateoxide in aMOSFET,primarily becauseityieldssuperiorSi-SiO2interface

--- Page 174 --- BASICSOF DEVICE FABRICATION 151 properties.Wetoxidation,ontheotherhand,proceedsatafasterrateandisthereforepre- ferred in forming the thicker barrier oxides. A photographofaproduction-lineoxidation/diffusionfurnace is shown in Fig.4.1.A simplifiedsketchofthefurnaceandinputsystemispresentedinFig.4.2along withashort pictorial descriptionof theoxidation process.Standard oxidation temperatures of800C to 1200Careproducedin thefurnace byresistanceheating.Thecoil-likeheatingelements are commonly arranged in three separately controlled zones.Theouter zones operateat a higher powerto compensate for heatlossesout theendsof thefurnace.Arelativelylong center zonecanthereby bemaintained at auniform temperaturecontrollableto±C.he tube down the centerof thefurnace is usuallymadeofclear fused quartz,although SiCand onto aslotted quartz boatand pushedintothecenterofthe furnace.During dryoxidation the oxygen gas is simplyfed into the backof the tube. Wetoxidation is performed by bubblingacarriergas(Aror N2)through water inaheated flaskor burningO2and H2at the inputto thetube.The latterapproach iscalledpyrogenic wetoxidation.Therequred time in the furnace depends on thefurnace temperature,the desired oxidationthickness, andtheSi surfaceorientation.(Otherfactorscanenterundercertainconditions.)Represen- tative dry and wetoxidation growthcurves are graphed in Fig.4.3.It shouldbe notedthat in a commercial system the wafer loading,insertion into the furnace,ramping of the fur- nace temperature,and gas control are all automated. Quartz tube Si wafers Insertion rod O_{2 Resistance-heated furnace (a) $O_2}or H{2}O Si wafer → Si wafer ~0.01-1.0μm SiO2 800℃-1200℃ (b) Figure4.2（a)Simplifiedschematicillustrationofanoxidationsystem.(b)Shortpictorialdescrip- tion of the oxidation process.

--- Page 175 --- 152 SEMICONDUCTORFUNDAMENTALS 10 1200℃dry 1.0 1100℃dry Oxide thickness (m) 1000℃dry 1200℃ wet 1100℃wet $900℃dry 1000℃wet 0.1 (100) 900℃wet 0.01 0.1 1.0 10 100 Oxidation time (hr) (a) Figure4.3 Dryand wetoxidation growthcurves.The SiOthickness formedon (a)(100)and(b) (111)siliconsurfacesasafunctionoftime.(FromJaeger[8,©1988byAddison-WesleyPublishing Co., Inc. Reprinted by permission of the publisher.) 4.1.2 Diffusion Solidstatediffusionisahistoricallyimportantandstillwidelyusedmethodforintroducing dopant atoms into a semiconductor lattice. The basic process is sketched in Fig.4.4.The semiconductorcrystal,aSi waferinFig.4.4,isexposed toasolidliquid,orgaseous source containing the desired impurity. Areaction at the wafer surface establishes a supplyof dopant atoms immediately adjacentto the semiconductor crystal.Atelevated temperatures the atoms slowly diffuse into regions of the semiconductor not protected by theoxide. Diffusion takes placeofcoursebecause the concentrationofimpurity atoms is greater outside thecrystal than inside thecrystal.The surface doping concentrations producedby this method are very high (up to10²/cm),and thus the surface region takes on then-or p-typecharacter associated with the diffusing atoms.It should be noted that diffusion also takes place intotheSiO2,albeitata relatively slowrateforthecommon dopants.This means the SiO2canprotect theunderlying Si foronly alimited periodof time.Foragiven dopant the masking time dependson theoxide thickness,the diffusiontemperature,and the background doping.

--- Page 176 --- BASICSOF DEVICEFABRICATION 153 10 1200℃ dry 1.0 1100℃dry Oxide thickness (um) 1000℃dry 1200℃wet 1100℃wet $900℃dry1000℃wet0.1$900℃wet (111) 0.01 0.1 1.0 10 100 Oxidation time (hr) (b) Figure 4.3 Continued. Diffusion is commonlyperformed in anopen tube system identical inconstruction to that usedforoxidation. Diffusion temperatures are also similar,rangingfromroughly 900Cto1200C.Theonlymajor diference is the introductionofadopant source inplace oftheoxidizingambient.Thespecificexampleofphosphorus difusionemployinga liquid source isschematically illustratedinFig.4.5.ThepicturedNcarriergasis passedthrough abubblercontaining phosphorus oxychloride (PoCl)that is a liquid at room temperature. The N2picks up vaporsfromthe liquidsourceandcarriesthemintothe furnace tue.The small percentageofoxygensimultaneouslyadmittedintothefurnacereactswiththePOCl to deposit P2OonthesurfaceoftheSiwafer,andSiinturnreacts withPOliberating phosphorus atoms that diffuse into thesilicon.Because undesirablesurface compounds tend to form at the higher diffusion temperatures,the foregoing is often combined with a second step where the dopant source is cut off,the furnace temperature increased,andthe already-introduced impurities driven deeper into the semiconductor at a more rapid rate. When a two-step process is used,the portion with the source present is called the pre- deposition,the latter portion with the source removed,thedrivein. Neglecting second-ordereffects,the impurity concentration versus position inside the semiconductor after the predeposition and drive-in steps can be computed from

--- Page 177 --- 154 SEMICONDUCTORFUNDAMENTALS n-type dopant impurity Concentration Diffused n-dopant Original p-dopant $Si{O}_2$ S-d x 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 T=900℃C-1200℃ $n+$ p-Si Figure 4.4 The basic diffusion process. Si wafers POC{{3 $O_2$ $NFigure4.5Schematicillustrationofphosphorusdiffusionusingaliquidsource.$N{1(x,t1)=N0erfc(x/2√D_,t1 … after predep (4.2a) {2(x,t{2}=N e-(x12√D{2{2 after drive-in (4.2b) D22} D{11 The subscripts1and2referto thepredepositionand drive-in,respectively,isthe distance into the semiconductor as measured from the semiconductor surface,N(x,t)the impurity

--- Page 178 --- BASICSOF DEVICE FABRICATION 155 $10^{}$ 10^{2\} $10^{}$ Phosphorus concentration (cm3) $108$ Predeposition Drive-in N_{1}{,{} N{2}x}{2) $10^7$ $10^{}$ $10$ $10$ 0 1 2 3 4 5 x(μm) Figure4.6Computedphosphorusdiffusionprofiles.（N0=10²1/cm3D=2.58×10-14cm2/ concentrationata depthxaftera given timetN0theconcentrationatx=0 during the predeposition,Dthediffusioncoefficientforthegivenimpurityandfurnacetemperature, and t the time durationof the diffusion step. Thecomplementary error function,erfc,is a function cited in mostmathematical handbooks. Theresults of a samplecomputationcor- responding to a phosphorus predeposition at1000C for 10minutes followed by a drive-in at1200C for 30 minutes are presented in Fig.4.6. 4.1.3 lon Implantation Ion implantationaffordsanalternativemeansof introducing dopantsandotheratoms into the near-surface region of a semiconductor. With the need for shallower junctions,lower- temperature processing, and more precise control,ion implantationhas become a work- horse manufacturing process. In ion implantation an impurity is introduced intothe

--- Page 179 --- 156 SEMICONDUCTOR FUNDAMENTALS High-voltage Resolving Acceleration y-scan x-scan terminal aperture tube plates plates Wafer Lens Beam (target) mask Analyzer magnet Faraday cup Ion source Figure4.7 Simplifiedschematicofanionimplantation system.(From RunyanandBean[9], ©1990 by Addison-Wesley PublishingCo.Inc.Reprinted by permission of the publisher.) semiconductor by creating ionsof the impurity,accelerating the ionsto highenergiesrang- ing from5keVto1MeV,and then literally shooting theions into the semiconductor.As one might suspect,the implanted ions displace semiconductor atoms along their pathinto the crystal.Moreover,the ionsthemselves do not necessarilycome toreston lattice sites. Afollow-upanneal(heatingof the semiconductor),an integral partof theoverallprocess, is therefore necessary to remove the crystal damage and “activate” implanted dopants. Asimplifiedschematic ofan ion implantation system and photographsof a commer- cialimplantationend-stationare presented inFigs.4.7 and4.8,respectively.Ionsofthe desiredimpurity are produced in the ion source shown at the extreme left inFig.4.7. The ions are next accelerated into the mass analyzer where unwanted ions,also produced inthe ion source,are removed. Theresulting ion beam isthenacceleratedto thepreset operating potential,focusedandfinallyscannedoverthesurfaceofthewafer.Scanningis accomplishedelectrostatically,bymechanicallymoving the wafer,or by acombination of the two methods.Anelectrical contact to the waferallows a flow of electrons to neutralize the implanted ions.A veryprecise determinationof thetotal numberof implanted ions/ cm²,called thedose and given thesymbol ,is obtained by integrating the target current over the time of the implant. Theconcentrationprofileproducedby ionimplantation has thegeneralformofa Gaussian distribution function and is mathematically described by $N(x)=$ \2({)$ e -(1/2)[(x-Rp)/△Rp]² (4.3)

--- Page 180 --- BASICS OF DEVICE FABRICATION 157 Figure 4.8 Photographs of an ion implantation end-station at the Intel Fab-9 fabrication facility. The top photograph shows the two ports associated with the station.The left port is open and awaiting the loading of wafers onto the wafer carousel,and ion implantation is taking place in the closed right port.The bottom photograph is a close-up of the wafer carousel and therobot arms thatautomatically load and unload the carousel.(Photographs courtesy of Intel Corporation.)

--- Page 181 --- 158 SEMICONDUCTORFUNDAMENTALS Forthosefamiliar withstatistics,Rpis themeanand △Rpis thestandard deviationof the P distributionfunction.Intheionimplantation literature,Rand △Rarecalledtheprojected range and straggle,respectively.These parameters vary with the implant ion and the sub- strate material and are roughlyproportional to the ion energy. Sample computed distribu- tions for phosphorus implantation into Si at various energies are shown in Fig.4.9. Like diffusion,portionsof a Si wafer can bemasked during ion implantation. Thin filmsofSiO2,SiN4,andlessfrequently,photoresistandAlhavebeenusedforthispur- $10^2$ Si $10^{}$ 10{8 (cm^{3)10^{ Phosphorus concentration 1016 100 keV 15 75 14 50 keV 13 25 1012 0 0.1 0.2 0.3 0.4 x(μm) Figure4.9 Computed phosphorusimplantationprofilesassuming aconstantdoseof104/cm2.

--- Page 182 --- BASICSOF DEVICE FABRICATION 159 pose.Basicallytheimplantedionsarestopped inthemaskingmaterial beforereachingthe underlying silicon. Ionimplantationofersanumberofadvantagesoverdiffusion.Itisintrinsicallylow- temperature process,withtheimplantation itselfusually being performed at room tempera- ture.Thefollow-upannealisoftensomeotherhigh-temperaturestep intheICfabrication, such as a subsequentoxidation.If a separate anneal is required,it can beperformed at temperaturesas lowas600Cunderoptimumconditions.Inanyevent,undesirablespread- ing of a concentration profile is minimized using ion implantation.Ion implantation also affords veryprecisecontroloftheimpurityconcentration,andvirtuallyanyatomcanbe implanted intoanygivensubstrate.An interestingrecentdevelopmentis theimplantation ofhigh dosesofoxygenintosilicon,producingaSiOlayer belowthesurfaceofthe wafer andtherebyleading tosilicononinsulator(SOI)structures.Finally,notefromFig.4.9that implantationtypicallyyieldsveryshallowconcentrationprofiles.Implantshavethuscome to replacethepredepositiontepindifusionsandareidallsuitedornumberofmodern device structures requiring extemely shallow junctions. 4.1.4Lithography In discussing diffusion and ion implantation,we talked aboutSiO2or someothermasking materialcoveringportionsofawafer.Selectiveremovalofathinfilmatprescribedregions acrossthesurfaceofa wafer,thepatterningofinsulatorsandmetalsrequiredinforming ICs, is achieved through a process called lithography. Photographs of a production-line lithography station are reproduced in Fig.4.10. Major stepsinthe lithographyprocessare depictedinFig.4.1lusing thepatterningof anSiO2filmasanexample.Thetop surfaceoftheSiO2-covered waferis firstcoated with anultraviolet-lightsensitivematerialcalledphotoresistwhichis suppliedasa liquid.A few drops of the liquid are placedonthe wafer,andthe wafer is spun at high speeds to produce a thinuniform coating.After spinning,a short”prebake”at relativelylowtem- peratures (80C-100C) is performedto drive solventoutof the resist andto improve surface adhesion.The hardened photoresist is similar to a photographic emulsion. The nextstep is to exposetheresist through a“mask”using UV-lightas illustratedin Fig.4.11(b).Themaskhere isacarefullypreparedglassorquartzphotoplatecontaininga copyofthepatterntobetransferredtotheSiO2film.Darkenedregions in theemulsionon the maskblock theUV-light.Regionsofthe photoresistexposed to the lightundergo a chemicalreaction which varies withthetypeofresistbeingemployed.In negativersiss the areas where the light strikes become polymerized and more difficult to remove.When placed in a developer the polymerized regions remain,while the unexposed regions dis- solve and wash away.The net resultafter development is pictured on the right-hand side of Fig.4.1l(c).Positiveresistscontainlargeamountsofasensitizerthatdramaticallyslows down the dissolutionrateoftheresist in analkaline developer.This sensitizerbreaksdown when exposedto lightleading to the preferential removal of the exposed regions as shown on theleft-handsideofFig.4.11(c).Asaninformationalaside,negativeresist was widly used in earlyIC processing.Positive resist is nowthemaintype in use because it affords better small-geometry control.

--- Page 183 --- 160 SEMICONDUCTORFUNDAMENTALS NikonNSR Figure 4.10 Production-line lithog- raphy stationconsisting of anSSI150 coat and develop system (center-left in upper photograph)and a Nikon steper (center-right in the upper photograph). The topof the SSI150 system is shown more clearly in the lower photograph. TheSSI150automaticallyperforms the coat, prebake, develop, and harden op- erations described in the text. The inter- vening pattern exposure is performed in the Nikon stepper. Without human inter- vention,the wafer moves along the input track oftheSSI150,enters the Nikon stepper,exits the Nikon stepper, and moves back along one of the two output tracks of the SSI 150. (Photographs courtesy of Intel Corp.)

--- Page 184 --- BASICSOF DEVICEFABRICATION 161 Resist \SiO_{2}$Si\uparrow (a)fra$ UV Mask SiO{2}$Si(b)PositiveresistNegativeresistSiO2$ $Si{O}_2$ !S Si (c) $Si{O}_2$ $Si{O}_2$ Si Si (d) Figure4.11Majorsteps inthe lithographyprocess:(a)Applicationofresist;(b)resistexposure through amask;(c)afterdevelopment;d)afteroxideetchingandresistremoval.(FromJaeger8], ©1988 by Addison-Wesley Publishing Co.,Inc.Reprinted by permissionof the publisher.)

--- Page 185 --- 162 SEMICONDUCTORFUNDAMENTALS The final setof steps involves transferring the pattern to the SiO2film.Immediately followingexposureanddevelopmentthephotoresistisgivenahardbakeheatingfo 30minutes at 120C-180C,to further increase adhesion to the wafer and to improve resistanceto thesubsequentetch.An acidetch,buffered HF in thecaseof SiO,is then used to dissolve unprotected regionsof theunderlying film.Lastly,thephotoresist,having completedits job,isstripped away.This is accomplishedbyusing a chemical solutionthat swellsand liftstheresistor byoxidizing(burning)theresist inanoxygen plasma system called an asher. The limitsof theUV-basedlithographyprocessjust describedare beingpushedbythe ever shrinking dimensions of modern devices.UV sources with shorter wavelengths and specialcompensating techniqueshave expandedthesmalldimension limit.Nevertheless, lithographic systemsemploying x-rays are being developed andare likely tocomeon-line in the near future. 4.1.5Thin-Film Deposition Toconnect device structures to the“outside world”requires the depositionand patterning of a metal layer.Infact,complexICs have threand sometimes fourelectrically isolated metallizationlayers.Electricalisolationofthemetallayersinturnrequiresthedeposition ofintervening dielectric layers.Thinfilmsarealsodepositedtopreventtheinterdiffusion of materials and to protectthe deviceor circuitfromcontamination.In-use methodsfor depositing the required films are reviewed next. Evaporation Evaporation isoneof theolder and morestraightforward methodsof thin-film deosition. As envisioned inFig.4.12,the material to beevaporated is placed inoron a resistance- heated sourceholderinside a vacuum chamber.To evaporate Al,for example,a short piece of Al wire would be placedon a tungsten filamentorboat.Thesubstrate on whichthefilm is to be deposited is also positioned inside thechamberfacing thesource.Thechamber is then evacuated,power supplied to theholder,and the source vaporized. Because ofthe reduced pressure,thesource material travels unimpeded to thesubstrate and deposits as a thin film. Generally speaking,hot-filamentevaporation is subjecttomoderately highlevelsof contamination. Electron-beam evaporation,a variationof the process where the source is heatedbyanelectronbeam,eliminates contaminationbut generatesdevice-degrading x-rays.Consequently,evaporation is seldom used in theproduction-linefabricationof modernICs,although it is stillextensively used inmaking simple devices where thecited problems are of minimal concern. Sputtering Sputteringlike evaporation,is performed in a vacuum chamber. Thesource material and the substrate(wafer)areplacedonopposing parallelplates connectedto a high-voltage power supplyas pictured in Fig.4.13.During a depositionthechamber is first evacuated

--- Page 186 --- BASICS OF DEVICE FABRICATION 163 Wafer Evaporating material I To vacuum pump Figure 4.12 Hot-filament evaporation. Vacuum chamber Cathode shield Wafers Cathode (source material) Anode Heater Argon High voltage inlet To vacuum pump Figure4.13 Schematicofad.c.sputtering system.Thesource materialcovers thecathode while the wafer is mountedon the system anode.(From Jaeger[8],©1988 by Addison-Wesley Publishing Co., Inc.Reprinted by permission of the publisher.)

--- Page 187 --- 164 SEMICONDUCTOR FUNDAMENTALS of airandthen a low-pressureamountof sputtering gas,typically Ar,isadmitted intothe chamber. Applying an interelectrode voltage ionizes theAr gasandcreatesa plasmabe- tween theplates.Sincetheplatecovered withsourcematerial is maintained at a negative potentialrelativeto thesubstrate,Ar+ionsareacceleratedtowardandinto the source cov- ered plate. The impacting Ar+ions in turn cause sourceatoms ormolecules to beejected from this plate. Being neutral,theejectedatomsormolecules readily travel to the substrate where they depositto form thedesiredthinfilm.Ad.c.power supplycan be used when depositingmetals,and anRF supply is necessary whendepositing insulating films.When sputtering compounds it is sometimes necessaryto introduce a gasofoneofthe compo- nents toassuretheformationofanear-stoichiometricfilm.Providinglow-temperature, low-contaminationfilms withanacceptable throughput,sputteringhasbecomethechief commercial method of depositing Al andother metals. Chemical Vapor Deposition(cVD) In chemical vapor deposition the thinfilm is formedfromoneor more gas phase compo- nents.Either a compound decomposesto form the filmor a reaction between gas compo- nents takesplacetoformthefilm.InvariablytheCVDreactions are surface catalyzed, preferentiallytakingplaceonthesurfaceof wafers insertedintothegas stream.In-use CVD processesfall intooneofthreegeneralcategories. Theyareatmospheric pessure (APCVDor simplyCVD),low-pressure (LPCVD), and plasma-enhanced (PECVD)pro- cesses. Atmospheric pressure depositions can be performed inrelatively simple systems. Low-pressureoftenofferscomparablekinetics withimproved uniformityandless gascon- sumption.In plasmaCVDtheelectrons in the plasma impart energy to the reaction gases, thereby enhancing the reactionsand permitting very lowdeposition temperatures. CVDreactors come in a varietyof shapesand configurations; an example configura- tion employed in AP/LPCVD depositions is pictured in Fig.4.14.CVD processing is ru- tinelyused toproducethemaskingand intermetallicdielectricfilmsrequired in thefor- mationofcomplexICs.PolycrystallineSi,whichfunctionsasapseudo-metalwhenhavily doped, isalsodepositedemploying atmosphericand low-pressureCVD.APCVD,LPCVD, and PECVDareall typicallyused at some point inanICprocess flow. 4.1.6Epitaxy Epitaxy is a specialtypeofthin-layerdeposition. Whereas depositions describedinthe preceding subsection yieldeitheramorphous or polycrystalline films,epitaxy produces a crystallinelayerthatisanextensionoftheunderlyingsemiconductorlattice.Theword formedfromthevapor-phase decompositionofsilicontetrachloride(SiCl4)or a silane compound(SiH4,SiH2Cl2,SiHCl3)inareactor verysimilartothoseemployedinCVD, additionalSi literallygrowsfollowing thelatticepatternofthepre-existing crystal.The doping oftheepi-layer,intentionallydistinctfromthatofthesubstrate,iscontrolledby introducingadopantcontaininggassuchasphosphine(PH),diboraneB2H6),orarsine

--- Page 188 --- BASICSOFDEVICEFABRICATION 165 Gas Gas input input Susceptor Lamp module Quartz lamps Wafers Infrared sensors Exhaust Figure4.14Cutaway viewofaradiantlyheatedbarrel-type reactoremployed in bothCVDpro- cesses and Si epitaxy.Wafers positioned verticallyonthecentral susceptor are heated by thelight from the surrounding quartz lamps.Process gasesenter at thetop,flow across the wafers,and exit coaxiallyatthebottom.(From Deacon10,©1984byLakePublishing.Allrightsreserved.Reprinted by permission of the publisher.) (AsH)during the deposition.Standard whole-waferepitaxy mustofcourse becompleted early in the process flow,priortothe growthor depositionofany permanent surfacefilms. We shouldalso notethat starting Si wafers are sometimespurchased from the wafer sup- plier withthe desired epi-layer already in place. 4.2 DEVICEFABRICATIONEXAMPLES The purposeofthis sectionisto ilustrate how the individualfabricationprocesses are combined and sequencedto producesolid state devices.Twoexamples are presented: The first describesthefabricationofasimple pnjunctiondiode.The secondoutlinesthemaking ofacomplexIC.Bothpresentationsare intentionallyqualitative innature,includingonly major processing steps and a minimum of processing details.

--- Page 189 --- 166 SEMICONDUCTORFUNDAMENTALS 4.2.1 pn Junction Diode Fabrication Figure 4.15graphically summarizesthe majorprocessing steps in theformationofa pn junctiondiode.The startingpointisa flat,damage-free,single-crystalSiwafer.It is assumedthat a preclean has removed allparticulates,organic films,andadsorbed metal ions from the semiconductor surfaces.For this particular illustration we furtheras- sume the wafer is p-type,having beenuniformly doped with boron during the formation of the crystal. Theinitial steps in the process flow are inpreparationfora subsequentphosphorus diffusion.First a thermaloxide is grown that willserve as a diffusionbarrier.Theoxide thickness must be comfortably greater than the projected masking thickness.Step 2is a lithographyprocess performed toopen”diffusion”holes in theoxidethatwilleventually becomethe positionsof the pnjunctiondiodes.SpecificallynotethattheStep2illustration in Fig. 4.15 assumes the use of a positive photoresist. After a properclean-up the wafer is next inserted into a phosphorus predeposition furnace and subsequently into a phosphorus drive furnace. The netresult,as picturedin Step3in Fig.4.15,is theformationofn+-p junctions insurfaceregions not protectedby the oxide.(The+inn+indicatesa very high doping.)By wayofclarification,some oxy- gen is required during the phosphorus predeposition as noted in the diffusion discussion. Also,the drive may be performed in anoxidizing atmosphere to minimize out-diffusion. Thus,to reopentheoxide holes,a subsidiary lithography step,not shown in the simplified process flowofFig.4.15,is usually necessary after diffusion. Thefinalstepsfacilitateconnecting the device to the“outsideworld.”Sputteringor possibly evaporationofAl yields a thinmetal filmover theentire surface of the wafer as pictured in Step4ofFig.4.15.Alithographyprocess,Step5inFig.4.15,isthen performed to remove excess metal external to the area of the diffused junction. Normally a low- temperature(≤500C) anneal would also be performed to promote a low-resistance contact betweenthemetal andSi. Withthecompletionof themetallizationcontact,the diodes become functional across the wafer. To produce commercial diodes,a diamond-edged saw would be used to cutthe wafer into pieces containing a single device.(A waferpiece containing a single deviceor ICis called a die.)Alarge areametalcontact is thenmade to thebackof the die,alead attached to thetop surfacecontact,and thedevice encapsulated inprotective plasticor hermetically sealed in a metallic package. 4.2.2 Computer CPU Process Flow Inthis subsection we examine a fabrication process flow that has been used by IntelCor- poration to produce computer CPUs andotherICs.The process flow description and ass- ciated figuresarea directexcerpt,reprinted with permission,fromComponents Quality and Reliability1991/192,IntelCorporation,©1990.No changes have been made tothe excerpted materialexceptforarenumberingofthefiguresandamodificationof thefigure captions.Figure4.16,reproducedfromthecitedreference,displayssimplifiedblock dia-

--- Page 190 --- BASICS OF DEVICE FABRICATION 167 Step Process Visualization (0) Start p-Si $Si{O}_{}$ p-Si (1) Oxidation $Si{O}_2$ Mask (2) Lithography #1 Positive resist $Si{O}_2$ $Si{O}_2$ $Si{O}_2$ p-Si S-d Phosphorus Phosphorus (3) $Si{O}_2$ $Si{O}_{}$ $SiO$ $Si{O}_2$ diffusion $n$ p-Si p AI Metallization $Si{O}_2$ $Si{O}_2$ (4) (sputter Al) $n$ p Mask Resist IV (5) Lithography #2 $Si{O}_2$ $S_i{O}_2$ $SiO2$ $Si{O}_{}$ $n^{}$ $n$ P p Figure4.15 Graphical summaryof the majorprocessing steps in theformationofa pnjunction diode.

--- Page 191 --- 168 SEMICONDUCTORFUNDAMENTALS STARTING SILICON POLYCIDEDEPOSITION MASK WAFER

• IMPURITY DOPING BOND PAD INITIAL OXIDATION MASK PATTERNING MASK POLYCIDEPATTERNING ALLOY +OXIDATION WELLIONIMPLANT WAFERBACKSIDE +DRIVE MASK PREPARATION SILICON-NITRIDE N-CHANNEL SOURCE- ELECTRICAL TEST DRAIN+IONIMPLANT PROCESS CONTROL MASK MASK OUTGOING QUALITY MONITOR SILICON-NITRIDE PATTERNING P-CHANNEL SOURCE- DRAIN ION IMPLANT POLYRE-OXDOPED FIELD THRESHOLD GLASS DEPOSITION ION IMPLANT WAFER SORT OPERATIONS FIELD MASK SILICON-NITRIDE CONTACT REMOVAL ETCH ASSEMBLY MFG. GATE OXIDATION METAL DEPOSITION TRANSISTOR THRESHOLD MASK DIVISION ION IMPLANT CLASSTEST MASK METAL PATTERNING OXIDE ETCH PASSIVATION CUSTOMER Figure4.16 Simplified block diagramofthe single poly,single metal,CHMOSprocess flow. gramof theoverallprocessflow.Figure4.17,whichiscorrelated withthetext description of the component processes,shows cross-sectional sketches of theIC structure at various stages of construction. Although mostof thetermsemployed in the process description will be familiartothe reader,afewneedto beclarified.Forone,thefabricationsequence isreferredto as a single poly, single metal,CHMOSprocess flow.”Poly”is shortfor polycrystalline Si.CHMOS stands for Complementary High-density Metal-Oxide-Semiconductor—a typeof fabrica- thatone levelof heavily doped polycrystalline Siandone levelofmetal are used tocontact

--- Page 192 --- BASICS OF DEVICE FABRICATION 169 and interconnect theMOSdevices. Textreferences arealso made to“wet”and”dry” etches.WetetchingisthefamiliardissolutionofamaterialsuchasSiO2,inaliquidchemi- cal bath. Dry etching is theremovalof a materialusing a plasma-enhanced gas-phase reaction. Finally,thefollowingprocessflowdescriptionisintendedtobe“lookedover,“not digested.At thispoint in the development,the reader is notequipped and not expected to understand the intricacies of either the processing or the IC structure. 1.StartingSiliconMaterialandWellDefinition[Fig.4.17(a)]TostartCHMOSde- vice production150mm(6”high-resistivity,（100orientationsinglecrystal,ptype (Boron doped)silicon wafers are used.P-type siliconisrequiredtocreate n-channel transistors.To create p-channel transistors,necessary forCHMOS devices n-type(a- senic or phosphorous doped)siliconregions (n-wells)are implanted. The wafer is masked,then implanted to create p-type and n-typesilicon regions on the same wafer. The n-well providesthebackgrounddoping for thep-channeltransistors whilethe p- type, startmaterial(protectedfromtheimplant bythe unexposedphotoresist)serves as thebackground dopingforn-channeltransistors.Ahightemperature drive cycle completes the formation of the well by thermal dopant transition. 2.FieldThresholdImplantandFieldOxidation[Fig.4.17(b)].Nitride(SiN4)isde- posited,masked,thenetched.Theetched nitrideregions definethe locationofthefield thresholdionplant[Fig.4.17(b)]andthelocations whereoxideSiO2is permittedto growduringfieldoxidationFig.4.17)].Areaswherentrideremainsmaskprevent) oxide growth.Theseregions are where transistors willbebuilt.Thethick(approxi- mately600o)fieldoxideisolatesadjacenttransistorsinordertopreventelectrical interactions.After fieldoxidation,the nitride mask is removed. 3.GateOxidation[Fig.4.17(d)].Athinthermaloxide is grown across the wafer.The portionsofthisoxideremainingafter subsequentprocessing willprovidetherequired gateoxide forthe MOStransistors.Device performance isclosely related to the growth of a dense,high-quality gate oxide. 4. Transistor ThresholdIonImplant[Fig.4.17(d)].Theboronimplantadjuststhe threshold voltage(Vt)of p-channeland n-channel devices to the desired level.Thick fieldoxide prevents the boron from penetrating into the isolationregions. 5.Polysilicon-to-DiffusionContacts[Fig.4.17(e)].Openingsaredefined inthethin gate oxideregion wherecontact between polyconductorsand diffusedregions in the silicon substrate (buried contacts) is required.

--- Page 193 --- 170 SEMICONDUCTORFUNDAMENTALS BORON $S_{₃}3N$ RESIST FIELD IMPLANT WELL WELL SILICON SILICON (a) START MATERIAL & WELL DEFINITION (b) FIELD THRESHOLD IMPLANT BORON $S_{₃3N}$ ois$ FIELD OXIDE GATE OXIDE P FIELD THRESHOLD IMPLANT IMPLANT WELL WELL SILICON SILICON (C) FIELD OXIDATION (d) TRANSISTOR THRESHOLD ION IMPLANT SILICIDE POLY RESIST P P +N SILICON WELL SILICON WELL (e) POLYSILICON TO DIFFUSION CONTACTS (f) POLYCIDE DEPOSITION Figure4.17Cross-sectionsthroughtheCHMOSstructureatselectpointsintheprocessfow.

--- Page 194 --- BASICSOF DEVICEFABRICATION 171 PHOSPHORUS RESIST POLYRE-OX SILICIDE RESIST POLY P P P-CHANNEL $N+$ N-CHANNEL S/D’S SOURCE DRAINS WELL WELL SILICON SILICON (h) N-CHANNEL SOURCE/DRAIN (g) POLYCIDE ETCH IONIMPLANTS& POLYRE-OX DOPED GLASS RESIST BPSG P P P $N+$ N+ WELL WELL SILICON SILICON (i) DOPED GLASS DEPOSITION CONTACT MASK ETCH PLASMA OXIDE RESIST 6XYNITRIDE METAL 0 09 P P+$ P+ N+ P +N P WELL WELL SILICON SILICON (k) METAL DEPOSITION& DEFINITION (1) PASSIVATION Figure 4.17 Continued.

--- Page 195 --- 172 SEMICONDUCTORFUNDAMENTALS 6.Polycide Deposition,Impurity Doping,Patterning andOxidation[Fig.4.17(f)].A polycrystallinesiliconlayer,about1500Åthickisdepositedovertheentirewaferby CVD techniques. The polysilicon is subsequently doped with phosphorousvia gas- source diffusion techniques forconductivity.Thephosphorous is diffused through the poly andinto the substrate,minimizing the resistanceof the buried contact.About 1500Åofsilicideisconformallydepositedonthedopedpolysilicon.Silicideisused in smallgeometry transistorstoincrease device performance speeds.The polycidestack is defined byplasma dryetch techniques withresistacting as themask[Fig.4.17(g)]. Thepolycide stack is oxidized to protect the transistor gates. 7.Sourceand DrainImplantandPolyRe-Ox[Fig.4.17(h)].The remaining polycide stack,thinoxideover thepoly,fieldoxideand resist provide amask for p-channel and n-channelsource-drain implants whichoccurs atthispoint.(This techniqueiscom- monly known as a”self-aligned source/drain”process,since the source and drainare aligned directly to thegate,which definestheir location relative to thechannel area) All remaining polyoxide is then removed and the poly re-ox step performed.There- ox step is a high-temperature thermaloxidation step which drives the source-drain implantintothesilicon,provides ahigh-integrity dielectricon the polysilicon （essen- tial in a double poly process) and grows an oxide on the exposed source-drain regions to prevent dopantout-diffusion during subsequent phossphosilicateglass deposition and processing. 8.DopedGlass Deposition[Fig.4.17(i)].Adoped(boron,phosphorousor both)silicate glass layer,5000Åto 10,000Å thick,is deposited via low-temperatureCVDtech- niquesto provideelectricalisolation betweenthe subsequentmetalconductor linesand the underlying polycidegatesand active device structures.The glass isthen prepared for subsequent masking steps. 9.Contact MaskandEtch[Fig.4.17(j)].Windows in the resist are exposed and opened to definethelocationofmetal-to-polycideand metal-to-siliconcontactholes.Depend- ing on the process,contactholes are wet,dry,or wetand dry etched.In a wet/dry combination,the wetetchgives a shallow slope nearthetopof thecontact.This im- proves metal stepcoverage.Theremaining doped siliconglass is thenremovedbyan isotropic dry etch which leaves the contact sidewalls vertical. 10.MetalDepositionandDefinition[Fig.4.17(k)].Metalconductorlayer issputter- deposited into the waferand defined.Themetal electricallyconnects the transistorsto the outside world.The voltage applied by thesemetal lines turns the transistors”on” and“off.” 11.Passivation[Fig.4.17(l)].PlasmaenhancedCVDtechniquesare usedto deposit a dualfilmpassivationlayerovertheentire wafer surface.The dual layerconsistofan

--- Page 196 --- BASICSOF DEVICE FABRICATION 173 oxynitrideunderneatha plasmaoxide. Theoxynitrideprovides long-termfieldreli- ability,bybeing designedtobeacontaminationandmoisture barrier.Thelatteris particularlycritical ifthediceareto beassembledinplasticpackages,plastic itself being a poor moisture barrier.The plasmaoxide layer provides handling protection for the wafer and individual devices. Mostoftheremainingproduction-lineandtesting stepscited intheFig.4.16block diagramare in preparation for eventualICdie separation and packaging.Thephotograph ofa sampleCPUdie,along withanidentificationofitsfunctionalunits,are displayed in Fig.4.18.The pictured PentiumTM Processor die,fabricatedfollowing an upgraded version of the describedprocessing,is approximately1.5cm squareand contains 3.1million transistors. Instruction Fetch Branch Pediction Code 07 Cache Instruction, Code Decode TLB Clock Driver BusInterface Superscall data Integer TLB Execution Pipelined units Floating Data Point cache Figure 4.18 TheIntelPentium TM Processoranditsfunctionalunits.(Photographandfunctional unit identification courtesy of Intel Corporation.)

--- Page 197 --- 174 SEMICONDUCTORFUNDAMENTALS 4.3SUMMARY Theoverview ofsilicon devicefabrication hopefullyprovided insightintotheconstruction and physical natureof device structures.First,common component processes wereexam- ined,includingoxidationdifusion,ionimplantation,lithography,evaporationsputering chemical vapor deposition,and epitaxy. The fabricationof a simple pn junction diode and amoderatelycomplexICprocess flow were then considered to illustrate how the individual processes are combined and sequenced in producing actual structures.

--- Page 198 --- R1 Part I SUPPLEMENTANDREVIEW R1 ALTERNATIVE/SUPPLEMENTALREADINGLIST Recommended Readings Type Relevant Author(s) Level (A =Alt., S= Supp.) Chapters For Semiconductor Physics Ferendeci A/S Advanced Undergraduate/ 1-6 Introductory Graduate Neamen A Undergraduate 1-6 Pierret S Introductory Graduate 1-6 Streetman A Undergraduate 1-4 Tyagi A/S Advanced Undergraduate/ 1-5 Introductory Graduate For Device Fabrication Jaeger S Undergraduate All Runyan and Bean S Professional All Sze (editor) S Professional All (1)A.M.Ferendeci,Physical Foundationsof Solid Stateand Electron Devices,McGraw- Hill, New York, © 1991. (2)R.C.Jaeger,Introduction to MicroelectronicFabrication,Vol.Vin theModular Se- ries on Solid State Devices edited by G.W.Neudeck and R.F.Pierret,Addison-Wesley, Reading, MA, © 1988. (3)D.A.Neamen,SemiconductorPhysicsand Devices, BasicPrinciples,Irwin,Home- wood, IL, © 1992. (4) R.F.Pierret,Advanced Semiconductor Fundamentals,Vol.VIin the Modular Series on Solid State Devices edited by G.W.Neudeck and R.F.Pierret,Addison-Wesley,Read- ing, MA,© 1987. 175

--- Page 199 --- 176 SEMICONDUCTORFUNDAMENTALS (5)W.R.RunyanandK.E.Bean,SemiconductorIntegratedCircuitProcessing Tech- nology, Addison-Wesley, Reading, MA, © 1990. (6)B.G.Stretman,olidtateElectronicDevices4theditionPrenticeHallEnlewood Cliffs, NJ, ©1995. (7) S.M.Sze(editor）,VSLI Technology,2ndedition,McGrawHill,NewYork,188. (8)M.S.Tyagi,Introduction to Semiconductor Materialsand Devices,John Wiley& Sons,NewYork, © 1991. Other Recent Texts (1)A.Bar-Lev,SemiconductorsandElectronicDevices3rdedition,PrenticeHall,Inc., New York, © 1993. (2)D.H.Navon,SemiconductorMicrodevicesandMaterials,Hol,RinehartandWinston, New York, © 1986. (3)D.L.PulfreyandN.G.Tarr,IntroductiontoMicroelectronicDevices,PrenticeHall, EnglewoodCliffs, NJ,© 1989. (4)C.T.Sah，FundamentalsofSolid-StateElectronics,WorldScientific，Singapore, © 1991. (5)J.Singh,Semiconductor Devices,anIntroductionMcGraw-Hill,NewYork,©194. (6)S.M.Sze,SemiconductorDevicesPhysicsandTechnologJohnWiley&Sons,Nw York, © 1985. (7) E.S.Yang,Microelectronic Devices,McGrawHill,NewYork,©198. (8) M.Zambuto,Semiconductor Devices,McGraw Hill,NewYork,© 1989. (↓ Reader-added textts)

--- Page 200 --- SUPPLEMENTANDREVIEW 177 FIGURESOURCES/CITEDREFERENCES (1)W.ShockleyElectronsandHolesinSemiconductorsLitonEducationalPublishi R1 Inc., © 1950. (2) S.M.SzePhysicsof Semiconductor Devices,2ndeditionJohnWiley& Sons,Nw York, © 1981. Verlag, Berlin, © 1982. (4) C.Jacoboni,C.Canali,G.Ottaviani,andA.A.Quaranta,“AReviewofSomeCharge TransportPropertiesofSilicon,Solid-StateElectronics,20,77(17). (5) P.M.Smith,J.Frey,andP.Chatteree,“High-FieldTransportofHolesinSillicon， AppliedPhysicsLetters,39,332(Aug.81). (6)D.K.Schroder,Semiconductor Material and Device Characterization,John Wiley& Sons, New York, © 1990. (7)R.F.Pierret,Advanced Semiconductor Fundamentals,Vol.VI in the Modular Series on SolidStateDviceseitedbyG.W.NudeckandRF.PrreAdion-WyR- ing, MA,© 1987. (8)R.C.JaegerIntroductiontoMicroelectronicFabricationVol.VintheModularSe- ries on SolidState DeviceseditedbyG.W.NeudeckandR.F.PierretAddison-Wsley Reading, MA, © 1988. (9)W.R.Runyanand K.E.Bean, SemiconductorIntegratedCircuit Processing Tech- nology, Addison-Wesley, Reading, MA, © 1990. (10)T.Deacon”SiliconEpitaxy:AnOverview,MicroelectronicManufactuind Testing, September 1984. (↓Reader-added references)

--- Page 201 --- 178 SEMICONDUCTOR FUNDAMENTALS REVIEWLISTOF TERMS Definingthefollowingtermsusingyourown wordsprovidesarapidreviewofthePrtI material. (1) amorphous (41) conductivity (2) polycrystalline (42) band bending (3) crystalline (43) diffusion (4) lattice (44) diffusion coefficient (5) unit cell (45) recombination (6) ingot (46) generation (7) carrier (47) band-to-band R-G (8) electron （48) R-G centers (9) hole (49) E-k plots (10) conduction band (50) direct semiconductor (11) valence band (51) indirect semiconductor (12) band gap (52) photogeneration (13) effective mass (53) absorption coeficient (14) intrinsic semiconductor (54) low-level injection (15) extrinsic semiconductor (55) equilibrium (16) dopant (56) perturbation (17) donor (57) steady state (18) acceptor (58) minority carrier lifetime (19) n-type material (59) minoritycarrier diffusion length (20) p-type material (60) quasi-Fermi level (21)n+(orp+)material (61) dry oxidation (22) majority carrier (62) wet oxidation (23) minority carrier (63) predeposition (24) density of states (64) drive-in (25) Fermi function (65) diffusion mask (26) Fermienergy （or level） (66)i ion implantation (27) nondegenerate semiconductor (67) dose (28) degenerate semiconductor (68) projected range (29) charge neutrality (69) photoresist (30) extrinsic temperature region (70) negative resist (31) intrinsic temperature region (71) positive resist (32) freeze-out (72) sputtering (33) drift (73)CVD (34) scattering (74)LPCVD (35) thermal motion (75)PECVD (36) drift velocity (76) epitaxy (37) saturation velocity (77) die (38) current density (78) poly (39) mobility (79) wet etch (40) resistivity (80) dry etch

--- Page 202 --- SUPPLEMENTAND REVIEW 179 PARTI—REVIEWPROBLEMSETSANDANSWERS The followingproblem sets were designed assuming a knowledge—at times an integrated R1 knowledge—of the subjectmatter inChapters 1-3ofPart I.Thesets could serve as a review or as a meansof evaluating thereader’s masteryof the subject.Problem Set A was adaptedfrom a one-hour “open-book” examination; Problem Set B was adapted froma one-hour”closed-book”examination.An answer key is included at the endof the problem sets. ProblemSetA Special Instructions: (1) You may employ anOPENbook in answering theProblem Set A questions. (2)Unlessintructedthrwisemloyamtrsccuatetotreplacesinerrmin numerical computations. Problem A1 AnswerthequestionsthatollowmakinuseoftheunitcllortheSicrytallticeepro- duced here: A D D X B C (a)If,as pictured,theoriginofcoordinates is located atthe lower backcorneroftheunit cell,what are the Miller indices of the plane passingthroughthe points ABC? (b)WhataretheMiller indicesof the direction vectorrunning fromtheoriginof coordi- nates to the point D? (c) Intermsofthe latticeconstant a,whatisthe distance betweennearestneighbors inthe Si lattice? (d)ModifyingthepicturedSiunitcell,indicatehowone would visualizeadonor.

--- Page 203 --- 180 SEMICONDUCTOR FUNDAMENTALS Problem A2 R1 riumconditionsatT=300K.p=10-6sec.Completethefollowingtablecharacterizing the wafer. Parameter Value Units Explanation $EG$n_i}$1010cm-3$n_{op0EF-E$\rho (chargedensity)$μ_{} ρ(resistivity) Lp D_ Problem A3 (a) Ontheenergy banddiagram provided below,indicatethe usual positioningof he fol- lowing energy levels: (i)E…instrinsic Fermi level (ii)ED. . .donor energy level (iii) EA.. acceptor energy level (iv)ET…trap or R-Gcenterenergy level (v)EF for a degenerately doped p-type material Addcomments as necessary to forestall any misinterpretationofyour graphical answer. $Ec$ $Ev$ (b)ASi device is maintained underequilibriumconditions atT=300K.Given theelec- tric fieldinside the deviceas is pictured next,NA=101/cm3for0≤x≤,and ND=1016/cm3for≤x≤,drawtheenergy band diagramforthe device.Include E,E,EandacarefullypositionedEF onyour diagram.Also indicate howyou arrived at your answer.

--- Page 204 --- SUPPLEMENTANDREVIEW 181 8 x_{ x_{b $xc$ 0 →x (c) Whatisthelectrostaticpotentialdropacrossthedevicedescibedinpart(b);iwha is V(x=xc)-V(x=0)? Record your work. Problem A4 AnND=10/cmSisamplemaintainedatroomtemperatureissubjecttotwoflashesof lightfromastroboscope.Eachflashcreates Δ=10/cm3holesuniformlythroughout the Si sample.The firstflashoccurs att=0and thesecondoccurs att=t,where is comparabletoτTheflashesoflightthemselvesareofinfinitesimallyshortduraioncom- pared to p. (a) Sketch(based on qualitative reasoning)the expected general variationof ∆p(t)asa function of time. ∆p(t) 2△_{ t 0 tf (b)Dolow-levelinjectionconditionsprevail atalltimesinsidethe Si sample?Explain. (c) Startingfrom theappropriate differentialequation,solvefor p(t)forallt≥0. (d) Assuming ∆n(t)=∆p()insidethebar,determinethefractionalchangeintheSi sampleconductivityimmediatelyafterthefirstflashoflight.Give bothasymbolicand numerical value for the desired[(0+)-(0-)]/(0-).0+and 0-referto times just before and just after the first flash,respectively. Problem Set B Special Instructions: (1) Problem Set Bquestions are to be answered with the bookCLOSED.

--- Page 205 --- 182 SEMICONDUCTOR FUNDAMENTALS (2)In answeringmultiple-choicequestions,choosethebestavailableanswerSelectonly one answer per question. (3)Employ=1010/cm3andkT=0.0259eVinnumericalcomputationsinvolvingSi samples maintained at 300 K. 1. General Information (1)Asshownintheaccompanyingfigureacubicunitcellhasoneatomcenteredhalfway down eachofthe vertical edgesofthecellandoneatompositionedinthemiddleof thetop and bottom faces of the cell.How many atoms are there within thecell? (a) 4 (b) 2 (c) 1 (d) 1/2 (2) Whatisthenameofthelatticecharacterizedbytheunitcell inQuestion(1)? (a) Simple cubic (b) bcc (c) fcc (d) Diamond (3) Determine the Miller indices of the plane pictured below. (a) (123) (b) (123) 3a (c) (321) (d) (321) (e) (632) -2a y a x

--- Page 206 --- SUPPLEMENT AND REVIEW 183 (4) Whichof thefollowingdirections is not perpendiculartothe100]direction? (a) [0ī1] (b) [032] R1 (c) [001] (d) [100] (5） In a particularsemiconductortheprobabilityofelectronsoccupying statesatanenergy kTabove the bottomofthe conductionbandise-10.Determine the positionof the Fermi level in the given material. (a) E=c (b) E-F ≈9kT (c) Ec-EF=10kT (d)EF=E+kT (6)Whichofthefollowing is the dominantcarrier scatteringmechanism inlowly doped (high-purity) silicon at room temperature? (a) Carrier-carrier scattering (b) Lattice scattering (c) Impurity ion scattering (d) Piezoelectric scattering (7) Whichof thefollowing sketches best describesthe DN versus NDdependenceofele- trons in silicon at room temperature? D{n D{ D{n D{n N{ N{D N{}$ -ND (a) (b) (c) (d) (8)Using the bonding model,indicate how one visualizes a donor. (9)Using theenergy banddiagram,indicate howone would visualize thefreeze-outof electrons on donor sites as T→0K. （10)Theresistivityofntypematerialistypicallsmallerthantheresistivityofcomrly doped p-type material. Explain why. (11)Usingtheenergy banddiagram,indicatehowone visualizesrecombination via R-G centers. (12) Define drift velocity.

--- Page 207 --- 184 SEMICONDUCTORFUNDAMENTALS I1.I Interpretation of Energy Band Diagrams Asilicondevicemaintained at300Kis characterized by thefollowingenergyband da- gram.Use the cited energyband diagram in answering Questions(13)to(20). E{ $E$ E_{/3 Ev Hole T x 0 x_{ }$L/3$x_{2}$ 2L/3 L (13)Theelectrostatic potential(V)inside thesemiconductor is as sketched below. V V V V V x x x x x (a) (b) (c) (d) (e) (14) Theelectricfield()isassketched below. 8 8 8 8 8 x x x x x (a) (b) (c) (d) (e)

--- Page 208 --- SUPPLEMENTANDREVIEW 185 (15) Do equilibrium conditions prevail? (a) Yes (b) No (c) Can’t be determined (16) The semiconductor is degenerate (a) Near x = 0 (b) For L/3≤x≤2L/3 (c) Nearx= L (d) Nowhere (17) Atx=x2，p=? (a) 7.63×106/cm³ (b) 1.35×1013/cm3 (c) 1010/cm^³} (d)1.72×$1016/c{m}^3（18)Theelectroncurrentdensity（J）flowingat=is(a)Zero(b)$μnn₁E_/LL (c) -μnn₁EG/L (d) DN[n(x2)-n(0)]/L （19)Theholedrifcurrentdensity（Jdrif）lowingatis (a) Zero (b) μ_pn₁{/L (c) -μpn₁EG/L (d) qpND(kT/q)/L (20)Thekineticenergyofthehole shownonthediagrmis (a) Ev (b) -_/3 (c) _3$ (d) Zero

--- Page 209 --- 186 SEMICONDUCTORFUNDAMENTALS III.ProblemSolving Auniformly doped n-typesilicon barof length Lismaintained atroom temperature under ∆pn0≤n0,and there are no“otherprocesses”(including photogeneration)occurring in side the bar. Δp{n（O）=△p{n0Δp_n(L）=0Silicon 0 L (21) Determine p0,the equilibrium hole concentration. (a) P_{0≈1010/cm^{³ (b) p0 \101{1}/}cm3$(c)p0\approx$10³}/cm^{³ (d) Not given (22) Doeslow-levelinjectionexistundersteadystatecondition? (a) Yes (b) No (c) Can’t tell (23) The equation (simplest form possible) one must solveto determine ∆() is d^{2}}{} (a) 0 = {d{^{2}dΔp{}Δ_{n}$(b)1pp$a△p{na{2△}{}Δ{}$ (c)

D{}$1e$ax^{2} ∂p{n a²2△}_{ (d)

D{p}$\partial t$∂x^{2}d^{}}{}Δ_{n} (e) 0 = {_} d_{^{2} TP

--- Page 210 --- SUPPLEMENT ANDREVIEW 187 (24) Thegeneral solutionfor ∆(x)will be(L2= Dpp) (a) Aexp(-x/Lp)+Bexp(x/Lp) R1 (b) A exp(-t/{p) (c) A+Bx (d) Aexp(-x/Lp) (25) The boundary conditions to be employed in solving for Δpn(x)are (a) ∆pn(0)=∆pn0 and ∆pn（∞)=0 (b)∆pn（O)=101/cm3and∆pn（∞）)=0 (c) ∆pn（0)=∆pn0 and∆pn(L)=0 0=()d pueu/0=(0)“（p) Answers-Set A Problem A1 (a) (111) (b) [211] (c) (√3/4)a (d)OneoftheSiatomsintheunitcellistobereplacedwithanon-Siatomexhbiting extra bond (representing a weakly bound 5th valence electron). Problem A2 Parameter Value Units Explanation $EG$ 1.12 eV 1010 cm- 3 $n_{}$ no$1017cm-3$n{0=ND(ND≥n₁) Po10{ $cm-3$P{ =n{2DEF-E_}$0.417eVEF-E=kTln(ND/n)=(0.0259)ln(1017/1010)\rho (chargeChargeneutralityprevailsunderthe0coul/c{m}^{3}density)specifiedconditions.801c{m}^2/V-secFromFig.3.5$μ_{n$

--- Page 211 --- 188 SEMICONDUCTORFUNDAMENTALS ρ(resis- 1 1 0.078 ohm-cm tivity) qμnND(1.6×10-19)(801)(10{17) R1 LP 2.93×10-3 cm L=√Dpp=√(8.57)(10-6) DP$8.57c{m}^{²}/secP=(kT/q)p=(0.0259)(331)ProblemA3(a)Eislocatednearmidgap,whileFmustbepositionedbelowE+3kT.EcED-E,ETEA$E_v}$EF(b)Underequilibriumconditions,weknowthatEFisposition-independent.Since=（1/q)(dE/dx）=(1/q)(dE/dx）=（1/q)(dEdx),fromtheversusxplot,wededucethatthebandsareflatfor0\le x\le x_{3}and\le x\le Inthex\le x\le xregionthebandsmustexhibitaconstantnegativeslope.Also,for0\le x\le a，E-EF=kTln(NA/n）=（0.0259)ln（1016/1010)）=0.358eV；for\le x\le c，EF-E_1=kTln(ND/n）=0.358eV.Combiningthecitedinformation,oneconcludestheenergybanddiagramisassketchedbelow.0.358eVEcT0.358eVEi$ Ev}$ 1 1 q q q

--- Page 212 --- 189 SUPPLEMENTANDREVIEW Problem A4 (a) Δp(t) R1 2△pf$△pfT\Delta pf$ o tf (b)Yes.∆max≤2Δ=2×1010/cm3≤n0=N=101/cm3.Thus△p≤n0asrequired for low-level injection. (c)Inthegivenproblem,theminoritycarrierdifusionequationsimplifiesto dΔ}_{n} △P_{n}=}0 + 1p Tp ForO≤t≤t.. . ∆p(0)=∆f …boundary condition ∆p(t)=Ae- t/p …general solution A = Δf∆p(t)=∆ e-t/p answer Fort≥t{. . . uonpuoApunoq·（1-∂+[∇=₂d+∇=3) ∆p(t)= Ae-(t-t)/p ...general solution A=∆p(1+e-τp) ∆p(t)=∆(1+e-/τp）e-(t-t）/τp answer (d) σ=1/ρ=q(μnn +pp)(0-)=qnD . .n0 =NDandn_0≥P0 d（d+“m）b+Nb=[(d∇+0d）+（u∇+0u）“]b=（+0)0 ${△}_f$ Δ_σ=q（μn+p)Δf μ_} $\sigma (0-)$qμ{nD$

1 + μ_{n $ND$ answer symbolic =(1 458 1010

1.34× 10-5 answer numeric \345 $\text{0}\text{sr}$

--- Page 213 --- 190 SEMICONDUCTORFUNDAMENTALS Answers-Set B (1) b. ..1/2×(2face atoms)+1/4×(4edge atoms)=2 (2) b (3) e.. .1, -2,3 -intercepts 1,-1/2,1/3

• 1/intercepts (632) —indices (4)d…Anydirectionwithanonzerofirstintegerwillnotbeperpendiculartothe[100] direction. (5) b…From the statement of the problem, we conclude 1 f(Ec+kT)= = e-10 1+e(Ec+kT-EF)/kT E-F=kT[ln(e10−1)−1]≈9kT (6) b (7)c…For nondegenerate dopings DN=(kT/q)n andn versus NDhas the formof sketch (c).(See Fig. 3.5.) (8) DONOR with 5 valence electrons (9) $Ec$ ED Ev (10) 1 ρ = qμ{nD .n-type 1 p = $q_p$ ..p-type Typically,p<nfor a given semiconductor andtemperature. Thus ρ(p-type)is greater than ρ(n-type) if ND = NA.

--- Page 214 --- SUPPLEMENTANDREVIEW 191 (11) Ec$----ET$Ev (12)Driftvelocityistheaverageelectronorholevelocityresultingfromanappliedelectric field.(One mightalso mention that at lowfields =,whereis the carrier mobility.) (13) d…Vhas the same shape as the “upside-down”of the bands. (14) e…is proportional to the slope of the bands. (15) a…EF is invariant with position. (16) c…EF approaches Enearx=L. (17) b… E-EF=G2-G/3=E/6 p=n;e(E-EF)/kT=1010e1.12/(6×0.0259)=1.35×101³3/cm³ (18) a . . .Under equilibrium conditions JN= 0. (19) b... JPldrift= qμp At x= x_{_} p= n_i}$(Ei=F)$1dE{}1△$1(EG/3)$E}$8=xpbq\Delta x$ 9 (L/3) qL Pldrif=μ_nEL (20) c. ..K.E.=E(L)-Ehole=EG/3 (21)c…n0≈ND=101/cm³ P0=n²/n0=10²0/1015=10³/cm³ (22) a. ..∆pn|max=∆pn0≤n0

--- Page 215 --- 192 SEMICONDUCTORFUNDAMENTALS (23)e…Steadystateconditionsandtheabsenceofphotogenerationalowonetoelimi- natethe/tandGtermsintheminoritycarrierdiffusionequation(Eq.3.54b). (24) a…This is Solution No.1cited in Subsection 3.4.3. (25)c

--- Page 216 --- ＰＡRTIIＡ JUNCTION DIODES pn 一

--- Page 217 ---

--- Page 218 --- 5 pn Junction Electrostatics In PartII we primarily examine devices whoseoperation is intimately tied to theoneor morepn junctionsbuiltintothestructure.Considerableattentionis initiallygiventothe chartermember of the device family,the pn junction diode. The diode analysis in PartIIA is of particular importance because it establishes basic conceptsand analytical procedures that are of universal utility. Special pn junction diodes are described at theendof PartIIA in the“read-only”chapteron optoelectronic diodes.In PartIIBtheconceptsand proce- dures are appliedand somewhat expanded intreating another important memberofthe family,thetwo-junction,three-terminal,bipolar junctiontransistor(BJT).Deviceswith more thantwo pn junctions are brieflyconsidered in thechapteronPNPN devices.The concluding chapter inPartIIBtreatsmetal-semiconductorcontactsandtheSchottky di- ode, a “cousin” of the pn junction diode. Acomplete,systematic deviceanalysis is typically divided intofourmajor segments. Providingthefoundationfortheentireanalysis,thefirsttreatsthecharge density,theelec- tricfield,andtheelectrostaticpotential-collectivelyreferred toastheelectrostatics— existing insidethe device underequilibriumandsteadystateconditions.Subsequentseg- ments are inturn devotedtomodelingthesteady state(d.c.)response,the small-signal(a.c.) response,andthetransient(pulsed)responseof thedevice.The pn junctiondiode analysis to be presentedfollows the notedfour-step development.In this chapter we takethe first step by exploring the electrostatics associated with a pn junction. 5.1 PRELIMINARIES 5.1.1Junction Terminology/IdealizedProfiles Suppose for the sakeof discussion that a pn junctionhas beenformed by diffusing a p-typedopantintoauniformly doped n-type wafer.The assumed situation is pictured in Fig.5.1(a).Inthe near-surfaceregion where the in-diffused NA>ND,the semiconductor is obviously p-type. Deeper in the semiconductor where Np>NA,the semiconductor is n-type.Clearly,the dividing line betweenthetwo regions,known formally asthe metallur- gical junction,occurs at the plane in the semiconductor where ND-NA=0. Note thatonly the net doping is relevant in determining the positionof the metallur- gical junction.Onlythe netdoping concentration is likewise required inestablishingthe electrostatic variables.Thus,insteadof presenting superimposed plotsof NAand NDversus xas in Fig.5.1(a),it is more usefultocombine NAand NDinformation into a single 195

--- Page 219 --- 196 pn JUNCTION DIODES NA NA(diffused) 10 ND ND(original) 1 →x 1 1 1 P N ↑ Metallurgical junction 1 (ND-NA=0) 1 1 1 1 1 (a) $ND-NA$ ↑ 1 1 1 1 1 1 1 1 1 (b) Figure5.1Junctiondefinitions:(aLocationofthemetallugicaljunctionb)dopingprofilea plot of the net doping versus position.

--- Page 220 --- pn JUNCTION ELECTROSTATICS 197 $ND-NA$ $ND-NAA$ ↑ x x Actual profile Actual profile Step junction Linearlygraded junction idealization idealization (a) (b) Figure.2Idealizeddopingprofles:a)Stepjunction(b)linearlgradedjncion. ND-NAversusxplotasillustratedinFig.5.1(b).Aplotofthenet dopingconcentration as a function of position is referred to as the doping profile. When incorporated into theanalysis,theactual doping profiles createdby the com- monplace diffusionand ion implantation processesdrasticallycomplicate the mathematics. Itbecomesdifficult toobtain and interpret results.Fortunately,only the doping variation in the immediate vicinityof themetallurgical junction is of prime importance.Surprisingly accurate results can beobtained using rather idealized doping profiles. The two mostcom- monidealizationarethestepjunctionandthelinearlygradedjunctionprofilesgraphically defined inFig.5.2.Themore appropriateof thetwoidealizationsdependsontheslopeof the actual doping profile at the metallurgical junction and the background doping of the starting wafer. The step junction is an acceptable approximation to an ion-implantation or shallow diffusion into a lightly dopedstarting waferwhereas the linearly graded profile would bemoreappropriatefor deep diffusions intoa moderateto heavily doped starting wafer.Inmost pn junctionanalysespresentedherein,wearbitrarilyinvokethestep junc- tion idealization to minimize the mathematical complexity of the analysis. 5.1.2Poisson’s Equation Poisson’s equation is a well-known relationship from Electricity andMagnetism.In semi- conductorworkitoftenconstitutesthestartingpointinobtainingquantitativesolutionsfor the electrostatic variables. Thethree-dimensional version appropriate for semiconductor analyses is ∇.8= ρ (5.1) K{ε0

--- Page 221 --- 198 pn JUNCTION DIODES In one-dimensional problems where=Ex,Poisson’s equation simplifies to d8 ρ (5.2) dx $K_{ε0 Kis thesemiconductordielectricconstantand ε0is the permittivityoffreespace.ρ,pre- viouslyassociated withresistivity,is understoodtobethecharge density(charge/cm)in analyses involving theelectrostatic variables.Assuming the dopants to betotally ionized, the charge density inside a semiconductor is given by ρ= q(p-n+ND-A) (5.3) Eq.(5.3)wasoriginallypresentedasthefirst portionofEq.(2.23)in derivingthecharge neutralityrelationship.Thecharge density isidentically zerofarfromany surfaces inside a uniformly doped semiconductor in equilibrium. However,ρis often nonzero and a func- tion of position under less restrictive conditions. Lastly notefrom Eq.(5.2)thatρis proportional to d ldx in one-dimensional prob- lems.The general functional form ofρ versus x can therefore be deduced from an-xplot by simply noting the slope of theplot as a function of position. 5.1.3QualitativeSolution Prior to performing a quantitativeanalysis,it is always useful to have a general ideaas to the expected formof the solution.Basedonthe Subsection 3.2.4 discussion leading to the derivationof the Einstein relationship,we already knowthere should be band bending and an internalelectric fieldassociated withthe inherently nonuniform doping ofa pn junction diode.Let us assume a one-dimensional step junction and equilibrium conditions in seek- ing to determinethegeneralfunctionalformofthepotential,electricfield,and charge density inside the diode. Ourapproach willbeto firstconstruct the energyband diagram fora pnjunction diode underequilibrium conditionsandthen to utilize previously established procedures in de- ducing theelectrostatic variables.For theassumed stepjunctionit is reasonable to expect regions farremovedfrom the metallurgical junction to be identical in character to an iso- lated,uniformly doped semiconductor.Thus,theenergy band diagrams for the regions far removedfrom the junction are concluded to beofthe simpleform shown inFig.5.3(a). Underequilibriumconditions weknow in additionthatthe Fermi level is a constantinde- pendentofposition.This leadstoFig.5.3(b),wherethediagramsofFig.5.3(a)areproperly aligned to the position-independentFermilevel. Themissing near-junction portionofthe Fig,5.3(b)diagramiscompleted inFig.5.3(c）byconnecting theEE,andEendpoints on the two sidesof the junction.Although the exact form of the band bending nearthe

--- Page 222 --- pn JUNCTION ELECTROSTATICS 199 $ND-NA\uparrow NDx-NA$Ec$EF$ $Ei$ Ev} p-side n-side (a) $Ece$ Ev} (b) Ec EF$ Ei Ev (c) Figure5.3 Step-by-step constructionof theequilibriumenergy band diagramfor a pn junction diode.(a)Assumed step junction profile and energy band diagrams for the semiconductor regions far removedfrom the metallurgical junction.(b)Alignmentof the part (a) diagrams to theposition- independent Fermi level.(c) The completed energy band diagram.

--- Page 223 --- 200 PnJUNCTIONDIODES metallurgical boundary is notknown,it isreasonabletoassumethevariation ismonotonic in nature,with a zero slopeatthetwoendsofthecentralregion.Figure5.3(c)isofcourse the desired equilibrium energy band diagram for a pn junction diode. Itis now a relatively simple matter to deduce the functional form of theelectrostatic variables.Forone,referring to Subsection 3.1.5onbandbending,theV versus x relation- ship musthave the same functionalformas the“upside-down”ofE（or Eor E).This leads to theV versus x dependence sketched in Fig.5.4(b)withVarbitrarily setequal to zero on the far p-side of the junction.Next,the versus x dependence shown in Fig.5.4(c) is obtained byrecording the graphical derivativeofEcas a functionof position.Finally, the generalfunctionalformofρversussketchedin Fig.5.4(d)is deducedfromthe slope of the -x plot. PerhapsthemostinterestingfeaturesoftheFig.5.4solutionarethevoltagedropacrs the junction underequilibrium conditionsand theappearance ofcharge nearthe metallur- gicalboundary.The”built-involtage（Vb）willbegivenseparateconsiderationinthe next subsection.Ofimmediate interest is the regionofcharge near the junction pictured in Fig.5.4(d).The question arises,Where does this charge come from?Answering this ques- tion provides considerable physical insight. Suppose thatthe p-andn-regions were initiallyseparated as pictured in Fig.5.5a). Charge neutrality is assumedto prevail inthe isolated,uniformly doped semiconductors. In the p-material thepositive hole charges,the +‘s in Fig.5.5,balance the immobile acceptor-sitecharges shown as’sin Fig.5.5.Likewise,in then-materialtheelectronic charge()everywhere balances the immobilecharge associated with the ionized donors (田).Next suppose astructurallyperfectconnection ismadebetweenthepand n materials as envisioned in Fig.5.5(b).Naturally,sincethere are many more holeson the p-sidethan on the n-side,holes beginto diffusefrom the p-sideto then-side an instant afterthe connection is made.Similarly,electrons diffuse from then-side to the p-sideof the junc- tion. Although theelectrons andholes can move to theopposite side of the junction,the donors and acceptors are fixed in space.Consequently,the diffusing awayof thecarriers from the near-vicinityof the junction leavesbehind an unbalanced dopant site charge as shown inFig.5.5(c).This is thesourceof thechargearoundthemetallurgical junction, nicely correlating with the previously deduced p versus x dependence redrawn in Fig.5.5(d).The near-vicinityof themetallurgicaljunction wherethere is asignificantnon- zero charge is called the spacecharge region or depletion region. The latter name follows fromthefactthatthecarrierconcentrationsintheregionaregreatlyreducedor deeted. We should also mention that the build-up ofcharge and the associated electric field contin- ues untilthe diffusion of carriers across the junction is preciselybalanced by the carrier drift.The individualcarrier diffusionand driftcomponentsmustofcourse cancel tomake JN and Jp separately zero underequilibriumconditions. ActuallyfromtheFig.5.4a)energybanddiagramonecanconcludeonlythatthemagnitudeoffirstincreases from zero on the p-sideof the junction,reaches a maximum near the metallurgical boundary,and then decreases againto zerofaronthen-sideofthejunction.Thepseudo-lineardependencesketchedinFig.5.4c)rlecsan advance knowledge of the quantitative solution.

--- Page 224 --- pn JUNCTION ELECTROSTATICS 201 $qbi$ Ec EF $EiEv$ (a) V Vbi →x - x{pq 0 $x_{}$ (b) 8 ↑ →x (c) a← →x (d) Figure5.4 Generalfunctionalformof theelectrostatic variables in a pn junctionunderequilib- and (d) charge density as a function of position.

--- Page 225 --- 202 pnJUNCTIONDIODES P N 曰 曰⊕ 田 田 田 田 曰 曰⊕ 曰⊕ 田 田 田 田 + 曰⊕ 曰⊕ 田 田 田 田 曰 + 曰 曰⊕ 田 田 (a) ← 曰 曰 曰 田 田 田 田 曰⊕ 曰④ 田 田 田 田 曰⊕ 曰⊕ 田 田 田 田 田 田 田 + (b) ← 8 曰⊕ 曰⊕ 日⊕： 曰 田 田 田 日 田 田 田 田 曰： 曰 田 田 田 田 曰⊕ 曰⊕ 曰 田 田 田 田 (c) ρ + →x (d) Figure5.5 Conceptual pn junction formationand associated charge redistribution.(a)Isolated p and n regions.(b)Electrons andholes diffuse to theopposite sideof the junction moments after joining the p and n regions.(c)Charge redistribution completed and equilibrium conditions re-estab- lished. (d) Previously deduced charge density versus position.(⊕-holes,-ionized acceptors, -electrons,and田-ionized donors.)

--- Page 226 --- pn JUNCTION ELECTROSTATICS 203 5.1.4 The Built-in Potential(Vbi) The voltage dropacrossthe depletionregionunderequilibriumconditions,knownasthe built-inpotential（Vb),isa junctionparameterof sufficientimportance tomeritfurther consideration.Weare particularly interested inestablishing a computationalrelationship for Vi.Working toward the stated goal,we consider a nondegenerately-doped pn junction maintained under equilibrium conditions with x =0positioned at the metallurgical boundary. The endsof the equilibrium depletion region are taken to occur at —xp and n on the p-and n-sides of the junction respectively (see Fig.5.4b). Proceeding with the derivation, we know dV 8= (5.4) dx Integrating across the depletion region gives {n ${}_{(}x)$ 8 dx = dV= V(n)-V（-p)=Vbib$(5.5)-xp$JV(-x{) Furthermore,underequilibriumconditions, dn JN=qμnn+N =0 (5.6) dx Solving forin Eq.(5.6)andmaking useofthe Einsteinrelationship,weobtain D{N dnldx kTdnldx = 3 (5.7) μ_{n}$nq9nSubstitutingEq.(5.7)intoEq.(5.5),andcompletingtheintegrationthenyields$xn$kT$fn(x)$dnkT$n(x{}V_{bi$ 8dx ln (5.8)

= n(-x{) -x_{p q J{n-x{$nqForthespecificcaseofanondegeneratelydopedstepjunctionwhereNDandNAarethen-andp-sidedopingconcentrations,oneidentifies$n(x=D$(5.9a)$n^{fra{}{2}n(-x{) =$(5.9b)N$

--- Page 227 --- 204 Pn JUNCTION DIODES and therefore $(NAD$ kT \_bi}= ln (5.10) n^{fra{}{2} q It is useful to perform a samplecomputation to gauge the relative magnitude of the built-involtage.ChoosingNA=ND=10/cm3andaSidiodemaintainedat300K,one computesV=（0.0259)ln（10/10²0)0.6V.Thisisatypicalresult.Innondegenerately doped diodesVi<E/q,orViislessthanthebandgapenergyconvertedto volts.Non- degeneratlydopedGei,andGAsdiodesmtinedtoomtematurexhibiti less than0.66V,1.12V,and1.42V,respectively. Therelationship between Vi andEGalludedto inthepreceding paragraph is nicely explainedbyanalternativeViderivation basedontheenergyband diagram.Referingto Figs.5.4(a) and (b), we note bi=V(n）-V（- (5.11a) 1 (5.11b) q or 1 V_{bi

一 [(EF)p-side+(F-)n-side] (5.12) q Clearly,foranondegeneratelydopeddiodeboth（E-F)p-sideand(F-n-side ae less thanEG/2,making Vbi<EG/q.Moreover,in a nondegenerately doped step junction under equilibrium conditions, (E₁-EF)p-side=kTln(NA/n) (5.13a) (EF −Ei)n-side =kTln(ND/n) (5.13b) Substituting Eqs.(5.13)into Eq.(5.12)and simplifying yields Eq.(5.10).Note that,while Eqs.(5.13)andhence Eq.(5.10)arevalidonlyfornondegenerate dopings,thereareno doping-related restrictions on the validityof Eq.(5.12).

--- Page 228 --- pn JUNCTION ELECTROSTATICS 205 (C) Exercise 5.1 P:Most real diodes are very heavily doped onone side of the junction.In computing thebuilt-in voltageofp+-nandn+-pstepjunctions,itiscommonpracticetoasume that the Fermi levelontheheavilydopedside is positionedat the bandedge;ie., EF=Eina+materialandE=inann+material.Makingthecitedassump- tion,compute and plot Vi as a functionof the doping（NAor ND)on the lightly doped sideofSi p+-n and n+-p step junctions maintained at 300K.The plot is to covertherange10114/cm3≤NAor ND≤1017/cm{3. (E-F)p-side assumed=E-E=/2 (EF − Ei)n-side = kT ln(Dn;) Substituting into Eq.(5.12),which is validfor arbitrary dopinglevels,werapidly conclude EG kT a(N V_{bi 2q + ln q n_i} Forn+-p junctions,NA simplyreplaces ND,yielding a computationally equivalent relationship.TheVbicomputational programandresultantplot(Fig.E5.1)follow: MATLAB program script… %VbiComputation (p+/n and n+/p junctions) %Constants EG=1.12; kT=0.0259; ni=1.0e10; %Computation ND=logspace(14,17); Vbi=EG/2+kT.*log(ND./ni); %Plotting close semilogx(ND,Vbi); grid axis([1.0e141.0e170.751]) xlabel(‘NAor ND(cm-3)); ylabel(‘Vbi (volts)’) text(1e16,0.8,‘Si,00K’) text(1e16,0.78,‘p+/n and n+/p diodes’)

--- Page 229 --- 206 pn JUNCTION DIODES 1 ………… …… ……………… … ………… ………… …… … 0.95 ………… …………… … 0.9 Vbi (volts) 0.85 Si,300K 0.8 p+/n and n+/p diodes 0.75 1014 1015 1016 1017 NA or ND (cm-3) Figure E5.1Built-in voltage inSip+-nandn+-pstepjunction diodesat300K. 5.1.5 The Depletion Approximation Toobtainaquantitativesolutionfortheelectrostaticvariables,itwillbenecessaryto solve Poisson’s equation. The depletionapproximationfacilitatesobtaining closed-form solu- tions to theequation.Appearing in a myriadofdevice analyses,the depletion approxima- tion is farand away themost important andmost widelyencounteredof thesimplifying approximations used in the modeling of devices. To understand why the depletionapproximation is introduced,considerthe one- dimensionalPoissonequation rewrittenas Eq.(5.14).Thedoping profile,ND-NA,ap- pearing in the equation de ρ (5.14) dx $Ks0$K{sε0 is assumed to be known.However,to write downρ as a function of x,solve the differential equation for versus x,and eventuallyobtain V versus x,one mustalso have explicit expressions for thecarrierconcentrations as a functionof x.Unfortunately,the carrier concentrations in ≠0 regions,like the pn junction depletion region, are not specified priorto solving Poisson’sequation.Rather,pand ninthe depletionregionare functionsof the potential as is obvious from anexaminationofthe Fig.5.4(a)energy band diagram. Although exactclosed-form (usuallycomplex)solutions do exist in certain instances,the

--- Page 230 --- pn JUNCTION ELECTROSTATICS 207 depletionapproximationprovides a simple,nearlyuniversal wayofobtainingapproximate solutions without prior knowledge of the carrier concentrations. We havealready established the basis forthe depletionapproximation.Aprominent featureofthequalitativesolution wastheappearanceofanonzerochargedensitystraddling the metallurgical junction.Thischargearises because thecarriernumbers arereduced by diffusionacross thejunction.Thecarrier”depletion”tendsto be greatest in the immediate vicinityof the metallurgical boundary and thentailsoff as one proceeds away from the junction. The depletion approximation introduces an idealizationof the actual charge dis- tribution. Theapproximation has twocomponents thatcan be statedas follows:(1)The carrierconcentrations are assumed to be negligible comparedto the netdoping concentra- tioninaregion-x≤x≤nstraddlingthemetalurgicaljunction.（2)Thechargedensity outsidethedepletionregionistakentobeidenticallyzero.Thedepletionapproximationis summarizedpictoriallyinFig.5.6(a）and isillustratedassuming astepjunctionin ρ=q(ND−A) ρ=0 ρ= 0 P N x - x{q 0 $x_{}$ (a) ρ ↑ + - x{p 0 →x $x_n$ Exact Depl.Approx. (b) Figure5.6(a)Pictorial summaryofthedepletionapproximation.(b)Illustrationoftheapproxi- mation as applied to a step junction.

--- Page 231 --- 208 pn JUNCTIONDIODES Fig.5.6(b).Whenthe depletionapproximation is invoked,theone-dimensionalPoisson equation simplifies to (ND- d K{0 …-p≤x≤x (5.15a) dx 0 . ..x≤-xp andx ≥x_n (5.15b) Note that,except forthe valuesof-xand ,the charge density is totally specified by invoking the depletionapproximation.Moreover,thecharge density willhave precisely $ND-NA\rho \pi \uparrow$- x{$+\to x\to x$x_{}$(a)$ND-NAA ↑ - x{r + →x x x_{ (b) Figure5.7Relationshipbetweenthedopingprofleandthechargedensitybasedonthedepletion approximation. (a) and (b) are two separate examples.

--- Page 232 --- pn JUNCTION ELECTROSTATICS 209 the same functionalform as ND-NA within the depletion region.Figure 5.7is presented to emphasize this last pointby showing theρversus plots associated with sampledoping profiles of a moderately complex nature. 5.2 QUANTITATIVEELECTROSTATICRELATIONSHIPS With the preliminaries out of the way,we proceed tothe main event: development of quan- titativerelationshipsfortheelectrostaticvariables.Thedevelopmentprimarilydealswth thestep junctionandispresentedindetailnotonlytoobtainthedesiredrelationshisu also to establishthe derivational procedures thatcan beappliedto other doping profiles. We begin by carefully specifying the system underanalysis,turn to the step junction under equilibrium conditions,and thenconsiderthe modifications required when there is an ap- pliedbias.Thesectionconcludes with a concise derivation and examinationof theelectro- static relationships appropriate for a linearly graded junction. 5.2.1 Assumptions/Definitions Major featuresofthe pn junction diode subjectto analysis are identified in Fig.5.8.All variablesare taken to be functions onlyofx,thecoordinate normal to the semiconductor surface.The device is thus said to be”one-dimensional”;obvious two-dimensional effects associated with the lateral ends ofa real device are assumed to be negligible.Fortheelec- trostaticanalysis,x=0is positioned at the metallurgicalboundary.Externalcontactsto the endsof thediode are specifiedtobe“ohmic”in nature.By definition,a negligible portionofanexternallyappliedvoltage appearsacross anohmic contact.Notethatthe 1 p-silicon +

Ohmic contacts P n-silicon n-silicon P 1 N Area(cm^{) 1 1 x=$ 0 x (a) (b) Figure5.8(a)Sketchofaphysicaldiode.(b)One-dimensionaldiodesubjecttoanalysisincluding the applied voltage,coordinate,and contact specifications.

--- Page 233 --- 210 Pn JUNCTIONDIODES symbolVAisusedfortheappliedvoltage.ThesubscriptAdistinguishestheappliedvoltage from the internal junctionvoltage.IntheinitialdevelopmentVAis setequalto zero,or equivalently,the device is assumed to be in equilibrium. 5.2.2 Step Junction with VA=0 Solution for p Weconsiderastepjunctionunderequilibriumconditions.NAisdrawngreaterthanNDin the Fig.5.9(a)sketchofthedopingprofile forthe sakeof illustration.As summarized in Fig.5.9(b),invokingthedepletionapproximationyieldsthecharge densitysolution .. .-xp≤x≤0 (5.16a) ρ

qND ...0≤x≤xn (5.16b) 0 . ..x≤-xp a ndx ≥x {n (5.16c) Thevaluesofxand xpare notknown at this pointbut willbe determinedlaterinthe analysis. Solution for Substitutingthecharge densitysolutionintoPoisson’sequationgivestheequationstobe solved for the electric field. -qA/ε0 ·..-xp≤x≤0 (5.17a) 3p qND/ε0 …0≤x≤xn (5.17b) dx 0 . ..x≤-p an dx≥_ (5.17c) =0farfrom the metallurgical boundaryand therefore=0everywhereoutsideof the depletion region.Since must also vanish right at the edges of the depletion region,= Oatx=-xpand=0atx=nrespectively become the boundary conditions for the (5.17a)and(5.17b)differentialequations.Separating variablesandintegratingfromthe depletionregion edge to anarbitrarypoint x,one obtains for the p-side of the depletion region qNadx'} x (5.18) -xp$or8(x)=-..-xp\le x\le 0$ (5.19) K{0

--- Page 234 --- pn JUNCTION ELECTROSTATICS 211 ↑ND-NAA ND →x N- (a) ↑p $qND-xp$ →x x_} (b) 18 - x{p 0 $x_{}$ →x (c) ↑V →x - x{q $x_n$ (d) Figure5.9 Step junctionsolution. Depletionapproximationbasedquantitative solutionforthe electrostaticvariablesinapnstepjunctionunderequilibriumconditions(VA=0).(a)Stepjunction profile.b)Chargedensity,(c)electricfield,andd)lectrostaticpotential asafunctionofpositon.

--- Page 235 --- 212 pn JUNCTION DIODES Similarly on the n-side {^{d{= $x^n$qD} -d^'} (5.20) K{0 J(x) x or qND(x{n$ =(x)3

• x) …0≤x≤x (5.21) K{sε0 Theelectricfieldsolutionis plotted inFig.5.9(c).Thisresultisofcourseconsistent withthequalitativesolutionsketchedinFig.5.4(c).Withinthedepletionregionthefieldis always negativeandexhibits a linear variation withposition.It shouldbenotedthatin constructing Fig.5.9(c)theelectric field was taken to becontinuous atx=0;the p-and n-side solutions were simplymatched atthe metallurgical boundary.FromElectricityand Magnetism,weknow theelectric field willbe continuous across a boundary as long as a sheetofcharge doesnot liealong the interface betweenthetwo regions.Ifthe(5.19)and (5.21)expressionsfortheelectricfieldareevaluatedat=0andequated,thecontinuity of the electric field is found to require NAx =N{ (5.22) Forthose familiar with Gauss'law,the factthat=0outside the depletion region means the totalcharge within the depletion region must sum to zero,or the minus charge on the p-side ofthe junction must balancethe pluscharge onthe n-sideof the junction.Previous charge densityplots(Figs.5.4-5.7)haveallbeendrawn withthisfact inmind.Appliedto the step junction ρ versus x plot in Fig.5.9(b),the balance of charge requires the rectan- gular areas on the p-and n-sidesof the junction to be equal,or qNAxp=qNDn.Thus Eq.(5.22)maybeviewedalternativelyasareflectionofthefactthatthetotalchargewithin the depletion region must sum to zero. Solution for V Since&=-dVldx,theelectrostatic potential is obtained by solving ...-xp≤x≤0 (5.23a) dV K{ε0 dxqND$ …0≤x≤x (5.23b) With thearbitrary reference potential setequal to zero atx=-xp,and remembering the voltage drop is Vbiacross the depletion region underequilibrium conditions,Eqs.(5.23a) and (5.23b)are respectively subjectto the boundary conditions

--- Page 236 --- pn JUNCTION ELECTROSTATICS 213 V=0 at x=-xp (5.24a) V=Vbi at x=x_n$(5.24b)Separatinviablannrnromthedelioeioeetoarbraoneobtainsforthep-sideofthedepletionregionx(5.25)$K{ε0 -x{pq JO $V(x)=$ . ..-xp≤x≤0 (5.26) $2Ks0$ Similarly on the n-side of the junction {_{{}= ^{n (5.27) $JV(x)$K_{s0$xor$qNDV(x)= bi- （(x{n-$x^{}$ …0≤x≤xn (5.28) 2K{ε0 TheelectrostaticpotentialsolutiongivenbyEqs.(5.26)and(5.28)isplottedin Fig.5.9(d).TheV versus x dependence is quadratic in nature,with a concave curvature on the p-sideof the junction and a convex curvature on the n-sideof the junction.Paralleling the &-field procedure,Fig.5.9(d) was constructed simply by matching the p-andn-side solutionsat=0.Theassumedcontinuityoftheelectrostaticpotentialatx=Oisjustified because there is nodipole layer (closely spaced sheetsof plus and minus charge) along the metallurgicalboundary.Notethat if the(5.26)and (5.28)expressions for the potential are evaluated at x= 0 and equated, one obtains qNA $qND$ 2K{ε0 x p 2

V_{bi$ 2Ks0 x 2 n (5.29) Solution for and xp Theelectrostatic solution is not complete until the valuesofnandxare determined.In the courseof the development, we have already laidthe groundwork forobtaining the n-and p-sidedepletion widths.Specifically,andxpare theonly unknownsin Eqs.(5.22)

--- Page 237 --- 214 pn JUNCTION DIODES and(5.29).Eliminatingxpin Eq.(5.29)usingEq.(5.22)and solving theresultingequation for xn rapidly yields 2K{ε0N{A}$71/2$x }$

$ib$ (5.30a) $ND(NA+D$ 9 and N{{} 厂 2K_{$ND$ 7 1/2 x{

$N_A

$NA(NA+$ $bib$ (5.30b) q It also follows that 2K_{sε0_bi 71/2 W=x_{ +_{

N{ND (5.31) q W,thetotalwidthof the depletionregionbetterknown simplyasthedepletion width,is often encountered in practical device computations. Exercise 5.2 P:Perform a sample computation to gauge the size of Wand|Imax underequilib- riumconditions.Specifically assumea Sistep junctionoperated at 300KwithNA= 1017/cm3and ND=1014/cm3 S: For the given junction, kT N{N{D (10{7)(10{4) ${}_{b}ib$

ln =(0.0259) ln =0.656V n^{fra{}{2} (10^{20) q Making use of Eqs.(5.30),one computes $2K_{s}OVbi$ 71/2 (2)(11.8)(8.85×10-14)(0.656) 1/2 $x_{}$ qD 二 (1.6×10-19)(1014) L =2.93× 10-4 cm=2.93μm (} x{

$\text{n}A$ x_} =(10-3)xn=2.93×10-7cm and W=x{ +xp =x{n=$ 2.93 μm

--- Page 238 --- pn JUNCTION ELECTROSTATICS 215 Also $qND$ (4-01×062)(101)(61-0×91) 1max=|8(0)|= $x_{}$ K{{ (11.8)(8.85 ×10-14) 4.49 × $10{3 V/cm 二 Wcould have been computed directly using Eq.(5.31).However,we wished to make the additional pointthat in an asymmetrically doped junction (NA>NDor ND> NA)the depletionregion liesalmostexclusivelyonthe lightly dopedsideofthe metallurgical boundary. 5.2.3 Step Junction with VA≠0 The solution fortheelectrostatic variables must beextended to VA≠0operating conditions ifit is to beof practical utility.One solution approach would be to pedanticallyrepeat the derivations from the previous subsection with VA≠0.Fortunately, there is an easier approach. Considerthe diode in Fig.5.10 with a voltage VA≠0 applied to the diodeterminals. This voltage must be dropped somewhere inside the diode.However,in a well-made device a negligible portionof the applied voltage appears across the contacts to the device.More- over,underlow-levelinjectionconditionsreasonablecurrentlevels)theresistivevoltage drop across thequasineutral p-and n-regionsextending from thecontactstothe edges of the depletionregion willalsobenegligible.Theappliedvoltage mustthereforebe dropped acrossthedepletionregion.WhenVA>0,thisexternallyimposed voltagedroplowersthe potentialonthen-sideof the junctionrelative to the p-sideofthe junction.Conversely, when VA<0,the potentialonthen-sideincreasesrelativetothe p-side.Inother words, VA dropped here Negligible voltage drop(low-level injection) Negligible voltage drop(low-level injection) / Negligible voltage drop Negligible voltage drop (ohmic contact) (ohmic contact) P N +VA Figure5.10Voltagedropsinternaltoadioderesultingfromanexternallyappliedvoltae.

--- Page 239 --- 216 PnJUNCTIONDIODES the voltage dropacrossthedepletionregionandhence theboundarycondition at=, becomes bi-VA. Since theonly modificationto theformationoftheproblemis a change inone boundarycondition,theVA≠0electrostaticrelationships can beextrapolatedfromthe VA=0relationships by simply replacing allexplicitappearancesof Vbiby Vbi-VA. Makingteinutuo ≠0 solution for the electrostatic variables given in Eqs. (5.32)-(5.38). $For-x\le x\le \mathrm{0...}$qN{(x{2}+x) 8(x)= (5.32) K_{sε0 $q{}_{(}x+xx$ V(x)$(5.33)$2{s0$2K_s$ N{ 1/2 x {q

$NAA$ N{D (5.34) q + $ForO\le$x ≤x_{ . . $qND(xnx)8(x)=$ (5.35) K_{0 qND(x_{n}-x^} V(x)=bi-V{A-$(5.36)$2K{s0$2K0$NA}$1/2$x_n}$

$NDNA+$ N{D (5.37) q and 2K_{sε0N{}+ND$1/2W=(5.38)$N{D L 9 TopreventanimaginaryresultVisobviouslyrestrictedtoVA≤Viin Eqs.(5.34),(5.37), and (5.38).The formulation fails because a large current begins to flow,andthe quasineu- tral region voltage drops cannot be neglected,when VA approaches Vbi.

--- Page 240 --- PnJUNCTION ELECTROSTATICS 217 (C) Exercise 5.3 P:Constructa log-logplotofthedepletion width（W)versustheimpurityconcentra- tion(NAor ND)onthe lightlydoped sideofSip+-n andn+-p stepjunctionsmain- tained at300K.Includecurves for VA=0.5V,0V,and-10Vcovering therange 10{4/cm3≤NAorND≤10/cm3 S:The V associated withthep+-nandn+-p stepjunctions iscomputed using the relationshipestablishedinExercise5.1.Also,withthe junctionasymmetrically doped,the doping factor in the Eq.(5.38)expression for W simplifies to N{A+ND 1 $NAND$ $aN where NB is the doping(NAor ND)on the lightly doped sideof the junction.The MATLAB programfor the W versus doping computation and theprogram results (Fig.E5.3)follow: MATLAB program script.. % This program calculates and plots the depletion width vs impurity %concentrationinSilicon p+/n andn+/p stepjunctions at300K. % % Three plots aregeneratedcorresponding toVA=0.5V,0.0Vand-10V % % The Vbi relationship employed is Vbi=(EG/2q)+(kT/q)ln(NB/ni) % where NB is the impurity concentration on the lightly doped side. %Constants and Parameters T=300; % Temperature in Kelvin k=8.617e-5; % Boltzmann constant (eV/K) e0=8.85e-14; % permittivity of free space (F/cm) q=1.602e-19; % charge on an electron (coul) KS=11.8; % dielectric constant of Si at 300K ni=1e10; %intrinsic carrier conc.in Silicon at 300K (cm^-3) EG=1.12; % band gap of Silicon (eV) %Choose variable values NB=logspace(14,17); % doping ranges from le14 to le17 VA=[0.50-10]; % VA values set

--- Page 241 --- 218 Pn JUNCTION DIODES %Depletion width calculation Vbi=EG/2+kT.log(NB./ni); W(1,:)=1.0e4sqrt(2KSeO/q.(Vbi-VA(1))./NB); W(2,:)=1.0e4sqrt(2KSeO/q.(Vbi-VA(2))./NB); W(3,:)=1.0e4sqrt(2KSe0/q.(Vbi-VA(3))./NB); %Plot close loglog(NB,W,-);grid axis([1.0e141.0e171.0e-11.0e1]) xlabel(‘NA or ND (cm^-3)’) ylabel(‘W (micrometers)’) set(gca,‘DefaultTextUnits’,‘normalized’) text(.38,.26,‘VA=0.5V’) text(.38,.50,‘VA=0’) text(.38,.76,‘VA=-10V’) text(.77,.82,‘Si,300K’) text(.77,.79,‘p+/n andn+/p’) set(gca,‘DefaultTextUnits’,‘data’) 10 S1300K d+u pire ui/+d. 10V W(micrometers) VA=0 100 VA=0.5V 10-1 1014 1015 1016 1017 NA or ND (cm^-3) FigureE5.3Depletion widthat selectappliedbiasesasafunctionofdoping(Si,p-and n+-p step junctions, 300 K).

--- Page 242 --- pn JUNCTION ELECTROSTATICS 219 5.2.4Examination/Extrapolation of Results Having expendedconsiderableefortinestablishingtheresults,itisreasonableto spenda few momentsexamining theresults. Weare particularly interested in howtheelectrostatic variableschange as a functionof the applied bias.Examining the (5.34)and (5.37)rela- tionships forand n,weconcludethese widths decrease underforwardbiasing(VA> 0)andincreaseunderrevesebiasingV<).Thisconclusionisofcourseconsistentwith the Exercise 5.3 computation showing a smallerdepletion widthfor VA>0and a larger depletionwidthforVA<0.Thechangesinxandlikewisetranslateintochangesinhe electricfield.Asdeducedfrom Eqs.(5.32)and(5.35),asmallerxand xunderforward biases cause the E-fieldto decrease everywhereinsidethe depletionregion,whilethe largerxpandnassociated withreversebiasesgiverisetoalarger-field.Thisconclusion is also reasonablefrom a physical standpoint.A decreased depletion width when VA>0 means less charge around the junctionand a correspondinglysmaller &-field.VA<0,on theotherhand,createsa largerspacechargeregionand a biggerelectricfield.Similarly, the potentialgivenby Eqs.(5.33)and(5.36)decreasesatallpoints whenV>0and increasesatallpointswhenVA<0.Thepotentialhillshrinksinbothsizeand x-extent under forwardbiasing,whereas reversebiasing gives rise to a widerand higher potential hill.The foregoing discussion is graphically summarized in Fig.5.11. In Subsection 3.1.5 we established a procedure for deducing the form of the electro- static potential from a given energy band diagram. The procedure was appliedin Subsection5.1.3toobtainthequalitativesolutionfortheelectrostaticpotntialinsidean junctionunderequilibrium conditions.Having just describedand envisioned (Fig.5.11d) how thepotential changes as a functionofbias,we should beable to reverse the cited procedure toconstruct theenergyband diagrams appropriate for a pn junction under for- ward andreverse bias. Specifically,weknow whattheenergyband diagramlookslike underequilibriumconditions(redrawn in Fig.5.12a).Conceptuallytakingthe upside-down of the potential plotsand appropriatelymodifyingtheequilibrium energy band diagram— smaller depletionwidthandsmallerhillforforwardbias,largerdepletionwidthandlarger hill forreverse bias—yields thediagramsforforward andreverse biasrespectivelypictured in Figs. 5.12(b) and (c). Severalcomments are inorderconcerningtheVA≠0diagrams.Forone,theFermi level is omittedfrom the depletion region because the device is no longer inequilibrium and a single levelcannotbe used to describethecarrierconcentrations in thisregion.In fact,thelevelslabledandccupitheormroitionofteFrvl quasineutral regions,are actually majority-carrierquasi-Fermilevels.However,thedevia- tion fromequilibrium inthenondepleted portionsofthediodeis normallysmall,especially far fromthe junction,andit isthereforeacceptabletocontinueusingtheEFdesignation. Finally,bycarefullyinspectingthediagrams,it isreadilyestablishedthat EFp -EFn =-VA (5.39) Equation (5.39)suggests one mayconceiveof the diode terminals as providing direct ac- cess to the p-and n-endsof the equilibrium Fermi level.Conceptuallygrabbingonto the

--- Page 243 --- 220 pn JUNCTION DIODES 1 1 1 P 1 1 1 1 N 1 1 1 1 1 K V{A<0 K -W—→VA=0 K →V{>0 (a) ↑p VA x VA }0$1$V_{A>0$(b)$V{>0$18VA=0$0>$ →x (c) ↑V -+-VA<0 +V{ }=0 ----VA>0 →x (d) Figure5.11Effectofforwardandreversebiasingonthe(a)depletionwidth,(b)chargedensity, (c)electric field,and (d)electrostatic potential inside a pn junction diode.

--- Page 244 --- pn JUNCTION ELECTROSTATICS 221 $Ec$ EiEv}$(a)Equilibrium(VA=0)EcEFpE$ Ev} (b) Forward bias (VA>0) EFp $EcEi$Ev$ (c) Reverse bias (VA<0) Figure5.12pnjunctionenergy banddiagrams.(a)Equilibrium(VA=0),(b)forwardbias(V> 0),and(c)reverse bias(VA<0). endsof the equilibrium Fermi level,one progresses from theequilibrium diagram to the forward bias diagram by moving the n-side upward by qVAwhile holding the p-side fixed. Similarly,the reverse bias diagram isobtainedfrom the equilibrium diagram by pulling the n-sideFermileveldownward. (C) Exercise 5.4 Once a quantitative relationship has beenestablishedforthe electrostatic potential,it becomes possibletoconstructafully dimensionedenergyband diagram.The”Dia- gram Generatorprogram thatfollows draws theequilibriumenergy band diagram

--- Page 245 --- 222 pn JUNCTION DIODES user is prompted to input the p-and n-side doping concentrations.Run the program trying different NAandNDcombinations.It is informative to includeatleastone combinationeach where NA>ND,NA=ND,andNA≤ND.Theasymmetrical junctions areof particular interest because theresultant”one-sided”diagrams differ from those normally included in textbooks.The usermight also consider modifying the program so that it draws the energy band diagram for an arbitrary applied bias. MATLAB program script.. % Equilibrium Energy Band Diagram Generator %(Si, 300K, nondegenerately doped step junction) %Constants T=300; % Temperature in Kelvin k=8.617e-5; % Boltzmann constant (eV/K) e0=8.85e-14; % permittivity of free space (F/cm) q=1.602e-19; % charge on an electron (coul) KS=11.8; % Dielectric constant of Si ni=1.0e10; %intrinsic carrier conc.in Silicon at300K (cm^-3) EG=1.12; % Silicon band gap (eV) %Control constants xleft=-3.5e-4; % Leftmost x position xright=-xleft; % Rightmost x position NA=input (‘Please enter p-side doping (cm^-3),NA=’); ND=input (‘Please enter n-side doping (cm^-3),ND=’); %Computations Vbi=kTlog((NAND)/ni^2); xN=sqrt(2KSeO/qNAVbi/(ND(NA+ND))); % Depletion width n-side xP=sqrt(2KSeO/qNDVbi/(NA*(NA+ND))); % Depletion width p-side x=linspace(xleft,xright,200); （0=<x）（Nx=>x)（OSX*）/2v（x-Nx）CINb-qA）=XΛ Vx2=0.5qNA.(xP+x).^2/(KSeO).(x>=-xP&x<0); Vx=Vx1+Vx2; % V as a function of x VMAX=3; % Maximum Plot Voltage EF=Vx(1)+VMAX/2-kT*log(NA/ni); % Fermi level %Plot Diagram close plot(x,-Vx+EG/2+VMAX/2); axis ([xleft xright 0 VMAX]); axis (‘off);hold on plot (x,-Vx-EG/2+VMAX/2);

--- Page 246 --- pn JUNCTION ELECTROSTATICS 223 plot(x,-Vx+VMAX/2,‘w:); plot（[xleft xright],[EFEF],‘w’; plot（[00],[0.15VMAX-0.5],‘W-’); text(xleft1.08,(-Vx(1)+EG/2+VMAX/2-.05),‘Ec’); text(xright1.02-Vx(200)+EG/2+VMAX/2-.05),‘Ec’); text(xleft1.08,(-Vx(1)-EG/2+VMAX/2-.05),‘Ev’); text(xright1.02,(-Vx(200)-EG/2+VMAX/2-.05),‘Ev’); text(xleft1.08,(-Vx(1)+VMAX/2-.05),‘Ei’); text(xright1.02,EF-.05,EF’); set(gca,‘DefaultTextUnits’,‘normalized’) text(.18, 0,p-side’); text(.47,0,‘x=0’); text(.75,0,‘n-side’); set(gca,‘DefaultTextUnits’,‘data’) hold off 5.2.5 LinearlyGraded Junctions The linearlygradedprofile,as noted in thepreliminary discussion,is a morerealistic ap- proximationfor junctionsformedbydeepdiffusions intomoderateto heavily doped wa- fers.Redrawn in Fig.5.13(a),the linearlygraded profile ismathematicallymodeled by ND-NA=ax$ (5.40) where a has units of cm-4 and is called the grading constant. Since we seek aquantitative solutionfor theelectrostatic variablesassociated withthe linearlygraded profile,this subsectionmight be viewed as an unnumbered exercise—an illustrationofhowtheproceduresestablished inthestep junctionanalysisareappliedto anotherprofile.However,thereare featuresofthelinearlygradedanalysisthataresuffi- cientlydifferenttomeritspecialconsideration.Firstandforemost,the Eq.(5.40)profileis continuousthrough=0.Thisactuallysimplifiesthemathematical development.Itisnot necessaryto treat thep-andn-sidesof the depletionregion separately or to matchthe solutions atx=0.There isonlyone ρ,E,andVsolutionfor theentire depletionregion. Moreover, because theprofile is symmetrical aboutx=0,alloftheelectrostatic variables likewiseexhibit a symmetry aboutx=0.A somewhat complicating feature is the nonuni- form doping outside of the depletion region. From previous work we know the nonuniform doping meansthere is aresidual ρ,electricfield,and potentialdropexternal to thecentral depletion region.Weignore this fact,takingρ=0,=0,and V=constantoutsideof the depletionregion inthe developmenttobepresented.Finally,althoughVistaken to bea constatofeiot.. is still necessary and will be included at theend of theanalysis.

--- Page 247 --- 224 PnJUNCTION DIODES ND−NA 个 x (a) ↑p -W/2 W/2 (b) ↑8 -W/2 0 W/2 (c) ↑V →x -W/2 W/2 (d) Figure5.13 Linearly gradedsolution. Depletion-approximation-based quantitativesolutionfor theelectrostaticvariables in a linearlygraded junction.(a)Linearlygradedprofile.(b)Chargeden- sity,(c)electric field,and (d)electrostatic potential as a functionof position.

--- Page 248 --- Pn JUNCTION ELECTROSTATICS 225 Launching into theanalysis proper by invoking the depletion approximationyields xb\ .-W/2≤x≤W/2 (5.41a) =(x)d$...x\le -W/2andx\ge W/2(5.41b)NotefromFig.5.13(b)thatthechargedensityissymmetricalaboutx=0,andonemusthaveW$x_{ } }$ (5.42)

2 Substituting into Poisson’sequation,separating variables,andintegratingfrom the p-edge ofthe depletionregion where=0toanarbitrarypointxinsidethedepletionregionthen gives qa x^{2 W W 8(x)=$(5.43)$2Ks0 2 2 A linearρ versus x dependence naturally gives rise to a quadratic & versus x dependence. Next setting(x)=-dV/dx, separating variables,and integrating once again fom the p-edgeof the depletion region where V=0 to an arbitrary point x,one obtains the cubic relationship qa 5i2(2)+ 3) x- x^3} W W $V(x)=$ (5.44) V x M $6Ks\epsilon 022Tocompletethesolution,todetermineW,wenotethatthevoltagedropacrossthedepletionregionmustbeequaltobi-VA,orV(x）=Vbi-VAat=W/2.MakingtheindicatedsubstitutionsintoEq.(5.44)andsolvingforW,onefinds12K071/3W=(bi-$ VA)$(5.45)qaForfuturereferenceitisimportanttonotethatthelinearlygradeddepletionwidthvariesasthecuberootofVbi-VA,whereasthestepjunctiondepletionwidthvariesasthesquare$root of bi-V{A. Numerical computations of the linearly graded ρ,,and V versus x would of course require an expression for Vi.The Eq.(5.10)expression for Vi cannot be used because it was specificallyestablishedassuming a stepjunction.However,theVderivationuptoand including Eq.(5.8)places norestrictionson the doping profileotherthanrequiring thatthe

--- Page 249 --- 226 Pn JUNCTION DIODES doping be nondegenerate. Consequently,for nonstep junctions one merely needs tore- evaluaten(n）andn（-x),theelectronconcentrationsattheedgesoftheequilibrium depletion region. For a linearlygraded junction $n(n)equilbiu0(ND-A)|w02=aW0/2(5.46a)$n^{a{}{2}n^{fra{}{2}n^{fa{}{2}$n(-xp)equilibrium(5.46b)p(-xp)equilibriu-(ND-NA)|-W012$aW_o2$whereW0=W|A=0.SubstitutingEqs.(5.46)intoEq.(5.8)thengiveskT($ 2 2kT (dV_{bi$lnln(5.47)二$2n_{ 2n{ q q or, making use of Eq.(5.45), （12K_bV_{bi$2kTln厂a1/3(5.48)=二$2n_}$ q pb Equation（5.48)cannotbesolvedexplicitlyforVbibutmustbenumericallyiteratedto determine Vjfor a given grading constant. 5.3 SUMMARY The pn junctionelectrostatics covered in this chapter provides a foundationfor theopera- tionalmodelingof the pn junction diodeandother devices that incorporate pn junctions. Early in the development we defined terms such as profileand metallurgical boundary, introducedtheidealizedstepjunctionand linearlygradedjunctionprofilesused extensively in analyses,and referenced Poisson’sequation,whichoftenconstitutesthestartingpoint in obtainingquantitativesolutionsforthelectrostaticvariables.Otherpreliminaryconsidera- tions includeda qualitative solutionfortheelectrostatic variables basedonenergy band argumentsandthederivationofcomputationalrelationshipsforthebuilt-involtage.The depletionapproximation,themost importantand widelyencounteredof thesimplifying approximations used in the modeling of devices,was introduced and illustrated. Theestablishedformalism was initiallyapplied toobtain quantitative solutions forthe chargedenitylcticfildandlctrotticpotnilisidetpjuctioundqi rium(VA=0)conditions. Theanalysis was subsequently extended to VA≠0.The effect of an applied bias on theelectrostatic variables was carefully examinedand used to deduce the energy band diagramsforpn junctions underforward and reverse biasing. Finally, quantitativesolutionswereobtainedfortheelectrostaticvariablesinsidealinearladd pn junction. It is hoped thereaderhasacquired aqualitativefeelfortheelectrostatic situationinside

--- Page 250 --- pn JUNCTION ELECTROSTATICS 227 a pn junction.Moreover,withthe informationprovided,thereader,if desired,should abletoobtainquantitativesolutionsfortheelectrostaticvariablesassociated withother doping profiles. PROBLEMS CHAPTER 5 PROBLEMINFORMATIONTABLE Complete Diffi culty Suggested Short Problem After Level Point Weighting Description 5.1 5.2.5 1 10 (1 each part) True-or-false quiz 5.2 5.1.4 1 5 (a-2, b-3) Vi from energy band 5.3 5.1.4 2 13(a-2, b-3, c-3,d-2, e-3) Isotype step junction 5.4 5.2.2 2 12(a-2, b-3,c-2,d-2, e-3) Step jct.compute, NA~ND 5.5 5.2.2 2 12(a-2, b-3,c-2,d-2, e-3) Step jct.compute,NA≥ND 5.6 5.2.3 2 9(a-2, b-3, c-2,d-2) Compute using diary 5.7 5.2.4 3 25 (a-15, b-5, c-5) Step junction program 5.8 5.2.5 2 9(3 each sketch) Combination profile 5.9 ” 2 9(a-2,b-2, c-5) Exponential profile 5.10 ” 2-3 10(a-3,b-2, c-5) Modified-step profile 5.11 ” 3-4 18(a-8, b-2, c-8) PINdiode 5.12 ” 2-3 12(4 each part) Given V,find E,ρ,ND-NA 5.13 ” 3-4 25(a-15,b-5,c-5) Linearly graded program 5.14 ” 5 35(a-30,b-5) Exact solution 5.1True or false: (a) The step junction is an idealized doping profileused to model p+-n and n+-p junctions. (b) The ρ thatappears in Poisson’sequation is the charge density and has unitsof couls/cm3. (c)Thespacechagereioaboutthemtllucljunctionisduetoaple-upofectrns on the p-side and holes on the n-side. (d) Thebuilt-inpotentialistypicallylessthanthebandgapenergyconvertedtovolts. (e) Invoking the depletionapproximationmakes thecharge densityinsidethedepletion region directly proportional to the net doping concentration. (f)Ohmiccontacts reducethe built-in voltage dropacross ajunction.

--- Page 251 --- 228 pn JUNCTION DIODES (g)In solutionsbasedonthedepletionapproximation,themagnitudeoftheelectricfield reaches a maximum right at the metallurgical boundary. (h) Ifone hasa+-n step junction,ajunction where NA(p-side)≥ND(n-side),thenit follows thatx_p≤x. (i) The potentialhillbetweenthen-sideandthep-sideofajunctionincreaseswithfor- ward biasing. (Thedelionitinlinrlradejunctioiesas E-2kTon thep-side and EF=E-/4onthen-side. (a)Drawtheequilibriumenergy band diagramforthisjunction. (b）Determinethebuilt-in voltage（Vbi)givingbothasymbolicandanumericalresult. 5.3Consider the p1-p2”isotype”step junction shown in Fig.P5.3. NA(x$1$NA1$NA2$ →x 0 Figure P5.3 (a) Draw theequilibriumenergbanddiagramforthejunctiontakinthedopintoe nondegenerate and NA1>N A2 (b)Deriveanexessionothebulinvltgthateissrosstheuctiuner equilibrium conditions. (c） Makeroughsketchesof thepotential,electricfield,andcharge densityinsidethe junction. (d)Briefly describe the depletion approximation. (e) Can thedepletionapproximation beinvoked in solvingfortheelectrostaticvariables inside the pictured p1-p2 junction? Explain. 5.4ASistep junctionmaintainedatroomtemperature underequilibriumconditions hasa p-sidedopingofNA=2×101/cm3andann-sidedopingofND=101/cm.Compute (a) bi.b$(b)$x{ ，$x_{n} and }_W.

--- Page 252 --- pn JUNCTION ELECTROSTATICS 229 (c) atx= 0. (d) Vat x=0. (e)Makeskchesthrerouhltoaleofthechaedenitlcticfildn- trostatic potential asafunctionofposition. 5.5RepeatProblem5.4,takingNA=107/cmtobethep-sidedoping.Brieflycompare the results here with those of Problem 5.4. 5.6A Si step junction maintained at room temperature has a p-side dopingof NA=(in- structor-suppliedvalue) and an n-sidedopingof ND=(instructor-supplied value).The appliedvoltage VA=(instructor-supplied value).Working in theCommand windowand making use of the MATLAB diary function to record your work session,compute (a) _bi.b (b) x_{ ,} {} }andW (c) at x=0. (d)Vat x= 0. 5.7GivenanondegeneratelydopedsiliconpnstepjunctionmaintainedatT=300K: (a)Computeandpresentcoordinatedplotsoftheelectricfield()andelectrostaticpoten- tial（V）insidethejunctionasafunctionofposition（).AssumeNA=10/cm3ND= 2×1014/cm3,and VA=-20Vinperforminga samplecomputation. Suggestions (i) EmploytheMATLABfunction subplottoachievecoordinatedplotsoftheele- tric field and electrostatic potential. 1.18|x=0andVmax=1.1(Vbi-VA）, (ii)Use（min,0)and(0,Vmax),where min 二 astheendpoint-valuesofthelectricfieldandpotentialplots,respectively.Also, with x max =2.5max(n,x),maxaMATLAB function，use（-xmax，xmax)as the x-coordinate limits. (b)Modify the part (a)program so thatresults corresponding to multipleVA values,say VA=VA0/2n withn=0to3,aresimultaneouslydisplayed.Make aprintoutofyour results. (c) Modifythepart(b)programsothatinadditiontotheplots,oneobtainsanoutput list of relevant parameters andcomputational constants.The list is to includethe following quantities:NA,ND,VA,bi,，x,WEat=0,andVat=0.Printoutasample set of results. (d) Use your program to examine how & versus x and V versus x vary with the relative magnitude of the doping on the two sidesof the junction.Experiment,for example, withthefollowingcombinationsof（NA,ND）:（10/cm310/cm³）（101/cm3,1015/

--- Page 253 --- 230 Pn JUNCTION DIODES cm³，（101/cm3,101/cm³），（101/cm3,1016/cm³），（101/cm3,107/cm),（1018/cm 1016/cm),and（1016/cm104/cm).Howwouldyoudescribetheresultswheneither NA≥10ONDor ND≥100NA? (e)Useyourprogramtogenerate answerstoProblems5.4 and5.5.Check your computer- generated results against those obtained manually. 5.8Thedopingaround themetallurgicaljunctionofa specialdiode ispictured inFig.P5.8. Sketchtheexpectedchargedensityelectricfieldandlectrostaticpotentialinsiethe- ode basedonthe depletionapproximation. Properly scale and labelrelevant lengths.In- clude a few wordsof explanation as necessary to forestalla misinterpretation of your sketches. $ND-NA\to x$-NA Figure P5.8 5.9A pnjunctiondiode has the dopingprofile sketched in Fig.P5.9.Mathematically, ND-NA=N0[1-exp(-x)],where N0andα areconstants. ND-NA $N_o$ x Figure P5.9 (a)Give a concise statement of the depletion approximation.

--- Page 254 --- pn JUNCTIONELECTROSTATICS 231 (b)Invokingthedepletionapproximation,makeasketchofthechargedensityinsidethe diode. (c) Establishanexpressionfortheelectricfield,()insidethedepletionregion. NOTE:Theinterestedreadermay wishtocompletetheelectrostaticsolutionbyobtaining expressionsorcomputationalrelationhipsforV(xxandi.Beforewarned,how- ever,thataconsiderableamountofmathematicalmanipulationisinvolved. 5.10ApnjunctiondiodehasthedopingprofilesketchedinFig.P5.10.Maketheassump- tion that n>0for all applied biases of interest. $ND-NA\uparrow$ND}$ND/2\to xxo$ $-NA Figure P5.10 (a)What isthe built-in voltageacross thejunction?Justifyyouranswer. (b)Invokingthe depletionapproximation,sketch thecharge densityρversusxinsidethe diode. 5.11Thep-i-n diode shownschematically inFig.P5.11is a three-regiondevice witha middleregionthatisintrinsic（actuallylightlydoped)andrelativelynarrow.Assuminthe p-and n-regions to be uniformlydoped and ND-NA=0in the i-region: (a)Roughlysketchtheexpectedchargedensityelectricfieldandelectrostaticpotential insidethe device.Alsodraw theenergyband diagram for thedeviceunderequilibrium conditions. (b)Whatis thebuilt-involtage dropbetween the p-and n-regions? Justifyyour answer. (c) Establisuantittivltioniporthechrgedenitylcticfldlecrostatic potential, and the p- and n-region depletion widths.

--- Page 255 --- 232 pn JUNCTION DIODES P 1 N → x x x_i$22FigureP\mathrm{5.115.12}Theelectrostaticpotentialinthedepletionregionofapnjunctiondiodeunderequi-libriumconditionsisdeterminedtobe[1+sin()$V_{bibV(x)$ .-W/2 ≤x ≤W/2 2 (a) Establishanexpressionfortheelectricfieldasafunctionofpositioninthedepletion region (-W/2≤x≤W/2) and sketch (x)versus x. (b) Establishanexpressionforthechargedensityasafunctionofpositioninthedepltion region and sketch ρ(x) versus . (c) Invoking the depletion approximation,determineand sketch ND-NA versus xin the depletion region. ●5.13Given a nondegenerately doped siliconlinearlygradedjunction maintainedat T=300K: (a)Computeandpresentcoordinatedplotsoftheelectricfild)andelectrostaticpote tial(V)insidethe junction as a functionofposition(x).Employa dopant gradient computation.NotethatEq.(5.48)mustbeiterated todetermine Vifora given grading constant. (b)Modifythepart (a)program sothatresultscorrespondingtomultiple VAvalues,say VA=VA0/2withn=0to3,aresimultaneouslydisplayed.Makeaprintoutofyour results. (c) Change the part (b)program so that inaddition tothe plotsoneobtains anoutput list ofrelevant parametersandcomputationalconstants.The list istoincludethefollowing quantities:a,VA,Vb,WEatx=0,and Vatx=0.Printouta sample setofresults. (d)Use your program to examine how versus x and V versus x vary with gradient con- stants over the range 1018/cm4≤a≤10²3/cm{4. (e) Compare theresultsobtainedinthisproblem withthecorresponding step junction results obtained inProblem 5.7.

--- Page 256 --- pn JUNCTIONELECTROSTATICS 233 ●5.14Ifequilibriumconditionsprevail,itis possibletoobtainaclosed-formsolutionfor theelectrostaticvariablesinsideapnstepjunction withoutinvokingthedepletionapproxi- mation.The“exact”solution validunderequilibrium（VA=0)conditions is detailedas follows: p-side (x≤0) solution … nJ dU’ x JUO F(U{'，UP L{e }$kT18=F(U,UFP)$qD$\rho =q{n}_1(eUFP-U-eU-UFP+e-UFP-eP)n-side(x\ge 0)solution...nJd{U}^{\prime }xJUoF(U^{\prime }-UB1,UFN)$L{v$kT18=F(U-UB1,UFN)9$L{D$\rho =qn;（eUFN-U+UB1-eU-UB-UFN+e-UFN-N)where...K_s\epsilon 0T71/2$2q^{2}F(U1，U2)=[e{²（e-U+U1-1）+e-U{²（eU1-U1-1]1/²2} 2 ."F'-function UFP=ln(NA/n₁) ...NA is the p-side doping concentration UFN = −ln(D). . .ND is the n-side doping concentrationV_bib} UB= UFP U{N 二 $kT/q V U

$k/q$ U0is the normalized potential（U)atx=0.ThevalueofUisobtained bysolving the transcendentalequation:F(U,UP)=(U0U1,UFN).Thisrelationshipresultsfrom the fact that the electric field must be continuous at x=0.

--- Page 257 --- 234 Pn JUNCTIONDIODES (a)Using the precedingrelationships,constructcoordinatedplotsofthe electric field() and theelectrostatic potential,V=(kT/q)U,inside the pn junctionas a functionof position.Assume NA=101/cm3andND=2×1014/cm3inperformingasample computation.If available,also run theProblem 5.7(a)program utilizing an identical set of parameters.Compare and discuss the two sets of E,Vversus x plots. Suggestions (i） First determineU0employingtheMATLABfunction fzero withan initial guess ofU0=UBI/2. (ii）SteppingUfrom0.1toU0andutilizingtheintegralrelationship,computeU= V/(kT/q)versusonthep-sideofthejunction.Repeatforthe n-sidesteppingU fromU0 to UB1-0.1. (iii)Foreach valueofUinstepi),likewisecompute.KnowingversusUandU versus x allows one to construct an& versus x plot. (iv)InconstructingtheandVplots,followthesuggestionscitedinProblem5.7a). (b) Extend the part(a)computations to obtain the normalized charge density (p/q)versus x.On a single setofcoordinates,construct a sample plotof plq versus x as deduced fromboththeexactanddepletion-approximation-basedsolutions.Discussyourlot results.

--- Page 258 --- 6 pn Junction Diode: /-VCharacteristics This chapter is devoted to modeling the steady state response of the pn junction diode.The currentflowing throughthediodeas a functionof theapplied d.c.voltage,the I-Vchar- acteristic,isqualitativelyandquantitativelycorrelated totheinside-the-deviceprocesses andparameters.Theinitialdevelopmenttreatsthe”idealdiode”and workstowardthe derivationof theidealdiodeequation—a simplewell-knownI-Vrelationship.Although some of the idealizationsmay notberealized inpractice,theideal diode development permits unobstructedinsight into theoperationofthedeviceandprovides a relatively simple starting pointformoreexacting analyses.Aftercomparing the idealtheory with experiment,we next focus on adjustments to thetheory to correctobvious discrepancies. Several deviationsfromtheidealaresystematicallyidentifiedandexplained,and appropri- ate modifications are introduced.Finally,we presentanalytical supplementsto the usual analysis that will be particularly useful in subsequent chapters. 6.1THEIDEALDIODEEQUATION As noted inthechapterintroduction,the idealdiode isveryusefulforprovidinginsight andforfacilitating a base-levelanalysis. TheI-Vcharacteristicsoftheidealdiodeare modeledbytheidealdiodeequation.Derivationof theequationisostensiblythetaskof this section,although it shouldbe understoodthat the analytical procedures,subsidiary results,and insightestablished in the process arereally of prime importance.Making use of the pn junctionenergyband diagrams constructed inChapter 5,we first pursue a quali- tative “derivation”oftheidealdiodeequation. Thisexercise illustratesthepowerand utilityofthe energyband diagram,yielding the generalformof the desired resultwithout writingdownasinglemathematicalrelationship.Preparing forthequantitative derivation, we next detail oursolution strategy.This is followed by the actual mathematical manipu- lations leading to theidealdiodeequation.Aprobing examination of thefinal and subsid- ary results concludes the section. 6.1.1Qualitative Derivation To set the stage,so to speak,consider the equilibrium energy band diagram for a pn junc- tion shown in Fig.6.1(a).Thegroupsof dots()andcircles(O)addedtothefigure crudelymodelthecarrierdistributiononthetwo sidesofthe junction.On thequasineutral 235

--- Page 259 --- 236 pn JUNCTIONDIODES Ec $EFEiOOOEvOO(a)Equilibrium(VA=0)NI$E{cEFn $Ei$ O Ev} O O $Ip$ P N I $VA>0$ (b) Forward bias (VA>0) Figure6.1 pnjunctionenergyband diagram,carrierdistributions,and carrieractivity in the near vicinityofthe depletionregionunder(a)equilibrium(VA=0),(b)forwardbias,and （c)reversebias conditions.(d) Deduced form of the I-V characteristic.

--- Page 260 --- pn JUNCTION DIODE: I-VCHARACTERISTICS 237 -IN O 1 $Ec$ $EFn$ $Ei$ $Ev$ O I_} (c) Reverse bias (VA< 0) ←Exponential →VA ↑ Constant (d) Figure 6.1 Continued. n-sideof the junction there are a large numberof electrons and a fewholes.The pyramid- likearrangementofdots(first introduced inFig.2.17)schematicallyrepresentstheroughly exponential decrease in the electron population as one progresses upward into the conduc- tion band.Conversely,on the quasineutral p-sideofthe junction there are a high concen- trationof holesand asmall numberofelectrons.Theholepopulationdropsoff inaroughly exponential fashion as one moves downward into the valence band.

--- Page 261 --- 238 pn JUNCTIONDIODES Theenvisionedelectronsandholeshavethermalenergyandareofcourse moving around inside the semiconductor.Concentrating firston then-sideelectrons,we seethat mostofthesecarriershaveinsufficientenergyto”climb”thepotentialhill.Excursionsinto the depletionregionmerelyresult inthe lower-energy carriersbeingreflected backtoward the n-side quasineutral region.However,there willbe some high-energy electrons that can surmountthe hillandtravel overto the p-sideof the junction.What we have been describ- ing shouldberecognizedasthedifusionoflectronsfromthehigh-electronpopulationn- side of the junction to the low-electron population p-side of the junction. Whereas electronson then-side see a potentialbarrier,electronson the p-side arenot restricted in any way.Ifa memberof the smallelectron populationon the p-side happens to wander into the depletionregion,it willberapidlysweptoverto theothersideofthe junction.Naturally,this p-ton-side driftcurrentpreciselybalancesthen-to p-side diffu- sion current underequilibrium conditions.The holesituation is completelyanalogous.The few p-sideholesthat have sufficientenergy to surmountthe potential energybarrierand gain entry to then-side of the junctionare preciselybalanced by n-side holes wandering into the depletion region and being swept over to the p-side of the junction. Aware of the primary carrieractivity in the vicinity ofthe junction,let us now consider theforwardbiassituationpicturedinFig.6.1(b).Themostsignificantchangerelativeto zero bias is a lowering of the potential hillbetween the p-and n-sidesof the junction.The same numberofminoritycarriersarestillwandering intothe depletionregionandbeing sweptoverto theothersideof the junction. However,with the potentialhilldecreased in size, more n-sideelectrons and p-side holescan now surmount the hilland traveltothe oppositesideof the junction.This gives risetobothanelectroncurrent(I）and a hole current (Ip)directedfrom thep-side to then-sideof the junction.Notefrom thecircuit sketchedbelow theFig.6.1(b)energyband diagram thatthe deducedcurrent(I=IN+ )flows in theproper directionforaforward biaseddiode.Moreover,because thepoten- tialhilldecreases linearlywiththeappliedforwardbiasandthecarrierconcentrationsvary exponentially asone progresses away from the band edges,the numberofcarriersthat have suficientenergy to surmountthe potential barriergoes up exponentially withVA.Thus,as summarized in Fig.6.1(d),the forwardcurrent is expected to bean exponentially increas- ing function of the applied voltage. Thereverse bias situation is described by the energy band diagram in Fig.6.1(c).Rela- tive to equilibrium,themajoreffectof thebias is to increasethe potentialhillbetweenthe p-and n-sides of the junction. Whereas some n-side electrons and p-side holes can sur- mountthe hillunderequilibrium conditions,even a very smallreverse bias,anything greater than a few kT/q in magnitude,reduces the majority carrier diffusion across the junction to a negligible level.The p-sideelectrons and n-side holes,on theother hand,can still wanderinto the depletionregionand be swepttotheothersideofthejunction.Reverse biasing thus gives rise to a currentflow directedfrom the n-side to the p-sideof the junc- tion.Being associated withminoritycarriers,thereversebiascurrent is expectedtobe extremely small inmagnitude.Noteinadditionthattheminoritycarrier driftcurrentsare not affectedbythe heightofthepotentialhill.It is the numberof minoritycarriers wander- ing into the depletion region persecondthat determines the current flow.(The situation is

--- Page 262 --- pn JUNCTION DIODE: I-VCHARACTERISTICS 239 similartoawaterfall.Thewaterflowingoverthefallsisindependentoftheheightofthe falls.)Therefore,assketchedinFig.6.1(d),thereversecurrentisexpectedtosaturate— become bias independent—oncethemajority carrier diffusion currents arereducedtoa negligiblelevelatasmallreversebias.Ifthereversebiassaturationcurrentistakentobe -0,theoverall I-V dependence is concluded to be of the general form I=I0(eV{/Vref-1) (6.1) Equation(6.1)isidenticaltotheidealdiodeequationifVefis setequal tokT/q. In additiontoessentiallyyieldingtheideal diodeequationtheforegoinganalysisvery nicely explainshow a solid state diode manages to rectify a signal;i.e.how the diode passes alargecurrent whenforward biased and a verysmall current whenreversebiased. Forwardbiasingreducesthe potentialhillbetweenthetwosidesofthejunction,permitting large numbersofmajoritycarriers tobeinjectedacrossthedepletionregion.Reversebi- asing increasesthepotentialhill,cuttingoffmajoritycarrierinjectionandleavingonlya residual current supplied by minority carriers. Once,aftercompleting thequalitative derivationandfeeling rather smug aboutthe insightprovided,theauthor wasasked,“Yes,but,doesn’t the injectionof majority carriers under forwardbiasandtheextractionofminoritycarriers underreversebiascause acharge build-up insidethedevice?”The immediateanswer is thatsteady stateconditions were assumed intheanalysis and a charge build-up,or a changeof anytype,does notoccur under steady state conditions.The question,however,has deeper implications.Theauthor, concentratingsolelonthecarrieractivityintheimmediatevicinityofthedepletionren hadfailedtoprovideanoverallviewofcarrieractivity insidethe device.Itis theoverall view that explainshowinjectedandextractedcarriers are resuppliedandthestatusquo maintained. In presenting the“bigpicture,“wetakethe diodeto bereversed biasedfor illustrative purposes andrefertothecompositeenergy-band/circuit diagram shown in Fig.6.2.The capacitor-likeplatesattheouterendsoftheenergybanddiagramschematicallyrepresent the ohmiccontacts to the device.The major reverse-biasactivity in the immediate vicinity of the depletionregion,minoritycarriers wandering intothedepletionregionandbing swept to theother sideofthe junction,is againpicturedonthe diagram. Addedtothe diagram areElevels associated withR-Gcenters.Wheneveranelectrononthep-side moves to the n-side,it is replaced by anelectron generatedthroughoneof the R-Gcenters. As pictured justto theleftof thedepletionregion inFig.6.2,anelectronfrom thevalence band jumps up to the R-G centerand then into the conduction band. Similarly, whenever a minoritycarrierholeis sweptfrom then-sidetothep-sidethehole is quicklyreplaced bythecarriergenerationprocess.Theelectrons falling downthehillto the n-side,andthe electronssimultaneously generated during the n-sidereplacementof lostholes,in turn give rise to an excessofmajority carrierelectrons on then-sideof the junction.(Twoextra electrons arepicturedadjacent to the depletionregiononthen-sideof the junctionin Fig.6.2.)Theextramajoritycarrierelectrons setupa localelectricfieldthatpushesadja- cent electrons towardthecontact.Withgreatrapiditythisdisplacement propagates until

--- Page 263 --- 240 PnJUNCTIONDIODES I Er

0→0→ →0→

、 、 、 ET ← ↑ ← ← Figure6.2 Compositeenergy-band/circuit diagramproviding anoverall viewof carrieractivity inside areverse-biased pn junctiondiode.Thecapacitor-likeplates attheouterendsoftheenergy band diagram schematically represent the ohmic contacts to the diode. the entire in-line groupof n-side electrons movesslightly toward thecontact.Electrons equalin numberto theexcess and immediatelyadjacenttothecontactare pushedintothe contact andoutinto theexternalcircuit.+ Theholeactivityonthe p-sideof the junction is similar. The excess appearing at the depletionregionedge causes in-line holes throughout the quasineutral p-region to moveover slightly.A numberof holes equal tothe excess are pushed into thecontact where theyrecombine withelectrons from themetal. Thisrecom- bination may be viewedas completing thecircle,eliminating the extra electrons pushed into the external circuit on the n-side ofthe diode. Asa somewhatunrelatedobservation,it is interesting tonotefromFig.6.2thatboth electrons andholescontribute to thecurrentthrough thedepletionregion,whilethehole current dominates far fromthe junction on the p-side of the device and the electron current dominates far from the junctionon the n-side of the device. Thetotal current throughthe diode must beconstant,butthecomponentelectronandholecurrentsobviously vary with position inside the diode. The phenomenon just describedmight belikenedto thereal-lifesituation where latecomersto apartyenter a room absolutelycrammed withpeopleItisassumedtherearetwodoorsatoppositeendsoftheroom.Byforcing their way into one door,the newcomers cause a like number of people to be pushedout the other door.

--- Page 264 --- Pn JUNCTION DIODE: I-VCHARACTERISTICS 241 6.1.2 Quantitative Solution Strategy Going into a football game,coachesalways have a game plan,a strategyfor winningthe game. Herein we developand explain the strategy used in thequantitative derivationof the ideal diodeequation.Althoughmathematical steps in the derivationare relatively fewand quite straightforward,thestrategy underlying the steps is rather involvedand needs tobe clearly understood inapplying thesolutionapproach tootherproblemsand in implement- ing modifi cations to the theory. General Considerations We begin by listing the basic assumptionsmade intheanalysis.Additional assumptions will be necessary as the strategy unfolds.) (1)Thediode is beingoperated under steady stateconditions. (2) A nondegenerately doped step junction models the doping profile. (3) The diode is one-dimensional. (4) Low-level injection prevails in the quasineutral regions. (5)Thereare noprocessesotherthandrift,diffusion,andthermalrecombination-genera- tion taking place insidethe diode.Specifically,GL=0. The precedingassumptions seem reasonablesincethey were allexplicitlyorimplicitly invoked in establishing the pn junction electrostatics. Let us nextconsiderthe generalrelationships availableforcomputingthe current. They are: I = AJ (A = cross-sectional area) (6.2) J=JN(x)+J(x) (6.3) dn JN = qnn+N (6.4a) dx dp$ = qp - (6.4b) dx Equation(6.3)reflectsthefactthatthetotalcurrentdensity isconstantthroughoutthe diode, buttheelectronandholecomponents vary with position.Equations (6.4) are the one-dimensional versionsofEqs.(3.18).Clearly,ifexactanalytical solutionsfor ,n, and p versus x were available,we would need to proceed no further. However,only an approxi- mate three-region electrostatic solution is available,as schematically summarizedin

--- Page 265 --- 242 pn JUNCTION DIODES 8=0 8≠0 8=0 -x{p ↑ $x_{}$ Depletion region Quasineutral p-region Quasineutral n-region Figure6.3 Diode electrostatic regions. Fig.6.3.Wedonotethattheconditionsrequiredfortheuseoftheminoritycarrierdiffusion equations,including&≈0andlow-level injection,are satisfied inthequasineutralregions of the diode. Quasineutral RegionConsiderations Undertheassumedsteadystate conditions withGL=0,theminoritycarrier diffusion equations appropriate for the p and n quasineutral regions are d^{}_{n{} ${\Delta }_n$ O=DN .. .x≤-xp (6.5a) $d{x}^2$ τn d^{2}^ $\Delta pn$ 0= D{P . ..x≥x_{n (6.5b) $d{x}^2$ Moreover,since=0anddn/dx=dp0dx=0,Eqs.(6.4)forthecarriercurrentdensi- ties in the quasineutral regions simplify to $d{\Delta }_n$ JN =qDN .x≤-xp (6.6a) dx dΔ_{n p=-qDP$ …x≥xn (6.6b) dx Weknowthegeneralsolutionto Eqs.(6.5)and it is a trivialmattertocomputethe carriercurrentdensitiesfromEqs.(6.6).Unfortunately,thecurrentdensity solutionsso obtained arelimitedto non-overlapping segmentsofthediode.Wecanonly determine JN(x)in thequasineutralp-regionandJp(x)in thequasineutraln-region.To solve forJ using Eq.(6.3),there mustbeatleastone pointinsidethe diode whereoneknowsboth

--- Page 266 --- pn JUNCTION DIODE: I-VCHARACTERISTICS 243 overlapping JNandJpsolutions,solutionspossiblyextrapolatedfromthequasineutral regions. Depletion RegionConsiderations The full-blowncontinuityequations,Eqs.3.46),mustbeusedinseekingsolutionsforthe carrier currents withinthe≠Odepletionregion.Underthepreviouslyspecifiedassump- tions, the continuity equations simplify to 1 dJ_N$ ∂n 0

(6.7a) q xp 1e thermal R-G 1d_ ∂p$ =0 + (6.7b) xpb e thermal R-G Suppose the additional assumption is now made that thermal recombination- generation is negligible throughout the depletionregion; i.e.nthermal RG and in Eqs.(6.7)yieldsdJN/dx=0anddJp/dx=0.JNand Jpare therefore determined to be constants independentof position inside the depletion region under the stated assump- tion. Theconstancyof thecarrier currents throughoutthe depletionregion(including the edges) in turn allows one to write JN（-xp≤x≤n）=J（-x) (6.8a) J(-x≤x≤n）=Jp(xn) (6.8b) JN(-xp)andJp(n）canbededucedofcoursefromthequasineutralregionsolutions evaluated at the edgesof thedepletionregion.Summing the JN and Jp solutions in the depletion region then gives J=JN(-p)+() (6.9) Clearly a solutionstrategy has beenformulated:to solve fortheminoritycarriercurrent densities inthequasineutralregions,evaluatethecurrent densitiesat the depletionregion edges,addtheedgecurrentdensitiestogether,andfinallymultiplyby Atoobtainthe current. The rather critical assumption thatthermal recombination-generation is negligible in the depletion region can be viewed as a defining propertyof the idealdiode. There is

--- Page 267 --- 244 pn JUNCTION DIODES absolutely noapriorijustificationfortheassumptionotherthanitleadstoasimplesolu- tion forthetotalcurrent flowingin the diode.The validityoftheassumptionas farasreal diodes are concerned will beconsidered when comparing theory and experiment. BoundaryConditions There is one more matter toaddress.Two boundary conditionseach are required in solving Eqs.(6.5a)and(6.5b)for Δnand Δpnin the p-andn-side quasineutral regions.In par- ticular,as summarizedinFig.6.4（a),Δnandpnmustbespecifiedattheohmiccontacts and at the edges of the depletion region. At theOhmicContacts The ideal diode isusuallytaken to be a”wide-base”diode,ora diode whosecontactsare several minority carrierdiffusion lengthsor more from the edgesof the depletionregion. In a wide-base diodeanyperturbationin thecarrierconcentrationscreated attheedgesof the depletionregionwilldecayto zerobeforereaching the contacts. Thecontacts may effectively be viewed as being positioned at x= ±∞.Thus,in the mathematical derivation P N ↑ ∆np(p-contact）∆np（-xp)）p(n) ∆pn(n-contact) (a) Ec Nd EFn EFp Fp Ev -{p $x_n$ 1 (b) Figure6.4 Boundary-conditionrelatedconsiderations.(a）Boundarypositionsandrequiredval- ues.b)Approximatevariationofthequasi-Fermilevelswithpositioninsideforward-biaseddiode.

--- Page 268 --- pn JUNCTIONDIODE: I-VCHARACTERISTICS 245 the boundaryconditions to beemployed are ∆np(x→-∞)=0 (6.10a) Δpn(x→+∞)=0 (6.10b) At the Depletion Region Edges Toestablishthe boundaryconditions at theedges of the depletion region, we make use of the quasi-Fermi levelformalism.Equations (3.72)are the defining equations for theelec- tronquasi-FermilevelN,andholequasi-Fermilevel,p.Iftheleft-andright-handsides of the exponential versionsof Eqs.(3.72a)and (3.72b) are multiplied together,oneotains np=n²2}e(-p)/kT (6.11) The Eq.(6.11)relationship is validthroughoutthe diode underarbitraryoperating condi- tions. Generally speaking,one does notknow the variationof the quasi-Fermi levels as a functionof positionpriorto solving for thecarrierconcentrations inside the diode.How- ever,asenvisioned inFig.6.4(b),itis reasonabletoassumetheFNandFplevels willvary monotonicallyfrom Fpfaronthe p-sideof thejunctionto nfaronthen-sideofthe junction.NotefromFig.6.4(b)thatthemonotonicvariation in the levels in turn makes =qVA at all points inside the diode.If the equal sign in the preceding expression for F-pis assumed to hold throughoutthe depletion region,one concludes np=n_{}{k . .-x ≤x≤ { (6.12) Eq.(6.12)has beenreferredto as the“lawof the junction."Evaluating Eq.(6.12)at the depletion region edges veryrapidlyleadsto thedesiredboundaryconditions.Specifi- cally,evaluating Eq.(6.12) at the p-edge of the depletion region gives n(-xp)p(-p)=n（-p)NA=n²2eqVA/k (6.13) or n_\fra{}{2}n(-x) =$e_A/kT(6.14)$N{}$and$n^{fra{}{2}∆n{p-x{=$(eVA/kT-1)(6.15)$NA}$

--- Page 269 --- 246 pn JUNCTION DIODES Similarly, $n(xn)p()=p(n)D=n2eV(6.16)or$n^{fra{}{2}p(xn =eq{A/kT (6.17)N_{}$and$n^{fra{}{2}Δp{n(=$(eVA/kT-1)(6.18)$N{}$ The assumption that FN-Fp=qVthroughout the depletionregion,orequivalently, assumingthequasi-FermilevelsareconstantatF=Fnand=pwithinthedeple- tion region,is obviously central toobtaining the depletion-edge boundary conditionsand criticalto theoverallanalysis.Rather lengthysubsidiaryanalysesandacomparisonwith experiment indicate the assumption is typically justified. “Game Plan”Summary Toobtainananalytical solutionfor thecurrentflowing in anidealdiodeas a functionof the applied voltage, proceed as follows: (1) Solvetheminoritycarrierdiffusionequations (Eqs.6.5)employing the(6.10)and (6.15/6.18)boundaryconditionstoobtain ΔnandΔpin thequasineutralregions. (2)Compute the minority carrier current densities in the quasineutralregions using Eqs. (6.6). (3) Per Eq.(6.9),evaluate thequasineutralregionsolutions forJN(x)andJp(x)atthe edges of the depletionregionand thensum thetwo edge-currentdensities.Finally, multiply the result by the cross-sectional area of the diode. Assuming Fand Fp are approximatelyconstantacross the depletionregion isequivalenttoassuming JN=0 authors inan alternative derivationofthe depletion-edge boundaryconditionsthatcloselyparallels theVideri- vation in Subsection 5.1.4.Like the constant quasi-Fermi assumption,and contrary to statements in some texts, there is really no simple a priori justification of the JN=0 and Jp=0 assumption.

--- Page 270 --- Pn JUNCTION DIODE: I-VCHARACTERISTICS 247 6.1.3 Derivation Proper In implementing thesolutionprocedure,let usfirst work withholesonthequasineutraln- sideof thejunction.To simplify themathematicsin solving Eq.(6.5b)itisconvenientto shifttheoriginofcoordinatestothen-edgeofthedepletionregionasshowninFig.6.5a). In terms of the translated x’-coordinate, we must solve d^{2}^}{} ${\Delta }_n$ 0=D{$...$x{ ≥0 (6.19) $d_x^2$ Tp subject to the boundary conditions ∆pn(x{→∞)=0 (6.20a) n^{ra{}{2}Δpn(x^{0 = (eqVA/kT-1) (6.20b) $ND$ Eq.(6.19)isoneofthespecial-casediffusionequations listedinTable3.2.Thegenerl solution (solution no.1 in Table 3.2) is ∆pn（x′）=A₁e-x/LP+A2exLp . . .x’ ≥ (6.21) where P =√DP (6.22) - x{p x_} →x →x{$0(a)$-x_{x n$\to x$x^{}$ 4 0 (b) Figure6.5Graphical definitionof the(a)x-and(b)x”-coordinate systems.

--- Page 271 --- 248 PnJUNCTIONDIODES can besatisfiedisforA2tobeidenticallyzero.WithA=0,applicationoftheEq.(6.20b) boundarycondition yields A₁=Δpn(x’=0).Wetherefore conclude n^{ra{}{2} Δp{n{x{c)= (egVA/kT-1)e-x’/Lp …x^{ ≥0 (6.23) D} and dΔ_{nD{n2} Jp(x{）=-qDP b= (egVA/kT-1)e-x/Lp …x’≥0 (6.24) $d{x}^{}$ L{pD On the quasineutral p-side ofthe junction with the x”-coordinate as definedin Fig.6.5(b),one obtains the analogous solutions n^{ra{}{2} (eqVA/kT- 1)e-x/LN …$x"\ge 0(6.25)$N{A} and dΔ_n{p D{n{}{2} JN(x)=-qDN b=$(eqVA1kT-1)e-x/LN...x"\ge 0(6.26)dx"$L{NN_{}$AllthatremainsistoevaluateEqs.(6.24)and(6.26)atthedepletionregionedges,sumtheresults,andmultiplybyA.Wefind$n^{}{2}$DN$ $JN(x=-xp)=JN(x”=0)

b LN $NA$ (eqVA/kT-1) (6.27a) Dn^{a{}{2}$ (x=）=Jp（x′=0)

q L{pN{}$ (eqVA/kT -1) (6.27b)

--- Page 272 --- Pn JUNCTION DIODE: /-VCHARACTeRISTICS 249 and (:Dn_\frac{}{2} D{_ n^{{}{2} 1 =AJ=qA (eqVA}kT-1) + (6.28) L{ NA} L{p $NB$ or I =I0(eqV{A/kT -1) (6.29) (cid:DN n_{ra{}{2} D{n^{{}{2}vb=0$+(6.30)$LN $N_{}$ L{pN{D}$Equation(6.29)istheidealdiodeequation.ItisalsosometimesreferredtoastheShockleyequation.6.1.4ExaminationofResultsItisworthwhileatthispointtopauseandexaminethefinalandintermediateresultsofthederivation.Wehopetobecomemorefamiliarwiththeresultswhilesimultaneouslygainingdeeperinsightintotheoperationofthepnjunctiondiode.Workingbackwardthrouhthederivation,wefirstexaminetheidealdiodeequationandtheassociatedsaturationcurrent.Subsequentlyweinvestigatethecarriercurrentsandcarrierconcentrationsinsidethedi-ode.Severalexercisesarealsopresentedtosupplementthediscussion.Ideal1-VThemajorfeaturesofthepredictedI-VcharacteristicsaresummarizedinFig.6.6.ForreversebiasesgreaterthanafewkT/q,afewtenthsofavoltatroomtemperature,theexponentialvoltagetermintheidealdiodeequationbecomesnegligibleandI\to -.Accordingtotheidealdiodetheory,thissaturationcurrentwouldbeobservedforreversevoltagesofunlimitedmagnitude.ForforwardbiasinggreaterthanafewkT/q,theexpo-nentialtermdominatesandI\to 0exp(qVA/kT).Reflectingtheexpectedexponentialde-pendence,theforward-biascharacteristicsareoftenplottedonasemilogscaleasillustratedinFig.6.6(b).Since$ln(I)= ln(0)+ 9 …if VA > few kT (6.31) kT$ 9 the ideal theorypredictsa VA>0semilogplotthathasa linearregionslopeofqkTand an extrapolated intercept of ln(0).

--- Page 273 --- 250 pn JUNCTION DIODES I $ln(l)loegA/kT\leftarrow Slope=q/kT|VA|>fewkT/q$ln(I0) →VA -8 4-10 →VA (a) (b) Figure6.6IdealdiodeIVcharacteristics:(a)linearplotidentifyingmajorfeatures;b)forward- bias semilog plot. TheSaturationCurrent Tworather significantobservations canbemadeconcerning the saturationcurrent.First, the size of Io can vary by manyorders of magnitude depending on the semiconductorused to fabricate the diode.This strong material dependence enters through the n²factor in the I0expression.Atroomtemperaturen=10/cm3inSiwhilen10/cminGe.Thus, Ge diodesare expected toexhibitareverse-bias saturationcurrentroughly106times larger than that of comparable Si diodes! The second observation relates to asymmetrically doped junctions.The I expression has two terms that vary inversely with the dopings respectively on the p-and n-side of the junction. Because of the cited doping dependence,the term associated with the heavily doped side of p+-n and n+-p junctions becomes negligible;i.e., D{pn^}} … p+-n diodes (6.32a) L{pD and $Dnn2$ ..n+-p diodes (6.32b) LA} In essence,one has to consideronly the lightly doped side of such junctions in working out the diode I-Vcharacteristics.We alsofound one canallbut neglect the heavily doped side ofasymmetrical junctions incomputing the depletion widthandotherelectrostatic vari-

--- Page 274 --- pnJUNCTIONDIODE:/-VCHARACTERISTICS 251 ables.These similar conclusions suggest that,as a general rule,the heavily doped sideof an asymmetrical junction can be ignored in determining theelectrical characteristics of the junction.If an asymmetrical junction is specified in a problem statementor atthe be- ginningofananalysis,itshouldbeunderstoodthattheheavilydopedsideisto beignored in completing the problem or analysis. We mightmentionthat it is rather fortuitous thatthecurrentcontributionfrom the heavily doped sideof asymmetrical junctions is negligible.Inmostreal diodesthe p+-or n+-sidedopingis degenerate.Ifthecurrentcontributionfromtheheavilydopedside were significantcomparison withexperiment wouldnecessitate amodificationof theideal diode theory to account for the degenerate doping. Exercise 6.1 P: Two ideal p+-n step junction diodes maintained at room temperature are identical except that ND1 1=101/cm3andND=101/cm.ComparetheIVcharacteristics of the two diodes; sketch both characteristics on a single setof axes. S: For p+-n diodes $Dnr$ L{ D Also D{P D{ (kT/q)μp Lp V Tp V p p2,sincetheRGcenteconcentrationsaretakentobeidenticalinthetwo diodes.Moreover,although the semiconductor material is not specified in the prob- lemstatement,thereisonlyasmall dference betweenthe ND=10/cm3andD= 1016/cm3mobilities in most materials (see Fig.3.5).Thus $\text{o}$ $N$ ${\mu }_{p}1$ ≤10 \or2} N{{1}{P} The diode1current is approximately ten times larger thanthediode2current forall applied voltages (see Fig. E6.1).

--- Page 275 --- 252 pn JUNCTION DIODES 1 ① ② →VA ② ① Figure E6.1 (C) Exercise 6.2 P:AnidealSi p+-n step junctiondiode is maintained at 300K.Thediodehasa cross-sectional areaA=10-4cm² andτp=10-6sec.Using the empirical-fitrela- tionship fortheholemobilityintroducedin Exercise3.1,writea MATLAB program that computesand plots the idealI-Vcharacteristicof the diode. Then-side donor doping is to beconsideredan input variable.Employa linear plot with the MATLAB axis functionsetto[-1,0.2,-20,50],where 0isthereversebias saturation current.Generallyuse yourprogram to explorehowtheidealdiodecharacteristic varies as a function of the semiconductor doping. S:The program thatfollows was written to handle multiple doping inputs.Enclosing your input in square brackets,simply type thedesired doping values separated by same general doping dependence noted in Exercise 6.1.However,examining the nu- merical lo values sent by the program to the Command window,one finds a notice- able mobility dependence becauseof the higherassumed dopings. The usershould also take noteof the extremely small size of the computed saturation currents. MATLAB program script… %Variation of Ideal-Diode I-V with semiconductor doping. %Si step junction,T=300K. %In responseto the“ND=”prompttype [ND1ND2…]to input %multiple doping values. %Initialization andUniversal Constants clear k=8.617e-5; q=1.6e-19;

--- Page 276 --- Pn JUNCTION DIODE: 1-VCHARACTERISTICS 253 %Device, Material,and System Parameters A=1.0e-4; ni=1.0e10; taup=1.0e-6; ND=input(Inputthe n-sidedopingconcentration,ND=’); T=300; %Hole Mobility Calculation NAref=2.35e17; μpmin=54.3; μp0=406.9; ap=0.88; p=pmin+pO./(1+(ND./NAref).^ap); %Themobilitycalculation hereassumesthe holeminoritycarrier %mobility isequal to the holemajority carriermobility. %I-V Calculation VA=linspace(-1,0.2); DP=k.T.p; LP=sqrt(DP.taup); IO=q.A.(DP./LP).(ni^2/ND) I=IO.’(exp(VA./(k.T))-1); %Plotting Resullt close plot(VA,I); grid; ymin=-2I0(1);ymax=5I0(1); axis([-1,0.2,ymin,ymax]); xlabel(‘VA(volts)’);ylabel(‘I(amps)’); %Adding axes,key xx=[-10.2];yx=[00; xy=[0 0];yy=[ymin,ymax]; hold on plot(xx,yx,-w’,xy,yy-w); j=length(ND); fori=1:j; yput=(0.70-0.06*i)ymax; yk(i,1)=yput;yk(i,2)=yput; text（-0.68,(0.69-0.06i)ymax,[‘ND=‘,num2str(ND(i/cm3’); end xk=[-0.8-0.7]; plot(xk,yk); text(-0.74,0.75ymax,‘Si,300K’; hold off

--- Page 277 --- 254 Pn JUNCTION DIODES x10-16 25

Si,300K 20

ND=ie+16/cm3 ND=2e+16/cm3 15 ND=5e+16/cm3 ND=1e+17/cm3 I (amps) 10 5 0 -5 -10 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 VA(volts) Figure E6.2 CarrierCurrents A sample plotof the carrierand totalcurrent densities as a functionof position insidea forward-biaseddiodeis displayed inFig.6.7.Areverse-bias plotis essentiallyidentical exceptallcurrent densitiesare negative.Steps in theconstructionof the Fig.6.7plotareas follows:Examining Eqs.(6.24)and(6.26),one concludes the minority carriercurrent den- sities in the quasineutral regions decay exponentially away from theedges of the depletion region.Within thedepletion region,JN and Jp per Eq.(6.8)are nextdrawnconstant at their respective depletion-edge values.Thetotal currentdensity inthe depletionregionisthen just the graphical sumof JN and Jp.Since the totalcurrent density is constant everywhere inside thediode,the value deduced forthedepletion region can beextended throughoutthe diode.Finally,themajority-carriercurrentdensitiesinthequasineutral regionsareobtained by graphically subtractingtheminority-carriercurrentdensitiesfrom thetotalcurrent density. The constructionofFig.6.7shows thatthereissufficientinformationavailableto deduce the current densities everywhere inside the diode.Moreover,theplot helpsone visualizethenatureof theJN(x)andJp(x)solutions.Thegeneralformof the solutions,we mightnote,is consistent withtheobservationmade at theendofthequalitative derivation. Whereas electronsandholesbothcontributeto thecurrentthroughthe depletionregion, the hole currentdominates far fromthe junctionon the p-sideof the deviceand the electron current dominates far from the junction on the n-side of the device.

--- Page 278 --- Pn JUNCTION DIODE: I-VCHARACTERISTICS 255 Current densities J \p$JNJN、$0 - x{q $x_n$ →x Figure6.7 Carrierandtotalcurrentdensities versuspositioninsideaforward-biased pnjunction diode. CarrierConcentrations Equations(6.23)and(6.25)specifythedeviationfromequilibriumof the minority carrier concentrations in the quasineutral regions.From these solutions we conclude forwardbi- asing increasesthecarrierconcentrationsovertheirrespectiveequilibriumvaluesandta reverse biasing lowers the concentrations below theequilibrium values.In either case,the perturbations decay exponentially as one proceedsaway from theedges of the depletion region.Moreover,after several diffusion lengths theperturbationseffectively dieoutand n²/NAasx→-∞and pn→pn0=n²/NDas→+∞independentof the appliedbias Because low-level injection is assumedto prevail in the quasineutral regions,we can also assert that themajoritycarrierconcentrations intheseregionsareeverywhere approxi- matelyequaltotheirequilibriumvaluesregardlessoftheappliedbias.lotsoftheforwad- and reverse-biascarrierconcentrations incorporating the foregoing informationare shown in Fig.6.8.Note thatthe exponential decaysof Δnand Δpnshowup as straight lineson the forward-bias plot becausethecarrierconcentrationsare beingplottedon a logarithmic scale. Underforwardbiasing themajority carriersareinjected inlarge numbersoverthe potentialhilltotheother sideof the junction.Onceontheothersideof the junction,the injected carriers become minority carriers and are progressively eliminated by recombina- tion astheyattempttodiffusedeeperintotheregion.（Thesituationandsolutionareallbut identical to thatofSampleProblem No.2considered inChapter3.)The netresult,pictured in Fig.6.8(a),is a build-upof minoritycarriers in thequasineutral regions immediately adjacentto theedgesofthe depletionregion. Thebuild-upofexcess minority carriers adjacenttothedepletionregion is aconsequenceof forwardbiasing,whichwillproveto be important in subsequent analyses. Underreversebiasing the depletionregionacts like a”sink”forminority carriers, draining thecarriersfromtheadjacentquasineutralregionsas pictured inFig.6.8(b).A

--- Page 279 --- 256 PnJUNCTIONDIODES n or p (log scale) ↑ | Pp nnPnn{n{2}N{DD}{} -x{p $x_n$ (a) n orp (linear scale) n_{2 P_{n \r{{D} n{ n{{{2} $\text{n}$ x

• x{pxn$ (b) Figure 6.8 Carrier concentrations inside a pn junction diode under (a) forward biasing and (b)reversebiasing.Thecross-hatchingidentifies excessminoritycarriers.Notethat (a)is a semilog plot while (b) is a linear plot.NA>ND was assumed in constructing the sample plots. reversebiasofonly a fewkT/q effectivelyreducesto zero the minority carrier concentra- tions attheedges ofthe depletionregion.Largerreversebiaseshavelittleeffectonthe carrier distributions.(Thisisconsistent withthefactthatthecurrent,whichis directly related to the slopeofthecarrier distributions at the depletionregionedges,saturatesfor appliedbiases greater than afewkT/q.)Overall,reverse biasing gives rise to a relatively smalldeficitofminoritycarriers inthe near vicinityofthe depletion region.

--- Page 280 --- Pn JUNCTION DIODE:I-VCHARACTERISTICS 257 Exercise 6.3 P:FigureE63isdimensionedplotofthstadstatecarierconcenrationinsie a pn junction diode maintained at room temperature. (a) Is the diode forward or reverse biased?Explain how you arrived at your answer. (b)Dolow-levelijectioncondtionsprvailinthequasinuraleionsofth Explain how you arrived at your answer. (c) Determine the applied voltage, VA. (d) Determine the hole diffusion length, L n or p (log scale) ↑ P_ 10^1}n_n}$1010^{}$10^8$Pn$ n{p 10^{ -103 →x 2.1×10^-{2}cm - x{q $x_{}$ 1.6×10^-{2}m Figure E6.3 S:(a) The diode is forwardbiased. There is a pile-upor minority carrierexcess (∆n>0and Δp>0) at the edges of the depletion region. (b)Low-levelinjectionconditionsdoprevail.ΔpnnnandΔnppeverywhere inside the quasineutral regions. (c) Wecanmake useofeitherthe depletionedge boundaryconditionsorthe”lawof thejunction”todetermine VA.Specifically,solving Eq.(6.12)for VAgives m$9$V{A}=$ -x{≤x≤ {

--- Page 281 --- 258 pn JUNCTION DIODES Evaluating the VA expression at the n-edge of the depletionregion and noting np→ n{²2}forx→±∞, we cmp \sr{{} n_{n(xn)) 1^{5 $VA$ ln$=(0.0259)ln(0.3Vq$n{n（∞)pn) \0{0 (d) Equation (6.23) can be rewritten ∆pn（x’）=∆pn（x’=0）e/p Inthe near vicinity of the depletion region edgeΔpn=pn,giving $pn(x^{\prime }）=pn(0)e-xLporpnx)$ $x^{}$ ln

p{0) Lp and $x^{}$ 1.6×10-2 Lp =3.47 × 10-3cm

$Pn(0$ 7 1010 ln ln p{n(xc) $\text{0}8r$ Exercise 6.4 Whenreverse biasedgreaterthan a fewkT/q,thecurrentflowing ina diodeisequal to q times the numberofminority carriers per second that wander intothe depletion regionandareswepttotheothersideofthe junction.Understeadystateconditions the numberofminoritycarriers thereby extracted per secondfrom the p-and n-sides of the junction must be preciselyequal to the numberof minority carriers generated per secondin thequasineutralregions.Inother words,thereverse-bias saturtion current may bealternatively viewedas arising fromminoritycarriergenerationinthe quasineutral regions,with Iρbeing equal to q times the numberof minority carriers generated per second. P: Suppose theactualreverse-biasminoritycarrier distributions in thequasineutral regions areapproximated by squared-off distributions as pictured in Fig.E6.4.All minoritycarriersaretaken to bedepletedaminoritycarrierdiffusion length toeither

--- Page 282 --- pn JUNCTION DIODE: I-VCHARACTERISTICS 259 sideofthe depletionregion.Remembering thatlow-levelinjectionprevails inthe quasineutral regions,and referring to theproblem introduction,derive an expression forthereverse-bias saturationcurrent basedontheapproximatedistributions. n or p Lp $n_{\text{2}}$ P{n NT $\text{rn}D$ n{ ← n{2 \rn{}$\to x$- x{pqx_{}$FigureE6.4distributionsisue\Delta n_{}$ n^{2}A} …-LN-xp≤x≤-x 1e thermal Tn τn$R-Gand\partial p$Δ_{n}n^{2D$...x\le x\le xn+LP1ethermalTpTpR-GGenerationistakingplaceinvolumesofALNandALponthep-andn-sidesofthejunction,respectively.Thus)+q(A1)i))(nN)$0 =q(AL ) T n D{n^{2} =qA + $A$ L{p{} One obtains the usual I0 expression! The approximation introduced in this problem,replacing an exponential distri- bution with a square distribution extending overone decay length (the same area being enclosed by both distributions),is often a useful analytical tool.

--- Page 283 --- 260 PnJUNCTIONDIODES 6.2 DEVIATIONSFROMTHEIDEAL Amoderatelylong listofassumptions was involved inderiving theidealdiode equation. Some of the assumptions were made without any a priori justification.It is therefore rea- sonable to expect that a careful examination ofexperimental I-Vcharacteristics willreveal discrepancies.Herein we first compare experiment and theory to identify themajor devia- tions fromthe ideal.Subsequently we treatspecific deviations,noting underlying causes and providing requiredmodifications to the idealtheory.Hopefullythe reader willnotbe dismayed bythe seemingly band-aidapproach in developing an acceptable theoretical de- scriptionof the diodeI-Vcharacteristics.Theiterativeapproachformulating a simple theory,comparing theory andexperiment,revising thetheory, comparing the revised theory withexperiment,and soon—is common engineeringand scientificpractice intreat- ing real problems of a complex nature. 6.2.1Ideal Theory Versus Experiment TheI-Vcharacteristic derivedfromaSidiodemaintainedatroom temperatureanddis- playedona0.1μA/divisionlinear-scale plotis reproduced inFig.6.9.For themost part thecharacteristicexhibits a formconsistent withtheoretical expectations.Theforward cur- rent is arapidlyrising functionofthe applied voltage,andthe reverse current is vanishingly small over a better partof the probed voltage range.However,there is one obvious eature not modeled by the idealtheory:A large reverse-bias currentflows when the reverse volt- age exceeds a certain value.This phenomenon is referred to as breakdown.Although the voltage where breakdown occurs can vary over several orders of magnitude,the phenome- non is common toall pn junction diodes.Breakdown constitutes a major deviationfrom the ideal and isaddressed in Subsection6.2.2. Todetermine whetherthereareadditionaldeviations,it is necessary to take a closer look attheI-Vdata.A semilog plotoftheforward-bias datafromtheFig.6.9device is displayed inFig.6.10(a).Reverse-bias datafrom the deviceare replotted inFig.6.10(b) employing an expandedcurrent scaleof50 pA/division.Data for both plots were obtained using a HP4145BSemiconductor Parameter Analyzer. The noticeable fluctuations inthe extremely smallreverse bias current are caused by noise inherent in the device andmea- surement system. Examining theFig.6.10(a)forward-bias data,we notethatthecurve exhibits theex- pected q/kTslopeforapplied voltages betweenapproximately0.35Vand0.7V.However, there are obvious deviations outsidethe cited voltage range.At forwardbiases in excessof 0.7Vthe slope progressively decreases or the characteristic”slopesover.”Thisdeviation, more or less to beexpected,is related to the highlevelofcurrentflowing in the device when VA→Vbi and is treated in Subsection6.2.4. Forvoltages below0.35Vthecurrentlevelsare quitesmalland thedeviationisclearly ofadifferentorigin.Inaddition,theobservedcurrentisfarinexcessoftheexpectedvalue and the slopeofthe characteristic curiously approaches q/2kT.These facts suggest we may be observing a current component eitheroverlooked or neglected in the derivationof the ideal diode equation.This suspicion is furtherenhanced by an examinationofthe reverse bias data.Forone,the device has an estimated I010-14A,some three orders of magni-

--- Page 284 --- PnJUNCTIONDIODE:I-VCHARACTERISTICS 261 1.0 0.8 0.6 1(A) 0.4 0.2 -50 -40 -30 -20 -10 0.20.4 0.6 0.81.0 -0.2 VA(volts) -0.4 -0.6 -0.8 -1.0 Figure6.9 LinearplotofthemeasuredI-Vcharacteristicderivedfromacommerciallyavailable Si pn junction diode maintained at room temperature.The plot permits a coarse evaluationofthe diodecharacteristic.Notethechange in voltage scale in going from forward toreverse bias. tude smaller than the reverse currentobserved at VA=-5V!Moreover,the Fig.6.10(b) characteristic does not saturate as predicted by the ideal theory. Rather,the current continu- allyincreases withincreasing reversebias.As willbeverified inSubsection6.2.3,the cited reverse-bias and small forward-bias deviations are caused by theadded current aris- ing from carrier recombination-generation in the depletion region. Carrier R-G was as- sumed to be negligible throughoutthe depletion region in deriving theideal diode equation. Although feature details do varyfrom device to device,and other nonidealities may be encountered,thedata displayed inFigs.6.9and6.10arefairlyrepresentativeof theI-V characteristics derivedfrom commerciallyavailableSipn junction diodesmaintained at room temperature.Similarcharacteristics are exhibited by GaAs diodes. Ge diodes main- tained atroom temperatureandSi diodesoperated at elevatedtemperatures,on theother hand,areoften foundto morecloselyapproachtheideal,specificallyexhibiting saturating reverse-bias characteristics.An explanationof this difference in behavior is included inthe R-G current discussion.

--- Page 285 --- 262 pn JUNCTION DIODES 100 $10^{-}2$10-4 1(amps) 10-6 q/kT 10-8 10-10 q/2kT 10-12 0 0.2 0.4 0.6 0.8 1.0 A (volts) (a) VA(volts) 0- -40 0- -20 -10 0 0 -100 -200 I(pA) -300 -400 -500 (b) Figure6.10 Detailedplotsof the measured I-Vcharacteristicderivedfrom acommercially avail- able Si pnjunction diode maintained atroom temperature.The Fig.6.9 and Fig.6.10characteristics are from the same device.(a) Semilog plotof the forward-bias current versus voltage.(b) Expanded scale plot of the reverse-bias current versus voltage.

--- Page 286 --- pn JUNCTION DIODE: /-VCHARACTERISTICS 263 6.2.2 Reverse-Bias Breakdown Althoughreferred toas“breakdown”the large reverse currentthatflows whenthereverse voltageexceedsacertainvalueiscompletelyreversibleprocess.Thatisbreakdowndoes not damage the diode in any way.The current must be limited,ofcourse,to avoid excessive heating.Theabsolute valueof thereverse voltage wherethecurrent goesoff to infinity is known asthebreakdownvoltageandisgiventhesymbolBR.PracticalVBRmeasurements typicallyquote the voltage where the currentexceeds a preselected valuesuch as 1μAor 1mA.The expected breakdown voltage of planar (one-dimensional) p+-n and n+-p step- junction diodesas a functionof the nondegenerate-sidedoping is plotted inFig.6.1for select semiconductors.AtagivendopinBRtendstoincrease withthebandgapofthe semiconductorusedtofabricatethediode.Ofgreatersignificance,thedopingonthe lightly doped sideof thejunctioncanbeusedto vary VBRfromafew volts tooverathousand volts. Notethatthe doping dependenceabovethe dashedlineinFig.6.11isrouhly described by 1 V_{ BR 80 N{{7} (6.33) where NB is the doping on the lightly doped sideof the junction. 1000 GaAs Si 100 V{BR (volts) Ge 10 1 $1014$10{s$ 1016 1017 1018 NA or ND(cm-{3)}) Figure6.11 Breakdown voltage as a function of the nondegenerate-side doping in planar p+-n and n+-p step-junction Ge,Si,and GaAs diodes.Avalanche is the dominant breakdown process for dopings above the dashed line.T=300K.(After Sze[1],©1981by John Wiley & Sons,Inc.Re- printed with permission.)

--- Page 287 --- 264 pn JUNCTIONDIODES From a theoretical standpoint,breakdown is directly related to thefailureofthe“no other processesassumptioninthederivationoftheidealdiodeequation.Infacttwo “other processes”—avalanching and the Zener process—can cause the breakdown cur- rent.Avalanchingistypicallythedominantprocess,withtheZenerprocessonlybecoming important whenbothsidesofthe junctionare heavily doped.In whatfollows, we delve into the physical nature of the two processes and provide process-specific information. Avalanching Working up to a physical descriptionof the avalanching phenomenon,let us first consider a reverse-biased diode where Visrelativelysmall andfar belowthebreakdown voltage. Ideally,thereversecurrentflowing inthediodeis duetominoritycarriersrandomlyenter- ing the depletionregionandbeingacceleratedbytheelectricfieldintheregiontotheother sideofthe junction.Incrossing the depletionregionthecarrieraccelerationis notcontinu- ous but is interruptedbyenergy-losing collisions withthe semiconductor lattice,as envi- sioned in Fig.6.12(a).Sincethemeanfree pathbetween collisions is10-6cm,and a median depletion widthis10-4cm,acarriercanundergotenstothousandsofcollisions in crossing the depletionregion.Thusatsmallappliedreversebiasestheenergy lost bythe carrierspercollisionisrelativelysmallTheenergytransferredtothelatticesimplycauses lattice vibrations—there is justlocalizedheatingthat isreadily dissipated. With increasing reverse biastheamountofenergy transferred tothe semiconductor lattice percollisionsystematically increases.Approachingthebreakdownvoltage,theen- ergytransferred percollisionbecomessufficienttoionizeasemiconductoratom.By”on- ize” we mean the collision frees a valence electron from the atom,or causes an electron from the valence bandtojumpintotheconductionband,therebycreatinganelectron-hole pair.Thisphenomenon,calledimpact ionization,was previously visualizedin Fig.3.15(f). Theaddedcarrierscreatedbyimpactionizationareimmediatelyacceleratedbytheeectc fieldin thedepletionregion.Consequentlytheyandtheoriginalcarriersmakeadditional collisionsandcreateevenmore carriers as envisioned inFig.6.12(b).Theresult isa snow- balling creationofcarriersverysimilartoanavalancheofsnowon amountainside.Atthe breakdown voltagethecarriercreationandreversecurrenteffectively gooff to infinity. Acouple ofclarifying comments are inorderconcerning the preceding discussion and avalanching.Forone,theFig.6.12(b)energybanddiagram isnotandrealisticallcould notbe,drawn to scale.Ifthe breakdown voltage were10oV,for example,the distance betweenthep-andn-sideFermilevels wouldhavetobeapproximately100timesthe EminusEband gapdistance.Second,it is importantto notethatavalanchebreakdown does notoccur sharply atVA=-BR.Whenthereverse-bias characteristicsareexam- ined carefully as in Fig.6.10(b),theyare foundtoexhibita sloping approachto break- down. There is considerable carrier multiplication a few volts before reachingbreak- down and even some carriermultiplication at voltages far belowbreakdown.Thereason is thatthe distance between collisionsis arandom variable statistically distributed aboutthe mean value.Thuscarrierscanoccasionallygain sufficientenergyto have an ionizingcollisionat voltagesfarbelowbreakdown.Ata fewvoltsbelowbreakdownthe

--- Page 288 --- PnJUNCTION DIODE: I-VCHARACTERISTICS 265 —0 Ec Ec EFn EFn Ev} Ev}(a) Small reverse bias (b) |VA|→V{BR Figure6.12Carrieractivityinsidethedepletionregionofareverse-biasedpnjunctiondiodewhen (a)VA≤VBRand(b)V→VBR.Carriermultiplicationduetoimpactionizationandtheresultant avalanche is pictured in (b). numberofcarriers gaining sufficientenergyto have an ionizingcollisionbecomesquite large.Theincrease incurrentassociatedwiththecarriermultiplication ismodeled byintro- ducing a multiplication factor,M.IfI0istakentobethe current withoutany carrier multiplication, then M (6.34)Io$ An empirical fit to experimental data gives 1 M= [V (6.35) m 1 一

--- Page 289 --- 266 pn JUNCTION DIODES where m takes on a value between 3 and 6,depending on the semiconductor used to fabri- catethediode.Please note thatthemultiplication factor can be used to correct theideal diode equation to account for carriermultiplication and avalanching. We next seek to explain the VBR dependence noted in the breakdown introduction. From thequalitative descriptionof avalanching,it was concludedthatbreakdown occurs when thecarriers gainan ionizingamountofenergy in travelinga lattice-scattering mean free path.This shouldbetrue independentof the junctiondoping.A specificenergy gain over a given distance,however,corresponds to a specific electric field.Inother words, breakdown occurs whentheelectricfield in the depletionregionreaches some critical value,CR,essentially independent ofthe junction doping. Considering a step junction,employing Eqs.(5.35)and(5.37),and evaluating theelec- tric field atx=0,we find $qND$ 2q $NNDD$ 1/2 (0)=- $K\epsilon 0$x _{}$二一(6.36)Next,squaringtheprecedingexpressionandmakinguseofthefactthat(O)\to cRwhenbi-VA\to bi+BRBR，weobtain$82}$2q$N{AN{DVBR (6.37) K0NA+N Since &CR is independentof doping,theright-hand sideof Eq.(6.37) must likewise e independentofdoping.Theright-hand sideofEq.(6.37)willbe independentof dopingif NA+NDD V_BR 8 (6.38) $NAND$ or for asymmetrically doped junctions 1 V{BBR N_BB} (6.39) The Eq.(6.39)result is not preciselyasobserved experimentally,but it is acceptably close given the simplicity of the argument. Concerning the VBR band gap（or semiconductor) dependence，we again note that breakdown occurs when thecarriers gain an ionizing amountofenergy in travelinga lattice-scattering mean free path. Therequired ionizationenergyobviously increases with increasing EG,but the meanfree path is found to varyonlyslightly for the semiconductors cited inFig.6.11.Thus,consistentwithFig.6.11,thebreakdownvoltage is expectedto progressively increase in going from Ge to Si to GaAs. One more dependence is worthmentioning.The VBR due to avalanching is found to increaseas thetemperature goes up. As noted in the mobility discussion in Subsec- tion3.1.3,lattice scattering increasesas thetemperature increases.Increasing lattiesat-

--- Page 290 --- Pn JUNCTIONDIODE:I-VCHARACTERISTICS 267 tering means a smallermean free path,a largercriticalelectricfieldforavalanching,and hence a higher breakdown voltage. Finally,the discussion has focused on planar step-junctiondiodes.Separate breakdown curves applyto linearlygradeddiodes.Moreover,step-junctiondiodes formed by diffusion or ion-implantation through a mask always have curved, nonplanar lateral edges. For a givenapplied voltagetheelectricfield is greater in the nonplanarregionthanin theplanar part of the device.Hence breakdown occurs sooner in the nonplanar regions,lowering the V_{p}$ BR ofthe diode. This”curvature”effect becomes more pronounced as the depthof the junction decreases.Specialedge modifications have been developed to minimize the effect. (C) Exercise 6.5 P:Computeandmakeplotofthemultilictionactoversus/BRhen m=3andm=6.Orienttheaxessoareverse-current-likeplotisobtained;i.e.,plot Mincreasing downwardandVAl/VBR increasing totheleft. Discuss theplotted results. S:The computer program scriptand resultantplot(Fig.E6.5)follow.Using MATLAB, the author wasonly able toobtain thedesired axes orientation byplotting-Mversus -A/BR.Theplotisnotedtobequitesimilartotheleft-handsideofFig.6.10(b). A perhaps surprising2× increase in current is predicted for applied voltages less than 80% and90%of the way to VBR for m=3 and m=6,respectively.There is already acarrierenhancementof~10%(M=1.1)at0.45VBR whenm=3 and at 0.65VBR when m= 6. MATLAB program script… %Exercise6.5..Multiplication factor %Initialization close clear %M calculation x=linspace(0,.99); %x=|VA|/VBR M3=1./(1-x.^3); %M when m=3 M6=1./（1-x.^6); %M when m=6 %Plotting result plot(-x,-M3,-x,-M6); grid axis([-1 0-100]) xlabel(’-|VA|/VBR’) ylabel(‘-Multiplication factor’) text(-0.8,-2.5,‘m=3’) text(-0.95,-1.5,‘m=6)

--- Page 291 --- 268 pn JUNCTION DIODES 0 -1 m=6 -2 m= 3 -3 -Multiplication factor -4 -5 -6 -7 8- -9 -10 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 -IVAI/VBR Figure E6.5 Zener Process Zener process isthe namegiven to theoccurrenceof“tunneling”inareverse-biaseddiode. Tunneling is thefirstphenomenon we haveencountered that isof a purely quantum- mechanicalnature.Ithasnoclassicalanalog.Thegeneralnatureandbasicfeaturesof tunneling can beunderstood with theaidofFig.6.13.The particlepictured inthe figure is taken to bepositionedonthe left-handsideofa potentialenergybarrier.Theheightofthe barrier isassumedto begreaterthanthekineticenergyofthe particle.Classically,theonly way a particlesuchasanelectron couldmove to theothersideofthebarrier wouldbeto gainadditionalenergyand goover the topof thebarrier.Quantummechanicallythere is another waythe particlecangettotheothersideofthebarrier:Itcan gothrough the barrier. Tunneling is going”through”a potentialenergybarrier.It shouldbeemphasized that the particleenergyremainsconstantduring the process.Also(thinkingclassically)theparticle and the barrier are not damaged in any way. There are two major requirements for tunneling to occur and be significant: (1)Theremust befilledstatesononesideofthebarrierandemptystatesontheotherside ofthebarrieratthesameenergy.Tunnelingcannottakeplace intoaregionvoidof allowed states. (2)The widthofthepotentialenergybarrier,inFig.6.13,mustbeverythin.Quanum- mechanical tunneling becomes significantonly ifd<100Å=10-6cm.

--- Page 292 --- PnJUNCTION DIODE: I-VCHARACTERISTICS 269 Potentialenergy E barrier ↑ Particle →x d Figure6.13 General visualization of tunneling. Ecp $Evp$ Filled states Empty states -EcnE_vn$ -Barrier kW→1 Figure6.14 Visualization oftunneling in areverse-biased pnjunction diode. Visualization of the Zener process,tunneling in a reverse-biased pn junction diode,is presented in Fig.6.14.The particlesdoingthetunneling are valencebandelectronsonthe p-sideofthe junction.The potentialenergy barrierclassicallyrestrictingtheseelectronsto the p-sideof thejunction is outlined in thefigure.Tunnelingtakes place when theelectrons pass throughthe barrierto emptystates at the sameenergy intheconduction bandonthe n-sideofthe junction.Thegreaterthereverse bias,the largerthe numberoffilled valence- electron states on the p-side placed opposite emptyconduction-band states on the n-side, and hence the greater the reverse-bias tunneling current. For tunneling to be significant,the barrierthickness,roughly the depletionwidthin the caseof the pn junction diode,must be<10-6cm. Referring to Fig.E5.3,wefind Si diode depletion widths<10-6 cm necessitate dopings in excessof107/cmonthe “lightly”dopedsideof the junction.Thus the Zenerprocess is importantonly in diodes that are heavily dopedonbothsidesof the junction.Thebreakdown voltageof the diodes is correspondingly small.The Zenerprocess makesasignificantcontributiontothebreak-

--- Page 293 --- 270 pn JUNCTION DIODES down current in diodes where VBR<6EG/q（~6.7Vin Si at 300K）and dominates in diodes where VBR<4EG/q(~4.5Vin Si at 300K). Two experimental observations can be used to distinguish between avalanching and Zener processbreakdown.Firstwhereasthe VBR associated withavalanching increases withincreasing TVBR decreases withincreasingTifthe Zenerprocess is dominan.Sec- ond,thebreakdown characteristics associated with the Zener process are very”sof.”The currentexhibits a veryslowapproachtoinfinityeven whenobservedonarelatively coarse scale. Historically,the Zener process was thefirst proposed to explainreverse-bias break- down.All diodes specificallyfabricatedtomake use of thebreakdowncharacteristic came to be known as Zener diodes. The name continues to be applied to all diodes making use of the breakdown characteristic,even thoughthose withbreakdown voltages in excessof 6E/q are functionally avalanche diodes. 6.2.3TheR-GCurrent In comparing theory andexperiment,a current far in excess of that predicted by theideal diode theory was foundto exist at small forward biases and allreverse biases in Si diodes maintained at room temperature. Theobserved”extra” current arises from thermal carrier recombination-generation in the depletion region that was assumedto be negligible in the derivationofthe ideal diode equation.Howthermal R-Gin the depletion region gives rise to an addedcurrentcomponentcanbeunderstood withtheaid ofFig.6.15.First consider the reverse-bias casemodeled in Fig.6.15(a).Heretofore inallquantitativeanalyses,and even in thequalitative derivationof the ideal diode equation, we associated the reverse current withthe minoritycarriers wandering into the depletionregion fromthe two sides of the junction. When the diode is reverse biased,however, the carrier concentrations in the depletionregion are reducedbelow their equilibrium values,leading to the thermal generationofelectronsandholesthroughouttheregion.Thelargeelectricfieldinthe depletionregionrapidly sweeps the generated carriers intothequasineutralregions, therebyadding to thereverse current.Forwardbiasing increasesthecarrierconcentrations in the depletion region above their equilibrium values giving rise to carrier recombination in theregion.As envisioned in Fig.6.15(b),onecaneffectively viewtheresultingadded forward currentas arising fromthecarriersthatcannotmakeitoverthe potential hillbeing partiallyeliminated via recombinationat R-Gcenters in the depletionregion. SeekingtoestablishaquantitativeexpressionfortheaddedcurrentRG,arisingfrom thermal recombination-generation in the depletion region, we note that the net R-Grate is the same for electrons and holes under steady state conditions.Moreover,for every elec- tron-hole pair createdor destroyed in the depletion region per second,oneelectron per secondflows into oroutof the diodecontacts.Summing either theelectrons or theholes created/destroyed throughout the depletion region per second andmultiplying byq should Useoftheterm depletionregion in thepresentcontext sometimesleads to confusion.Itmust beremembered that thecarrierconcentrationsinthedepletionregionare notzeroorautomaticallyless thantheireuilibium values,but are merely small compared to the background doping concentrations.

--- Page 294 --- pn JUNCTION DIODE: /-VCHARACTERISTICS 271 Ideal diode current R-G current $Ec$ EFn R-G current -ET Ev} Ideal diode current (a) Reverse bias Ideal diode current R-G current $Ec$ EFn ET Ev} Ideal diode current (b) Forward bias Figure6.15TheR-Gcurrent.Visualizationoftheadditionalcurrentresultingfromarverse- bias generation and(b)forward-bias recombination in the depletion region. thereforegive the magnitudeof theaddedcurrentflowing in thedevice.Proceeding as indicated and accounting for the polarity of the current yields {n ∂n b-=- dx (6.40) -x{p$ 1e thermal R-G

--- Page 295 --- 272 Pn JUNCTION DIODES Note that,becauseof the conditions prevalent in the depletionregion,the familiar special- case RGrlationshintherlG∆/isnotapplicable.Ratherthegeneal- case result initially presented as Eq.(3.35)must be employed: ∂n $np-n_2$ (6.41) 1e thermal 二 p(n+n₁1）+n（p+p1) R-G and f_} np - n^{^} $R-G=qAdx(6.42)-xpp(n+n1）+n（p+p_{1}1)ForreversebiasesgreaterthanafewkT/q,thecarrierconcentrationsbecomequitesmallthroughoutmostofthedepletionregion.Withthecarrierconcentrationsbeingng-ligible(n\to 0,p\to 0),theintegralinEq.(6.42)isreadilyevaluatedtoobtain$q_An_$

20 W …reverse biases > few kT/q (6.43) where To 2 1 一 Toni n_} + Tn P} n{

2 1 一 n −ET)/T (6.44) For forward biases thecarrierconcentrations cannot be neglected andeven an approximate evaluationoftheEq.(6.42)integralbecomesratherinvolved.WemerelynotethatGi expected to vary roughly as exp(qVA/nk),1<η≤2,for forward biases greater than a few kT/q.2] Typically,the expectedηis closeto2,and thecombined forward and reverse bias dependence is approximately described by q_An (eVA/kT-1) W (6.45)

2τ0 1+ V_{bib √ T_{TP e9V{}12kT kT/q 20 With the introduction of a second current component,it is common practice to refer to thecurrentdescribed bytheidealdiodeequationas thedifusioncurrentIDF.The diffusion current expression is rewritten here to facilitate a comparisonof the two

--- Page 296 --- pn JUNCTION DIODE: I-VCHARACTERISTICS 273 components: (d:n}2}D{nn}{2}vb =}{}$+$pr{B}$(eqVA/kT-1)(6.46)ThetotalcurrentflowinginthediodeisofcoursejustthesumofthediffusioncurrentandtheR-Gcurrent:I=IDIFF+R-G(6.47)Wearenowinapositiontoexplainearlierexperimentalobservations.InSidiodesatroomtemperatureqAnW/2\tau 0\ge I0andtheR-GcurrentdominatesatreversebiasesandsmallforwictersiGirorionlotherneversaturates,butcontinuallyincreaseswithincreasingreversebias.TheforwardbiasRGvariesasexpV/2kT)forVA>fewkT/q,consistentwithexperimentalobservationsatsmallforwardbiases.WithincreasingforwardbiasestheIDIFFcomponentwhichin-creasesmorerapidlwithvoltageeventuallyovertakestheRGomonenleadingtotheqkTreonomiloplotofheorwrdbiascctisic.cuseF$αn²2}hile ee torto semiconductor. Whereas qAn;W/20>I0in Si and GaAs diodes at room tempera- ture,the larger nofGe typicallymakes >qAn;W/20inGediodes at roomtemperature. Also, since _0n² while the reverse bias IR-G ∞n,the reverse bias diffusion component of the current will increase at a faster rate with increasing temperature. The diffusion com- ponenteventuallydominates at a sufficientlyelevated temperature.The n;dependenceex- plains the experimentalobservations thatGe diodesmaintained at room temperature and Si diodesoperated atelevatedtemperatures areoftenfoundtomoreclosely approachthe ideal, specifically exhibiting saturating reverse-bias characteristics. (C) Exercise 6.6 Ifthewktofclteexle expressioioedheoljucioiocuroorwe than a few kT/q may be approximately modeled by I =I01eVAkT+02eqVA/2kT ↑ ↑ IDIF $IrG$ whereboth0and2aretakentobeconstantsindependentofvoltage.

--- Page 297 --- 274 Pn JUNCTION DIODES P:(a)Compute 01 and02fora Si p+-n step junction diodemaintainedat 300K withA=10-4cm²，ND=101/cm3,andτn=p=τ0=10-6sec.Inthe2 computation,assume thattheexpV/2k)terminthedenominatorofthe-G expression is much greaterthanunityand set VA=Vbi/4 inevaluatingvoltage- dependent terms. As in previous exercises compute Vbi using Vbi=(E2q)+ (kT/q)ln(ND/n₁). (b) Repeat part(a)foraGep+-nstepjunction diode maintained at300K.Assume a setofparametersidenticaltotheparta)Sidiodeexceptthat150cms n=2.5×1013/cm3,andK=16. (c) Making useofthesimplifiedcurrentrelationshipintroducedinthisproblemcon struct a semilogplotofI versusVAthatsimultaneously displays the forward-ias characteristics of thepreviously described Si and Ge diodes.Limit the plot axes to 0V≤VA≤1Vand10-1²A≤I≤10-A.Begin the calculationat VA=0.1V. Discuss your results. S: Thecomputed 0and2 values forthetwo diodes are summarized in thetable presented below.The part (c)computerprogramandplotfollowthe table.The and I2values are stored inmemoryafterrunning the listedcomputerprogramand were accessed from the MATLAB Command window. Asexpected,20fortheSidiode while01>2fortheGediode.Because of thesmall bandgapand corresponding large nlarge forwardcurrentsflowinthe Ge diodeatrlativlysmall voltagesakinitllbuimossibletoobrvethe component.Conversely,boththeIDIFFandR-Gcomponents areclearlyevidenton the silicon diode plot. Silicon Germanium $0$ 5.38×10-16A 6.23×10-9A ${}_{o}r$ 2.08×10-13A 8.34×10-10A MATLAB program script… % Comparisonof forward bias I-VA for Si and Ge diodes at 300K. % This program uses a simplifiedformula for the current: %I=I01exp(qVA/(kT))+I02exp(qVA/(2kT)). % Also, Vbi=EG/2q+(kT/q)ln(ND/ni).

--- Page 298 --- pn JUNCTION DIODE:/-VCHARACTERISTICS 275 %Initialization close clear % Constants T=300; % Temperature in Kelvin k=8.617e-5; % Boltzmann constant eV/K e0=8.85e-14; % permittivity of free space (F/cm) q=1.602e-19; % charge on an electron (coul) KS=[11.816]; % Dielectric constant [SiGe] ni=[1.0e102.5e13]; %intrinsic carrier conc. at300K[SiGe] μp=[437 1500]; %hole mobility [SiGe] EG=[1.120.66]; % band gap [Si Ge] % Given Constants A=1.0e-4; %cm^2 ND=1.0e16; %cm^(-3) taun=1.0e-6; % seconds taup=1.0e-6; % seconds %101 DP=kT.p; LP=sqrt(DP.taup); I01=qA.(DP./LP.ni.^2./ND); %I02 Vbi=EG/2+kT.log(ND/ni); W=sqrt(2.KSeO/(qND).Vbi); I02=qA.ni/sqrt(tauntaup).W.(kT).(3.Vbi./4); % Currents for bothSilicon(ISi)andGermanium(IGe) VA=linspace(0.1,1); ISi=I01(1).exp(VA.(kT))+I02（1).exp(VA./(2kT)); IGe=I01(2).exp(VA./(kT))+I02(2).exp(VA./(2k*T)); % Plot semilogy(VA,ISi,VA,IGe,-); grid axis（[011.0e-121.0e-3]); xlabel(‘VA(volts)’); ylabel(‘I(A)’); text(.7,1.4e-9,T=300K’); text(.7,4.0e-10,‘ND=1.0e16/cm^3’); text(.25,1.4e-5,‘Ge’); text(.48,1.4e-8,‘Si);

--- Page 299 --- 276 pn JUNCTION DIODES 10-3 10-4 10-5 G 106 10-7 ()I Si 10-8 10-9 T=300K ND=1.0e16/cm^3 10-10 10-11 …… ……… 10-12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 VA(volts) Figure E6.6 Exercise 6.7 A generalizationoftheforward-biasmodeling expression in Exercise6.6 iscom- monly introduced inanalyzing experimental data.Specifically,oneemploys I=01eqVA/n₁kT +I{2eqVA/n₂2kT ↑ ↑ IDIFF where nand n₂ are arbitrary constants determined by a matchtothe near-ideal and R-Gcurrentrelated regionsofthe forward-bias characteristics,respectively. P:Determine the 01,02,n₁,and n₂that provide anoptimalmatch to the forward biasI-V data presented in Fig.6.10(a). S:FigureE6.7isareproductionofFig.6.10(a)withstraightlines drawn throughthe two linear regions onthe plot. Based on the generalized modeling expression （and similar to Eq.6.31),themathematical descriptionof the straight lines is $ln(I)=ln(0j)+9VA$ …j=1,2 $n;$

--- Page 300 --- pn JUNCTION DIODE: I-VCHARACTERISTICS 277 TheVA=0interceptthereforeyieldsthejvalues,andthenvaluescanbecom- puted from VA_{2}-V{ $nj$

(kT|q)ln(I2/11) where 1and2 are any two points along the straight lines.Weconclude $01\text{1}0-14A$n_{1} 1.10{2}10-1{2A n_{2}=} 1.5 TheIDIFFcomonentisalmostuncharacteisticallyidal.Theobservedc isonthelowsidebut withinthenormallyobservedrange.201,asexpeced. 100 10-2 10-4 (amps) 10-6 q/kT 10-8 10-10 q/2kT 10-12 0.2 0.4 0.6 0.8 1.0 VA (volts) Figure E6.7 6.2.4VA→VbiHigh-CurrentPhenomena A significantcurrent beginsto flow inthe diode whentheappliedforward bias approaches Vbi.Alargecurrent inturncreatesconditions inconsistent withanumberof idealdiode assumptionsandapproximations.Forone,theassumptionthatalloftheappliedvoltage is dropped across the depletion regionbecomes questionable.IfI=0.1A,for example,a

--- Page 301 --- 278 Pn JUNCTIONDIODES resistanceoutsidethedepletionregionofonly1ohmwouldgiverisetoasignificant voltage dropof0.1V.Alarge currentmay alsoproduceahigh levelof injection.The effectsand modeling of these deviations are considered next. Series Resistance Thequasineutral regions have an inherent resistance determined by the doping and dimen- sions of theregions.Althoughquite small in a well-madedevice,there is also aresidual resistance associated with the diode contacts.These combine to form a resistance,Rs,in series withthecurrentflowacross thejunctionasenvisioned inFig.6.16(a).At lowcurrent levels thevoltage dropacross theseriesresistance,IR,is totally negligiblecomparedto the applied voltagedropacrossthedepletionregionbetterknownasthe”junction”voltage, V.Under thecitedcondition,V=VA,as assumedintheelectrostaticandidealdiode - x{q $x_n$ I P N VA} |V}→Rs$ (a) log(l) △V △V Slope-over region Slope = Rs q/kT slope →VA → I (b) (c) Figure6.16Identificationanddeterminationoftheseriesresistance.a）PhysicaloriginoR. (b) Forward-bias semilogplotused to deduce ΔVversusI.(c)ΔVversusIplotused to deduce Rs.

--- Page 302 --- PnJUNCTION DIODE: I-VCHARACTERISTICS 279 derivations.AtcurrentlevelswhereIRbecomescomparabletoVA,howeve,theapplied voltage dropappearing acrossthedepletionregion isreducedto V=VA-_s$ (6.48) Effectively,partoftheappliedvoltageis wasted,a largerappliedvoltage is necessaryto achievethesame levelofcurrentcomparedtotheideal,and thecharacteristicsslopeover as illustrated inFig.6.10(a). To correctfor the series resistance,one merely replaces VAby V=VA-IRsin previouslyderivedI-VArelationships.Sincethediffusioncurrenttypicallydominatesat the current levels whereIRs becomes important, we can write I=I0eqV/kT=I0eq(A-Rs)/kT…VA→Vbi (6.49) Technically Eq.(6.49)is a transcendental equationthatcannot besolved for I as a function ofVA.However,I versusVAis readilyestablished bychoosing a Vvalue,computingI from Eq.(6.49),and then computing VAfrom Eq.(6.48).Thecomputation doesrequire a knowledge ofRs.Rs canbe determinedfrom experimental data as outlined in parts (b) and (c)ofFig.6.16.Working with a forward-biassemilog plot similar toFig.6.10(a),one extends the ideal-diode partoftheplotintotheslope-overregionand notestheΔVvoltage displacement betweenthetwocurves as a functionof I.Since △V=VA-V=R,the slopeofthelinethroughaplotoftheΔVversusIdatayields. Exercise 6.8 P:Crudelyestimatetheseries resistanceofthe diodeexhibiting theI-Vcharacter- istics presented in Fig. 6.10(a). S: Ablow-up of the slope-over region is needed to producea decent ΔV versusI plot.We can,however,estimate R using the ΔVat a given I.The Fig.6.10(a)I-V curve terminatesata currentofabout80mAwhenVA=0.85V.Referringto Fig.E6.7,wefind thatan extensionof theq/kTregionon theplot passes through 80mAatroughly0.78 V.ThusΔV≈0.07Vand R=△V/I≈0.07/0.08~1ohm. High-Level Injection Thelow-level injectionassumptionmade inthe derivationof the ideal diode equation be- gins to fail when theminoritycarrierconcentrationatthedepletionregionedgeonthe lightly doped sideof the junctionapproaches thedopingconcentration.InSiatroom tem- peraturethistypicallyoccursatappliedvoltagesafewtenthsofavoltbelowVi.Afurther increaseintheappliedvoltagegivesrisetohigh-levelinjection.Underhigh-levelinjection

--- Page 303 --- 280 pn JUNCTION DIODES boththeminority carrierand themajority carrierconcentrations adjacent to the depletion region are perturbed,as pictured in Fig.6.17(a).Themajority carrier concentration must increase to maintain approximate charge neutrality in the quasineutral regions.An analysis ofhigh-level injection leads to a predictedcurrent varying roughlyasexp(q/2kT).Inother words,one expects a high-current q/2kTregion on a semilog plotof the forward bias I-V characteristicsas sketched inFig.6.17(b).Thepredictedhigh-currentq/2kTregionis seldom observed,however, because it is obscured by the slope-over associated with se- ries resistance. We shouldnote,nonetheless,thattheenhanced carrier concentrations as- n or p (log scale) 4 High-level injection Pp nn Pn$np$ →x - x{q x_} (a) log(l) Caused by high-level injection q/2kT slope Ideal region →VA (b) Figure6.1High-levelinjction.Carrieconcentationsundehi-levelinjecionconiton. (b) Predicted effect on the observed characteristic.

--- Page 304 --- pn JUNCTION DIODE: 1-VCHARACTERISTICS 281 sociated withhigh-levelinjectioncanreducetheobservedseriesresistance.Thereduction inresistivitresultingromhighlevelsofcarrieinjectionireferredtoasconductivity modulation. Exercise 6.9 P:Themeasured I-Vcharacteristic ofa Si diodemaintained atroom temperature is crudely sketched inFig.E6.9.Note thatthecurrent scale is logarithmicfor forward biasand linearforreverse bias.Nonidealitiesexhibited bythecharacteristicareiden- tifiedbycapitalletters.Variouspossiblesourcesforthedeviationsfromtheidealare listed totherightofthesketch.IdentifythecauseofeachnonidealIVfeature;place the proper source number(s) adjacent to the letters on the sketch. log(l)

1. Photogeneration E 2.Thermal recombination D in the depletion region Ideal
2. Avalanching and/or region Zener process -V_A$ C
3. Low-level injection VA$ 5.Depletion approximation
4. Thermal generation in the ↑ depletion region B 7.Band bending A
5. Series resistance
6. V_{A>V_bib
7. High-level injection -1 Figure E6.9 S: A3,B6,C2, D10, E8 6.3 SPECIALCONSIDERATIONS Thetwo topics addressed in this section areofa supplemental nature as far as the basic d.c. responseof the pn junction diode is concerned.Considerations associated with the topics couldbe delayeduntilabsolutely required later in thetext, butthat wouldtendtode-

--- Page 305 --- 282 pn JUNCTION DIODES emphasize their importance by submerging them in a larger context. Also,a linkage of ideas and concepts critical to a greater depthof understanding becomes more difficultto establish.Thechargecontrolapproach,addressedfirstisa sortof”big picture”method of analysis thatoftenallows a reasonablyaccurate approximate solution with a minimum of mathematics. The narrow-base diode,considered inSubsection 6.3.2,is both a special deviation from the idealand a conceptual link to the bipolar junction transistor. 6.3.1ChargeControl Approach In the charge control approach,thebasiccarrier variable is thecharge associated withthe minoritycarrierexcess（ordeficit)withinanentirequasineutralregion.Tobespecific, consider aforward-biasedp+-njunction diode.LetΔpn(x,t)bethe minoritycarrierexces in then-side quasineutral region at a given timet and at a point x,n≤x≤∞.The total excess hole charge,Qp,within the region is then ∞ $Q=qA$ Δp{n(x，t)dx (6.50) $Jxn$ Equation(6.50)also appliestoreverse bias withanegativeQpbeing interpretedasacarrier deficit.Qp is of course independent of x,but is potentially a function of time. Still considering a p+-n diode,we next seek a relationship that describes theoverall behaviorofthis combinedminority carrier charge in then-sidequasineutral region.A logical placeto start is theminoritycarrierdiffusionequation,Eq.(3.54b)which descrs theoverallbehavioroftheminoritycarrierholeconcentrationatagivenpoint.Specifically, with G_=0, aΔp_{ a^{2Δ}_{ Δp_{$

D{$(6.51)1e$∂x^{2}$pSinceina\approx 0region$aΔp{n{ =-q (6.52) ∂x we can alternatively write∂(q△pna_}qΔ_{n$ (6.53) ∂t xe Tp

--- Page 306 --- PnJUNCTIONDIODE:I-VCHARACTERISTICS 283 Being interested in thecombinedbehavioroftheqΔpn(x,t),let us integrate alltermsin Eq.(6.53)over the volume(=A∫dx)of the n-side quasineutral region.After somerear- rangement, we arrive at a -[A$d$b}$\infty =-A${p(∞)$(6.54)2p$J_n}x_n$ThebracketedquantitiesarerecognizedtobeQp.Moreover,$J{p(∞)$-AdJP=-AJP（\infty )+AJ（）=AJp(n）\equiv AJDF=iDFF(6.55)$J{p(n)$Theholecurrentgoestozerofaronthen-sideofthejunction,makingJp(\infty )=0.SineusedforthefinalresultinEq.(6.55)becausethecurrentispermittedingeneraltobeafunctionoftime.MakingtheindicatedsimplificationstoEq.(6.54)yields$dQ{P$=iDIF$Q{$(6.56)pTpTheresultofourmanipulations,Eq.(6.56),hasaverysimpleinterpretation.Isaystherearetwowaystochangetheexcessholechargewithinaregion:Holescanflowintooroutoftheregion(iDiFF),ortheexcesschargecanbemodifiedbyrecombination-generationwithintheregion（-Qp/\tau p).Eq.(6.56)iseffectivelyacontinuityequationfortheexcessholecharge.Thechargecontrolapproachfindsapplicationinbothsteadystateandtransientanaly-ses.Toillustrateitsuse,considerap+-njunctiondiodeundersteadystateconditions.InthesteadystatedQp/dt=0,DIFF=IDIFF,andEq.(6.56)reducesto$Q{$(6.57)$I DIFF Tp Now suppose a solution was not available for Δpn(x).We do know the valueof pnat x=n(x=0)from the Eq.(6.18) boundary condition and we suspect the solution,if obtained,wouldbe exponentialwitha decay constantofa diffusionlength.Thesefacts suggest anapproximationsimilartothat introduced in Exercise6.4.We equate Qptothe excess holechargeassociated withasquared-offdistributionextendingadiffusionlength

--- Page 307 --- 284 Pn JUNCTION DIODES intothen-sideandequaltotheΔnvalueatthedepletionregionedge.Therefore,without actually solving for Δpn(x),we estimate n^{ra{}{2} QP≡q(ALp)Δpn(x′=0)=qALP (eqVA/kT -1) (6.58) N{D giving Q_ $LP$ n_{fra{}{2} DIFF

qA ND} (eVA/kT-1) (6.59) p p Since Lp/τp=Dp/Lp,Eq.(6.59)is recognized tobetheusualresultfor the diffusion current in the p+-n diode. 6.3.2 Narrow-Base Diode Current Derivation Inthe ideal diode derivationthe diodecontacts were assumed to be several minority carrier diffusionlengthsormorefromtheedgesof the depletionregion. This givesrise tothe ∆np(-∞)=0and ∆pn(∞)=0 boundary conditions.For diodes whose width onthe lightly doped side is equal to thesubstrate or waferthickness,the wide-base diode assump- tion is typically justified.Theassumptioncomes intoquestion,however,ifthediodeis fabricated,forexample,in a thin epitaxial layer.A diode where the widthof the quasineu- tral regionon the lightly dopedsideof the junction ison-the-order-oforless than a diffu- sion length is known as a narrow-base diode. To determine the diffusioncurrentflowing in a narrowbase diode,it is necessaryto revise the ideal diode derivation. We take the diode under analysis to be ap+-n step junc- tion diode identical inevery way to an ideal diodeexceptthatthe widthof thequasineutral n-region is possiblycomparableto or less than a minority carrier diffusionlength.As graphicallydefined in Fig.6.18,is the distancefrom themetallurgicaljunctiontothen- side contact,and x=c-the distance from the n-edge of the depletionregion to the n-sidecontact.The n-sidecontact is assumedtobeohmic,makingΔpn=0atx=x.In general,theminoritycarrierconcentrationatacontactafinite distancefromthedepletion regionedge dependsontherateof carrierrecombination-generationatthecontact.At a well-madeorohmic contact,therecombination-generationrate is highandtheminority carrier concentration is maintained near its equilibrium value. Paralleling the derivation of the ideal diode equation, we must solve d^{2}^}} ΔP_{} 0=D{P …0≤x{≤x{c (6.60) $d_x^2$

--- Page 308 --- Pn JUNCTION DIODE: -VCHARACTERISTICS 285 0 _nx c →x +d N →x{ 0 $x_c$ Figure6.18Specificationofthen-sidecontactpositioninanarrow-basediode.isassumedto be comparable to or less than Lp. subject to the boundary conditions n^{fra{}{2} $∆pn(x=0)=$ (eqVA/kT-1) (6.61a) $N$ ∆pn（x{′=xc）=0 (6.61b) As in the ideal diode derivation, the general solution is ∆n（x{）=A₁e-xLp+A2ex P …0≤xc≤xc (6.62) HerehwevrAisnotrequiedtoani.Rtr,applinthebounaryconditiosives ∆pn(O)=A₁1+A2 (6.63a) O=A_₁e-xcLP+A2excP (6.63b) Solving Eqs.(6.63)for AandA2,andsubstitutingbackintothegeneralsolution,we conclude e(xc-x^{)/Lp-e-(xc-x’)/Lp Δpn(x{）=∆pn(0)( ..0≤x^‘≤xc(6.64) exc/Lp-e-xc/Lp or more compactly in terms of the sinh function sinh[(xc-x’)/LP] ∆pn（x^{¹）=∆pn(0) …0≤x^{≤x^{c$ (6.65) sinh[xc/]

--- Page 309 --- 286 Pn JUNCTION DIODES where e^{5-e-5 sinh(ξ) (6.66) 2 Finally, d△_{n IDIFF=AJP(x′=0)=-qAD (6.67) $d{x}^{}$ x{0 leading to DIFF=I′(eqVAT-1) (6.68) D{n^{{}{2}$cosh(x{c}_{)}(6.69)LPNdsn(cPwhere$e^{+ e-$cosh(\xi )\equiv (6.70)2LimitingCases/Punch-ThroughToassistintheexaminationandinterpretationofresults,itisusefultonote\xi ..\xi \to 0(6.71a)sinh(\xi )\to$e^{}$...\xi \to \infty (6.71b)$\fr2 }\ra{2}$1+..\xi \to 0(6.72a)2cosh(\xi )\to$e^{}$ …ξ→∞ (6.72b) Consider first the limit wherex→∞.Ifx→∞orx/Lp≥1,the ratio of the sinh terms in Eq.(6.65)simplifies to exp(-x’/Lp)and Eq.(6.65)reduces to Eq.(6.23)in the ideal diode derivation.Similarly,cosh(x/Lp)/sinh(x/Lp)→1,and Eq.(6.69)reducesto the standard ideal diode result for a p+-n junction.In essence,the narrow-base diode analysis may be viewedas a generalizationoftheidealdiodeformulationvalidforquasineutral regions of arbitrary thickness.

--- Page 310 --- pn JUNCTIONDIODE: I-VCHARACTERISTICS 287 Althoughmathematicallyreassuringthewide-base limit leadstonothingnew.Itisthe oppositelimitwherex→0that provestobemoreinteresting.Forone,ifx/Lp≤1,the simplifies to ∆n(x{)=∆p() 1 (6.73) x The perturbedcarrier concentration becomesa linear functionof position,just astraight line connecting thetwo endpointvalues set by the boundaryconditions.The linear depen- dence is adirectconsequenceofnegligiblethermalrecombination-generationineion much shorterthanadiffusionlength.In fact,the Eq.(6.73)resultcouldhave beenobtained muchmore readily byneglectingthe R-Gterm（-Δpn/τp）in theoriginalminoritycarrier diffusion equation.Thisobservation is justification for henceforth neglectingthethermal R-Gterm in the minority carrier diffusionequation when the quasineutral width is small compared to a diffusion length. Asecondsmall-widthobservationinvolvesthereverse-bias current.In the limitwhere x/Lp≤1,cosh（x/Lp）/sinh（/Lp）→Lp/xand→qA（n²/ND）Dp/x）.Thewidthof the quasineutral region,=c-n,decreases with increasing reverse bias because of the growingdepletionwidth.ItthereforefollowsthatDIFF(V<0-I1/xdoes not saturate,but systematicallyincreases withtheappliedreverse bias.Ifcis sufficiently small,thereis alsothe possibilityofx→0.Thesituation whereanentiredevice region becomes depleted is referred to as punch-through. Basedon the Eq.(6.68)/(6.69)result, IDIFF(VA<0)→-∞ifx→0.However,becauseofthedepletionapproximationused in defining the widthand natureof thequasineutral region,thetheoreticalformulationbe- comes invalid in the extreme x→0limit.Experimentally and in more precise theoretical formulations,thediffusioncurrentremains finiteatthe punch-through voltage in anarrow- base diode,providedtheelectricfieldinsidethedeviceisinsufficienttoproduceavalanche breakdown. Exercise 6.10 P: A planar Si p+-n step junction diode maintained at room temperature has an n- side dopingofND=10/cm3andann-side thicknessof=2μm.Invoking the depletionapproximation,determinethepunch-throughvoltage; i.e.determie the voltage required to completely deplete the n-side of the diode. S:Atthepunch-through voltage=,wherefrom Eq.5.37 2K_{s$71/2-VA)(Vbibi$x_{}$ qNDD

--- Page 311 --- 288 pn JUNCTION DIODES Thus 2K{ (Vbi-VAx {fra{}{2}$n=x^2$ 2 c $qNDD$qND = {bib V_A 2K{ε0 x$ c (1.6×10-19)(1016)(2×10-4)² = 0.92 - -29.7V (2)(11.8)(8.85×10-14) iwasreadromFg..nub.14 Inspecting Fig.6.11,we note that avalanche breakdownoccurs at VBR=55V in a comparable wide-base diode. Thus,punch-through will take place prior to break- down in the given narrow-base diode. 6.4SUMMARYANDCONCLUDINGCOMMENTS The chapter was devoted to modeling the steady state response of the pn junction diode. Webegan withaqualitative descriptionofdiodeoperationusing theenergy band diagram asa visualizationtool.Thereverse-biascurrent inanideal diodeis associated withminority carriers wandering into the depletionregionand being accelerated to theopposite sideof the junction.Forwardbiasing lowers the potentialhillbetween the twosidesof thediode and enhances majority carrier injection across the junctionand into the opposite-side quas- ineutralregion. There is ofcourse no carrier build-up anywhere inside the device under steadystate conditions.Recombination-generationactsto stabilize theminority carrier concentrations,while a very rapidrearrangement and interchange withthe contactsmain- tain the constancyof the majority carrier concentrations. Considerable attention was next given to the formulation,derivation,and examination ofthe ideal diode theory. Although the predictedI-Vcharacteristic exhibitsserious defi- ciencies when carefullycompared with experiment,theideal diode development provides valuable insight into theinternaloperationofthe diodeand ananalyticalbase formore accurate formulations.Even with its limitations,thetheoryandextensionsofthetheoryare widely used indiodeandotherdeviceanalysestoobtainmanageable firstorderpredictions. Theideal diodeequation,it shouldbecautioned,isonly a smallpartof theidealdiode theory. The theory encompasses a clear understanding of the solution strategy and approxi- mations,the mathematicalsteps to follow inestablishing a quantitativeresult,and predic- tions relative tocarrier concentrations,carrier currents,material dependenciesdoping de- pendencies, and so on. As alreadyacknowledged, a carefulcomparison between the ideal diode and experi-

--- Page 312 --- pnJUNCTIONDIODE:I-VCHARACTERISTICS 289 mentalI-Vcharacteristics revealed several deviationsfrom the ideal.For a Si diodemain- tained at room temperature,the deviations included reverse-bias breakdown,excess current levels undersmallforwardbiasandallreversebiases,and areducedI-Vslopeor slope- overat forwardbiasesapproaching Vi.Breakdown wasassociated withavalanchingorthe Zener process,the excess current withthermal carrier recombination-generation in the depletionregion,andthereducedIVslopeathighforwardcurrents withthevoltagedrop across internal seriesresistancesor possibly high-levelinjection.Eachofthe deviation- causing phenomena was described physically, and the more important phenomena were modeledanalytically.Informationof special note includedVBR due to avalanching varies BR roughlyastheinverseofthedopingconcentrationonthe lightlydopedsideofthejunction, carriermultiplication leads to an enhanced currentfar belowthe breakdown voltage,the R-Gcurrentexhibits an expected dependenceapproaching exp(qVA/2kT)under forward biasing and is proportional to Wunderreversebiasing,theR-Gcurrent varies asn;while the diffusioncurrent varies as n²,the voltage dropacross the internal series resistance reduces thevoltage dropacross thejunctionto V=VA-IR,andhigh-levelinjection gives riseto an expected high-current exp(qVA/2kT)operational region that is usuallyob- scured by the slope-over associated with series resistance.The development,it shouldbe noted,concentrated on the step junction.Analytical modificationsmay be required in treat- ing other profiles. Inthe special considerations section we introduced ananalyticalapproachknown as charge controland generalizedtheideal diodeanalysistoaccommodate base widthsof arbitrarythickness.Whenthequasineutral width is smallcompared to a diffusion length, the perturbed carrier distribution was notedto become linear.Forextremelysmallbase widths,thereisalsothepossibilityofpunch-throughorcompletedepletionofthe naow base at a reverse-bias voltage less thanV BR Although we have covered the highlights in this summary, thereareotherpiecesof information-terms, techniques,approximations, equations-that are partof the overall information package. The PartII Supplementand Review does contain a Review Listof Termsgroupedbychapter.Asfarasequationsareconcerned,theideal diodesolution(Eqs. 6.29/6.3)thelawofthejunction(Eq.6.2)thedepleion-edgeboundaryconditonEq 6.15/6.18),and thereverse-biasR-Gcurrentexpression(Eq.6.43)might becommittedto memory. The book should be retained for looking up the remainderof theequations. PROBLEMS CHAPTER6 PROBLEM INFORMATIONTABLE Complete Diffi culty Suggested Short Problem After Level Point Weighting Description 6.1 6.1.4 1 10 (1 each part) Ideal diode quiz 6.2 6.2.4 1 10 Compare actual-ideal I-V

--- Page 313 --- 290 pn JUNCTION DIODES 6.3 6.1.1 1 9 (3 each part) Band diagram sketches 6.4 ” 2 8 Forward-bias bigpicture 6.5 6.1.2 2-3 10 Derive Eq.(6.15)using JN=0 6.6 6.1.3 2 10 (5 each part) Typical I values using diary 6.7 ” 2 16(a-10, b-5, c-1) IdealI-Vvariation withT 6.8 ” 3 16(a-3, b-10, c-3) Photodiode I-Vderivation 6.9 ” 1 5 Photodiode I-Vplot 6.10 6.1.4 2 10(a::c-2,d-4) Deduce info from conc.plot 6.11 ” 3 12 Deduce, plot carrier conc. 6.12 ” 3 12(a-9,b-3) pn diode T-sensor 6.13 6.2.2 2 6 (2 each part) BR-related calculations 6.14 2 10(a-8,b-2) Breakdown I-V(M0) 6.15 6.2.3 3 12 Breakdown I-V(MIR-G) 6.16 ” 3 8 Determine elevated T-point Modified IR_G 6.17 ” 3 10 6.18 6.2.4 3 9(a-6,b-3) R-related computations 20（10-line fit, 8-least 6.19 ” 3 Deduce oj,n;from -V sq. fit, 2-discuss) 6.20 ” 2-3 15 Deduce Rsfrom I-V 6.21 ” 4 40(a-2, b-10,c::e-8,f-4) Multiple nonideality I-V 6.22 6.3.2 3 10 Pseudo-narrow-base diode 6.23 ” 3 15 Narrow/wide diode combo 34(a-2,b-2, c-4,d-2, 6.24 ” 4 Multiple considerations e-12, f-12) 6.1 Ideal Diode Quiz Answerthefollowingquestionsasconciselyas possible.Assumethequestionsrefertoan ideal diode. (a)Theforward-biascurrentisassociated with whattypeofcarrieractivit? (b)Thereverse-biascurrentisassociatedwithwhattypeofcarrieractivit? (c)Why isthereverse-biascurrentexpectedtobesmallinmagnitudeandtosaturateat a small reverse voltage? (d)Underreversebiasingwhatprocessesoccurinthequasineutralregionsadjacenttothe depletionregionedges?Isitdriftanddiffusion,diffusionandrecombination,genera- tion and diffusion, or generation and drift?

--- Page 314 --- Pn JUNCTIONDIODE:I-VCHARACTERISTICS 291 (e)Whycan’ttheminoritcarrierdfusionequationsbeusedtodeterminetheminority carrierconcenrationsndcurrentinthedepletionrgionofa (f)Whatapriorijustificationis thereforassumingrecombination-generationis negl- gible throughout the depletion region? (g) What exactly is a”wide-base”diode? (h) What is the “law of the junction”? (i) Given a semilogplotoftheforward biasI-Vcharacteristic,how doesone deter- mine I? (j） Trueorfalse:Thereverse-biasaturtioncurrentmaybealternativelyvieweda from minority carrier generation in the quasineutral regions. 6.2Compare (preferably with theaidofsketches)theI-Vcharacteristic derivedfroman actual Si pn junctiondiode maintained atroom temperature andthetheoretical idealdiode characteristic.Ineffecting yourcomparison,carefullynoteand(as necessary)clarifyall deviationsfromtheideal.Finallybrieflyidentifythecauseofeachdeviationfromtheideal theory. Devote no more than a sentence or two to the cause of each deviation. 6.3Sketchtheenergy band diagram foran ideal p+-n step junctiondiode showingthe carrier activity in and near the depletion region when (a) V_{ }=0.$ (b) V_{A> 0. (c) V_{A<0. 6.4Constructacompositeenergy-band/circuitdiagramanalogoustoFig.6.2that provides an overall view (the“big picture”view)of carrieractivity inside a forward-biased pn junction diode. 6.5Referring to theViderivation inSubsection5.1.4,showthat the Eq.(6.15)boundary condition can be derived bycontinuing to assume JN=0whenVA≠0.Use the Eq.(5.10) result to eliminate Vbi in your expression for n(-x). 6.6Consideringann-p silicon stepjunction（NA=101/cm3,n=10-6sec,A= 10-3cm²)tobeideal,performcalculationstodeterminetypicalcurrentlevelsexpected from thedevice under differenttemperature andbiasing conditions.Invoke theMATLAB diary function to record your work. (a) Compute the ideal diodecurrentatT=300K when(i) VA=-50 V,(i)VA= -0.1V,(ii)VA=0.1V,and(iv)VA=0.5V. (b) Assuming τn does not vary significantly with temperature,repeat part (a) forT= 500K.

--- Page 315 --- 292 Pn JUNCTION DIODES 6.7In this problem we wish to explore how the ideal diode I-V characteristic varies as a functionoftemperature.Consideran idealSi p+-n step junctiondiode with an area A= 10-4cm2,ND=1.0×1016/cm3andaroomtemperaturep=10-6sec. (a)Usingtheempirical-fitrelationshipfortheholemobilityspecifiedin Exercise3.1,the n;fit relationshipcitedinExercise2.4(b,andassumingτto betemperature-indepen- dent,construct a MATLAB programthat computes and plotstheidealI-Vcharacteristic ofthe diode with temperature TinKelvin being consideredan input variable.Printout a sample plotsimultaneously displaying thecharacteristicsassociated withT=295K, 300K,and 305K.(Toclearly display both thereverse andforward characteristics, limit the plotaxes to-1V≤VA≤0.2Vand-20≤I≤50.Use the larest0 to set the current limits when displaying severalI-Vcurves on the same set of coordinates.) (b)Appropriatelymodifyyourpart（a)programtoobtain asemilogplotofI0versusTfor 300 K≤T≤400K. (c)Comment on the part (a) and (b)results. 6.8Photodiode/Solar Cell A pn junction photodiode is just a pn junction diode that has been specially fabricated and encapsulatedto permit lightpenetrationintothe vicinityofthemetallurgical junction. Commerciallyavailablesolarcells are in essence large-area pn junction photodiodes de- signed to minimize energy losses.The general formofthe similar I-Vcharacteristics ex- hibitedbyphotodiodesand solarcells isreadilyestablishedby astraightforwardmodifi- cation of the ideal diode equation. Consideranideal p+-n step junctiondiode where incident light is uniformlyabsorbed throughoutthe device producing a photogenerationrateofGLelectron-holepairsper cm-sec.Assume that low-level injection prevails. (a) What is the excess minority carrier concentrationonthe n-side a large distance (x→∞)from the metallurgicaljunction.[NOTE:∆pn(x→∞)≠0.] (b)Theusualidealdiodeboundaryconditions(Eqs.6.15and6.18)stllholdattheedges of the depletionregion. Using therevised boundary condition established in part (a) and Eq.(6.18),deriveanexpressionfortheI-Vcharacteristicof thep+-n diode under thestatedconditionsof illumination.Asinthederivationoftheidealdiodeequation, ignore allrecombination-generation,includingphotogeneration,occurring in thede- pletion region. (c) SketchthegeneralformofthephotodiodeIVcharacteristicstaking inturnGL=0, GL=GL0,GL=2GL0,andGL=4GL0.Assumethe lightintensity ishighenough to significantly perturb the characteristics when GL=GLo.(NOTE:Problem6.9can be substituted for this part of the problem.)

--- Page 316 --- pn JUNCTION DIODE: /-VCHARACTERISTICS 293 6.9IVcharacteristicsofasolarcellareapproximatelymodeledbytherelationship, I= I0(eqVA/kT−1)+L whereIisthecurrent duetolight.ILis always negativeand a voltage-independentcon- stantfor agiven levelofillumination.Constructaplotillustratingthegeneral natureofthe solarcellcharacteristics.SettingT=300K,simultaneouslyplotIversusVAforthe assumed valuesofL0=0,-1,-2,and-4.Limit VAto-0.5V≤VA≤0.1V. 6.10FigureP6.10is adimensionedplotofthesteadystatecarrierconcentrations inside a pn step junction diode maintained at room temperature. n or p (log scale) ↑ $n_n$ Pp101} 10^{4 106 $10^{}$ n{p 10^{3}P{n10^{$\to x$- x{pqx_{n}$ Figure P6.10 (a)Is the diode forward or reverse biased?Explain how you arrived at your answer. (b)Do low-level injectionconditionsprevail in thequasineutralregionsofthediode?Ex- plain how you arrived at your answer. (c)What are the p-side and n-sidedoping concentrations? (d) Determine the applied voltage, VA. 6.11Avoltage VA=23.03(kT/q)isbeingappliedtoastepjunctiondiode withn-and p- sidedopingsofNA=1017/cm3andND=1016/cm3,respectivelyn=1010/cm3.Makea dimensioned log(p and n) versusx sketchofboththe majorityand minority carriercon- centrations inthequasineutralregionsof thedevice.Identifypoints10and20diffusion lengths from the depletion region edges on your sketch.

--- Page 317 --- 294 pnJUNCTION DIODES 6.12As was confirmed graphically in Problem6.7,the I-Vcharacteristics of the pn junc- tion diodeare very sensitive to temperature.Itshould not betoo surprising then that there are silicon diodes soldcommercially astemperature sensors.To use the diode to measure temperature,it is typicallyforward-biased withaconstant current sourceandVAismoni- tored as a functionofTas shown inFig.P6.12.Makethe assumptionthe diode is being operated inthe forward biasrange whereI=I0exp(qVA/kT).Also take the intendedrange of operation to be 0≤T(C)≤100C. I = constant + $VA(T) Figure P6.12 (a)WithI as an input variable,modify the Problem6.7(b)programtocompute and plot VA versus Tfor0≤T(C)≤100°C.Printout a sample plot usingI=10-4A. (b)Consideringthesensitivity（dVAdTinmVC)ofthesensor,woulditbeprefeabl monitor the temperature usingI=10-4AorI=10-A?Support your answer. 6.13Givenaplanarp+-nSistepjunctiondiodewithann-sidedopingofND=1/cm andT=300 K,determine (a) The approximate VBR of the diode. (b)The depletion width at the breakdown voltage. (c) The maximummagnitudeoftheelectric field inthedepletion regionat the breakdown voltage. ●6.14Letus pursue themodelingofcarriermultiplicationand avalanche breakdown in an otherwiseidealdiode.As noted in the text,Eq.(6.35)can be used tocorrect the ideal diode MIO.MoreovertheexpectedBRduetoavalanchingvariesroughlyasN0.75.Areason able fitto the Fig.6.11dependence for Si diodes is VBR ≤ 60(NB/1016)-0.75

--- Page 318 --- pn JUNCTIONDIODE: I-VCHARACTERISTICS 295 where NBisincm-and VBRisin volts.TaketheSidiodeunderanalysistobeanearly ideal p+-n step junction maintained at 300K. (a) ConstructaMATLABprogram toperformI-Vcomputations that includecariermul- tiplicationand avalanche breakdown.Specifically,compute and plot themodified re- versebiasII0 versus VAcharacteristiccoveringtherangefrom-VBRto0.(Note the I0 normalization.) Employ m=6in your computations and set the axis function parametersto[1.1*VBR,0-5,0].PrintoutasamplecharacteristictakngN 2×101/cm.Isyourplotconsistent with thecurvingapproachtobreakdownshown in Fig.6.10(b) and Exercise 6.5? (b)Use your programtoexplorehowthe diodecharacteristic variesas afunctionof m and the impurity concentrationon the lightly doped sideof the junction. ●6.15RepeatProblem6.14replacing I0 by thereversebias R_G.Normalize thecurrent axis to the magnitudeof the current at VA=-VBR/2.Run your program setting ND= 2×1016/cm3.Compare your plot with Fig.6.10(b). 6.16Thediffusioncomponentof the current is expectedtodominate inSidiodesat sufi- cientlyelevatedtemperatures.What is a sufficientlyelevated temperature?Toanswerthis questionsupposonehasaSipnstepjunctiondiodewhereN=10/cmτ and Lp≤10-² cmfor300K≤T≤500K.Determine the temperature where IDIFF= R-Gatareverse biasof bi-VA=BR/2. 6.17Inside a Si diodeat300Karegion of widthd containsthree times as manyR-G centersasadjoining regions.This specialregion,asenvisioned in Fig.P6.17liestotally inside the depletion region when the diode is zero biased.Derive an expression forthe R-Gcurrent(R-G)to beexpectedfrom the diode whenreverse-biased greaterthan a few kT/q volts. Record all derivational steps. —W—→ p+$N$-x{pq$\leftarrow \to$x }$ d Figure P6.17

--- Page 319 --- 296 pn JUNCTIONDIODES gible.T=300K. (a)Determinetheforward-biascurrentat whichtheappliedvoltagediffersfromtheideal by 10%. (b) Repeat part (a)with R_=20Ω 6.19Point-by-pointforward-bias I-Vdata from a1N757Si pnjunctiondiodemaintained atroomtemperature is listed in TableP6.19and plotted in Fig.P6.19.Followingthepro- cedureoutlinedin Exercise6.7,determine01,2,n₁,andn₂.Because the line-drawing approach is somewhat subjective,alsoapply a least squaresfit (availablein MATLAB)to appropriatesegmentsofthedatatodeterminethefitparameters.Discussyourresults. TableP6.19 VA(volts) I(amps) VA(volts) I(amps) 0.02 6.070×10-10 0.48 3.615× 10-6 0.04 1.203×10-9 0.50 5.200×10-6 0.06 2.165×10-9 0.52 7.576×10-6 0.08 3.508×10-9 0.54 1.122×10-5 0.10 5.417×10-9 0.56 1.711× 10-5 0.12 8.210×10-9 0.58 2.694× 10-5 0.14 1.183× 10-8 0.60 4.426× 10-5 0.16 1.730× 10-8 0.62 7.587× 10-5 0.18 2.449×10-8 0.64 1.359× 10-4 0.20 3.416× 10-8 0.66 2.531×10-4 0.22 4.764× 10-8 0.68 4.852× 10-4 0.24 6.501× 10-8 0.70 9.444×10-4 0.26 8.866× 10-8 0.72 1.841×10-3 0.28 1.209×10-7 0.74 3.518× 10-3 0.30 1.666×10-7 0.76 6.433× 10-3 0.32 2.305×10-7 0.78 1.103× 10-2 0.34 3.201×10-7 0.80 1.752×10-2 0.36 4.462×10-7 0.82 2.585× 10-2 0.38 6.285× 10-7 0.84 3.579× 10-2 0.40 8.845× 10-7 0.86 4.706×10-2 0.42 1.249× 10-6 0.88 5.941× 10-2 0.44 1.776×10-6 0.90 7.264×10-2 0.46 2.527×10-6

--- Page 320 --- PnJUNCTIONDIODE: I-VCHARACTERISTICS 297 $10^{-}1$10-3 (sdue) I $10^s$ $10^⁻{7 109 0 0.2 0.4 0.6 0.8 1 VA (volts) Figure P6.19 6.20Following the procedureoutlined inFig.6.16,determine theRsvalueofthe1N757 diodecharacterizedbytheroomtemperature data listed inTableP6.19andplottedin Fig. P6.19. ●6.21The model shown in Fig.P6.21 is sometimes employed as a generalized representa- tion of real pn junction diodes.The diode symbol is understoodto representthestandard modelof the device including the diffusionand R-Gcomponents of the junctioncurrent. The seriesresistor (Rs)accountsfornon-negligiblevoltage drops inthequasineutralre- gions and contactsofthe diodeunder high-currentconditions.The shuntresistor(RSH) accounts for a possible leakage current in the semiconductor external to the pn junction. Note that the current through the pn junctionandthe voltage drop across the junction are redefinedto be IandV,respectively.

--- Page 321 --- 298 Pn JUNCTION DIODES VA 1 _s$+R_{S}H$ Figure P6.21 (a)Ifthecurrentthroughthejunctionproperisdescibedbytherlationship, IJ=I01（e9Vj/n₁kT-1）+I02(eqVj/n2kT-1) where01and2areconstantsandtypicallyn≤1andn2,confirmthatthetotal currentthroughthe diode isdesriedb V_} I=I01（eqVj/n₁kT-1）+02(eqVj/n2kT -1)+ HSY withVA=V+IRs (b)ConstructcomputerogamthatcnbusedtoivestigatetheeffectofR,RH and the sizeofthecurrent prefactors（0and02)on the shape of theobservedfor- ward-biasI-VAcharacteristic.Inperforming thecomputation,itis besttoobtainI-VA data pairs associated withevenly stepped valuesofV.Limit yourplotted log(Ivs. VA outputto0≤VA≤1Vand10-10A≤I≤10-²A.EmployI01=10-13A， 2=10-9A,n₁1=1,n2=2,R=1Ω，SH=10²Ω,andT=300Kinyour initial calculation. (c) Employing a manuallycontrolledcomputational switch,modify your program sothat it computesand simultaneouslydisplaysfourI-VAcharacteristicsassociated with four differentvaluesofeitherRHR1or2.Usingtheamebasicseofprameers as specified in part(b),computeand record theI-VAcharacteristicsresulting fromthe following variation of parameters: (i）RSH=101²,10,10,and103Ω(01=10-13A,02=10-9A,R=1Ω); (ii）R=1,10,10²2,and103Ω(01=10-1A,I02=10-A,RSH=101²Ω);

--- Page 322 --- pn JUNCTIONDIODE:I-VCHARACTERISTICS 299 ()I02=10-9,10-10,10-11,and10-12A(101=10-13A,R=1Ω,RSH= $1022;(iv)01=10-13,10-12,10-11,and10-10A(02=10-A,Rs=1\Omega ，SH=$10^{2}Ω$.(d)FollowingtheprocedureestablishedinExercise6.6constructasubsidiarpramthatcanbeusedtocomputeand02fromfirstprinciplerelationships.Specificallyassumean+-pstepjunction.Jointhesubsidiaryprogramtoyourmainprogramand10-6sec.TheremainderoftheparametersshouldofcoursebeappropriateforSiatnT=300K.(e)Constructamodifiedversionofthepart(c)programthatdisplaysthecomputedresultsonalinearI-VAplot.Limitthelinearcurrentaxisto0\le i\le 10{-}^{2}A.ComputeandrecordlinearplotsoftheI-VAcharacteristicsresultingfromthevariationofparame-tersspecifiedinpart(c).Howdotheresultsherecomparewiththoseobtainedinpart(c)?(f)Generallycommentontheresultsobtainedinthisproblem.6.22Considerthespecialsiliconp+-nstepjunctiondiodepicturedinFig.P\mathrm{6.22.}Notethatp=\infty for0\le x\le and=0forx\le x\le Excludingbiasesthatwouldcausehigh-levelinjectionorbreakdown,developanexpressionfortheroom-temperatureI-Vcharacteristicofthediode.Assumethedepletionwidth(W)neverexceedsforallbiasesofinterest.$p= $0{=}_{}$ P+$NN+x0$x_{bx_{c$ Figure P6.22 6.23Reconsiderthespecial silicon p+-njunctiondiodepictured in Fig.P6.22.Insteadof =0inthe≤≤regiontakeτtobnonzerobutstillsuffcientlysmallsotha Lp≤xc-.Assumingthe depletion width（W)neverexceedsfor all biasesofinter- est, and excludingbiases that wouldcausehigh-level injectionor breakdown,derive an expression for the room temperature I-V characteristic of the diode. 6.24In modern device processing a procedure called denuding is used to reduce the R-G center concentration in the near-surface regionof devices.Aplanar Si p+-n step junction diode maintained at 300K has an n-side doping of ND=1016/cm3.As shown in Fig.P6.24,denuding has created areduced R-Gcenterconcentrationof NT in then-side

--- Page 323 --- 300 Pn JUNCTION DIODES regionfromχ=0to==2m.The R-Gcenterconcentration in theχ>xn-side region is NT2=1001. $NT$ $NT2=10NT$ $VA$ +d N N \fra } { } →x 0 x_{= 2μm Figure P6.24 (a) What is the junction built-in voltage (Vbi)? (b)What is the diode breakdown voltage(BR)? (c) Whatappliedvoltage is necessaryto expandthen-sidedepletionregionedgeto x = $x{bb} （d）Iftheminoritcarierlifeimesareτandτ2inthenearjunctionand>reions respectively,what istheτpp2ratio? (e) EstablisaexresioorexressionsorthedifusioncurrentIDFFlowinnte diode as afunctionoftheappliedvoltage. Tosimplifythedevelopment,asume L1>xandtakethen-sidecontacttobemanyLp2diffusionlengthsfromthedeple- tionregionedge.MakeasketchoftheexpectedIDIFF versusVAcharacteristic when the diode is reverse-biased. (f）Assumingτ01τp1for0≤x≤andτ02τp2for>，establishanexpresion or expressions for the R-Gcurrent(R-G)as a functionof theapplied voltage when the diodeisreverse-biasedgreaterthanafewkT/q volts.Makeasketchoftheexpected reverse-bias R-G versus VA characteristic.

--- Page 324 --- 7 pn Junction Diode: Small-Signal Admittance 7.1INTRODUCTION In this chapter we examine and model the small signalresponseof the pn junction diode. Asmall sinusoidal voltage()istakentobesuperimposedontheappliedd.c.biasgiving risetoana.c.current(i)flowing through thediode,as pictured in Fig.7.1.The a.c.response of a passive device likethe diode is characterized by specifyingthesmall-signaladmit- tance,Y=i/u.Response information can be alternatively conveyed through the use of a small-signal equivalentcircuit.The junctionregion responseof the pndiode hascapacitive (C)and conductive (G)components and in general exhibits anadmittanceof the form Y=G+jωC (7.1) where j=√-1and ω is theangularfrequencyof the a.c. signal in radians/sec. The correspondingequivalentcircuit,modeling theentire diodeand validforarbitrary d.c.bi- asing conditions,is shown in Fig.7.2.The arrows through the capacitance and conductance symbols in the figure indicatethatC andG arefunctionsof the applied d.c.bias.R,the series resistance introduced inChapter6,models the portionof the diodeoutside the junc- tionregion.Rs canlimit the performanceof the diode in certainapplications,butit is normally very small compared to the junction impedance except at large forward biases. Unless stated otherwise, we will henceforth assume Rs to be negligible. In establishing explicitexpressions for the admittance of the junction region,it is con- venient to divide the development into two parts.In the first we take the diode to be reverse- biased.When reverse-biased,the diodeconductance is small andY≈ jωC.Moreover,the reverse-bias capacitance is linkedsolely to majority carrieroscillations inside the device. The second part treats forwardbias where the conductance cannot be neglected andmi- nority carriers contribute to the overall response. 7.2 REVERSE-BIASJUNCTIONCAPACITANCE 7.2.1 General Information Reiterating thestatement in the introduction,whenreverse-biased,thepn junction diode becomes functionallyequivalentto acapacitor.Many“capacitorsinICs andothercir- cuitsareinfactreverse-biased pn junction diodes.Thepn junctiondiodedoesdifferfrom 301

--- Page 325 --- 302 pn JUNCTION DIODES $1+iPNva=V0sin\omega t++2$v_a$ VA Figure7.1Diodebiasingcircuit.vistheappliedsmall-signalvoltage;iistheresultanta.c. current. C Rs 1 1 1 G Y=jωC+G 1 Figure7.2 General-case small-signalequivalent circuit fora pn junction diode.Y=jωC+Gis the admittanceofthejunctionregion;Rmodels theseriesresistanceofthecontactsandquasineutral regions. a standardcapacitor in that the diodecapacitance monotonically decreases withincreasing reverse bias.The sample C-Vdata presentedinFig.7.3 illustrate the observed dependence. The general a.c. behavior of the pn junction diode under reverse biasing can beex- plained by examining what happens inside the diode as the a.c.signal runs through a bias cycle.With the a.c.signal superimposedonthed.c.bias,the total voltage dropacrossthe junction becomes VA +a. During the positive portion of the vacycle,the a.c.signal slightlyreduces thereverse bias across the junction,andthe depletion width shrinksbya smallamount,as envisioned in Fig. 7.4(a).The ρ-plot in Fig. 7.4(b),which was drawn assuming a step junction,emphasizes that there is a corresponding decrease in the charge on thetwo sides of the depletionregion.Whenthe a.c.signalreverses and goes negative, va now increases thetotalreversebiasacrossthe junctionto greaterthan VA,andthe depletion widthincreases slightlyaboveits steady statevalue.Adepletionwidthlarger than thesteady state value in turn givesrise toan increase inthe depletionregioncharge on the two sidesof the junction. Theoveralleffectof the a.c.signal maythus be viewedas a smalloscillationofthedepletionwidthaboutitssteady state valueandanassociated ∆ρ charge densityoscillation,as pictured inFigs.7.4(c)and (d).The ∆ρ-plots wereobtained

--- Page 326 --- PnJUNCTIONDIODE: SMALL-SIGNAL ADMITTANCE 303 150 125 100 75 C(pF) 。 50 25 0 -25 -20 -15 -10 -5 0 5 VA(volts) Figure7.3 Monotonic decrease inthe diodecapacitance withincreasing reverse bias.Sample C-Vdata derivedfrom a1N5472A abrupt junction diode. by subtractingthe d.c.charge density distributionin Fig.7.4(b)from theVA±vacharge density distributions. The small size ofva,typically a few tens of millivoltsorless,dictates that the maxi- mum displacementofthe ∆ρchargefrom theedgesofthe steadystate depletionregion willbeextremelysmall.Effectively,itlookslikeplusandminuschargesarebeingalter- nately addedand subtracted from two planes inside the diode separated by a width W. The described a.c. situation is physicallyidenticalto whattakes place inside a parallelplate capacitor.It is wellknown that the parallelplate capacitorexhibits a capacitanceequalto the dielectric permittivityof the material between the plates,times the areaof the plates, divided bythe distancebetweentheplates.Thediodecapacitanceisconcluded byanalogy to be K{εOA C_}$

(7.2) W Thecapacitanceassociated with the depletion widthoscillation is known as the junctionor depletion-layercapacitanceand is identified by thesubscriptJ.=Cifthediodeis reverse-biased,since there are noother significant charge oscillations.Also,we know W

--- Page 327 --- 304 pn JUNCTION DIODES —W— _{0 P N v_{a 0> 0> (a) 4 qN{ + →x $-q_{}$ (b) △ρ 4 _{0 →x (c) $v_a$ 0> →x (d) Figure7.4Depletion-layerchargeconsiderations.a)Depletionwidthand(b)totalchargedensity oscillationsinresponsetoanapplieda.c.signal.(c)a>0and(d)<0a.c.chargedensitis.

--- Page 328 --- PnJUNCTIONDIODE: SMALL-SIGNALADMITTANCE 305 increases withincreasingreverse bias.Cbeingproportionalto1/Wisthereforeexpected to decrease withincreased reverse biasing,in agreement withexperimental(Fig.7.3) observations. One pointmust beclarifiedconcerning theoscillationofthedepletion widthaboutits steady state value: Toachieve thisoscillation,majoritycarriers mustmove rapidly intoand outof the affected region in sync withthe a.c.signal.In otherwords,thecarriers areas- sumed to respond to the a.c.signal as if it were a d.c.bias. When this happens,the device is said to follow the a.c. signal quasistatically.In the discussionof the d.c.response,we noted the majoritycarriers in thequasineutralregions were capableof a rapidrearrange- ment,withmajoritycarriers being suppliedoreliminatedatthe contacts as required.The majoritycarrierresponse time inSi is typically~10-10 secor less,andthequasistatic assumption made here is routinely valid up to very high signal frequencies. 7.2.2 C-VRelationships The precise capacitance versus voltage dependence expectedfrom a given diode is esta- lished byreplacing Win Eq.(7.2)by the appropriate expression relating Wto the applied voltage.Intreating the junctionelectrostaticsinChapter5the WversusVArelationship was foundto vary withthedopingprofile.TheChapter5 analysis specificallygave 2Ksε0 71/2 W= V{bb i-V{)$...asymmetricalstepjunction(7.3)$q_{B and 12Ksε0 71/3 W= V_{bi -V{A)$...linearlygradedjunction(7.4)qawhereNBisthedoping（NAorND)onthelightlydopedsideoftheasymmetricalstepjunctionandaisthelineargradingconstant.AlthoughEqs.(7.3)and(7.4)couldbeseparatelysubstitutedintoEq.(7.2),itismoreconvenienttodealwithasinglegeneralizedrelationshipvalidforawiderangeofprofiles.WorkingtodevelopageneralizedWversusVArelationship,weconsideraprofilewithoneside(x<0)heavilydopedandtheconcentrationonthelightlydopedsidedescribedbythepowerlawNB(x)=bxm$ …x>0 (7.5) where b>0and m are constants for a givenprofile.Examplesofone-sided power-law profiles for select m-values are displayed in Fig.7.5.Note that m=0 and m=1corre- spondtotheasymmetricalstepjunctionandtheone-sidedlinearlygradedjunction,resc- tively. Also note that negative m-values are permitted.A profile where m<0and the doping decreases as one moves away from the junction is saidto be hyperabrupt.Hypera- brupt junctions can be formed by ion implantation or epitaxy.

--- Page 329 --- 306 pn JUNCTION DIODES $ND-NA$ND-NAND-NA$m=1m=0m=-1$ x (a) (b) (c) Figure7.5Exampleone-sided power-lawprofiles:(a) linearly graded,(b) step,and (c)m=-1 hyperabrupt. Inside a pn junction with aone-sided power-lawprofile,the depletion width depen- dence for m>−2 is [(m+ 2)Kε0 1/(m+2) W= (Vbib$

• VA)$(7.6)qbThederivationofEq.(7.6)isleftasanexercise.Form=0,b=NB—whereNBwithoutatrailing(x)isunderstoodtobeaposition-independentconstant—andEq.(7.6)reducestoEq.(7.3).Settingm=1andb=a/4inEq.(7.6)yieldsEq.(7.4).SatisfiedEq.(7.6)yieldsacceptableresults,weconcludeuponsubstitutingEq.(7.6)intoEq.(7.2)that$K{ε{AA C{ = (7.7) (m+2)K_$71/(m+2)V_A$ qb Alternatively,it is sometimes convenient to introduce K{AA CJO=CJ|VA=0 (7.8) = [(m+2)_ε0 (2+）/1 _bi$ qb Equation（7.4)isappropriateforatwo-sidedlinearlygraded junction.bmustbesetequaltoa/4insteadofato account for the difference in profiles.

--- Page 330 --- PnJUNCTIONDIODE: SMALL-SIGNAL ADMITTANCE 307 where, in terms of the capacitance at VA= 0, C30 C{_} = (7.9) $1/(m+2)$VA}$1$V_{bi$Apnjunctiondiodethatismanufacturedtotakeadvantageofthecapacitance-voltagevariationdescribedbyEqs.(7.7)/(7.9)iscalledavaractor.Varactorisacombinationofthewordsvariableandreactor,wherereactoralludestothereactance=1/j\omega Cofthedevice.Varactordiodesarewidelyusedinparametricamplification,harmonicgeneration,mixing,detection,andvoltage-variabletuning.Insuchapplicationsitisoftendesirabletoemployadiodeexhibitingthemaximumcapacitanceratiooveragivenvoltagerange.Thisfigureofmeritiscalledthetuningratio(TR).ExaminingEq.(7.9)forreversebiasessuchthat-VA/Vbi\ge 1,wefind$V{}1/(m+2) $C_{(}VA1$ TR= (7.10) C_{V{A{2 $V$ As is obvious from Eq.(7.10),the largest tuning ratios are derived from devices withthe smallest m-values-TR progressively increases in going from linearly graded (m=1),to step(m=0),to hyperabrupt(m<0) junctions. This should explain the special interest in hyperabrupt profilesand thecommercialavailabilityofhyperabrupt varactordiodes. (C) Exercise 7.1 P:Equation(7.7)can beusedtocomputeand plotfullydimensionedcapacitance- voltagecharacteristics;“universal”normalizedC-Vcurves are readily constructed using Eq.(7.9).To examine the general nature of the predictedC-Vcharacteristics, computeandplotnormalizedC/CJ0 versusVA/Vicurvesappropriatefor linearly graded,step,and m= -1hyperabrupt junction diodes. Limit the voltage axis to -25≤VA/bi≤0.Commenton theresults. S: The normalized characteristics were computed using Eq.(7.9).The MATLAB pro- gram scriptandoutputplot(Fig.E7.1)follow. Thestep junction curve is notedto providea fairlygoodmatchto theexperimentaldatapresented in Fig.7.3ifone assumes V~1V.Also,ifVA=0isusedforVA1,andassumingsimilar Vvalues, thetuningratio[C(A)/CVA2)]employinganyVA2isobviouslygreatestforthe hyperabruptdiodeandleastfor the linearlygraded diode,inagreementwiththetext discussion.

--- Page 331 --- 308 Pn JUNCTION DIODES MATLAB program script.… %Exercise 7.1..NormalizedC-Vcurves %Computation clear m=[10-1]; s=1./(m+2); x=linspace(-25,0); %x=VA/Vbi $y=[]; %y=CJ/CJ0 for i=1:3, y=[y;1/(1-x).^s(i)]; end %Plot close plot(x，y,-);grid axis([-25001]) xlabel(‘VA/Vbi’);ylabel(‘CJ/CJO’) text(-20,.42,‘linear (m=1)’) text(-20,.27,‘step (m=0)’) text(-20,.10,hyperabrupt (m=-1)’) 1 0.9 0.8 0.7 0.6 0.5 0.4 linear.(m=1). 0.3 step (m=0) 0.2 0.1 hyperabrupt (m=1） 0 -25 -20 -15 -10 -5 0 VA/Vbi Figure E7.1

--- Page 332 --- Pn JUNCTION DIODE: SMALL-SIGNAL ADMITTANCE 309 7.2.3Parameter Extraction/Profiling C-Vdatafrom pn junction diodesandother devices are routinely usedto determine device parameters,notablytheaveragedoping or dopingprofileonthelightlydopedsideofa junction.C-V measurements have become so commonplace in the characterizationand testingof devicesthatautomated systems areavailableforacquiring and analyzing the C-Vdata. Since themeasurement is an integral componentof parameter extraction,we begin here with a brief description of an automated C-V system. TheC-V system in the measurements laboratoryadministered bytheauthor is pic- tured schematically in Fig.7.6.TheMSIC-Vmeteristheheartofthesystem,emloying a15mVrms a.c.signalata probing frequencyof1MHz.Four capacitanceranges witha maximum valueof2 pF,20pF,200pF,and2000pF are availableforselectionfromthe front panel.Ad.c.bias supply inside the meter has twofull-scale rangesof ±9.999V, programmable in0.001Vincrements,and ±99.99V,programmable in0.01Vincrements. Operatorcontrolofthebiasing,thedatadisplaydatamanipulations,andhard-copyoutput to a printeror plotter are accomplished by software on the personal computer shown to the leftof theC-Vmeter in Fig.7.6. The probeboxto the rightof the meter is used ifthe deviceor devices under test are situated on a wafer.Thecircularchuck in theprobebox provideselectricalcontactto the backofthewaferanda wire probecontacts thedevice structure onthe topof the wafer.Thechuck can be resistance-heated underthe localcontrol ofthe832T-Controller.Afterprobecontact,thetopoftheprobebox is typically lowered to keeproom lightfrom perturbing the capacitance measurement.If the device undertest is encapsulated,the leads to the probe box are removed and thetest device is inserted into an adapterconnectedto the inputterminalsof the meter. We shouldmentionthemeter automaticallymeasuresandcompensatesforthestraycapacitanceassociated withcabling, probe box components, and the encapsulated-device adapter. Turning tothe interpretationof theC-Vdata,suppose thedeviceundertest is known to be an asymmetricallydoped abrupt junction.(The termabrupt is normallyused to de- scribeactual dopingprofilesapproximatelymodeled by the idealized step junction.For the assumed junction profile,m→0andb→NBin Eq.(7.7).Additionally,if bothsides of Eq.(7.7) are inverted and then squared,one obtains 1 2 2b-$(7.11)$C^{f{{}qNBKE0A^{}$ Equation（7.11)indicatesthata plotof1/C2versus Vshouldbeastraightline,witha Thus,assuming the area Aof the diode is known,NBis readily deducedfrom theslopeof the plot.Although perhaps stating theobvious,a straight line1/C}versus VAplotisalso confirmationthat the diodecan bemodeledas a step junction.A sampleanalysisofC-V data basedon the1/C2 versus VAplot is presented in Exercise7.2.

--- Page 333 --- Cooling water In/out Probe interface —Thermocouple Heater- Probe box —Spring-clip adapter ① ① ① ① ① min 。 Range pf Diode T-Controller C-V Meter 832 565 Reset H Q on/off ① ① ① ① ① ① 0000 Monitor Computer/Controller 310

--- Page 334 --- PnJUNCTIONDIODE:SMALL-SIGNALADMITTANCE 311 Exercise 7.2 P:Themanufacturer indicates the1N5472A is ann+-pabrupt junction diodewith an area A=3.72×10-cm².Encapsulation is noted to typicallyintroducea 2 pFstraycapacitance shunting thediode junction.Utilizingthe measuredN5472A C-VdatapresentedinFig.7.3,applythe1/Cversus VAplottechniquetoconfirm the abrupt natureof the junction and to determine the p-sidedoping concentration. Also quote the deduced value of Vbi. S:A1/C2versus VAplotofthe1N5472AC-Vdatais displayedinFig.E7.2.Prior toconstructingtheplot2pF was subtractedfromallmeasuredcapacitancevaluesto Fig.E7.2datapointsareseentolieinanalmostperfectstraightline.Thejunctionis definitelyabrupt.Performing a least squares fit to the data gives 1 =(6.89×1019)-(9.78×1019)VA C^{fra{}{3} whereCJis in farads andVAis in volts.Referringto Eq.(7.1),weconclude 2.5 2 1.5 $1/c^{}$ $10^{3}3$ 1 $p{F}^2$ 0.5 0 -25 -20 -15 -10 -5 0↑ 5 VA (volts) V_{i Figure E7.2

--- Page 335 --- 312 Pn JUNCTION DIODES 2 NA$

|d03b$2(6101\times 826)(\epsilon -01\times 220)(1-01\times 88)(811(61-0\times 91)=8.84\times 1015/c{m}^{3}and6.89\times 1019$Vbi= VA1{$

0.70V 9.78× 1019 Itshouldbe noted that the deducedVis lower thanone would expectfromthe NA9×101/cmp-sidedoping. The Vbi value deducedfromthe C-Vdata is subjectto serious extrapolationerrors and is sensitiveto doping variations in the immediate vicinity of the metallurgical junction. The foregoing plotapproachcouldobviously be extended to the linearly gradedand other profiles.However,this isseldom done. As it turns out,thedoping variation with positionon the lightlydopedsideofajunction can be deduced directlyfrom theC-Vdata without priorknowledgeaboutthe natureof the dopingprofile.Omitting the derivational details,we merely notethedopingconcentration versus position iscomputed using 2 N{B(x)= (7.12) qKsε0A²|d(1/C{2)dA| K{AA$ (7.13) x

C_} where x is the distance into the lightly doped sideof thediode asmeasuredfromthemet- allurgical junction.Notethatsubstituting the Eq.(7.11)step junctionrelationshipinto Eq.(7.12) yields the required position-independent result. The process of determiningthe doping as a function of position iscalled profiling.The profile determined using Eqs.(7.12)/(7.13)becomes inaccurate if the doping is a rapidly varying functionof position,and only a limited portionof the junction can be scanned. Moreover,the result tends to be”noisy” because the slopeor derivative of the C-Vdata is required.Nevertheless,C-Vprofiling is relatively simple to implement, typically yields useful results,andfinds widespread utilization. Software provided withautomated C-V systemsevenperformsthe(7.12)/(7.13)computationsanddisplaystheresultgraphically. A sample profile,thatof a hyperabrupt junctiondiode automaticallyprocessed by the Fig.7.6C-Vsystem,is reproduced in Fig.7.7.

--- Page 336 --- pn JUNCTIONDIODE: SMALL-SIGNAL ADMITTANCE 313 1E17 CONCENTRATIONCM-3 1E16 1E15 0 3 (um)x Figure7.7Dopingprofileofahyperabrupttuningdiode（C809).OutputfromtheFig.7.6C-V measurementsystem.isthedistancefromthemetallurgicaljunctionintothe lightlydopedside. 7.2.4 Reverse-BiasConductance All standardcapacitors exhibit acertain amountofconductance.The same is trueof thepn junction diode.Although predominantlycapacitive,thereverse bias admittance does have a smallconductive component. A few words are in orderconcerning the reverse-bias conductance. By definition,thedifferentiald.c.conductanceofadiode is just theslopeof theI-V characteristic,dIldV,at thed.c.operatingpoint.Ifthediodeisassumedtorespondtoan a.c.signal quasistatically,thenthea.c.conductance=△l/ΔV=dlldV=differential d.c. conductance.Limiting our considerations to frequencies where the diode canfollow the a.c.signalquasisticallyandintroducinthesmbolGfortheassociatedlowrqn conductance, we can write IP $G0=$ (7.14) P$

--- Page 337 --- 314 Pn JUNCTION DIODES Foranidealdiode where I=0[exp(VA/kT)−1], G0

$kT$ 9 $0eqA/kT 二 kT 9 (I+0) …idealdiode (7.15) When the reverse bias exceeds a fewkT/q volts in an ideal diode,I→-I0 and we see fromEq.(7.15)thatG0→0.This is consistent withthefactthatthereverse-biasI-V characteristicsaturatesandtheslopeoftheideali-Vcurvegoestozero.Ifthed.c.recom- bination-generation current dominates in the given diode then,for reverse biases greater than a few kT/q volts, qAn;w d qAn;W/2τ0 G0

$d_{}$ 20 (m+2)(bi-V{) …IR-G dominant (7.16) where use has beenmadeofthe Eq.(7.6)relationship for W.Whenthe R-Gcurrent domi- nates,Eq.(7.16)indicatesthere is a residualconductanceatallreverse biases,withthe precise voltage dependence varying with the doping profile of the junction. Regardless of the junction type or dominant current component,it should beempha- sized that Eq.(7.14)canalways be appliedto themeasured I-Vcharacteristic to deter- mine G{. Exercise 7.3 P:The diode exhibiting thereverse-bias I-Vcharacteristic displayed in Fig.6.10(b) has a measured junction capacitance of C=63 pF at VA=-10 V. The series resistanceofthe diode wasestimated tobe1ohminExercise6.8.Taking thed.c. operating point to be VA=-10Vandthe a.c.frequency tobef=100kHz,confirm that Rs and G=G0 may betotally neglected in modeling the admittanceof the device. S: Figure 6.10(b)can be used toestimate the low-frequency conductance. We find dI 40 pA $G_{0o}$

AP 18V

2.22 ×10-12S VA=-10V By way of comparison, at the operational bias and frequency, ωC=(2)(105)(6.3×10-11)=3.96×10-5S The capacitive component is clearlymuchgreater than the conductive component and Y ≡jωC_.

--- Page 338 --- Pn JUNCTION DIODE: SMALL-SIGNAL ADMITTANCE 315 Next,consider the series resistance.SinceRis in series withY,we need to compare|Z|=1/|Y|with .Here we find R{s≤1Ω and 1 2.52×104Ω ωC}

The reactance ofthejunction is muchgreater thantheseriesresistance.Forthegiven operationalconditions,the diode may beessentially viewed as a pure capacitor. 7.3F FORWARD-BIASDIFFUSIONADMITTANCE 7.3.1General Information Capacitance arises from charge oscillations insideof a device structure. The junction ca- pacitance introduced in the reverse-biasanalysis is caused bythe in-and-outmovementof majoritycarriersaboutthesteadystate depletionwidth.Theminoritycarrierconcentra- tionsalsooscillateabouttheedgesofthedepletion width inresponsetothea.c.signal.The minoritycarrier numbers are sominuscule underreverse-biasconditions,however,thatthe contributionto theadmittance is negligible.Forwardbiasing gives rise to nothing new as farasthemajority carriersare concerned.These carriers stillmove backandforthabout theedges of the depletion regiongiving riseto ajunctioncapacitance.In fact,the Crela- tionships developed in theprevioussectioncan beapplied withoutmodificationto forward bias.The something new underforward bias is a significantcontributionfromtheminority carrierchargeoscillation in responsetothe ac.signal. As notedduring theexaminationofidealdioderesults inSubsection6.1.4,forward biasing of the diode causes a build-upofminority carriers in the quasineutral regions im- mediatelyadjacent to the depletionregion. Thebuild-up becomes largerand largerwith increasing forwardbias.Inresponse to an a.c.signal,the voltage dropacrossthejunction ischangedtoVA+vaandtheexcessminoritycarrierdistributionsoscillateabouttheird.c. values as pictured in Fig.7.8(a).Thisresults inanadditionalcapacitance.Iftheminority carrierscanfollowthesignalquasistatically,thecarriersmovebackandforthinunison between the two straight lines inthe figure.However,thesupplyandremovalofminority carriers is notas rapidas thatofthe majoritycarriers.Atangular frequencies approaching the inverseoftheminoritycarrierlifetimes,theminority carrierchargeoscillationhas difficultystaying in sync withthe a.c.signal.Theresultisanout-of-phasespatial variation ofthe charge something like the undulating distributions sketched in Fig.7.8(a).Anout- of-phasecharge oscillation enhances theobserved conductance andreduces theobserved

--- Page 339 --- 316 pnJUNCTIONDIODES norp (log scale) ↑ pnnp\oran}{ }n_{^{2}$N$ → x - _{pq x_{n (a) 5 X CD GD (b) Figure7.8Thediffusionadmittance.(aMinoritycarrierchargefluctuation(greatlyexagrtd givingrisetothediffusionadmittance.b)Forward-biassmall-signalequivalentcircuitforthepn junction diode (series resistance assumed to be neglligible).

--- Page 340 --- pn JUNCTION DIODE: SMALL-SIGNAL ADMITTANCE 317 capacitance.Inother words,thecapacitanceandconductanceassociated withtheminority carrier oscillations are expected to be frequency-dependent. Becausetheminoritycarrier build-upabouttheedgesofthedepletionregioniscaused bythediffusioncurrenttheadmittanceassociatedwiththeminoritycarrierchargeoscil- lation is calledthediffusionadmittance,YD.In general,as we haveindicated, D=GD+jωCD (7.17) where CD andGDarethediffusioncapacitanceanddiffusionconductance,respectively. Theoverallforward-biasadmittanceisofcoursetheparallelcombinationof the junction capacitance and the diffusion admittance as noted in Fig.7.8(b). Measurementofthe forward-biasadmittance poses more ofa challenge thanthe re- verse-bias capacitance measurement. The larger d.c.currentthat flows when the diode is forward-biasedtendstoloaddown thedetectionandbiasing circuitry inalmostallcom- mercially available C-V systems.In measurements on a Si diode atroom temperature,a forwardbiasofonly a fewtenthsofa voltoften leadsto a fallacious resultor automatic terminationof themeasurement.One instrument thatcanbeusedforforward-biasmea- surements is the HP4284ALCRMeterpictured schematically in Fig.7.9.TheHP4284A withthe001optioninstalledcontainsaspecialisolationcircuitthatminimizestheloading problem for d.c.currents upto0.1A.A very versatile pieceofequipment,the HP4284A from 20 Hz to1MHz,variationof the a.c.signal levelfrom5mVto 20Vrms,setting of the d.c.bias between ±40V,andcapacitance detectabilityclaimedto rangefrom0.01 HP4284A Cp=180pF G=.082S rinter Insert device Figure7.9 The Hewlett-Packard4284ALCRMeterusedforforward-biasmeasurements.The attached printerconvenientlyprovidesa hard-copyrecordof the displayscreen data.

--- Page 341 --- 318 Pn JUNCTION DIODES femtofarad to10farads.The instrument basicallyperformssingle-pointmeasurements,but willautomaticallycyclethroughtenoperator-set valuesoffrequency,d.c.biasor..s nal level. 7.3.2Admittance Relationships Obtainingexplicitrelationshipsforthedifusionadmittanceisnotdifficultbutthemathe- matical manipulations canbecome rathertedious.In thestraightforwardbrute-force ap- proach,theidealdiodederivation issimplyrepeated withthed.c.quantitiesallreplaced by d.c. plus a.c. quantities. Separate solutions are then sought forthe a.c. quantities.Once a solution is obtainedfor the a.c.current as a function of the a.c.voltage,the admittance is computedfromY=ilv.Alternatively,the a.c.currentsolutioncanbeobtainedfromthe d.c.current solution by noting equivalencies in the two formulations. We will pursue the latter approach. The device under analysis is takento be a p+-n junction diode. With an a.c. signal superimposedonthed.c.biasthen-sideminoritcarrierdifusionequaiontobeslvedis ∂△pn(x,t a²Δpn(x, Δp{n(x,t

Dp (7.18) 1e $\partial x^2$ Tp Assuming the a.c.signal is a sine or cosine function, we can write ∆n（x,t）=Δpn（x）+p（,ω）jωt (7.19) where ∆pnisthetime-invariant（d.c.)portionofΔpn（x,tandpnistheamplitudeofthe a.c.component.Substituting the Eq.(7.19)expression for ∆pn(x,t)into Eq.(7.8)and explicitly working out the time derivative yields $d^{21}$ d^{2}} ${\Delta }_{}$ p\{} jωnejωt=
+D{P$ejwrej\omega r(7.20)$d_^{2}d_^{2}$TpThed.c.anda.c.termsinEq.(7.20)mustseparatelybalance.Thus,aftercollectingliketermsandsimplifyingthea.c.result,oneobtains$^{2}}}Δ_}$0=DP$ (7.21a) $d_x^{}$ τp $d^2$ $p$ =0 D$(7.21b)$dx^{}$ p/(1+jω) Equation（7.2la)isrecognizedastheusualsteadystateminoritycarrierdiffusionequation. Note that the a.c.version of the diffusion equation,Eq.(7.21b),hasexactly the same form except thatτpis replaced byτp/(1+jωp).

--- Page 342 --- pn JUNCTIONDIODE:SMALL-SIGNAL ADMITTANCE 319 In solvingEqs.(7.21)theusualboundaryconditionappliesat=∞i.e.Δpn（∞)= n（∞)=0.The boundary conditionat=n,however,becomes n^{ra{}{2} $\Delta pn(x=n）=\Delta pn（n）+p(）$ (eq(VA+va)/kT-1) (7.22) $ND$ or n^{fra{}{2} Δp{nx)= (eq_A/kT-1) (7.23) $N_0$ $n^{\text{fra}}2$ $pn()=$ eqVA/kT(eqva/kT-1) (7.24a) $N_{}$ n^{fra{}{2} $q_{}$ $\}$ kT eg_{A/kT .a≤kT/q (7.24b) vain the foregoing is understood to be the amplitude of the a.c.signal. a At this point we have yet to solve for anything. We have merely established the equa- tionand boundaryconditions to beemployed insolving forthe a.c.variables andcurrent. It is now a simplematter,however,to progress to the desired solution.Since the a.c.mi- nority carrier diffusionequation is identical inform tothe d.c.equationexcept p→ τ/(1+jωp),and sincethe boundaryconditionsareidenticalexcept [exp(qVA/kT) -1]→[(qu/kT)exp(qV/kT)],thea.c.andd.c.currentsolutionmust likewisebeidentical except for the cited modificationstoτpand the voltage factor.Forap+-ndiode we know D{n^}} D{P n^{{{2} DIFF= QA (eqVA/kT-1)=qA (eqVA/kT-1) (7.25) $LD$ V Tp $ND$ Thus,making the noted τand voltage factor substitutions,we conclude D{P}√1+jn^{fra{}{2}q_{}eq_A/k=q_{a}0eqVAk√1+j yb= P} (7.26)NB} V kT kT or in terms of the ideal-diode low-frequency conductance (Eq.7.15), idiff=G0√1+jωpa (7.27) and idiff Y{D

=G_0√1+jω$...p+-ndiode(7.28)v_a$

--- Page 343 --- 320 Pn JUNCTIONDIODES Forann+-pdiodeτp→τnin Eq.(7.28).Givenatwo-sidedjunction,G0must be separated into its n andp components and each multiplied by the appropriate √i+ jωT factor. Ifthe Eq.(7.28)diffusionadmittanceforap+-ndiodeisseprated intorealandimagi- nary parts and the result compared with Eq.(7.17),one finds GD = 2V1+ +) 1/2 (7.29a) Go (√1 + w2- 1/2 CD= (7.29b) a\qr2} GD andCD are notedtobefunctionsofboththe d.c.bias （throughG)andthesignal frequency.With G varying as exp(qVA/kT),the diffusioncomponents increase very rap- idly withincreasing forward bias.WhereasCdominatesthe a.c.responseatsmallforward biases,the diffusion capacitance surpasses and eventually overshadows the junction ca- pacitance as the forward bias is progressively increased. Relative to the frequency depen- dence，atlowfrequencieshere1,√1τ1ω/2n GD⇒ G$...\omega p\le 1(7.30a)CD\Rightarrow G_{}$ _ . . .ωp≤1 (7.30b) 2 10.0 GD \r{{} 1.0 CDCDO}$ 0.1 0.01 0.1 1 10 100 ωtp or ωn Figure7.10 Diffusioncapacitanceand diffusionconductancenormalizedto theirlow-frequency values asafunctionofωp（p+-n diode)or ωn（n+-pdiode).（CD0=G0τ/2)

--- Page 344 --- PnJUNCTION DIODE: SMALL-SIGNAL ADMITTANCE 321 Ifτp=10soexamfequencyindepnentresonedecibedbys..) is expectedforf≤1/(20πτp)≈16kHz.For signalfrequencies where ωp≥1,thecon- ductance increases while the capacitance decreases with increasing frequency.The relative change in GD and CD compared to their low-frequency values is graphed as a functionof ωp in Fig. 7.10. (C) Exercise 7.4 P:The1N5472An-p abrupt junctiondiodeyielding thereverse-bias C-Vdata plottedinFig.7.3andanalyzedin Exercise7.2hasazero-biasjunctioncapacitance ofC0=120pF.TheVi valueafordingthebestfittojunctioncapacitancedata was determined to be0.7Vin Exercise 7.2.FromI-Vdatafollowing theprocedureout- lined in Exercise 6.7,oneobtains 0=8×10-13Aandn,=1.22.Thep-side minority carrier lifetime is estimated to beτ=5×10-7sec. (a) Assuming ωn=0.01,but employing relationships validfor arbitrary valuesof ωTn,computeandplottheexpectedC,D,andC+CDversus VAforthegiven diode.Restrict VAto0≤VA≤0.65V.Specifically notetheapproximate voltage where CD = C_. (b) Repeatpart(a)setingωn=0.1,1,10,and100.Commentontheresults. S: The primary relationships employed in the computation are: (1 {{}C_{_}=$1/2$τ{n _{}1+ a2 1/2 C{ ${\omega }_{n2}$ and $G0=$ 9 0eVA/n₁kT kT TheCexpression is Eq.(7.9)withm=0.TheCDrelationship is Eq.(7.29b)with preplaced byτnand aτassociated withtheωexternal tothesquareroot.Gis the Eq.(7.15) low-frequency conductance with n,inserted into the exponent to account for the nonidealityof the diode. The MATLAB program scriptand the part (a) computational results are repro- ducedbelow. Actual forward-biasC-V data derivedfrom the1N5472A usingthe

--- Page 345 --- 322 Pn JUNCTION DIODES HP4284ALCRMeter(Fig.7.9)are includedon the plot.Thejunctioncapacitance is noted to dominate atlowforwardbiases,withthe diffusioncapacitance rising to C=Cat VA≤0.545V.Thesamecomputationalresults areobtained,asexpected, forall ωτn<1.Withincreasing ωngreater than ω=1,CD progressively d- creasesatallbiases andtheCD=Cpointshifts tohigherandhighervoltaes. CD=CatVA=0.575Vand0.62Vwhenωn=10and100,respectively. MATLAB program script… %Exercise7.4..Forward-BiasCapacitance %Computational constants clear CJO=120e-12; %farads Vbi=0.7; %volts Vth=0.0259; %Vth=kT/q in volts taun=5.0e-7; %seconds I0=8.0e-13; %amps n1=1.22; %ideality factor wt=input(inputtheangular-frequencylifetimeproduct); VA=linspace(0,0.65); %CJ Computation CJ=CJO./sqrt(1-VA./Vbi); %CD Computation GO=I0/Vthexp(VA./(n1*Vth)); CD=taun.*G0./(sqrt(2)*wt)*sqrt(sqrt(1+(wt)^2)-1); %Measured CD Data VAm=[0.10.20.30.40.420.440.460.480.500.520.540.560.58]; CDm=[1.31e-101.43e-101.61e-101.88e-101.97e-102.08-10. 2.23e-102.46e-102.76e-103.46e-104.40e-106.54e-109.38e-10]; %Plot close semilogy(VA,CJ,-r);axis([00.71.0e-101.0e-9];grid hold on;semilogy(VA,CD,-g);semilogy(VA,CJ+CD) semilogy(VAm,CDm-2e-12,o’) xlabel(‘VA(volts)’);ylabel(‘C(farads)’) %Key semilogy(0.127e-10,o);text(0.125,7e-10,.C-VData’) x=[0.10.2]; y1=[6.1e-106.1e-10];semilogy(x,y1,-y’);text(0.21,6.1e-10,CJ+CD’) y2=[5.2e-105.2e-10];semilogy(x,y2,—r);text(0.21,5.2e-10CJ’) y3=[4.3e-104.3e-10];semilogy(x,y3,-g);text(0.21,4.3e-10CD) hold off

--- Page 346 --- Pn JUNCTIONDIODE: SMALL-SIGNAL ADMITTANCE 323 10-9 ..C-V Data CJ+CD CJ CD (speej) 10-10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 VA(volts) Figure E7.4 7.4SUMMARY The chapter was devoted to examining and modeling the small-signal response of thepn junction diode.The discussion was divided intotwo parts corresponding to reverse-biasing and forward-biasing of the diode.When reverse-biased,the pn junction diode is function- ally equivalentto acapacitor.Thecapacitanceofthediodecan in factbecomputedusing the well-known parallel plate capacitor formula(Eq.7.2).The pn junction diodediffers from a standard capacitor in that the diode capacitance monotonically decreases within- creasing reverse bias.The reverse-bias junction capacitancearises physically from thein- and-outmovementof themajority carriersaboutthesteady state depletion widthinre- sponse to the a.c. signal.Reverse-bias diodes are employed as capacitors and variable capacitors(varactors) in numerous circuitapplications.Capacitance measurementsare used extensively inthecharacterizationandtestingof devices,particularlyindeterminingthe average doping or doping profileon the lightly doped sideof a junction. Therelevanceof the profile to theuseof the diodeas a varactor,and parameterextraction/profilingproce- dures, were noted during the course of the discussion. Whenthediode isforward-biased,there is a significantbuild-upofminoritycarriers in the quasineutral regions immediately adjacent to the depletion region. Theoscillation of the minoritycarriercharge in responseto the a.c.signal gives riseto anadditional admit- tance componentthedifusionadmitance.Anexpressionforthedifusionadmittance was established byappropriatelymodifying the ideal diode equation. The diffusionadmittance

--- Page 347 --- 324 Pn JUNCTION DIODES componentsare strong functionsof the d.c.bias,eventually dominating theobserved ad- mittance as the forward bias is progressively increased.At signal frequencies where w 1,theminoritycarriershavetroublefollowingthea.c.signal andtheresultingout-of-phase oscillationsenhancethediffusionconductanceattheexpenseofthediffusioncapacitance. PROBLEMS CHAPTER7 PROBLEMINFORMATIONTABLE Complete Diffi culty Suggested Short Problem After Level Point Weighting Description 7.1 7.4 1 16 (2 each part) Quick quiz 7.2 7.2.2 2 10 Derive Eq. (7.6) 7.3 2 12(a/b-10,c-2) Dimensioned C-Vplot *7.4 7.2.3 2 15 (plot-5, i-5, ii-5) Deduce NB，Vbifrom C-V 7.5 3 10 Derive Eqs.(7.12)(7.13) 7.6 ” 3 15 NB(x) versusx 7.7 7.3.2 1 5 Verify accuracy of Fig. 7.10 7.8 ” 2 10 (prog-8, questions-2) Relative size of ωCD，GD 7.9 ” 2 10 τ measurement 7.1 Quick Quiz. Answer the following questions as concisely as possile. (a)What is the physical origin of the junctioncapacitance? (b) Sketch them=-1hyperabrupt profileofan n+-pjunction. (c) Define quasistatically. (d) Define varactor. (e) Define profiling. (f） Make a sketchof the low-frequency conductance of an ideal diode showing both for- ward and reverse bias.Comment as necessary to forestall a misinterpretationof your sketch. (g) What is the physical origin ofthe diffusionadmittance? (h) Whydoes the diffusionconductance increase with increasingωpwhenωp≥1? 7.2Giventheone-sidedpower-lawprofiledescribedbyEq.(7.5,andgenerallyfollowing the proceduresoutlined in Chapter 5,derive Eq.(7.6).Assume a p-n junction where

--- Page 348 --- Pn JUNCTIONDIODE: SMALL-SIGNALADMITTANCE 325 7.3(a)Constructa computerprogramthat yieldsfully dimensionedreverse-biasC-V curves thatcan be compared directly with experimental data. The program istobe specificallydesignedforcomparison withthedatafromSip+-nabruptjunctiondiodes maintained at300K.The diodearea (A),the lightlydoped sideconcentration(NB, andthelarestreverse-biasvoltageofinterestmxaretobeinputvariables. (b)EmployingA=3.72×10-cm²andNB=8.84×101/cm3,compareyourprogram output withtheexperimentaldatapresentedinFig.7.3.DoestheViresultcitedin Exercise 7.2have any bearing on the agreement between experiment and theory? Explain. (c) How does the lightly doped side concentrationaffectthe junctioncapacitance?Sub- stantiate your answer. NOTE:Those seeking a greaterchallengemightconsidergeneralizingthecomputer program in this problem to handle anyone-sided power-law profile. * 7.4The1N4002isoneof the popular 4000-series general-purpose diodesused inauto- motiveandotherapplications.C-Vdatafrom a1N4002 p+-njunctiondiode is listed in TableP7.4.Before analyzing thedata,subtract3pFfromeachcapacitance valueto account for the stray capacitance shunting the encapsulated diode.Assuming the diode profile tobe abruptandA=6×10-cm2,applytheplotapproachdescribedinthetexttodetermine the lightlydopedsideconcentrationandthe“best-fit”Vi.Quotetheresultsobtainedby (i)“eyeballing”astraightlinethroughthedataand(ii)byperformingaleastsquarsfitto the data.(NOTE: The1N4002is notas ideallyabruptas the1N5472AofExercise7.2.Do not be surprised if your plot points deviate somewhat from a straightline.) TableP7.4 1N4002Reverse-BiasC-VData. VA(V) C(pF) VA(V) C(pF) 0.0 38.709 -5.0 15.548 -0.2 33.717 -6.0 14.599 -0.4 30.567 -7.0 13.834 -0.6 28.319 -8.0 13.189 -0.8 26.598 -9.0 12.639 -1.0 25.170 -10.0 12.163 -1.4 23.060 -11.0 11.746 -1.8 21.490 -12.0 11.373 -2.2 20.254 -13.0 11.037 -2.6 19.248 -14.0 10.734 -3.0 18.405 -15.0 10.458 -4.0 16.762 7.5Derive Eqs.(7.12)and(7.13).HINT:See,forexample,p.43in reference[3].

--- Page 349 --- 326 Pn JUNCTIONDIODES ●7.6Construct a computer program thataccepts C-V data input andoutputs a plotof the NB(x)versus x profile based on Eqs.(7.12)and (7.13).Test run your program using the 1N4002datainTableP7.4. 7.7VerifytheaccuracyofFig.7.10.Computeandsimultaneously plotthe diffusionca- pacitance and diffusionconductance normalized totheir low-frequencyvalues as a function of ωp.Limit thecomputation to0.01≤ωτp≤100. 7.8Whatis therelative sizeof thecapacitiveandconductivecomponentsofthe diffusion admittanceata given d.c.bias andsignal frequency?Toanswerthisquestion,firstexamine Eqs.（7.29)and(7.30)to determine the limiting valuesof ωCD/GDwhen ω≤1and ωτp≥1.Next,compute andplotωCD/GDversus ωpfor0.01≤ωp≤100.Doesyour plot approachthe correct limiting values? In words,what is theanswer to theoriginal question? 7.9Forward bias admittance measurements have been used to determine the minority car- rier lifetimeonthe lightlydoped sideofa junction.NotefromEqs.(7.30)that, given a p+-ndiode,CD/GD=τ/2whenωp≤1.Forward biasC-Vdatafroma1N572Adiode was presented in Exercise7.4.Thecorresponding GD-VAdata for forward biases between 0.50Vand0.58Vare listed inthefollowing table. Determine theapparentτn of the 1N5472An+-p diode at eachof the listed voltages assuming ωn≤1.Also quote the average n deduced from the data. VA(V) $GD(S) 0.50 2.00×10-4 0.52 3.90×10-4 0.54 7.15×10-4 0.56 1.33×10-3 0.58 2.28×10-3

--- Page 350 --- 8 pn Junction Diode: Transient Response At the beginning of the pn junction diode discussion, we noted that a complete,systematic device analysis was normallydivided intofourmajor segments:modeling of the internal electrostatics,steady state response,small-signalresponse,andtransientresponse.Inthis chapter we address the finalmajor segmentof the pn junction diode analysis,the transient or switching response.In a number of applications a pn junction diode is used as an elec- trical switch.A pulseofcurrentor voltage is typically used to switchthediodefrom for- ward bias,calledthe”on”state,to reverse bias,called the”off”state,and vice-versa.Of prime concern tocircuitand deviceengineers is the speed at which the pn junction diode can bemade to switchstates.Generally speaking,it is during the turn-off transient,going from theon totheoffstate,wherespeed limitations aremost significant.Thesubsequent development therefore concentrates on the turn-off transient.Moreover,the diodeunder analysis is assumedto beideal. Thisallows us to conveythe basicconcepts and principles of transient operation with a minimum of mathematical complexity. 8.1TURN-OFFTRANSIENT 8.1.1Introduction Considertheidealized representationofa switching circuitshown in Fig.8.1(a).Priorto t=0the diode is takento beforward-biased with asteady state forward current,F,flow- ingthroughthediode.Att=0theswitchinthecircuit israpidlymovedto therighthand position.For use in switchingapplications,one would likethecorresponding diode current to decreaseinstantaneouslyfromIFtothesmallsteady-statereversecurrentconsistentwith theappliedreverse voltage.Whatoneactuallyobserves is sketched in Fig.8.1(b).Instead ofa vanishinglysmall currentthereversecurrentimmediatelyafter switching iscompa- rableinmagnitudetotheforwardcurrentifVR/R~/RF.Subsequentlyhcurent throughthedioderemainsessentiallyconstantatthelarge-IRfora limitedperiodoftime before eventually decaying to the steady state value. The periodof time during which the reverse currentremains constant is known as thestorage time or storage delay time(t). The total time requiredfor the reverse currentto decay to10%of its maximum magnitude isdefinedtobethereverserecoverytime(),whiletherecovery time（)isthe difference betweenand.Thecitedtimescharacterizingthetransientarealso defined graphically in Fig.8.1(b). 327

--- Page 351 --- 328 pn JUNCTION DIODES RF RR i(t) + V_F} VR}$t=0(a)i(t)$IFt_{s}$t-0.1/R一11111$-\sr}$1${s}$11$-t$\to (b)UA(t)4VONt$s}$ 1 1 (c) Figure.1Theturnofftransient.）Idelizedrepresentationoftheswitcincircuit.bketch and characterization of the current-time transient.(c)Voltage-time transient.

--- Page 352 --- PnJUNCTIONDIODE:TRANSIENTRESPONSE 329 Thevariationofthe instantaneousdiodevoltage(vA)correspondingto the i-ttransient is shownin Fig.8.1(c).Specificallynotefromthe figurethat (i)the junctionremainsfor- wardbiasedfor0<t<eventhoughtheexternallyappliedvoltageissuchastoreverse- =0. In analyzing the transientresponse,we make theassumptionthat thebatteryvoltages (VFand VR)are large compared to the maximum forward voltage drop（VoN)acrosthe diode. Under the stated assumption V} VON V} $I$

(8.1a) R_ R} and VR+VA|O<1
Vs} I}

R{ Rs} (8.1b) Additionally,thequalitativeandquantitativeanalysestofollowfocusonthestorage delay portionofthetransient.Becausethedecayingportion isreadilydistortedby strayca- pacitance in themeasurementcircuit,it isthat has cometo bequoted as theprimary figure of merit in characterizing the turn-off transient. 8.1.2 Qualitative Analysis In looking at theturn-off transientfor thefirst time,a numberofquestions undoubtedly cometomind.Why is there adelay in going fromtheon-state to theoff-state?Or perhaps betterstated,what isthephysicalcauseofthedelay?Whatgoesoninsidethediode during the transient?Howisit the dioderemainsforward biasedfor0<t<teventhough the applied voltage is such as to reverse bias the diode? The rootcause of the delay in switching between the on andoff states is easy to iden- tify. Forward biasing of the diode,as we have noted previously several times,causes a build-up or storageof excess minority carriers in thequasineutralregions immediately adjacentto the depletionregion.Whenthediode isreverse biased,on theother hand,there is a deficitofminoritycarriers inthe near-vicinityofthedepletionregion.Simply stated, to progressfrom theon-statetotheoff-state,the excess minority carriers picturedin Fig.8.2must be removedfromthe two sides of the junction.Thestorage delay time derives its namefrom the fact that the majorityof the storedcharge is being removedfromthe diode during the t portion of the transient. As the charge control analysisof Subsection 6.3.1 indicated,removalof theexcess minority carrier charge in thequasineutral regions can beachieved in two ways.Forone, the carriers can beeliminated in place via recombination.Recombination is of course not instantaneous; severalminoritycarrier lifetimes wouldberequiredto go from theon-state to theoff-state if recombination were the sole meansof carrier removal.Theothermethod

--- Page 353 --- 330 pnJUNCTIONDIODES n or p (linear scale) Forward Forward p{ \np_$ReverseReverseS5\to x$-x{pqx_{n}$ Figure8.2Storedminoritycarriercharge leadingtothedelay inswitchingbetween theonandoff states.Thereverseand forward minoritycarrierconcentrationsare plotted simultaneouslyon alinear scale with thexcoordinates matchedatthe depletion regionedges.The break in the x-axis insidethe depletionregionacknowledges a difference in theforward-bias and reverse-bias depletion widths. Thecross-hatched areasidentifyminoritycarriers thatmust be removed forswitching to be complete. of reducingthecarrierexcess is by anetcarrierflowoutofthe region.Once thesustaining external bias is removed,theminoritycarriers can simplyflowback to theother sideofthe junction wheretheybecome majoritycarriers.Thisreverse injectioncouldconceivably occur ata veryrapidrate.Thetimerequiredtodrifbackacrossthe depletionregion is only W/v~10-10 sec,where v is the average drift velocity in the depletion region. However,the numberof minority carriersremoved per second is limited by the switching circuitry. Themaximum reverse current that can flowthough the diode is approximately VR/RR=R.ThesmallerR，theslowerthecarrierremovalrate.Averyrapidtransient couldbeobtained byreplacing RR with a short,but such a procedure would likelylead to a currentflow exceeding device specificationsand damage to the diode. Summarizing, there are two mechanisms,recombination and reverse current flow,that operate to remove the excess stored charge.Neither mechanism can safely remove the charge at a sufficiently rapid rate to beconsidered instantaneous.Hence one observes a delay in going from the on to the off state. We have yet to answer the question how it is the diode remains forward biased during theO<t<portionof the transient.Toanswerthequestion,considerthe progressive removal of the holeexcesson the n-sideofap+-n junction as envisioned in Fig.8.3.Note from thefigurethatduringthet<tstagesofthedecaytheminoritycarrierconcenration

--- Page 354 --- Pn JUNCTIONDIODE: TRANSIENT REPONSE 331 $pn(x)个t=0tincreasing$t = ts p{n0 →x x_} Figure8.3 Decay of the storedhole charge inside a p+-n diode as a functionof time for $0\le t\le s.attheedgesofthedepletionregion(x=xn）isgreaterthantheequilibriumvalue.InChapter6weestablisheddepletionedgeboundaryconditionsthattiedtheminoritycarrierconcentrationsatthedepletionregionedgestotheappliedvoltage.ThesesameboundaryconditionsapplyundertransientconditionswithVA\to \upsilon A.ThusauA>0indicatesthereisanexcessofminoritycarriersadjacenttotheedgesofthedepletionregion.Pertinenttothepresentdiscussion,thereverseisalsotrue—aminoritycarrierexcessabovetheequilib-riumvalueattheedgesofthedepletionregionimpliesthejunctionisforwardbiased.Orstatedanotherway,itistheresidualcarrierexcessattheedgesandinsidethedeple-tionregionthatmaintainstheforwardbiasacrossthejunction.Itisonlywhentheholeconcentrationat=dropsbelowtheequilibriumvaluethatthediodebecomesreversebiased.Finally,acommentisinorderconcerningtheslopeoftheFig.8.3curvesatx=.Inanidealp-niod=Ap（xn）-ADp=xor$d△_{n$ {slope of ∆pn(x)or pn(x)]} i (8.2) dx

versus x plot at x=x_{n AD_{ x=x{n

--- Page 355 --- 332 pn JUNCTION DIODES Allt>0 concentration curves must therefore slope upward at x=nbecausei<0. Moreover,theslopesat=mustbethesamefor allt>0curvesin Fig.8.3because i=-=constant during the0<t≤portion ofthe transient. Exercise 8.1 P:Use thequalitative insight gained into the dioderesponse to predict howkey fa- tors are expectedto affecttheobserved ittransient.Theaccuracyofthepredictions will be checked after working out the quantitative theory. The figuresafter the problem statementcontain abase-line sketchofanittran- sient.Using a dashed line,sketch the expected modification to the base-line transient if as indicated on the figures: (a) IF is increased to IF. (b) IR is increased to _. (c) p is decreased (made shorter). Explain how youarrived at the modified i-t sketches. i(t) i(t) i(t) 1 F} $I$ F$Ftdecreasedtt$-sr}- \sr }- s{}$ (a) (b) (c) S:The reasoningleading to thegraphicalanswers presented in Fig.E8.1 are as follows: (a) IncreasingIF increasesthestoredcharge insidethe diode.Since thestoredcharge is increased and theremoval rate is unchanged,it willtake longer to remove the stored charge.tis expected to increase. (b) Increasing IRincreases therate at which the stored charge is removedbythe reverse current flow.Thus in this case the storage delay time is reduced. (c)Ashorterminoritycarrierlifetimeincreases thecarrierrecombinationrateand will therefore decrease t.

--- Page 356 --- Pn JUNCTIONDIODE:TRANSIENT RESPONSE 333 i(t) i(t) i(t) $F$ $I$ pdecreased t t 1 -qr} -\sqr} -\s} (a) (b) (c) Figure E8.1 8.1.3 The Storage Delay Time Quantitative Analysis We seek a quantitative relationshipthatcanbeused to predictandcompute the storage delaytime,t.To simplify the analysis,we treatan ideal p+-n step junction diodeand make useof the charge controlapproach.Theelectroncharge storedonthe p-sideofa p+-n junctionis ofcourse negligiblecompared to theholecharge（Qp）storedonthe n-side.We also know thati=iDFF in an ideal diode andreferring to Eq.(6.56)in Subsection 6.3.1, $d{Q}_P$Q{P$ i (8.3)

一 p Working towardasolution,wenotethati=-=constantfortimes0+≤t≤, wheret=0+is an instant after switching.Thus Eq.(8.3)simplifiesto dQ{_ $Q_P$

一 IsR} + …0+≤t≤ts (8.4) p The Qp andt variables in Eq.(8.4)can be separated and an integrationperformedover timefromt=0+tot=.Weobtain QP(ts）_dQP s {1-=p (8.5) $OP(0+)R+QP{J}_0+$ giving =−pmn1+ p)) QP(s IR+QP(O+)/ =τ{pln$ (8.6) Q{P(0+)

--- Page 357 --- 334 pnJUNCTIONDIODES TheQp(0+)andQp(）appearing in Eq.(8.6)mustbe dealt withtocomplete the derivation.Qp(O+)provesto bereadilyexpressedintermsofknown parameters.Because charge cannotbeeliminated instantaneously,Qp(0+)=Qp(o-).However,priorto switch- ing,dQp/dt=0andi=IF.Itthereforefollows from Eq.(8.3)that $Q(0-{)}_{Q}P(0+$I$

(8.7) τp QP(),thestoredchargeremainingat=,poses moreofaproblem.Wishingtoerron theconservative side(ie.,obtainanestimateofthatis toolarge)we takeQp()tobe approximately zero.Eliminating Qp(0+)in Eq.(8.6)using Eq.(8.7)andsetting Q()= 0,we conclude $\stackrel{˚}{}$ ts=pln(1 (8.8) \sR} Equation(8.8)isnotedtobe intotalagreement withthequalitativepredictionsof Exercise8.1.increases with increasingIF,decreases withincreasing IR,and is directly proportionaltoτp.Asapointof information,amorepreciseanalysis,based on acomlete Δn（x，t)solutionandproperlyaccountingfortheresidualstoredchargeatt=,gives[] _s} 1 erf (8.9) Tp I} 1+ $IF$ Althoughconsiderablydifferent inappearance,theEq.(8.9)solution is likewisenotedto be in total agreement with the qualitative predictionsof Exercise8.1. Measurement In both Eqs.(8.8) and(8.9)is directlyproportionalto p.Moreover,theonlyother parametersaffectingtarethecurrentsIFandIRcontrolledbytheswitchingcircuitryThis suggests it should be a relatively simplematter to determine theminoritycarrier lifetime on the lightly doped side of an asymmetrical junction by measuring the turn-off i-ttran- sient,noting the storage delay time,and computing τpfrom Eq.(8.8)or Eq.(8.9). A measurement system that can be used to observe the transient response of diodes with lifetimes in theμsec range is pictured in Fig.8.4.The measurementcircuit in combi- nation with the Tektronix PS5004 d.c.power supply and FG5010function generator simu- lates theswitchingcircuitofFig.8.1(a).Tofirstorder,the d.c.biasapplied at thepower supply nodedetermines IF.Rapid switching of the voltage applied across thetest diode is

--- Page 358 --- Pn JUNCTIONDIODE:TRANSIENTRESPONSE 335 □ 0 0-0 TEK11401 11A32 Menu DM5010 PS5004 FG5010 touch Power supply 1 +V — 1000Ω 一 150Ω Function Scope generator Test 200μF 1 100Ω diode 一 1 一 一 Figure 8.4 Transient response measurement system. provided by the negative-going square-wave pulse derivedfromthe function generator. Because thecharge on theplatesof the capacitorcannot change instantaneously,the voltage drop across the capacitor must remain constant as the output of the function generator goes from0Vto the preselected negative value.Thisforces the voltageonthe diode sideof the capacitor to decrease byan amountequalto the peak-to-peak valueof the square-wave pulse.Subsequent current flow will tend to discharge thecapacitor,but thecircuit RC time constants are10-² secondsor greater,making the discharge negligible during a typical pulsing period.A voltage proportional to the instantaneous currentthrough thetest diode,

--- Page 359 --- 336 Pn JUNCTION DIODES Tek ←→ Cursor’s Wirsdowl 2.5V 500mV /div EA trig’d -2.5V -4μs 2us/div 16μs 2s/div Avg (L1) Main Avg# >32 Linear Fast @ 1024 pts -4.24μs 1MΩ DC 350MHz A11 Wfms Avg(L1) off Status Main Figure8.5 Samplecurrent-time turn-off transient.Output derivedfromthe Fig.8.4measurement system.(They-axisdisplayvoltage is directlyproportionaltotheinstantaneouscurrentthroughthe diode.)

--- Page 360 --- PnJUNCTIONDIODE:TRANSIENTRESPONSE 337 the i-t transient,is monitoredusing the Tektronix11401Digitizing Oscilloscope.Wave- forms displayedon the11401can be analyzed in place,orthe screendatacan be sentto a printer for subsequent examination. A sample measured i-t transient is reproduced in Fig.8.5.As extracted from there- sponsecurve,R/F≡1.0and≈5.0sec.One deduces aτp=ln(1+)= 7.2 sec employing Eq.(8.8) and aτ=22 μsec utilizing Eq.(8.9).A more accurate determinationof the lifetime,anda checkas to whether thetheoryproperlymodelsthe diode undertest,isobtained by varying theIR/IFratioandchoosing theτpyielding the bestfittothenormalized/versusIR/IFplotshowninFig.8.6.Thedashedandsolid lines inFig.8.6 werecomputed employing Eqs.(8.8)and(8.9),respectively.Experimental 10

下 、 1.0 0 O _sq} \a{ 0.1 0 0.01 0.01 0.1 1.0 10 IR/IF Figure8.6 Theoreticalandmeasuredstoragedelaytimesnormalizedto τversusthereverseto forwardcurrentratio.Thedashed line wascomputedusingEq.(8.8)andthesolidlineusingEq.(.9). Experimental data are from a1N91Ge diode()and a1N4002 Si diode(⊙).

--- Page 361 --- 338 Pn JUNCTIONDIODES data froma1N91Ge diodeanda1N4002SidiodehavealsobeenaddedtoFig.8.6.The Ge diode data can be veryclosely matchedto the moreexacting theory ifoneemploysτp= 14.5 μsec.TheSi diode data,however,deviates from the predicted dependence.Novalue ofτpcan bechosentofitboththeupperandlowerendsofthecurve.Sidiodes,itmust be remembered,are seldom ideal diodes as assumed in the derivationof Eqs.(8.8)and (8.9). 8.1.4 General Information We concludethe discussionof theturn-off transient with a fewobservations of a practical nature.Firstasageneralrule~τ（orτn).IncreasingtheR/Fratiodecreasesbelow Tp as is obvious fromFig.8.6,butmoreoftenthannot there are constraints that limitthe P sizeof theIR/IFratio.Anotherapproachtoachievearapid switchingresponse is to build diodes with shortminority carrierlifetimes.Sinceτ and pareproportionalto1/NT, where NTis theR-Gcenterconcentration,theminoritycarrier lifetimecanbe decreased by theintentionalintroductionofR-Gcentersduring thefabricationofthediode.The reductionof theminoritycarrier lifetime in Si devices is typicallyachievedby diffusing goldinto theSi.There is a limit,however,to theR-G center concentrationthatcanbe added to a diode.While a shorter lifetime makes formore rapid switching,it also propor- tionallyincreasestheR-GcurrentR-Gα1/0)—ahighR-Gcenterconcentration may increase theoff-statecurrenttounacceptable levels.R-Gcenterconcentrationsapproach- ing thedonororacceptorconcentrationsalsoaffectthe diodeelectrostatics.Inanyeven there is no need for pn junction diodes with extremely large R-G center concentrations. Other devices withfewer storedcarriers,suchasthebipolar junctiontransistorandthe metal-semiconductor diode (bothaddressed inlaterchapters),areavailable foruse when the application requires subnanosecond switching times. Finally, mention should be madeof the step-recovery or snap-back diode. The re- sponseofthestep-recovery diodeis specialinthatthetportionofthetransient is very short, ~1 nsec. With a storage delay time ~l μsec,the reverse current part of the i-t transient looks likea step,rapidly“snappingback”tothesteady state valueafterreaching t=t.Step-recovery diodes are used as pulse generators andhigh-order,single-stage har- monic generators. In fabricating the diodes,a narrow, lowly doped region is sandwiched between heavily dopedp andn regions.Formed by employing epitaxial techniques,the junctions in this p-i-n type structure are required to be veryabrupt.Thespecial doping profilecauses theminority carriercharge to be stored veryclose to theedgesof the deple- tion region.This facilitates almostcomplete removal ofthecharge by the end of the storage delay time.With littleadditionalchargetoberemovedafterreachingt=,thecurrent drops abruptly to the steady state value. 8.2 TURN-ONTRANSIENT The turn-ontransientoccurs when the diode is switched from thereverse-bias off-stateto the forward-biason-state.Thetransition can beaccomplished with a current pulse,a volt- agepulse,oramixtureofthetwopulses.Becauseofitssimplicityandutilizationinprac-

--- Page 362 --- PnJUNCTIONDIODE:TRANSIENTRESPONSE 339 i(t) _A(t) ↑ 个 VON $IF$ t t (a) (b) Figure8.7Turn-ontransientassumedtostartfromi=0:(a)currentpulse；(b)voltage-time response. tical circuits,we consider herein the case where a current pulse is used to switch the diode into the on state. When the diodecurrent is changed instantaneouslyfromthe prevailing reversebias value to a constant forward current IF,the voltage dropacross the diode,vA(t),monotoni- cally increasesfromthe OFFatt=0to VONatt=∞.Thefirststageoftheresponsefrom t= 0 to the time whenv =0 is extremely short in duration. The few minority carriers A needed to raise thejunction voltage to zero are rapidly injected across the depletion region. Majoritycarrierrearrangementalsoactsquicklytoshrinkthe depletionwidthtoits zer- bias value.The shortdurationofthefirst portionofthetransientallows us to actas if the diode were being pulsed fromi=0 toi=IF att=0as depicted in Fig. 8.7(a). Figure8.7(b)showsthecorrespondingvoltageresponseassumedtostartatvA=0. In seekingaquantitative solutionforA()≥0,we againtakethedevice underanal sis to be an ideal p+-n step junction diode and make use of the charge control approach. The envisioned growthof thestored n-side hole charge with time is pictured in Fig.8.8. NotefromthefigurethatQp=0att=0consistent withinitiatingthetransient atvA=0. Sincei=IFthroughouttheturn-ontransient,the Eq.(8.3)relationshipforthestoredhole charge reduces to dQ{P Q{$

If} 一 (8.10) 1p Tp Q{p(t) dQP dt’ = t$(8.11)$-Q_{P{$

Jo Jo or upon evaluating the Qp integral QP(t) (8.12) 0

--- Page 363 --- 340 Pn JUNCTIONDIODES P{n(x,) t=∞ t increasing t=0 Pn0$\to x$x_}$Figure8.8Build-upofthestoredholechareinsideap-ndiodeduringtheturnontransint.SolvingEq.(8.12)forQp(t)givesQp(t)=p(1-e-/)(8.13)UndersteadystateconditionsthestoredholechargeinanidealdiodeisQP=IDIFF\tau p=0\tau p(（eVA/kT-1）...steadystate(8.14)Asafirst-orderapproximationletusmaketheassumptionQp)duringtheturn-ontran-sientisdescribedbyEq.(8.14)withVA\to vA.Thisisequivalenttoassumingthebuild-upofstoredchargeoccursquasistatically.OnecanthenwriteQP(t)=10\tau p（eqvA(1)kT-1)(8.15)Equatingthe(8.13)and(8.15)expressionsforQp(t),andsolvingforvA(t),wearriveatthesolution$\fqra{}{}} (1 − e−1+) _{A(t) $ln1+(8.16)q$\or0 }$

--- Page 364 --- Pn JUNCTIONDIODE: TRANSIENTRESPONSE 341 The turn-on responsemodeled by Eq.(8.16) is similar to the turn-off response in that theoveralllengthofthetransientincreaseswithincreasingIFandτ.Thisistobemoreor less expected becausethechargerequired toreach thesteadystateisdirectlyproportional toIFandτp;i.e.Qp(∞)=IFpas deducedfromeither Eq.(8.10)or Eq.(8.13).Perhaps the mostinterestingfeatureoftheturn-ontransientisaninitialrapidrise invA(t),with vA(t)increasingtoalargefractionofVoNinaveryshortperiodoftime.If,forexample, T=300KandVON=0.75VorI/0=3.77×10²,vA(t）increasesto（0.84)VONafter onlyt=0.01τp.By wayofcontrastthefinalapproachtothesteadystateisconsiderably slower, requiring a period of time equal to several (C) Exercise 8.2 TheCPG (ConcentrationPlot Generation)program thatfollows is intended as avi- sualizationandlearningaid.TheprogramplotsoutcurvesofΔpn(xΔpnmaxver- sus x’/Lp=(x-xn)/Lpatselectt/forboththeturn-of andturn-on transients. Thecomputationsare basedonthe directsolutionofthetime-dependentminority carrierdiffusionequationforanideal pnstep junction[4.Theuserchooses thetype of plotto be displayedfrom anopening menu.Menuchoices are linear or semilog plots of theturn-off transientconcentrations and linearor semilog plotsof the turn- on transientconcentrations.In theturn-off plotstis steppedfrom0.1ttotin0. increments.Inthe turn-on plotstis steppedfrom0.1τto 2pin0.1τincrements. Theusermust specify theIR/IFratiowhena turn-offplot is desired.Weshould mention that Figs.8.3 and 8.8 were drawn basedon theCPGprogramoutput. Possible uses of the program are: (1) Visualize the stored-charge decay during turn-off. (2) Visualize the stored-charge build-up during turn-on. (3) Examineandcomparecorresponding linearandsemilogplot. (4)Confirmthatthe=slopeofthe linearturn-offcurvesareallthesamefor 0<t≤.(Aretheslopesthesameonthecorrespondingsemilogplot？Arethe =nslopesorcurvesonasemilogplotrelatedtothecurrent?) (5) ExaminetheeffectofIR/Fonthedecayofthestored charge. (6)Ascertain whythe approximatetresultofEq.(8.8)becomesless andless accu- rate asIR/IFis increased.(Lookat the storedchargeremaining att=tasa function of /.) (7)Checktheaccuracyofthequasistaticapproximationemployed inthe derivation of Eq.(8.16).Onasemilogplottheturn-oncurvesshouldallbeparalleltothe t=∞ curve if the build-up proceeds quasistatically.) (8) Comparethe turn-onvA(t)computedemploying Eq.(8.16)andtheνA()values deduced from the turn-on plot.

--- Page 365 --- 342 Pn JUNCTIONDIODES Hopefully the user will not be limited by thecited suggestions,but willfeelfree to experimenton his/her own. The user might also consider modifying the program to better display a givenoutput,to extend the computations,or to obtain a specific output such as vA(t) versus t. MATLAB program script… %Exercise 8.2—Turn-off/Turn-on ConcentrationPlot Generator %Determine type of desired plot clear close S=menu(Choose the desiredplot’OFF-Linear”OFF-Semilog,.. ‘ON-Linear’,‘ON-Semilog’); %Compute ts/taup if turn-off plot is desired ifs<=2, %Let Iratio=IR/IF and TS=ts/taup Iratio=input(PleaseinputtheIR/IFratio:IR/IF=); ifIratio==0,%Catch ifIR=0 TS=1; else TS=(erfinv(1./(1+Iratio)))^2; end else

end %Set values of X andTto be computedfor desired plot %X=x’/LP and T=t/taup ifs1|s2, X=0:0.03:3; T=TS/10:TS/10:TS; else X=0:0.03:3; T=[0.1:0.1:2]; end %Plot steady-state curve, set axes-labels y0=exp(-x); ifs1|s3, plot(X,y0,‘g’) axis([0 301]) else semilogy(X,y0,‘g’) axis([0 3 1.0e-31])

--- Page 366 --- Pn JUNCTION DIODE: TRANSIENT RESPONSE 343 end xlabel（‘x/LP’);ylabel（‘∆pn(x`,t)/∆pnmax) grid; hold on %Primary computations and time-dependent plots j=length(T); fori=1:j, A=exp(-X).(1-erf(X./(2sqrt(T(i))-sqrt(T(i)))); B=exp(X).（1-erf(X.(2sqrt(T(i)))+sqrt（T(i))); yon=(A-B)/2；%yon=∆pn(x,t)/∆pnmax during turn-on ifs==3, plot(X,yon); elseif s==1, yoff=exp(-X)-(1+Iratio).*yon；%yoff=∆pn(x,t)/∆pnmaxduringturn-off plot(X,yoff); else end ifs==4, semilogy(X,yon); elseif s==2, yoff=exp(-X)-(1+Iratio).*yon; semilogy(X,yoff); else end end; hold off 8.3SUMMARY In this chapter we examined theelectricalresponseand internalcarrier responseof pn junction diodes subjected to a large rapid change in theapplied voltage or impressedcur- rent,a change intended to switch the diodefrom the forward-bias on-state to the reverse- bias off-state,or vice-versa.During the turn-off transientthe excess minority carriers stored in the quasineutralregionsmust be removed before steady state conditionscan be re-estab- lished.Thediodeinitiallyremainsforwardbiasedand a large,constant,reversecurrent flows throughthe diode untilthe carrierconcentrations decrease to theirequilibrium values at the edgesof the depletion region. Thistakes place in a periodoftime known asthe storage delaytime,t.tis the primary figureofmeritused to characterize thetansient response of pn junction diodes. The storage delay time increases in relation to the initial storeof carriers,decreases with the rateof carrier removal by the reverse current,and is directlyproportionaltotheminoritycarrier lifetime.Thestorage timeisdecreasedbya- ing R-Gcenterstothesemiconductor during devicefabrication,and step-recovery diodes

--- Page 367 --- 344 Pn JUNCTION DIODES are produced byproperly tailoring the dopingprofile.Measurementof theturn-off transient canalsobeused to determinetheminority carrierlifetimeon the lightly doped sideofa junction.Overall,thefocusof thechapterhasbeenonestablishing a basic understanding oftransientoperationand providing physical insight,knowledgethat will prove very useful in treating one of the pre-eminent switching devices,the bipolar junction transistor. PROBLEMS CHAPTER 8 PROBLEMINFORMATIONTABLE Complete Difi culty Suggested Short Problem After Level Point Weighting Description 8.1 8.3 1-2 15 (a:c-2,d::h-1,i:j-2) Quick quiz 8.2 8.1.2 1 6 (2 each part) Interpret pn(x,t)plot 8.3 8.1.3 2 8 Improved Qp(t) approx. 8.4 8.2 2-3 10(a::c-3,d-1) Open circuit voltage decay 8.5 ” 2 6 Compute turn-on times 8.6 ” 2 10(a-3,b-7) Pulse$IF1to2>1 8.7 ” 2-3 10(a-6,b-4) Combined turn-off/turn-on 8.8 ” 3 15(a-10,b-5) Compare turn-on A(t) 8.1 Quick Quiz. Answer the following questions as concisely as possible. (a) Define storage delay time. (b) Define recovery time (t). Explain. (d)Whatistherootcauseofthedelayinswitchingfromtheon-statetotheoff-state? (e) Namethetwomechanismsthatacttoremovetheexcessstoredchargeduringtheturn- off transient. (f) Trueor false:If△pn(x,t)>0,A>0. (g) Trueorfalse:If>0,theslopeoflinearp(x,)versusplotmustbepositiven increases with x) atx=xn. (h) What is special abouttheelectrical and physical properties ofa step-recovery diode? (i) True or false: Increasing both IF andIR by a factorof2 willhave no effecton the storage delay time.Indicate how you arrived at your answer. (j) Trueor false: Recombinationactuallyacts toretardthebuild-upofstoredcarriers during the turn-on transient.—Indicate how you arrived at your answer.

--- Page 368 --- Pn JUNCTION DIODE: TRANSIENTRESPONSE 345 8.2Theholeconcentrationonthen-sideofapnstepjunctiondiodeatagiveninstantof time is as pictured in Fig. P8.2. $PnO$ x $Pno$ 2 $x_{}$ Figure P8.2 (a)Isthejunctionforwardorreversebiased?Explainhowyouarrivedatyouranswer. (b) Ifpn0=104/cm3andT=300K,determinevA. (c)Isthereaforwardorreversecurrentflowing throughthediode？Explainhowyouar- rived at your answer. 8.3TheapproximationQ()=0wasusedinderivingEq.(8.8).Researchershavesug- gested the alternative approximation,[5] $I$ Q_{P( s)= 1+R}R which leadsto therevised charge controlexpression [(1+{ t_{s=p$ 1+2 Determinewhethertherevisedtexpressionisasignificantimprovement.Constructaplot oft/versusR/IFsimilartoFig.8.6usingtherevisedchargecontrolexpressioninplae of Eq.(8.8).Briefly comment on the result. 8.4Anideal p+-n step junctiondiodecarrying aforwardcurrentIFis suddenlyoen- circuited att=0. (a) Sketchtheexpectedvariationof pn(x,t)versusx at progressivelyincreasing times afteropen-circuiting the diode.Check youranswerusing theCPGprogram in Exer- cise 8.2.) (b)Deriveanexpressionforthestoredholecharge,Q,insidethediodeattimes>0. Be sure to express Qp(0) in terms of known parameters.

--- Page 369 --- 346 Pn JUNCTIONDIODES (c)Assuming a quasistatic decayof the hole charge,derive an expression forvA(t).Take thevA voltagesof interestto begreaterthana fewkT/q;i.e.exp(quA/kT)≥1.Also use the factthatI/0=exp(qVON/kT)-1≤exp(qVON/kT)to simplyyourresult. (d)Does the part (c)result suggestanything?Explain.(ForfurtherinformationseeSub- section 8.5.2 in Schroder[3].) 8.5Anidealp+-nstepjunctiondiodeis switched withacurrentpulsefromI=0toI= 1mA att=0.Calculate the time necessaryfor the diode voltage to reach90% and95% ofits final value.Letτp=1sec and0=10-15A. 8.6Anidealp-nstepjunctiondiodeinitiallyforwardbiasedatispulsedtoaconstant currentofI2>IF1att=0. (a)Sketchtheexpectedviationofp)ersusatprogresivelyincreasingtimesafter t=0. (b)Assumingaquasistaticbuild-upofthestoredcharge,deriveanexpressionforA(). -1mA.After1secacurrentpulseisappliedtoswitchthediodebacktoanmA Assume the diode to be ideal withτp=1sec. (a)Sketch the i(t)through the diode as a function of time. (b) Establish an expressionfor A() at timest>1sec. 8.8a）Theturn-oncurvesdrawnbytheCPGprograminExercise8.2correspondo values stepped from 0.1to2 in 0.1 increments.Noting Δp{n(0, equA(t)/kT -1 eqON/kT −1 appropriatelymodifytheprogramtoobtainνA/ONatthesteppedvaluesoft/p.Let VON=0.5VandT=300Kinperformingasamplecomputation (b)Compare thevA(t)derivedfrom the part(a)exact solution with the Eq.(8.16)solution that was obtained by assuming turn-on proceeds quasistatically. The comparison as- suming VoN=0.5V andT=300K may be presented in either a plot or point-by- point format. NotethatI/0=exp(qVON/kT)-1.

--- Page 370 --- 9 Optoelectronic Read Only Diodes 9.1INTRODUCTION To round out the discussionof pn junction diodes,we consider in this chapter special diode structuresthatarespecificallydesignedandbuiltforopticalapplications.Manyofthe diodes to be discussed involve a semiconductoror semiconductors other than Si. The reader is therebygiven a glimpseofthelarger semiconductor picture wheremoreand morema- terialsare being used in the fabricationof sophisticated special-purpose devices.Semicon- ductor photodevices quitegenerallyfall intooneofthree functionalcategories.Two groups of photodevices convert photo-energy intoelectrical energy.If the purpose of the photo-to- electricalconversion is to detector determine informationaboutthe photo-energy,thede- vice iscalledaphotodetector.Ifthe purposeofthe photo-to-electricalenergyconversion is to produce electrical power,thedeviceiscalledasolarcell.Thethird typeof photodevice convertselectricalenergy intophoto-energyand includes lightemitting diodes(LEDs)and laser diodes. Herein we surveyone setof devicesfromeachofthe three categories:pn junctionandrelatedphotodiodes belongingto thephotodetector family,pn junctionsolar cells, and LEDs. Thecommercialmarketplacehasexperienceda virtualexplosionofoptoelectronic diodeapplications in recent years.Thebarcode reader nowcommon at check-outcounters, the digitaldiskreader inaudio systems,andthelaserprinter in theofficeallmake useof LEDsor laserdiodesas a photosource.Photodetectors in combination withLEDsor laser diodesare usedincircuitisolators,intruderalarms,andremotecontrols.Inaddition,the optical signal is generated using a laserdiode and detected using aphotodiode inmodern- ized telecommunication networksemployingopticalfibers.Solarcellsare usedtopower hand-heldcalculators,battery chargers,andcommunicationsatellites.Althoughslowto find significantuse inlarge-scale power generation,theU.S.NationalPhotovoltaics Pro- gramcallsforaninstalledsolar-cellcapacityof200-1000megawatts bytheyear200and 10,000-50,000megawattcapacityby2010-2030.Clearly,thevarietyand scaleofopto- electronic diode applications can only be expected to increase with time. As anaid in the discussion,the visibleandadjacent wavelengthregionsoftheoptical spectrum are identified inFig.9.1.Therelative responseofthe humaneye plotted inthe upperpartofthefigure is correlated withthemajorcolorbandsshown spanningthe roughly0.4 μmto0.7 mvisibleportionofthespectrum.Notethattherelationship between theoptical wavelength(λ) and the associatedphotonenergy（ph =hν)is 347

--- Page 371 --- 348 pn JUNCTIONDIODES Relative eye response (um)γ← 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Violet Noaipur Blue Green low -Ultraviolet→ Red Infrared- -Visible- ↑ ↑ ↑ ↑ ↑ →λ(m) 6H-SiC ZnSe GaP AlAs GaAs InP !S Figure9.1 Visibleandadjacentregionsoftheoptical spectrum(middle)correlated withtherela- tive eyesensitivity fornormal photopic vision(top)and wavelengths where thephotonenergy isequal to the 300 K band gap energyof select semiconductors (bottom).

--- Page 372 --- OPTOELECTRONIC DIODES 349 given by C hc hc λ= .c = speed of light (9.1) ν hv $E$ ph or,ifλis expressed in m and Eph ineV, 1.24 λ= Eph (9.2) The wavelengthat whichthe photonenergy isequaltothe semiconductorbandgapenergy, λG=1.24/EG,isalsocitedinFig.9.1forsomeofthemoreimportantoptoelectronic materials. 9.2 PHOTODIODES 9.2.1 pn Junction Photodiodes Apn junctionphotodiodeis justapn junction diodethathasbeen specificallyfabricated andencapsulatedtopermit lightpenetrationintothevicinityofthemetallurgicaljunction. Theabsorptionoflightinsidethediodecreateselectron-holepairs,aspictured inFig.9.2. Recombines before reaching junction $EcEp$Ev}$OEc-EFn$Ev$Recombinesbefore-LN$w→+ 下 reaching junction 1 1 P N Figure9.2Visualizationoflightabsorpionelectron-holecreationandthelight-inducedcurrent in a pn junction photodiode.

--- Page 373 --- 350 pn JUNCTION DIODES On average,minoritycarrierscreated in thequasineutral regions within a diffusion length ofx=-x（p-side)or=n（n-side）livelongenoughtodiffusetothedepletionregion. These carriers,and carriers photogenerated within the depletion region,are subsequently swept by the &-fieldto theopposite sideof the junction,thereby contributing anadded reverse-goingcomponenttothecurrentthroughthediode.Ifthephotogenerationrate() is assumed to beuniform throughoutthe diode,theaddedcomponent dueto light(IL) shouldbe equalto -q times theelectron-hole pairs photogenerated per second inthe volumeA(LN+W+LP),or =dark +L (9.3) with IL=-qA(LN+W+L)GL (9.4) Amodified derivationoftheidealdiodeequationandthe R-Gcurrentrelationshipwith GL≠0 confirms the foregoing result (see Problem 6.8). Examining Eq.(9.4),we note thatthe depletion width Win a pn junction diode is typicallysmallcompared to LN+Lp.IfWis negligible,ILbecomes independentofthe appliedbias.The light-onIVcharacteristicsarethereforeexpectedtobeessentiallyiden- tical to the darkI-Vcharacteristic,exceptthelight-oncurves are translateddownward, moved in the-I direction,along thecurrent axis.Moreover, because IGL,the down- wardtranslationofthecharacteristicsshouldincrease inproportiontotheintensityofthe incident illumination. The describedformof the photodiode I-V characteristics are illus- trated in Fig. 9.3. An importantcharacteristicof any photodetector is its spectralor wavelengthre- sponse—how the observed IL,for example,varies with the wavelength of the incident light.A spectralresponse curve representativeof Si pn junctionphotodiodes is reproduced in Fig.9.4.The pictured photodiode responselike thoseofallphotodetectors,spansonly a limitedrangeof wavelengths.Theupper wavelength limit in most photodetectors istied 1 0= VA$ GL=G_{ G{=2G0 GL=4G0 Figure9.3 Photodiode I-Vcharacteristics.

--- Page 374 --- OPTOELECTRONIC DIODES 351 1 0.8 0.6 0.4 0.2 ……… 0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Wavelength (um) Figure 9.4 Spectral responseofa Si pn junction photodiode. The photo-power incidentonthe photodiode was thesame forallwavelengths. Representative characteristic: Theresponse varies somewhat with diode construction. directly to the semiconductor band gap.Photonsareabsorbedandelectron-hole pairsare ph<EG, on the other hand, the semiconductor is all but transparent to the light.The semiconductor spectral response there- foreessentiallycutsof atλG=1.24/EG.ForSi,G=1.12eVat300Kand,consistent with Fig.9.4,one expects a minimal response at wavelengths greater than λ≈1.1m. Two reasons canbecitedfor the decrease in the spectralresponse at shorter wave- lengths.First,as is common practice,the photo-power was heldconstant in accumulating the Fig.9.4 data.Since the photonenergy increases with decreasing wavelength,the flx of photons striking thesemiconductorcorrespondingly decreases.Thus partof theredued response at shorter wavelengths is simply due to the fact that there are fewer photons tobe absorbed. The remaining decrease in the spectralresponse is associated with the nonuniform absorptionof light insidethe diode. During the discussionof photogeneration in Sub- section3.3.3,the lightintensity wasnotedtofalloffexponentially withdistance fromthe surface of the semiconductor.The decay constant corresponding to the inverse of the ab- sorption coefficient(1/)represents theaverage depthof penetrationoflightinto a mate- rial.Asλis decreasedbelow λG,α increases rapidly(see Fig.3.20)andthelightis absorbedcloserand closer to the semiconductor surface.Eventually the majorityof gen- erationtakes placeon the sideof the junctionadjacent to the surface.Carrier recombina- tion is greater near the surfaceand an increased numberof the photogenerated carriers

--- Page 375 --- 352 PnJUNCTIONDIODES recombinebefore theycandiffuse tothe depletionregion.The netresult is aprogressive reduction in the response with decreasingλ. Another property employed in photodetector characterization is frequency response— howrapidly the detector can respond to a time-varyingoptical signal.Inthis reardthe standard pnjunctionphotodiodeexhibitslimitedcapability.Photogeneratedminoritycar- riers must diffuse to the depletion region beforean electrical current is observed externally. Diffusion being a relatively slow process,the maximum frequency response of pn junction photodiodes is at best in the tensof MHz.This response is quite low compared to the frequency response attainable with photodiodes described in the next subsection. 9.2.2 p-i-n and Avalanche Photodiodes p-i-n Photodiodes Ap-i-n diode is a three-region structure in whichan”intrinsic(actually lighly doped)i- region is sandwiched between heavily doped p-and n-regions.In the p-i-n photodiode, shown schematically in Fig.9.5(a),an opening is made in the surface metallization to admit light,thetopsemiconductorreioniskeptverythintominimizeabsorptionintheregion, and thei-layerwidthisspecificallytailoredtoachievethedesiredresponsecharacteristics. Becauseofthelowdoping,thei-layer is totally depletedunder zerobiasor becomes depleted at smallreverse biases. Furthermore, the heavy doping of the outer p- and n- regionscausesthedepletion widths in theseregionsto be verynarrow.Thusaspictured in Fig.9.5(b),thedepletion widthinsidethedeviceisefectivelyequaltothei-layerwidth independentoftheappliedreverse bias.Theenergy bandsinFig.9.5(b) are linearfunctions of positionand the -field is approximatelyconstant in the i-regionbecauseofthe low semiconductor doping.It shouldalso be notedthat the heavy doping of theouter p-and n- regions means the minority carrier diffusionlengths in these regions willberelatively small.As a result,the greater partof the photocurrent flowing in a p-i-n photodiodearises from carriers generated in the central depletion region. Operationaladvantages of the p-i-n photodiode thathave made it one of themost widelyemployed photodetectors stem from the existenceandtailorabilityof thei-region. For one,the diode can be optimized for response at a given wavelength by making the i- layer widthequalto the inverseof theabsorptioncoefficient(1/)at the specified wave- length.Second,withmostof the photocurrentarising from lightabsorption in the i-region, the frequenyresponse isgreatly enhanced overthat ofa pn junctionphotodiode.The large -fieldin thedepleted i-regionleadstotherapidcollectionofphotogeneratedcarriers and a maximum frequency response, 1 1 $f_{m}axx$ carrier transit $W_{}_} (9.5) time across WI where Wis the width of the i-region and at is the saturation drift velocity(see Subsection3.1.2).Typicllyva10cm/sec.IfW=5mforexamplethenfmax 20GHz.Weshouldnotethat Wcannotbemadearbitrarilysmalltoimprovethefrequency

--- Page 376 --- OPTOELECTRONIC DIODES 353 2hv Insulator Antireflection coating +d$i$n+$(a)$EFp$EFn1$Ec 1 Ev}$ 1 1 -p+ i-layer 1 (b) Figure9.5pi-photodiode.Crosssection.)Revrse-biasenegybanddiagramemphaiig the i-region and picturing photogeneration. response. TheRCtime constant associated with the internal series resistance (Rs)and the junctioncapacitance（C=ε0A/W)increases withdecreasing Wandeventuallylimits the response time of the diode. The excellentfrequency response of the p-i-n photodiode makes it a prime candidate for use as the photodetector inoptical fibertelecommunications.By and large,silica-based optical fibersystems installed priorto1990operated at the1.3m wavelength wherechro- matic dispersionin the fibers is at a minimum.Moreadvanced high bit rate systems,on +Chromatic dispersionisaspreadingoutofa lightpulsecausedbydiferentwavelengthsoflighttravelina slightldifrentvlocities.Theuseofmultiwavlengtphotosourcesrequiresoperationatthechromaids- sion minimum.

--- Page 377 --- 354 pnJUNCTION DIODES 2.5 GaP AlAs 2.0 $x=0.45AlSb1.5InPGaAs1$EG(eV) 1.0 GaSb In0.53Ga0.47As 0.5 InAs 0.0 5.4 5.6 5.8 6.0 6.2 Lattice constant (Å) Figure9.6BandgapenergyversuslatticeconstantofselectI-VcompoundsandalloysTheline connecting two compounds specifies the EG versus latticeconstant for alloys madeof the twocom- pounds.Inmostcasesthelatticeconstantisalinearfunctionofthex-valueingoingfromcompoud- A to compound-B.Thus,for example，In0.8Ga02P wouldhave a lattice constantof 5.87Å- 0.2(5.87A-5.45Å)=5.79Å,and anEG=1.5eVis deducedfrom the lineconnectingInPand GaP.Solidand dashed linesidentifyalloysthatare directandindirectsemiconductors,respectively. theother hand,employ single-wavelength photosources and operate at1.55 μm,where fiber loss is at a minimum.In eithercase the wavelengthofoperation is beyond the1.lμm cutoff of Si,thereby necessitating the use of some other semiconductor. The material sys- temofchoiceforfiberopticapplicationsis thealloyIn053Ga047AsdepositedonanInP substrate. As shown in theFig.9.6bandgapplotofcommonII-Valloy systems, In0.53Ga0.47AshasanEG≡0.75eVG=1.65μm）andis latticematchedtoInP.Lattice matching,having exactlythesamelatticeconstant,facilitatesthe depositionofquality In0.53Ga0.47As layers on commercially available InP substrates. Figure 9.7 shows the cross sectionof an InGaAs p-i-n photodiode. The wide bandgap readilytransmittedtotheabsorbingIn0.53Ga0.47Asi-layer.DefectsintheInGaAslayer

--- Page 378 --- OPTOELECTRONICDIODES 355 Silicon nitride

$p^{}--$ InGaAsP i(n-)-In0.53Ga0.47As n -InP (buffer) n+ -InP (substtrate) Z 2 hv Figure9.7 Cross section of an InGaAs p-i-n photodiode. are minimizedbythepriordepositionofanInPbufferlayer.ThewidebandgapInGaAsP cap-layer is addedtoreduce surface-related darkcurrents.Finally,theinsulating silicon nitride layer generally protects the surface and minimizes surface recombination. AvalanchePhotodiodes Avalanche photodiodes are speciallyconstructed p-i-n, pn,or evenmetal-semiconductor (see Chapter14) photodiodes that are operated near the avalanche breakdown point. A standardSi avalanchephotodiodeconfigurationis displayed inFig.9.8.Oneobvious Passivation coating n contact Antireflection coating Guard n n+ Buu pp{+}$ pcontact Figure9.8Si avalanchephotodiode.(FromYang[6©1988byMcGraw-Hill,Inc.Reprintedby permission of the publisher.)

--- Page 379 --- 356 Pn JUNCTION DIODES special structuralfeature is the guardring aroundthe junction periphery. As notedin Chapter 6,junctioncurvature leads to earlybreakdownaboutthe junction periphery.The guard ring minimizes the edge breakdown problem.Uniformly doped low-defect material and minimization of defect creation during device processing arealsorequired toachieve uniform breakdown across the face of the junction. The primary advantageof the avalanche photodiode is a photo-signal gain leading to improvementinthesignal-to-noise(S/N)ratio.Asageneralrule,amplificationofasignal is accompanied byamplificationof the noiseandaddednoisefrom the amplifier.Signal gain therefore typicallyleads toareductionin theS/Nratio.Insideanavalanchephotodi- ode,howeveravalanchemultiplicationamplifies thephoto-signal withoutamplifyingthe typicallydominantreceivercircuitnoise.Thus there is an improvement in the S/Nratio until theadded avalanche-related noise becomes comparable to the circuit noise. Avalanche photodiodes made from InGaAson InPand from Ge provide alternatives to the p-i-n photodiode for use in fiber optic telecommunications. 9.3 SOLARCELLS 9.3.1 SolarCellBasics A varietyof device structures can andhave beenemployed in constructing solar cells.By far and away the mostcommon cells are in essence just large-area pn junction photodiodes. Solar cells are designedof course to minimize energy losses,whereas photodiodes are routinely designed to achieve a specific spectral response or a rapid time response.Design differencesnotwithstanding,thesolarcellI-Vcharacteristicsare ofthesamegeneral form as the photodiodecharacteristicsofFig.9.3.Note that power is derivedfrom the illumi- nated device if the d.c.operating point lies in the fourthquadrant whereI is negativeand Vis positive.Fourth-quadrantoperation can beachieved,for example,by simply placing a resistor in series with the illuminated solar cell. Understandably,thefourth-quadrantcharacteristic isofprimeinterest inevaluating and applying solarcells.Itisthereforecommonpracticeto showonlythefourth-quadrant portionofthecharacteristicsandtoorientthe-I axis upwardontheplotasilustrated in Fig.9.9.Figure9.9alsographicallydefinesthefollowing solarcellparametersofinteres: Vocthe open circuit voltage Isc.the short circuit current V，Im·.theoperatingpoinvoltageandcurrentyieldingthemaximumpower output. cisobviouslythemaximumvoltagethatcanbesuppliedbythecellforagivenphotoin- put,andIcisthemaximumcurrentthatcanbederivedfromthecell.ItfollowsthatP

max mm<.Inasemntsofolcperformnceoneoftenencnters P m} FF Ⅲ \c} max $\text{sc}$ (9.6)

--- Page 380 --- OPTOELECTRONIC DIODES 357 -I -_sc -m} Maximum power rectangle VA 0 V_m _o}c Figure9.9Invertedfourth-quadrantIVcharacteristicofsolarcellunderllumintionwithkey solar cell parameters identified along the coordinate axes. FFisknownasthefllfactorandisofcoursealwayslessthanunity.Theultimatemeasure of solarcellperformance,thepowerconversionefficiency(η),is determinedfrom the current-voltage parameters employing P m} FFIsCV{Oc$max(9.7)nPinPin{P}_{i}n$ where Pin is the photo-energy incident per second or input power. 9.3.2 EfficiencyConsiderations Akey issue with solarcells isconversionefficiency—convertingthemaximumamountof availablesolanegyintolctricalengyThehghetheefficiecyofthecllstheower the costandcollection area required toachieve a desired electrical output.Anumberof factorsenterintotheoverallefficiencyexhibitedbyacellandinvolvebothmaterialand design considerations. Any discussion of conversion efficiency logically begins with the outputfrom the sun. The spectral distributionofthesunsenergyreachingtheearth is plotted in Fig.9.10.The AM0(airmass zero)curve is the measured radiantenergy just outside the earth’s atmo- sphere and isof interest inorbiting satelliteapplications.TheAM1.5curve,normalizedto yieldatotalspectralpowerdensityof100mW/cmisrepresentativeofaverageterrestrial conditions in theUnited States.In eithercase,mostof the spectral power is in thevisible, with a long tailextending into the infrared. Whenconsidering materials for use in solar cells,the area under the spectral curve beyond the λG of a semiconductor yields the incident powerthatislostor wastedbecauseitcannotbeabsorbed.Si,withaslightly largercutoff wavelength,has an advantage in this regard compared to GaAs:20% and35% of the incidentenergy is notabsorbed in Si and GaAs,respectively.However,it wouldbe incor- photonabsorption atλ<λG wavelengths is examined,onefindsthatonly the EG portion

--- Page 381 --- 358 pn JUNCTION DIODES 2500 AM0(air mass zero,outside atm)-135mW/cm2 …… 2000 Spectral irradiance (W/m²-m) AM1.5 (average terrestrial U.S.)-100mW/cm2 1500 λ-GaAs 1000 500 Visible 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Wavelength (μm) Figure 9.10 Solar spectral irradiance. ofthephotonenerg is usedprofitably in producinelectron-holepairs.TheEph> portionof thephotonenergyaddsto thekinetic energyofthephotogenerated carriersand is eventually dissipated as heat.Calculations indicate that ~40% oftheabsorbed photon energy is unavoidably wasted, dissipated as heat,in Si.The corresponding loss isonly ~30% in GaAs because of its larger band gap.Clearly,a trade-off exists between the two cited loss mechanisms,giving rise to an optimum band gap where the energy conversion is at a maximum. Rather fortuitously,the band gapsof Si and GaAs,the semiconductors with the most advanced technologies, both lie very close to the theoretical maximum. Given a specific semiconductor material, the next task is to design and fabricate the solar cell to minimize further energy losses.Indiscussing device-related loss mechanisms, we willrefer to thehigh-efficiency Si solarcell pictured inFig.9.11.Observefirstofall thatthecontacttothetoplightincident)sideofthecellismadethroughnarrow“fngrs.” The fingers,allconnected togetheralongone edgeofthecell,are a design compromise. Zero-widthfingersorfingersonlyalongthecelledges wouldallowmaximum lightpene- tration to the underlying silicon.However,a series resistanceofonly a few ohms can seri- ously degradetheefficiencyofasolarcell.Thefartherapartthefingers,thelongerthe current paththroughthe narrow n-regionat the topof thecell,andthegreatertheseries resistance.Metallizationand contactresistances can become important if thefingers are too narrow.The chosen finger size and spacing are calculated to provide an optimum trade-

--- Page 382 --- OPTOELECTRONIC DIODES 359 Finger “Inverted”pyramids Oxide n p-silicon p× p+$ Rear contact Oxide Figure9.11 Schematicdiagramofahigh-efficiencySisingle-crystalsolarcell.FromGreen et al.[7],©1990IEEE.) off betweenresidual”shadowing”blockingofsomeofthelightthatstrikesthecell,and cell series resistance. Anotherpotentially significantlossmechanism is thereflectionoflightattheSi sur- face. Approximately 30% of the light striking a bare planar Si surfaceat normal incidence willbe reflected.Tominimize losses due to reflection,the top surfaceof solarcellsare typically“textured”andcovered with anantireflectioncoating.Texturing of the surface, theformationoftheinvertedpyramidsinFig.9.1decreasestherefectedlightbyforcing the lighttostrikethe Si surfacetwoormore timesbeforeescaping.Texturing isachieved byplacingthe Si inananisotropicetchingsolution,anetchthat preferentiallyremoves Si atoms along certain crystalline planes.Having an indexofrefraction intermediate between air andSi,the top SiO2layer in thepictured cell,orpreferably a deposited antireflection coating withoptimized parameters,furtherreduces thereflection.Solarcellsconstructed in the described manner have attained a net reflection of less than1%. Once the light has entered the semiconductor,the focus shifts to maximizing the light absorption.IntheFig.9.11solarcellthebottomsurfaceoxideandmetallizationeffectively formamirrorthatreflects lightbackintothesilicon.Long wavelengthlightliterally ping”dramatically enhances the long (λ~λG) wavelength absorption. Finally,the cellmust be designed andbuiltto collectas manyof the photogenerated minoritycarriers as possible.This necessitatesminimizing carrierrecombinationthrough- out the devicestructure.Very long minoritycarrier lifetimes are therule inmodern single- crystal Sicellsyieldingdiffusionlenthsgreaterthanthewidthofthecell.Withthetop and bottom surfaces also carefullyoxidized to minimize surface recombination,carriers generated almost anywhere in thecell volume have a highprobabilityof diffusing to the depletion region and being swept to the opposite side of the junction before they recombine.

--- Page 383 --- 360 Pn JUNCTION DIODES 9.3.3SolarCell Technology Solarcellsfall intothreegeneralcategories:thin-film,single-crystal,andconcentrator.The light-absorbing semiconductor inthin-filmcells is a deposited amorphousor polycrystal- line film. Thin-film amorphous Si cells are commercially available，whileCdTeand CuInSe2cellsareunderdevelopment.Almostallsingle-crystalcellsaremadeofeitherSi or GaAs.Thecell described in the preceding section is an exampleof a single-crystal Si cell.Single-crystalGaAscellsoftencontainAlGaAsorotheralloy layers,withtheactive GaAs layerbeing epitaxially depositedonGaAs,Ge,or Si substrates.Concentratorcells, designedtooperate under light intensitiesequalto10osunsormore,againprimarilyutilize Si and GaAs,although highlyefficientcells havealso beenconstructed usingInGaAsPand InP/InGaAs.Thegreatstridesmadeat improving solarcellefficiencies inrecentyears, including the best solarefficienciesattained to date ineachcategory,are summarized in Fig.9.12. Depositedthin-film solarcellsareof interest becausethey can bemadecheaplyand in large area configurations.Commercial amorphous-Si (a-Si)modules have beenproduced withareas upto1.2m².Amorphous-Sicells are producedby vapordepositingSiontotin- oxidecoated glass substrates.The tinoxide is atransparentconducting material that func- tions asthefrontorilluminated-sidecontact.Analuminumor silver layeris depositedover the a-Si to form the back contact. Thel μm thick amorphous Si is appropriately doped during deposition to achieve a p-i-n-type structure.Themaximum efficiency of a-Si cells inthelaboratoryis13%;theefficiencyofcommercialunits isoftenafactoroftwolower. In addition,the outputfrom a-Si cells decreases by10%-15% during thefirst yearof operation; exposure to light apparently breaks passivating bonds,giving rise toadditional traps andenhancedrecombination.Nevertheless,amorphous-Siaccounts for a significant fractionof worldwide solar-cell shipments,mostofthea-Sicells being destined for use in consumerproducts.Thin-filmCdTecells,havingrecentlyattainedanefficiencyof1%in the laboratoryand11% in thefield,are likely to provideeffectivecompetitionin the near future. 32 32% High-efficiency 28 concentrator cells (gallium arsenide 24 or silicon) 23% Flat-plate 20 single-crystal Eficiency 16 silicon cells 16% 12 Flat-plate thin film (amorphous 8 or polycrystalline) cells 4 0 197819791980198119821983198419851986198719881989199019911992 Figure 9.12 Maximumefficienciesoflaboratory solar cells as a function of time.[8]

--- Page 384 --- OPTOELECTRONIC DIODES 361 Concentrator cellsare obviously intendedforhigh-powerapplications. Thebasic premise leading to the developmentof thecells was that a mirroror concentrating lens system wouldbe less expensivethan an equivalent area of solarcells.Providing somewhat of a bonus,the highercurrentand voltage levels in concentratorcellsalso leads to more eficientcelloperation as is evidentfromFig.9.12.Thestackingofconcentratorcells— placing for example a Si cellor a GaSb(λG=1.7μm)cell under a GaAs cell to absorb wavelengths not processed by the GaAs—is expectedto eventually yieldefficiencies in excessof35%.Specialconcerns withthedesignandoperationofconcentratorcellsinclude heat dissipation,elevated device temperatures,and high current densities. 9.4LEDs 9.4.1GeneralOverview Basic operationof theLED is relatively eassy to explainutilizing informationabout semi- conductorsanddiodeoperationpresentedinearlierchapters.Weknowthatforwardbiasing a pn junction diode causes large numbersof majority carrierelectronson the n-side tobe injectedoverthereduced potential hillintothe p-sidequasineutralregion.Holesonthep side aresimilarly injectedinto then-sidequasineutralregion.These injected carrierssub- sequentlyrecombine.Asexplained in Subsection3.3.2,using E-k diagrams,thecrystal momentaoftheelectronsandholesaredecidedlydiferentinindirectsemiconductors like Si.Thismakes it difficult to conserve momentum in band-to-band transitions. The recom- bination in indirect semiconductors therefore takes place predominantly through R-G cen- ters, and the energy released during therecombinationprocess is dissipated as heat.In direct semiconductors likeGaAs,on theother hand,the crystalmomentaof theelectrons andholesareaboutthesame,andasignificantportionoftheinjectedcarriers iseliminated viaband-to-band recombination.As visualized in Fig.9.13,the energy released in the band- to-bandrecombinationprocessisintheformofphotonsanduponscapingfromtheode becomes the light produced by the LED. In the discussionof the carrier distributions in theconductionand valence bands $Ec$ EFn EFp Ev$ Figure9.13ProductionoflightinanLEDresultingfromcarierinjectioninaforward-iased junction diode and subsequent band-to-band recombination.

--- Page 385 --- 362 Pn JUNCTION DIODES (Subsection2.4.3),it was noted that the distributions peaked verycloseto the bandedges. Thedistributionmaximaare infactreadilyshownto bekT/2fromthebandedges in a nondegenerate semiconductor. Therecombinationofconduction bandelectronsandva- lence bandholes is therefore expected to produce a photon distribution with a spread in energies comparable to kTand a peak energy slightly greater thanEG.In some cases the photonsareactuallycreatedbyanelectronfallingfrom a bandgapcenterslightlybelow the conduction band edgeand/orexcitons are formed priorto recombination.(An exciton isan electronandaholethatbecomeelectrostaticallycoupledandfunctionasaunit) Whenthis happens,the peakphotonenergycanbeslightlylessthanEG.Inanyeven,the peak photonenergy istypicallyclosetoGandthepeakoutput wavelength is atroughy λG=1.24/E.Forthelightproducedtobevisible,theoutputlightwavelengthandthus λGmustlieintherange0.4m<λG<0.7μm,whichimplies1.77eV<<3.10eV. From theoperational descriptionand theenergy/wavelength considerations,we con- clude a semiconductor used to produce visible LEDs is subjectto three requirements. The semiconductor should bedirect,have a band gap energy intermediate between1.7eVand 3.10eV,and be amenable to the formationof pn junction diodes. Surprisingly,very few semiconductorsmeetall threerequirements.SiandGeareexcluded immediately.GaAs wouldbeidealexcept that its band gap is too small.Examiningeither Fig.9.1or Fig.., we conclude theIII-Vcompounds GaPand AlAs have band gaps in the desired range,but bothGaPandAlAs are indirect.Figure9.1also indicates theIV-IVcompoundSiChasa bandgapof thecorrect size.Unfortunately,SiCis alsoindirect.SeveralII-VIcomound semiconductors including ZnSecited in Fig.9.1are both directand have band gaps in the desired range.However,untilrecently it has beenimpossibleto form pn junctions inmost of theII-VIcompounds.Native defects in theII-VIcompounds tend to compensate either the n-or p-type dopants introduced to form a junction. With no elemental or compound semiconductors meeting the three requirements,it is understandable why semiconductor alloys and special “light-enhancing” centers have come to be employed in producing com- mercially available LEDs,as detailed in the next subsection. 9.4.2CommercialLEDs A summaryofrelevantinformationaboutcommerciallyavailableLEDsispresented in Table9.1.Individualtableentriesarediscussedbrieflybelow. GaAs0.6P0.4 GaAs is a direct semiconductor,but its band gap is too small to produce visible lightemis- sion.GaPhasaλGinthevisiblerange butisindirect.Individuallyneithercompoundmeets the LEDmaterial requirements.However,referring to Fig.9.6,a combinationof thesetwo compounds—the GaAs1_xPxalloy—is bothdirectandhas a light-producingEG for- values intherange0.28≤x≤0.45.GaAs0.6P0.4withan=0.4is specificallyused as the light-emittingmaterial because it yieldsanoutputthatappearsbrightestto thehuman eye.Theexternalefficiencyη=photo poweout/electricalpowerinofaGaAsxPx LED decreases byabouta factorof10asxis increased fromx=0.28to the direct-indirect transition atx=0.45.However,as xis increased,theoutput wavelength decreases and the

--- Page 386 --- OPTOELECTRONIC DIODES 363 Table9.1 Characteristics ofCommercial LEDs.(Adapted fromCraford[9].) External Effi ciency Performance Semiconductor Color Peakλ(m) η(%) (lumens/watt) EstablishedMaterials GaAs0.6P0.4 Red 0.650 0.2 0.15 GaAs0.35P0.65:N Orange-Red 0.630 0.7 1 GaAs0.14P0.86:N Yellow 0.585 0.2 1 GaP:N Green 0.565 0.4 2.5 GaP:Zn-O Red 0.700 2 0.40 Recent Additions AlGaAs Red 0.650 4-16 2-8 AlInGaP Orange 0.620 6 20 AlInGaP Yellow 0.585 5 20 AlInGaP Green 0.570 1 6 SiC Blue 0.470 0.02 0.04 GaN Blue 0.450 2 0.6 Luminousperormanceofvisible lightsourceismeasuredintermsofthelumensderivedfromthedeviceper wattofelectrical powerinput.Lumensarecalculatedbymultiplyingtheradiantoutput(in watts)imes the relative response of the human eye (Fig. 9.1). human eye response increases by a factor of~50.The product of the LED efficiency and the eye responseexhibits a maximum at=0.4,the valueused in producing commercial devices. The low-costredLEDs madeofGaAs0.6P0.4 werethefirstsolidstate lamps to be successfully mass marketed. They were used in the displays on watches and hand-held calculators manufactured in theearly1970s.Subsequently,of course, they were replaced by liquidcrystal displaysthatpermittedlongerbattery life.Exhibitingrather lowbright- ness,theGaAs0.6P0.4diodes primarilyfunctionas indicator lamps in indoorapplications. Figure 9.14(a)shows the cross sectionofaGaAs0.6P0.4LED.Starting with aGaAs substrate,a layer with x systematically increased or”graded”fromx=0tox=0.4is first deposited to minimize latticemismatch problems. Becauseof the large difference in the latticeconstantsofGaAs andGaAs0.6P0.4,direct depositionofGaAs0.6P0.4onGaAs would yieldamaterial withalargenumberofdefects.NexttheGaAs6P04layerandthenanother graded layer withx increasedfromx=0.4tox=0.6is added to the structure.Finally,Zn is diffused into the upper layers to form the p-sideof the junction.The increased xof the toplayerproducesawider bandgap,thusalowing lightgeneratedintheGaAs0.6P0.4ma- terial to pass through the top layer with minimum reabsorption.

--- Page 387 --- 364 pn JUNCTION DIODES Top p-contact $S_{3}N4$ VKA x graded 0.4→0.6 p-GaP:N n-GaAs0.6P0.4 x graded n-GaP:N n-GaAs{1-xP_x 0→0.4 n-GaP (substrate) n-GaAs (substrate) V//A n-contact, segmented to enhance (a) bottom surface reflection (b) p-GaP p-AlGaAs upper confining layer p-AlInGaP upper confining layer AlGaAs active layer AlInGaP active layer n-AlGaAs lower confining layer n-AlInGaP lower confining layer n-GaAs substrate n-GaAs (substrate) (or thick AlGaAs epilayer) (c) (d) Figure9.14 LEDcross sections:(a) GaAs0.6P0.4red LED,(b)GaP:NgreenLED,(c) AlGaAs “high-brightness”red LED,and(d)AlInGaPLED.(AdaptedfromCraford[9,10].) GaAS0.35P0.65:N,GaAs0.14P0.86:N,GaP:N Asrecorded in Table9.1,orange-red,yellow,and green LEDs are made fromGaAs−xPx materials with x-values ranging fromx=0.65 tox=1.Technically,allof these materials are indirect. However,doping with nitrogen as indicated by the”:N”designationmakes these materials pseudo-direct.Nitrogen,itshould be noted,is aColumn Velement like As andP.Because nitrogenreplaces anotherColumnVelement whenit is addedtothe GaAs-xPxlatice,itdoes notact likeastandarddopant.Noadditionalelectronsorholes are produced.Rather,the nitrogen introducesanelectroniclevel,anelectrontrappinglevel, approximately0.1eVbelowtheconductionbandedge.Thisspecialcenter iscalledan isoelectronictrap because it is formedfrom an element with the same (iso= same)valence structureastheatom itreplaces.Ofimportance to LEDoperation,anelectronattractedto

--- Page 388 --- OPTOELECTRONIC DIODES 365 the trap becomes highly localized in space,which inturn causes a large spread in the allowedmomentumof thetrapped electron.Matching themomentumofholes in theva- lence band,the trapped electron has a dramatically enhanced probability ofmaking a hole- annihilating transitionthatresults intheproductionof a photon.Inessence,the isoelec- tronictrapactsasaconduitfortherelativelyefficientradiativerecombinationofelectrons and holes. The GaAs0.35P0.65:NandGaAs0.14P0.86:NLEDs aresimilar inconstructiontothe GaAs0.6P0.4LEDexcept the devices are formedon GaP substrates.The GaP:N structure picturedinFig.9.14(b)isdiferentinthatasubstrategrading layerisnotrequiredandboth the active n-and p-regionsofthe diodeare formedepitaxially.Because the light resulting fromthenocrtiiothsrcueshk <G), thereis very litle reabsorption intheepitaxial layers.Moreover,theGaP substrate isalso transparentto theLEDlight.Thusevendownwardly directed lightis reflectedandmay eventually escape without being reabsorbed.This yieldshigher externalefficienciesand brighter devices.Theorange-red,yellow,and green LEDs describedhere,along withthe GaP:Zn-OredLEDdescribednext,presentlycommandthelargestshareoftheLED market. GaP:Zn-O ZnaloneactsasanacceptorandOaloneacts likea donor inGaP.Whenadded inaproxi- matelyequal numbersandafterproperannealing,however,the ZnreplacingGaand theO replacingPtendtoformonadjacentlatticecitesandfunction togetheras anisoelectronic trap.The Zn-O trapcomplex differs from Nin that the traplevel lies deeper in the GaP band gap,approximately 0.3eVbelowthe conductionband edge.Lightproduced bytran- sitions through the complex is red in color withλ =0.700 m.Fabrication ofGaP:Zn- peak OLEDs parallels that of the GaP:NLEDshown inFig.9.14(b). AIGaAs Introducedinthelatter halfofthe198Os,theAlGaAsLEDis arelative newcomertothe commercialmarketplace.Itissometimessoldasthe”highbrightness”redLED,ehiiting a luminance up to 20 times greater than the more common GaAsP:N and GaP:Zn-Ored LEDs.Notable applicationsof the AlGaAsredLED,whichhas capturedroughly10%of themarketincludesuseinthespoilerorthirdautomobiletail lightandas theheellighton athletic tennis shoes.Thecross sectionof anAlGaAs LEDis shown in Fig.9.14(c).Light isderivedfromrecombinationinthenarrowactiveregion whichhasan E≤1.V.The adjacentconfining layershavea slightly widerbandgaptominimizereabsorptionofthe generated lightandtomaximize extractionofrandomly directed lightfrom thestructure. In thebrightest AlGaAs devices,a thick (100-200 μm)epitaxial layer is grown underthe bottomconfining layer and theE<E GaAs substrate is removed.Note from Fig.9.6 ph that an AlGaAs alloy withEG<2eV is a directsemiconductor.Moreover,allAlGaAs alloys havealmost exactly the same latticeconstant,are latticematched,to GaAs.The combinationofminimalnonradiative recombinationdue to theperfectionof the lattice-

--- Page 389 --- 366 pn JUNCTIONDIODES matchedlayersandtheproductionoflightvia directrecombination（which isaboutan orderofmagnitudemoreefficientthanrecombination throughisoelectronic traps)leadsto the high-effi ciency/high-brightness of the structure. AllnGaP AlInGaPLEDs werefirstofferedforsale in1992.Involving a ratherunusualalloycon- sisting of threeColumnIIIelementsandoneColumnVelement,AlInGaPtechnology is still initsinfancy.Nevertheless,AlInGaPLEDsalready significantlyoutperformestab- lished orange,yellow,andgreen devices.Inthe long run,AlInGaPmay eventually become the dominanthigh-performanceLEDtechnologyforallcolorsfromredthrough green. Thecross sectionof an AlInGaPLEDis shown in Fig.9.14(d).Like the AlGaAsLED,the confining and active layers are lattice matched to the GaAs substrateand recombination in the active region is direct. Differentcolors are derivedofcourse from structures with differentactive layercompositions.GaPis presentlyused as thetop contacting/window layerbecause itprovidesa muchhigherelectricalconductivity thanthatattainablewith p-type AlInGaP. Sic TheSiCstoryisaratherinterestingone.Electroluminescence was firstreportedfromaSiC samplein1907thatundoubtedlycontainedbuiltinpnjunctions.SiCLEDs wereinnsely researched inthe1960sandbrieflyofferedforsalein theearly197Os.These initialoffer- lowing a breakthrough in the formationof high-qualitySiC substrates,and given theestab- lishedneedfor a blueLED,SiCLEDs were againofferedforsale in1990.Although significantly improvedto0.02%,theexternalefficiencyremainslow inthe newoffering becauseLEDoperationstillreliesonlow-probabilityradiantrecombinationinanindirect material. GaN GaNisa direct II-Vsemiconductor withanEG=3.36eV.The GaNblue LED was first offered for sale in April1994.Thesomewhat surprising developmentof theGaNLEDby a chemical company with no expertise in manufacturing semiconductor devices was made possible by a breakthrough in forming pn junctions in GaN andGaNalloys. The light- producing radiativerecombinationactuallytakesplace inanEG=2.75eVInGaNfilm sandwiched between wider-band-gap AlGaNlayers.It is projectedthat themoreeficient GaNLED willrapidly surpass the SiCLED in sales and become the blue LEDofchoice. Providing the“missing”colorcomponent,theGaNblue LEDmakes possible full-color LED-based outdoor video displays that operate at reasonable power levels. 9.4.3LEDPackagingandPhotonExtraction Across-sectional sketchof the standardLEDpackage is presented in Fig.9.15.The LED chip,typicallymeasuringapproximately250μm onaside,isplaced in areflective cavity.

--- Page 390 --- OPTOELECTRONIC DIODES 367 Epoxy dome Reflecting cup Lead Figure 9.15 Cross section of the standard LEDpackage. Thereflectivecavity is used because a portion,in somecasesa large portion,of the light actuallyescapesfromthesidesofthechip.Oneofthediodeleadsisconnectedtotheaviy while a wire bond ismadefromthetopcontacton thechip to the second diode lead.The chip and leadframe are embedded in anepoxy thatholds the leadframetogetherand(as will beexplained)facilitatesphotonextractionfromthechip.Thedome-like shapeofthe Getting generated light outof the LEDchip poses moreof aproblem thanone might suspect.We have already noted thatwiderband gap materialoverlays the light-emitting layersintheGAs06P04AlGaAs,andAlInGaPEDs.Thewidebndapmtlcs like a window,allowing light to reach the surface of the diode without being reabsorbed.A windowmaterial is notnecessary inthenitrogenorZn-OdopedLEDsbecause theemitted photons have anenergy at least 0.1eVbelow theband gap energyof the hostmaterial. Ideally it is also desirable to have EG>E materialbelow the light-emitting layer.Half ph ofthegeneratedlightis initiallydirecteddownwardandsomelightrelectsoffthetop surface. With G E materialbelowthelight-emittinglayer,the lightcanbounce ph around inside theentirestructure with a minimal probabilityof reabsorption,thereby dra- matically increasing thepercentageof the lightstrikingthesemiconductor surfaces. Just because light makes it to the semiconductor surface, however, doesn’t mean the light willexit the LEDchip.Let us review some basicoptics. When light traveling in a dielectricmediaofrefractiveindeximpinges upontheinterface withaseconddilectric mediaofrefractive index n2,thelight in generalwillbe partiallyreflectedandpartially

--- Page 391 --- 368 Pn JUNCTION DIODES n{2 $n_{}$ Figure9.16LightreflectionandtransmissionatanintrfacebetweentwodilectricmaterilsTe light is shown incidentat an angleθfrom the normal in material number1. transmittedaspicturedinFig.9.16.Fornormalincidence=0,thefractionofthelight transmitted,TN,is given by 4n_}TN=$(9.8)(1+n^{}$ wheren=n/n₂is the indexofrefractionratio.nis invariablygreaterthanunity (n_>n2)forlight being transmittedfrom asemiconductorintoanothermedia.Whenever n1>n2,lightincidentatanglesgreaterthanacriticalanglewillexperiencetoal internal reflection（T=0).Thecritical angle iscomputedfrom n_2} sinθc$ (9.9)

$n_{}$ Accounting for total internalreflection,assumingthe incident light isuniformly distributed in all anglesandrandomlypolarized,theusualcaseatthetop surface inLEDs,and sum- ming overallangles,theoverallfractionofthe lighttransmitted,Tisdeterminedtobe (sin θc){} 2 \ TN (9.10) 2 n(1+n c^} In performing a samplecomputationletusconsider lighttransmissionat theGaPair interface.GaPwith arefractive index of3.4 is fairlyrepresentativeofthe semiconductors usedin LEDs.Settingn₁(GaP)=3.4andn2(air)=1,one computes TN=70%.However, c=17.1°andT=3.0%!!Ahugepercentageof the light is subjectto total internal reflection.The epoxy, withan indexofrefraction intermediate between air and the semi- conductor,allows a muchlarger fractionof the lightto escapefrom the LEDchip. The conically shaped domeofthe LEDpackage in turnmakes it possible for muchof the light to arrive at the epoxy-air interface within the critical angle.

--- Page 392 --- PARTIIB BJTs AND OTHER JUNCTION DEVICES

--- Page 393 ---

--- Page 394 --- 10 BJT Fundamentals Chapters 5-8 were devoted to the detailed analysis of the pn junction diode.As a logical extensionof the diodeanalysis,the development next progressesfrom theone-junction/ two-terminal diodeto the two-junction/three-terminaltransistor.In thischapter we initiate the discussionof thebipolar junction transistor (BJT).Thechaptercontains a collageof introductoryBJTinformation—definitionof terms and symbols,qualitativeoperational concepts,keyrelationships,and so on.In combination,the information forms the required knowledge base preparatory to a detailed device analysis. 10.1TERMINOLOGY TheBJT is a semiconductor devicecontaining threeadjoining,alternately doped regions, withthemiddleregionbeingverynarrowcomparedtotheminoritycarrierdiffusion length in that region.pnp and npn junction transistors are pictured schematically in Fig.10.1.As indicated in the figures,the narrow central region is knownas thebase, and theoutertwo regions are referred to as theemitterandcollector.Itmightappearfrom a cursory inspec- tionof Fig.10.1thattheoutertwo regionsare interchangeable. However,in practical devicestheemitterhasa different geometryandis typically more heavily dopedthan thecollector.Interchanging thetwoterminalstherefore significantlymodifiesthe device characteristics. Standard circuit symbols for the pnpand npn versions of the bipolar junction transistor are presented in Fig.10.2.Symbolsforthe d.c.terminalcurrentsand voltages,pluscurrent and voltage polarities,are also noted in the figure.The“+”and”-”signs usedto visu- allyspecifythevoltagepolaritiesinFig.10.2areactuallyredundantthedoublesubscrip on thevoltage symbollikewise denotes thevoltage polarity.Thefirstletter inthedouble subscripttiietetrlndtheondriiiettrmlB for example,is the d.c.voltage drop between the emiter(+)and base（-).Observe that thepositivecurrentlowdirectionsspecifiedinFig.1.2areinsomecasescontraryoE convention, whichalways takes the currentflowing into a terminal to be positive.Given thechosenpolarities,howeverallterminalcurrents are positivequantities whenthetran- sistor isoperated inthestandardamplifying mode.Thechosenpolaritiestherebyavoid unnecessarycomplicationsand are much moreconvenient intreating the physicalopera- tion of the transistor. Althoughallthreecurrentsand voltagesshown in Fig.10.2areused in specifyingthe transistorcharacteristics,onlytwoofthecurrentsandtwoofthevoltagesare independent. Thecurrent flowing into a device must beequal to the current flowing out of a device,and 371

--- Page 395 --- 372 BJTs AND OTHERJUNCTION DEVICES Emitter Base Collector Emitter Base Collector E C E C P+$NP+NPNBB(a)pnp(b)npnFigure10.1Schematicrepresentationofthe(a)pnpand(b)npnBJTshowingdeviceregionsandtheterminaldesignations.IEVECIEVCE$lC + E C E C + + VEB $BB$ VCB VBE BB$ V{BC + + B B (a) pnp (b) npn Figure10.2 (a)pnpand(b)npnBJTcircuit symbols.The d.c.terminalcurrents,voltages,and reference polarities are also noted in the figure. the voltage droparound a closedloop must be equalto zero. Thus,by inspectionfrom Fig.10.2(a) or (b), IE= B +IC (10.1) and VEB +VBC+VCE=0 (CE=-VEC) (10.2) Iftwoofthetransistor currents or voltagesareknown,Eq.(10.1)or(10.2)canalwaysbe used to determine the third terminal current or voltage. Because more thanone current and one voltage are involved in theoperation ofthe BJT,the device characteristics are inherently multidimensional.For the descriptionof the characteristicsto betractable,itis necessarytofocusonthecurrents,voltages,and polari- ties of primary interest in a particularapplication.This is accomplished byspecifying the basic circuitconfiguration in which the device is connected and the biasing mode.

--- Page 396 --- BJT FUNDAMENTALS 373 \c} C IE E P +d E IE ICC B BB} VEC B B} VEC +d N P N (out) N (out) VEB VCB VEB VCB p{+ P (in) (out) (in) (in) B B E E C C (a) Common base (b) Common emitter (c) Common collector Figure 10.3 3Circuitconfigurations:(a）commonbase;(b）commonemitter;and(c）common collector. In most applicationsa signal isinputacrosstwooftheBJTleads,and anoutput signal is extractedfrom a second pairof leads.SincetheBJT hasonly three leads,one of the leads mustobviously be partofboth the inputandoutputcircuitry. The terms common base,commonemitter,andcommoncollectorareused toidentifytheleadcommontothe inputandoutputandto specifythecorrespondingcircuitconfigurationsilustratedin Fig.10.3.Thecommonemitter is themost widely employedconfiguration,withthecom- monbase finding occasionalutilization.Thecommoncollector is seldomifeverusedand willbehenceforthneglected.Inspecting Fig.10.3,weseethecircuitconfigurationsuc- cinctlyidentifiesthecurrentsandvoltagesof interest.In thecommonemitterarrangement, forexamlIandaeclrlthelvntoututviable.Idledkecoe commonbaseandcommonemitteroutputcharacteristics arepresented in Fig.10.4for future reference. 5 16 E=4mA B=40A 4 12 IE=3mA ${}_B=30A$IC(mA) 3 IC(mA) E=2mA 8 {B}=20A 2 E=lmA B}=10μA 4 1 E=0 ${B}0 0 0 0 10 20 30 40 50 60 0 5 10 15 20 -VCB(volts) VEC(volts) (a) Common base (b) Common emitter Figure10.4 Idealized sample sketchesof(a)thecommonbaseand(b)thecommonemitteroutput characteristics of a pnp BJT.

--- Page 397 --- 374 BJTs AND OTHER JUNCTION DEVICES Table10.1 Biasing Modes. Biasing Biasing Polarity Biasing Polarity Mode E-B Junction C-B Junction Saturation Forward Forward Active Forward Reverse Inverted Reverse Forward Cutoff Reverse Reverse Thebiasing mode helps to further specifytransistoroperation in a givenapplication byidentifyingthevoltagepolaritiesofprimary interest.Specifically,thebiasingmodetells one the polarityofthe bias(forwardorreverse) being appliedto thetwo transistor junc- tions. In all,there arefourpolarity combinations as summarizedin Table10.1and Fig.10.5.Active or forward active biasing,wherethe E-B junction is forward biased and the C-B junction reverse biased,is the most widely encountered of the operational modes. Almostalllinear signalamplifiers,suchasoperationalamplifiers,areactivemodebiased. Underactivemodebiasingthetransistorexhibitsitslargestsignalgainandsmallestsignal distortion.Saturation,when bothjunctions are forward biased,and cutoffwhenboth junc- tions are reverse biased,giverise respectivelytotheon-statehighcurrentflow,lowvolt- age)andoffstatelowcurrentflowhighvoltage)associated withoperationofthetransistor as a switch.In digitalcircuitstheselow-voltageandhigh-voltagestatescorrespondtothe mode,the C-B junction is forwardbiased andthe E-B junction is reverse biased.Effec- tively,therolesof theemitterandcollectorare thereby interchangedor inverted relative to the forward active mode. Throughoutmostof the developmentto follow,the deviceunderanalysis is takento be a pnp BJT.Although the npn BJT is used in a far greater numberofcircuit applications andIC designs,thepnpBJTis amore convenient vehicleforestablishingoperational principlesandconcepts.It is assumed thereader canreadilymodifythe pnp development and results so they are appropriate for annpn BJT. 10.2 FABRICATION An idealized pseudo-one-dimensionalmodel fortheBJTsimilarto Fig.10.1is usedboth inqualitative discussionsandtoobtainfirst-orderquantitative results.BJTfabricationis briefly described here to provide some insight into theactualphysical natureof the device structure and to indicate thecorrelation with theone-dimensional model.In many ways the fabricationofthebipolartransistor is justastraightforwardextensionofthediode fari- cation described inChapter 4.Themajor difference,of course,is that two pn junctions must be formed in close proximity. Cross sections of a typical discrete,double-diffused pnp transistor and an integrated

--- Page 398 --- BJT FUNDAMENTALS 375 VEB(pnp) VBE(npn) 个 Active Saturation VCB(pnp) V{BC(npn) Cutoff Inverted (a) Saturation $IC$ ↑ $B0$ Active B}0 Cutoff VEC(pnp) 0= VCE(npn) {>0 Cutoff Inverted active Saturation (b) Figure10.5(a)CombinationsoftheBJTinputandoutputvoltagesresultinginthefourbiasing modes.)RgionsoftheBJTcommonemiteoutputchaactisticsasociatedwiththefoubasing modes. circuit npntransistorareshownrespectively inFigs.10.6(a)and(b).Starting witha heavily doped p+-Si wafer,the first step in forming the discrete pnp is the depositionof a high- resistivity(lightlydoped)p-type Siepilayer.Thislayer,typically5to10 m thick,ulti- matelycontains thetransistorproper. The starting wafer merely providesmechanical sup- port and a low-resistance pathto thecollectorcontactonthebottomof the wafer.After growing an oxide and opening oxide windows,a phosphorus or an arsenic diffusion is

--- Page 399 --- 376 BJTs ANDOTHERJUNCTIONDEVICES E B B B SiO_{2}$B不n+n+Narrow(5\mu m)nnp-epilayerPWide(200m)p+-substrate不ce$ C (a) B B C Substrate B E C SiO_{2}$ Narrow(~5 μm) 不 p+ p+ p n+ n+ p+ n 业 n-epilayer n +-buried layer Wide (~200μm) p+-substrate 业 1 (b) Figure10.6 Crosssections and simplifiedmodelsfor(a)atypicaldiscrete,double-diffusedpnp BJTand (b)an integrated circuit npn BJT. performed yielding the n-type base.With the p-typedopant boronreplacingphosphorusor arsenic,theprocedureisnextrepeatedtoformthep+-emitterFinally,aluminumis epos- ited on topof the water and patterned tocreate the emitterandbase contacts.It shouldbe noted thatthe geometrical features in Fig.10.6 are not drawn to scale.A better ideaas to the truerelative sizeof thestructure isobtained by expanding the figure by abouta factor of50 in the horizontal direction. The npn integrated circuit transistor pictured inFig.10.6(b) is similarlyfabricated by performing a p-type diffusion and then an n-type diffusion into a high-resistivityn-type epilayer.In theICcase,however,theepilayer has a doping type opposite to thatof the substrateand diodeisolationof thenpn transistor is completed by a deep p-type diffusion around thetransistor periphery.Themost negative potential in thecircuit is appliedtothe p-typesubstrate.Beingatahigher potential,the n-epitaxialregion is thereforeelectrically isolated by a reverse-biased pn junctionfromother deviceson theICchip. In moread-

--- Page 400 --- BJT FUNDAMENTALS 377 -W_B$-Emitter-Base\to -Collector+dNP(a)$WEB{}$\leftarrow WCB\to -W$Ec$Ei(b)EF$E_v}$ V V → x (c) 8 8 ↑ ↑ (d) ρ ρ ↑ Figure10.7 Sketchesof theelectrostatic variables in a pnpBJTunderequilibriumconditions. (a) Depletionregions,(b)energyband diagram,(c)electrostatic potential,(d)electricfield,and () charge density.Thetransistor regions are assumed to be uniformly doped with NAE≥NDB>NAc.

--- Page 401 --- 378 BJTSANDOTHERJUNCTIONDEVICES vanced designs the lateral pn junctions have been replaced by silicon dioxide isolation.The n+buried layer shown in Fig.10.6(b)is formed by diffusion prior to depositing the epi- layer.The buried layer servesas a low-resistance pathbetweentheactivecollectorregion of the transistorand the top-side collector contact. 10.3 ELECTROSTATICS Paralleling thepn junction diodeanalysis,we begin theBJTanalysis by looking intothe electrostatic situation insidethetransistor.Fortunately,underequilibriumandnormalop- erating conditions,theBJTmay be viewedelectrostaticallyas nothing more than twoin- dependent pn junctions.The pn junction electrostatics established in Chapter 5—relation- shipsforthebuilt-inpotentialchargedensity,electricfield,electrostaticpotentiald depletion width-may beapplied separately,withoutmodification,to the E-BandC-B junctions. Thus,for example,assuming the transistorregions to be uniformly doped and taking NAE(emiter doping)≥NDB(basedoping)>NAc(collector doping),the usual case in a standard pnp transistor,the equilibrium energy band diagram and electrostatic vari- ablesbasedonthe depletionapproximation are concludedto be as pictured in Fig.10.7. Upon examining Fig.10.7,note that the depletion widthsareconsistent withtheas- sumed NAE N}$ DB

NAC doping profile. Specifically,almostallofthe E-B depletion width（WEB）lies inthebase while mostof theC-Bdepletion width（WCB）liesinthe collector.WCB

W EB since the lightly doped side of the C-B junction has a lower doping than the lightlydopedsideof theE-B junction.Also notethat WBis thetotal widthofthe B base andWis the standardsymbol for the portionofthe basethat is not depleted;i.e.,for a pnp transistor, W=W_B-nEB-$nCBB (10.3) where xnEB and nCB arerespectively theportionsof the E-BandCBdepletion widths lying inside the n-type base.The Win BJT analyses,referredtoas thequasineutral base width, should notbe confused with the diode depletion width that was associated withthe symbol Win earlier chapters. Exercise10.1 P:Draw the energy band diagram characterizing:(a)an npn transistor under equilib- rium conditions;(b)a pnp transistor under active mode biasing.Take the transistor regionsto beuniformly dopedand assume a standarddoping profile where theemit- ter doping > base doping >collector doping. S:(a) Given the complementary nature of npn and pnp transistors,the energy band diagram for an npn transistor can beobtained from the equivalent pnp diagramby merely flipping thepnp diagram upside down. Thus,either by flipping Fig.10.7(b) upside down or by appropriately combining the equilibrium band diagrams ofan np

--- Page 402 --- BJTFUNDAMENTALS 379 and a pnjunction,theequilibriumenergy band diagram foran npntransistor iscon- cluded to have the form shown in Fig.E10.1(a). Ec EF Ev} Figure E10.1(a) (b) Underactive mode biasing the E-B junction is forward biased and the C-B junc- tion isreversebiased.Thus,referringtoFig.10.7(b),theFermilevelonthebase side of the E-B junctionisraisedrelative to theemittersideFermilevelandbothhe EBdepletiowidtandpotentialillaredceased.Convrselyrvrseiasi C-Bjunction lowers the Fermi levelonthebasesideof thejunctionrelativeothe collector-side Fermi level andboththeC-Bdepletion widthand potentialhillare increased. The deduced energy band diagram is sketched in Fig.E10.1(b). Ec $Ei$ EF Ev} Figure E10.1(b) (C) Exercise 10.2 P:Paralleling Exercise 5.4,construct a MATLAB (computer）program that draws the equilibrium energy band diagram for a nondegenerately doped Si pnp or npn BJT. Assumethatthe transistor regionsare uniformly doped and theBJT is maintainedat room temperature. S:ABJT”diagram generatoris includedon diskas file BJT_Eband. Thestudent- producedcode for theprogram is reproduced in Appendix M. As supplied,the

--- Page 403 --- 380 BJTs ANDOTHER JUNCTION DEVICES program draws theequilibriumenergy band diagramfora pnp BJT with NAE=1018/ cm³3，NDB=1016/cm3NAC=101/cm,andWB=1μm.Thedopingparameters and total base width can be modified by changing the DOPINGand WIDTHentries on program lines 5 and 6.n-type dopings are input as negative values. 10.4 INTRODUCTORYOPERATIONALCONSIDERATIONS To provide insight into theoperationofthe bipolartransistor,letusconsider apnp transis- torunderactivemodebiasingandinitiallyfocusontheactivityoftheholesinandadaent to the base region.As pictured inFig.10.8,the primarycarrieractivity in the vicinityof the forward-biased E-B junction ismajoritycarrier injectionacross the junction intothe opposite-sidequasineutralregions.Naturally,thep+-nnatureofthejunctionleadstomany more holes being injectedfrom theemitter into the base thanelectrons being injected from the base into theemitter.Thekey to transistoraction is what subsequently happens tothe carriers thatare injected intothe base.Ifthequasineutral base width were much largerthan a minoritycarrierdiffusion length,theinjectedholes would simplyrecombine in then- type baseand there would be no interaction between the two junctions.The structure would be nothing more than two back-to-back pn junctions. However,by definition,the BJTis a structure where the base is narrow compared to a minoritycarrier diffusion length.Thus, the vastmajorityof injectedholes diffuse completely throughthequasineutral base and entertheCBdepletionregion.TheaceleratingelectricfieldintheCBdepletionregion thenrapidly sweepsthesecarriersintothecollector.Thenarrow widthofthebasethereby leads to a couplingof the E-BandC-B junctioncurrents,withcarrieractivity being de- cidedly differentfrom two back-to-back pn junctions. In additiontociting thecarrieractivity underlying transistoraction,theforegoing helps to explain the naming of the transistor regions. When active mode biased,the emitter functions as a sourceof carriers,emitting carriers intothebase.Conversely,thecollector portionofthereverse-biasedC-Bjunctionactslikeasink,collectingtheemittedcarirs after they pass through the control or base region. $Ec$Ei}$EF$ $Ev$ O O Figure10.8 Carrieractivity inapnpBJTunderactive mode biasing.

--- Page 404 --- BJTFUNDAMENTALS 381 $W_{}$ P{+$—W-PE$\c{}I IC E ← C IE$IcICn$IEnB{1B{2BB3↓B}$ E-B B-C depletion region depletion region Figure10.SpatialvisualiztioofthedifusioncurrentsflowininpnpBJTunderaciemde biasing. For a more complete picture,let us next expand the discussion with the aidof Figs.10.8and10.9toincludeallofthediffusioncurrents flowing intheBJT.(Paralleling the diode analysis,theR-G currents associated withthe depletionregions are initially neglected.)InFig.10.9the holediffusioncurrentflowing across the E-B junction,the currentassociated withtheholesinjectedintothebase,isidentifiedasEp.Similarly,the hole diffusioncurrent flowing across theC-B junction,a currentalmost exclusivelyresult- ing from the injectedholes thatsuccessfullycross thebase,is labeledICp.Asindicad previously,very fewof the injected holes are lost by recombination in the base of a well- made transistor. Thus, p Thetotal emitter andcollectorcurrents are ofcourse E= EP +En (10.4) and C=Cp +n (10.5) IEnisthecurrentassociatedwithelectroninjectionfromthebaseintotheemitter.Corr- spondingtothedifusioncurrentontheheavilydopedsideofapnjunctionEnEpn arises fromtheminority carrierelectrons inthecollectorthat wanderinto theC-Bdeple- tionregionandaresweptintothebaseBeingareversebiascurrenCnICp.Sinceboth the emitter andcollectorelectron componentsare smallcompared to the respective hole components, and since IEp,it foloshatIE.Tisisaw-nownproprty of the transistorterminalcurrents.IB,ontheother hand,is popularlyknown to be small comparedtoIE andIC under active mode biasing. This is consistent with thefact IB=IE-IC.Adirect inspectionof Fig.10.9also leads to the conclusion thatIBis expectedtoberelativelysmall.ThethreecomponentsofthebasecurrentB=nB ICn,andIB2equaltothecurrentflowingintothebasetoreplaceelectronslostbyrecom- bination with holes injectedfrom the emitter, are all small. Finally,a few words are inorderon how the BJT manages to amplify a signal.When connected in thecommonemitterconfiguration,theoutput current isIC,the input current

--- Page 405 --- 382 BJTs ANDOTHERJUNCTIONDEVICES Hole current- large OUT $p-$ B-C E-B junction junction small IN Electron current 1 Figure10.10Schematic visualizationofamplificationinapnpBJTunderactivemodebiasing. is IB，and the d.c.current gain isIC/B.B，anelectroncurrent ina pnpBJT,andI, predominantly a holecurrent,are tied togetherthrough theoperationof the E-B junction; thatis,increasingIB proportionallyincreases I.Controlof the largerICby the smaller B is made possibleby the coupled junctionarrangementthat physically divides thesmall electronand large hole currentscrossing the E-B junction into two separate currentloops (see Fig. 10.10). 10.5 PERFORMANCEPARAMETERS Several parameters are commonly usedto characterize theperformanceofa BJTasan amplifier.Theyincludetheemitterefficiencyandthe basetransportfactorwhichrelateo the internal operationof the device,and the d.c.currentgains,whichrelate to the external operation.Inthis section we surveythedefinitionsand theinterrelationshipsofthese im- portant performance parameters in preparation for the quantitative analysis to be presented in the next chapter. EmitterEfficiency Increasing the holecurrent injected across the E-B junction inFig.10.10 whileholding the totalemittercurrentconstant wouldclearlydecreasetheelectroncurrentand increase the overallcurrentgain.Thus,thinkingofIE as theemitterinputandEp as theuseful emitter output, the emitter eficiency is defined to be IEp γ

$Ie$

… pnp BJT (10.6) Obviously 0≤γ≤1and the current gain in a BJTis maximized by makingγ ascloseas possible to unity.

--- Page 406 --- BJT FUNDAMENTALS 383 Base Transport Factor The fractionof theminoritycarriers injected into the basethatsuccessfullydiffuseacrss the quasineutral widthofthe base andenter the collector is known as the base transport factor（α).Ina pnpBJTthenumberofcarriers injectedinto the base fromtheemitter is proportionaltoEptheresidualnumberofcarriersenteringthecollectorisproportionalto $Icp$ (10.7) αT 二 . . .pnp BJT Note that0≤a≤1.Thesmallerthelossof injectedcarriers viarecombination inthe quasineutralbase,thesmallerthedegradationofBJTperformanceandthe larger.Here again maximum amplificationoccurs when the performance parameter,α,is as close as possible to unity. Common Base d.c.CurrentGain When connected in the common base configuration,the active mode（-VcB>0) por- tionof theoutputcharacteristics sketched in Fig.10.4(a)isaccuratelymodeledby the relationship I= αdcE +CB0 (10.8) wheredcisthecommonbased.c.currentgainandICBoisthecollectorcurrentthatflows whenIE=0.Making useofEqs.(10.6)and(10.7),we canalternatively write p= P=E (10.9) and C=C+Cn=γT+Cn (10.10) Comparing Eqs.(10.8) and (10.10),we conclude αdc =αT (10.11) and $CBO=Cn$ (10.12)

--- Page 407 --- 384 BJTsANDOTHERJUNCTIONDEVICES Eq.(10.11)issignificantinthatitrelatestheexternalgainoftheBJTtotheinternal per- formance parameters.Also note from Eq.(10.11)that 0≤dc≤1. Common Emitter d.c.CurrentGain Whenconnectedinthecommonemitterconfiguration,theactivemodeportionoftheout- put characteistics(eeFigs.10.4band10.5b)isapproximatelymodeledbtherelationship C=βdcB+CE0 (10.13) where βdcisthecommonemitter d.c.currentgainandICEois thecollectorcurrentthat flows whenIB=0.AsecondrelationshipbetweenIand Bcan beestablishedby substi- tutingIE=+Binto Eq.(10.8).Oneobtains IC=αdc(C+B）+CBO (10.14) or, after rearranging and solving for C, $P_{-}$a_c{B ICB{O$(10.15)$- Comparing Eqs.(10.13) and(10.15),we conclude αdc { c==$(10.16)andCBOCEO=-a(10.17)AlsonotefromEq.(10.13)that$\c}$\beta dc$

(10.18) B} if,as is the usualcase,ICeo is negligiblecompared toICatthe specifiedoperating poin. Equation(10.16)issignificantbecauseitindicatesthatβdccanalwaysbededucedonce αdc is known.Sincedistypicallyclosetounityand≥B,aβdc≥1istobeexpected based on either Eq.(10.16)or Eq.(10.18).

--- Page 408 --- BJT FUNDAMENTALS 385 10.6 SUMMARY In this chapter we covered very basic informationabout the bipolar junction transistor. Muchofthecircuit-related information was probably alreadyfamiliar to thereader.We notedterminalcurrentand voltage symbols, polarities,andrelationships,reviewed the BJTcircuitconfigurationsand biasing modes,and cited the generalformof theoutput characteristics.Transistorfabrication wasbrieflydescribedtoobtaininsightinto theactual physical nature of the device structure and to indicate the correlation with the one-dimen- sional modelappearing in subsequent analyses.Only a minimal effort was also required to establish a rathercomplete pictureof device electrostatics.Underequilibrium and normal operating conditions,theBJTcanbeviewedelectrostatically as nothing more thantwo independent pn junctions. Some insight into how the device works was provided by ex- amining thecarrieractivityandcurrentsinsidethestructureunderactivemodebiasing. Finally,parameters used to characterize the performance of the device as an amplifier—the emitterefficiency,the base transport factor,and the external current gains—were defned and interrelated. PROBLEMS CHAPTER 10 PROBLEM INFORMATIONTABLE Complete Diffi culty Suggested Short Problem After Level Point Weighting Description 10.1 10.6 一 10 (1 each part) Quick quiz 10.2 10.1 1 4 (2 each part) Mode tables 10.3 10.3 2-3 22 (a-4, b-6, c:e-4) pnp equilibrium electrostatics 10.4 ” 2-3 22(a-4, b-6, c:e-4) npn equilibrium electrostatics pnp active mode 10.5 ” 2-3 10(a-4, b-6) electrostatics npn active mode 10.6 ” 2 8 (2 each sketch) electrostatics 10.7 10.4 3 12 (6 each figure) pnp carrier activity/currents 10.8 ” 3 12 (6 each figure) npn carrier activity/currents 11(a-1, b-1,c-3,d-2, 10.9 10.5 2 pnp parameter calculations e-2,f-1, g-1) 11(a-1, b-1, c-3,d-2, 10.10 ” 2 npn parameter calculations e-2,f-1, g-1) 10.11 ” 3 10 ICB0 thought problem

--- Page 409 --- 386 BJTs ANDOTHERJUNCTIONDEVICES 10.1 Quick Quiz. Answerthefollowingquestionsasconciselyas possible. (a)Whatisthenameofthecircuitconfgurationwhereistheinputcurrentandisthe output current? (b)WhatisthenameofthecircuitconfigurationwhereICistheoutputcurrentandVEis the output voltage? (c)Name the four biasing modes. (d)Whatisthepurposeofthe“buried layer”found inintegratedcircuitBJTs? (e) In astandard pnp BJT,what are therelative sizesof NAE,NDB,and NAc? (f）Whatis the standardsymbolfor thewidthofthedepletionregion ina pn junction diode?Whatis thestandardsymbolforthequasineutral widthofthebase inaBJT? (g) Thebaseregion inaBJTisnarrow.Whatistheprecise definitionof”narrow”? (h)Why is it necessary for the base region in a BJT to be narrow? (i) Define in words (noequations)what is meant by”emittereficiency. (j） Define in words what is meant by “base transport factor.” 10.2Completethetablesbelowbyindicatingthepolarity+or-)oftheinputandoutu voltages associated with each of the four biasing modes. (a) pnp (b) npn Mode VEB VCB$ModeVBEBE$V{BCBC$ Active Active Inverted Inverted Saturation Saturation Cutoff Cutoff 10.3ASipnpBJTwithNAE=5×1017/cm,NDB=105/cm,NAC=1014/cm,and WB=3mismaintainedunderequilibriumconditionsatroomtemperature. (a)Sketchtheenergy banddiagram for the device,properly positioning the Fermi level in the three device regions.(The Exercise10.2BJTEband program mightbe usedto check your answer.) (b)Skechthelcrosttipotntialtinintheemitrrion)thci field,and（ii)thechargedensityasafunctionofpositioninsidetheBJT. (c) Calculate the net potential difference betweenthecollectorandemitter. (d) Determine the quasineutral width of the base. (e) CalculatethemaximummagnitudeoftheelectricfieldsintheEBandCBdepletion regions.

--- Page 410 --- BJT FUNDAMENTALS 387 10.4ASinpnwithNDE=1018/cm3,NAB=1016/cm3,NDC=101/cm3,andWB= 2m is maintained under equilibrium conditions at room temperature. (a)Sketch theenergy band diagram for the device properly positioning the Fermi level in the three deviceregions.(The Exercise10.2BJT_Ebandprogrammightbeusedto check your answer.) (b)Sketch(i)theelectrostaticpotential,settingV=0intheemitterregion(i)theelctrc field,and(ii)thechargedensityasafunctionofpositioninsidetheBJT. (c) Calculate thenetpotentialdifference betweenthecollectorandemitter. (d) Determine the quasineutral width of the base. (e) Calculatethemaximummagnitudeoftheelectricfields intheE-BandCBdepletion regions. 10.5BiasesofEB=0.5VandVCB=2VareappliedtotheProblem10.3BJT. (a) Sketchtheenergy banddiagramforthedevice,properlypositioningtheFermi level in the three device regions. (b) Superimposedon the respective sketches completed inresponse to Problem10.3, sketch theelectrostaticpotentialelectricfield,andchargedensityasafunctionof position inside the biased BJT. 10.6ForatypicallydopedSinpntransistorsketchtheenergy banddiagram,electrostaic potential,electricfield,andchargedensityinsidethedeviceasafunctionofpositionnder active mode biasing. 10.7ApnpBJTis saturationbiased withIC>0.ConstructfiguressimilartoFigs.10.8 and10.9showing thecarrieractivityanddiffusioncurrentsinsidethetransistor. 10.8AnnpnBJTisbiasedintocutoff.ConstructfiguressimilartoFigs.10.8and10.9that show thecarrieractivityand diffusioncurrents insidethetransistor. 10.9 GivenapnpBJT whereEp=1mA,En=0.01mA,=0.98mA,andn= 0.1 A, calculate: (a) α (b)γ (C) E，C，B (d) αdc and βdc (e) ICBO and CEO (fCpi fixed.What effectdoestheICpincrease haveon dc？Explain. (g)Enisicaedwlllotcrntcomonentmainfixedhct IEn increase have on dc? Explain.

--- Page 411 --- 388 BJTs ANDOTHERJUNCTIONDEVICES 10.10 Given an npn BJT whereEn=100A,Ep=1μA,n=99A,andCp= 0.1 μA, calculate: (a) αT (b)γ (c) E，C，B (d) αdc and βdc (e) ICB0 and ICE0 (f)ICnisincreasedtovaluecloserto100Awhileallothercurrentcomponentsremain fixed.What effect does the ICn increase have on βdc?Explain. E{ increase have on βdc?Explain. 10.11ExplainwhytheCBogivenbyEq.（10.12)isdevoidofaholecomponent.Specif- callyhouldnttherebcomotsciaedwtmioitcairholesithet wander into theC-B depletion region andend up being swept into the collector?

--- Page 412 --- 11 BJT StaticCharacteristics Thischapter is primarilydevoted tomodeling the steady-state responseofthebipolarjunc- tion transistor. The development is dividedintothreemajor segments.Buildingonthe introduction inthepreceding chapter,wefirstperformanidealtransistoranalysisthat closelyparallels the ideal diodeanalysis.Baselinerelationships are establishedforcom- puting theBJTperformance parameters andforconstructing theBJT static characteristics. The second portionof the chaptercompares theidealtheory withexperimentand syste- matically examines the noted deviationsfrom the ideal.Someof the deviations are ex- pected fromthediodeanalysis whileothersare uniquetothetransistor.Thethirdandfinal segmentprovides informationaboutthe polysilicon emitter BJTand the heterojunction bipolartransistorHBT).Thesearerecentlyimplementedspecialtransistorsthatoffer- formance improvements over the standard BJT. 11.1IDEAL TRANSISTORANALYSIS 11.1.1 Solution Strategy The derivationoffirst-orderrelationships for the BJTperformance parametersand terminal currents involves assumptions and solutionprocedures very similar tothose employed in the derivationof the idealdiode equation. There isofcourse an increase in complexity associated with the existence of a third quasineutral regionof finite extentand the simul- taneous handling of three (E,B,C)current-voltage expressions. We summarize herethe generalsolutionstrategy.Includedare listingsofthebasicassumptions,the material pa- rametersemployed inthe derivation,thedifferentialequationsto besolved along withthe associatedregionalboundaryconditions,andcomputationalrelationshipsforthecurrents and parameters of interest. Basic Assumptions (1)The device underanalysis is takento be a pnpBJTwithnondegenerate uniformly doped emitter, base, and collector regions—the E-BandC-B junctions are modeled as step junctions. (2) The transistor is being operated under steady-state conditions. (3) The transistor is one-dimensional. (4)Low-level injectionprevails in thequasineutralregions. (5) Therearenoprocessesotherthandrifdifusion,andthermalrecombination-genera- tiontakingplaceinsidethetransistor.Specifically,GL=0. 389

--- Page 413 --- 390 BJTs AND OTHER JUNCTION DEVICES (6) Thermalrecombination-generationisnegligiblethroughoutthe E-BandC-Bdeple- tion regions. (7)Thequasineutralwidthsoftheemiteandcollectoraremuchreaterthantheminority carrierdiffusionlengthsintheseregions.(Theemitterandcollectorareeffectively taken to be semi-infinite in extent.) Note thatthermal recombination-generation is not ignored in thequasineutral base region and that initially no restrictions are placedon the widthof the base. Notation The symbols formaterial parameters tobeemployed in the analysis are listed byregion in Fig.11.1.Note that regional subscripts (E,B,C)are used to identify the minority carrier parametersinaregionasopposed tothepreviouslyemployedcarriertype(np)subscripts. This is necessitated by the existenceof twotransistorregions with the same doping type. Inaddition,forconsistency innotation,a singleregionalsubscriptis usedinsteadofa doublesubscripttoidentifythedopingconcentrations.E0=n/NE,pB0=n/NB,and nCo=n/Ncarecompactsymbolsfortheequilibriumminoritycarrierconcentrations in the emitter,base,and collector,respectively. DiffusionEquations/BoundaryConditions Under the stated assumptions,theminority carrier diffusionequations can be used to solve fortheminoritycarrierconcentrations inthequasineutralregionsofthetransistor.The boundary conditions neededtocompletethe solutions are analogous in form tothoseem- ployedintheidealdiodederivation.Specificallysincethequasineutralwidthsoftheemit Emitter Base Collector p+$ N P

+→x + 0 W + →x^{}$001111i1iiNE=NAE$NB=NDBB$NC=NACDE=DNDB=DPD$ DC= DN$TE=$ $d_{2}a$ tc={n$LE=LNd7=7LC=LNnE0=np0$ PBO =P_nOnCo= np0$ Figure11.1 Coordinate systemsand material parameter symbols (in bold)employed in the ideal transistor analysis.

--- Page 414 --- BJT STATIC CHARACTERISTICS 391 terandcollectoraretakentobemuchgreaterthanminority carrierdiffusionlengthsin these regions,the perturbedemitterandcollector carrierconcentrations (∆nE and ∆nC) must vanish as one moves away from the E-B and C-B junctions,respectively.Likewise, the“lawofthe junction”(Eq.6.12)canbeinvokedto give theboundaryconditions at the E-Band C-B depletion region edges.Appropriately expressed in terms of the coordinate systems defined in Fig.11.1,the region-by-region equationsto besolved andthecorre- sponding boundary conditions are summarized below. Emitter Region The diffusion equation to be solved is d^{}_{n Δ_ne =0 DE (11.1) $d_2$ TE subject to the boundary conditions ∆nE(”→∞)=0 (11.2a) ∆E(“=O)=nEO（eEB/k-1) (11.2b) Base Region The diffusion equation to be solved is d^{2B ΔP{BB$0=DBB$ d_^{2} B subject to the boundary conditions ∆B(O)=PBO（eVEB/kT-1) (11.4a) ∆B（W)=BO（eVCB/kT-1) (11.4b) CollectorRegion The diffusion equation to be solved is ${}_{}^{2}nc$ Δ_nc$0=Dc(11.5)$d_{^{2}$ TC subject to the boundary conditions ∆nc(x{→∞)=0 (11.6a) ∆nC(x’=0)=nCO（eqVCB/kT-1) (11.6b)

--- Page 415 --- 392 BJTs ANDOTHERJUNCTIONDEVICES Computational Relationships Once solutions have beenobtainedfor ∆nE,∆pB,and Δnc,the electron and holecurrent components flowing across the E-BandC-Bjunctions—theEn,Ep,Cn,and ICpintro- duced in the previous chapter-—are readily computed.For example, with recombination- generation assumedto be negligible in the E-B depletion region,the IEn in a pnp transis- tor can beequated to theminority carrier diffusion current in the emitterevaluatedat x {}=} 0,}orr dΔne $En=-qADE(11.7)$dx”}{}0$Similarly,$d△p_{$E=-qA{D}_{B}B(11.8)dx$x 0d△_{$Ice=-qA{D}_{B}B$ (11.9) dx x=\ and $d\Delta nc$ $n=qADC(11.10)$dx’} {0$Finally,expressionsfortheperformancefactorsandtheterminalcurrentsareestablishedemployingtheforegoingsolutionsfortheinternalcurrentcomponentsandthefollowingrelationships:$IEp$(11.11)\gamma =1EP+IEn(sameas10.6)$\e}$(11.12)\alpha T$IEP (same as 10.7) (11.13) $\alpha dc=\gamma \alpha (sameas10.11)$βc=1-ac αdc (11.14) (same as 10.16) E=EP+En (11.15) (same as 10.4)

--- Page 416 --- BJT STATIC CHARACTERISTICS C =CP +n (11.16) (same as 10.5) B=IE−C (11.17) (same as 10.1) 11.1.2 General Solution(WArbitrary) Implementing thesolution procedure outlined in the preceding subsection,we perform here the mathematical manipulations leadingtoexpresionsfor theperformanceparameters and theterminal currents.Before proceeding,thereadermay findit helpful to reviewtheideal diode derivation in Subsection6.1.3andthe treatmentofthe narrow-base diode inSubsec- tion 6.3.2. Emitter/Collector Region Solutions In theemitterquasineutralregionthegeneral solutionoftheminoritycarrierdiffusion equation (Eq.11.1) is ∆nE（”）=A₁e-x/LE+A2exLE (11.18) Because exp(x”/LE)→∞asx”→∞,the Eq.(11.2a) boundary condition can be satisfied onlyifA2isidenticallyzero.WithA2=0,applicationoftheEq.11.2b)boundarycondi- tion yields A₁1=△nE(x”=O).Thus ∆nE(x”）=nEO（eEBkT-1）e-x/LE (11.19) and applying Eq.(11.7), DeEn= qA nEO(eqEB/kT -1) (11.20) LE Similarly,inthecollectorquasineutralregionthesolutionofEq.11.5)subjecttothe Eq. (11.6) boundary conditions yields ∆nC(x’）=nCO（eqVCB/kT-1)e-x/LC (11.21) Utilizing Eq.(11.10),one then concludes Dc$b-=$ nCo(e{CB/kT-1) (11.22) Lc

--- Page 417 --- 394 BJTSANDOTHERJUNCTIONDEVICES As should have beenanticipated,except for notationaldifferencesand the minus sign in Eq.(11.22),thecurrentsolutions here areidentical to thep-sidecontribution to theideal diode equation.The minus sign arises in Eq.(11.22)because the collectorcurrent is defined tobepositiveforcurrentflowingfromthen-basetothep-collector;iethepositiveflow direction is defined opposite to that in the diode. Base Region Solution Whereas theemitterandcollectorsolutionsarenothing morethan one-sidedidealdiode solutions,thebasereionsolutionandtheassociatedcurentsareexpectedtodiferrom previousesultsMahmaticallythisexpctaiontmromtheobservationstatthebase is finiteinextentandtheperturbedcarrierconcentrationdoesnotnecessarilyvanishat eitherx=0orx=W. Proceeding withthe derivation,we notethatthegeneralsolutionto the base region diffusion equation (Eq.11.3)isof the usual form, ∆B(x)=A₁e-x/LB+A₂exLB (11.23) However, applying the Eq.(11.4) boundary conditions gives ∆B(O)=BO（eEB/kT-1)=A₁1+A2 (11.24a) ∆B（W）=BO（eCB/kT-1)=A₁e-W/LB+A2eW/LB (11.24b) IfEqs.11.24)aresolvedforAandA,andtheresultingexpressionssubstitutedbackinto the general solution, we obtain e(W-x)/LB-e-(W-x)B ∆B(x)=∆B(O) eW/L_B -e-W/_B ex/_B-e-x/_B +∆{B(W) (11.25) eW/LB -e-W/LB or more compactly in terms of the sinh function sinh[(W-x)/LB] sinh(x/L_B) ∆B(x)=∆B(O) +△{B(W (11.26) sinh(W/LB) sinh(W/LB) where e^{5-e-5 sinh(ξ) (11.27) 2

--- Page 418 --- BJT STATICCHARACTERISTICS 395 Please note that ifVCB=0,∆pB(W)=0and Eq.(11.26)reduces tothe narrow-base diode solution,Eq.(6.65).This must bethecase,of course,since thetwo problem specifications areidentical when∆pB(W)=0. With the ∆pB(x)solution in hand, we can ow compute the hole currents flowing across the E-B andC-B junctions. Applying Eqs.(11.8) and(11.9),and explicitlydis- playing the voltage dependencies of ∆pB(0)and ∆pB(W),we conclude D_{BB}$cosh(W/LB)1Ep=qA(eqVEB/kT-1)-PBO$L_{B}$sinh(W/LB)sinh(W/LB)(11.28)$DB$1cosh(W/LB)(eqVcB/kT-1)$IcP$

qA LB PBO$sinh(W/LB)(eEB/kT1）一sinh(W/LB)(11.29)where$e^{+e-$cosh(\xi )=(11.30)2PerformanceParameters/TerminalCurrentsInestablishingrelationshipsfortheperformanceparameters,thetransistorwasimplicitlyassumedtobeactivemodebiased.Underactivemodebiasing,VEB>0andB<0inapnptransistor.Thisbiasingcombinationmakesthe[exp(qVCB/kT)-1]terminEqs.(11.28)and(11.29)smallcomparedtothe[expVEB/kT)-1]term.Thus,neglectingthe[expCB/kT)1]termintheEpandCpexpressions,substitutingthecurrentexpres-sionsintoEqs.(11.11)and(11.12),andnotingnE0B0=NB/NE,weobtain1(11.31)\gamma =DE$LBNBB}$sinh(W/LB)1+$BLE $N$ cosh(W/LB) 1 (11.32) αT cosh(W/LB)

--- Page 419 --- 960 BJTs ANDOTHERJUNCTION DEVICES Also 1 αdc =γa =$(11.33)(ciDE{L}_{BB}$ $N_B$ cosh(W/L_B) + sinh(W/L_B) and 1 1 βdc (11.34) 二 1 (:DE $L_{}$ NB} 1 cosh(W/LB)+ sinh(W/LB)-1 αdc Expressionsforthetotalemitterandcollectorcurrentsare nextobtained by simplyadding their respective n and p components. E =qA(En D{BB}$cosh(W/LB)(eqVEB/kT-1)nEo+PBO$ ${}_B$ sinh(W/LB)) (cidDB (eqVclkr -1) 1 PBO$sinh(W/LB)(11.35)1(eqVEB/kT-1)PBOsinh(W/LB))(D$ D{B}cosh(W/LB) (eq_{B/kT -1)1 nco + PBO (11.36) l Lc LB sinh(W/LB)) Finally,ifdesired,anexplicitexpressionorthebasecurentcouldbeestablishedemploy- ing B= IE-IC. Exercise 11.1 P:(a) Showthattheexpressionsfortheemitterandcollectorterminal currentsreduce to the ideal diode equation in the limit where W>LB. (b) Show that setting VCB=0in the Eq.(11.35)expression for the emitter current leads to the I-Vrelationship established for the narrow-base diode. S:(a)In the limit where W≥ LBor W/LB→∞, cosh(W/LB) eW/_B+e-W/LB →1 sinh(W/LB)

eW/LB-e-W/L_B

--- Page 420 --- BJT STATIC CHARACTERISTICS 397 and 1 2 →0 sinh(W/LB) eW/LB -e-W/_B Whentheabove limitingvaluesforthehyperbolicfunctionsaresubstitutedintoEqs. (11.35) and (11.36), the result is (cid:DE D{BB}b=$ne0$ + PBO$(eqEB/kT-1)LE$LBD{B}$(eqCB/kT-1)nco+PBO$ Lc $L_{}$ Except for differences in notationand theminus sign inthe IC relationship,the fore- going are clearly ideal diodeequations. Aspreviously explained,theminus sign arises from the n-to-p definitionof positive current flow in the C-B junction. The manipulations hereconfirm anearlier statementthata structure where W>LBis nothing more than two back-to-back diodes. (b) IfVCB=0,the[exp(qCB/kT)-1]term vanishesand Eq.(11.35)reduces to DBB} cosh(W/LB)) (eVEB/kT -1) nEO + PBO LE $LB$ sinh(W/LB)) Given the p-n nature of the E-B junction,the term involving nEo (the term arising from IEn)may be neglected.Except for notational differences,the above equation is then identicalto the narrow-base result[Eqs.(6.68)and(6.69)].Ingeneral,withthe collectorandbasetiedtogthrorCB0theidaltrnsistoisfunctionallyequiva- lent to the previously analyzed narrow-base diode. 11.1.3 SimplifiedRelationships(W≤LB) Therelationships developed in the last subsectionconstitute a complete solutionforthe transistorparametersand currents.Theycould beuseddirectly in performing calculations and constructing characteristics.However,simplified ormodified versions of the solution help to provide insight and allowone to more readily manipulatethe solution.In this sub- section we examine the simplifications resulting from thefact that W≤LB in a typical transistor. Thecited narrow widthrequirementcouldhave been imposed muchearlier inthe developmenttosimplifythesolutionfortheminoritycarrierdistributioninthequasineutral base.Thiswouldhaveautomaticallysimplifiedall subsequentexpressions.Unfortunately,

--- Page 421 --- 398 BJTs AND OTHER JUNCTION DEVICES the earlyapplicationof the narrow widthrequirementprecludes examiningother limiting case solutions,does not permit investigating special case (W~LB)solutions,and, most importantly,leads to thelossofsignificantterms in some expressions.Weshouldalso mention that more compact relationships couldhave been developedthrough the use of additionalhyperbolicfunctions suchascoth()=cosh()/sinh(ξ).It was feltpreferable, however,to dealexclusively withthesinhandcoshfunctions.In whatfollows,wemake use of the sinhand cosh small-argument limits first noted in the narrow-base diode discus- sion and repeated here for the reader’s convenience. sinh(ξ)→ξ …ξ≤1 (11.37) and $ra2$ cosh(ξ)→1+ ..ξ≤1 (11.38) 2 ∆p_{}(x)in the Base Since0≤x≤W,the（W-x）/LBand x/LB arguments in Eq.（11.26)will be much less than unity forall valuesofxifW/LB≤1.In thespecifiedlimitallthesinh()functionsin Eq.(11.26)can therefore be replaced by their arguments and one obtains ∆pB(x) ≡ B(0)(1−) (11.39) △p{B(x) △B(O) →x △B(W) 0 W Figure11.2Perturbedcarrierconcentration(pB-B0)in the baseofa pnpBJT.Thepictured distributioncorrespondstoctivemodebiasingEB>0,B.

--- Page 422 --- BJT STATIC CHARACTERISTICS 399 or x ∆B(x)=∆B(0)+[∆B（W)-∆B(O)] (11.40) $\text{ve}$ Eq.(11.40)indicates thatthe perturbed carrier concentration in thequasineutralbase becomes asimplelinear functionofpositioninthe W≤LBlimit.Thestraightline distri- butionrunsfrom∆B(0)=B0[exp(VEB/kT)−1]atthebaseedgeoftheE-Bdepletion regionto △pB（W）=Bo[exp(qVCB/kT)-1]atthe baseedgeof theC-Bdepletionregion as picturedinFig.11.2.Itis lefttothereaderasaproblemtoverifythatthedistribution remainsfairlyclosetolinearevenforW/LBratiosaslargeasunity. Performance Parameters If we keepat mosttermstosecondorder in W/LB when imposing the W/LB≤1limit,the general solutions forthe performanceparameters(Eqs.11.31-11.34) simplify to 1 γ= (11.41) DEN\B}$ W 1+ 1 αT

2() (11.42) 1 2 1+ 2 1 αdc (11.43) 2 DE NBB} Wv$1++DB$ $N$ LE 2 1 $\beta dc=$ (11.44) (W) 2 DE \B} W 1 + $DBB$ NE LELE 2 LBB Examining Eq.(11.41),we notethat,consistent withpreviousqualitative discussions,an NE≥NBleads toanemittereficiencynearunity.Likewise inagreement withourunder- standing of transistoroperation,Eq.(11.42)indicates a W≤LB is required to achieve a base transport factor close to unity. The Eq.(11.41)-Eq.(11.44)resultsare sufficientlysimplethatsamplecalculations can be readilyperformed.Suppose a pnp BJToperated at 300K is characterized by the

--- Page 423 --- 400 BJTs ANDOTHER JUNCTIONDEVICES Table11.1 SampleSetofAssumedandComputed MaterialParameters for apnpBJT Operated at 300 K. Emitter Base Collector NE=1018/cm{3 NB=$1016/c{m}^3$ NC=10{5/}cm3} E=263 cm²}/V-sec B=437cm²/V-sec μC= 1345cm²2/V-sec DE=6.81cm²/sec DB=11.3cm²/sec DC=34.8cm^²}/sec E = 10-{7sec B =10-6 sec C =10-6 sec LE=8.25×10-4cm LB= 3.36×10-3cm LC= 5.90×10-3 cm WB=2×10-4cm materialparametersrecordedinTable11.1andaWWB.Substitutingthelisted material parameters into the performance relationships gives DE $N_{BB}$ W 1.46× 10-3

$DBNELLE$(1{)$2=1.77\times 10-32B)$ and γ= 0.9985 αT= 0.9982 αdc = 0.9968 dc= 310 Eventhoughemitterdopingsare typicallygreaterthantheassumed101/cm（roughlythe upper limit imposed by the nondegenerate natureof the theory),thecomputed resultsare fairlyrepresentativeofactualtransistors.Inmodernhigh-performancetransistors,however, αis likelytobeclosertounityandthegainprimarilylimitedbytheemittereficiency. Exercise 11.2 P:The Eq.(11.40)linear dependence of ∆pB(x)in the W≤LB limit should nothave come as a surprise. As previously encountered in Problem 3.17 and the narrow-base diode analysis,the negligible thermalrecombination-generation inregions much

--- Page 424 --- BJT STATIC CHARACTERISTICS 401 shorterthana diffusionlengthautomaticallyleads toa linear dependence.Specifi- cally showthatEq.(11.40)isthesolutiontotheminoritycarrier diffusionequation ifthermalecombiationgentionistakentobenegligibleinthequasinutalbase of a pnp transistor. S: With thermal recombination-generation assumed to be negligible,the Eq.(11.3) base-region diffusion equation reduces to d^{}{_BB 0= $d_x^{}$ which has the general solution ∆B（x）=A₁+A2x Applying the Eq.(11.4) boundaryconditions then yields ∆B(O)=A{1 and ∆B（W)=A₁+A2W=ΔpB(O）+A2W or A2=[∆B（W)-∆B(O)]/W Finally,substitutingthededuced AandA2backintothegeneralsolutiongivesthe expected result, ∆B（)=∆B(O）+[∆B（W)-∆B(O)] Exercise 11.3 P: Consider a pnp transistor(W<LB）biased into saturation.Employing a linear scale and using the same setof coordinates,sketch the perturbed minority carrier distribution normalized to theequilibrium carrierconcentration (Δn/nor △p/po)in the quasineutral regions of the emitter, base, and collector.

--- Page 425 --- 402 BJTs ANDOTHERJUNCTION DEVICES S: Equations(11.19)and(11.21)allowus towrite ∆nE(x”)/nEO=（eEB/kT-1)e−-x”LE ∆nC(x’)/nCO0=（eqVCB/kT-1)e-x’/LC When saturation biased,VEB>0and VCB>0.Thus the distributions in theemitter and collector are similar.Moreover,the distributions have the same general form as the carrierexcess in a forward-biased pn diode. There is a pileupofthe carriers immediately adjacent to the depletion regionedge and an exponential decay of the carrier concentration as one proceeds deeper into the quasineutral region. At the edges of the quasineutral base, we infer from the Eq.(11.4) boundary conditions that ∆B(O)BO=（eEB/kT-1) ∆B(W)/PBO=（eCB/kT -1) With EB>0andB>0,thereisacarrierexcessatbothedgesofthequasineutral base.Moreover,note that the normalized carrierexcess at thebase edge and the emitteredgeoftheEBdepletionregionarepreciselyequal.Similarly,ΔB(W)B0 =∆nC(x=0)/n0.Finally,we know the distribution in the base must be approxi- mately linear because W≤LB. The desired sketch is concluded to have the general form displayed below. E B C △PB (EB >VCB > 0) ΔnE PBO$\Delta nc$ nEO $nco$ →x -W- Exercise 11.4 P: Work out an expression for the base transport factor using the Eq.(11.40) approxi- mation for △pB(x).Comment on the result.

--- Page 426 --- BJT STATIC CHARACTERISTICS 403 S:SincetheapproximateΔpB(x)solutionisa linearfunctionof dΔB(x)/dxisthe same forallxinthequasineutralbase.Itthereforefollows from Eqs.(1.8)and (11.9) that Cp = Ep and $\{$ αT IEP An α=1implies noneoftheinjectedcarriersare lost byrecombination incrossing the quasineutralbase. Theresultis consistent withtheassumptionof negligible recombination-generation in the base region,thekey assumptioninvoked inthe Exercise11.2derivationofEq.(11.40).Clearly,termsoforder(W/LB)²are lost in computing the performance parameters using the linear ∆pB(x)relationship. 11.1.4Ebers-Moll Equations andModel In this subsection we shiftour interestfrom the performance parameters to the terminal currents.We beginbyre-examining Eqs.(11.35)and(11.36).IfVB is setequalto zero in Eq.(11.35),theremaining partoftheequationhasthe generalformof anidealdiode equation fortheE-Bjunction.Likewise,if VEBis setequalto zero in Eq.(11.36),the remaining partof the equation has the general form of an ideal diodeequation for the C-Bjunction.Itisreasonablethereforetothinkoftheprefactorto[exp(VEB/kT)-1]in Eq.(11.35)and the prefactorto [exp(qVCB/kT)-1]in Eq.(11.36)as effective diode saturation currents and to introduce (cid:DE DBB}$cosh(W/LB)$b=$LEneo$ + $LB$ PBO sinh(W/LB)) (11.45a) (cid:DC D{BB}$cosh(W/LB)$R0=A( nco + PBO (11.45b) Lc $L_{BB}$ sinh(W/LB)) Next compare the VCB terms intheIE andICequations. Sincetheprefactorof [exp(qVCB/kT)-1]inEq.(11.35)is smallerthanR0,theprefactorof[exp(qVCB/kT)- 1]in Eq.(11.36),one isled to view theVCB term in Eq.(11.35)as thefractionofthe R0[exp(qVB/kT)-1]currentfromtheC-Bjunctionthat makesitto theE-Bjunction. Analogously, the VEB term in Eq.(11.36) may be viewed as the fraction of the F0[exp(qVEB/kT)-1]currentfromtheE-Bjunction thatmakes itto theC-Bjunction. This suggests the introduction of D{BB}$PBO$αFO=αRRO=qA (11.46) LB sinh(W/LB)

--- Page 427 --- 404 BJTs ANDOTHERJUNCTION DEVICES $I$ Ise} E C F=FO(eqVEB/kT-1) R=RO(eVCB/k-1) IE Ic αRI} αFIF B Figure11.3 Large signalequivalentcircuitforapnpBJTbasedontheEbers-Mollequations. where αF and αRare the forward and reverse”gains,respectively.αRR0is to be used in Eq.（11.35)andαI0inEq.(11.36).Rewriting Eqs.(11.35)and(11.36)explicitlyincor- porating the newly introduced parameters yields E=IFO(eVEB/kT−1)−αRRO(eVCB/kT-1) (11.47a) C=FF0(eEB/T-1)-RO0(eCB/kT-1) (11.47b) Equations(11.47) areknownastheEbers-Mollequations.Computer-aidedcircuit analysis programs,such as SPICE,often use these equations,or a modified form of the equations,to solve for d.c.operating point variables and to construct the BJT device char- acteristics.Although established herefrom first principles,the Ebers-Mollequations may alternatively be viewed as empirical relationships where thefour parameters are deduced from a match to experimental data.Actually,only three of the parameters need to be deter- mined,sincethefourth canalways becomputedfrom the FIFo=RRo requirementof Eq.(11.46). From acircuit viewpointthe Ebers-Mollequations andIB=IE-Iare nothingmore than nodeequations forthethreeBJTterminals.Associating circuit symbols withthein- dividualterms in theEbers-Mollequations leadsto thelarge signalequivalentcircuitin Fig.11.3.(Thereader shouldconvince himself/herself thatthe equations and theFig.11.3 circuit are indeed equivalent.)Theequivalentcircuit is sometimes useful in circuitanalyses and in treating more complex device structures such as the PNPN device family. Exercise11.5 P:Showthatthe forward current gain in the Ebers-Moll formulation is equal tothe common base current gain; i.e.,αF=αdc.

--- Page 428 --- BJT STATIC CHARACTERISTICS 405 S: Dividingtheleft-andright-hand sidesofEq.(11.46)bythecorrespondingsides of Eq. (11.45a) gives $DBB$ $PBO$ vb LB sinh(W/LB) αF

D{BB}$cosh(W/LB)qAOu+PBO$ $LE$LB sinh(W/LB)) 1 (:DE L_{BEO cosh(W/LB)+ sinh(W/LB) DB LEBO Since nE0B0=B/NE,the secondformoftheabove F equationandthe(11.33) αdc equation are seen to be identical. Exercise 11.6 P:In Chapter10it was pointedoutthattheactivemode portionof thecommon-base output characteristics is accurately modeled by the relationship C=αcE+CBO (same as 10.8) Show thatthe Ebers-Mollequations canbe manipulatedtoobtain anequivalent expression. S:Underactivemodeoperation（VEB>0,CB<0,exp(CB/kT)≤1)theEbers- Moll equations reduce to E=FO(eqEBkT−1)+αRRO IC=FFO(eEB/k-1)+IRO Solving the IE equation for the voltage factor yields IE-αR{{ (eqVEB/kT −1)= $IfF$ which when substituted into the IC equation gives IC =FE +(1-αFαR)RO

--- Page 429 --- 406 BJTs AND OTHER JUNCTION DEVICES Since F=c,theaboveandtheoriginallycitedequationareseentobeidentical ifone identifies ICBO=（1-αFαR)RO. (C) Exercise11.7 P:Utilizingthe Ebers-Mollequationsandsubsidiaryrelationships，constructa MATLAB(computer)program thatcalculates and plots theinputandoutput charac- teristics of a pnpBJToperated inboththecommon baseand the commonemitter configurations.In performing a sample computation,assume room temperatureop- eration,adeviceareaA=10-4cm,W=WBandmaterial parameterssimilarto those listed in Table11.1. S:Thecommonbase inputcharacteristic is aplotofIE versus VEB withtheoutput voltage VCBtakingon select stepped values.Convenientconstructionofthecharac- teristicrequiresanequationwhereEisexpressedasafunctionofVEBandCB,or tion,Eq.(11.47a),is preciselyofthis form and may be used directly to compute the common base input characteristic.To constructany of the other characteristics,how- ever, it is first necessary to combine and/orrearrange the Ebers-Mollequationsto obtain a computationalequationof theproperform. Thealgebraicmanipulations, similar to those in Exercise11.6butsomewhatmore involved,yieldthe setofcom- putational relationships: (1)Commonbase input[IE=IE（EB，CB)] IE=FO(eVEB/kT−1)-αRRO(eCB/kT-1) (2) Common baseoutput[C=IC(CB，IE)] IC=αFIE-（1-FR)RO(（eVCB/kT-1) (3)Commonemitterinput[B=B（EB，VEC)] B=[(1-)FO+（1-R)ROe−EC/T]eEB/kT -[(1-F)FO+(1-αR)RO] (4) Commonemitteroutput[=IC（VEC,B)] (αFFO-ROe-EC)[B+（1-F)FO+（1-αR)IRO0] $= +RO- (1-αF)FO+（1-αR)ROe-VEC/kT

--- Page 430 --- BJT STATICCHARACTERISTICS 407 TheMATLAB programBJT, includedondiskand listed inAppendix M,com- putes theBJTcharacteristicsemployingtheforegoing relationships.Thedesired characteristic is chosen from anopening menu.After running the BJT program with thesamplesetofparametertheusemaywishtoinvestigatetheeffctofcaing the device parameters.Constants,material parameters,and the W=WB Ebers-Moll parameters are specified or computed in a subsidiary program, BJTo.Modify the appropriate BJT0program linetochangethedopings,lifetimes,base width,and/or device area. The BJT0 program automatically computes the mobilities,diffusion co- eficient,ddiffuionlsbedoteiputopngndliftim.B

2.5μmNB=1.5×1016/cm3andN=1.5×10/cm,valuesslightlydifferent from those in Table11.1,were incorporated in the as-provided program to afford a closer match to theexperimental characteristics presented in the next section.Axis specifications,it shouldbe noted,areoptimizedforthe sample setof parametersand may have to be revised if the parameter set is modified. 11.2 DEVIATIONSFROMTHEIDEAL As demonstrated in Exercise11.7,theidealtransistorrelationshipsestablished inthepre- ceding sectioncan be used to generate theexpectedformof theBJTstatisticcharacteristics. In this section we compare the ideal-device characteristics withmeasured characteristics and explore reasons for the noted deviationsfromthe ideal.Wefirstconcentrateonthe underlying phenomena leading to differences in the general formof the predicted and ob- served characteristics.Subsequent consideration is given to phenomena,such as base resis- tance and depletion-region recombination-generation, whichaffectthe precise degreeof numerical agreement between the computed and observed values. 11.2.1 ldeal Theory/ExperimentComparison ideal-theorycharacteristicsispresentedinFigs.11.4and1.5.Figure11.4displayscom- mon-base inputandoutput characteristics;Fig.11.5displays commonemitter inputand outputcharactristics.Theexpeimentalcharactristicsweredeivedfrom2p BJT maintained at room temperature. The theoreticalcurves were generated assuming W=WBandemployingtheas-providedBJTprogramdescribedin Exercise17. The predicted andobservedcommon-base input characteristicsof Figs.11.4(a)and (b) areseento beof thesamegeneral character;namely,the inputcurrent()exponentially increases withincreasininputvoltageEB.Thetheoreticallycharacteristicsareesen tiallyindependentof theappliedoutputvoltage(VcB).Experimentally,however,theinput current at a giveninput voltage noticeably increases withincreasingly negative values of VCB Turning to thecommon baseoutputcharacteristics displayed in Figs.11.4(c)and(d), we findgenerallyexcellentagreement betweentheoryand experiment.In bothcasesthere

--- Page 431 --- 408 BJTs ANDOTHERJUNCTIONDEVICES THEORY EXPERIMENT 5 5 VCB=-20V 4 0= 4 (v)}$33allVCB\le \mathrm{02211000.40.50.60.70.80.40.50.60.70.8}VEB(volts)VEB(volts)(a)(b)554IEstep=1mAS5E=4mA433S5B=3mAIE=2mA22S5$E=1mA 1 1 S5 I_B=0 IB=0 0 0 0 10 20 30 40 50 0 10 20 30 110 120 -VCB(volts) -VCB(volts) VCBO 4 (c) (d) Figure11.4 CommonbasepnpBJTcharacteristics:(a)idealtheoryand（b)experimentalinput characteristics;(c)idealtheoryand(d)experimental output characteristics.Roomtemperature data derived froma2N2605 pnp BJTand primarilyrecorded employinga Hewlett-Packard4145BSemi- conductor Parameter Analyzer. The -VcB≥110Vportionof(d)is an idealized sketchofthe characteristics observed on a curve tracer.The theoretical curves were generated assuming W=W and employing the computer program constructed for Exercise11.7. is arapid increase in the output current(C)starting at sightly negative-VcB values,and a nearlyperfect leveling-offorconstancyoftheoutputcurrentfor-VcB>0.Theoretically there is no limitto the -VcB voltage that can be applied to the transistor. Ontheother hand,as might be expectedfromthediodeanalysis,thereal-device output voltage is limited to a maximum value of VcB0 by some sort of breakdown phenomenon. Thecommon emitter inputcharacteristics shown in Figs.11.5(a)and (b)are,forthe mostpart,inexcellentagreement.TheVEC=0characteristicappearsbyitselfinboththe

--- Page 432 --- BJT STATICCHARACTERISTICS 409 THEORY EXPERIMENT 20 20 0S= 15 15 VEC=0 10 VBC=0 VEC>few kT/a (^}$10550。-\mathrm{50.40.50.60.70.8}-\mathrm{50.40.50.60.70.8}VEB(volts)VEB(volts)(a)(b)543$B=10μA $c(mA)$c}m$3B=7.5\mu A/Bstep=2.5\mu A22B=S\mu AB=2.5\mu A1${}=0 $B\0 0 0 20 40 60 80 100 0 0 20 40 60 80 ↑ 100 VBC (volts) VEC(volts) VCEO (c) (d) Figure11.5Commonemitterpnp BJTcharacteristics:(a)idealtheoryand(b)experimentalinput characteristics;c)idealtheoryand(d)experimentaloutputcharacteristics.Roomtemperaturata derived from a2N2605pnp BJT and recordedemploying a Hewlett-Packard 4145B Semiconductor puter program constructed for Exercise11.7. theoretical and experimental plots,whilecharacteristicsassociated with anoutputvoltage of VEC>fewkT/q aregrouped togetherand displaced positivelyalong theinput voltage (EB) axis. The greatest deviation betweentheory and experiment canbefound in thecommon emitteroutputcharacteristicsdisplayedinFigs.1.5c)andd).Firstwhereasthetheoreti- cal characteristicsare perfectlyflat(VECindependent)underactivemodebiasing,there is a very perceptibleupward slopeto theexperimental curves even atrelatively lowoutput

--- Page 433 --- 410 BJTs ANDOTHERJUNCTIONDEVICES voltages.Moreover,atlargeoutput voltages the experimentalcharacteristics turn sharly upward.Finally,some sortof breakdown phenomenon clearly limits the maximum low- current output voltage,VCE0,that can be applied to a transistor. Overall theagreementbetween the ideal theoryand experiment is surprisingly good. Deviationsfrom the idealthatmustbe explainedinclude the strong dependenceofthe common base inputcharacteristicsontheoutput voltage,the initial quasilinear slopeand subsequentupwdcurvtueofthcommoemitteouutchactisticndtheCBo and VCEo limitations imposed by breakdown phenomena. 11.2.2Base WidthModulation By setting W= WBinthecomputationsleadingto thetheoreticalcharacteristicsin Figs.11.4and11.5,weimplicitly assumed Wto be aconstantindependentof theapplid voltages.In truth,thequasineutral widthof the base is nota constant independentof the appliedbiases.Changing VEBand/or VCBchangesthedepletion widths abouttheE-Band/ or C-B junctions,thereby shrinking or expanding W as illustrated in Fig.11.6.Note that even small variations in the depletion widths can be significantbecause of the narow physical extentofthebase.This phenomenon,the variationof Wwiththeapplied voltaes, is known as base width modulationor theEarly effect (namedafter J. Early whofirt recognized the phenomenon).The practical importanceof base widthmodulation canbest be appreciated by referring to Eqs.(11.41)-(11.46).Inspectionof these equations reveals the performance andEbers-Moll parameters areallfunctionsof W. Thus,theyalso vary withtheappliedvoltages.Twoof thedeviationscitedintheprevious subsectionaredirect results of this bias dependence. Re-examining the equation used to generate the common base input characteristics, we can write D{BB}$cosh(W/LB))PBO(eqVEB/kT-1)+LEneo$ + L{B PBO sinh(W/LB)) L_B sinh(W/LB) (11.48) assuming-CB>fewkT/q.Moreover,ifonekeepsonlylowest-orderterms in the W/LB ≤1expansionofthehyperbolicfunctionsandneglectsthetypically small DEnE0/LEterm, the defining equation for thecommon base input characteristics reduces to DBB$Vb=a$ BO e _EB/kT (11.49) W When the reverse biasappliedto the C-B junction increases,theC-Bdepletionwidth increasesandWdecreases.Thus,accordingto Eq.(11.49),IEatagiven VEBisexpecedto CBoanCrebriedormoteRCnRCymoloftmloyeintpfi cation sheetstoidentifytheeroinputcurrentbreakdownvoltagesBVCBoandBVCEO found in the early device literature have subsequently been supplanted by the noted symbols.

--- Page 434 --- BJT STATIC CHARACTERISTICS 411 $VEB>0$ VCB<0 E-B depletion C-B depletion region region P{ N P +d N P Increasing -VCB$|w\to 1$ $W_B$ |←W_{B→ Figure11.6 Base widthmodulation.Decrease in Wresulting froman increase in thereversebias applied across the C-B junction. increase with increasing-CB.Thisispreciselythe behaviorexhibited by theexperimen- tal characteristics presented in Fig.11.4(b). Base widthmodulation is also responsibleforthequasilinear increase in thecommon emitteroutputcurrentrecorded inFig.11.5(d).Underactivemodebiasing the Exer- cise11.7equationusedto generatethe theoretical common-emitteroutputcharacteristics reduces to C=βdcB+CEO (11.50) (same as 10.13) where, as previously established (Eq. 11.44), 1 βdc 二 DE $N_{BB}$ W 2 + 2 B} Since W decreases with increasing reverse bias across theC-Bjunction,βdc systematically increases withincreasing VEC.Thus,withbase widthmodulationtaken into account,the IC ata givenIBunder active modebiasing is likewise expectedto increase with increasing VEC.ItshouldbenotedthatbasewidthmodulationdoesnotexplainthesharpupturninI as the output voltage approaches VCEo The remaining experimental characteristics—thecommon base output characteristics and the common emitter input characteristics—were found to be in fairly good agreement withthetheoreticalcharacteristicsconstructedassumingW=WB.Thequestionmayarise: How is it that thecommon base output characteristics and the commonemitter input char- acteristicsare relatively insensitive to base widthmodulation?To answerthisquestion,we note thattheactive mode portionof the common base outputcharacteristics is accurately modeled by (see Exercise 11.6), C= dcE+CBO (11.51) (same as 10.8)

--- Page 435 --- 412 BJTs AND OTHERJUNCTION DEVICES where, as previously established (Eq.11.43), 1 αdc DE $NB$ W 1 w 2 1+ + 2 $LB$ In a well-made transistor the W-dependent terms in the denominator of the expression are small compared to unity. As a result,even large changes in W giverise to an almost imperceptiblechange inαdc.Likewise,theICBo appearing in Eq.(11.51)canbe shown to be essentially W-independent (and ismoreover negligible inmost instances).Base width modulation is thereforeconcluded to have littleeffecton thecommonbase outputcharac- teristics.Itisleft as an exercise to showthata similarargument can beappliedtothe common emitter input characteristics. A comment is inorderconcerning how the transistor dopings and thebiasing mode affect thesensitivitytobase widthmodulation.Inastandard transistorNE>NB>NC. Thusalmostallof the E-Bdepletionregion lies in the baseandmostoftheC-B depletion region lies in thecollector.Underactivemode biasing the E-B junctionexperiences only a small forwardbiasandplaysanegligiblerole inmodulating thequasineutral widthofthe base. The C-B junction,on theother hand,is routinely subjected to a large reverse bias. Theeffectofbase widthmodulationis minimized,however,because mostoftheC-B depletionregionliesinthecollector.Theoppositeistrue ifthe sametransistorisoperated in the inverted mode.In the invertedmode a reverse bias is appliedacross the E-B junc- tion,theassociated depletion width extends almost exclusively into the base,and the stan- dard transistor is inherently quite sensitive to base widthmodulation. 11.2.3Punch-Through Punch-through might be viewed as base widthmodulation carried to theextreme.Specifi- cally,itrefers to the physical situation wherebase widthmodulation hasresultedin W→0.That is,the base is said to be punched through when the E-BandC-B depletion regions touch inside the base as illustrated in Fig.11.7.Once punch-through has occurred, the E-BandC-Bjunctionsbecomeelectrostaticallycoupled.As pictured in Fig.11.8,an increase in-Bbeyond the punch-through pointlowers theE-Bpotential hillandallows E-B depletion C-B depletion region region 个 +d P W=0 _B→} Figure11.7 Visualizationof punch-through(W→0.

--- Page 436 --- BJT STATIC CHARACTERISTICS 413 E B C Increase $in-VCcBBChangeinducedherebyincreasein-CBFigure11.8Energybanddiagramexplanationofwhyapost-punch-throughincreaseintheoutputvoltagegivesrisetoalargeincreaseintheoutput(collector)current.alarge(exponentiallyincreasing)injectionofcarriersfromtheemitterdirectlyintothecollector.Onlyaslightincreasein-VBgivesrisetoalargeincreaseinIC.Ifitoccurs,punch-throughlimitsthemaximumvoltage(CB0orVCE)thatcanbeappliedtoatransis-tor.Wesay"ifitoccurs”becausearapidincreaseinoutputcurrentatoutputvoltagesapproachingVCBoorVCEocouldalternativelybecausedbyavalanchemultiplication.Simplystated,thephenomenonthatoccursfirstdeterminesthe“breakdown"voltage.Exercise11.8andT=300K,determinetheC-Bjunctionvoltagethatwillleadtopunch-through.S:Ingeneral,restatingEq.(10.3),W=W_B-$EB−$CBwhereforstepjunctionsasinferredfromEq.(5.37)L$2_{NE$)(b(EB)EB;1/2Vbi(EB)$k}ln( (:x_{n BB$

L q $NB(NE+B$ 二 q n^{fr{}{2} 2K_{ NC$(b(CB)-CB);1/2kT$NC{BB}_}B=$ln（$NB(NC+BBn^{{{2}$ q q L

--- Page 437 --- 414 BJTs ANDOTHERJUNCTION DEVICES At the punch-through voltage, W=0 and one can write 2_sε$NC$ 1/2 O=W_B-xnEB $NB(NC+NBB$ q or on solving for VCB (W_{B-xnEB^} VCB = bi(CB) … at punch-through 2K_{ε00 NC$ q NB(NC+NB) Computations employingtheparameters in Table11.1give Vbi(EB)=0.835V, Vbi(CB)=0.656 V,_nEB =3.29 × 10-5 cm,and a punch-through voltageof VCB =-235V. 11.2.4 Avalanche Multiplication and Breakdown Common Base Ifa BJTisconnected in thecommon base configurationandactivemode biased to a larger and larger-VCB,thecarrieractivity withinthe C-Bdepletion regionoftheBJT is very similar to thatoccurring in a simple pn junction diode.A growing numberof the carriers crossing theC-Bdepletionregiongain enoughenergytocreateadditionalcarriers bythe impact ionization of semiconductor atoms. A point is eventually reached where a carrier avalancheoccursandthecollectorcurrentincreases rapidlytoward infinity.Intermsof carrier multiplication,M→∞ at the breakdown point, where M is the multiplication factor introduced in Subsection6.2.2.Assuming avalanche breakdown takes place before punch- through,the maximum magnitudeof the voltage that can be applied to theoutputof the BJT undercommonbase operation,VCBo,is clearly just thebreakdown voltageofthe C-B junction. Common Emitter Thecommonemittercase is farmore interesting.Theoutput voltage applied toatransistor connected in the common emitter configuration is VEC= -VCB.Under active mode EB biasing,theE-B junction is forward biased andVEB is typicallyquite small,lessthana VEC-CBitisreasonabletoexpectCCBHowverexminngtheexperi mental characteristics from the same device displayed inFigs.11.4(d)and11.5(d),one findsaVCE0consideablysmalerthanVCB0;CE090V,whereasVCBO120V！ This unexpectedresultcan be explained qualitatively with the aidof Fig.11.9.The initial injection of holes into the base, labeled in the figure, leads to the holes labeled enteingthebaesdeofheCBdeinon.AoutBuctonibiase far below breakdown,a fewof these holes gain a suficientamountofenergy to impact ionize semiconductor atoms and create extra holes and electrons ②.The added holes drift

--- Page 438 --- BJT STATICCHARACTERISTICS 415 Ec Ei EF Ev} ③ 4 2 0 1 5 Figure11.9 Step-by-step explanationofthe carriermultiplicationandfeedbackmechanism; initialholeinjection;1injectedholesenteringtheCBdepletionregion;2electron-holepir generation by impact ionization; ③ generated electrons being swept into the base; ④excess base electrons injected into the emitter; 5holes injected from the emitter into the base in response to the step 4 electron injection. along withtheinjectedholes intothecollector;theaddedelectrons aresweptinto thebase 3.The extra electrons in the base createamajoritycarrierimbalancethat must beelimi- nated.The most efficientmethod of removing the excess would be for electrons to simply flowoutofthe base lead.This is precisely what happens whenthe BJTisconnectedin the common base configuration.When the transistor is connected in the common emittercon- figuration,however,the basecurrent is held constant during the measurement of the com- monemitteroutputcharacteristic—theextraelectrons are notallowed to flowout thebase lead.Consequently,thecarrierimbalance in the base canonlybealleviated byinjecting electrons from the base into the emitter 4.As was emphasized in the early operational discussion of the transistor,electron and hole injection across the E-B junction are inti- matelytied together.Everyelectron injectedfrom thebase into theemitter is accompanied by the injectionof IEp/En addedholes from the emitter into the base 5.Thus for each additionalelectron-holepaircreated intheC-Bdepletionregionbyimpact ionization, EpEn+1=1/(1-γ)≥d+1additionalholesflowintothecollector. Effectively,thecarriermultiplication in theC-Bdepletionregion is internallyampli- fied under common emitteroperation.Moreover,the process we have just described is regenerative;i.e.it feeds backon itself.Withaddedinjectionthere isaddedcarriermulti- plication intheC-Bdepletionregionandan even largerenhancementofthecollector current.(Alternatively,the βdcof the transistor may be viewed as being enhanced bythe process.)Thisallleadstoaninstabilitypointwhere→ataVEcfarbelowtheava- lanche breakdown voltage of the C-B junction.

--- Page 439 --- 416 BJTs ANDOTHERJUNCTIONDEVICES To modelthe process quantitatively,we note thatthe injection-related componentof the collector current in a pnp BJTincreasestoICp=MTEp,or α→Mα and αdc→ Mαdc,whencarriermultiplication in theC-Bdepletionregion is taken intoaccount.In other words,previousrelationshipscanbeadjustedtoaccountforcarriermultiplicationby replacing αd andα（=αc）by Mαc and MαF. Referring to Eq.(10.15),thesimplifiedexpressionfor thecommonemitteroutput characteristics under active mode biasing transforms to Mα_{dc$ICBO$_{}IC$

1-M{a + $1\text{-}Mad$ (11.52) M→1/αdTypicallyisonlyslightlyless thanunity.Mneed therefore increaseto only slightly greater thanunitybeforereaching theM=1/αc point. This is expected to occur at an output voltage far below the M→∞ breakdown voltage of theC-B junction. Specifically,makinguseofEq.(6.35)with VBR=VCBO, M\VCEO 1 1 (11.53) m αdc 1 一 $CBO$ or VCBO$CEO=VCBO(1-dc)lm=(11.54)\beta dc+1)1m$ where as previously defined,3≤m≤6.Fromthelatterformof Eq.(11.54),it is imme- diately obvious that VCE0< VCBO. In thecontext of BJT operation the mechanism we have just described and modeled must be viewed as detrimental. The maximum voltage one canapplyto thetransistor is reducedbecause ofthecarriermultiplicationandfeedback.However,inotherdevice contexts the mechanismplays a positive,if notcrucial,role. Forone,as explainedin Chapter13,the increase in carriermultiplicationandfeedback withbias helpsto trigger switchingfromtheoff-state to theon-state intheSiliconControlledRectified(SCR)and other PNPN devices. Themechanism is also used to amplify the photosignal in a photo- transistor.Phototransistors （or photo-BJTs)are BJTsconstructed so thatincident lightcan penetrate to theC-B junction.Whenoperated as a photodetector,the photo-BJTis typi- (Some phototransistors are even produced without a base lead.)As depicted in Fig.1.10, roughlyβ+1carriersenterthecollectorforeachelectron-holepaircreatedbytheab- sorption of a photon in the C-B depletion region.

--- Page 440 --- BJT STATIC CHARACTERISTICS 417 $Ec$ $E_i$ EF E_v} Photon →βdc +1 holes Figure 11.10 Photosignal amplification in a phototransistor. Exercise 11.9 P:Two pnp BJTs are similarexcept NB≥Nin Transistor A while NB≤NCin Transistor B.The doping profiles in the two transistors are graphed below. 1018 1018 1018 log doping (cm{-³) 1016 1016 1014 P N P P N P E B C E B C Transistor A Transistor B (a) Whichtransistorisexpectedtohavethe largerpunch-through voltageunderactive mode biasing? Explain. (b)IflimitedbyavalanchebreakdownoftheC-B junction,whichtransistor will exhibit the larger VCB0? Explain. S:(a) Transistor A willhave the larger punch-through voltage under active mode junction(EBwillbepreciselthesameinbothtransistors.Ontheotherhand,the

--- Page 441 --- 418 BJTS ANDOTHER JUNCTION DEVICES depletionwidthassociatedwiththereverse biasedC-Bjunctionreferringto Eq.5.37) is 2K_{s0$Nn)(VbiCB)-1/2..inTransistorA$q{{BN{B}$and2K\epsilon L1/2{(}_{b}i(CB))$x{n} B q_{B …in Transistor B Since(N/NBTransistoA=1/100andVbi(CB)A<bi（CB)|BCBinTransistorA≤ x} /10in TransistorB for the same applied-VCB.Assumingthebase widths(WB) are the same in thetwotransistors,alarger-VCBisclearlyrequiredtocompletely deplete the base of Transistor A. (b) Transistor A will exhibit the larger VCBo if avalanche breakdown of theC-B junction is the limiting phenomenon.Theavalanche breakdown voltageof anasym- metricallydoped pn junction is determined by the doping onthe lightly dopedside ofthejunction.ThismeansthatN=10/cm3andNB=1016/cm3arethedoings varies roughly as the inverse of the doping on the lightly doped side ofa pn junction (seeFig.6.11orEq.6.33)TransistorAisconcludedtoexhibitthelarger CBO (C) Exercise 11.10 P:Modify the Exercise11.7computerprogram so that base widthmodulation and carriermultiplication dueto impact ionizationcan be included in thecomputation of theBJT characteristics.As a sample applicationandemploying the set of parameters specified in Exercise 11.7,compare the commonemitter output characteristics ob- tained when(i)base widthmodulationandcarriermultiplicationare ignored,(ii) onlybase widthmodulation is included,and(iii)both base widthmodulationand carrier multiplication are taken into account. S: The MATLAB program BJTplus,included on disk and listed in Appendix M,per- mits theoptional inclusionof base widthmodulationand carriermultiplication due to impact ionization whencalculating the BJT characteristics. Subprogram BJTO, which inputs constants and material parameters,and subprogram BJTmod, which performs computations relatedto base width modulation,are run-time requirements. The requestedcomputational results,thecommonemitteroutput characteristics p- gressively showing theeffectsof base widthmodulationand carriermultiplication, are displayedon thefacing page.Note that base width modulation leads to a moderate

--- Page 442 --- BJT STATICCHARACTERISTICS 419 quasilinearrise in theoutputcurrent withincreasing outputvoltage.Alsoconsistent with the text discussiontheadditionofcarriermultiplication gives rise toa sharp upturn in theoutput current at output voltage approaching VCEo.Thecomputed char- acteristicstaking intoaccountboth effectscompare very favorably withtheobserved characteristics presented in Fig.11.5(d). 4 Bstep=2.5μA 3 IC(mA) No base width modulation. 2 No carrier multiplication. 1 . 0 0 20 40 60 80 100 VEC(volts) 5 IBstep=2.5μA 4 IC(mA) Base width modulation 3 included. No carrier multiplication. 2 1 0 0 20 40 60 80 100 VEC(volts) 5 Bstep=2.5μA 4 IC(mA) Both base width 3 modulation and carrier multiplication included. 2 一 0 : 0 20 40 60 80 100 VEC(volts)

--- Page 443 --- 420 BJTS AND OTHER JUNCTION DEVICES 11.2.5 Geometrical Effects Up to this point in theanalysis we have assumed the bipolar transistor to be”one-dimen- sional,”withallcurrents primarilyrestrictedtoflowing inonedirection.Fromthecross sectionsofactualBJTspresented inFig.10.6,itisobvious thecurrentflowpaterns must be decidedlymore complex.This leads to several deviations from the idealof a geometrical nature.Included under”geometrical effectsare internal voltage drops directly associated with the three-dimensionalcharacterof the current. Emitter Area ≠ Collector Area In the ideal model,carriers injected from the emitter into the base are assumed to travel in a straight lineto thecollector. Thecurrentflow path inanactualtransistor can have a considerablelateralcomponent. This is illustrated inFig.11.11(a)using thecrosssection of a discrete planar transistor. The spread in current can beapproximately taken into ac- count byemploying different effective areas for the emitter and collector. Effective emitter area $Si{O}_{}$ B ↑ B $N+$ $p+$ +N N Hole P current P lines Effective area P of collection C (a) SiO{}}$EBIE$BB}$\uparrow B$N_{+}$P++N$IB}$+N$BB}$ P P rc C (b) Figure11.11Illustrationofgeometricaleffectsinadiscreteplanartransistor.(a)Currentspread- ing andcrowding.(b)Baseandcollectorseriesresistances.The dashed lineregionin(b)identifies the”intrinsic transistor,“the”heart”of the device.

--- Page 444 --- BJT STATIC CHARACTERISTICS 421 C ↑_c rc \C^} 7 VEC 1 1 1 1 ​ P 1 B B{} $V^{\prime }ECIntrinsicNtransistorrB$ 1 1 1 P{+$11V^{}$ 1 EB VEB 1 E’ 1 1 1 rE IE E E Figure11.12BJTequivalentcircuitincludingtheemitter（base（andthecollector（r) series resistances. Series Resistances Thebasecurrentflowing betweenthebasecontactandthe“heart”ofthetransistor（the dashed line region in Fig.11.11b)must pass through a resistivebulk region. There is also a small resistanceassociated withthe basecontact itself.Thismeansthevoltage dropacross the EBjunctionis somewhat less thantheVEBterminal voltage.Althoughtypicallyquite small inmagnitude,the voltage difference isoften significantbecausethe E-B junction is forward biased under active mode operationand theemitter current is an exponential func- tionof the junctionvoltage. Toaccountforthevoltagedifference,one introduces a base series resistance,B,equaltothesumofthebulkandcontactresistances.IfBis theonly seriestrcouco replaced by VEB,，where VB is computed from B EB V} EB 一 $BBB$ “heart”orcentral portionofthetransistoroftencalledthe”intrinsictransistor.”Aspic- tured in Fig.11.12,individual series resistances connecttheconceptual internal terminals ofthe intrinsic transistortotheexternalterminalsof the physical transistor.Ineachcase the series resistanceisthe sumof bulkandcontactresistances.The voltages atthe terminals ofthe intrinsic transistor are deducedfrom theapplied voltages byaccounting for theIr dropacrossthe relevant series resistance(s).The valuesof theseriesresistances are usually deduced from subsidiary measurements. CurrentCrowding We havealreadynotedthe basecurrentflows laterallyfromtheheartofthetransistortothe base contact.Italsoflows laterallyacrossthefaceoftheemittergivingrisetoavoltage drop in the base from thecentertotheedgesof theemitter.The voltage applied across the

--- Page 445 --- 422 BJTs ANDOTHERJUNCTION DEVICES B Figure11.13Interdigitatedcomb-likecontactstructureemployedinpowertransistorstominimize current crowding.(Top view of the transistor.) E-Bjunctionandtheassociatedinjectioncurrentfromtheemitterto thecollectorare therefore greatest attheedgesof theemitter.A higherdensityofcurrentlines was placed neartheemitteredges inFig.11.11(a)to illustrate theeffect.Thisphenomenon,a larger currentflowaroundtheemitterperiphery,is calledcurrent crowding.Currentcrowding and theonsetof localizedheatingcanbecome particularlydetrimentalathighcurrentlev- els.To minimize the problem,power BJTs are constructed with emitters that havea large periphery-to-area ratio.This is achieved by connecting several narrow,long,emitter con- tacts intoacomb-like structureand interdigitating theemittercontacts withbase contacts as sketched in Fig.11.13. 11.2.6Recombination-GenerationCurrent Thermal recombination-generation was assumed to be negligible throughoutthe E-Band C-B depletion regions in the idealtransistor analysis.Contrary to theassumptionand by analogy with the diode,one might expectto find significant R-G current components in a Si BJT maintained at room temperature. Indeed,under active mode operation with a small forward biasappliedto the E-B junction,the R-Gcurrentfrom the E-B junction islikely to be thedominantcomponentof the base current. Thisleads to a loweremitterefficiency and a reduced gainat lowcurrent levels (smallVEB).Also,when the input isopen circuited, the reverse-bias R-Gcurrentfrom theC-B junctionadds to and is likely to dominatethe observedoutputleakagecurrentICB0orICE).Unlikethediodehowever,theidealtran- sistor characteristics are only nominallyaffected by the inclusion of the R-G current.Un- dermostactive-modeoperating conditions,the inputbiasand current(IBorIE)aresuch thatthediffusioncomponentdominatestheinputcurrent.Moreover,atotherthan lowinput current levels,theoutput currentextractedfrom thereverse-biasedC-B junction is com- prised primarilyof the injected (diffusion dominated)current from theemitter. TheemiterefficiencyinapnpBJTbecomesEpEpnRG）whentheRGcurrentisincluded.

--- Page 446 --- BJT STATIC CHARACTERISTICS 423 11.2.7GradedBase A short descriptionoftransistorfabricationemploying successive diffusions to formthe baseandemitterwaspresentedinSection10.2.Eachofthedifusionsgivesrisetoadoant distributionthatfallsoffmoreorlessexponentiallywith distance into the semiconductor. Arough sketchof theresultantemitterandbase dopingprofilesfollowing thedouble dif- fusion process is shown in Fig.11.14.Thepoint we wish to make isthatthe doping within thebaseisnotconstantasassumedintheidaltransistoranalysis.Ratherthebaedping is typically astrong functionof position,decreasingfrom amaximum at theE-B juntion to a minimum at the C-B junction. The base is said to be”graded”or to possess a graded (non-constant) doping profile. Whentreating semiconductorfundamentals,we establishedthefactthat a doping variation withpositionwillgive rise toabuilt-inelectricfield.Itfollows theremustbea built-inelectricfieldthroughoutthequasineutralwidthof thegradedbase ina double- diffusedtransistor.Seekingadditionalinformationaboutthebuilt-inelectricfield,let NB（x)bethe donor doping in the baseofapnp BJT.By definitionρ=q(p-n+B） 0,and therefore n(x）=NB(x),within the quasineutral portion (0≤x≤W)of the base. Moreover,under equilibrium conditions dn JN=q{nn+N =0 (11.55) dx which when solved for & yields DN$dn/dxkTd{N}_B(x)/dx8(x)=...0\le x\le W(11.56)$μ{}$nq$N{x)$Doping(logscale)NE$N{B}$i11NC$ 1 1 1 Distance 1 1 from surface 1 8 1 P{ N P + →x 0 W Figure11.14Sketchofthedopingprofilesinsideadouble-difusedtrnsisr.

--- Page 447 --- 424 BJTs ANDOTHERJUNCTION DEVICES NB(x)=B(O0)e-x/xdf$(11.57)and$dN{B(xN{)N{Bx)$e-x/xdiff=(11.58)dx$xdiff $x_{d}iffFinally,substitutingintoEq.(11.56),wearriveattheresultkT/q...0\le x\le W(11.59)JIPXTheexponentialdopingvariationgivesrisetoaconstantelectricfieldinthequasineu-tralregion.Themagnitudeofthebuiltinfieldcanbequitesubstantial.IfforexamlethedopingdecreasesbytwoordersofmagnitudeinspanningabasewidthofWB=2m,then$x^diff =4.34×10-cmand≈600V/cm at roomtemperature.More importantly,how- ever,thefieldis directedsoastoassistthetransportofminoritycarriersacrossthequasi- neutral widthofthebase. This holds for boththeemitter-injected holes in a pnptransistor and theemitter-injected electrons in an npn transistor.(<Oin an npn transistor.)The drift-enhancedtransportofcarriers acrossthebase decreasesthetransittimeand thereby reduces recombination in thebase.Thereducedrecombination inturn increases the trans- port factorand the current gains above the values computedfrom the ideal theory. The built-inelectricfieldandattendantdecrease in thecarrier transit timealsoleadtoan im- proved high-frequency response. The graded doping in the base is one deviation from the ideal that is clearlybeneficial. 11.2.8 Figures of Merit Weconcludethesectionwithabriefexaminationofspecial”figuresofmerit.”Theseare plots that can bequickly interpretedto determinetheextentof deviationsfrom the ideal andotherinformationabout thetransistorunder test.Researchpapers on new or revised BJT structures invariablycontain one or bothof the figures to be examined. The first,called a Gummel plotand illustrated in Fig.11.15,isa simultaneous semilog plotofIBandIcasafunctionoftheinputvoltage VEB.Theoutput voltage issetatsome convenient value,typically afewvolts,duringthemeasurement.Ideally,boththeIBandI plots should be straight lines with the same q/kTslope. In reality,as discussedin Subsection11.2.6,the R-Gcurrenttypicallycontributes totheobservedbase currentat smallEB biases.AtlargeB biasesthecolectorcurentplotbegintoslopeover.The slope-over ismost likely causedbyhigh-levelinjection,possiblyaggravatedbycurrent crowding.Paralleling the diode,seriesresistanceis a probablecontributing cause ifasi- multaneous slope-overoccurs in both the IBandIC plots.Note thatassuming ICEo is neg- ligible，theratioofICtoBatagivenVEByieldscatthechosenoperatingpoint.The

--- Page 448 --- BJT STATIC CHARACTERISTICS 425 Ideal High-level injection, current crowding, IC and B (log scale) $Icand/orseriesresistance\beta dcB$ Ideal R-G component VEB Figure11.15 Sketch illustrating the usual formand interpretationof theGummelplot. deducedβdisideallyconstant independentofB.Duetononidealities,however,theI to IB ratio clearly decreases at both low and high values of EB. The second figureof merit is a log-log plotof βdc versus IC withthe output voltage heldconstant.A sampleplot derivedfrom experimental data is presented inFig.11.16. 1000 100 βdc 10 1 0.1 10-9 $10-8$10-7 10-6 $10-5$ 10-4 $10-3$10-{ $10^{-}$IC(A) Figure11.16Measuredcommonemitterd.c.currentgainasafunctionoftheoutputcurrnt.Data derived from a 2N2605 pnp BJT. EC=5V.

--- Page 449 --- 426 BJTSANDOTHERJUNCTIONDEVICES Truthfully,thisplotismerelyacontinuousrecordofthesame βdinformationthatcanbe deduced in a point-by-point fashionfrom theGummel plot.Again βdc is seen tofall offat lowIC(smallVEB）values becauseof the increasing importanceof R-G.AthighI(large VEB)valuesthedecreaseingainiscaused byhigh-levelinjection,currentcrowding,ris resistance,or a combination of the listed effects. 11.3 MODERNBJTSTRUCTURES This section is primarily intended toexpose the readertorecentand ongoing developments relatedto thecontinuedevolutionof thebipolar junctiontransistor. The discussionof “modern”BJT structures also provides an opportunity to make use of the previously estab- lishedconcepts andinformation,particularly those associated with nonidealities. The structuresto beexaminedare thepolysiliconemitterBJT and the heterojunctionbipolar transistor(HBT).Theformer structure is presently employed in a numberof integrated circuits,includingthemostrecentversionsofpersonalcomputerCPUs.1The com- applications.[12] 11.3.1PolysiliconEmitter BJT Complementary MOSorCMOSis presently the preferred large-scale ICtechnologybe- cause ofits low power dissipationandhighpacking density.[Chapters16-18are evoted to MOS(metal-oxide-semiconductor) fundamentals and devices.] ABJT,however, can supply several timesthedrive currentof anequal size MOStransistor.A larger drive cur- rent translates into the faster switching of large capacitive loads.To takeadvantage of this fact,modifiedICdesigns have beenintroduced where bipolartransistorsare incorporated into the high-load speed-sensitive portions of predominantlyCMOScircuitry. The hybrid bipolar-CMOStechnology is referred to as BiCMOS.Anidealized cross sectionof the npn BJT formed by the Intel 0.8 μm BiCMOS process is reproduced in Fig.11.17. Carefullyexamining Fig.11.17,you willnotethata polycrystallineSifilmlies e- tween themetallicemitter contactandthesilicon proper. Theemitter is formed by depos- iting a polysiliconfilmovertheintendedemitterarea,ndopingthefilmwitharseniand then diffusing someof the dopantfrom thefilm ashort distance into the underlying crys- tallineSi.TheresultingemitterconfigurationashallowlayerintheSiproperconacted a similarly doped polysilicon thin film,has been named a polysilicon emiter. Polysilicon emittersoffer several advantages overemitters formed in the usual manner. From a fabricationstandpointthepolysliconemittercontactsareespeciallysuitedforp- ducing theshallowemitter/base junctionsrequired inmodernICs.Inaddition,theyare compatible with self-aligned processing techniquesthatminimize the parasiticresistances and capacitances of a BJT.Of particularrelevance to the present discussion,there is also an operational advantage. Specifically,the commonemitter current gain of a polysilicon emitterBJTismuch largerthanaBJTwithashallowall-crystallineemitterofequivalent thickness.

--- Page 450 --- BJT STATICCHARACTERISTICS 427 At base Alemitter Alcollector contact stack contact stack contact stack Polysilicon Encroachment Salicide LDDoxide spacer Salicide of adjacent p-well Oxide Implanted p+ Implanted n+ p^Base Thin oxide film base contact collector contact Silicon portion of emitter n-Well(Collector) p-Epi p+ Substrate Figure11.17 Idealizedcrosssectionofthetriple-diffusednpnpolyemitterBJTproducedbythe Intel0.8m BiCMOSprocess.(FromGuptaand Bohr11.Reprinted withpermission.) Before pursuing theoriginofthecited polysilicon emittergainenhancement, we need toexplainthattheemitters inmodernBJTsareroutinelymadeverythin toachievehigh operating speeds.Unfortunately,reducingtheemitterthicknessultimately leadsto areduc- tioningain.Inhigh-performanceBJTsthequasineutralwidthoftheemitter（WE)canbe less thanaminoritycarrierdiffusionlength（LE).IfWE/LE≤1inastandardall-crystalline emitter,recombinationmaybeneglected intheemitterand,likeinthebase,theminority carrier distribution becomesa linearfunctionofpositionaspictured inFig.11.18(a).As- sociated withthemodifiedminoritycarrierdistributionintheemitter,thereis asignificant increaseinthebase-to-emitterinjectioncurrentunderactivemodebiasing.Theemiter efficiencythereforedecreasesandthegainlikewisedecreases.Quantitatively,asverifiedin Problem11.11,WE≤LE simplyreplaces LEintheEq.(11.41)-(11.44)performancepa- rameter relationships. Theuseofa polysiliconemittercompensates in part forthe decrease in gain due to the shallownatureoftheemitter.Althoughtheactualsituation israthercomplex,anddiferent theoriesapplydependingontheprecisenatureofthepolysilicon/Siinterface[3,tofirst order the polysiliconportionoftheemittermaybemodeled asmerely low-mobility silicon. Alowermobility inthepolysiliconisreasonable,sinceone wouldexpectadditionalcarrier scattering fromthegrainboundaries,the disorderedregions betweenthe smallcrystalsor “grains”inthepolycrystallinefilm.Assumingthequasineutralwidthsofboththepoly- crystallineandcrystallineportionsoftheemitteraremuchsmallerthantherespectivemi- noritycarrierdiffusionlengths,recombinationwillbenegligiblethroughouttheemitter, andtheminoritycarrierdistributionsinboththepolycrystallineandcrystallineportionsof

--- Page 451 --- 428 BJTs ANDOTHERJUNCTIONDEVICES Emitter Base Collector ΔPE PEO N P N →x ←WE→ ←W→ (a) All-crystalline shallowemitter Emitter Base Collector polySiSi △PE 1 PEO 1 1 1 1 Ni} P N →x ←WE 1 ←W→ (b) Polysillicon emitter Figure11.18 Normalized minoritycarrier distribution underactivemode biasing in (a)anall- crystallineemitterBJTandb)apolysiliconemitterBJT.Quasineutral widthsinthebaseandmiter are assumed to be much smaller than the respective minority carrier diffusion lengths. the emitter willbe linear functionsof position.Atthepolysilicon/Siinterface in annpn BJT,theminoritycarrierholeconcentration（orequivalently△pE）andtheholecurrent mustbecontinuous.With differentmobilities in thetworegions,thecontinuityof the hole current in turn requires d△PE1= qDE2 $d△PE2$ $qDE1$ …1-polysilicon; 2-Si (11.60) dx dx or d△PE1 De2} d△PE2μE2d△PE2}$(11.61)xp$DeL$ dx μE1 dx Equation(11.61)indicates the linearslopeof theminoritycarrierconcentrationmustbe greaterinthepolysiliconportionoftheemitterasilustratedinFig.11.18b).Comparing Figs.11.18(a)and(b),we note thatthe netresultofemploying a polysiliconemiter is a

--- Page 452 --- BJT STATICCHARACTERISTICS 429 reduced ∆pE slope at theemitter edgeof the E-Bdepletionregion,andthus asmaller currentback-injectedfromthebaseintotheemitter.Asmallerbasecurrentofcoursesig- nifies improved emitter efficiency and enhanced gain. 11.3.2 Heterojunction Bipolar Transistor(HBT) In Section11.2 we noted several deviations from the ideal that were associated one way or another withthe base doping.Specificallybase-widthmodulation,theextrinsicbaseresis- tance(B),andthevoltagedropacrossthefaceoftheemitterleadingtocurrentcrowding arealldecreased byincreasinthebasedoping.Unfortunatelyincreasingthebaseoping also decreasestheemitterefficiency,which in turn decreases the gainof astandard transis- tor.There is an unavoidabletrade-off between gainandthe noted deviations from theideal in designingastandard bipolartransistor. The situationisdifferent in a heterojunction bipolartransistor where,as a generalrule the base is themost heavily doped region.A heterojunction is,in general,a junction be- tweentwodissimilarmaterials.However,whenappliedtothe HBT,it is understoodtobe a junction between two dissimilar semiconductors.Theemitter in an HBT is made from a wider band gap semiconductor than the base.In most semiconductor combinations this sets upabuiltin barrierimpedingforward-biasinjectionfromthebasetotheemitterandgiving rise to a high injectionefficiency even when NE≤NB.Althoughbeing touted here as a “modern”BJT structure,the heterojunction idea was actually described by W. Shockley intheinitial patentonthebipolar junctiontransistor filedin1948！Thesuccessfulfabri- cation of HBT structures only became possible,however, with the more recent develop- ment of sophisticated semiconductor film growthtechniquessuch as molecular beam epi- taxy (MBE)[14]. Akey issue in dealing with heterojunctions is the bandalignment.Band alignment is howthe respectiveconduction bandedges and valence band edges of thetwo semiconduc- tors line up relative to eachotherat the interfacebetweenthetwo materials.Although alignments for systems of interest are usually determined from experimental observations. Examplealignmentsfor someof themostextensivelyresearchedsystems are pictured in Fig.11.19.Itshouldbeemphasizedthat theFig.11.19diagramsare nottrueheterojunction band diagrams,butmerely informational diagrams drawn assuming charge neutralitypre- vails throughoutthe semiconductors.Bandbending in thecomponentmaterialsexists ad- jacenttoan actualheterojunction.Weshouldalsomentionthat ineachofthepicturedcases the component semiconductors have essentially identical latticeconstants.Employing two semiconductors with thesame lattice dimensionsallows the heterojunction to be formed with aminimum of interfacial and bulk defects.Almostall HBTs have beenmade utilizing lattice-matched systems. Animportantexceptionisthe Si/Si-xGex/SiHBTthatemploysapseudomorphicSi_xGexlayerforthebase. Pseudomorphiclayersarethinflmstypicall10thatconformtothelaticepatternoftheunderg substrate.Such layers must be keptthin becauseif grown thicker than acertaincritical value,thestressed ilm relaxes,creating numerous defects and a latticeconstant differentfrom thatof thesubstrate. For information about thecurrentstatusandrecentapplicationsof Si/SiGe HBTs,see D.L.Harameetal.”Si/SiGe Epitaxil- Base Transistors,” IEEE Trans.on Electron Devices,42,455-482(March1995).

--- Page 453 --- 430 BJTs ANDOTHERJUNCTIONDEVICES $Ec△EC=0.263eVEc$E{c$△EC=0.27eVEc$ 1 1 0.66eV 1.82 eV 1.42 eV 1.42eV 1 1 Ev} △E=0.49eV -Ev} $Ev$ $Ev$ △E=0.132eV (a) Al0.3Ga0.7As/GaAs (b) GaAs/Ge Ec △EC}=0.3eV Ec △EC}=0.2eV Ec Ec 1 ↑ 1.35 eV 1 1 1 1 1 0.75 eV Ev} 1.46 eV 1 1 1 1 1 1 1.35eV △Ev=0.4eV Ev} 1 $Ev$ △Ev=-0.19eV $Ev$ (c) InP/In0.53Ga0.47As (d) Al0.48In0.52As/InP Figure11.19Heterojunctionbandalignmentsforthe(a）Al03Ga07As/GaAs,(b）GaAs/Ge, (c)InP/In0.53Ga0.47As,and（d）A10.48n0.52As/InPlattice-matchedsystems.(DatafromTiwariand Frank[15].) AlGaAs/GaAs is themost extensivelyresearchedandtechnologicallyadvancedofthe heterojunction systems.Theidealizedcrosssectionofan Al03Ga07As/GaAs Npn HBTis shown in Fig.11.20.(Byconventionan uppercase letter,the Nin Npnis usedtoidentify the wider band gap sideof a heterojunction.) The required n+-GaAs subcollector,n-GaAs collector,p+-GaAs base,and N-Al03Ga0Asemitterfilmsaredeposited in successionon asemi-insulating(S.I.)GaAs substrate.The S.I.GaAs substrateandtheS.I.sidewalls,the latter formedby deepprotonimplantation into the depositedlayers,dielectrically isolate the HBTfromother deviceson the wafer. The structure is completed by theappropriate etching of layers and the depositionofmetal contacts. Ifthe E-Bheterojunction is assumed to be abrupt,the Al03Ga0.7As/GaAs Npn HBT can be modeled to firstorder by the equilibrium energy band diagram in Fig.11.21(a). Except forthewideremitter band gap and the banddiscontinuities at the E-B interface the abrupt-junction HBTdiagram is seen tobenearly identical totheenergy band diagram of anequivalently doped BJT.Naturally,since NE≤NB≥NC,theE-BandC-Bdepletion regions lie almostexclusively in theemitterand collector,respectively.Althoughmodern deposition techniques are capableof growing heterojunctions that are abrupt on an atomic scale,it is very commonto enhance device performance by intentionally growing a compositionallygraded junction; i.e.thecomposition is slowly changedfromGaAsto

--- Page 454 --- BJT STATICCHARACTERISTICS 431 E B C n-GaAs cap S.I. N-Al0.3Ga0.7As S.I. $p^{}-GaAsprotonimplanted)n-GaAsimplanted)$n_{}- GaA$Semi-insulatingGaAssubstrateFigure11.20IdealizedcrosssectionofanAl03Ga07As/GaAsNpnHBT.$Ec$Ev$ (a) $Ece$ Ev} (b) Figure11.21 EquilibriumenergybanddiagramfortheAl0.3Ga0.7As/GaAs NpnHBTassuming the E-B heterojunction is (a)abrupt and (b)compositionally graded. Al0.3Ga0.7Asover adistanceofa fewhundredangstroms.Thiseliminatesthesharpband discontinuitiesand gives rise to themodifiedenergyband diagram in Fig.11.21(b). Thebasicoperational analysis ofan HBTwith a compositionallygradedE-B junction is essentiallyidenticaltothatofastandardBJT.InfacttheBJTanalyticalresultsmaybe

--- Page 455 --- 432 BJTS ANDOTHERJUNCTIONDEVICES takenoverdirectly ifappropriatemodifications are madeto accountfor the differentn valuesofthe HBTemitterand base.Specifically,Eq.(11.31)and laterrelationshipswere writtenin asimplifiedformbynotingnE0pBo=NB/NEinastandardBJT.InanHBT, however,nEOBO=（n/NE）/n/NB);theNB/NEinBJTrelationshipsmustbereplaed by（NB/NE)(n/n).AssumingW≤LB,theemitterefficiencyofthe gradedHBTthen becomes 1 (11.62) γ

DE W N_B} $n\text{fra}2$ 1+ D{BB LE $NE{n}_B$ 2 TheNE≤NBinan HBTisoffset bythefactthatniBofthenarrow gap baseis muchgreater than niEof the wide gapemitter.Even though NB/NE100,forexamplethen/nratio in a typicalAl0.3Ga07As/GaAs HBTis10-5,yielding a theoretical emitterefficiency very close to unity. As a practicalmatter it should be noted thatemitterefficiencies in actualHBTs tendto bebelowthevaluecomputedfromEq.(11.62).Theemitterefficiencyisloweredbya surface regionsof theE-B junction periphery.Proper surface passivation in GaAsdevices poses more of a problem than in Si devices.Nevertheless,HBTs are now produced with very respectable gains and unmatched high-frequency performance. 11.4 SUMMARY Inthis chapter we tookonthetaskofmodeling the steady-state responseof the bipolar junctiontransistor.Initiallypermittingthequasineutral widthofthebasetobearbitrary,an idealtransistoranalysis parallelingtheidealdiodeanalysis was performedtoobtaigeneral relationshipsfortheminoritycarrier distribution inthebase (Eq.11.26),theperformance parameters(Eqs.11.31-11.34),andtheterminalcurrents(Eqs.11.35and11.36).Subse- quently the generalizedrelationships weresimplified,making use of the fact that the quas- ineutral widthofthebaseistypicallymuchless than aminoritycarrierdiffusion lengthin a well-madetransistor.Theresulting expressionfor theminority carrier distribution in the base(Eq.11.40)was noted to be a linear function of position. The performance parameter relationshipsappropriate for a well-made transistor were recorded as Eqs.(11.41)-(1.44). Simplifiedhandlingoftheexpressionsfortheterminalcurrents wasaccomplishedbyintr- ducing the Ebers-Moll model (Fig.11.3) andequations (Eqs.11.47). Several deviationsfrom theidealwere noted in comparing theory and experiment. Bias-dependent variationof thequasineutral widthofthe base,referredto as base width modulation,causes a numberof deviations including thequasilinear upward slantofthe commonemitter output characteristics.Punch-through (total depletionof the base) or ava- lanche breakdownof theC-B junction,whicheveroccurs firstlimits the maximumoutput voltage that can be appliedto the transistor underactive-mode common-base operation.As faras active-modecommon-emitteroperation is concerned,carriermultiplication and an

--- Page 456 --- BJT STATIC CHARACTERISTICS 433 internalfeedbackmechanismcancombineto make VCE0<VCBo.Othercited phenomena leading to deviationsfrom the ideal include the seriesresistance externalto the”intrinsic” transistor,thevoltage drop in the baseacross thefaceof theemitter giving rise to current crowding,recombination-generation inthe E-BandC-Bdepletionregions,andthe graded doping of the base.Special note was also madeof theGummeland βdc versus IC plots that are often presented as a graphical measure of transistor performance. Finally,the sectiononmodernBJT structures servedthedual purposeof introducing specialtransistorstructuresthepolysiliconemitterBJTandtheHBT,whilesimultanusly providing sample practical use of previously established concepts and information. PROBLEMS CHAPTER11 PROBLEMINFORMATIONTABLE Complete Diffi culty Suggested Short Problem After Level Point Weighting Description 20 (1 each part except 11.1 11.4 1 Quick quiz b-2,c-3,j-2, 1-2) 11.2 11.1.1 2 10 (2 each part) Deduce info given I-plot 11.3 11.1.2 3 18 (1/2 each equation) npn equation modifications 11.4 11.1.3 3 17 (1 each answer) Current and parameter calc. 11.5 ” 2 10 △B(x) plot，W/LB varied 11.6 ” 2 10 △B(x) plots for four modes 11.7 ” 1 12 (3 each part) Sketch carrier distributions 11.8 ” 2 10(a-1, b-1, c-2,d-1, e-5) Deduce info from dist. plots 11.9 ” 2-3 12（γ-5，α-5,βdc-2) Parametric effect table 11.10 ” 3 12 (3 each plot) βdc VS.B，NB，τE，NEplots 11.11 ” 3 12(a-4, b:e-2) Shallow emitter 11.12 11.1.4 3-4 14/”diode” (a-6,b::e-2) “Diodes” 11.13 ” 3 10 (5 each part) Apply Ebers-Moll circuit 11.14 ” 3-4 25(a-5,b-10,c-10) Derive computational eqs. 11.15 11.2.2 3-4 15 (a-10, b-5) Emitter input W-sensitivity 11.16 11.2.3 2 8 Punch-through calculation 11.17 11.2.4 2 8(a-3, b-3, c-2) Compare two transistors 11.18 ” 2 10 Determine Early voltage 11.19 ”” 3-4 20/”diode” (a-10,b-10) Plot “diode” I-VA 11.20 11.2.8 2 12(a-10,b-2) Gummel and gain plots 11.21 ” 3 10 VEC dependence of gain plot

--- Page 457 --- 434 BJTS ANDOTHER JUNCTIONDEVICES 11.1 Quick Quiz (aIntheidealtransistoranalysis,whatassumptionsaremadeabouttheregionalE,BC doping concentrations and quasineutral widths? (b)RoughlysketchtheminoritycarrierdistributioninthebaseofapnpBJTundractive mode biasing whenW/LB≤1and W/LB≥1. (c) ApnpBJT with W/LB≤1is active modebiased.Given ∆Bmax=10B0，make sketchesof∆B（x),ΔB（)B0,andB（)in thebase. (d) What are typical sizesforγ,α,αdc,and？ (e) EstablishageneralexpresionforB=B(EB,CB)intermsofthe Ebers-Moll parameters. (f) (g) Referringto Fig.11.5(d),whatisthecauseofthequasilinearslopein thecharacteris- tics atthe lowerVECvalues?What isthecauseofthesharpupward curvatureinthe characteristics as EC→VCEO? (h）ItisestablihedthatVCB0ofaBJTislimitedbypunch-through.Nevertheesssuen claims VCE0<VCB0 in the given device.Is this possible?Explain. (i) What is the cause of “current crowding”? (j) Citethebeneficialeffects associated with agradedbase. (k) What is a Gummel plot? (1) Whatisthecauseofthedecreasein dcatlowandhighICvaluesasexemplifedy Fig.11.16? (m)WouldtherebeanyadvantagetoconstructingaSiBJTwheretheentireemittrime of polysilicon? Explain. (n) Define “heterojunction.” (o)What are the differences between a BJT and an HBT? 11.2ThelectronandholecurrentsinsidepnpBJTbiasedintheactivemodeareloed inFig..lltceeeencedtoteoleurntjciobae Determine: (a)The emitter efficiency (γ); (b) The base transport factor (α); (c)Thecommon emitter d.c.current gain (d); (d) Thebase current (IB). （e) Forthegiventransistoristherecombination-generationcurrentarisingfromtede- pletionregionsnegligibleasassumedintheidealtransistoranalysis?Expli.

--- Page 458 --- BJT STATICCHARACTERISTICS 435 Hole current I 1666.0 0.999 0.001 0.001 ←$10^{-}6$ x E B C Figure P11.2 11.3IndicatetherequiredrevisionstotheequationsinSubsections11.1.1and11.1.2ifthe device underanalysis is taken to bean npn BJT. 11.4Samplecomputations(preferablycomputer-aided)are to be performed to gaugethe magnitudeof the BJT currentcomponents and the size of the performance parameters. Assume roomtemperatureoperation,a pnpBJT,active modebiasing with EB=0.7V andVCB=-5V,W=WB,A=10-cm,andmaterialparametersaslistedinTable11.1 of Subsection11.1.3.Employing the relationships developed in the ideal transistor analysis (Subsections11.1.1and11.1.2),calculate: (a) cosh(W/LB),sinh(W/LB),andW/LB(all tofive significantfigures); (b）IEp，En，Cp，andCn; (c) IE，C，andB; (d）B1,B2,andB3 asdefinedin Subsection10.4; (e) γ,αT,αdc，andβdc. 11.5Examinehowtheminoritycarrierdistribution inthebaseofa pnpBJTvaries with the W/LBratio. Taking VCB=0sothat △pB(W)=0,constructa multicurveplotof ∆B(x)/ΔpB(0)versus x/Wcorrespondingto WLB=10,5,1,0.5,and0.1.Notethat0≤ ∆B(x)/∆pB(0)≤1and0≤x/W≤1.Commentontheresultantplot. 11.6Graphically illustratehowtheminoritycarrierdistribution inthebaseofaBJTva- ries withthebiasing mode.Constructfourindividual“demonstration”plots (subplotsin MATLAB)ofΔpB(x)pBo versus x/Wcorrespondingrespectivelytoeachof thefour biasing modes.Forthepurposesofthe illustration,assume W/LB≤1and restrict ∆pB(O)pBoand ΔB(W)pB0tobe≤10Foraisscalingemployin0ax = 1,min=-2,and max=10.Appropriately label the four plots.

--- Page 459 --- 436 BJTs AND OTHER JUNCTIONDEVICES 11.7Theequilibriummajorityandminoritycarrierconcentrations inthequasineutralre- gionsofaBJTare shownasdashed lines inFigs.P11.7(a)andP11.7(b),respectively.These figures are intendedto be linear plots, with a break in the x-axis depletion regions to ac- commodate the different depletion widthsassociated withthe various biasing modes.Note thatthecarrierconcentrations in the three transistorregionsare notdrawnto theproper relative scale,butonlyqualitativelyreflectthefactthat NE≥NB>NC.Employing solid lines and remembering the figures are intendedto be linear plots,sketch themajorityand minoritycarrierdistributionsintherespectivequasineutralregionsoftheW≤LBtransistor under (a) active mode biasing; (b) inverted mode biasing; (c) saturation biasing; (d) cutoff biasing. n_{norp PEO nBO PCO 0550 550 → x E B C (a) Majority carrier concentrations nor_n$ nco PBO nE0 0550 550 x E B C (b) Minority carrier concentrations Figure P11.7

--- Page 460 --- BJT STATIC CHARACTERISTICS 437 11.8The ∆nE/nE0,△pBPB0,and ∆ncnC0distributionsinthequasineutralregionsofa pnp BJT are as sketched in Fig. P11.8.Determine: (a) The polarity of VB; (b) The polarity of VCB; (c) The magnitude of VCB; (d) The biasing mode; (e) Repeat parts (a)-(d) if the device is an npn BJT. 10 △P{B{B}$\Delta nc$\oC$ ∆nE/nEO -1 →x E B C Figure P11.8 11.9Completethetablebelowbyindicating whetherthenotedchange in a BJTdevice parametericrasesdcreases,orhasnoefectonthelistederormanceparmrs. Change Effect onγ Effect on α Eff ect on βdc Increase WB Increase B Increase NB Increase E Increase NE

--- Page 461 --- 438 BJTs ANDOTHERJUNCTIONDEVICES 11.10LetthelifetimesanddopingsspecifiedinTable11.1bethereferencevalues NBO,andsoon.Making useofthemobilityfitrelationships quotedinExercise3.1,con- structfourindividuallog-logplots（subplotsinMATLAB)ofversusτB/B0,B/ E/E0,andNE/NE0.Ineachcase vary theindependent variablefrom0.1to10.Assume T=300Kand W=WB=2μm inallcomputations.Neglectthe factthat theemitter is degenerately dopedover a portionof the specified range. Do yourresults here agree with the answers to Problem11.9? 11.11Theemitter inmodernBJTsisoftenmade verythintoachievehighoperating speeds.In this problem we examine the effectof employing a“shallow”emitter on the performance parameters.Considerthepnp BJTpicturedinFig.P11.11where,likethe base,theemitter isoffinite width.Let WEbe thequasineutral widthof theemitterand assume ∆nE=0 at the metallic emitter contact(x”=WE). E B C p{+$NP$x {}←$++\to xWE00WFigureP11.11(a)ParallelingtheideltrnsistornalysisobtinrevisedexrsionsoranEn(b)EstablishrevisedexpressionsfortheperformanceparametersanalogoustoEqs.(11.31)-(11.34).(c)EstablishtherevisedexpressionsfortheperformanceparameterswhenW/LB\le 1and$WE/LE≤1. (d) Referring to the part(b)and(c)answers,howareγ and βd affected as WEis syste- matically decreased? (e)MakeasketchsimilartoFig1.2showingtheminoritycarrierdistributioninshal- low(WE/LE≤1)emitter under active modebiasing. 11.12Whenoneof thetransistorterminalsisleftfloating ortwoof the terminals are con- nected together,the transistor becomes a diode-like two-terminal device.The six possible “diode” connectionsare pictured inFig.P11.12.Answerthequestions thatfollowafter choosingaspecificconnectionforanalysis.Itis suggestedthatatleastoneopen-ciruit connection (a,c,or e) and one short-circuit connection (b,d,or f)be analyzed.

--- Page 462 --- BJT STATIC CHARACTERISTICS 439 I E B C I E B C +d N P + P{+} N P VA VA (a) (b) I 一 +d N P + +d N P VA VA (c) (d) 1 I +d N P + P{+ N P + VA VA$ (e) (f) Figure P11.12 (a)DerivetheI-VArelationshipforthe“diode”byappropriatelymanipulatingandcom- bining the Ebers-Mollequations.I should beexpressedonly in termsof VA andthe Ebers-Moll parameters. (b) Developexpressionsfor∆pB(O)B0and∆B（W)B0intermsofVAandtheEber Moll parameters. (c) Simplifythepart(b）resultsbysetingF=α=αand0=R0=0.(Thesimli- ficationsherewouldbevalidforatransistorwherethematerialparametersoftheemit- ter and collector are identical.) (d) Utilizingthepartc)relationshipsandassumingW≤LB,sketchtheminoritycarrie distribution in the base of the transistor if VA>kT/q. (e) Repeat part (d) for a reverse bias where −VA≥kT/q. 11.13ThecommonemitteroutputcharacteristicsofapnpBJTforsmall valuesofECa picturedinFig.P11.13.Thed.c.operatingpointofthetransistorliesatthebounday between activemode and saturation modebiasing as illustrated in the figure. (a)ReferintoFig.11.3inSubsection1.1.4drawthesimplifiedlargesignalequient circuit for the transistor at the given operating point. (b)Employingthesimplifiedequivalentcircuitofpart（a)or workingdirectly withhe Ebers-Mollequations,obtainanexpressionfor VEC at the specifiedoperatingpoint. Your answer should be in termsofIBand the Ebers-Moll parameters.

--- Page 463 --- 440 BJTs ANDOTHERJUNCTIONDEVICES Ic$\uparrow B=constant1\to VEC-d.c.operatingpointFigureP\mathrm{11.1311.14}ThecomputationalequationsfortheBJTcharacteristicsquotedinExercise11.7wereestablishedbyappropriatelycombiningand/orrearrangingtheEbers-Mollequa-tions.Performthealgebraicmanipulationsnecessarytoderivethequotedequationfor(a）Thecommonbaseoutput[C=IC(VCB，E)];(b）Thecommonemitterinput[B=B（EB，VEC）)];(c）Thecommonemitteroutput[C=C(VEC，B）)].11.15Inmodernhigh-performancetransistors,thegainisusuallylimitedbytheemitterefficiency,whichimpliesDENBW/DBNELE>（1/2)(W/LB）2.(a)Showthatthecommonemitterinputcharacteristicsofsuchdevicesareexpectedtoberelativelyinsensitivetobasewidthmodulation.(b)Sketchtheexpectedformofthecommonemitterinputcharacteristicsifthetransistorgainisnotlimitedbytheemitterefficiency;i.eDENBWDBNELE～(1/2)(W/LB)\mathrm{2.11.16}HowistheExercise11.8resultmodifiedifVEB=0.7V?11.17TwopnpBJTsareidenticalexceptthattheemitterandcollectorregiondopingareinterchangedasillustratedinFig.P\mathrm{1.17.101710171015}$1015 1014 $10^{4 P N P P N P E B C E B C Transistor A Transistor B Figure P11.17

--- Page 464 --- BJT STATIC CHARACTERIStICS 441 (a)Whichtransistor isexpectedtohave thegreateremitter eficiency?Explain. (b)Whichtransistorwillexhibitthegreatersensitivitytobasewidthmodulationunder active mode biasing? Explain. (c)Iflimitedbyavalanche breakdownoftheC-B junction,whichtransistor willexhibit the larger VCB0? Explain. 11.18ForVEc≥20VthecharacteristicsdisplayedinthemiddlefigureofExercise1.10 canbecloselymodeled byalinearrelationshipoftheformI=B(VEC-V)B,whereB andV are constants.Working withthemiddlefigure(the“base widthmodulationin- cluded”figure),or preferably working withnumerical values deducedfromtheBJTplus program itselfdetermine the valuesofBand Vrequiredtofiteachof thefourB>0 characteristics.NotethatVreferredtoasthe“Earlyvoltage,“isthevoltageinterceptthat wouldresultifthelinearregionofacharacteristicis extrapolatedintotheVEC axis.Com- ment on your numerical results. 11.19OneormoreI-Vrelationshipsfor diode-likeoperationof the transistor were es- tablished inanswering Problem11.12(a).Adaptthe BJTplusand subsidiaryprogramsof Exercise11.10to generate plotsof theI-VAcharacteristics.TheBJT0program provided for Exercise11.10may be used withoutmodification to inputthe device parameters for samplecomputations.Ifnecessaryforclaritydisplaytheforward andreverse biasportions of the characteristics on separate plots. (a）InitiallyconstructtheI-VAprogram(s）assumingW=WBand neglectingbothbase width modulation and carrier multiplication. Make a hardcopyof the resulting characteristics. (b)Subsequently include both base widthmodulationand carriermultiplication.Be sure to take into accountcarrier multiplication in the E-Bdepletionregion whena large reverse bias is appliedacross theE-B junction.Make ahardcopyof samplecomputa- tions; compare the part (a) and (b)results. 11.20Gummel andGainPlots. (a)Construct a theoreticalGummelplotsimilarto Fig.11.15employingthe devicepa- rameters specified in the BJT0 subprogramof Exercise11.10.(The parameters in the BJTO subprogram provide a reasonable match to the 2N2605 pnpBJT characteristics displayedinFigs.11.4and11.5.)Assumeactive modebiasinganda VCB<0suchthat the VCB terms in Eqs.(11.47)can be neglected.Let theIandIB=IE-C versus VBcaccmtom.4)ei teristics. Approximately accountfor the recombination-generation current arising from the E-B depletion region by adding R-G=O2（e9EB/n2kT-1) to theideal-device base current.Use2=10-14Aand n=1.5in calculating the corrected basecurrent.Toapproximatelyaccountforhigh-level injectionandother

--- Page 465 --- 442 BJTsANDOTHERJUNCTIONDEVICES collectorcurrentnonidealities,multiply theideal-devicecollectorcurrent by thecor- rection factor 1 $\Gamma =1+e9(VEB-0.75)/2kTi.e.,calculatethecorrectedICemployingIC=IC(ideal).ShowboththeidealandcorrectedcharacteristicsonyourGummelplotmakingappropriateuseofdashedandsolidlines.Limitheplotoutputto10-9A\le 1\le 101Aand0.3V\le B\le 0.8V.(b)Expandthepart(a)programtoadditionallydisplaya\beta dcversusICorgainplotsimilartoFig.11.16.UsetheMATLABpausefunctiontointerruptexecutionoftheprogrambetweenplots.Settheaxeslimitstoobtainaplotcoveringtherange10-\le \beta dc\le 11.21IncreasingtheVECvalueemployedinacquiringdcversusIdatainvariablyleadstoasmallbutperceptibleincreasein\beta dcovermostoftheICrange.SampleexperimentaldataarepresentedinFig.P\mathrm{11.2.}FormulateandsupportanexplanationastothecauseoftheobservedVECdependence.1000-2N2605pnpBJT\beta dcVEC15V7VEC=5VVEC=10V100$\0- $10-s$10^-5r10-fr} $10-3$10- 10-1 $Ic(A) Figure P11.21

--- Page 466 --- 12 BJT Dynamic Response Modeling MajorapplicationsoftheBJTinclude use as a small-signal amplifier in wide-band or tuned circuitsand useasa switch in digital logiccircuits.Inthischapter we examinethefirs- ordermodelingof thesmall-signalresponseand the transient (switching)responseofthe transistor. The small-signal response is routinely described ormodeled in terms of a small- signalequivalentcircuit.Small-signalequivalentcircuitsareemployed,forexample,to calculate amplifier signal gain, input impedances,andoutput impedances.Inmodern ap- plicationsthedesireforhigh gainandlargebandwidthleadstothealmostexclusiveuseof the BJT in thecommonemitterconfiguration.Consequently,thecoverage hereinempha- sizes thedevelopmentofequivalentcircuitsappropriateforthenotedconfiguration.Rela- tivetothetransientesponsedigitallectronicsandotherswitchingapplicationsrequ the BJT to be switchedrapidly between thecutoff“off”-stateand thesaturation“on”- state.Anexaminationofthe switching responseand asimplifiedanalysisoftheobserved time delays are presented in the second section of the chapter. 12.1SMALL-SIGNALEQUIVALENTCIRCUITS 12.1.1Generalized Two-PortModel ConsiderFig.12.1(a)wheretheBJTis assumedtobeconnected inthecommonemitter configurationand is modeledas a”blackbox”with twoports.Note thatfor the present analysis(andcontrarytothepnpBJTanalysesinChapter1)allcurrentsaredefinedto be positive whenflowing intoa deviceterminal.Likewise,allvoltages arereferencedto theterminal commonto the input andoutput.With the BJT assumedto beconnected in the commonemitterconfiguration,thed.c.inputcurrent,thed.c.outputcurrent,thed.c.input voltage,andthed.c.outputvoltagearerespectivelyIB,C,BE,andVCE.Thecorspod- ing a.c.quantitiesare i,icbe,andve.In the preceding chapter,it wasestablishedthat IBcouldbe written as an exclusive functionof theapplied d.c.voltages;thatisIB= IB(VBE,VCE).Whena.c.potentialsbe andvce aresuperimposedonthed.c.terminal voltages，the input current quite generally becomes B(VBE+be,VCE+)= B(BE，CESimilarlyitisreadilyestablishedthatthed.c.outputcurrencanlo be expressed as an exclusive functionof the d.c.input andoutput voltages;thatisI= IC(VBE，E).ThuswhenνbeandearerespectivelyaddedtoVBEandVE,theotal outputcurrentanalogouslybecomes(BE+be，CE+e）=C(BE,CE）+. 443

--- Page 467 --- 444 BJTs ANDOTHER JUNCTION DEVICES iB=B+ib$iC=1c+$ B C + + in VBE + Ube BJT VCE+Uc ou E E (a) i{b ic B C + 8210be + Ube 811 822 Uce 812ce E E (b) Figure12.1 (a)TheBJTviewed as a two-port networkand connected in thecommonemiter configuration.(b)Low-frequency small-signal equivalentcircuit characterizingthe a.c.responseof the BJT. The foregoing was primarily a statement of fact. We now make a critical assumption; namely, we assume the transistorcanfollow the changes in potential quasistatically so that B（BE+e，CE+）=B（BE+νbe，CE+e）andiC（BE+νbe，CE+e）= C(VBE+be,CE+e).This allows us to write B（VBE+be，VCE+e）=B(BE，CE）+ib (12.1a) C(BE+be,CE+υc）=C（BE，CE）+c (12.1b) or =IB(BE+be,VCE+e）-B（VBE,VCE) (12.2a) i=IC（VBE+νbe,VCE+νce）-C(VBE，VCE) (12.2b) Next,expanding the first term on the right-hand sideof Eqs.(12.2)ina Taylor seriesabout the d.c.operatingpoint,andkeepingonlyfirst-ordertermsintheexpansionhigher-ordr

--- Page 468 --- BJTDYNAMICRESPONSE MODELING 445 Ube and ce terms are assumed negligible),one obtains a_BB} a_BB} B(BE+be,CE+e）=B（BE，VCE）+ BE VCECE Ube + VCEVBE Uce (12.3a) a\c a\c C（VBE+υbe，CE+）=C（BE，CE）+ ∂BE VCEC Ube + CE VBE Uce (12.3b) which when substituted into Eqs.(12.2)gives _BB} $a_B$ i_{b ∂BE VCE Ube + VCE VBEB Uce (12.4a) a\c} ic

BE VCEC Ube + CEVBE vce (12.4b) Dimensionally the partial derivatives in Eqs.(12.4) areconductances.Introducing a_sqB} a_BB} _BB} aBB} g1 BE VCEVCE EBVEC 812 VCE BEB I ∂ECVEB 介 介 介 介 (12.5a,b) npn pnp npn pnp

• B}in +B}oout +B}in +B}outa\c a\c g{21 VBE VCECE = EB VECEC 822 三 VCEVBEB EC VEBB 介 介 介 介 (12.5c,d) npn pnp npn pnp +Ic\in +Cou +Icin +C out we can then write ib=811be+812ce (12.6a) ic=821be +822ce (12.6b) Equations(12.6a)and(12.b)may be viewed as the a.c.current nodeequations forthe baseandcollectorterminals,respectively.Theseequationsindicatethe iandicurrents flowingintotheterminalsdivideintotwocomponents.Onecomponentbe attheinput

--- Page 469 --- 446 BJTs ANDOTHERJUNCTIONDEVICES and g22uce attheoutputisjustthecurrentthroughaconductanceconnectedacrosste givenport.Thesecondcurrentcomponent（g2uce attheinputand g2be attheoutput)is controlledbythevoltageattheoppositeportand islogicallyassociated witha current generator.Wethereforeinferthesmall-signalcircuitconsistentwiththederivedequations is as pictured in Fig.12.1(b).Thereader should verify thatthe nodeequations fortheB andCterminals inFig.12.1(b)areindeedEqs.(12.6a)and(12.b),respectively.] Severalcommentsare inorderconcerning theFig.12.1(b）result.First,thesmall- signalequivalentcircuit isunderstoodtobeapplicableatlowoperationalfrequencies where capacitive effectsassociated withthetransistor junctions can be neglected. Second,al- though aviableequivalentcircuit,analternativeformofthecircuittobeestablishedinthe next subsectionis normallypreferred in practicalcomputations.Lastly,it shouldbe noted that the developmentleading to Fig.12.1(b) was of a purelymathematical nature.The Fig.12.1(b)resultcouldbereadilymodifiedtoapplytootherBJTconfigurationsoreven to other three-terminal devices by simplychanging theidentityofthetwo-portterminals. Theonly tie toa specificdeviceorconfigurationis throughtheconductance definitions.If desired,Eqs.(12.5)and thed.c.currentrelationshipsestablishedinChapter1lcouldbe employed toobtain explicit relationships for the small-signal conductances. 12.1.2 Hybrid-Pi Models In performing small-signalanalyses,the Hybrid-Piequivalentcircuit is by farandaway the most frequently usedmodel. Themodel has several advantages thatmake it particularly attractivetocircuitdesignengineers.Notably,themodelparametersarereadilyrelatedto the d.c.operatingpointvariablesandthetemperature variationofthe parameters iseasily deduced.Simplifiedand complete versionsof thebasic Hybrid-Pimodel are shownrespec- tively inFigs.12.2(a）and(b).Thecitedlow-frequencyequivalentcircuitsapplytothe from the Fig.12.2(b)π-arrangementof circuitelements with“hybrid”(a combinationof conductance and resistance) units. gmis thetransconductance,a measure of the forward voltage gain,r。theoutput resistance,rthe input resistance,and rthe feedthrough resistance. The simplifiedmodelofFig.12.2(a),whichfindsextensive use infirst-orderanalyses, can be derived directlyfrom the generalized two-portmodel.Underactive-modebiasing the Eqs.(11.47) Ebers-Moll relationships for a pnp BJT simplify to E ≡FOeVEB/kT …VEB >0,VCB<0 (12.7a) C ≡FOeVEB/ …VEB > 0,VCB<0 (12.7b) and B=IE-C≡(1-F)FOeVEB/kT (12.7c)

--- Page 470 --- BJTDYNAMICRESPONSE MODELING 447 ic B C + + Ube $r\pi$ 8m{bb Uce E E (a) $ib$ ic B ru C + 8mbevce vbe $r\pi$ ro E E (b) Ccb H B B’ C’ C rb rμ rc Ceb $r\pi$ ro gm{be E’ re E E (c) Figure12.Hybrid-Piequivalentcircuits.()Simplifiedandb)completelow-frequencHybrd- Pi equivalentcircuitsthatmodeltheintrinsictransistorunderactive-modecommon-emitteropera- tion.(c)High-frequencyequivalentcircuitincludingtheparasiticseriesresistances.

--- Page 471 --- 448 BJTs ANDOTHERJUNCTION DEVICES Ifbase-widthmodulationisassumedtobenegligible,Fand0areconstantsindependent of bias and BB} 8{12 ≤0 (12.8a) ∂VEC VEB ∂c 8{22 ∂VEC VEB ≤0 (12.8b) Withg12=g220weobtainthedesiredresul;Fig.12.1(b)reducestopreciselytheme formasFig.12.2（a).Wealsoconcludegmg1andr1/g1.Explicitlyevaluating andrπ using Eqs.(12.7)yields a\c q{c 8m$\partial VEBVECkT(12.9a)11$_c$\beta dc(12.9b)$rπ$(\alpha B/EB)|VEC$q_B/kT I{B 8m 8m Note that gmandrare readilyevaluated given aknowledgeof the d.c.operating point. The four-element low-frequencymodel ofFig.12.2(b) is used when base-width modu- lation cannot be ignoredoramore precise analysis is to be performed.Although somewhat more involvedthan the simplifiedmodel derivation,the Fig.12.2(b)circuit parameterscan be related to the two-portmodel parameters in a straightforward fashion.It is leftasan exercise to show that in general $m=821-812 (12.10a) r0=1/(g22+g₁2) (12.10b) r=1/(8₁1+812) (12.10c) r=-1/g_12 (12.10d) Typically g2is several ordersofmagnitudesmaller thantheother two-portmodel conduc- tances and may be neglected incomputing gm,,and.Equations（12.9)mayofcourse be used to evaluate gm and rif g12is much less than g21 and g1. Finally,themodel shown in Fig.12.2(c) is appropriate for useat higherfrequencies. Includedinthemodelaretheparasiticseriesresistances(,rande)thatconnectthe intrinsictransistorto theexternaldeviceterminals(seeSubsection11.2.5).Cb andCeb

--- Page 472 --- BJTDYNAMICRESPONSEMODELING 449 respectively modelthecollector-baseand emitter-base pn junction capacitances.Under active-modebiasingtheCbassociated withthereverse-biasedC-Bjunction is primarily ajunctioncapcitcewhileCbassociatedwittheorwad-biaedEBjunctionil both junctionanddiffusioncomponents.Theappropriatejunctioncapacitancerelation- shipsfromChapter7maybeused without modiicationinevaluatingCcb andCb.How- ever,thediffusioncomponentofCb mustbemodifiedasoutlined inProblem12.7.An additionalcapacitancebetweentheC andBterminals and acapacitancefrom theC’ter- minalto a substrate terminalmay be necessarytoproperlymodeltransistors incorporated inICs.ItshouldbenotedthattheFig.12.2(c)modelistypicallyvalidatfrequenciesupo roughly500MHz. The modelbecomes increasingly inaccurateabove500MHz because it does nottake intoaccountthe delays in signal propagationacross the various device regions. 12.2 TRANSIENT(SWITCHING)RESPONSE TheBJTfindsextensiveutilizationas anelectronicswitchinapplicationsrangingfrom simple discrete-device circuits to complex IC logic circuits.When compared to a pn junc- tiondiode,theBJToffersintrinsicallyfasterswitching speedsandtheadvantageofathird lead,which greatlyfacilitates theswitching process.Like the pn junction diode,a time delay inprogressing betweenthe”on”and“off”statescanbeattributedforthemostpart to the build-up or removal of excess minority carriercharge. Since switching speed is a majorconcern inmanyapplications,it is reasonable to examine the switching process in some detail. 12.2.1QualitativeObservations Forthe purposes of the discussion, we assume a switching circuitof the form pictured in Fig.12.3(a).Vccsuppliestheemitter-to-collectord.c.bias,andRistheoutputloadresis- tor.Switching is accomplished by pulsing the input power supply between positiveand negativevoltages.Understeady-state conditions with<0,the BJT is biased intothe cutoff mode,making IC≤0.In the described statethe transistor is”off”and its operating point liesnear thebottomoftheloadlineshown superimposedonthecharacteristics in Fig.12.3(b).Conversely,understeady-state conditions with>0,theappliedvoltages are typically such that both junctions are forwardbiased.Operating in the saturation mode the transistor is in the”on”state withIC=Vcc/RL. Referringto Fig.12.4let us now examinethe transientresponse itself.We begin with the turn-on transientthattakesplaceafteris pulsedfrom-to+Vatt=0.With VsassumedtobemuchgreaterthanvEB,thebasecurrentas shown inFig.12.4bjumps uptoB≈Vs/Randremainsconstantatthis valueaslongasv>0.Thecorresponding idealized iresponseissketchedinFig.12.4(c).Priortot=0thetransistoriscutoff biased andtheminoritycarrierdistributioninthequasineutralbaseischaracterizedbyplot(i)of Fig.12.4(d).Followingthet=0turn-onpulse,minoritycarrierholes beginpilingupin thebase[plot(ii)ofFig.12.4(d)].iin turnincreasesroughlyinproportiontotheslopeof

--- Page 473 --- 450 BJTs ANDOTHERJUNCTIONDEVICES ic C RL $iBPNVccB$R_{s}$P+$v_{s}$\uparrow iE$v_{s} V_s} E t $-5_s$ (a) ↑ $B=VS/RsVCC/RLON$IB= Constant OFF →VEC 0 VCc (b) Figure12.3BJTswitchingfundamentals.(a)Idealized switchingcircuit.(b)Load linewiththe “on” and“off”steady-stateendpoints superimposedon thecommonemitteroutputcharacteristics. ∆pB(x,t）evaluated atx=W. The increase in i tracks the build-upofminority carriers inthe quasineutralbase as long as thetransistor remainsactive mode biased.Eventually, however,ΔB(0,)=(n²/NB)exp(qEB/kT)reaches the point whereEB= VEC or UCB=0,and the transistor subsequently becomes saturation biased.Once saturation bi- ased, the minoritycarrierbuild-up is completed as pictured inplot(iii)ofFig.12.4(d). Consistent with thecited plot,iincreases only slightly during this portionofthe transient to finally attain its on-state value.

--- Page 474 --- BJTDYNAMICRESPONSEMODELING 451 $v_s$ V_s 0 t $-\_sq$ (a) $iB$ \_{/_s 0 1 -ξVs/Rs 1 1 (b) 1 1 1 ic 1 VCC/RL 1 1 1 1 1 1 1 1 1 t 0 (c) ΔPB △PB △PB △PB △PB → x x → x →x 0 W 0 W 0 W 0 W 0 W (i) (ii) (iii) (iv) (v) (d) Figure12.4BJTswitching characteristics.(a)Appliedinputvoltage.(b)Inputcurrent（iB)asa functionof time.(c)Output current(i)as a functionof time.(d)Approximate minority carrier distribution in thequasineutral base at various times during thetransientresponse.

--- Page 475 --- 452 BJTs ANDOTHERJUNCTIONDEVICES The icturn-offtransientalsoexhibitstwo distinct phases.Thesituation insidethe base of the transistor during the two phases is approximatelymodeled byplots (iv)and(v)of Fig.12.4d)respectively.Duringtheinitialphase,occurringimmediatelyaftervisulsed from+Vsto-Vs,iremains approximately constant while the stored charge associated with saturation biasing is removed from the quasineutral base. Thetime delay here isob- viously similarin naturetothe storage delaytimeencountered inthe pnjunctionanalysis. The second turn-off phase begins when sufficientstored charge has beenremoved sothat the transistor becomes active modebiased.Once active mode biased,iC monotonically decreases inproportiontothestoredchargeremaininginthequasineutralbase.Ultimaly the totalremovaloftheminoritycarriercharge leadstocutoffbiasingand theoff-state. Althoughofsecondaryimportance,itisinterestingtonotefromFig.12.4(b)thattheiB response duringturn-offisessentiallyidenticaltothei-tturn-offtransientofthepnjunc- tion diode.The basecurrent abruptlyreversesandremainsconstantatB=-V/Runtil the BJTentersthecutoffmode.Withincreasing reverse bias across the E-B junction,B subsequently decreases to approximately zero. 12.2.2 ChargeControlRelationships In performing aquantitativeanalysisoftheBJT transientresponse,we willemploythe chargecontrolapproach.eforelaunchinintotheanalysisitisnecesarytofirstestablish the charge control relationships specifically applicable to the BJT. Thepn junctionchargecontrolresults established inSubsection6.3.1can beapplied directlyto thequasineutral baseof theBJTwithonlyminormodifications.Working witha pnp BJT,making required symbol changes,and accounting for the finite width (o≤x≤ W)of the quasineutral base, we infer $d{Q}_B$ $Q_B$ =-A d_p(12.11) p J_{p(0) 一 TB where Excess hole charge wQ_B

=qA ΔB(x，t)d$ (12.12) in quasineutral base Jo IfE≤AJp（O)andi=AJp（W),whichisequivalenttoassumingtheemitterandcollector currentsare dominated bythe holediffusioncurrentsrespectivelyflowing across the E-B and C-B junctions, we conclude -A dJ= A[J(O)-Jp（W）]=E-C=B (12.13) JJp(O)

--- Page 476 --- BJTDYNAMICRESPONSE MODELING 453 and $d{Q}_{BB}$ Q_BB} $i$ (12.14) 1p B Note as anaside that if Eq.(12.14) is appliedundersteady-state conditions,dQB/dt= 0and iB→IB=QB/B.This resultmay be interpretedasfollows: Thereare a totalof QB/q excess minoritycarriers in the quasineutral base.Theaverage lifetimeofanexcess minoritycarrier isB.Thusevery B secondstheQBexcess storeof charge is eliminated byrecombination.Understeady-state conditions acurrentequaltotheeliminatedcharge per second,B=QB/B,mustflowinto thebase toreplacetheeliminated carriers.The noted result is valid providedthe current associated withrecombination inthequasineutral base is thedominantcomponentofIB;that is,carrier injectionfrom thebase intothe emitterandotherbasecurrentcomponentsarenegligibleasimplicitlyassumedinthederi- vation of Eq. (12.14). Continuing the developmentof charge controlrelationships,letus nextexaminethe collector current.Consistent with previous approximations,we can write ∂△B(x,t C =AJp(W)=-qADB (12.15) e $x=Ifweassume\Delta pB(x,)hasthesamefunctionalformas\Delta pB(x)understeady-statecondi-tions,thenintheW\le LBlimitx|w∆B(x,t)=∆B(O0,)+[∆B(W,t)-∆B(0,)](12.16)whichunderactivemodebiasingsimplifiesto(12.17)Thus,takingtheBJTtobeactivemodebiased,△pB(x,t)$ ΔpB(0,t (12.18) ∂x W x=w W qAW Q_B}= A Δ{B(0, (12.19) Jo 2

--- Page 477 --- 454 BJTs ANDOTHERJUNCTIONDEVICES and a△_{B $qA{D}_B$ $Q_B$ c=-qADB B ∂x W △{B(0, t)===(W^{2}{2}DB) (12.20) x=w or OB} ic (active mode) (12.21) τt$where$W^{} τ{t $rB$ (12.22) Assuming QB(t)can be determinedby solving Eq.(12.14),Eq.(12.21)then provides a simplemeansof deducing the instantaneous collector current. The τparameter finds extensive use in approximateanalysesandisknownasthebasetransittime.Itisinterpreted as theaverage time takenbyminoritycarriers to diffuseacrossthe quasineutralbase.A- cording to Eq.(12.21),onemay viewthecollector currenttobetheresultof thestored charge in thequasineutral base dropping into thecollectorevery τseconds.Also notethat βdc=CB=TB/₁ 12.2.3 Quantitative Analysis Turn-on Transient Duringtheturn-ontransientiB≤V/RsIBB=constantand dQ_B} = IBBBQ_BB}$ (12.23) 1p TB Thegeneralsolutionofthe(2.23)differentialeqation is QB()=BB+Ae-B (12.24) where Ais anarbitraryconstant.Toavery goodapproximation,QB(0)=0.This istrue sincethetransistoriscutoffbiasedatthebeginningof thetransient.SubstitutingOB() 0into Eq.(12.24)whent=0,and solving forA,yieldsA=-IBBτB.Thus QB(t)=BB（1-e-t/B） . . .t≥0 (12.25)

--- Page 478 --- BJT DYNAMICRESPONSE MODELING 455 and Q_{B(t BB{B (1-e-/B) . . .0≤i≤t, (12.26a) ic(t) =VCc=\cc ...t≥ } (12.26b)R{{L} Therise time,t,introduced above is understood to be the time period over which the BJT is active mode biased. The turn-on solution is illustrated graphicallyon the left-hand sideofFig.12.5.Exhib- iting an exponentialdependence,QB(t)increases smoothlyfrom zeroto its maximum valueofQB（∞）=BBB.iC（t）tracksQB(）untiltheBJTbecomessaturationbiasedat t=.Consistent with the qualitative observations,iC(t)is subsequently assumed to re- mainconstantatiC(）)=VcC/RL=ICC.SettingC(）=ICCinEq.（12.26a),onecan solve for to obtain 1 7 t{=B (12.27) Icc} 1 $BBB$ Q_B(t) Q_{B(t) BBBB$=0$Icc$=11011t$0 t (a) ic(t) $iC(t)\zeta =0$lcc ζ=1 1 1 1 0 t 0M t t} $tsd$ (b) 介 介 Turn-on Turn-off Figure12.5 Transient response solution.(a)Storedcharge in thequasineutral base and (b)col- lectorcurrent as a functionof time. The turn-on transient is shownon the leftand the turn-offtran- to iB≤0 and iB=-BB during the turn-off transient.

--- Page 479 --- 456 BJTs ANDOTHERJUNCTION DEVICES In agreement with intuition,t,canbedecreasedby increasingIBB（faster stored-charge build-up)or decreasingIcc (less active-mode charge storage).We also note that substitu- tionoftheEq.（12.27)expressionforinto Eq.（12.25)yieldsQB()=Iccτ₁.This value for the stored chargeattheactive-mode/saturation-modetransition point was appropriately positionedalong theQB(t)axis inFig.12.5(a)and willberequired incompletingtheturn- off analysis. Turn-off Transient During the saturation-modeandactive-mode portionsofthe turn-off transient, -/RBBcosnhechagereinequnrlbsethrfro the charge control relationship d{Q_{B} =-BBBQ_B}$(12.28)1pTBwhichhasthegeneralsolutionQB(t）=-BB+Ae-B(12.29)Aisanarbitraryconstantandt=0hasbeenresettothebeginningoftheturn-offtransien.Sincethetotalstoredchargeatthebeginningoftheturn-offtransientisjustequaltothestoredchargeattheendoftheturn-ontransientQBturn-oQB（\infty turn-onBBB.SubstitutingQB(0)=BB\tau BintoEq.（12.29)whent=0,andsolvingforA,yieldsA=(1+\xi )BB\tau B.Wethereforeconclude0=1[3-B_2+I）a=（）(12.30)and...0\le t\le sd(12.31a)$ic(t)=Q_B(t) $BB{B Ps1=1 [3-B-+I] (12.31b) Thestorage delay time,sd,introducedabove is understood to bethe timeperiodover which the BJT is saturation-mode biased. Theturn-offsolutionis illustrated graphicallyontheright-handsideofFig.12.5for theassumed valuesof=0andξ=1.ξ=0correspondsto=0andiB≤0duringthe turn-off transient;=- andiB=-BBifξ=1.Itisimmediatelyobvious from Fig.12.5that boththeconstant(saturation-mode)andfalling(active-mode)portionsofthe iC transientare decreased when>0.This is to be expected since aniB<0aids the withdrawalof storedchargefromthequasineutralbase. At the saturation-mode/active-

--- Page 480 --- BJTDYNAMICRESPONSE MODELING 457 modetransitionpointandQB|turnoQBu-oncτSubstitutingthe transitionpoint valuesintoEq.(12.31b)and solving ford,weobtain 1+
sd = {Bln$(12.32)$\cc}+\ BB{B SinceIccτBBB<1,Eq.(12.32)confirms thatsd decreases whenξ>0.Generally speaking,decreasingτB（fasterrecombination)decreasingBBlesstotalstoredcharge) and increasingIcc(relatively less saturation-mode charge) alsodecreasesd 12.2.4PracticalConsiderations The foregoing analysis was highlysimplified to provide physical insightandto achieve a solutionofthe generally correctform withoutbecoming bogged down in an excessive amountofmathematics.Readers interested in a more precise solution are encouraged to consulttheBJTsupplementalreferencecitedat theendofPartII.Amorerealistic sketch ofthe ictransientresponse is presentedinFig.12.6.Otherthan a"roundingneartransi- tion points,theonlymajor difference between Fig.12.6and earlieriC plots is an added time delay,t,at thebeginning ofthe transient. Theaddedtime delay arisesfromthe charging of the E-B junction capacitance in goingfrom cutoff to the active mode. Figure12.6alsoillustratesthemeasurement-baseddefinitionsoftherisetime),the storagedelaytime(d),andthefalltime（),parametersoftenfoundindevice specifica- tion sheets. It shouldcome as no surprise that variousmethods have beenemployedto speedup theturn-onandturn-offtransients.Acommonproceduretospeed up theturn-ontransientv_{s}$t$iC(t) 1cc 60 0.1Icc t 0 →k → $td$ t {sd $tf Figure12.6 Actualformofthecollector-currentswitchingtransientandthemeasurement-based definitionsof,d,and.Theinput voltage waveformisincludedforreferencepurposes.

--- Page 481 --- 458 BJTsANDOTHERJUNCTIONDEVICES C C E B Schottky diode □ $Si{O}_2$ $n$ +u Ohmic Schottky p contact apoip B n n+collector buried layer (a) (b) E Figure 12.7 Schottky diodeclamped npn transistor.(a）Schematic circuitrepresentation. (b) Physical implementation. isto add acapacitorto theinputcircuitry,suchas placing acapacitorin parallel withRin Fig.12.3(a).Thecapacitivedischargeafter→Vprovidesamomentarypulseofcurrent that accelerates the build-up of stored charge andtherebyreduces the time to saturation. A speed-up of the turn-off transientcanand has beenachieved by adding recombina- tion-generation centers to the BJTbase. Theadditionof R-G centers,such as gold in S, reduces TB and therefore increasestherateof carrierremovalfrom the base.In another method to speed up the turn-off transient,a Schottky diode is connected between thecol- lector and base as shown inFig.12.7.Thisarrangement is knownasa Schottky diode clamp.The Schottky diode,a device formed from arectifying metal-semiconductorcontact (see Chapter14),conducts at a much lower forward-bias voltage than a pn junction.Con- sequently,when thetransistorenters the saturationmode,theSchottky diode begins to conductand”clamps” theC-B junctionvoltage at arelatively low positive value.This greatlyreducesthesaturation-modebuild-upofstoredcharge.Moreover,thereis verylile minority-carrier charge storage ina Schottky diode.With less charge to be removedfrom theBJTand verylittlechargestoredintheSchottkydiode,theturn-offtimeis dramaticall reduced. 12.3 SUMMARY Inthis chapter we first examined the small signal response and then the transient or switch- ing responseofthebipolarjunctiontransistor.Asmall-signalequivalent circuitfortheBJT connected in thecommon emitter configuration,a generalized two-portmodel validat low frequencies,was established employing a purely mathematical argument.The development was of interest in itselfbecause it can beapplied withonlyminor modifications to essen- tially anythree-terminal device.Subsequently,the popular Hybrid-Piequivalentcircuit was

--- Page 482 --- BJT DYNAMIC RESPONSE MODELING 459 introducedandrelated to thetwo-portmodel.Simplifiedandhigh-frequency versionsof the Hybrid-Pi circuit were also reviewed. Because of the importanceof switching speed in many applications,the transientre- sponse was examined in some detail.Thetransiet response wasfirst describedqualita- tively,concentratingonthe build-upandremovalofexcessminoritycarriers inthe quasi- neutral base. Similar to the pn junction diode, storedcharge must be added orremoved in progressingbetweentheof-state（cutoff)andtheon-state(aturation).Chargecontrole- lationshipsspecificallyapplicabletotheBJTwereextestablishedassumingforsimplicity thatrecombinationinthequasineutralbaseandminoritycarrier diffusionatthequasineu- tralbaseboundaries totally dominatedtheobservedterminalcurrents. Theconceptofa base transittime,theaverage timerequiredfor diffusing minoritycarriers to cross the quasineutral base,was introduced in thecourse of the development.Quantitative expres- sions forthe storedcharge,thei transient,andcharacteristic timeperiods were derived for both the turn-on and turn-off transients.A concluding subsection cited practical consid- erations including methodsthat have beenemployedtospeed up the transientresponse. PROBLEMS CHAPTER12 PROBLEM INFORMATIONTABLE Complete Diffi cult Suggested Short Problem After Level Point Weighting Description 12.1 12.3 1 10 (1 each part) Quick quiz 12.2 12.1.1 2-3 8 Derive common base circuit 12.3 12.1.2 1 5 Determine g_m,π 12.4 ” 3 8 Confirm Eqs.(12.10) 12.5 ” 4 20 Calculate H-Pi parameters *12.6 ” 4 20(a-14, b-6) Derive/compute f 12.7 12.2.2 3-4 10 E-B diffusion capacitance 12.8 12.2.3 2 5 EstimateCcτ/BBB 12.9 ” 2 8 (4 each part) t,and tsd plots *12.10 12.2.4 3 10(a-6, b-4) tf derivation and plot 12.1 Quick Quiz Answer the following questions as concisely as possible. (a) In words,what isthequasistaticassumption?How is the assumptionutilized in the derivation of the generalized two-port model? (b)Why are there separate definitions for the npnand pnp gijconductances?

--- Page 483 --- 460 BJTs ANDOTHERJUNCTION DEVICES (c)What is theoriginof the“Hybrid-Pi”name? (d) State the names of the following Hybrid-Pi parameters:gm,ro,π,and rμ μ (e)Whatisthejustificationforaddingcapacitorstothelow-frequencyHybrid-Pimodel to obtain the high-frequency model? (f）DescribewhathappenstoB,iCandtheminoritycarrierconcentrationintheuasi- neutral baseoncethe transistorenters the saturationmode during theturn-ontransient. (g)Giveboththewordandmathematicaldefinitionsofthebasetransttime.” (h)How are τB and τ,related to βdc? (i）Why doesaniB<0(>0)duringtheturn-offtransientgiverisetoareductionin both the storage-delay andfall times? (j）What is a Schottky diode clamp,and what purposedoes it serve? 12.2Establishasmall-signalequivalentcircuitanalogoustoFig.12.1(b)appropriaefor thecommonbaseconfiguration.Provide asummaryofthe parameter definitions similarto Eqs.(12.5). 12.3Thecommonemitteroutputcharacteristicsofa2N2605pnpBJTwere displayedin Fig.11.5(d).ThesimplifiedHybrid-Piequivalentcircuit istobeusedinmodelingthe low- frequency a.c.response of the transistor.Suppose IB=5μA and VEC=10Vatthe d.c. operating point.Determine the valuesofgmand rto beemployedin the Hybrid-Pimodel. 12.4Performthecircuitanalysisandmathematicalmanipulationsrequiredtoconfirm Eqs.(12.10). 12.5The MATLAB programBJTplus,provided as a solutionto Exercise11.10,computes B=B（VEB，VEC)andC=C(EC，B）withandwithouttakingintoaccountbaswidth modulation.Starting from scratchor borrowing from the BJTplus and subsidiaryprogram code,write a MATLAB(computer)program thatautomaticallycalculates the valuesof the Hybrid-Pi[Fig.12.2(b)]parameters.Provideforalternativecomputations withandwithout taking into accountbase-widthmodulation.Forsimplicity,letthe input variablesbeV EB and VEC.Employ the same device and material parameters as those found in the BJTO subprogram.(The MATLAB scripts for bothBJTO and BJTplus are listed in Appendix M. Run your program with VEC=10Vand V chosen so that IC ≤ 1mA. Present EB computationalresultsforgmrrandrbothwitandwithoutaccountingforbae- μ width modulation.Alsocompute gmand rusing Eqs.(12.9).Discuss yourresults. 12.6The unity beta frequency,,isoftenquotedasa figureofmerit inassessingthehigh- frequencyresponseofaBJT.By definition,is the signalfrequency at whichiib|=1 when the BJTcommonemitteroutput is a.c.short-circuited(ve=0).Silicon BJTs in- tendedforuseinRFandmicrowaveapplicationsareavailable withan=10GHz.“Gen- eral purposeSi transistors typically have an fin the100’sofMHz.

--- Page 484 --- BJTDYNAMICRESPONSEMODELING 461 (a) Making useof the high-frequency Hybrid-Pi model,derive a general expression for i/b when e= 0. when d.c. biased such thatV =0.68 V,VEC=10 V,B=5.57A,andIC= EB 1.00mA.Thehigh-frequency Hybrid-Pi parameters forthe2N3906atthegiven d.c. operating point are tabulated below. Parameter Value 8m 3.86×10-2S ro 2.00×10{4Ω $r\pi$ 4.65× 103Ω 3.59×106Ω $ru$ Ceb$23.6pF$Ccb$2.32pF10\Omega$r_{}$2.8\Omega$rc re 0 12.7Showthatthe diffusion componentofCbfora pnpBJTunderactivemodebiasing is given by (} 2 D{B}n_{a{}{2}$ 2 π 2 Cebldiff

3 (kT/q) $\text{n}B$ eqVEB/kT 3 $kTlq E 3

$8^m$ HINT:CbldiisthesameasthediffusioncapacitanceCD,tobeexpectedfromanarrow- base diode.NotingtheIDIFFflowing inanarrow-basediodeisgivenbyEqs.(6.68)/6.69), revise thediffusionadmittance derivation in Subsection7.3.2toobtain theCDfora narrow- base diode. The revision requires useof the expansion,ctnh(ξ)=cosh(ξ)/sinh(ξ)= （1/)（1+²/3)ifξ≤1.Makeappropriatesymbolidentificationsintransferringtheresult to the BJT. 12.8Refertothe Fig.12.3(b)characteristic.Suppose VcC/RL=5mA,V/R=30μA, andIB was stepped in 5 μA increments to obtain the IB= constant curves. Estimate ICcτ/BBτBforswitchingbetweenthe“off”and“on”points shown inthefigure.

--- Page 485 --- 462 BJTsANDOTHERJUNCTION DEVICES ●12.9Rise andStorage-Delay Times (a）Letx=IccτBBB.EmployingEq.（12.27)andnoting0≤x≤1,constructaplot of t_{/_B versus x. (b） EmployingEq.（12.32)constructaplotofdBversussimultaneouslydisplaying curves corresponding toξ=0 andξ=1. 12.10Fall Time (a) Deriveanexpressionforthefalltime,,usingthemeasurements-baseddefinition presented in Fig.12.6and the Eq.(12.31b)iC versus trelationship. (b) Plott/TBversusχ=ICCBBBwhenξ=0andξ=1.Notethat0≤x≤1.Discuss your results.

--- Page 486 --- 13 PNPNDevices Read Only In earlier chapters we progressed through theanalysesof thetwo-region,one-junctiondi- odeand the three-region,two-junctiontransistor.Containing fouralternatelydopedreions and three interacting pn junctions, PNPNdevices,or thyristors,embody the next(and final) increment in junctioncomplexity. Thyristors,oftenlargeareadevicespackaged inmetal studs foroptimumheatdissipation,findextensiveuseashigh-powerrectifiersandelec- tronic switches.Althoughtypicalratingsareconsiderablylower,thyristorsareavailable withcurrentratings inexcessof5kAandvoltageratingsexceeding10kV.Forthemost part,weconcentrate hereinonthebest-knownandleadingmemberofthePNPNdevice family,thesiliconcontrolledrectifier(SCR).Onesectionisdevotedtodescribingthe physicalnatureofthe deviceandtheratherinteresting“negativeresistance”orbistable switchingcharacteristics.Asecondinvestigatestheinternaloperationofthedevicegiving riseto theswitchingcharacteristics.Practicaldetailsrelatedtoturn-onand turn-offare considered in a third.Theconcluding sectionbriefly surveysothermembersof thePNPN device family. 13.1 SILICONCONTROLLEDRECTIFIER(SCR) Thesiliconcontrolledrectifieris pictured schematicallyinFig.13.1(a).TheSCRcontains fouralternatelydopedregionslabeledP1throughN4inthefigure.J12J23andJ4identify the P1-N2,N2-P3,and P3-N4junctions,respectively.Theouter P-regionandthecontact totheregioniscalledtheanode(A);theouterN-regionandthecontacttotheregionis calledthecathode(K).TheinnerN-and P-regionsarereferred toas bases.Thethird device terminal,thegate(G),is connectedtotheP3base.Weshouldinterjectthata similartwo- terminalPNPNdeviceconstructedwithoutagateleadisknownasthePNPNdiodeor Shockleydiode.NotethatVAkisthevoltageappliedbetweentheanodeandcathode,AK thecurrentflowing into theanode,andIG thecurrent flowing into the gate. TheSCRis typicallyfabricated bystarting witha properlythinned,lowlydopedn- typeSiwafer(ND~5×101/cm)theinteriorofwhicheventuallybecomesthe N2base. Next,p-typedopant is diffused into the twosidesof the wafer,yieldingalmost identical J12andJ23junctions.Finally,n-typedopantisdiffusedintoonesideofthewaferforming the heavily doped N4-cathode.Theresulting doping profile is sketched inFig.13.1(b).As mightbeinferredfromthefabricationdescription,andasemphasized inFig.13.1(b),the N2and P3base regionsareofmoderate electrical width.Unlike the wide-base diode where the widthsof the P-and N-regions are assumed to be much greater than a minority-carrier diffusionlength,ortheBJTwherethe widthofthebase is routinelymuchlessthana 463

--- Page 487 --- 464 BJTs ANDOTHERJUNCTIONDEVICES G IG IAK}$AP1N2P3KVAK$J_2}J{3J{3q3}$(a)$10^{0 10{v Doping (cm^3 810I 1018 1016 1016 1014 1014 P1 N2 P3 N4 k W{{2 $W_{3}LN$ (e.g., 250m) (e.g., 25 m) (b) Figure13.1Thesiliconcontrolledrectifier(SCR).(a)Schematicrepresentationshowingthedop- ing regions, junctions,device terminals,and current plus voltage variables.(b) Typical doing profile. minority-carrier diffusion length,the widthsof theSCR baseregionsmust be roughlycom- parable to a diffusion length. This,as we willsee,is critical to theproper operationof the SCR. The general formof the characteristics derivedfrom an SCR is sketched in Fig.13.2. The VAK<0portionofthe characteristicsisIGindependentand very similarto thatofa reverse-biased pn junction diode. The device conducts little current until breaking down at V in SCR work is referred to as the reverse-bias holding or blocking BR voltage.Theforward-biascharacteristics,on theotherhand,are quite distinctive.IfIG= 0,systematically increasingVAkfromzerocauses lttlecurrenttoflowthrough thedevice untilVAKecedsF,themximumorwrd-bisblockivoltageOnceFis ex ceeded, however, the device switches from the high-impedance blocking mode to the low- impedance conducting mode.Operationthen liesalong the forward-bias diode-likechar-

--- Page 488 --- PNPN DEVICES 465 IAK IGincreasing -V_{BRBR → VAK V{BFB$ G IG IAK K A 1 Figure13.2 SCRcharacteristics and circuit symbol. acteristic near VAK=0.Applying anI>0lowers theforward-bias blocking voltae and permitsthedevicetoentertheconducting modeatalowerappliedVAK.VBR andVBF,it should be noted,can be in the hundredsoreven thousandsof volts,whilethe voltae drop in the conducting mode is routinely on theorderof a volt. 13.2SCROPERATIONALTHEORY We wishto explaintheobserved SCRcharacteristics.Tosimplifythe initialconsiderations letI=0.WithI=0,theoperationoftheSCRisequivalenttothatofthePNPNdiode. The question arises as to how to begin the analysis.A reasonable approach would be to start withthesimplestpossiblemodeland subsequentlyconsidermore sophisticated modelsas requiredtoobtainagreementwiththeobservedcharacteristics. Thesimplest possible modelfor theSCR wouldbethree pn junction diodes in series as envisioned in Fig.13.3(a). Working with the diode model,we notethat if VAk<0,the middle diode becomes forwardbiasedandmay bereplaced witha shortcircuit.Thisleads to the simplifiedmodel of Fig.13.3(b).Furtherobserve from Fig.13.1(b)thatthe dopingonthe lightly doped side ofthe ctoi1/cm，ltheonotheltloeo junctionis1014/cm.TheJ34junctionthereforebreaksdownatarelativelysmallVAK< O,and theSCRreverse-bias characteristic is predictedtobebasicallythe sameas there- verse-biascharacteristicoftheJ12pnjunction.Inasimilarmanner,ifVAk>0,theJ12and J34diodesbecomeforwardbiasedandmaybereplaced withshortcircuitsleadingtothe simplifiedmodelofFig.13.3(c).Consequently,theforward-biasedSCRcharacteristicis

--- Page 489 --- 466 BJTs ANDOTHER JUNCTIONDEVICES A Jq} $J22$ $J3$ K K VAK (a) J_q2} J{34q A K VAK<0 (b) A J{}3} K VAK>0 (c) Figure13.3Thediodemodel.（a）Arbitrary VAK;(b）VAK<0;(c)VAK>0. predictedtobe basically the sameas the reverse-bias characteristicof the J23 diode.The J_2diodedominatingtheVAk<0characteristicandtheJ23diodedominatingtheVAK> Ocharacteristicare in turn expected to havesimilarcharacteristics because of thealmost identical dopingprofilesof thetwo junctions.Inother words,the SCRcharacteristic based on the diode model is concluded to be nearly symmetrical aboutVAK =0 and of the form pictured in Fig.13.4. Although providing insightand yielding the correct general form of both the reverse- bias andforward-biasblocking characteristics,thediodemodelfails to predictthe forward- bias switching characteristics.Moreover,according to the diode model,VBF is approxi- matelyeualtoBR，whichinturnisproimtlyqultothebreakonvoltgofhe J12（orJ)junction.TheJ2breakdownvoltageisunderstoodtobethesmallerofthe IAK -V_{BRBR →VAK V{BFB$ Figure13.4 Predicted SCR characteristics based on the diode model.

--- Page 490 --- PNPN DEVICES 467 E B C A A P1 N2 P3 G G N2 P3 N4 K K C B E (a) (b) avalanche breakdown voltage and the punch-through voltage required to totallydeplete the lowly dopedN2region.Asitturnsoutthe blockingvoltagespredicted bythediodemodel are upper limits; theobserved blocking voltages are invariably somewhat smaller. Thecritical defectofthediodemodel is that it does nottake into accountthe interac- tionbetween thethree junctions.An interaction must exist because the widthsof theN2 and P3baseregionsare comparableto therespective minoritycarrierdiffusion lengths. Correctingthedefect,theinteractionbetween junctions is taken intoaccountinthetwo- transistormodel.Inthetwo-transistormodelillustratedinFig.13.5,theSCRisconceptu- ally subdivided into two interconnected transistors.The P1-N2-P3regions form apnp transistor and the N2-P3-N4regionsform an npn transistor.The Plregion is identified as the emitterof thepnptransistor,andthe N4regionis taken to betheemitterof the npn transistor.Shorting wiresconnect theregions thatare common to thetwotransistors.Please note forfuturereference that theBJTs in themodelare intrinsically low-gaintransistors. Becausethebase widthsareonlycomparabletothediffusion lengths,the transport factors andthereforethegains willbe considerably smallerthanunity undernormaloperating conditions. Let us now use the two-transistor model to explain how the SCR manages to switch from the high-impedanceblocking mode to thelow-impedance conducting mode.Fig- ure13.6 is included to assist in the discussion.Assume the SCR is in theblocking mode withIG= 0 and VAK>0.With VAK 0bothtransistors are active-mode biased and majoritycarriersfromthePlandN4emittersareinjectedintotheadjacenttransistorbases (①inFig.13.6).Afractionof the injectedcarierssucessfullydifuseacrosthebase region ② and enter the base of theother transistor r3.Forming an excess of majority carriers thatcannot be shunted into anexternal terminal,the carrierscoming from theother transistor lead to an increase in the base-to-emitter injection, which in turn givesrise to addedemitter-to-baseinjection④.Theseaddedcarriersfollowthesamepathastheorigi- nally injected carriers leading to even more injection.Clearly the process feeds backon itselfor isregenerative.AtlowappliedVAKtheprocess is stableandthere is negligible Except for the sourceofthe majoritycarriersentering the basetheprocess describedhere is identicaltothe regeneativercessintroducedinSubsction11.2.4toexlantereductioninCofBJTcomaredtoCBo

--- Page 491 --- 468 BJTs ANDOTHERJUNCTION DEVICES P N P A 1 ② 3 VAK>O$11111113K②NPNFigure13.6Useofthetwo-transistormodeltodescribetheregenerativeprocessthatleadstoswitching.Initialcarrierinjection.2Diffusionacrossthequasineutralbase.3Injectedcar-riersenterthebaseoftheothertransistor.4Additionalinjectioninducedbythemajoritycarrierexcessinthebase.carrierbuild-upabouttheJ23junction.However,withincreasingVAKasignificantcarrierbuild-upreducesthereversebiasandultimatelyforwardbiasestheJ23junction.(Remem-ber,apnjunctionisforwardbiasedifaminoritycarrierconcentrationinexcessoftheequilibriumvalueexistsattheedgesofthedepletionregion.)Intermsofthetwo-transistormodel,thetransistorsbecomesaturationbiased.TheSCRentersthelow-impedancecon-ductingmodewhenallthreejunctionsbecomeforwardbiased.TheforegoingqualitativeargumentsoutlinedtheinternalprocesswherebytheSCRswitches,butitwasrathervagueaboutdeterminingthecriticalswitchingvoltage.Actuallyquantitativeconsiderationsemployingthetwo-transistormodelarenecessarytopindownthepreciserequirementforswitching.LetusagainassumetheSCRisintheblockingmodewithIG=0andVAK>0.Beingactive-modebiasedundertheassumedconditions,thetransistorsinthemodelcanbeapproximatelyrepresentedbythelargesignalequivalentcircuitsshowninFigs.13.7(a)and(b).ThesecircuitsfollowfromtheEbers-Mollequiva-lentcircuitofFig.11.3.WhenactivemodebiasedtheRRcurrntgeneratorintheEbersMollcircuitisnegligible,IF\approx IE,andR\le -R0.IftheFig.13.7（a)and(b)equivalentcircuitsareconnectedaccordingtothearrangementinFig.13.5,theresultisFig.1.7(c).InFig.13.7c)subscript1isassociatedwiththepnptransistorandsubscript2withthenpntransistor.EquatingthecurrentflowingintoandoutoftheE2nodeinFig.13.7(c)yieldsIAK=\alpha 1AK+RO1+2AK+RO2(13.1)or$RO{1+R{O2}IAK=$(13.2)$1-(a{1{2})$

--- Page 492 --- PNPN DEVICES 469 αFIE αFIE IE $IEECEC$IrOR}IROR}$BB(a)pnpBJT(b)npnBJT$α₁EIAK} C1 AIRO$ E1 ${\alpha }_{1}AK+O0$ B2 α2i{2 B1 IAK} IRO2} K C2 E2 $\alpha 2AK+O22$ (c) SCR Figure13.7Simplifiedlarge-signalequivalentcircuitsfor(a)anactive-modebiasedpnptransis- tor,(b)anactive-mode biased npntransistor,and(c)an SCRin theblocking mode withI=0and VAK>0. From Eq.(13.2) one infers that the critical switching voltage is reached when α1+a21.Asstressedearlier,BJTsinthemodelareintrinsicallylow-gaintransistors; that is,theSCRisconstructedsothatα₁+2<1at small VAK>0.Duetoa varietyof mechanismstobedetailed shortly,+2progressivelyincreases withincreasingV AK Thus eventually +2approaches unity and the SCR switches into theconducting mode. The mechanismsthatcause+2toincrease withincreasingVAKwere alldescribed in the BJTdiscussion. They include base-widthmodulation,an increase in the emitter ef- ficiency,and carriermultiplication intheC-B junctions.In general,onecan write αdc=MyαT (13.3)

--- Page 493 --- 470 BJTs ANDOTHERJUNCTION DEVICES where,reviewing previous definitions, is the base transport factor, theemitter efi- ciency, andMthe carrier multiplication factor. When VAk>0and the SCR is in the blocking mode,the J23 junctionisreverse biased.Mostofthe depletionregionaboutthe J23 junctionextends into thelightlydopedN2baseof thepnptransistor. Thus increasing VAK}$causesconsiderablebase-widthmodulationandacorrespondingincreasein\alpha .ThediscussiononBJT“figuresofmerit”inSubsection\mathrm{11.2.8}pointedoutthatrecombination-generationcurrentreducestheBJTemitterefficiencyatlowinjectionlevels.IncreasingtheinjectionlevelincainintheCRithreorealoexctedtoincrsetheinjc-tionefficiency.Finally,iftheincreasein\alpha andareinsufficienttocause\alpha +\alpha 2toapproachunitcaiermultiplicationintherevers-based3junctionwillultimatelytrig-gertheSCR.Obviously,aspreviouslasserted,thebreakdownvoltageoftheJ3juncionmarkstheupperlimitoftheforward-biasblockingvoltage.Theonlyremainingexperimentalobservationtobeexplainedisthedecreaseintheforward-biasblockingvoltagewithincreasingI>0.Thereasonforthedecreasemaybeobvioustothereaderinlightofthepreviousdiscussion.Verysimply,apositivegatecurrentintotheP3regioncausesmajoritycarrierstobeaddedtothebaseoftheN2-P3-N4transistor.InjectionfromtheN4emitteristherebyenhancedandtheregenerativetrigeringprocessbecomesunstableatalowerVAK.Typicallyonlyafewmilliampsofgatecurentarenecessarytoturnonamulti-ampereanodecurrent.13.3PRACTICALTURN-ON/TURN-OFFCONSIDERATIONS\mathrm{13.3.1}CircuitOperationToprovideacompletepictureofSCRoperation,itisusefultoexamineturn-onandturn-offfromanexternalorcircuitpointofview.ConsiderthedevicebiasingconfigurationandVAKcuilolinoningIAK$ ↑ A VAA/R_{L ON P1 RL IG N2 P3 VAA G IH $T} GOFF VAK K VAA V{BFB (a) (b) Figure13.8(a）VAK>0SCRcharacteristicsandsuperimposedloadline.(b）Assumeddevice biasing configuration.

--- Page 494 --- PNPNDEVICES 471 the SCRis initiallybiased in theblockingmodeatthe”off”pointin part (a)ofthe figure. One way to switch thedevice”onistoincrease theapplied voltage VAA.Thiscauses a paralleldisplacementoftheloadline in the directionof increasing VAK.Whenthe load line no longer intersects theblocking-modecharacteristic the SCRturns on.Alternatively, with VAasshownIGcanbeincreasedromerountilthehihimpedanceportionofthe IG≠0curve[corresponding,forexample,totheG2curveinFig.13.8(a)]nolongerinter- sects the load line. With no high-impedance intersection point,the deviceoperation then switchesto the pictured”on”point.It should be notedthatonce theSCR“latcheson”,the gate currentcan beturnedoffandtheSCRwillremain in theconductingmode. Ifbeing operated at the”onpoint shown in Fig.13.8(a),theSCR can be turnedoff by decreasingVAA.OnceIAKisdecreasedbelowIH,thelow-impedanceconductingmode can no longer be maintained and the device reverts to the blocking mode. Theoretically, an IG<0orextractinggatecurrentcanalsobeusedtoturnoff theSCR.However,thecharge removalratefromthegatedP3regionmustbefasterthanthechargeentryrate intothe region.This necessitatesaspecial devicedesigntohandletherelativelyhighgatecurrents. AnI≈10A,for example,wouldtypiclly berequired to turn off an AK=100A. Devices speciallydesigned to provide IG turn-off are calledGTO(gate turn-off)SCRs. 13.3.2Additional Triggering Mechanisms Thus far we have describedtriggering initiatedbytheelectrical injectionof majoritycar- riers intothe baseregionsof the SCR.Therequired carrier injection can also be achieved by shining a sufficiently intense pulseof hv>EG lighton the device.Provision must be madeofcourse toallow lightpenetration intoa baseregion,usuallytheP3region,ofthe device.Thegreatestadvantageofphoto-triggering is that it permitscompleteelectrical isolationof the device input circuit.A specially fabricated SCR designed to accept a photo- input iscalleda light-activated SCR(LASCR)or light-activated switch(LAS). SCRs can also be turned on by increasing the device operating temperature. The for- wardblocking voltage decreases with increasing temperature. Raising theoperating tem- perature thereforehas the sameeffect as applying anG>0.Thetemperature dependence is quite strong,limiting themaximum operating temperatureof SCRs to roughly125- 150°C. Becauseof its extremely high gain[AK（on)/G(turn-on)~104-106],the SCRis susceptibleto inadvertent switching dueto noise pulses in thegate circuitry.Thiscan e remediedattheexpnseoftiggeringnitivitybyusingtheshortedcathodeconfiguration described inthenextsubsection.An unexpected rise inthe devicetemperaturecan likewise lead to inadvertent switching. Packaging of high-power devices in metal studs optimizes heat dissipation and minimizes temperature fluctuations. 13.3.3Shorted-CathodeConfiguration Ifconstructed asdescribedin Section13.1,₁+α2,andhence VBF and theforward-bias characteristics in general,areextremely sensitive to the dopings and widthsof thebase regions.Onlya slightvariation inthese parameters leadsto a significantvariationinthe

--- Page 495 --- 472 BJTs ANDOTHER JUNCTION DEVICES K $IG G A N4 B P3 1 1 1 1 1 A 1 A 1 N2 1 1 1 1 1 1 1 I 1 1 P1 1 1 A Figure13.9 The SCR shorted-cathode configuration. devicecharacteristics.Historicallythefirstcommercialventureseekingtofabricateand sellPNPNdiodes was a financialfailure because theturn-onvoltage could notbecon- trolled within acceptabletolerances. The now-standard shorted-cathode configuration picturedinFig.13.9was subsequentlyintroducedto obtainreproducible switching characteristics. The keyfeatureof the shorted-cathodeconfiguration is the extensionof the cathode metallizationovertheP3region. Theextensionaddsanohmiccontactto theP3regionat a pointremotefromthegate.AtlowIAK currentlevels,theP3/N4 shortessentiallyelimi- nates injection fromthe N4region,andthegainofthe N2-P3-N4transistor iseffectively zero.However,athighercurrentlevelsthe lateralflowofcurrentbeneaththe N4region creates a forward-bias P3-to-N4 voltage drop that increasesfrompoint A to pointBin Fig.13.9.WhenthepotentialatthepointBreachesacritical value,~0.7Vorso,signifi- cant N4to P3 injection takes place near the point B.Inrapidsuccession,the IAK current then increases,the injecting portion ofN4 widens from the point B toward the point A,and the gainof the N2-P3-N4transistor rises abruptly. Shortly thereafter,injection is uniform across the face of the N4 region,1+α2→1,and the SCR switches into the conducting mode.Note thattheshorted-cathode arrangement makes it possibletoobtain reproducible switchingcharacteristicsbysimplycontrollingtheresistanceoftheP3regionbeneaththe cathode. 13.3.4di/dtanddu/dtEffects Alarge,short-duration IG pulse is sometimes used to minimize the time required to turn on an SCR.However,if the gate current increases too rapidly in a shorted-cathodeSCR, the injection current across the P3-N4 junction may rise to excessive levels near the gate before the injection becomes uniform across the faceofthecathode.Theconcentrationof current near the gate edge ofthe junctioncauses localized heating,which in turn can lead tothermaldevicefailure.Thedescribedfailuremechanismisknownas dildtburnout. Obviously,care must be takento limit the dildt of the turn-on pulse.

--- Page 496 --- PNPN DEVICES 473 Theapplicationofa pulsedanode-to-cathodevoltage also gives rise to a significant transienteffect.Specifically,theSCRconductsprematurely;theforward-biasblocking voltageislessiftheanodevoltageisappliedatarapidrate.Infactitisobservedtahe reductionintherise timeofan applied voltage pulse leadstoa systematic decrease inthe voltage requiredto turnon an SCR.Prematureconduction likewise occurs under high- frequency a.c.operationor if the device is exposed to a high-voltage noise spike. This phenomenon is referred to as the duldt effect. The duldteffectisdirectlyrelatedtothechargingoftheJ23junctioncapacitance.The depletionwidthabouttheJ23junctionwidensastheanode-to-cathodevoltageisincreased positively.Theassociatedmajoritycarriercurrent inducesadditional injectionfrom the P1 and N4 emitters and thereby enhances thegainsof the componenttransistors.The increased gain accountsforthepremature conduction.Thegreaterduldt,thegreatertheenhanced gain,the greater the reduction in the blocking voltage. 13.3.5 Triggering Time For a PNPNdevice to switchfrom theblocking mode to the conducting mode,it is first necessary forcarriersinjectedfrom theanodeandcathodetocrosstheadjacent base.The average time takenby minoritycarriers todiffuseacross a quasineutralbaseregion was notedtobe W²/2DBinChapter12.If W2and W₃aretherespective quasineutral widthsof the N2andP3base regions,thenthecorresponding transit timesacross theregionsare t₁=W/2Dpand=W/2DN.Asafirst-orderapproximation,thetriggeringtimeis taken to be the geometrical mean of the two transit times: W_{2}_3} $tOn≈√__1} (13.4)

$2\text{DD}$ TheimportantpointtobederivedfromthetOresultisthatthebase widthsshouldbe small forrapid switching.However,the smallerthe base widths,the largerthecomponent transistor gains and the smaller the punch-through voltages.Smaller base widths are there- fore expected to lower the blocking voltages and decrease the power-handling capabilityof the device.Clearly,there is a trade-offbetween high-power handling capabilityand raid switching. 13.3.6Switching Advantages/Disadvantages Both the SCRand BJT can functionas electronic switches.Both the SCRand high-power versionsof theBJTare usedinpower-controlapplications.Derivedinlarge partfromthe preceding discussionandfollowing Navon1,thecomparativeadvantagesand disadan- tages of the SCR are summarized below. Advantages (1) The SCRrequires verylittlegatecurrenttoturnon very largeanode-to-cathode currents.

--- Page 497 --- 474 BJTs AND OTHER JUNCTION DEVICES (2) The SCRcan block both polarities of an a.c.signal. (3)TheSCRhasaveryhighblockingvoltagecapabilitycombinedwithalowvoltagedop in the conducting mode. (4)UnliketheBJT,theSCRis notsubjectto currentcrowdingwhenoperating inthe conducting mode under steady-state conditions. Disadvantages (1)Whereaseliminationof thebasecurrentturnsoff theBJT,SCRscannotbeturnedoff by setting I_= 0. (2) SCRs cannot operate at high frequencies. (3) SCRs are prone to turn on by noise voltage spikes. (4)SCRs have a limitedtemperature range of operation. 13.4 OTHERPNPNDEVICES There are a numberofotherPNPNdevices soldcommercially inadditiontothosealeady mentioned. We briefly survey four of the more common structures. Aschematic representationofthe dual-gateSCRorsiliconcontrolled switch(SCS)is shown in Fig.13.10.TheSCScan betriggered byeither gateandoffers greaterflexibility to the circuit designer.Devices thatexhibita symmetricalresponse to positiveand nega- tive applied voltages aresaidtobe“bilateral.Theidealizedcrosssectionanda sketchof the characteristics exhibitedby a bilateral diodeor DIACare presented inFig.13.11. Figure13.12displaysthecross sectionandcharacteristicsofthebilateralSCRorTRIAC. A P1 N2 G2 G1 P3 N K Figure13.10 Schematicrepresentationofthedual-gateSCRorsilicon-controlledswitch(SCS).

--- Page 498 --- PNPN DEVICES 475 A IAK}$NPPNNVAKPPNK(a)(b)Figure13.11TheDIAC(DiodeAC).(a)Idealizedcrosssectionand(b)generalformofthedevicecharacteristics.IA$ A1 G $IA$ N N P VA N EEEE=5 P N A2 (a) (b) Figure13.12 The TRIAC(Triode AC).(a)Idealizedcross sectionand(b)generalformofthe device characteristics. As is obvious from its cross section,the DIACis just an integrated versionof twooppo- sitelyorientedPNPNdiodesconnected in parallel.Similarly,the TRIACfunctions as two oppositelyorientedSCRsconnected in parallel.Thebilateral devicesare especiallysuited for use in a.c.power-control applications. Lastly,there is the so-calledprogrammable unijunctiontransistor (PUT).Contraryto its name,thePUTis notaunijunctiontransistor!Aunijunctiontransistor(UJT)is a three-

--- Page 499 --- 476 BJTs ANDOTHER JUNCTION DEVICES IE E B2 VBB VE Rs$AIEA\uparrow$R_{2}$PPGGNNPP$R_{}$NN\to VEK$Vp∞VBB K RL B1 (a) (c) (b) Figure13.13PUT（programmable unijunction transistor).(a)Structure;(b)circuitconnectionto simulateUJTcharacteristics;(c)I-Vcharacteristics ofa PUT. lead device withtwoohmiccontacts attheendsofa bar-likeresistive region.Athird lead is connected toone sideofa pn junction formedalong the bar.TheUJTexhibits switching characteristics grosslysimilartothoseof theSCR.As illustratedin Fig.13.13(a),thepro- grammableunijunctiontransistor is actuallyananode-gatedPNPNdevice.Differing from a conventional SCR,thegate contact in a PUTis made to theN2base insteadof the P3 base.When connected as shown inFig.13.13(b),thePUTis functionallyequivalent to a UJT,yielding characteristics likethose sketchedinFig.13.13(c).Thecharacteristics can be modified by simplychanging theexternal RandR2resistors,thereby givingrisetothe “programmability””of the structure.

--- Page 500 --- 14 MS Contacts and Schottky Diodes Themetal-semiconductor (MS)contact plays a very important roleofone kindor another inallsolid-state devices.When intheformof a non-rectifyingorohmiccontact,itis the critical linkbetween the semiconductorandtheoutsideworld.TherectifyingMScontact, referred to as the Schottky diodeor the MSdiode,is found in anumberof device structures andisanimportantdevicein itsownright.Physicallyandfunctionally,thereareclose similarities betweentheMSdiodeand theasymmetrical(p-n orn+-p)step junction diode.Indeed,a large portionof the pndiode analysis can beapplied directly to theMS diode with only minor modifications. We initiate theMSanalysis by establishing the equilibrium energy band diagram for anidealcontact.Withtheaidofthediagram,onecanreadilydistinguishbetweenrectiying and ohmic contacts.The next sectiontreats theelectrostatics,I-Vcharacteristics,a.c.re- sponse,andtransientresponseofthe Schottkydiode.Thechapterconcludes with a presen- tation of select practical informationaboutMScontacts. 14.1 IDEALMSCONTACTS AnidealMScontact hasthefollowingproperties:(1）Themetalandsemiconductorare assumedtobe in intimate contacton anatomic scale,withno layersof any type (suchas anoxide)between thecomponents.(2)There is nointerdiffusionor intermixingofthe metal and semiconductor.(3) There are no adsorbed impurities or surface charges at the MSinterface. The initialtask at hand is to construct theenergy band diagram appropriate foranideal MScontact underequilibriumconditions.The surface-included energy band diagrams for the individual,electricallyisolatedmetaland semiconductorcomponents are picturedin Fig.14.1.Flatband （zerofield)conditionsareassumedtoexist throughout the semicon- ductor.Inbothdiagramsthe vertical linewheretheenergy bandsare abruptlyterminated is meant to represent a surface.The cross-hatching onthe diagrams identifies allowedstates that are nearly completely filled with electrons. Severalkeyenergiesandenergy differences arereadilyintroduced withtheaidof Fig.14.1.The ledgeatthetopofthe vertical linedenotestheminimumenergy anelectron must possesstocompletelyfreeitselffromthematerial andiscalledthevacuumlevel,E. Theenergy difference between the vacuum leveland the Fermienergy is knownas the 477

--- Page 501 --- 478 BJTSANDOTHERJUNCTIONDEVICES E0 E0 χ Φs$\Phi M$ $Ec$ $s\text{qra}$ EEv}$ Metal Semiconductor Figure14.1Surface-includedenergy band diagrams for ametal(left)andn-type semiconductor (right). workfunction(Φ)ofthe material.The metal workfunction,ΦM,is an invariantfundamen- tal propertyofthespecified metal.The valueof ΦMranges from3.66eVformagnesium to 5.15eVfor nickel.The semiconductor workfunction,Φs,is composedof two distinct parts; that is Φs=χ+(c-)FB (14.1) Theelectronaffinity,(E0-E)surface,isaninvariantfundamentalpropertyofthe specified semiconductor.χ=4.0eV,4.03eV,and4.07eVfor Ge,Si,andGaAs,respec- tively.Conversely,(E-F)FB,theenergydifferencebetweenEandEFunderfatband or zero field conditions,is a computable function of the semiconductor doping. Suppose now the ΦM>Φs metal and n-type semiconductor of Fig.14.1 are brought together to forman ideal MScontact.Let us assume the contact formation is accomplished almost instantaneouslyso thatthere is negligibleelectrontransferbetween thecomponents duringthecontactingprocess.Ifthisbethecase,thenaninstantafterformationtheerg banddiagram forthecontact willbeaspicturedinFig.14.2(a).In this figure theisolated energybanddiagramsare verticallyalignedtothecommonEreference levelandsimply abutted at the mutual interface.Itshould beemphasizedthat ΦM and are materialcon- stants and remain unaffected by the contacting process. Since EFs≠FM,the MS contact characterized by Fig.14.2(a) isobviously not in equilibrium.Under equilibrium conditions the Fermi level in a material or a group of ma- terials in intimate contactmust be invariant with position (seeSubsection3.2.4).Conse- quently,a short timeaftertheconceptualcontactformation,electrons willbegin transfer- ring from the semiconductor to the metal given the situation pictured in Fig.14.2(a).The net loss of electrons from thesemiconductor creates a surface depletion regionand a grow- ing barrier to electron transfer from thesemiconductor to the metal.This will continue until the transferrateacross theinterfaceisthe same inbothdirectionsandEFis thesame

--- Page 502 --- MS CONTACTS AND SCHOTTKY DIODES 479 $\Phi M>\Phi s$ 1 1 n-type n-type Metal 1 Metal semiconductor 1 semiconductor 1 1 1 E0 E0 1 1 xe SΦ 1 WΦ ΦM$xve$ Φs Ec Srac}$EFME$ EFM $Ec$ S\ra} Ev} EiE_\v}$(a)(c)下EEEEEv$ (b) (d) Figure14.2 Energyband diagrams for ideal MScontacts between a metal and an n-type semicon- ductor:ΦM>Φsystem (a)aninstant aftercontactformationand(b)underequilibriumconditions; ΦM<system（c)aninstantatercontactformationandd)underequilibriumconditions. throughoutthestructure. The netresult,theequilibrium energy band diagram foran ideal ΦM>Φsmetalton-typesemiconductorcontactis shown inFig.14.2(b).Indrawingthis figure,extraneouslinessuchastheEreferencelevelandtheportionoftheverticalsurace line above Ehave been removed.Also note that ΦB = ΦM-χ. . .ideal MS(n-type) contact (14.2) where ΦB is the surface potential-energy barrierencountered byelectrons with E=EFin themetal.Finally,iftheentireargument isrepeatedforametaland-typesemiconductor where ΦM<Φ,oneobtainstheequilibriumenergyband diagram shown inFig.14.2(d).

--- Page 503 --- 480 BJTs AND OTHER JUNCTION DEVICES I n-type VA Metal semiconductor (a) 1 EESS EFM $E_{}$ $Ev$ (b) 1 EFM E{c S\ra}}$E$ Ev} (c) VA (d) Figure14.3 Response of the ΦM>Φ(n-type)MScontactto an applied d.c.bias.(a) Definition of currentandvoltagepolarities.(b)Energy banddiagramand carrieractivity whenVA>0. (c) Energy band diagram and carrier activity when VA<0.(d) Deduced generalformof theI-V characteristics.

--- Page 504 --- MS CONTACTS AND SCHOTTKY DIODES 481 Let us nextqualitatively examinetheeffectofbiasing thetwoMSstructuresin Fig.14.2.As specified inFig.14.3(a),the semiconductor is taken to be grounded andVA A appliedtothemetal.ThecurrentIis definedtobepositive whencurrentflows fromthe metal to the semiconductor. Consider first the ΦM>ΦsMScontact.ApplyingaV>0as in Fig.14.3(b)lowers EFMbelowEs,reducesthebarierenbyelectronsinthesemionductor,andtherefore permitsanetflowofelectronsfromthesemiconductortothemetal.IncreasingVAleadsto arapidlyrising forwardbiascurrent,sincean exponentially increasing numberofelectrons fromthe semiconductor areableto surmountthe surface barrier.Ontheotherhand,apply- ingaVA<0raisesEFMaboveEFsaspicturedinFig.14.3(c).Thisallbutblocksthefow of electrons from the semiconductor to the metal.Someelectrons in the metal willbe able to surmountthe ΦB barrier, butthe associated reverse-bias current shouldbe relatively small.Moreover,since ΦB is ideally the same for all reverse biases,the reverse current is expected toremainconstantafterthe reverse biasexceeds a few kT/q volts.Clearly,we have just describedrectifying characteristics similarto that displayedby a pn junction diode.Theideal n-type semiconductortometalcontact where ΦM>Φsis identiedas an MS diode. Theresponseto an applied bias isconsiderablydifferent forthe ΦM<sMScontact. There is nobarrierofanykind intheFig.14.2(d）structureforelectronflowfromthe semiconductor to the metal. Thus even a smallV>0 gives rise to a large forwardbias current.Underreverse biasing there is a smallbarrierfor electronflowfromthemetal to the semiconductor,butthebarrieressentiallyvanishes if thereversebiasexceedsafew tenths of a volt.Large reverse currents are expected at relatively smallreverse biases, and the reverse current definitely does not saturate. Thebehavior here is obviously non- rectifying or ohmic-like. The overall conclusion is thatan idealMScontact formedfrom ametal and an n-type semiconductorwillerctifyiconttifΦ>ndanohmiclikeconttifΦ< .ParallelargumentsappliedtoanidealMScontactformedfromametaland ap-type semiconductoleadtotheconclusionthatthecontactwillberectifingifΦ<Φsand ohmic-likeif M>Φ.TheseconclusionsaresummarizedinTable14.1.It shouldbere- emphasizedthatallresultsandconclusions inthissectionare contingentupontheMS contact being ideal.Required modifications imposed by theoften nonideal natureofreal MS contacts are discussed in Section14.3. Table14.1 ElectricalNatureofIdealMSContacts. n-type p-type Semiconductor Semiconductor $\Phi s$ Rectifying Ohmic Ohmic Rectifying

--- Page 505 --- 482 BJTS AND OTHER JUNCTION DEVICES Exercise14.1 P:(a)Constructtheequilibriumenergy banddiagramappropriateforanidealptpe semiconductor to metal contact where ΦM<Φs. (b) Repeat part (a) when ΦM>Φ_s. (c) Verify thatan ideal MScontact formedfrom ametal and a p-typesemiconductor will berectifyingifΦM<Φandohmic-likeifΦMΦ. (d) Establishanexpressionforthebarrierheight,ΦBFM-linterfaceofthe rectifying p-type contact. S:(a)/(b)The“prescription”for drawing the equilibrium energy band diagrames- tablished inthetext canbesummarized asfollows:(i）Drawthesurface-included energy band diagrams for the individualcomponents.ii)Verticallyalign the dia- grams to the common Ereference level,and join the diagrams at the mutual inter- face.(ii)Withoutchanging the interfacialpositioning of the semiconductorbands, move thefield-free semiconductor bulk (theregion farfromthe interface)up ordown untilEFis constanteverywhere.（iv)Appropriatelyconnectup the E,E,andEat the interface with thefield-free positioning of the bands in the semiconductor bulk. (v) Eliminate extraneous lines. Following thecitedprescriptiononeobtains theequilibriumenergy band dia- grams shown below. Metal p-Semiconductor Metal p-Semiconductor Ec $Ec$ $E_i$ E{ $E$ Ev} Ev} SΦ>WΦ() (b) ΦM >Φs (c) Hole flow under bias must be examined to determine whether the given MScon- tacts are rectifying or ohmic. Empty electronic states in the metal,which decrease exponentially withenergy belowthe Fermi level,can be thoughtof as holes for the purposeofthediscussion.FortheΦM<contact,thereisclearlyabarriertoole flow in both directions underequilibrium conditions.Moving EFM upward elatie to EFs reduces the barrier to hole flow from the semiconductor to the metal. The resulting S→Mhole current is expected to increase exponentially with increased separation between E FM and Es.Reversing the bias blocks hole flow from the semi- conductor to the metal,leaving only a saturating hole current from the metal to the

--- Page 506 --- MS CONTACTS AND SCHOTTKY DIODES 483 semiconductor. The ΦM<Φcontact is obviously rectifying.For the ΦM>Φs contact, there is nobarrier to hole flowfrom the semiconductor to the metal.More- over,thesmallbarrier toholeflowfromthemetaltothesemiconductor vanishes if EFM is movedonly slightly downward relative to s.The ΦM>Φcontact is concluded to beohmic-like,thereby completing the required verification. (d) Since clinterfac -FM= ΦM- it follows that ΦB=FM-vlinterface=(-E)-(clinterface-M) or ΦB=G+-Φ …ideal MS(p-type) contact 14.2 SCHOTTKYDIODE Having establishedthebasicnatureof therectifyingMScontact,we undertakehere amore quantitativeanalysisoftheSchottky(MS)diode.Followingtheusualoutline,theanalysis includesasurveyofthed.cacandtransientcharacteristicsprecededbyanexamination ofthe deviceelectrostatics.Strong parallels withthe pn junction diode permit arelatively condensed presentation.Throughout the discussion we take the semiconductor to be n-type and uniformly doped. Theassumedcurrentand applied voltage polarities are as specifid in Fig. 14.3(a). 14.2.1 Electrostatics Built-in Voltage Like in the pn junction diode,there is a voltage dropor built-in voltage across theMS diodeunderequilibriumconditions.Referring toFig.14.4(a),thebuilt-in voltage(Vi)is readily deduced to be 1 ΦB-(Ec - EF)FBB bi{}$ (14.3) q where as previously noted ΦB=ΦM-for an ideal MS(n-type)contact.

--- Page 507 --- 484 BJTs AND OTHER JUNCTIONDEVICES ^{bb ΦB下 E (E{-EF)FB Ei Ev + →x 0x_ }$(a)\rho \uparrow$qND →x $x_{}W$ δ-function of negative charge (b) 8 ↑ 0 W → x (c) V W 0 →x -V_bi$ (d) Figure 14.4 Electrostatic variables in an MS(n-type）diode under equilibrium conditions. (a)Equilibriumenergybanddiagram.b)-(d)Charge densityelectricfield,andelectrostaipoten- tial as a function of position.

--- Page 508 --- MSCONTACTSANDSCHOTTKYDIODES 485 ρ,E,and V Afurther examinationofFig.14.4(a)revealsthere is a depletionofelectrons inthe semi- conductoradjacenttothe MS interface.Completely analogous to the pn junction diode,the depletionregion can be viewed as extending a distance into the n-type semiconductor, with a net positive chargearising fromthedonors in the depleted region.Unlike thepn junction,however,there is no negative p-side acceptor charge to balance the positive n- sidedonor charge.Rather,an essentiallyδ-functionof negativecharge (extraelectrons) piles upinthemetalimmediatelyadjacenttotheinterface.Thecharge densityinsidethe structure is thereforeconcluded to beroughlyas pictured inFig.14.4(b).Formallyinvok- ing the depletion approximation, we can write {qD …0≤x≤W (14.4a) =d 0 …x>W (14.4b) Notethat xncan beequated to the depletion widthW,since the depletionregion is totally contained inside the semiconductor. Given theδ-function natureof thechargeonthemetalsideof the interface,itauto- maticallyfollows that&=0andV=constantinsidethemetal.Consequently,no further consideration need be given to the metal. Onthe semiconductor sideof theMScontact,theelectricfieldand charge density are related through Poisson’s equation (Eq.5.2); namely, d8 ρ $qND$ …0≤x≤W (14.5) dx K{ε0 K{ε SeparatingvariablesandintegratingEq.14.5)fromanarbitrarypointinthedepletion region tox=Wwhere=0,we obtain 0 W qND d8^{'= dx^'} (14.6) K_{sε0 JE(x) Jx$or$qND$8(x)=$ (W - x) . . .0 ≤x ≤W (14.7) K{0 Plotted in Fig.14.4(c）,theEq.(14.7)result isidenticaltotheEq.(5.21)solutionforthe E-field on the n-sideof ap+-n step junction if nis equated to W.

--- Page 509 --- 486 BJTs ANDOTHERJUNCTIONDEVICES Seekingthesolutionforthelectrostaticpotentialinthesemiconductor,wenextnote $dV$ qN_ dx -8= K{0 (W-x) …0≤x≤W (14.8) Once againseparatingvariablesandintegratingfromanarbitrarypointinthedepletion region tox=Wwhere the potential is arbitrarily set equalto zero,we obtain 0 w qNDdV^{ =$(W-x^{\prime })d{x}^{\prime }(14.9)V(x)$JxK{sε0 or $qND$ $V(x)=-$ (W - $x)$ …0≤x≤W (14.10) 2K_{ε At first glance thesolutionforV(x)wouldappear to differfrom that in the pn junction analysis.Actually,the solutionsaretotallyequivalent.In treatingthe pnjunction,the po- tential was set equal to zero on the far p-sideof the junction; an MSdiode solutionof precisely thesame form wouldresult if themetal were the V=0reference.Withthe analysis focusing on a single semiconductor region as in theMS diode, however, the surface-side edge of the semiconductor bulk is the preferred reference point. Underequilibriumconditionsthepotential dropacrossthedepletionregionis -Viatx=0,andthe V(x) versusx dependence is as sketchedin Fig.14.4(d).IfVA≠0, then Vbi→bi-VAandV=-(bi-VA)at=0.The simple Vbi→Vbi-VA replacement assumesof course that the back contact to the diode is ohmicand that theIR potential dropacross the semiconductor bulk is negligible. Depletion Width SinceV(0)=-(Vbi-VA),evaluatingEq.(14.10)atx=0gives qN_$-(bi-VA)=-$^{}$(14.11)$2K{s0 Thus,identical to the situation in ap+-n step junction, 2K{sε 1/2 W= (14.12) qNDD$

--- Page 510 --- MSCONTACTSANDSCHOTTKYDIODES 487 Exercise 14.2 P:Copper is depositedon a carefullyprepared n-type silicon substrate to form an ideal Schottkydiode.ΦM≈4.65eV,χ=4.03eV,ND=101/cm3,andT=300K. Determine (a)\Φ_{B} (b) _b{ib, (c) WifVA= 0,and (d)||maxif VA=0. S:(a)ΦB=ΦM-=0.62eV −kTnn)=0.56(.( $E$ (1016 (b)(E-F)FB= ≡0.20eV 2 \0\0} 1 [ΦB-(E-F)FB]= 0.42V $V_{b}ib$

q 2K_{s$71/2(2)(11.8)(8.85\times 10-14)1/2(c)W=$ (Vbib$

VA)$

(0.42) $qNDD$ (1.6×10-19)(1016) = 0.234 μm $qND$ (1.6×10-19)(1016)(2.34×10-5) (d)|8|max=||x=0|= W= K{0 (11.8)(8.85 ×10-14) =3.59×10{⁴V/cm 14.2.2 1-VCharacteristics WhereastheMSdiodeelectrostatics andthegeneral shapeoftheMSdiodeI-Vcharacter- istics are very similartothoseof a pn junctiondiode,thedetailsof the d.c.currentfloware decidedly different.In ap+-n diode,as reviewed inFig.14.5(a),thedominantcomponents ofthe currenttypically arise from recombination in the depletionregion undersmallfor- ward biases and hole injection from the p+to the n-side of the diode under larger forward biases.Theelectron injectionfrom the lighter doped nto the p+-side is always negligible. In an MS(n-type)diode,as pictured inFig.14.5(b),the recombinationandhole-injection currentsstillexist.However,becauseof therelatively lowpotentialbarrier seenbyelec- trons in the semiconductor,electron injectionfrom the semiconductor into the metal rou- tinely dominates theobservedcurrent.Stated another way,theelectroninjection leads to a very large forward bias current before the recombination and diffusion (hole-injection) currents become important.Thesituation underreverse bias is similar.Electronflowfrom the metal to thesemiconductoras previously pictured in Fig.14.3(c)totallydominates the

--- Page 511 --- 488 BJTs ANDOTHER JUNCTION DEVICES Negligible Ec EFn EFp —ET oooo I{-G -Ev}F (a) p+-n junction diode Dominant q(V_bi-{$-$E{c$S$ET -Ev}$ → + →x 0 W (b) MS diode Figure14.5 Negligibleanddominantcurrentcomponentsinaforward-biased(a)p+-njunction diode and (b) MS diode. observedcurrent.Thereverse-biasholediffusioncurrentandtheR-Gcurrentassociated withcarrier generation inthedepletionregionaretypicallynegligible.Reflectingthefact thatminority carriers inthe semiconductor normallyplay an insignificantrole in determin- ing theI-Vandother characteristics,theMSdiode isoften said to be a”majoritycarrier device.” The currentresulting frommajority carrierelectron or hole injectionover the potential barrier in anMSdiode isreferred to as thethermionic emission current. Toestablisha quantitative expression for the thermionic emission current,we treat an n-type device and focus initiallyon electron injectionfromthe semiconductor into themetal. The x-coordi- nate is assumed to be normal to the MS interface and directed into the semiconductoras shown in Fig.14.5(b). Consider an electron entering the depletion region fromthe semiconductor bulk. The electron is capableof surmounting the surface barrierandcrossing into the metal ifit has

--- Page 512 --- MS CONTACTS AND SCHOTTKYDIODES 489 avxvelocitydirected toward the interface such that 1 KE m *2x} ≥q(Vbi-A) (14.13) x 2 n or 71/2 29 ${|}_{x}1\ge min$ (V_bib$(14.14)m\ast =Supposetherearen(vx)electrons/cminthesemiconductorbulkwithanegativelydirectedvxsufficienttosurmountthebarrier.ParallelingthedriftcurrentderivationinSubsec-tion\mathrm{3.1.2},thecurrentassociatedwiththissetofelectronswillbeSo\to M,x=-qA{v}_{x}n{(}_{)}(14.15)Summingoverallelectronsintheconductionbandwithxvelocitiescapableofsurmount-ingthebarrierthengives-vmin$ $so\to M=-qA$ ν_{xn(_x)d (14.16) -∞ Fora nondegenerate semiconductor itcan be shown tha1] (4πkTm2 n e(EF-Ec)/kTe-(m/2kT)2 (14.17) $h^3$ Substituting Eq.(14.17)into Eq.(14.16),integrating,andsimplifyingtheresult,one obtains ISO→M=AAT²e-ΦB/kTeqVA/kT (14.18) where （m\ A*= (14.19) m{o and 4πqm0k^{}$

= 120 amps/(cm^²-K^2) (14.20) h{3 TheconstantA was introduced in arelated analysisofelectronemissionfrom metalsand has since become known as Richardson’s constant.

--- Page 513 --- 490 BJTs ANDOTHER JUNCTION DEVICES Electronscrossingtheinterfaceintheoppositedirectionfromthemetalintothe semi- conductor always seethe same potential barrier,ΦB.Consequently, Mo→s(VA)=M→S(VA=0) (14.21) Moreover,underequilibriumconditionstheM→SandS→Mcurrentsacrossthebarrier must precisely balance, or IMO→s(VA=0)=-So→M(VA=0)=-AAT²e-ΦB/kT (14.22) ThetotalcurrentatanarbitraryVAisofcoursegivenb I=So→M+M→s=So→M+Mo→s(=0) (14.23) Combining Eqs.(14.18) and (14.22),we therefore conclude I= Is(eqVA/kT-1) (14.24) I=AAT2e-ΦB/kT (14.25) S TheEq.(14.24)/(14.25)result isclearly theMSdiode versionof theideal diodeequa- tion.For forward biases greaterthan a fewkT/q volts,theexponentialterm in Eq.(14.24) dominates andI→Iexp(qVA/kT).Forreverse biases greaterthan a few kT/q volts,the exponential term becomes negligible and thecurrent is predicted to saturate atI=-I. As formulated,the theory implies I wouldremainconstant at -I forreverse biasesof unlimited magnitude. Representative experimentalI-VcharacteristicsderivedfromaMBR040MSdiode are displayed inFig.14.6.ForforwardbiasesofVA≤0.35V,experiment(thesolidline curve inFig.14.6a)and theoryare inalmost perfect agreement.Over therange 0.V VA≤0.35Vtheslopeoftheforwardbiassemilogplotis veryclosetoqkT.Likeinthe pn junctiondiode,the decreaseinslopeat largerforwardbiases istypicallycaused byan appreciable voltage drop across the bulk series resistance. Turning to thereverse-bias characteristic in Fig.14.6(b),we note two significant de- viations fromtheideal.First,asshouldhavebeenanticipated,andagainparalleling thepn junction diode,a breakdown phenomenon limits themaximum magnitude of the reverse bias volage.IntheabenceofegeefecttheexpctedBRduetoaalanchinginanMS diode is essentially identical to thatof an equivalently doped p+-norn+-p diode.Second, thereverse-biascurrent doesnotsaturate. Differing from the pn junction diode,theob- served behavior here cannot be attributed to therecombination-generationcurrent. Rather the systematic increase in the magnitudeof thereverse-biascurrent is primarily theresult of a phenomenon known as Schottky barrier lowering.ΦB is not a bias-independent con- stant as assumed in the ideal theory, butdecreases slightly with reverse biases of increasing magnitude. Specifically, ΦB = ΦBO-△ΦB (14.26)

--- Page 514 --- MSCONTACTSANDSCHOTTKYDIODES 491 $10^{-}$ $10-3IDIFF$10^-5 q/kT I(A) 10^-} $R-G$10^-9} 10-11 0 0.2 0.4 0.6 0.8 1.0 VA(volts) (a) VA(volts) -50 -40 -30 -20 -10 0 0 -0.2 -0.4 I(μA) -0.6 -0.8 -1.0 (b) Figure14.6 Measured I-V characteristics derivedfrom a MBR040MSdiode:(a)forward bias; (b)reverse bias.Thedashedlinesin(a)aretheoreticalestimatesofthediffusion(IDFF)andecom- bination-generation(R-G)currentsflowingin thediode.Theexperimentaldata wereobtainedm- ploying an HP4145B Semiconductor Parameter Analyzer.

--- Page 515 --- 492 BJTs AND OTHER JUNCTION DEVICES where ΦBo is the barrier height when =0 at the MS interface and q/8s 71/2 △Φ

q (14.27) B $4\pi K\epsilon 0s,theelectricfieldatthesemiconductorsurface,iscomputedusingthepreviouslyestab-lishedelectrostaticrelationships.SinceIvariesasexp(-B/kT),evenasmalldecreaseinBivesetoadiloticableicseinthemagitudeoftherv-biacWeshouldpointoutthatatreversebiasesapproaching-BR,avalanchemultiplicationmayplayaroleinenhancingtheobservedcurrent.inalliilin4landIRG.Teretoeiclestimteoftedifuioncuenndcombiniontflowinginthegivendiode.AspreviouslyindicatedtheIFFandRGcomponentsaeentobetotallynegligiblecomparedtotheobservedthermionicemissioncurrent.(C)Exercise14.3P:Assumethe0.1V\le VA\le 0.35Vportionoftheforward-biascharacteristicinFig.14.6(a)canbemodeledbyanequationoftheformI=IeqVA/n1kTwhereIandn,areconstants.AccordingtoEq.(14.24),the"idealityfacorshouldbeequaltounitybutisusuallydeterminedtobeslightlygreaterthanunity.Employingthepoint-by-pointdataprovidedinthetablebelow,performaleastsquaresfittodeterminethevaluesofIandnthatyieldanoptimummatchtothegivenMSdiodecharacteristics.VA(volts)(amps)\mathrm{0.104.047}\times 10-\mathrm{70.152.792}\times 10-\mathrm{60.201.890}\times 10-\mathrm{50.251.263}\times 10-\mathrm{40.308.084}\times 10-\mathrm{40.354.487}\times 10-3S:Inlogarithmicformthefitequationbecomes$ln(1)=ln(s)+ $n_1$

--- Page 516 --- MS CONTACTS AND SCHOTTKY DIODES 493 Associating VAwith xandln(I）with y,the MATLAB polyfit function can beused to perform therequiredleast squaresfitto theexperimental data.The best fit values derived from the program listed below are [n=1.03 and Is=1.02×10-8amps$.MATLABprogramscript..I=[4.047e-72.792e-61.890e-51.263e-48.084e-44.487e-3];VA=[\mathrm{0.10.150.20.250.30.35}];y=log(I);c=polyfit(VA,y,1);slope=c(1);formatcompactn1=1/(0.0259\ast slope)Is=exp(c(2))\mathrm{14.2.3}a.c.ResponseAsmalla.c.signalsuperimposedonad.c.reversebiasgivesrisetoachargefluctuationinsidethediodeaspicturedinFig.14.7.Avariationinthedepletionwidthaboutitsequi-libriumvalueandtheassociatedchangeinchargeinsidethesemiconductorbalancefluc-tuationsinthe\delta -functionofchargeattheMSinterface.Majoritycarriersmoverapidlyinandoutofthesemiconductortofacilitatethevariationinthedepletionwidth.WiththechargefluctuationtakingplacealongtwoplanesseparatedbyadistanceW,thedescribeda.c.situationisphysicallyidenticaltowhattakesplaceinsideaparallelplatecapacitor.Onecanthereforewritebyanalogy$K{{AA$C=(14.28)Wor,makinguseofEq.(14.12),$KsεOA C= (14.29) 2K_{sε0 1/2 ($V_{b}i-V)$ qN{DD$forauniformlydopedn-typesemiconductor.AlsonoteforfuturereferencethatifbothsidesofEq.(14.29)areinvertedandthensquared,oneobtains12(14.30)$Cc^{ra{}{2}$二$qND_εA2(Vbi-VA)

--- Page 517 --- 494 BJTs ANDOTHER JUNCTIONDEVICES +v_{a -$+VA-11111n-type11111\leftarrow W\to$qND →x 0> Figure14.7ChargefluctuationsinsideanMS(n-type)diodeinresponsetoanappliedac.signal. $|v_{a|≤Vbi-VA. Reverse biasC-Vdata derivedfrom a commercialMBR040MSdiodeare presented in Fig.14.8(a).The data wereobtained using the measurement set-up previously de- scribedin Subsection7.2.3and schematicallypictured inFig.7.6.Variationofthecapaci- tance withtheappliedd.c. bias is generally as expected.Consistent withthe predicted slowerrateasthereversebiasis increasedinmagnitude.Figure14.8(b)providesamore detailed verificationof thetheory.The1/C²versus VAplotof the experimental dataexhib- itsanearlystraight line dependence in agreement withEq.(14.30).Per Eq.(14.30)the semiconductordoping couldbe deducedfrom the slopeof the straight line fitted to theplot pointsandVifrom theextrapolated1/C²=0intercept.Inaddition,once both the semi- conductor doping and V; are known,ΦBcan be computed employing Eq.(14.3). Although thereversebiasbehavior isessentially identicalto thatofanasymmetrically doped pn junction diode,the forwardbias a.c.responseofthe MSdiode is significantly different.In theMSdiode the diffusioncomponentof the current is typically negligible. Thus there is verylittleminority carrierinjectionandstorage withinthesemiconductor. Since it is storedminoritycarriers thatgiverisetothediffusionadmittance,theMSdiode does not exhibit a diffusion capacitance or diffusion conductance. There is of course aforward-bias depletion-region capacitance,a potentially large parallelconductance G=dIldVA,anda bulk series resistance(Rs）thatmust be included under certain circum- stances.However,even at a.c.frequencies routinely approaching or into theGHz range,C and G remain frequency independent.

--- Page 518 --- MSCONTACTSANDSCHOTTKYDIODES 495 60 50 40 C(pF) 30 20 10 L 1 1 0 -12 -10 -8 -6 -4 -2 0 2 VA (volts) (a) 7 6 5 $1/c^{}$ 4 $10^3$ 3 $p^2$ 2 1 0 -12 -10 -8 -6 -4 -2 0 ↑ 2 VA (volts) V_{bi (b) Figure14.8(a） SampleC-V data derivedfroma MBR040MSdiode.(b)1/C² versus VAplot constructedfrom the experimentalC-Vdata.[Note:Tocorrectfor theencapsulation-related stray capacitance shunting theMS diode,3.4 pF was subtractedfromallmeasured capacitancevalues before constructing the part (b)plot.]

--- Page 519 --- 496 BJTs ANDOTHERJUNCTION DEVICES Exercise 14.4 P:Working directly with theplotted data in Fig.14.8(b),estimate the barrier height (ΦB) inside the MS diode. Assume an n-type device,A=1.5×10-3cm, and room temperature operation. S:From the extrapolated1/C²interceptof the straight line drawn through theplot points,we roughly estimate bi =0.6V The slope of the straight line is approximately 6×10-3/pF^{2 slope = -5.5×10²{0/F^{2-V 10.3+ 0.6 and therefore 2 ND

qKSε0A^{2|slope| 2 (1.6×10-19)(11.8)(8.85×10-14)(1.5×10-3)²(5.5×1020) =9.7×$105/c{m}^{3}Noting$EG}$(Ec—F)FB2$n_{i}$ =0.20 eV we conclude, making use of Eq.(14.3), ΦB = bi+(E-F)FB≌0.6+0.2=0.8eV 14.2.4 Transient Response A very rapid transientresponse is the most distinctive characteristicof theMSdiode.In pn junction devicestheexcess minoritycarriers stored in the quasineutral regionsof the semi- conductormust beremovedbefore the devicecanbeswitchedfrom the forward-biason-

--- Page 520 --- MSCONTACTSANDSCHOTTKYDIODES 497 C Schottky diode B E Figure14.9 Schematic circuitrepresentationofaSchottky-diode-clamped npnBJT. statetothereverse-biasoff-state.InanMSdiodethereisverylittleminoritycarierine- tion and storage withinthesemiconductor becausethe diffusioncomponentof thecurent is typicallynegligible.Thereverse recoverytimeofcommercialMSdiodes isroutinely onlya few nanoseconds.TheMSdioderesponsetime can infact be limited,notbythe stored charge,but bythe internalRCtimeconstantassociated withthejunctioncapacitance and thebulk seriesresistance.Smallarea devices,sometimescalledhotcarrier diodes, are commerciallyproduced with a maximum C≤1pFand sub-nanosecondresponse times. Asfirst noted intheChapter12discussionofthe BJTtransient response,the MSdiode has beenusedtospeed uptheBJTturn-oftransient.Thearrangement initiallypicturedin Fig.12.7(a)and shown again in Fig.14.9,an arrangement where an MS(Schottky) diode is connected between the base andcollectorof a BJT,is referred to as a Schottky diode clamp.Basically,when the transistorenters the saturation mode during the turn-on transient at a relatively low forward bias.Inother words,use is madeof the fact that the MSdiode conducts at a lowerforwardbias than a pn junction.Since theC-B junction is heldata relatively lowvoltage,thereisminimalcharge storage intheBJT.Withlesschargetobe removedfrom theBJTand very littlechargestored inthe Schottkydiode,theturn-offtime is thereby significantly reduced. 14.3 PRACTICALCONTACTCONSIDERATIONS 14.3.1RectifyingContacts At the beginning of the chapter,anMScontact was definedto be ideal if the metaland semiconductor were in intimate contacton an atomic scale,there was no intermixing of components,and there wereno adsorbedimpuritiesor surfacechargesat theMS interface. Unfortunately,even though great progress has beenmade in recent years toward achieving

--- Page 521 --- 498 BJTS ANDOTHERJUNCTIONDEVICES theideal MScontact,realMSstructures invariably turnoutto benonideal.Silicondevices, for example,are likely tocontain a thin (5Å-25 Å)oxide layerbetween the metal and semiconductor.A nativeoxide layerforms almost instantly when Si is exposedto theat- mosphere.Arsenic precipitates are believed to formon GaAs surfaces.Allowed electronic states thatcharge and dischargeareknown to existonessentially allsemiconductorsur- faces. In extreme cases,thecitednonidealitiescanleadto anill-functioningornon- functioningMSdiode. Theminimized nonidealities inmodern-day structures,however, primarily affect the ΦB barrier height characterizing the contact. ContrarytoEq.(14.2),ΦB≠ΦM-in mostreal n-typediodes.Asimilarstatement can bemadefor p-typediodes.InSi,GaAs,and the majority ofother semiconductors, surfacecharges tend tofixor”pin”theequilibriumFermi level at a specificenergy within the surface bandgap.Becauseof this pinning effecttheobserved barrierheightnormally variesonly slightly withthemetalused tofabricate thediode.Regardlessof themetal employed informing aGaAs MSdiode,forexample,ΦB≤2E/3=0.95eVfor n-type devicesand ΦB≌E/3=0.47eVfor p-type devices.Since ΦBcannot be predicted from a prior knowledge of ΦM and χ,measurements must be performed to accurately determine BforvenytmnaronrureAnCbsedproc thatoutlinedinExercise14.4isoneofthemorepopularmeasurementtechniques.Itshould beemphasizedthat theformalism developed intheprevious sectionstillappliesprovided one employs the measured ΦB in all relevant expressions. 14.3.2OhmicContacts Metal-semiconductorcontacts that have a low impedance regardlessofthebiasing po- larity are anessential partof justabouteverymodern device structure.Althoughmetal- semiconductor combinations whereΦM<Φ(n-type)and ΦM>Φs(p-type)ideallyyield ohmiccontacts,itfollows fromthebarrierheightcomments in thepreceding subsection that thenotedmetal-semiconductorcombinationscouldproducerectifying contactsin practice. As a case in point, because the surface Fermileveltends to be pinnedat E-2EG/3in GaAs,the depositionofany metal on n-type GaAs forms a barrier-type contact.Itisreasonablethen to ask,“Howareohmiccontactsachieved in practice? Ohmic contacts are usuallyproduced in practice by heavily doping the surface region ofthe semiconductor immediately beneath the contact.In Si processing,forexample,an n+on n region as shown in Fig.14.10(a) would becreated priorto depositionof the metal. The reason this procedure leads to a low-impedancecontact can be explained with theaid ofFig.14.10(b).Theequilibriumbarrier heightis tofirstorderunaffected byanincrease in the semiconductor doping. However,the depletion width,andhence the widthofthe barrier, systematically decrease with increased semiconductor doping. When the semicon- ductor dopingexceeds~107/cmsignificanttunnelingcantakeplacethroughthethin upperportionofthebaier.Fordopingsexceedng10/cmtheentirebarrierbecomes so narrow thateven low energy majority carriers can readily transfer betweenthe semicon- ductor andmetalvia the tunnelingprocess.Inotherwords,althoughthebarrierexists,it effectivelybecomestransparentto carrierflow whenthecontact is formedonaheavily doped semiconductor.

--- Page 522 --- MS CONTACTS AND SCHOTTKY DIODES 499 metal SiO_2} $n$ n-Si (a) Low doping Moderate doping High doping (b) Figure14.10Ohmiccontactformation.(a）Heavy dopingof the semiconductor beneaththeMS contacttofacilitateohmiccontactformation.(b)Emissioncurrentsacrossabarrier-typecontactasa functionofdoping.Theemission is shown varying from solelythermionic emission at low semicon- ductordopingstopredominantlyfieldemission(tunneling throughthe barrier)athighsemiconductor dopings. Whereas heavy doping of the underlying semiconductor is akeystep inohmic contact formation,it is seldomtheentire story.Annealingorheatingthe device structure isrou- tinely necessary to minimize thecontactresistance.Aluminum is themost widely used contacting metal in thefabricationof discreteSidevices.Heatingto approximately475C for a few minutes in a nitrogen atmosphere allows the aluminum to penetrate the native oxide layeronthe Si surfaceandfacilitates a certainamountofbeneficialAl-Si interdiffu- sion.In contacts formed over shallow p+-n or n-p junctions, however,nonuniform inter- diffusionoverthecontactareacan lead to penetrationof thealuminum through the junction and junction shorting.Pictured schematically in Fig.14.11(a),the nonuniform penetration of Alinto theSi is calledspiking.AsmallpercentageofSi is sometimesaddedtothe deposited Alfilm to inhibitSidiffusionfromthesubstrate andtheresultant spiking. Anotherconsideration is thestabilityoftheohmiccontact during subsequent process- ing.Aluminum is not an acceptable contact material in complex integrated circuits where additional processing at temperatures in excess of 500C must be performed after contact formation.Rather,hightemperaturestability is achieved by employing themetal-likesilli- con compounds (silicides)formed withmembers of the refractory metal family(Mo,Ta, Ti,W).Titanium silicide,TiSi2,is presently in widespread use as acontactmaterial.ATi film depositedovertheSicontactarea is convertedtoTiSi byheating inan inertatmo- sphere. Since Si is consumed in theprocess,the silicide-silicon interface moves a short

--- Page 523 --- 500 BJTs ANDOTHERJUNCTIONDEVICES Al $Si{O}_2$ Si n (a) TiNTiTiNAI To Metal other “stack” W TiN devices (tungsten Insulator Insulator “plug”) TiSi2 n{2}} Si substrate (b) Figure14.11（a）VisualizationofspikingbeneathanAl-Sicontact.（bIllustrationofcontactand interconnect metallization representative of that found in modern ICs. distance intothe Si,thereby eliminating surface defectsand minimizing contamination. Thenetresult is a clean,planar,temperature-stable,ohmiccontact.Figure14.11(b)illus- trates the TiSi2contactandinterconnectmetallizationrepresentativeofthatfoundin modern ICs. 14.4SUMMARY The physical properties of an ideal MScontact were specified and the methodology was presented forconstructing the energy band diagram characterizing a given metal-semicon- ductor system.Energy band diagrams corresponding to different combinations of the ΦM M and Φs workfunctions were employed to probe the basic nature of the contacts.Ideally,the combinations Φs(p-type)>ΦM>Φs(n-type)and Φ(p-type)<ΦM<Φs(n-type) were concluded to yield rectifying and ohmiccontacts,respectively.Operation of the rec- tifying contact,called a Schottky diode or MS diode whenutilized as a device,was next examined in greater detail.The device electrostatics and reverse-bias a.c.response of the MS diode were found to beall but identicalto thatof a comparably doped p+-norn+-p junction diode.Differing from the pn junction diode,the thermionicemission currentas- sociated withmajority carrier injectionoverthesurface barrier was identified as the domi- nant d.c.currentcomponent.Because the diffusioncurrent is typically smallbycompari-

--- Page 524 --- MS CONTACTS AND SCHOTTKY DIODES 501 son,therelatedminoritycarrierinjectionandstoragearelikewisesmall.Asconsequence, the MSdiode doesnotexhibitasignificant diffusionadmittanceand switches between the on-andoff-statesatanextremelyrapid rate.Althoughthebasic descriptionof deviceop- eration is unaffected,theΦBbarrier height inactualSchotky diodes is seldomequaltothe ideal device value.Moreover, theohmicor rectifying natureof anMScontact maybe differentfromthatexpected fromideal-contactconsiderations.Anohmiccontact is usually produced in practice by heavilydoping thesurface regionof the semiconductor immedi- atelybeneaththecontact. PROBLEMS CHAPTER14 PROBLEMINFORMATIONTABLE Complete Diffi culty Suggested Short Problem After Level Point Weighting Description 14.1 14.4 1 10 (1 each part) Quick quiz 14.2 14.1 2 10 (5 each part)/comb. Draw/use band diagram 14.3 14.2.1 2 8 (2 each part) Parameter computation 14.4 ” 2 8 W versus ND plot 14.5 ” 4 30(a-18,b-8, c-4) Autodraw band diagram 14.6 14.2.2 2 8 Confirm Eq.(14.18) 14.7 ” 2-3 15(a-4, b-5, c-4,d-2) MS photodiode 14.8 ” 1 8 Effect of Rson I-V 14.9 ” 2 8 Injection ratio 14.10 ” 3 15 (a-10,b-4, c-1) Schottky barrier lowering 14.11 14.2.3 2 12 C-V data analysis 14.12 ” 4 20(a-12, b-6, c-2) Linearly graded MS diode 14.1 Quick Quiz (a)Whatare thedifferences between an MSdiode,a Schottky diode,and ahotcarrier diode? (b） AnidealMScontactisformedbetweenametalandasemiconductor withΦM=. Under whatconditions willthecontactbeohmic-like,andunder whatconditions will the contact be rectifying? (c) Thesolutionsfortheelectrostatic variables ρ,E,andVonthe lightlydoped sideofa p+-norn+-p junctiondiodeandinsidethesemiconductorofacomparablydoped MSdiodeareallbutidentical.Arethereanysignificantdiferencesintheelectrostatic formulations? (d) Name the dominant current component in MS diodes.

--- Page 525 --- 502 BJTs ANDOTHERJUNCTIONDEVICES (e) Somehow the IMo→s componentof the current in an MS(n-type) diode was determined withoutadetailedmathematicalanalysisofelectroninjectionfromthemetalintothe semiconductor. How was this accomplished? (f) ExplainwhyMSdiodes do notexhibita diffusion capacitanceor diffusioncon- ductance. (g) Explain whyMSdiodes switchveryrapidlyfromtheforward-biason-statetothe reverse-bias off-state. (h)What is a “Schottky diode clamp”? (i） Describetheusualprocedurefollowedinformingpracticalohmiccontacts. (j)What is “spiking”? 14.2AnumberofidealMScontactsareformedonGeSi,andGaAssubstratesmaintained atroomtemperature.ForoneormoreoftheMSparametercombinations listedbelow, (a)Draw theequilibriumenergybanddiagramcharacterizingtheidealMScontact. (b) PresentanargumentsimilartotheonesummarizedinFg.14.3toconfirmtheohmic or rectifying nature of the contact. combinationA:ΦM=4.75eV,χ(Ge)=4.00eV,ND=1016/cm3. combinationB：ΦM=4.75eV,χ（Ge）=4.00eV，N=101/cm³3. combinationC:ΦM=4.00eV,χ(Si）=4.03eV,ND=101/cm3. combination D:ΦM=4.25eV,χ(Si）=4.03eV,NA=1016/cm3. combinationE:ΦM=4.75eV,χ(GaAs)=4.07eV,ND=1016/cm3. combinationF:ΦM=4.75eV,(GaAs）=4.07eV,NA=107/cm3. 14.3Anidealrectifyingcontactisformedbydepositinggold（ΦM=5.10eV)onanND= 101/cmdopedsiliconsubstratemaintainedatroomtemperature.Calculate (a) Φ_B, (b) V_{i (c)Wunder equilibrium conditions, (d)max inthesemiconductorunderequilibriumconditions. 14.4Construct a plotof the equilibrium depletion width versus the NDdoping concenra- tion insiliconMSdiodesmaintained atT=300K.Vary NDover the range104/cm≤ N≤107/cmandincludecurvescorrespondingtoB=0.5eV0.6eV,and0.7eV. 14.5（a）Paralleling Exercise 5.4,constructaMATLAB(computer)programthatdraws the equilibrium energyband diagram for a nondegenerately ND-doped siliconMSdiode maintained atroom temperature.Let B and the NDdopingofthe semiconductor be input parameters.Be careful to exclude parameter combinations that would yield anon- rectifying contact. (b)Generalizethepart(a)program sothat italsodrawstheequilibriumenergyband dia- gram for NA doped silicon diodes.

--- Page 526 --- MS CONTACTS AND SCHOTTKY DIODES 503 (c) Revisethe(a)/(b)programssotheycanbeusedtogeneratetheequilibriumenergy band diagrams for GaAs MS diodes. 14.6Fillinthemissingsteps inthederivationofEq.(14.18).Accepting Eqs.(14.16)and (14.17) to be accurate,performandrecord themathematicalmanipulationsleadingto Eq.(14.18). 14.7MSPhotodiode AnND-dopedsiliconMSdiodeisilluminatedtherebygeneratingelectron-holepairsinside the semiconductor. (a)Ifthediodeterminalsaretakentobeshort-circuited,drawtheenergybanddiagram forthe device and picture what happens to the photogenerated carriers created inthe semiconductor nearthe MS interface.Withthecurrentand voltage polarities as speci- fied in Fig.14.3(a),what is the polarityof theshort-circuit photocurrent? (b)Ifthe diodeterminals aretakentobeopen-circuited,drawtheenergybanddiagram characterizing the illuminated device.Rememberthetotalcurrentmustbe identically zero underopen-circuit conditions. Explainhow you arrived at the form of your diagram. (c) Assumethelightisuniformlabsorbedthroughoutthesemiconductorproducing a photogenerationrateofGLelectron-holepairspercm-secandgivingrisetolow-level injection.Foranarbitraryappliedbiasandollowinthesimlifiedprocedureoutlind in Subsection9.2.1,derive anexpressionforthephotocurrent(IL)flowing in theMS photodiode. ited by an illuminatedMSphotodiode.Is your sketchconsistent withtheanswersto parts (a) and (b)? 14.8Thequasineutral portionofthe semiconductorandtheohmicbackcontactof Schottky diodes introduce aresistance Rsin series withthecurrentflowing acrossthe rectifyingMSjunction.Graphicallyillustatetheefectoftheseriesesistanceonthedode I-Vcharacteristic.TakingI=10-A,constructasemilogplotoftheforward-biasI-V characteristics whenR=0,0.1 Ω,1.0Ω,and10 Ω.Limit yourplotted outputto 0≤VA≤0.6Vand10-9A≤I≤10-1A. 14.9Theminority-carrierinjectionratioisoftencitedasaquantityof interest incharac- terizing MSdiodes.Bydefinition,it isthe numberofminoritycarriersinjected intothe semiconductor permajoritycarrier injectedfromthe semiconductor into themetal when the deviceisforwardbiased.Mathematically,theratioisjustIDIFFITE.DIFFandITEare respectively the diffusionand thermionicemission currents flowing in the MSdiode.Esti- mate theminority carrier injection ratio in a silicon MSdiode where A*=140 amps/cm2- K²2，ΦB=0.72eV,ND=1016/cm,τp=10-6sec,andT=300K.IDIFFin thegiven diodemaybeequatedtotheIDIFFlowingina-nstepjunctiondiodewithequivalent P device parameters.

--- Page 527 --- 504 BJTs AND OTHER JUNCTION DEVICES ●14.10In this problem we wishto examine the magnitude and effectof Schottkybarrier lowering. Parameters willbeemployedsimilar to those of theMS diode yielding the Fig.14.6 characteristics. (a)GivenasiliconMSdiodewithND=10/cmoperatingatroomtemperature,compute and plot △ΦB versus VAfor-50V≤VA≤0. (b）ComputeandplotI（VA）/（VA=0)versus VAfor-50V≤VA≤0. (c)Comment on your results. ●14.11Asubsetof the1/C²versus VAdataplotted in Fig.14.8(b)isreproduced in Table P14.11.Analyze the data employing a least squares fit to determine V,ND,and ΦB.Assume A=1.5× 10-3cm.Compare your results with the approximate values obtained in Exercise 14.4. Table P14.11 -VA(volts) 1/C²}（1^²/farad²) 1.09 0.953 2.08 1.494 3.07 2.035 4.06 2.579 5.05 3.125 6.04 3.673 7.03 4.217 8.02 4.763 9.01 5.320 10.00 5.890 14.12ThedopingprofileinsidethesemiconductorcomponentofanMSdiodeis linearly graded; i.e.,ND(x)=ax. (a)Derive solutions for ρ, , V, and Winside the semiconductor. (b) Indicate how Vi is to be determined and computed. (c)Establish an expression for the junction(depletionregion)capacitance.

--- Page 528 --- R2 Part I SUPPLEMENTANDREVIEW ALTERNATIVE/SUPPLEMENTALREADINGLIST Type Author(s) Level Relevant Chapters (A-Alt., S-Supp.) General References Streetman A Undergraduate 5-7,11 Neamen A Undergraduate 7-10 Tyagi A/S Advanced Undergrad 6-10,12-14,18 Introductory Grad For Selected Topics R2 Navon A Undergraduate 12 (PNPN) Sah S Undergraduate to 7(BJT, HBT and Graduate (variable) PNPN) Yang A/S Undergraduate 3 (pn electrostatics) 6 (BJT technology) 7 (MS diodes) (1)D.H.Navon,SemiconductorMicrodevices and Materials,Holt,Rinehartand Winston, New York,© 1986. (2) D.A.Neamen,Semiconductor Physicsand Devices,Basic Principles,Irwin,Home- wood,IL,© 1992. (3)C.T.Sah,FundamentalsofSolid-StateElectronics,WorldScientific,Singapore, ©1991. (4) B.G.SteetmanSolidtateElectronicDevices4theditionPrenticeHall,Enlewood Cliffs,NJ,©1995. 505

--- Page 529 --- 506 JUNCTION DEVICES (5)M.S.Tyagi,IntroductiontoSemiconductorMaterialsandDevices,JohnWiley & Sons, NewYork, © 1991. (6) E.S.Yang,Microelectronic Devices,McGraw Hill,NewYork,©1988. FIGURESOURCES/CITEDREFERENCES (1) S.M.Sze,PhysicsofSemiconductor Devices,2ndedition,JohnWiley&Sons,Nw York, © 1981. (2) C.T.Sah,R.H.Noyce,and W.Shockley,“Carrier Generationand Recombinationin P-NJuconNuonCcicocRE4,2t.. (3)D.K.SchroderSemiconductoraterialandDeviceCharacterizationJohnWlend Sons, New York, © 1990. (4)R.H.Kingston,“SwitchingTimesinJunctionDiodesandJunctionTransistors,“Pro- ceedings IRE,42,829-834(1954). (5) R.H.Deanand C.J.Neuse,“ARefinedStep-Recovery Techniquefor MeasuringMi- norityCarrierLifetimesandRelatedParameters inAsymmetricp-nJunctionDiodes R2 IEEETransactionsonElectron Devices,ED-18,151-158(Mach1971. (6)E.S.Yang,Microelectronic Devices,McGraw Hill,NewYork,©198;pae383. (7) M.A.Green,A.W.Blakers, J.Zhao,A.M.Milne,A.Wang,and X.Dai,“Character- izationof23-PercentEfficientSiliconSolarCells”IEEETransactionsonElectronDe- vices, 37,331-336 (Feb. 1990). (8) Photovoltaics:ProgramOverview FiscalYear1992,report produced by theNationl Renewable EnergyLaboratory,DOE/CH10093-190DE93000055,March193. (9)M.G.Craford,“LEDsChallengetheIncandescents,”IEEECircuitsandDevices,8, 24-29 (Sept. 1992). (10)M.G.Craford,“Recent Developments inLight-Emitting-Diode Technology,”IEEE Transactions onElectron Devices,ED-24,935-943(July1977). BiCMOS Technology,SolidState Technology,35,33(Oct.1992).Also see theoher articles in the special serieson BiCMOSProcessing,editedby D.Gupta and M.T.Bohr, started in Solid State Technology, June1992.

--- Page 530 --- SUPPLEMENT ANDREVIEW 507 (12)M.Lundstrom,“III-VHeterojunctionBipolarTransistors,“Chapter1inHeterojunc- tion TransistorsandSmallSizeEfectsin Devices,editedbyM.WillanderStudentliteratur and Chartwell Bratt, © 1992. (13)I.R.C.Post,P.Ashburn,andG.R.Wolstenholme,“PolysiliconEmittersforBipola Transistors:AReviewand Re-EvaluationofTheoryandExperiment,IEEETransactions on Electron Devices,39,1717 (July1992). (14)ForinformationaboutMBE,consultthedevicefabricationreferences listedintheR1 mini-chapter. (15)S.TiwariandD.J.Frank,“EmpiricalFittoBandDiscontinuitiesandBarrierHeights (16)D.H.Navon,SemiconductorMicrodevicesandMaterials,Holt,Rinehartand Win- ston, NewYork,© 1986; pp.410,411. (17) See,forexample,S.M.Sze,Physicsof Semiconductor Devices,2ndedition,John Wiley and Sons,New York,©1981; pp.255,256. REVIEWLISTOFTERMS R2 Defining thefollowing terms using yourown wordsprovides arapidreviewofthe PartII material. (1) wide-base diode (19) profiling (2) narrow-base diode (20) junction capacitance (3) ideal diode (21) diffusion admittance (4) law of the junction (22) turn-off transient (5) quasineutral region (23) storage delay time (6) breakdown (24) reverse recovery time (7) impact ionization (25) reverse injection (8) avalanching (26) step recovery diode (9) multiplication factor (27) photodetector (10) Zener process (28) solar cell (11) tunneling (29) LED (12) conductivity modulation (30) p-i-n diode punch-through (in narrow-base diode) (31) avalanche photodiode (14) high-level injection (32) fill factor (15) quasistatic (33) shadowing (in solar cells) (16) hyperabrupt (34) texturing (in solar cells) (17) varactor (35) concentrator solar cells (18) abrupt (36) isoelectronic trap

--- Page 531 --- 508 JUNCTIONDEVICES (37) total internal reflection (69) heterojunction (38) BJT (70) band alignment (39) emitter, base,collector (71) lattice-matched (40) common base (72) graded junction (in HBT) (41) common emitter (73) Hybrid-Pi equivalent circuit (42) active mode (74) transconductance (43) saturation mode (75) base transit time (44) cutoff mode (76) rise, storage delay,and fall times (45) inverted mode (77) Schottky diode clamp (46) buried layer (78) thyristors (47) quasineutral base width (79) SCR (48) emitter efficiency (80) anode,cathode, gate (49) base transport factor (81)blocking voltages (50) common base d.c.gain (82) two-transistor model (51) common emitter d.c.gain (83) GTO SCR (52) performance parameters (84) LAS (53) Ebers-Moll equations, model (85) dildt burnout (54) forward gain (86) duldt effect (55) base-widthmodulation (87) DIAC,TRIAC (56) Early effect (88) PUT (57) punch-through (in BJT) (89) Schottky diode (58) regenerative (90) vacuum level R2 (59) phototransistor (91) metal workfunction (60) intrinsic transistor (92) electron affi nity (61) base series resistance (93) thermionic emission (62) current crowding (94) Richardson’s constant (63) graded base (in BJT) (95) Schottky barrier lowering (64) Gummel plot (96) hot carrier diode (65) BiCMOS (97) Fermi-level pinning (66) shallow emitter (98) field emission (67) polysilicon emitter (99) Al spiking (68) HBT (100) silicide PARTII—REVIEWPROBLEMSETSANDANSWERS The following problem sets were designed assuming a knowledge—at times an integrated knowledge—of the subject matter in Chapters 5-11of Part II. The sets could serve as a review or as a means of evaluating the readers masteryof the subject. Problem Set Ais adaptedfrom a one-hour”open-book”examination; Problem Sets BandC are combina- tions of select problemsfrom”closed-book”examinations. Theanswers to ProblemSets A and Bare includedattheendofthis section.Answers are not providedforProblem SetC so that it can be used as a homework set nintegrated-knowledge-typequestionsonthe pn junction diode.

--- Page 532 --- SUPPLEMENTANDREVIEW 509 Problem Set A Problem A1 A pnjunctionhas thedopingprofile sketchedbelow.Throughoutthisproblem,asumethe carrier concentrations maybe neglected (n=0,p=0)in the0≤x≤x;regionofthe diode. $ND-NAA$N{}$0$xi$\to x$-N{BB(a) What isthe built-in voltage acrossthejunction?Justifyyouranswer. (b) Invoking the depletionapproximation,make asketchofthe charge density inside the diode.Label significantρ and x values. (c) Obtainananalyticalsolutionfortheelectricfild,()atllpointsinsidethedepe- R2 tion region（-xp≤x≤n).Show all work and make a sketch of the deduced (x) versus x. (d) Ina standard pn stepjunction NAxp=NDn.How aren andxrelated here? (e) Draw theenergybanddiagramforthediodeunderequilibriumconditions.Clearly identify the points=0and=on your diagram.Also indicate howyour diagram differs from that of a simple pn step junction where NA=ND=NB Problem A2 Twosiliconp+-nstepjunctiondiodesmaintainedat300Karephysicallyidenticalexcept forthen-sidedoping.Indiode#1,ND=104/cm3;indiode#2,N=101/cm3.Compare the operationof the two diodes by answering the questions thatfollow. Diode #1 Diode #2 VAP+$NVAP+$ N ∠ND=10{4/cm³3$ ND=1016/cm³ (a)Which diode willexhibitthe largerbuilt-in voltage(V)?Explain. (b) Which diode will exhibit the larger breakdown voltage(VBR)?Explain.

--- Page 533 --- 510 JUNCTIONDEVICES (c) Whichdiode willexhibitthelargerjunctioncapacitance（)atagivenreversebias voltage when|VA|>V?Explain. (d)Ifthediodesare assumedto beideal andreverse biased ataVAl>few volts,which diode will support the largerI?Exlain. (e) Ifthediodesarenotassumedtobeideal,whichdiodewillsupportthelargerwhen reverse biasedataVA|>few volts?Explain. (f)Which diode willexhibit the larger diffusion capacitance(CD)ata given applied for- Explain. (g)Whichdiodewillexibitthelarerstoragedelaytime)undertransientconditionsif the IF/IR ratio is the same? Explain. Problem A3 (a)The diagram below pictures the major carrier activity in a pnp BJT underactive mode biasing.Constructa similar diagrampicturing themajor carrieractivity inthesame pnp BJT under inverted mode biasing.(Include a few words of explanation as neces- sary to forestall a misinterpretation of your diagram.) (b)ThecturedBJTiscidntllconnctedupbackwrdothtthecollectorfunctions as theemitterandtheemitterfunctionsas thecollector.Inthebackwardconnection R2 with VCB> 0 and V <0the device exhibits a lower gain and is more sensitive to EB base-width modulation.(i) Explain why the backward connection leads to a lower gain. (ii) Explain whythe backwardconnectionleadstoa greater sensitivityto base-width modulation. -EcEi$EF$ -E_v$ Problem Set B Problem B1 Apnjunctiondiodehasthedopingprofileshowninthefollowingsketch.Mathematically, ND-NA=N0[exp(αx)-1],where N0andα are constants.

--- Page 534 --- SUPPLEMENTANDREVIEW 511 $ND-NA$ →x -N_0}$(a)Giveaconcisestatementofthedepletionapproximation.(b)Invokingthedepletionapproximation,makeasketchofthechargedensityinsidethediode.(c)Establishanexpressionfortheelectricfield(),insidethedepletionregion.(d)Indicatehowyouwouldcompletetheelectrostaticdevelopmenttoeventuallyobtainanexpressionforthedepletionwidth,W.Beasspecificaspossibleabouttheequationstobesolvedandtheboundaryconditionstobeemployed.Do{n}^{\prime }twastetimeactuallyperformingthemathematicalmanipulations.Organizeyouranswerintosteps—step1,step2,etc.ProblemB2R2TheenergybanddiagramgivenbelowcharacterizesaSistepjunctiondiodemaintainedatroomtemperature.NotethatE（-\infty )=(+\infty ).Also,+x=2\times 10-4cm,A==（ps-）-/u\epsilon =（p-d）-=0=="u\epsilon -459c{m}^2/V-sec,K=11.8,and\epsilon 0=8.85\times 10-14F/cm.$EG/4 EFp 下 Ec ↑ -EF{n E{} E_{{} E_{v}$\to x$- {$0$x_{}$ (a) What isthe magnitudeofthereverse-bias voltage(VA)beingappliedtothediode? Explain how you arrived at your answer. (b) Determine Vi,the built-in voltage. (c) Compute the recombination-generation current flowing through the diode at thepic- tured bias point. (d)Computethediffusioncurrentflowingthroughthediodeatthepicturedbias point.

--- Page 535 --- 512 JUNCTION DEVICES (e) WhatwillbethejunctioncapacitanceC)exhibitedbythediodeatthepicturedbias point? (f)Onalinearscale,sketchtheminoritycarrierconcentrationsversuspositionforthe x≤-xp andx≥n portions of the diode. (g)WouldyouexpectthedevicetoexhibitsignificantdiffusioncapacitanceCD)atthe applied bias point? Explain. (h)Ifthediode werepulsedfromthe picturedbiaspointtoalargerreversebiasat=0, wouldyouexpecttoobserveacurrenttransientcharacterizedbyastorage delaytime t?Explain. ProblemB3 Two Sipnptransistors,BJT#1andBJT#2,areidenticalexcept WB>WB2.NE≥NB> NCandW≤LBinbothtransistors.Underthesameactivemodebiasingconditions,which transistor will exhibit (a) the larger emitter efficiency? Explain (b) the larger base transport factor? Explain. (c) the larger βdc?Explain. (d) the greater sensitivity to base-width modulation? Explain. R2 (e) the larger punch-through voltage? Explain. Ifthemaximumoutputcurrentisassumedtobelimitedbycariermultiplicationand- lanching, which transistor will exhibit (f) the larger VCBo?Explain. (g) the larger VCE0? Explain. Problem Set C Problem C1 (a)Whichofthefollowing assumptions is notinvokedin deriving theidealdiode equation? (i) No recombination-generation in the depletion region. (ii) Low-level injection. iii)Narrow-basediode;iethenandpquasineutralwidthsaremuchlessthathe respective minority carrier diffusion lengths. (iv)No”other”process;ie.nophotogeneration,avalanching,tunneling,et. (b) Underreversebiasingandsmallforwardbiasingthedominantcurrentcomponent in most Si pn junction diodes maintained atroom temperature is whichof the following? (i) The diffusion current. (ii)The R-G current. (iii) The ideal-diode current. (iv) The drift current.

--- Page 536 --- SUPPLEMENTAND REVIEW 513 (c) Which of the following statements is incorrect? (i) pn junction breakdown is a reversible process. (ii)FortheZenerprocesstooccur inapn junctiondiode,thedepletion widthmust be very narrow (≤10-6 cm). (iii) The avalanche breakdown voltage varies roughly as the inverse of the doping concentration on the lightly doped sideof p+-n and n+-p junctions. (iv)Inoemidtoomtemtureavlncigitheomiatess causing breakdown if BR ≤4.5V. (d) Whichofthefollowingstatementsaboutthejunctioncapacitance（C)iscorrect? i)ThemumbleClurR (iii)Cvanishes under forward biasing. (iv)Cisaitedphysicllwituctutionsintheminoritcarieconcnrations at the edges of the depletion region. ProblemC2 Minoritycarrierconcentration versus position plotsand sketchesareoftenusedto describe the situation insidesemiconductor devices.A linear plotof the minority carrierconcentra- tion on the n-sideof two ideal p+-n diodesmaintained atroom temperature is pictured below.The n-side doping (ND)and thecross-sectionalarea(A)are the same in bothdiodes. Assume low-level injection conditions prevail. R2 $Pn$ A Diode A Diode B $n_2$ $\mathring{D}$ →x x_} (a) Thediodesare[(i)forward biased,(ii)ero biased,(i)reverse biased]. (b) The magnitudeof the biasappliedto Diode Bis [(i)larger than,(ii)the same as, (iii) smallerthan] the magnitudeof the bias appliedto Diode A. (c) Themagnitudeofthed.c.currenfowinthroughDiodeBisisignificantly largerthan,(ii)roughlythesameas(iii)significantlysmallerthan]themagnitudeof the d.c. current flowing through Diode A. (d) Thebreakdownvoltage（BR)ofDiodeBis[(i)significantlylargerthan,ii)roughly the same as,(iii)significantly less than] the breakdown voltageof Diode A.

--- Page 537 --- 514 JUNCTIONDEVICES (e)DiodesAandBare testedinthe same switching circuit.I/Ris thesame forboth diodes.Whichdiode willexhibitthe larger storage delaytime(larger)?[(i)DiodeA; (ii) Diode B;(iii)will beessentiallythe same for both diodes.] Problem C3 The steady-statecarrierconcentrationsinsideapnjunctiondiodemaintainedatroomtem- perature are as pictured below. n or p (log scale) ↑ ppnn}pnn{p$R2\to x$- {x_{ }$ (a) Isthe diodeforward orreverse biased?Explain how you arrived at your answer. (b)Dolow-levelinjectionconditions prevail insidethediode?Explainhowyou arrivedat your answer. (c)Qualitatively,whatisthephysicalrelationshipbetweenthepile-uporstoreofminority carriers near the depletionregion edges and the diffusioncapacitance(CD)? (d)Qualitativelwhatisthephysicalrelationshipbetweenthepile-uporstoreomit carriersnearthedepletionregionedgesandthe storagedelaytime()observed during the turn-off transient? ProblemC4 Thereverse-bias current-voltage(I-A)junctioncapacitance(C-VA),and turn-off tran- sient (i-t)characteristics derivedfromap+-nSistep junction diode maintained atroom temperatureare sketchedto theright.Afterreproducing thefigures,answer the following questions by addinga dashedlineto theappropriatecharacteristic.Notethatan answerof no effect (adashed linethesameasthegivencharacteristic)is possible.In such cases write no effect. (a-c) Roughly indicatehowtheI-VA,C-VA,andi-t characteristics aremodifiedif the n-side doping(ND)is increased by a factorof2.Allother parameters remain the same.

--- Page 538 --- SUPPLEMENTANDREVIEW 515 (d-f）RoughlyindicatehowtheI-VA,C-VA,anditcharacteristicsaremodifiedifthe minoritcarrierlifetimeonthenside)andtheefctivedepletion-rgiongenea- tionlifetime()areincreasedbyafactorof2.Allotherparametersremaintesam. (a) (d) 1 ↑ →VA →VA (b) (e) 5 5 ↑ R2 VA (c) (f) i(t) i(t) 1 $IF$ $IF$ t 1 -q} -q}

--- Page 539 --- 516 JUNCTIONDEVICES Answers—Set A Problem A1 (a） ParallelingthederivationinSubsection5.1.4,onecanwriteVbi=(kT/q）ln[n(n)/ n（-xp）].Heren（n）=n(（∞)）=NBandn（-p）=n（-∞）=n²/NB.Thus kT Nr{{} 2kT $V_bi

ln $nra2$ ln q q (b) ρ ↑ $qNB$ -x_{po →x $x_i$ $xn$ -q_{BB R2 $-q{N}_{BB}$ (c) K_{ . ..-xp≤x≤0 d8 ρ 0 …0≤x≤x; dx K{ε qN_BB} …$x_1\le x\le xn$ $K\epsilon 0$ {x qN_B}qN_B}$\leftarrow ，xp8(x)=$ (x+x) . ..-x ≤x≤0 -xpK{0K{0qN_}$ 8(x) = constant =(0)

K{sε{ x_{p …0≤x≤x; $(xn=0$ \Nx} $q{N}_B$ d{^{= dx^{’ 8(x)= (x_ }x) ...x₁≤x≤x{nJ8(x) K{sε0 K{0 $Jx$

--- Page 540 --- SUPPLEMENTANDREVIEW 517 8 $-p$ 0 $x_i$ $xn$ x (d) The(+) and(-)charge areas on the part (b) ρ-plot must be equal, or (-qNB）(-p)）=qBxn-x₁）⇒xp=_{n-xi Alternatively,the&-field mustbecontinuousat=x,giving qB} qNBB} K{ x_{p

K_{0 (x_n -x1) x{-_ (e) R2 Ec EF $Ei$ $Ev$ 0 $x_i$ Aconstantenergybandslopeorstraight lines in the0≤x≤xregionis theonly differ- ence.Notethat(E-F)|p-side=(EF-E)n-side sinceNA=ND=B Problem A2 (a) Diode #2 bi=(1/q)[(E-|p-side+F-E|n-side].（E-)|-sideisthesameforboth diodes.(EF-En-side=kTln(ND/n;) is largerfor the heavier doped diode.Thus i2i.NoethatpimliesdgeneorerhvyongThusthend relationship,Vi=（kT/q）ln(NAND/n²)cannotbeusedtocompute Vbi.

--- Page 541 --- 518 JUNCTIONDEVICES (b) Diode #1 VBR>1000VifND=1014/cm3,whileBR≈60VifN=10/cm. (c) Diode #2 $KS\epsilon OA$C__} = W For a p+-n step junction 2K_{sε 2K{ε$1/2W=(Vbi-VA)$ (-VA) qNDD$qNDD$ wherethelatterformof theWequationfollowsfromthefactthatVA>Vb.Thus, W₁>W₂andC_2>C. (d) Diode #1 For anideal p+-n diodereverse biasedtoVA>few volts,we can write D{n{2} -0 =-q \rD} R2 Now1/ND1 >1/ND2.AlsO D{ D{_$D$ KT μp d7 V V qp Sincepdecreases withincreasing ND,Dp/L>DP2/L2.Bothofthecitedfactors in the I expression are such as to make o1>102. (e) Diode #1 In areal Si diodemaintainedat300Kthereverse current is typicallydominatedby the recombination-generation current component. n_iR-G=-qA W …given -VA >few volts 2 Fromthe Eq.(6.44）definitionofτ0weinferτ0α1/N（bothnandτare proportional to 1/N).Only the doping is specified as beingdifferent in the two diodes,not N. Thus 0is the same forboth diodes.However,W>W2from part (c).Consequently, |R-G>|R-G{2

--- Page 542 --- SUPPLEMENTANDREVIEW 519 (f) Diode #1 Assuming thediodesareforward biased into theidealregionofoperation,CDαG0 (I+0)α0.G0isthediodelow-frequencyconductance.However,01>from part (d). Thus C1>D2. (g) Same for both diodes Examining either theapproximate solution[Eq.(8.8)]or themoreaccurate solution [Eq.（8.9)]for,oneconcludes1=2iftheI/Rratioandτp1/Narethesame forthetwodiodes.TheI/Rratiois noted tobethe same in theproblem statement.N and therefore τ are alsoinferredtobethe same inthetwo diodes,sinceonly theN doping is specified as being different. Problem A3 (a) Ec Ei R2 EF Ev 。 oo oo (b)(i)Theefficiencyoftheinjectingcollectorisfarbelowthatoftheemitterand dc 二 γa is thereby degraded.The E-B junction has NE≥NB,making hole injection into thebasemuch greater than electroninjectionfrom thebase into theemitter underactive modebiasing.Ontheother hand,N<NB,givingrise to significant electroninjectionfromthe base into thecollectorunder“backward”(inverted) operation. (ii) When usedasanamplifier,thelarger depletionregionis associatedwiththe reverse-biased junction. WiththeC-B junction reverse biased underactive mode operation, most of the depletion region extends into the collector because NB>N.However,whentheEBjunctionisreversebiased,asisthecaseunder “backward”operation,mostofthe E-B depletionregion extends into thebase because NE≥NB.Much larger variationsof Wforequivalent biasing conditions are expected under backward operation.

--- Page 543 --- 520 JUNCTIONDEVICES Answers—Set B Problem B1 (a)Inthedepletionapproximationonemakesthefollowingsimplifingassumption andnare much lessthan|N-NA|ina-x≤x≤regionaboutthe metallurgical junction.(ii)ρ=0 elsewhere. (b) ρ -x{p →x $xn$ (c) d8 qN_} (eax - 1) …-xp≤x≤x_{ dx K_{ε0 K{0 8(x) q_0} $x$ $q_o$ [e^ax{‘/a-x’}|{p R2 =，3P K_{s $(eax-1)$d^x{== K{0 Jo J-x{pq_o(eax- e-a{) -(x+x 8(x)=K{ε0$[\alpha (d)Step1:SolveforV（x)inthe-xp\le x\le ndepletionregion.dV=-8(x);V=0atx=-xdxStep2:Twoequationswithn,x,andViasunknownsareobtainedbyimposingtheboundaryconditions，（n）=0andV（xn）=Vbi-VA.OnceVisknown,thesetwoequationscanbesolvedforthexn，x,andW=+xresultingfromagivenVA.Inthisparticularproblem,thesolutionwillhavetobeob-tainednumerically.Step3:VisobtainedbyaprocedureparallelingthelinearlygradedanalysisinSubsection\mathrm{5.2.5.}ProblemB2(a)VA=-$ 1 (EFp $EFn=$ EG} =-0.56V 2q q

--- Page 544 --- SUPPLEMENT AND REVIEW 521 $E_G$ =0.56V 2q b (C) -G=-qA =-1.6×10-10A kT (d) DN =$\mu n=(0.0259)(1352)=35.02cm2/secqLN=\surd DNn=[(35.02)(10-6)]12=5.92\times 10-3cmkTDP$ μp=(0.0259)(459)=11.89cm²/sec q LP=√Dpp=[(11.89)(10-6)]²=3.45×10-3cm NA=ND=n;eEG/4kT=(1010)(e1.12/(0.0259)1)=4.96×1014/cm (d:n}D{n{)$IDiff=-0=-A$L{}$+$L{BD}$=-(1.6\times 10-19)(10-3)[(R2451\times 10-)(40)+3.45\times 10-3=-3.02\times 10-13A$K{ε{AAKε{AA (11.8)(8.85×10-14)(10-3) (e) C_} 5.22 pF W $x_n+$ 2×10-4 (f) n or p ↑ n{p $n_{{}^2}$ $n^2$ p_{n $\text{rn}A$ $\mathring{D}$ Note: LN>LP →x - x{p $x$ (g) No.There is no storeof minority carriers adjacent to the depletion region. (h)No.Again,there is no storeofminoritycarriers adjacentto thedepletionregion.(The device is already reverse biased prior to applying the voltage pulse.)

--- Page 545 --- 522 JUNCTIONDEVICES ProblemB3 (a) BJT #2 IfWB1>WB2,thenW>Wunderthesamebiasingconditionsand 1 1 γ_{1}$二DE$N_BB}_s{{}$ <γ{

DEN_{BB}_s2}$1+1+$DB$LE$LE 1 1 (b) BJT #2 αT₁

1 2 <α2 1 (2} 2 1+ -2 L{B 1+ 2 一 LB Physically,fewercarriers are lost via recombination incrossing the narrower base. (c) BJT #2 βdc=αdc/（1-dc),weconcludedc2>βdcl1. (d) BJT #2 Changes in bias give rise to the same ΔWin the two transistors.However, with W1>W₂,ΔW/W₂>ΔW/W，making BJT 2moresensitivetobase width modulation. R2 (e) BJT #1 Because WB>WB2andchanges inbiasgiverise tothesameΔW,it will clearly take a larger appliedVCB to completelydeplete the wider BJT 1base under active mode biasing. (f) Same for land #2 VCBOis equalto theBRof theC-Bjunction.The avalanche breakdown voltage of the C-B junction shouldbethe same forbothtransistorssince the junction dopings are the same. (g) BJT #1 Accordingto Eq.(11.54),VCE0=CB0dc+1)/m,where3≤m≤6. Now,asnoted inansweringpart(c),dc<d2.Thus,CE01>CE02.

--- Page 546 --- ＰＡＲＴII FIELD EFFECT DEVICES

--- Page 547 ---

--- Page 548 --- 15 Field Effect Introduction— The J-FET and MESFET 15.1GENERALINTRODUCTION Historically,the fieldeffect phenomenon was thebasis for the first typeofsolid-statetran- sistor everproposed.Fieldeffecttransistorspredatethebipolar junctiontransistor by ap- proximately20years.Asrecorded inaseriesof patentsfiledinthe1920s and193s,J.E. Lilienfeld in theUnitedStatesandO.Heilworking in Germany independentlyconceived a transistor structure of the form shown in Fig.15.1.The device workedon the principle thata voltageappliedtothemetallicplatemodulatedtheconductanceoftheunderying semiconductor,which in turnmodulated thecurrent flowing between ohmic contacts A and B.This phenomenon,where the conductivity of a semiconductor is modulated by an elec- tricfieldapplied normalto the surfaceof the semiconductor,has been named the field effect. Theearly fieldeffecttransistor proposals wereofcourse somewhat ahead of their time. Modern-day semiconductormaterials werejustnotavailableand technological immaturity, in general,retarded the developmentoffield effect structures for many years.A practical implementation had to await the successful developmentofother solid-state devices,no- tablythebipolar junctiontransistor,inthelate194Osandearly1950s.Thefirstmodern- dayfieldeffectdevice,the junctionfieldeffecttransistor(J-FET),was proposedandana- lyzed byW.Shockley[in1952.Operational J-FETs were subsequently builtby Dacey C Aluminum A B Semiconductor (Cu2S) Metal contact Figure15.1 Idealization of the Lilienfeld transistor. 525

--- Page 549 --- 526 FIELDEFFECTDEVICES Source Z Drain p-gate 1 Depletion 2a region 1 n-channel 1 Depletion region p-gate + 三 $V_{}$ + V{ Figure15.2 Schematicdiagramofthejunctionfieldeffecttransistor.(From DaceyandRoss[3, ©1955 AT&T.Reprinted with permission.) and Ross2]in1953.As pictured schematically in Fig.15.2(which was adaptedfroman earlyDaceyandRosspublication)pnjunctionsreplacedthemetallicplateintheLilie- feld structure,the A and Bcontacts becameknown as thesourceand drain,and the field- effect electrode was namedthe gate.In theJ-FETit is thedepletionregions associated with the pn junctionsthatdirectlymodulatethesemiconductorconductivitybetweenthesource and drain contacts. A key componentofpresent-day microelectronics,the metal-oxide-semiconductor fieldeffecttransistor(MOSFET)achievedpracticalstatus during theearly1960s.The now-familiar planar versionof the structure pictured in Fig.15.3,a structure with thermally grownSiO2functioningasthegateinsulatorasurface-inversionchannelandislands dopedoppositetothesubstrateacting asthesourceand drain,wasfirstreportedbyKahng and Atalla[4]in1960.Followingintense developmentefforts,commercialMOSFETspro- duced by Fairchild Semiconductorand by RCAbecame available in late1964.Figure15.3 originallyappearedonthecoverpageofthe1964applications bulletin describingtheeary FairchildMOSFET[5]. Amajor development occurring in the latter half of the196Os was the inventionby Dennard[]of theone-transistor dynamicmemorycellused in therandomaccessmemory (DRAM).The DRAMcell(see Fig.15.4)is an integrated combinationofacharge storage element (a capacitoror pn junction)and a MOSFET utilized as a switch.A quote from a recentreview article8]best underscores the significanceof the inventionandits impact. “The1-TDRAMcell is now the mostabundantman-madeobject onthis planetearth.” The early197Os sawthe introductionofstillanother significant fieldeffectstructure— thecharge-coupled device(CCD).Physically,aCCDmight be viewed as a MOSFETwith

--- Page 550 --- FIELDEFFECTINTRODUCTION—THEJ-FETANDMESFET 527 + + DS s$Gate$SiO_{2}$SourceDrainPPNBasesubstrateFigure15.3CrosssectionofaMOSFETwithcorrectbiasingpolarity.(FromMacDougall[5].)Polysilicon2AluminumIntermediateoxidePolysiliconlGateloxidePoly-polyoxideGate2$n^{} drin Field p-silicon oxide oxide Field Capacitor Access implant transistor Figure15.4SchematiccrosssectionofaDRAMcell.Theplanar,single-capacitorcellofthetype shown was employed in16K-256Kmemories.(From Wynsand Anderson[7,©1989IEEE.) a segmented gate. As pictured in Fig.15.5,the proper applicationof biases to the CCD gates induces the systematic movement or transfer of stored charge along the surface chan- nel and into the device output. The image sensing element in thecamcorder,a combination home TVcameraandvideo recorder,is a CCDcontaining a two-dimensionalarrayof gates. Stored charge is produced in proportion to theoptical image incidenton the two- dimensional gate array. The electrical analog of theoptical image is subsequently trans- ferred to the output and converted into an electrical signal. The invention,development,and evolution of field effect devices continues to the pres-

--- Page 551 --- 528 FIELDEFFECTDEVICES (a) Input Gate Transfer electrodes Gate Output $P_{}$ $P_{}$ $P_3$ $n$ n+ p-silicon substrate $Si{O}_2$ (b) Figure15.5 Firstcharge-coupleddevicecomprising eight three-phaseelementsand input-output gates anddiodes,shown(a)in planview and(b)schematically in itscross-sectional view.(From Tompsett et al.[9]Reprinted with permission.) ent. Notably, in progressing through the198Os,complementary MOSorCMOSbecame the integrated circuittechnologyofchoice.Figure15.6showsacut-away viewof a funda- mental building block in CMOS circuitry,the CMOS inverter.The term”complementary denotes the factthat,unlikeotherMOS circuit technologies,bothn-channel (electron cur- rent carrying)MOSFETsand p-channel (holecurrent carrying)MOSFETs are fabricated on the samechip.AlthoughCMOScametothe forefront inthe198Osbecauseof its lower power dissipationandother circuit-relatedadvantages,the change was evolutionary— 1960s and was even mass produced as part ofLED watch circuitry in972. Whereas the use ofCMOSexemplifiescontinuing evolutionary changes,there is also continuing field effect device inventionand development. Themodulation dopedfield ef- fect transistor(MODFET)pictured in Fig.15.7is anexcellentexample.Although theelec-

--- Page 552 --- 529 A1 CVD $Si{O}_2$ Thermal SiO_{2}n {}n {}p{+}$ p + n p p+ n-channel MOSFET p-channel MOSFET Figure15.6 Cut-away schematic viewofa CMOSinverter.(FromFonstad[11,©1994by McGraw-Hill,Inc. Reprinted by permission of the publisher.) Source Gate Drain Dielectric passivation nGaAs Epitaxial AlGaAs layers Inplant isolation 2 DEG Substrate Undoped GaAs Semi-insulating GaAs Implant isolation Figure15.7 Perspective viewof the basic MODFETstructure.(From Drummondet al.12], © 1986 IEEE.) tron transport propertiesofGaAs are superior to silicon,GaAs technology lacks a quality homomorphic (GaorAscontaining) insulatorcomparableto SiO2.Moreover,the deposi- tion ofother insulators on GaAs has consistently failedto produce devices withacceptable electricalcharacteristics. Theemergenceofsophisticateddepositiontechniques suchas molecular beamepitaxy(MBE),however,hasmade it possible to growlayersofdifferent lattice-matched semiconductorsontopofeachother. Thealloy semiconductor AlGaAs appearing inFig.15.7hasa larger band gapthanGaAsandessentiallyassumes therole

--- Page 553 --- 530 FIELDEFFECTDEVICES of”insulator”intheMODFETstructure.TheMODFET,toutedtobethefastestthree- terminal semiconductor device,is finding increased usage in high-frequency applications. PartII,spanning fivechapters,is intendedto providean introductory descriptionof fieldeffectdevicesand deviceoperation.Thecoremembersof thefieldeffectfamilyare examined in some detail to expose thereader to relevant terms,concepts,models,and analyticalprocedures.Theremainderofthis chapter is devoted to the junctionfield effect transistor(J-FET)andthecloselyrelatedMESFET(metal-semiconductorfieldeffecttran- sistor). These devices are viewed as conceptualbridges between the junction devices in PartIIandthe“pure”fieldeffect devicesconsideredlater inPartIII.TheJ-FET/MESFET discussion,moreover, serves as a firstrun-through on terminology,analytical procedures, andthelike,therebymakingthe subsequent discussioneasier to understand.To permit flexiblecoverageof thegeneral topicarea,however,the latterchaptersare self-contained, with no specific references to the J-FET/MESFETpresentation.Chapters16and17pro- videa baselinecoverageofMOSstructuralanddevicefundamentals.Thetwo-terminal MOS-capacitoror MOS-C is thestructural heartofallMOSdevices.In Chapter16we definetheidealMOS-Cstructure,presentaqualitativeandquantitative descriptionofthe structure inthestatic state,andexaminethe capacitance-voltage characteristics derived fromthedevice.Theoperationaldescriptionandanalysisofthebasic transistorconfigu- ration,the long-channel enhancement-modeMOSFET, are presented inChapter17.Fi- nally,Chapters18and19containMOS-relatedpracticalinformation.Thenatureandeffect of deviations fromtheidealare described inChapter18,while”read-only”Chapter19 provides an introductory discussion of small-dimension effects in MOSFETs.A brief sur- vey of modified and developmental FET structures is also included in Chapter19. 15.2 J-FET 15.2.1 Introduction Asmentioned intheGeneral Introduction,the junctionfieldeffect transistor (J-FET)was first proposedand analyzedby W.Shockley in1952.Across sectionof the basic device structureisshowninFig.15.8.IntheJ-FETtheapplicationofagatevoltagevariesthep- junctiondepletionwidthsandtheassociatedelectricfieldin the directionnormaltothe G (Gate) Depletion region A p+} S N D (Source) (Drain) G Figure15.8 Cross sectionof the basic J-FET structure.

--- Page 554 --- FIELD EFFECT INTRODUCTION—THEJ-FETAND MESFET 531 S G D $Si{O}_2$ Isolation $N^{+}$ p^{+ $N$ uoisnJip p{+ p{+ N-epi p{+$ Si-substrate G Figure15.9 Perspective view ofamodern epilayer J-FET. semiconductor surface.Changes in the depletion widths,in turn,modulate the conductance between the ohmic source and drain contacts. TheJ-FETwasinitiallynamedtheunipolartransistortodistinguishitfromthebipolar junction transistor and toemphasize thatonlyone typeofcarrier was involved in theop- erationof thenew device. Specifically,for the structure pictured in Fig.15.8,normal operationofthetransistorcanbedescribedtotally intermsoftheelectrons flowinginthe n-regionfromthe sourceto the drain.The source(S)terminal gets its name from the fact thatthecarrierscontributing to thecurrentmovefromthe externalcircuitintothe semi- conductor atthis electrode. The carriers leave the semiconductor,or are”drained”from the semiconductor,at the drain (D)electrode.The gate is so named becauseof its control or gating action. The modern version of the J-FET shown in Fig.15.9,although some- what different in physicalappearance,is functionallyequivalent to theoriginal Shockley structure. 15.2.2Qualitative TheoryofOperation Toestablish the basic principlesofJ-FEToperation,we willassume standard biasing con- ditions and treat the symmetrical,somewhat idealized, Shockley structure of Fig.15.8. Given ann-typeregion between thesourceanddrain,standardoperationalconditions pre- vail in theJ-FET when the topand bottom gates are tied together,VG≤0,andV≥0,as illustrated in Fig.15.10.Note that withV≤0,the pn junctionsarealways zeroorreverse biased.Also,Vp≥0ensures that the electrons in the n-region move from the source to the drain (in agreement with the naming of theS and Dterminals).Ourapproach here willbe to systematically change the terminal voltages and examine what is happening insidethe device.

--- Page 555 --- 532 FIELDEFFECTDEVICES G S D N \fra{{≥ 0V_{}0$ 工 G Figure15.10Specificationofthedevicestructureandbiasingconditionsassumedinthequalita- tive analysis. First supposethatthegateterminalis grounded,V=0,andthedrainvoltageis increased in small steps starting fromV=0.AtVp=0（rememberVis also zero)the device isin thermalequilibriumandaboutallone seesinside the structure aresmall deple- tion regions about the topandbottom p+-n junctions (see Fig.15.1la).The depletion regions extend,ofcourse,primarily intothe lightly doped,centraln-regionofthe device. SteppingVDto smallpositivevoltages yieldsthe situationpictured inFig.15.11(b).A current,ID,beginstoflow into the drainand throughthe nondepleted n-region sandwiched between the two p+-n junctions.The nondepleted,current-carrying region,we might note, is referredtoasthechannel.ForsmallV,thechannel looksandactslikeasimpleresistor andtheresulting variationofID withVDis linear(see Fig.15.12a). WhenVDis increased above a few tenths ofa volt,the device typicallyenters a new phaseofoperation.To gaininsight into therevisedsituation,refertoFig.15.11(c),where an arbitrarilychosenpotentialof5Vis assumedto exist atthe drainterminal.Since the source is grounded,it naturallyfollowsthat somewhere inthechannelthepotentialtakes on the valuesof1,2,3,and4volts,withthe potential increasing as one progressesfom the source to the drain.The p+sides of the p+-n junctions,however,are being held at zero bias.Consequently,theapplied drainbiasleads indirectly to areverse biasingof thegate junctionsand an increase in the junction depletion widths.Moreover,the top andbottom depletionregions progressively widen in going downthe channelfrom the sourcetothe drain (seeFig.15.1ld).Stillthinkingof thechannel region (the nondepleted n-region)as a resistor,but no longer a simple resistor,one would expectthe loss of conductive volume to increase the source-to-drain resistance and reduce the ΔIDresulting from a given change indrainvoltage.This is preciselythesituation pictured inFig.15.12(b).Theslopeofthe ID-VD characteristicdecreases at larger drainbiases becauseof the channel-narrowing effect. Continuing to increasse the drain voltage obviously causes the channelto narrowmore and more,especiallynearthedrain,untileventuallythetopandbottomdepletionregions touch in the near vicinityofthedrain,as pictured inFig.15.11(e).Thecomplete depletion of thechannel,touchingof thetopandbottom depletionregions,is an important special conditionand is referred to as”pinch-off.”When thechannel pinchesoff inside the device, the slopeof theID-VDcharacteristic becomes approximately zero (see Fig.15.12c),and

--- Page 556 --- FIELD EFFECT INTRODUCTION—THE J-FETAND MESFET 533 P_+} N Depletion region $P_{+}$ (a) Current flowregion ID$ (b) 0 5V 1 2 3 4 (c) Widens (d) -Pinch-off (e) VDsal D △L K → L (f) Figure 15.11 Visualizationof various phases of VG=0J-FEToperation.(a） Equilibrium (D=0,V=0);(b)smallVDbiasing;（c）voltagedropdownthechannelforanarbitrarilyassumed VD=5V;（d)channelnarrowingundermoderateVDbiasing；（e)pinch-off；（f）postpinch-off (VD>Dsat).

--- Page 557 --- 534 FIELDEFFECTDEVICES ID Linear →VD (a) ID$ ↑ Slope-over due to channel narrowing →VD (b) ID ↑ Pinch-off point IDsat ↑ Approximately level for {D > VDsat 1 1 1 1 1 →VD VDsat (c) Figure15.12 Generalformof theID-VDcharacteristics.(a) Linear,simple resistor,variation for very small drain voltages.(b) Slope-over at moderate drain biases due to channel narrowing. (c) Pinch-off and saturation for drain voltages in excess of VDsat·

--- Page 558 --- FIELDEFFECTINTRODUCTION—THEJ-FETANDMESFET 535 the drain biasatthe pinch-of point is giventhespecial designationVDsat.Fordrainbiases in excessofVDsat,theID-VDcharacteristicsaturates,thatis,remains approximatelycon- stant at the IDsat value. Thestatements presented withoutexplanation in thepreceding paragraphare totally factual.TheID-VDcharacteristic does leveloff or saturate whenthe channel pinchesoff. Atfirst glance,however,thefactsappeartoruncontrarytophysicalintuition.Shouldnot pinch-offtotallyeliminateanycurrentflow inthechannel?Howcanoneaccountforthe factthatDotagesinecessofhavesentillyoefctontherncurnt In answer to the first question,let us suppose ID at pinch-off was identically zero.If IDwere zero,there wouldbenocurrentinthechannel atanypointand thevoltage down the channel wouldbethe same as at V=0,namely,zero everywhere.If the channel potentialiszeroeverywhere,the pnjunctions wouldbezerobiasedandthechannelinturn wouldbecompletelyopenfromthesourcetothedrain,clearlycontradicting theinitial assumptionofa pinched-off channel.Inotherwords,acurrentmustflowin theJ-FETto induce and maintainthe pinched-offcondition.Perhaps theconceptual difficultyoften en- counteredwithpinch-ofarisesfromthenedforalargecurrenttoflowthroughadepleion region.Remember,depletionregionsare nottotally devoidof carriers.Rather,thecarrier numbersare just smallcompared to thebackgrounddoping concentration(NDor NA)and may still approach densities~10²/cm3 or greater. Moreover,the passage of large currents throughdepletionregionisnotunusualinsolid-statedevices.Forexample,alargecurn flows through the depletionregion in a forward-biased diodeandthrough both depletion regions in a bipolar junction transistor. WithregardtothesaturationofIDfordrainbiasesinexcessofDathereisver simplephysicalexlnioWhetheribiasiincrseaboveVtenco portionofthechannelwidensfromjustapointintoadepletedchannelsectionΔLinexent. AsshowninFig.15.11(f),thevoltageonthedrainsideofthe△Lsectionis VD,whilethe voltage on the source side of the section isV sat.In oterwd,theaplieddanoltae inexcessofVDsatV-VDaisdroppedacrossthedepletedsectionofthechannel.Now, assuming △L≤L,the usual case,the source-to-pinch-off regionof the device willbe essentiallyidenticalinshapeandwillhavethesameendpointvoltages(zeroandVDsatas were presentatthestartofsaturation.Iftheshapeofaconductingregionandthepotential appliedacross theregiondo notchange,thenthecurrentthroughtheregion mustalso remain invariant. This explains theapproximateconstancyofthe draincurrent forpost- pinch-offbiasing.NaturallyifLiscomparabletoLthenthesamevoltagedrop(VDsat) willappear across ashorter channel section(L-ΔL)andthepostpinch-offID will in- creaseperceptiblywithincreasingVD>VDsat.Thiseffectisespeciallynoticeableinshort channel (small L)devices.] Anotherapproachto explaining the saturation of the ID-VDcharacteristics makes use of an analogous situation ineveryday life,namely,a waterfall.The water-flow rateover a fallsiscontrollednotbytheheightofthefallsbutbytheflowratedowntherapidsleading to the falls.Thus,assuming an identical rapidsregionthe water-flowrateat thebottomof the twofalls pictured inFig.15.13 isprecisely thesame,eventhoughtheheightsofthe fallsaredifferent. Therapidsregion is analogous to the sourcesideof thechannel in the J-FET,thefallspropercorresponds tothepinched-off ∆Lsectionatthedrainendof

--- Page 559 --- 536 FIELDEFFECTDEVICES Water Water L Rapids Falls Figure15.13 The waterfalls analogy. thechannel,andtheheightofthefallscorrespondstotheVD-VDsatpotentialdropacross the ∆L section. Thus far we have establishedthe expected variationof ID with Vwhen V=0.To complete the discussion we needto investigate theoperationof the J-FET when V<0.It turns outthatVG<0operation is very similartoVG=0operation withthreeminor modifications.First,ifV<0,thetopandbottom p+-njunctions arereverse biasedeven whenVD=0.Areverse biason thejunctions increases the widthofthe depletion regions and shrinks the V=0 widthof the channel.Consequently,theresistanceof thechannel increases at agiven VD valueand the linear portionof theID-VDcharacteristic exhibits a smaller slope when VG<0(seeFig.15.14a).Second,because thechannel is narrower at V=0,thechannelalsobecomespinchedofatasmallerdrainbias.Therefore,aspictured inFig.15.14(b,atandIDsawhen<0aresmallerthanVaandIDsatwhen 0.Finally,notethatforsufficientlynegativeVbiases itis possibleto depletetheentire channeleven with VD=0(seeFig.15.14c).Thegate bias,VG=Vp,wherethegate voltage first totallydepletestheentirechannel withVDsetequalto zero,isreferred toasthepinch- off gate voltage.For≤Vpthedraincurrentisidenticallyzerofor all drainbiases. 15.2.3 Quantitative /D-VRelationships Wanted:a quantitativeexpressionfor thedraincurrentasa functionof theterminalvolt- ages; that is,ID=D(D，VG). DeviceSpecification.Theprecise device structure,dimensions,andassumedcoordinate orientations are as specified inFig.15.15.The y-axis is directed down thechannelfrom thesource to the drain while the x-coordinate is oriented normalto the p+-n metallurgical junctions,Lis the channel length,Zis the p+-n junction lateral width,and2a is the dis- tance betweenthetopandbottommetallurgical junctions.Notethaty=0andy=Lare Ifthedrainbiasismadeverylarge,thep+-njunctionsinthe vicinityofthedrain willeventuallybreak down, leading toa veryrapidincrease inIDwithVDforany gatebias.This breakdown has beenomittedfromallofthe theoretical characteristics sketched herein.

--- Page 560 --- 537 $D$

V{} 00$ V{} 0$ IDsat V_{<0$ $_V_{{0$ VD VD$ VDsat (a) (b) V{} 0 0$ V_{}V$ (c) Figure15.14Modificationofthe-VDcharacteristics when V<0.(a) Decrease in thelinear slopeofthecharacteristics forsmalldrain voltages.(b)Decrease in thesaturationcurrentand satu- ration drain voltage.(c)Gate pinch-off. slightlyremovedfrom the source and draincontacts,respectively.V(y)isthe potentialand W（）=p（=ottom（y）isthejunctiondepletionwidthaanarbitrarypoinyinte channel. W(y) lies almost totally in the n-region because of the p+-n nature ofthe junctions. Basic Assumptions.(1)The junctions are p+-n step junctions and the n-region is uniformly doped with a donorconcentration equal to ND.(2) The device is structurally symmetrical about thex=a plane as shown in Fig.15.15,and the symmetry is maintained by operating the device with the same VG bias appliedto thetop and bottom gates.(3)Current flow is confined to the nondepleted portions of the n-regionand is directed exclusively in the y- direction.(4) W(y)can be increased to at least a width a without inducing breakdown in thep+-njunctions.(Weimplicitlyassumedthistobethecase inthequalitativediscussion.) (6)L≥a. --- Page 561 --- 538 FIELDEFFECTDEVICES V_G Z $v}$ D S N y=0 y=L VG (a) VG 0 W(y) P+$ V(y) 2a x W(y) V_{$ (b) Figure15.15Devicestructuredimensions,andcoordinateorientationsassumedinthequantita- tiveanalysis.(a)Overalldiagram.(b)Expanded viewofthechannelregion. For drainand gate voltages belowpinch-off,0≤V≤VDsat and0≥V≥Vp,the derivationofthedesiredID-VDrelationshipproceedsasfollows:In generalonecan write uΔNb+gu"b=^ (15.1) Within theconductingchanneln≈NDandthecurrent is flowingalmostexclusively inthe y-direction.Moreover,withn≈ND,the diffusion componentof the current (qD∇n) should be relatively small.Underthe cited conditions Eq.(15.1)reduces to dV JN = Ny=qnND=-qD (in the conducting channel) (15.2) dy --- Page 562 --- 539 Since thereare nocarrier sinksor sources inthe device,thecurrentflowing throughany cross-sectionalplanewithinthechannelmustbeequaltoID.Thus,integratingthecurrent densityoverthecross-sectionalareaoftheconductinchannelatanarbitrarypointyields ID = -∫JNydxd2=z ${2a-W(y) a dV$ J{y dx= 2z $q_{nD$ dx (15.3a) w(y) JW(y) dy dV =2qZμ_nND_a$ (15.3b) dy The minus sign appears in the general formulaforIDbecause ID is defined to bepositive in the-ydirection.Use wasalsomadeofthefactthatthestructureis symmetricalabout the x = a plane. Remembering that IDis independentofy,onecan recast Eq.15.3b) intoamore useful form by integrating ID over the lengthof the channel.Specifically, L D dy=IDL=2qZnNDa (15.4) V(0)=0 or 2qZμ_nND_a$ VD W(V) $ID$ 1 一 dV (15.5) L Jo a To proceed any further, we need an analytical expressionfor Was a functionofV. It shouldberecognizedthattheelectrostaticprobleminsidetheJ-FETisreallytwo- dimensional in nature. To obtain an exact expression for Was a function of V would neces- sitate the solutionof Poisson's equation taking into account both the x and y variationof the electrostatic variables. Fortunately, with L≥a (basic assumption 6),the y-direction dependenceoftheelectrostatic variables(potential,electricfield,andcharge density)in the J-FETdepletionregions is expected to be a muchmoregradualfunctionof position than the x-direction dependence of the same variables.This allows us to invoke the gradual channel approximation,an approximationthat is encountered frequently in the analy- sis of field-efctdevices.SpecificallyddyisnelectedinsolvingPoissonseuation for the electrostatic variables in the FETdepletion regions.In essence,a one-dimensional x-direction solution is obtained for each channel y-value. Fortheproblemunder consideration,invoking the gradual channelapproximation simply means that we can approximate W at every point y using Eq.(5.38),the one- dimensional expressionestablishedin Section5.2.Thus,for thegiven p-n stepjunctions, $2K_{s$ 1/2 2K_{s{$ 71/2 W(V)= $qNDD$ (_{bib$ - $V{)$ = qN{DD$ (bi+V-V{) (15.6) --- Page 563 --- 540 FIELD EFFECTDEVICES where,as is evidentfrom Fig.15.15(b),VA=V-V(y)is the applied potential drop across the junction at a given point y. It is next convenient to note that W→a when VD=0(V=0)andV=Vp.Thus substitutinginto Eq.（15.6)yields $2K_{s$ 71/2 a = qN{D$ (15.7) and W(V) 1/2 (15.8) $bi- V_$ a Finally,substituting W(V)/a from Eq.(15.8) into Eq.(15.5)and performing the indicated integration, one obtains $2qZ_nNa$ {VD-$ 2-3 3bi−vp)[(VD+V 3/2 3/2 $\oe$ L $V_{bib$ V{_$ 一 $V_bi-V{}$ $for 0≤V D≤VDsat VP≤V ≤0$ (15.9) Equation（15.9)couldbe simplifiedbyintroducing G0=2qZμnNDa/L;physically,G0is the channel conductance one would observe if there were no depletion regions. We have retained 2qZμnNDa/L in the text expressions so that the major parametic dependencies are immediately obvious. It should be reemphasized that the foregoing development,and Eq.(15.9)in particular, applyonlybelowpinch-off.InfactthecomputedIDversusVDforagiven VG actually begins to decreaseifVDvaluesinexcessof VDsatare inadvertently substitutedinto Eq.(15.9).As pointedout inthe qualitative discussion,however,IDis approximately con- stantifeces.Tofrtosticofrioofthrctrit can be modeled by simply setting D|VD>DSa|=DaD (15.10a) or 3/2 IDsat L (15.10b) --- Page 564 --- FIELD EFFECTINTRODUCTION—THEJ-FETAND MESFET 541 TheIDsatrelationshipcanbesimplifiedsomewhat bynotingthatpinch-ofatthedrainend of the channel implies W→a when V(L)=VDsat.Therefore,from Eq.(15.6), 2K_{ε0 b + Dsat- v 1/2 a = qNDD$ (15.11) Comparing Eqs.(15.7) and (15.11),one concludes Dat = -V_ (15.12) and 2qZμ_nN{D_a$ 3/2 IDsat 二 L (15.13) TheoreticalID-VDcharacteristicscomputedusing Eqs.(15.9)and(15.13)arepre- sented in Fig.15.16.Forcomparison purposes a sample setof experimentalcharacteristics are displayed in Fig.15.17.Generallyspeaking,thetheory doesareasonablyadequate job of modeling the experimental observations. A somewhat improved agreement between ex- perimentandtheorycan beachieved by liftingtheassumptionofnegligiblevoltage drops in the regionsof the device between theactive channel andthe source/draincontacts (see Fig.15.18).Exercise15.3examines the required revisions to the theory when the source and drainresistances are taken intoaccount.Finally,itshouldbepointedoutthatinsatu- rationmost J-FETcharacteristics canbecloselymodeled bythesimple relationship $DSat = ID0(1 - V_/V){ where IDO0 = IDsat|V{=0 (15.14) Althoughappearingtotallydifferentthan Eq.(15.13),thesemi-empirical"square-law relationshipof Eq.(15.14)yields similar numerical resultsand is mucheasier to use in performing first-ordercircuitcalculations wherethe J-FET is viewed as a“black-box." Equation(15.13),ontheotherhand,is indispensable ifone wishesto investigatethede- pendence of the J-FET characteristics on temperature, channel doping,or some other fun- damental device parameter. --- Page 565 --- 542 FIELD EFFECTDEVICES $V_{} 0$ 1 1.0 7 Dsat- Dsat line 1 0.8 1 1 1 VG=-0.5V $D$ 0.6 / $\ra{DO}$ 1 / 0.4 VG=-1.0V / / 0.2 / VG=-1.5V VG=-2.0V 0 1 2 3 4 5 VD (volts) Figure15.16NormalizedtheoreticalD-VDcharacteristicsassumingVb=1VandVp=-2.5V. DO =IDsatV{G=0· 5 0 4 -0.5 3 $ID$ VG(volts) (mA) 2 01- 1 -1.5 -2.0 0 1 2 3 4 5 VD(volts) Figure15.17SampleexperimentalD-VDcharacteristics.CharacteristicswerederivedfromaTI 2N3823 n-channel J-FET.) --- Page 566 --- 543 $Rs$ $P+ RD S N WD P+ ↑ y=0 y= L (V= IDS) (V=VD-DRD) Figure15.18ModifiedmodelfortheJ-FETincludingtheresistanceofthesemiconductorregions between the ends of the active channel and the source/drain terminals. Exercise 15.1 P:Ann-channel Si J-FETis to beconstructed with achannel doping of ND=1016/ cm3.Assuming room temperature operation,determine the maximum junction- to-junction half-width(a)that can beemployed inconstructing the J-FET. S:Let WBRbethedepletion widthwhenthepnjunctionsin theJ-FETarebiasedto thebreakdownvoltage.Forthe J-FETto exhibit saturating characteristics, we must have a≤W BR.AsreadfromFig.6.11,VBR55VforthegivenND=101/cm³ p+-n junctions. Also，Vbi=0.92V,as inferredfrom the figure associated with Exercise 5.1inSubsection 5.1.4.Thus 2Kε{0$ 71/2 (2)(11.8)(8.85×10-14) 71/2 $amax = W_BR qNDD$ (Vbi+ BR)$ = (1.6×10-19)(1016) (0.92+55) or $amax =2.70 m (C) Exercise 15.2 P:Constructaplotcomparing theIDsatD0 versus V/Vpcomputed using thephe- nomenological relationship (Eq.15.13)and the square-law relationship (Eq.15.14). Assume Vb=1VandVp=-2.5Vor,equivalently,setb/Vp=-0.4.Howdoes the comparison vary with the Vbi/Vpratio? S: Setting VG=0in Eq.(15.13)yields $2qZμ_n_a$ {[A）-](- 3/2 $-V{-$ IDO$ 二 L --- Page 567 --- 544 FIELD EFFECTDEVICES Thus in the phenomenological theory (after some rearrangement), IDsat IDO$ 二 V−1)[1-) 3/2 2 -1 -3 According to the square-law relationship, VG 2 IDO$ = 1 - $V{_}$ A comparative plotof the twoIDsatDo relationships and the scriptof the MATLAB program used to generate the plot is reproduced below. Given the differ- ence in functional forms,the agreement is quite good.By changing the z= V/ value in the computerprogram,theagreement is found to improve with increasing |b/Vp|. MATLAB program script... %Exercise15.2..Comparisonof IDsat Relationships %Computational parameters clear z=-0.4; %z=Vbi/VP x=linspace(0,1); %x=VG/VP %P-Theory(y=IDsat/ID0) :(()((I-z)(x-z))-I)(I-z)*（ε）--X=wnN Den=-1-(2/3)*(z-1)*(1-(z.(z-1)).^(1.5)); yP=Num./Den; %Square-law Theory yS=(1-x).^2; %Plotting result close plot(x,yP,x,yS,--');grid xlabel('VG/VP');ylabel('IDsat/IDO'); text(0.38,0.4,'Square-Law') text(0.27,0.3,'Eq.(15.13)') text(0.8,0.83,'Vbi/VP=-0.4') --- Page 568 --- FIELD EFFECTINTRODUCTION—THE J-FETANDMESFET 545 1 0.9 Vbi/VP=-0.4 0.8 0.7 0.6 IDsat/ID0 0.5 Square-Law 0.4 Eq.(15.13) 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 VG/VP Exercise15.3 P:Show that thesourceand drainresistances at theendsoftheactivechannel(R and RD in Fig.15.18) are appropriately taken into account by replacing VD with VD-ID(R+RD)and VGwithVG-DRinEqs.(15.9),(15.12),and(15.13). S:Withthevoltagedropsatthechannelendstakenintoaccountthechannel voltages aty=0andy=Lare V(0)=IDRandV(L)=V-IDRD,respectively.Inserting therevised voltage limitsintoEq.(15.4)and likewise modifyingEq.(15.5),wefind 2qZμnNDa D-IDRD 厂 W(V) ID$ 1 dV (15.5') L $J{{s$ a Since the Eq.(15.8) expressionfor W(V)/a remains unchanged,the integralin Eq.(15.5)isreadilyevaluated.Theintegrationyeld --- Page 569 --- 546 FIELD EFFECTDEVICES $2qZ_nNa$ {D - ID(Rs+ RD) $Io$ = L 3/2 (15.9') $V_bi-V_$ $(IDR{s+Vb-Va)3²$ 3/2 Vbi- V_p$ TurningnexttothemodificationofE.15.12),wenotethat whenD=Dat W→a,V(L)= VDsat−IDsatD,andfromEq.(15.6), 2K_{ε0$ 1/2 (bi+ Dsat-DsaD -) (15.11') a = $qNDD$ However, 2Ks0 7 1/2 (V_{bi$ hi − Vp)c (15.7) D $qND$ and clearly, Dsat-DatD= - (15.12') Finally,settingVD=VDsaandID=IDsatinEq.（15.9),andsimplifyingthe result using Eq.(15.12'),one obtains $2qZμ_nNDa$ {-V-DmRs-3Vb-Vp)× IDsat (15.13') L $DsRs+Vb-V)}} 3/2 Vbi-V_$ Notethatreplacing VDbyVD-ID(R_+RD) andVGbyVG-IDRsin Eqs.(15.9),(15.12),and(15.13) doesindeedyieldEqs.(15.9),(15.12),and (15.13'), respectively. A few wordsare inorderconcerning theforegoing results.First,it isobvious thattheprimedequationscannotbe solvedforIDorIDsatasaclosed-form funcion --- Page 570 --- 547 of VD and VG.Nevertheless,numerical iterationtechniques can beused toreadily calculatetherevisedcurrent-voltage characteristics. Second,thisproblem could have beenrapidlycompleted by(i)changing theintegration variable in Eq.(15.5') to V'=V-DRand(ii)requiring theoriginal andrevised versionsof Eq.(15.5) to have the same general form. The change-of-variable approach,however, is not very informative. 15.2.4 a.c.Response Thea.c.responseoftheJ-FET,routinelyexpressedintermsoftheJ-FETsmal-signal equivalentcircuit,ismostconvenientlyestablishedbyconsidering thetwo-portnetwork showninFig.15.19(a).Initiallywerestrictourconsiderationsto lowoperationalfrequen- cies where capacitive effects may be neglected. Let us begin by examining the device input.Under standard d.c.biasingconditions the input port between the gate and source is connected across a reverse-biased diode on the insideof the structure.A reverse-biased diode,however,behaves (tofirstorder) likean opencircuitatlowfrequencies.Itisstandardpractice,therefore,tomodeltheinputtothe J-FETby an open circuit. G D In J-FET ino S S (a) G D + + $2_}$ $8m_gq}$ 8d $_}$ 一 S S (b) $C_{gd$ G HH D + + $^_g}$ $C_{}$ $8m_g$ gd $v_e}$ S S (c) Figure15.19(a） The J-FET viewedasa two-portnetwork.(b)Low-frequencyandc)high- frequencysmall-signalequivalentcircuitscharacterizingtheac.responseoftheFE. --- Page 571 --- 548 FIELDEFFECTDEVICES Attheoutput port the d.c.draincurrenthas alreadybeenestablished tobe a function ofVDandV；thatis,ID=ID(VD,VG).Whenthea.c.drainandgate potentials,vandυ g' arerespectivelyaddedto the d.c.drainand gate terminalvoltages,VDandVG,the dain currentthroughthestructureismodifiedtoD(D,)+,whereiisthea.c.component ofthe draincurrent.Provided the devicecanfollowthea.c.changes in potential quasistat- ically,which is assumed to be the case at lowoperational frequencies,one can state +ID（VD，V）=D（VD+，VG+g) (15.15a) and =D（VD+,VG+g）-D(D，G） (15.15b) Expanding the firsttermontheright-hand sideofEq.(15.15b)ina Taylor series aboutthe d.c. operating point,andkeeping onlyfirst-orderterms in theexpansion(higher-order terms are negligible), one obtains

$D$ D（VD+，VG+g）=ID（V，G) + $\text{D}$ v_}+ V{GlVD v8 (15.16) VG which when substituted into Eq.(15.15b) gives$ ∂} i_}

v_d} + VGVD$(15.17)DimensionallythepartialderivativesinEq.15.17)areconductances.Introducing$}8d$ D VG=constantant …the drainor channelconductance (15.18a) \D} 8m \_{ …transconductance or mutual conductance (15.18b) VD=constant we can then write (15.19) Eq.(15.19)may be viewed as the a.c.-current nodeequation for the drain terminal and,by inspection,leadstotheoutputportionofthecircuitdisplayedinFig.15.19(b).Since,as

--- Page 572 --- FIELD EFFECT INTRODUCTION—THE J-FETAND MESFET 549 Table15.1 J-FETSmall-SignalParameters.Entriesinthetable wereobtainedbydirectdiffer- entiationof Eqs.(15.9)and(15.13).G0=2qZμnNDa/L. Below pinch-off(VD≤VDsat) Above pinch-off(V D ≥VDsat) 8a= G0[1 - (YD+Vb-V 1/2 $g_{=}0$V_{bib V{_ 一 8m = Go(VD+V-V 1/2 _{bib V_G 1/2 8m= G01-( 1/2 一 V{_ V_{bi

$V_p$ concluded earlier,thegate-to-sourceorinput portionofthedeviceissimply anopen circuit, Fig.15.9(b)isthedesiredsmall-signalequivalentcircuitcharacterizingthelow-freqeny a.c. response of the J-FET. The gmparameter in themodelingoffieldeffecttransistorsplaysaroleanalogouso the gain parameters (α and β)previously encountered in the BJT analysis.As its name indicates,g may be viewed as either the device output admittance or the a.c.conductance of the channel between the source and drain.Explicit g and gmrelationshipsobtained by direct differentiationof Eqs.(15.9)and(15.13),using the Eq.(15.18)definitions,arecata- logued inTable15.1.Notethatthegd=0resultinTable15.1foroperationoftheevice under saturationconditions isconsistent withthetheoreticallyzeroslopeoftheID-D characteristics whenV≥V Dsat Athigher operating frequencies often encountered in practical applications,the Fig.15.19(b)circuitmustbemodifiedtotakeintoaccountthecapacitivecouplingbetween the gateandthe drain/source.Being pn junctions,the J-FETgate junctions are in general representedbyasmall-signalequivalentcircuitsimilartothatpresented inFig.7.2.How- ever,since the J-FET gate junctions are normallyreverse biased,theadmittance of the junctions is adequatelyrepresented by the depletion region capacitance. This capacitance must bepartially connectedbetweenthe gateand sourceandpartially connectedbe- tween the gateanddrain. Theresulting high-frequencyequivalent circuit is picturedin Fig.15.19(c). Exercise15.4 P:Using thelow-frequencyequivalentcircuitof Fig.15.19(b),and assuming he J-FET is saturationbiased,showthatthesourceand drainresistancesat thendsof theactivechannlgiverisetoaeducedefectivetransconductance 8m $8^s$ $1+m$

--- Page 573 --- 550 FIELD EFFECT DEVICES S: Under saturation conditions g→0.Open-circuiting g and adding Rand RDto the proper nodes in Fig.15.19(b) yields the working equivalent circuit displayed in Fig.E15.4.As deducedfrom Fig.E15.4, i=gmg=gm（g-R_s) Thus 8m ia 二 $1+m$ $8^m$ $vgwhere8m$ ${}^m$

$1+gm$ It is interesting to note that RD does not affect thetransconductance if g=0.Rs, on the other hand,causes a decrease in the effective transconductance of the J-FETif mR_s is comparable to unity. i_} RD G D + 8m^_s} + v_{r{{} _g} Rs 一 S S Figure E15.4 15.3 MESFET 15.3.1 General Information Thesimilarity between MS(Schottky) diodes and pn junction diodes was noted inChap- ter 14.In particular,a depletionregion, which can be modulated by theapplied voltage, also forms beneath a rectifying MScontact.It shouldtherefore come as no surprise that field-effecttransistorscanbebuiltwithrectifyingmetalsemiconductorgates.Asimplified perspective view ofa GaAs MESFET and an idealizedcross sectionof the structureare pictured in Fig. 15.20.

--- Page 574 --- 551 S G D 1 1 +N N-GaAs +N Semi-insulating GaAs (a) G Depletion region 1 S D N Substrate (b) Figure15.20 The MESFET.(a) Simplified perspective view of a GaAs MESFET.(b)Idealized MESFETcross section.(Notethatthe totaln-regionthickness in a single-gate structure is takento be a.) As mightbe inferredfromFig.15.20,theMESFETs in commercialproductionare GaAs-basedn-channel structures. TheMESFETs are primarily used in high-frequency applications where GaAs is preferred over Si because of its superiorelectron-transport properties.Historically,difficultieswereencounteredinfabricatingmetal-insulator-GaAs FETs withacceptable characteristics.MESFETfabrication was developed as an alternative and is presently the most matureof GaAs fabrication technologies. GaAs MESFETs are the heartofmonolithicmicrowave integrated circuits(MMICs) and are also sold as dis- crete devices foruse in amplifiersandoscillatorsthatmustoperateatfrequencies in excess of 5 GHz. There are two basic types of MESFETs—the depletion-mode or D-MESFET and the enhancement-mode or E-MESFET.Cross sections of the two types of MESFETs are shown

--- Page 575 --- 552 FIELDEFFECTDEVICES S D G $n+$ n- $n+$ Zero-bias Semi-insulating depletion Implanted GaAs region regions (a) D-MESFET S D G n-$ V/A Ti/Pt/Au(Schottky contactto GaAs) Zero-bias Implanted AuGe/Ni Ohmic contact Semi-insulating depletion region GaAs region Passivation layer(insulator) (b) E-MESFET Figure15.21 GaAs MESFET structure types.(a) Depletion-modeor D-MESFET;(b)enhance- ment-modeor E-MESFET. inFig.15.21.Similarto thepreviouslydescribed J-FET,a bias appliedto the gateof a D-MESFETfurther depletes the subgate regionand reduces the channel conductance. The E-MESFET,on theother hand,is fabricated so thatthebuilt-in voltageassociated withthe metal-semiconductorcontact issufficientto totally deplete thechannel.Aforwardbias must be applied to the E-MESFET gate to reduce the depletion width and obtain a channel current. Thepreponderanceof applications including MMICs involveanalog circuitry and make exclusive useof D-MESFETs.Digitallogic circuits incorporate both D-and E-MESFETs. It should be notedthat highoperationalfrequencies necessitate the fabricationof de- vices with extremely shortchannel lengths.The gate oractive channel length in commer- cial devices is routinely l μm. The discrete GaAsMESFETslisted in the1993 Hewlett-Packard catalogue of communications components,for example, are quoted to have a nominal gate length of 0.25 m. 15.3.2Short-Channel Considerations Given the structural similarities between the MESFETand J-FET,theID-VDtheorydevel- oped for the J-FETcould be applied withonly minor modifications to the MESFET pr- vided all derivationalassumptions are satisfied.Unfortunately,someofthederivational assumptions are atbestquestionable forthe typical short-channelMESFET.Forone,the L>a assumption and the associated gradual channel approximation are clearly suspect. Second,theelectricfieldintheFETchannel isimplicitlyassumedtobesufficiently lowso

--- Page 576 --- FIELDEFFECTINTRODUCTION—THEJ-FETANDMESFET 553 that thecarrier drift velocity is given by v=oy，where|ois the usual low-field mobility.However,ifadrain voltageofV=1VisappliedacrossachannellengthofL= 1m,theaverage magnitudeoftheelectric field in thechannel will be |y|=VD/L= 104V/cm.Since|increasesasone progresses down thechanneltoward the drain,the magnitudeoftheelectricfieldnearthedrainwillbeevenlargerthanthecitedaverage. Referring toFig.3.4inPartI,wefindtheelectrondriftvelocity inSi is significantly differentfromtheextrapolatedlineardependenceatanelectricfieldof104V/cm.Failure ofthelow-fieldassumptionoccurs at anevensmaller-valueinGaAs. In thefollowing discussion,weexaminethremodificationstothelongchanneltheory that havebeenproposed inthedevice literature.Eachoftheapproximate shortchannel modelshasitsvaliditylimitsandutilityundercertainconditions. VariableMobilityModel The nonlinear variationofd with as thecarriers progress down theMESFETchannel Eq.(3.3) withβ=1 adequately models the v versus dependence,one can write μ_{

(15.20) 1+μ{0 U_sat and |μ{0l $\d}$ μ(8)≡ (15.21) 8 μ_0 1+ V_sat where()isthefield-dependentmobilitandatisthesaturationdrifvelocity.If() with0=-n and→8=-dVldy is now substitutedforμnin Eq.（15.1)of the long-channel J-FETderivation,andsubsequentequationsareappropriatelymodified,one ultimately obtains D(long-channel) $ID$ = .. .0≤VD ≤VDsat (15.22) $μ{D$ 1+ $_s\atL ID(long-channel) is theID computed using Eq.(15.9). Above pinch-off,D(long- channel)→IDsat(long-channel)and V→V-p.Inaddition,2amust bereplaced by ainthe long-channelcurrentequationsiftheMESFETisasingle-gatestructureaspicturd in Fig. 15.20. Examining Eq.(15.22),we notethatthedraincurrent is alwaysreducedrelativeto the long-channel case.This is to beexpected since()≤μn.Also,VD/Linthe denominator --- Page 577 --- 554 FIELDEFFECTDEVICES ofEq.（15.22)isjust.ItisthereforereasonabletoidentifynVD/L=nasthe long-channel result ifv<v sat Overall,the variablemobility modelprovides an acceptable descriptionof the non- linear v versus Edependence in theFETchannel and is adequateas a first-ordercorrec- tionto thelong-channeltheory whentreatingFETsofmoderatechannellength(typically L≥10μm).However,themodeldoes not account for thefailureof the gradual channel erly describe the observed characteristics of short-channel MESFETs. Saturated Velocity Model Currentsaturation in long-channel devices isalways caused by apinching-offorconstric- tionofthechannel near the drain.Analternative mechanism can give rise to current satu- ration in short-channel devices where the gradual channel approximation is no longer valid. Supposeinashortn-channeldeviceis sufficientlylarge sothatv→satata pointy<LintheFETchannel.With=at,thecurrentfowingatthepointywillbe I(y₁）=qsatNDZ[a-W(₁）] (15.23) ThecontinuityofthechannelcurrentrequiresIy）tobethesameatallpointsinthe channeland,inparticular（y>y）=y1).Since=sataty=y₁,therecanbeno furtherincrease inv at pointsy>y₁.Itthereforefollows from Eq.（15.23)that,to satisfy theI(y>y₁）=I(₁）requirement,W（y)must be constantfor all y₁≤y≤Lasenvi- sioned inFig.15.22.Intotalcontrastto theassumptionmade inthegradualchannelap- proximation,thedescribedbehaviorcanoccuronlyifthereisanappreciablefieldinthe subgate depletionregion, with someofthefield linesterminating on charges externalto the gatedregionas illustrated inFig.15.22.Moreover,eventhoughthechannel isonly partiallyconstricted,thedraincurrentthroughthe device pictured in Fig.15.22 willhave saturated.Theapplicationofa largerdrainbias willmerely cause the ypoint in thechan- G E-fieldlines S D y= 0 y=y_1$ y=L n-GaAs Substrate Figure15.22 Approximate representationof the depletion region (shaded area)andelectricfield in a short channel MESFET where the drift velocity has reached its maximum value (v=vsal)at the point yin the channel. --- Page 578 --- 555 nel to movecloserto the source.In general,draincurrent saturation(ID=IDsat occurs whenthechannel pinchesof or when→saaty=L. If velocity saturationoccurs ina MESFET withachannel lengthofonlyafewmicrons or less,then the depletion width inthey₁≤y≤Lregionof the device will differbyonly a small amountfrom the depletion width aty=0.For the envisioned special situation, W(y₁)canbereplacedbyW(0)in Eq.(15.23).This leadstotheapproximation IDsat = I(₁)≡qatDZ[a-W(0)] (15.24) with 2K{ε0 71/2 V_G 1/2 W(0)= ({bib$ - V {) =a 一 (15.25) qN{DD$ $V{bib$ $V_{p$ Note that W(O),the depletion widthaty=0,is computed assuming the gradualchannel approximationcan beappliedto thesourceendof thechannel. The saturation drain cur- rentscomputed using Eqs.(15.24)and（15.25)arefound tobe infairlygood agreement with experimental results derivedfrom short-channel (L~1μm)GaAs MESFETs. Two-Region Model Although usefulthesaturated velocitymodelprovidesonlyan expressionforIDsat.The two-regionmodel,on theotherhand,providesanentirecharacteristicconsistent withthe saturated velocity model. Inthe two-region model theanalysis is broken into two parts corresponding to the two spatialor drift velocity regions pictured in Fig.15.22.For0≤y≤ythe gradual channel approximation and long-channel theory are assumed to hold with v=μothroughout the region. The v model is applied to the y₁≤y≤Lportion of the channel.The = sat transition pointistakentooccuratthe-value whereat.Naturlly,the long-channlthorispliethrouotthechannlifyat Parallelingthelong-channeltheory,theID-VDcharacteristics are computedusing Eq.(15.9)orthesinglegatequivalentwenV≤.LikewiseonesetsIV>Va = $ID| D=Vsa$ IDsat.However, in general VDsat≠VG-V and Eq.(15.13) is not used to 三 ,= atoytatthedinendothechannl.Iftelongchannlrltiohipsare solved forV(y)in thechannel(seeProblem15.3),theresulting expression differentiatd withrescttoton,nL）uatoDey sat' one obtains 3(bi−vp)[(+-V 3/2 3/2 $V_{bi-$ $V_bi-}$ V_{p$ SsatL= (15.26) $VDsa+Vb_VG}$ 1/2 -1 V_bi-V{$ --- Page 579 --- 556 FIELDEFFECTDEVICES 0.5 0.4 VG=0 0.3 $\o}$ VG=-0.5V $\ra{DO}$ 0.2 $VG=-1.0V VG=-1.5V 0.1 VG=-2.0V 0 0 0.4 0.8 1.2 1.6 2 VD(volts) Figure15.23 NormalizedtheoreticalID-VDcharacteristicsofashortn-channelMESFETbased on the two-region model. Vbi=1V,Vp=-2.5V, sat=-5×10³V/cm,and L=1μm.The draincurrentisnormalizedtothe=0saturationcurrentID）ofanequivalentlong-channelFET VDsatforagivenVandseofdeviceparametersisdeterminedfromanumericalsolution of Eq.(15.26). A sample setof theoreticalID-VDcharacteristics basedonthe two-regionmodelis displayed inFig.15.23.An at =-5×103V/cmappropriateforGaAsn-channel sat MESFETs andL=1μm was assumed in establishing the characteristics.For comparison withlong-chnnlresultthechacteisicswrenormliedtoID0=0compted using the single-gate version of Eq.(15.13). ExaminingFig.15.23 we note thatVDsatformostVG valuesoccurs ata significantly lower voltageandID/IDisreducedrelativetothecomparablelong-channelcharacteristics presented inFig.15.16.Also,IDsat exhibits a VG dependenceapproximately described y Eq.(15.24).Although somewhat crude inappearance witha noticeable discontinuity in slope at VD=VDsat,thecharacteristicsresultingfromthetwo-regionmodelaffordarea- sonablefirst-ordermatchtotheobservedshort-channelcharacteristics.Itshouldbeac- knowledged,however,thatanaccurate modeling ofthe short-channel characteristics inher- ently requiresthe numericaltwo-dimensional solutionoftheelectrostaticandcurrent equations. --- Page 580 --- FIELD EFFECTINTRODUCTION—THEJ-FETAND MESFET 557 15.4 SUMMARY Themodulationof theconductivityof a semiconductorby theapplicationof an electric field normalto thesurfaceof the semiconductor isknown as the fieldeffect.Thechapter began with a general historical introduction to thefieldeffect familyof devices. There- mainderof thechapter,devoted to theJ-FETandMESFET,provided a specific introduction state J-FEToperation wasexamined bothqualitativelyandquantitatively,withthe quanti- tative analysis including theclassiclong-channelID-VDderivationbasedon the gradual channel approximation.The channel conductance(g)and the transconductance(gm)were noted to be key parameters inmodeling the a.c. response of the J-FET. The MESFET discussion emphasized short-channel considerations and the required modifications to the steady-state long-channel J-FETanalysis. PROBLEMS CHAPTER15 PROBLEMINFORMATIONTABLE Complete Difi culty Suggested Short Problem After Level Point Weighting Description 15.1 15.4 1 10 (1 each part) Quick quiz 15.2 15.2.3 2 9 (3 each part) Compositristor 15.3 " 3 10(a-7, b-3) Relationship forV(y) 15.4 " 2 5 sat from ID maximum 15.5 " 3 25(a-3,b-5,c-3,d-5, J-FET operation with e-2, f-7) $V{_B}=00$. 15.6 " 3 12(a-3,b-9) Linearlygradedchannel 15.7 " 4 25 Rsand RDincluded ID-VD 15.8 15.2.4 2 9(a-6, b-3) Pinch resistor 15.9 " 3 12(a-3,b-6, c-3) $f\ max}$ * 15.10 3 18(a-12,b-6) m T-dependence 15.11 15.3.2 3 8 Back-gated MESFET 15.12 " 2-3 8 Derive Eq.(15.22) 15.13 " 2 5 Derive Eq. (15.26) 15.14 " 3 18(a-14,b-2, c-2) Two region ID-VD --- Page 581 --- 558 FIELD EFFECTDEVICES 15.1 Quick Quiz. Answer the following questions as concisely as possible. (a) Define "field effect." (c) Forap-channel J-FET(aJ-FETwithn+-p gating junctions and a p-regionbetween the sourceanddrain)does the draincurrentflowintooroutof the draincontactunder normal operating conditions? Explain. (d) (e) What is meant by the term"pinch-off"? (f) What is the mathematicaldefinition ofthe drain conductance?of the transconductance? (g) ofa J-FETundersaturationconditions.Assumethed.c.characteristicsofthedevice are similar to those shown in Fig. 15.16.) (h) WhatdoMESFET, D-MESFET,and E-MESFET standfor? (i) Whyisthemagnitudeoftheelectricfieldinthechannelofconcerninmodelingshort- channel MESFETs? ()Statedconciselywhatisthepimarydiferencebtweenthelong-channlandthetwo- region short-channel ID-VDtheories? 15.2In this problem you willbe asked various questions about the device shown in Fig.P15.2.The device,whichmightbecalledacompositristor (compositetransistor)is formed from a uniformly doped n-type bar.Ohmiccontacts are made to the top and bottom ofthebarandare connectedto theoutside worldthroughleads DandB,respectively. p+-n stepjunctionsare formedon thetwo sidesof the barandareconnectedto theoutside world through contacts E andC.As shown in the figure,d is the separation between the twop+regionsandListhelaterallengthofthepregions.Toreceivefullcredit,youmust indicate your reasoning in addition to answering each of the following. D P{+ N $p{+$ → E L C 一 k—d→$ B Figure P15.2 --- Page 582 --- FIELD EFFECT INTRODUCTION—THEJ-FETAND MESFET 559 (a)Given:D-Btiedtogetherd≤Lp,whereLpistheminoritycarrierdiffusionlengthin then-region；EB>0;VCB<0.Problem:Sketchthecurrent(C)flowingoutofthe CcontactasafunctionofVBifIEisheldconstantatvariousdiferentvalues. (b) Given: E-C tied together; D-B tied together; d Lp.Problem: Sketch the current flowing into the E-Cleads as a functionof thevoltage appliedfromE-Cto D-B. (c) Given:E-Ctiedtogether;d<2WBR，whereWB is the depletion width when the pn BR junctions inthecompositristorare biasedtothe breakdown voltage;Lisalmostequal to thetotallengthofthebarbetweenDandB;VDB>;VEB<0.Problem:Sketch thecurrentID）flowingintothecontactDasafunctionoftheVDBvoltageif EB is heldconstant at various different values. 15.3AsshowninFig.15.11(c),thevariationinvoltagedownthelengthoftheJ-FET channel is typically a nonlinear function of position. (a)Deriveanexpressionthatcanbeusedtocomputethepointinthechannel(thatis,y/L) where a given channel voltage0≤V(y)≤VDisexpected tooccur.HINT:LetL→y andVD→V(y)in Eq.(15.4),solvefory,evaluate theremaining integral,and then form the ratio y/L from your result. (b） Assuming VG=0,VD=5V,Vb=1V,andV=-8V,calculatey/LforV（y）=1, 2,3,and4V.How do thecomputed y/Lvalues compare with thepositioning of the voltages shown in Fig. 15.11(c)? 15.4IfEq.（15.9)is usedtocompute IDasafunctionof Vfor a given VG,andif VDis allowedtoincraseaboveDatonefindsIDtobeapakedfunctionofVDmaximiing VDat.TheoeoiuggconwaoetabliEq.rl VDat.icloce points ofafunctioncanbeusedtoderiveEq.(15.12)directlyfromEq.(15.9). 15.5Supposeas shown in Fig.P15.5,the bottomgate leadofa long-channelJ-FETis tied to the source and grounded. VG≤00$ p+$ S N D $V{≥0$ VGB=0$ Figure P15.5 --- Page 583 --- 560 FIELD EFFECT DEVICES (a）SketchanoutlineofthedepletionregionsinsidetheVGB=0devicewhenVGismade sufficiently large to pinchoff the channel withV=0. (b) IfVp(the pinch-off gate voltage)=-8V whenthetwo gates are tied together,Vbi= 1V,and assuming p-n step junctions,determine Vp(the top gate pinch-off voltage) when VGB=0and VD=0.Isyouranswerhereconsistent withthesketchin part(a） Explain. (c) AssumingV<G<0,sketchanoutlineofthedepletionregionsinsidethedevice when the drain voltage is increased to the pinch-off point. (d) Deriveanexpressionthatspecifies VDsatintermsofVT，i,andVforVGB=0 operation.(Youranswer shouldcontainonlyvoltages.Make no attemptto actually solve for VDat.) （e）Inlightofyouranswerstoparts（c）and（d)willVDatforVGB=0operationbgreatr thanorlessthanthe VDsatfor VGB=GToperation?Explain. (f)Derive anexpression for IDas a functionof VD andVGT analogousto Eq.(15.9). 15.6AJ-FETisconstructed withthegate-to-gate dopingprofileshown inFig.P15.6. Specificallyassume the p+-regiondoping is muchgreater than themaximum n-region doping and make other obvious assumptions as required. ND-N{A}=N{}$ x $N{-N}$ 0 D 2a x P + Figure P15.6 (a)EstablishanexpressionforthedepletionwidthW)associatedwithoneofthelinarly graded junctions.LetVAbetheappliedvoltage dropacross the junction.Itmaybe helpful torefer to Subsection 7.2.2ortoreview the linearly graded junctionanalysis in Subsection 5.2.5.) (b)Neglectingtheμndoping dependenceandassumingthatthe left-handandighhand gates are tied together,appropriately modify thetext J-FET analysis to obtain the be- low-pinch-offID-VDrelationshipforthislinearlygradedjunction.（Caution:More than the W(V)la expression must be modified.) 15.7GraphicallyillustratetheeffectoftheRandRDresistancesontheIDVDcharacter- istics.Referring to Exercise15.3,write a MATLAB(computer）program that can beused to calculateandplotthe J-FETID-VDcharacteristics for arbitrary valuesof RsandRD.Nor- --- Page 584 --- FIELDEFFECTINTRODUCTION—THEJ-FETANDMESFET 561 malize allIDvaluestoID0=IDsatVG=0ascomputedfromEq（15.13).（Thenormalized currentrelationshipsshouldinvolveonlyIDDorIDsatD0,D,G,andtheparameters Vbi,VP，G0R,andG0RD,whereG0=2qZnNDa/L.）RunyourprogramassumingVbi= 1Vand Vp=-2.5V;successively set G0R=G0RD=0,0.1,and0.5.Compareyour results with Fig.15.16.Comment on the comparison. 15.8Bipolarintegrated circuits occasionally make useofpinch resistors. Thetwo- terminalpinchresistor isessentiallya J-FETwiththegate(s)internallyshortedtothe source.Exhibiting a voltage-dependent nonlinearresistance,the devicefindsuse in appli- cations thatrequire large valuesofresistance,but where the precise values are notcritical. (a) Assumingthetextdevelopmentcanbeused withoutmodification,establishgeneral expressions for the d.c.conductance (G=1/Resistance =I/V)and the a.c.conduc- tance (g=dl/dV)of the pinch resistor. (b)Compute the valueof the d.c.and a.c.resistances(R=1/Gandr=1/g)atanapplied voltageof VDat/2.EmployZ/L=1,a=0.5μm,ND=1016/cm3,Vbi=1V,and V =- 2V. 15.9The maximum frequencyofoperationorcutof frequencyof theJ-FETis given by 8m $f_ maxx$ = $2πC$ where CG is the capacitance of the pn junction gates. (a)Derivetheaboveexpresionorbypresentinanargumntanalogoustothatound in Subsection 17.3.2. (b)Show that for a J-FETone can write $8m$ $qμ_{} ^{}$ $\_m_ x$ $2πC_$ $2πK_ε0L{2}$ (c) GivenasiliconJ-FETwith ND=1016/cm,a=0.5μm,andL=5m,computethe limiting value of the cutoff frequency. * 15.10Inthis problem we wishto explore the temperature dependenceoftheJ-FET transconductance. (a)Making useof parametrictemperature informationorrelationshipsfromearlierchap- ters,compute andconstructa log-log plotofgm(T)normalized to gm(300K)versus T(K)for225K≤T≤475K.Ata minimum,include data points at 50 K intervals. Set V=0andtake thedevicetobesaturationbiased(V≥VDsat).Assumeann- channelsilicondevice withND=1016/cm3,NA（ofthep+regions）=5×10/cm, and a=0.6m.[NOTE:Neglecting the very small change indevice dimensions with --- Page 585 --- 562 FIELDEFFECTDEVICES dent.Separately,however,Vi and Vpare temperature dependent (b) Assuminggm(T)/gm(300 K)αT(K)−n，determine n. Also，superimposed onthe part (a) plot and spanning the same temperature range,plotμn(T)/μn(300 K)versus T.Briefly discuss yourresults. 15.11Some n-channel MESFETs are built on p+substrates so they can be“back-gated”; i.e.,atopMSdiode gate is paired with a bottom p+-n gate.Let ViTbethe built-in voltage ofthetop(MS)gate,biB thebuilt-involtageof thebottom（pn)gate,VGTthetop gate voltage,VGB thebottom gate voltageV thetopgate voltagerequiredto pinchoffthe channel when VGB=V=0,2a the top-to-bottom junction width,andLthe lengthofthe channel as defined by the shorter MS gate.Invoking the two-regionmodel, derive an ex- pression for the below-saturation ID as a function of VD. 15.12 Derive Eq.(15.22). 15.13Utilizingthesolutionto Problem15.3(a）andfollowingtheprocedureoutlinedin the text, derive Eq.(15.26). 15.14(a)Constructa MATLAB（computer）program tocalculateandplotthe n-channel FETID-VDcharacteristicspredictedby the two-regionmodel.NormalizeIDto the VG=0saturationcurrent(D0）ofanequivalent long-channel FET.Take Vbi,V， sat' and Lto be input parameters.Run and check your program employing the parameters used to generate Fig. 15.23. (b） RunyourprogramemployingVi=1Vp=-2.5Vat=-104Vcmcharc- Fig.15.16. (c) For Vbi=1Vand Vp=-2.5V,atwhat valueofaLdoesthelong-channeltheory begin to fail？Define“fail”asareductionofthecomputedIDsatD0to less than0.95 $when V {_{}= 00 --- Page 586 --- 16 MOS Fundamentals Themetal-oxide (SiO)-semiconductor (Si) or MOSstructure is,without a doubt,the core structure in modern-day microelectronics. Even ostensibly pn junction type devices incorporate the MOS structure in some functional and/or physical manner.A quasi-MOS device,as noted in the Section15.1GeneralIntroduction,wasfirst proposed inthe1920s. Thedawn ofmodern history,however,is generallyattributedto D.KahngandM.M.Atalla whofiledforpatentson theSi-SiO2basedfieldeffecttransistorin1960.The MOSdesig- nation,itshouldbenoted,is reservedforthetechnologicallydominantmetal-SiO-Si system. The more general designation,metal-insulator-semiconductor (MIS),is used to identify similar devicestructurescomposedofan insulatorotherthanSiO2or a semicon- ductor other thanSi. Thischapter is intendedto serve as anintroductiontoMOSstructuralanddevice fundamentals. The two-terminal MOS-capacitor or MOS-C is both the simplestof MOS devices and the structural heartof allMOS devices.We begin with a precise specification ofthe"ideal”MOS-Cstructure.Energybandandblock-charge diagramsare nextcon- structedandutilizedtoqualitativelyvisualizethecharge,electricfield,andbandbending inside theMOS-C under staticbiasing conditions.Quantitative relationships for theelec- trostatic variables inside the semiconductorare thendevelopedandsubsequently related to thevoltage appliedtothemetallicgate.Capacitance isofcourse theprimaryelectrical observable exhibited by an MOS-capacitor. The MOS-C capacitance-voltage(C-V)char- acteristics are importantnotonlyfrom a fundamental butalsoapractical viewpoint.Inthe final sectionof the chapter,our knowledgeof the internal workingsoftheMOS structure is used to explain and analyze the normallyobserved form of the MOs-CC-Vcharacter- istics. Thechapter concludes with an examinationofcomputedideal-structure character- istics,comments about measurement procedures,andotherrelevantC-Vconsiderations. 16.1IDEALSTRUCTUREDEFINITION As pictured in Fig.16.1,the MOS-capacitor is a simple two-terminal device composed of a thin(0.01m-1.0m)SiO2layersandwiched between asiliconsubstrateandametallic fieldplate.Themostcommonfield plate materials are aluminumand heavilydoped poly- crystallinesilicon.Asecondmetalliclayerpresentalong thebackor bottom sideofthe Heavily doped Siismetallicinnature.Polysilicongates,usedextensively incomplexMOSdevice stucturs, are deposited byachemical-vaporprocessandthenheavilyn-orp-dopedbyeitherdiffusionorionimplantaion. 563 --- Page 587 --- 564 FIELDEFFECTDEVICES V_G $SiO_{2}$ 0+ 0 Si x Figure16.1 The metal-oxide-semiconductor capacitor. semiconductor providesanelectricalcontactto thesilicon substrate. Theterminalcon- nectedtothefieldplateandthefieldplateitselfarereferredtoasthegate;thesilicon terminal,whichis normally grounded,is simplycalledthebackorsubstrate contact. TheidealMOSstructure hasthefollowing explicitproperties:（1)themetallicgate is suficiently thick so that itcanbeconsideredan equipotentialregionunder a.c.as wellas d.c. biasing conditions;(2)theoxide is a perfect insulator with zero currentflowing throughtheoxide layerunderall static biasing conditions;(3)thereare no charge centers located in the oxide or at the oxide-semiconductor interface;(4)the semiconductor is uni- formlydoped;5)thesemiconductorissuficientlythick sothatregardlessoftheapplied gatepotential,afield-freeregion(theso-calledSi“bulk)isencounteredbeforereaching the backcontact; (6)anohmic contact has been established between the semiconductorand themetalonthebacksideofthedevice;(7)theMOS-Cisa one-dimensionalstructure withallvariablestakentobeafunctiononlyofthex-coordinate(seeFig.16.1);and (8) ΦM=Φ=χ+(E-)FB.Thematerial parameters appearing in idealization8 were previously introduced in Section14.1and will bereviewed in the next section. Allofthe listedidealizationscanbeapproached inpracticeandtheidealMOSstruc- ture is fairlyrealistic.Forexample,theresistivityofSiO2canbeashighas1018ohm-cm, andthe d.c.leakage currentthroughthe layer is indeed negligible for typicaloxidethick- nesses and applied voltages.Moreover,even very thin gates can be considered equipoten- tial regionsandohmicbackcontacts are quiteeasy to achieve inpractice.Similar state- ments canbe made concerning mostoftheotheridealizations. Specialnote,however, should bemadeof idealization 8. The ΦM=Φsrequirementcould beomitted and will in factbeeliminated inChapter18.Therequirementhasonly beenincludedatthis pointto avoid unnecessarycomplications in the initial descriptionof thestatic behavior. --- Page 588 --- MOSFUNDAMENTALS 565 16.2 ELECTROSTATICS—MOSTLYQUALITATIVE 16.2.1Visualization Aids EnergyBand Diagram Theenergyband diagram is anindispensableaid invisualizing the internal statusofthe MOSstructureunderstaticbiasingconditions.Thetaskathandistoconstructthediagram appropriatefortheidealMOS structure underequilibrium (zero-bias)conditions. Figure16.2shows the surface-includedenergy band diagrams forthe individualcom- ponentsof theMOSstructure.Ineach case the abruptterminationof the diagram in a verticalline designates a surface.Theledgeatthetopoftheverticalline,knownasthe vacuumlevel, denotes the minimum energy()anelectron must possess to completely free itself fromthematerial.The energy difference betweenthevacuum leveland the Fermi energy inametal isknownasthemetal workfunction,ΦM.Inthesemiconductortheheight of thesurfaceenergybarrier is specified intermsof theelectronaffinity,x,theenergy difference between the vacuum leveland theconduction bandedge atthe surface.χ is used insteadofE-EF because the latter quantity is nota constant in semiconductors, but varies asa functionofdopingand bandbending near the surface.Notethat(E-F)FBis theenergydifferencebetweenEcandEFintheflatband(FB)orfield-freeportionofthe semiconductor. The remaining component, the insulator,is in essence modeled as an intrin- sic wide-gap semiconductor where the surface barrier is again specified in terms of the electron affinity. Theconceptualformationof theMOSzero-bias band diagramfrom theindividual components involves a two-stepprocess.First the metal and semiconductorare brought togetheruntiltheyarea distance xapartand thetwo-component system is allowedto equilibrate.Once the system is in equilibrium themetal and semiconductor Fermi levels E0$ $E0 x 4 $Ec$ X ΦM x (E-F)B Ec$ EF $E$ EF $E_{$ $E_v}$ $E$ Metal Insulator Semiconductor Semiconductor with band bending Figure16.2 Individualenergy banddiagramsforthemetal,insulator,and semiconductorcompo- nents oftheMOSstructure. The diagramlabeled“semiconductor with band bending”defines (E-)FBandshowsχtobeinvariantwithbandbending.Thevalueofχ,itshouldbeemphasized, is measured relative to E at the semiconductor surface. --- Page 589 --- 566 FIELDEFFECTDEVICES E0 xi$ ΦM}$ X Φ' x^{} M E EF C E F $E_v}$ Figure16.3 Equilibrium energy band diagram for an ideal MOS structure. must be at the same energy(see Subsection3.2.4).Moreover,the vacuum levelsoftheM and S components must also be in alignment because we have specified ΦM=Φs.The foregoing implies that there are no chargesor electric fieldsanywhere in the metal-gap- semiconductor system.Next the insulatorofthickness is insertedinto the emptyspace between themetaland semiconductorcomponents.Given thezeroelectricfieldin the gap,theonlyefectofinsertingtheinsulatoristoslightlylowerthebarrierbetweentheM andScomponents.Thus theequilibriumenergy band diagram forthe idealMOS structure is concluded to be of the form pictured in Fig.16.3. BlockCharge Diagrams Complementary in nature to the energy band diagram,blockcharge diagrams providein- formationabout theapproximate charge distribution insidetheMOSstructure. As just noted in theenergybanddiagram discussion,thereare no chargesanywhere insidethe ideal MOSstructure underequilibriumconditions.However,when a bias is appliedtothe MOS-C,charge appears withinthemetal and semiconductor near themetal-oxideand oxide-semiconductor interfaces.A sampleblock charge diagram is shown in Fig.16.4. +Q Charge x Position -Q M 0 S Figure16.4 Sampleblock charge diagram. --- Page 590 --- MOS FUNDAMENTALS 567 Note thatno attempt ismadeto representtheexactcharge distributions inside thestructure. Rather,asquared-offorblockapproximation is employed and hence theresulting figure is calledablockchargediagram.Blockcharge diagrams are intendedto bequalitative in nature; themagnitudeand spatial extentof the charges shouldbe interpreted withthis fact in mind.Nevertheless,because the electric field is zero in the interior of both the metal and the semiconductor (see idealization 5),the charges within the structure must sum to zero according to Gauss's law.Consequently,inconstructing blockcharge diagrams,thearea representing positive charges is always drawnequalto the area representing negative charges. 16.2.2 EffectofanAppliedBias General Observations Beforeexamining specific-casesituationsit is usefultoestablishgeneral groundrulesas tohowone modifies the MOSenergy band diagram inresponseto an applied bias.Assume normaloperatingconditions where theback sideof theMOS-Cis groundedand letVbe the d.c.bias applied to the gate. WithVG≠0 we note firstof allthat thesemiconductorFermienergy is unafected by the bias and remains invariant (level on the diagram) as a functionof position.This is a directconsequenceof theassumed zero currentflowthroughthe structure underallstatic biasing conditions.In essence,the semiconductor always remains in equilibrium indepen- dentof the bias applied to the MOS-C gate. Second,as ina pn junction,the applied bias separates the Fermienergies atthetwoendsof the structureby an amountequaltoqG; that is, EF(metal)−(semiconductor)=-qG (16.1) Conceptually,the metal and semiconductor Fermi levels may bethoughtof as"handles connected to theoutside world.Inapplying a bias,one grabsonto the handles and rear- ranges the relative up-and-downpositioningofthe Fermilevels. Theback contactis groundedand the semiconductor-side handlethereforeremainsfixed in position. The metal-sidehandle,on theotherhand,is moveddownwardifVG>0and upwardif $V{<0 $ Since thebarrier heightsare fixedquantities,themovementof themetalFermilevel obviouslyleads in turn to a distortion inother featuresof theband diagram.The situation is akin tobendingarubber dolloutof shape.Viewedanother way,VG≠0causes potential drops andE(E）band bending interior to thestructure. No band bending occurs,of course,in the metal because it is an equipotential region.In theoxide and semiconductor, however,the energy bands mustexhibitan upward slope (increasing E going from thegate towardthe backcontact)when V>0andadownward slope when V<0.Moreover,the applicationofPoissonsequation totheoxide,taken tobean ideal insulator with nocarriers or charge centers,yields d oxide/dx=0 and therefore oxide = constant. Hence, the slope --- Page 591 --- 568 FIELD EFFECTDEVICES of theenergy bands in theoxide is aconstant—Ecand Eare linearfunctionsof position. Naturally,band bending in the semiconductor is expectedto be somewhatmore complex in its functional form,but per idealization 5,must always vanish(→0)before reaching the back contact. Specific Biasing Regions Given the general principles just discussed,it is now arelatively simple matterto describe the internal status of the idealMOSstructure under variousstaticbiasing conditions.Tak- ing the Sisubstratetoben-type,considerfirst theapplicationofapositivebias.Theappli- cationofV>0lowersEFinthemetalrelativetoEFinthesemiconductorandcausesa positiveslopingof theenergybands in both theinsulatorand semiconductor.Theresulting energy band diagram is shown inFig.16.5(a).Themajorconclusiontobe derived from Fig.16.5(a)is thattheelectronconcentrationinsidethesemiconductor,n=nexp[(F- E)/kT],increases asone approaches theoxide-semiconductor interface. This particular situation,where themajoritycarrierconcentration is greaternear theoxide-semiconductor interface than in thebulk of the semiconductor,is known as accumulation. When viewedfroma charge standpoint,theapplicationofV>0 places positive charges on theMOS-C gate.Tomaintain a balanceof charge,negativelycharged electrons must be drawn toward the semiconductor-insulator interface—the same conclusion estab- lished previously by using the energy band diagram.Thus the charge inside the device as a function of position can be approximated as shown in Fig.16.5(b). Consider next theapplicationof a small negative potential to theMOS-C gate.The applicationofasmallV<0slightlyraisesEinthemetalrelativetoEFinthe semicon- ductorand causes a smallnegativesloping of theenergybands inboththe insulatorand semiconductor,asdisplayed inFig.16.5(c).From the diagram itisclearthattheconcentra- tionof majoritycarrierelectronshasbeen decreased,depleted,inthe vicinityoftheoxide- semiconductor interface.A similar conclusion results from charge considerations.Setting V<0placesaminuschargeonthegate,whichinturnrepelselectronsfromtheoxide- semiconductor interfaceand exposes the positively charged donor sites. The approximate chargedistributionisthereforeasshowninFig.16.5d).Thissituation,wherethelectron and hole concentrations at theoxide-semiconductor interface are less thanthe background doping concentration (NAor ND),isknown forobvious reasons as depletion. Finally,supposealargerand largernegative biasis appliedtotheMOS-Cgae.AsV is increased negativelyfrom the situation pictured inFig.16.5(c),thebands atthesemicon- ductor surface will bend up more and more. The holeconcentration at the surface(p)will likewise increase systematicallyfromless thann; when E(surface)<E,to n; when E（surface)=EF,to greater thannwhenE(surface)exceedsE.Eventually,the hole concentration increases to the point shown in Fig.16.5(e)and (f),where E(surface)-E(bulk)=2[EF-E(bulk)] (16.2) --- Page 592 --- MOS FUNDAMENTALS 569 +Q Accumulation ( {0) x -Q M S (a) (b) Ionized donors +Q Depletion small x -Q (c) (d) Holes +Q Onset of inversion a x Q (e) (f) Holes lonized donors +Q Inversion x $(^{{}$ Q (g) (h) Figure16.5Energy banddiagramsandcorresponding blockchargediagramsdescribingthestatic state in an ideal n-type MOS-capacitor. --- Page 593 --- 570 FIELD EFFECTDEVICES and ps=neE(surface)EkT=nelE-E(bulk)k=bulk= (16.3) Clearly,whenp=NDforthe special appliedbias V=Vthe surface is no longerde- pleted.Moreover,forfurtherincreases innegativebias(<),exceedsnblk=ND and the surface region appears to change in characterfrom n-type to p-type.In accordance with thechange incharacterobservation,theV<Vsituation wheretheminoritycarrier concentrationat the surface exceeds thebulkmajoritycarrierconcentration is referredto as inversion.Energy band andblockcharge diagramsdepicting the inversion condition are displayed inFig.16.5(g)and (h). If analogous biasing considerations are performed foran ideal p-type device,there- sults willbeas shown inFig.16.6.Itis importanttonotefrom thisfigurethatbiasing regions in a p-type device are reversed in polarityrelative to the voltage regions in an n- type device;that is,accumulation in a p-typedeviceoccurs when V<0,and so forth. In summary,then,one can distinguish three physically distinct biasing regions-ac- cumulation,depletion,andinversion. Foranidealn-type device,accumulation occurs whenV>0,depletion when V<V<0,andinversion when V<V.Thecited voltage polarities are simply reversedfor an ideal p-typedevice.No bandbending in the semiconductor or flat band at VG=0marks the dividing line between accumulationand depletion.ThedividinglineatVG=Vissimplycalledthedepletion-inversiontransition Energy band diagram Applied dc $V_{} 00$ $V_{<00$ $V>V{>0$ V_{>V$ voltage i ！ 1 +Q +Q 1 +Q Exposed i acceptors Charge 1 1 1 diagram 0- i i -Q 1 i 1 -Q i 1 1 - i i -- i 1 Electrons Name Flat band Accumulation Depletion Inversion Figure16.6 Energy band and block charge diagrams for a p-type device under flat band, accu- mulation, depletion, and inversion conditions. --- Page 594 --- MOS FUNDAMENTALS 571 point, withEq.16.2)quantitativelyspecifyingtheonsetofinversionforbothn-andp- type devices. Exercise16.1 P:Constructlineplots（withVGplottedalongthe x-axis）that visuallyidentifythe voltage ranges corresponding to accumulation,depletion,and inversion in ideal n- and p-type MOS devices. S: The "plots” shown below are in essence a graphical representation of the word summary givenat theendofthepreceding section.Notethatacc,depl,and invare standardabbreviations for accumulation,depletion,and inversion,respectively. INV DEPL ACC VG n-type $VT}$ 0 ACC DEPL INV → VG p-type 0 VT$ 16.3 ELECTROSTATICS—QUANTITATIVEFORMULATION 16.3.1Semiconductor Electrostatics PreparatoryConsiderations Thepurposeofthis section is to establishanalyticalrelationshipsforthecharge density (),theelectricfield(),andtheelectrostatic potentialexistinginsideanidealMOS-C under staticbiasing conditions.Thetask is simplifiedby noting thatthemetal is anequi- potential region.Charge appearing near the metal-oxide interface resides only a few Ang- stroms（1Ångstrom=10-cm)intothemetaland,toahigh degreeofprecision,maybe modeled as a δ-functionofcharge at theM-O interface.Since by assumption there is no charge intheoxide(idealization3),themagnitudeof thecharge in the metal is simply equal to the sumof thecharges inside the semiconductor.Also,as noted previously, with no charges intheoxide,itfollows thattheelectricfieldisconstantintheoxideandthe potentialisa linearfunctionofposition.Inotherwords,solving fortheelectrostatic ari- ables insidean idealMOS-Cessentiallyreduces to solvingfortheelectrostatic variables inside the semiconductor component of the MOS-C. The mathematical descriptionof theelectrostatics insidethe semiconductor is esta- --- Page 595 --- 572 FIELDEFFECTDEVICES lished inarelativelystraightforward manner beginning with Poisson'sequation.In thefol- lowing analysis we invoke the depletionapproximation toobtain a first-order closed-form solution.Thedevelopmentcloselyparallels thepresentation inthe pn junctionanalysisof Chapter5.Itshouldbe pointedout,howeverthatanexactsolution is possible intheMOS- C case. Theexactsolutionstemsfrom simplificationsassociated withthefact thatthe semiconductor in an idealMos-C is always in equilibrium regardless of the applied d.c. bias.For reference purposes the exact solution is presented in Appendix B. In performing theanalysis(x)is takento bethe potential inside the semiconductor at a given point x;xis understood to be the depth into the semiconductor as measured from theoxide-semiconductor interface[see Fig.16.7(a)].The symbol ,instead ofV,is used in MOS-C work to avoid possible confusion with externally applied potentials.Inaccor- dance withidealization 5,the electric field (=-dpldx)is assumed to vanishasone proceeds into the semiconductor substrate.Following standardconvention,thepotential is chosen to be zero in the field-free region of the substrate referred to as the semiconductor bulk. evaluated at theoxide-semiconductor interface (atx=0)is given the special symbol, s,and is known as the surface potential. Figure16.7(b)indicateshow (x)is relatedto band bendingon theenergy band dia- gram. As shown (and consistent with Eq.3.12), 1 (x)= -[E₁(bulk)-E(x)] (16.4) q and 1 φs$ [E(bulk)-E(surface)] (16.5) q Figure16.7(b) also introduces an importantmaterial parameternamely, 1 f$ = 一 [E(bulk)-F] (16.6) q F is clearly related to the semiconductor doping.Forone, the sign of F indicates the doping type; directly from the definition one concludes F>0if the semiconductor is p- type and F<0if the semiconductor is n-type.More importantly,the magnitude of F is functionallyrelatedtothedopingconcentration.Givenanondegenerate Si substrate main- tained at or near room temperature,we know from Chapter 2 that Pbulk=ne[Ei(bulk)−EF]/k=NA. . .ifNA≥ND (16.7a) nbulk=n;e[EF−E(bulk)]/kT=ND. . .ifND≥NA (16.7b) --- Page 596 --- MOSFUNDAMENTALS 573 φ φs Oxide Semiconductor →x 0 (a) Ec$ φb -Ei ↑q -EF Ev$ |→Bulk →x 0 (b) Figure16.7Electrostaticparameters:(a)Graphicaldefinitionofand s.(b)Relationshipbe- tween (x) and band bending; graphical definition of . Thus, combining Eqs.(16.6)and (16.7)yields kT $ln(NA/n₁)$ . .. p-type semiconductor (16.8a) q $ = kT $ln(ND/n₁) ... n-type semiconductor (16.8b) q Extensive use will be made of the s and F parameters throughout the MOS discus- sion.Ourimmediate interest inthese parameters involves theiruse inquantitatively speci- fyingthebiasing state insidethe semiconductor.Clearly,underflat bandconditions s=0.Moreover,substituting Eqs.(16.5) and (16.6) into Eq.(16.2),one concludes $φ_s}= 2$ at thedepletion-inversion transition point (16.9) --- Page 597 --- 574 FIELDEFFECTDEVICES With F>0in a p-type semiconductor,itfollows that s<0 if the semiconductor is accumulated,0<s<2 ifthe semiconductor is depleted,and φs>2if the semi- conductor is inverted.For an n-type semiconductor theinequalities are merelyreversed. Exercise16.2 P:(a)Construct line plots （with splottedalongthe x-axis）that visually identify the surface potential ranges correspondingtoaccumulation,depletion,and inversion in ideal n-type and p-type MOS devices. (b) ForeachoftheF,sparametersets listedblowindicatethedopingtypeand the specified biasing condition.Also draw the corresponding energy band diagram and block charge diagram that characterize the static stateoftheideal MOS system. $ φs$ (i) $kT/{$ = 12, $kT/q$ = 12 _$ φs$ (ii) -9, =3 $kT/q $kT/{q _$ φs$ (iii) -9, -18 $kT/q$ = $b1I\}$ φ}$ φs$ (iv) 15, =36 $kT/q$ $kT/q φ_$ φs (v) -15, =0 $\k/q$ $kT/q$ S:(a)Convertinthedicussionattheendoftheprecedinsubsctionintoaphi- cal representation yields INV DEPL ACC → φs F < 0(n-type) 2\$ 0 ACC DEPL INV → φs F>0 (p-type) 0 2$ --- Page 598 --- MOS FUNDAMENTALS 575 (b) Biasing Energy Band Block Charge Set Doping Condition Diagram Diagram +Q (i) P Depletion EF -Q EF (ii) n Accumulation Holes (iii) n Depl /Inv Transition $EF$ (iv) P Inversion Electrons EF (v) n Flat Band M O S --- Page 599 --- 576 FIELDEFFECTDEVICES Delta-Depletion Solution The approximateclosed-form solution for theelectrostatic variables based in parton the depletionapproximation is convenientlydivided into three segments corresponding tothe three biasing regions of accumulation,depletion,and inversion. Let us firstconsideraccumulation.Figure16.8 displayscharge densityand potential plots constructed using theexact solutionfound in Appendix B.After verifying the general correlation between the Fig.16.8 plots and the p-bulk semiconductor portionof the dia- 400 300 ρ -20 $q}}$ 200 Accumulation A layer $k}$ 100 -10 0 0 1 0 0.5 1.0 0.5 1.0 x(μm) x(m) (a) Accumulation (φs=-6k/q) x(μm) x(m) 0 0.5 1.0 0.5 1.0 0 0 1 $\rb}$ ρ -1.0 1 $k$ φ 10 Depletion region -2.0 20 (b) Middle of depletion(Φs=ΦF=12kT/q) Figure16.8 Exact solutionfor thecharge densityand potential insidethe semiconductor como- nent of an MOS-Cassuming ΦF=12kT/q andT=300K(kT/q=0.0259 V).(a)Accumulation 24kT/q）,and(d)heavilyinverted(s=2F+6kT/q=30kT/q).Theρ-diagrams were drawnona linear scale and the + axes oriented downward to enhance the correlation with the diagrams sketched in Fig.16.6.The dashedlinesonthe part(b)through(d) p-plots outline the depletion approximation version of the charge distribution. --- Page 600 --- MOS FUNDAMENTALS 577 x(μm) x(μm) 0 0.5 1.0 0.5 1.0 0 0 $qN}$ -1.0 A $kT_$ 10 -2.0 20 (c) Onset ofinversion （φs=2ΦF=24kT/q) x(m) x(μm) 0 0.5 1.0 0.5 1.0 0 0 -1.0 10 ρ $qN}}$ A -200 $k}$ φ 20 Inversion -300 30 layer -400 40 (d) Deepintoinversion（φs=2Φ+6kT/q=30kT/q) Figure16.8 Continued. grams sketched in Fig.16.6,specifically note from Fig.16.8(a)thatthe charge associated with majority carrier accumulation resides in an extremely narrow portion of the semi- conductor immediately adjacent to the oxide-semiconductor interface. By comparison, the depleted portionofthe semiconductor undermoderate depletionbiasing shown in Fig.16.8(b)extends muchdeeper into the semiconductor.Given the narrow extentof the accumulation layerit wouldappear reasonableas afirstorderapproximation to replace the accumulation charge with a δ-function of equal charge positioned at the oxide- semiconductor interface. Indeed, we have just described the delta-depletion solutionfor accumulation. Because of the assumedδ-functionof charge at x=0,it automatically follows thattheelectricfieldandelectrostaticpotentialareidentically zeroforallx>0 underaccumulationbiasing inthe delta-depletionsolution.This is clearly somewhatinac- curate,but acceptableas a first-orderapproximation. --- Page 601 --- 578 FIELDEFFECTDEVICES Turning next to inversion,we notefromFig.16.8(d)that like theaccumulation lyer charge,the charge associated with minority carrier inversion resides in an extremely nar- row portion of the semiconductor immediately adjacent to the oxide-semiconductor inter- face.Moreover,in comparing the depleted semiconductor regions when s=F (middle of depletion),φs=2（onsetof inversion),and φ=2F+6kT/q (inversion),wefind the depletion width increases substantially with increased depletion biasingbut increases only slightly once the semiconductor inverts. Based onthe firstof the foregoing observa- tions,theactual inversionlayercharge is approximatelymodeled in the delta-depletion solution by a δ-functionofequal charge positioned at the oxide-semiconductor interface. Toaccountforthesecondobservation,itisadditionallyassumedtheδ-functionofcharge added in inversionprecisely balances thechargeaddedto theMOS-C gate. As a conse- quence,inthedlta-depltionsolutionorinvrsionbiasesthedeletiongioch x>0electricfield,and thex>0electrostatic potential remainfixed at their s=2F values. In other words,the inversionbias solution is established by merely adding a δ-function of surface charge to the solution existing at the endof depletion. Theremaining biasing region to beconsidered is depletion.In the standard depletion approximation theactual depletioncharge is replaced with a squared-off distribution ter- minated abruptly a distance x= Winto the semiconductor. Assuming a p-type semicon- ductor and invoking the depletion approximation,one can write ρ= q(p-n+D-NA)=-qA(0≤x≤W) (16.10) Poisson's equation then reduces to d8 $qNA}$ (0≤x≤W) (16.11) dx $K{ε$ $Kε$ Thestraightforward integrationof Eq.（16.11)employing theboundarycondition=0at x= Wnext yields Φp qNA$ 8(x)= dx = $K_{ε0$ (W-x) (0≤x ≤W) (16.12) Asecond integration with =0 at=Wgives $qNA$ φ(x)= (W- x){}$ (0≤x≤W ) (16.13) $2Ksε0$ The final unknown in theelectrostaticrelationships,the depletion width Wis determined from Eq.(16.13)by applying the boundary condition =s atx=0.We obtain $qN_ W^{}$ (16.14) 2K_{s0$ --- Page 602 --- MOS FUNDAMENTALS 579 and therefore 2K{s0$ 1/2 W= φs$ (16.15) $qN{}$ Taken together,Eqs.(16.10),(16.12),(16.13),and(16.15)constitutethedesireddepletion bias solution.For an n-bulk device NAin the preceding equations is replaced by-ND. Beforeconcluding,special note shouldbe madeof the depletion width, W,existing at the depletion-inversiontransitionpoint.Inthe delta-depletionformulation,Wisof course the maximum attainable equilibrium depletion width. Since W= W when s=2F,simple substitution into Eq.(16.15) yields $2K_$ 71/2 $W_{T}$ = (2\f)$ (16.16) qNA}$ A plotof W versus doping covering the typical range of MOSdoping concentrations is displayed in Fig. 16.9. $10^{}$ Si,T=300K $10{\$ $W{(μm) $10^{$ 10-2 $10{1 $\0{sr}$ $10{$ $10{1}$ $1\0{}$ NA or ND(cm^-{3 Figure16.9 Doping dependenceof the maximum equilibrium depletion width inside silicon de- vices maintained at 300 K. --- Page 603 --- 580 FIELD EFFECTDEVICES 16.3.2Gate Voltage Relationship In Subsection16.3.1the biasing state was described in terms of the semiconductor surface potential,.Results formulated in this manner are dependentonlyon the properties of the semiconductor.s,however,is an internal system constraint or boundary condition. It is the externallyapplied gate potential,VG,that is subjectto directcontrol. Thus,ifthe resultsof Subsection16.3.1 are to beutilized in practical problems,anexpressionrelating VG and s must be established. This subsection is devoted to deriving the required relationship. We begin by noting that VGin the idealstructure is dropped partlyacross theoxide and partly across the semiconductor, or symbolically, G=semi +Δ0x (16.17) Because =0in the semiconductor bulk,however,the voltage drop across the semicon- ductor is simply ∆φsemi =φ(x= 0)= φs (16.18) The taskof developing a relationshipbetween VGand s is thereforereducedtothe prob- lem of expressing Δx in terms of s. As stated previously(Subsection16.2.2),in anideal insulator with no carriersor charge centers d{_ox}=0$ (16.19) dx and dφox = constant (16.20) = xp Therefore 0 二 8 xdx=x$ (16.21) OX -x0 ox ox where xis theoxide thickness. The next step is to relate Eox to the electricfield in the semiconductor. The well-known boundary conditionon thefields normalto an interface between two dissimilarmaterials requires (Dsemi - Dox)O-S interface= QO-s (16.22) --- Page 604 --- MOS FUNDAMENTALS 581 whereD=εisthedielectricdisplacementandQosis thecharge/unitarealocatedat the interface.Since Qos=0in theidealized structure (idealization3), D = D$ semilx=0 (16.23) ox $K_{s}$ &s 8 (16.24) ox Ko and $K_s}$ $Δ_φ{oxx$ $x_{$ (16.25) Ko}$ Kis the semiconductor dielectric constant;K,theoxide dielectric constant;and &,the electricfield in thesemiconductorat theoxide-semiconductor interface.Finally,substi- tuting Eqs.(16.18)and(16.25)into Eq.(16.17),andrecognizing that Es isa knownor readily determined function of s,we obtain $K_s}$ $x_0s$ (16.26) Ko If the results of the delta-depletion solution are employed, a combinationof Eqs.(16.12) and (16.15) gives 2qNA φs 1/2 $8_s=$ (16.27) $K{ε$ and K_s$ $2qNA$ + Ko xo $K{_0$ sφ} (0≤φs≤2φF) (16.28) V Ifthedelta-depletionformulationisinvoked,theδ-function layersofcarrierchargeattheO-Sinterfacewould technicallycontributeaQosunderaccumulationandinversionconditions.However,=0forallaccumulation biases and =2Fforall inversion biases in the delta-depletionsolution.In thecitedformulation,therefore the VG-srelationship we are deriving wouldonly be used in performing depletion-biascalculations. SinceKs=11.8forsiliconandK0=3.9forSiO2，weconcludefrom Eq.（16.24)thatx=3inan MOS system with nochargeat the Si-SiO2interface.All energy band diagrams in this chapter should be consistent with this observation. --- Page 605 --- 582 FIELDEFFECTDEVICES 4 Exact 3 Delta-depletion VG (volts) 2 1 $Vt$ φs L $kT/{q}$ -10 HHHHH 0 10 20i 30 40 i $2\r}$ $k}$ -4 Figure16.10 Typical interrelationshipbetweentheapplied gatevoltage and thesemiconductor surface potential;HH delta-depletion solution, exact solution(=0.1 m, NA=$10^{5/cm3,T=300K). TheVG-s dependence calculatedfrom Eq.(16.28)employing a typical setof device parameters is displayed in Fig.16.10.Also shown is the corresponding exact dependence. The figure nicely illustrates certainimportant featuresofthegate voltagerelationship.For one, s is a ratherrapidly varying function of VG when the device is depletion biased. However,when the semiconductor is accumulated (s<0) or inverted (s>2F),it takes a large change in gate voltage to produce a small change in .This implies thegate voltage dividesproportionally between theoxideand the semiconductor under depletion biasing. Underaccumulation and inversion biasing,on theother hand,changes in the ap- plied potential are dropped almost totally across theoxide.Also note that the depletion bias region isonlyslightlygreaterthan1volt inextent.Sincethecharacterof thesemiconductor changes drastically in progressing fromone sideof thedepletion biasregion to theother, we are ledtoanticipateasignificantvariationintheelectricalcharacteristicsoveraraher narrow range of voltages. --- Page 606 --- MOS FUNDAMENTALS 583 Exercise 16.3 P:An MOS-Cis maintained atT=300K,=0.1 m,and the silicon doping is NA=10{1/cm3}Compute: (a) F in kT/q units and in volts (b) Wwhen φs=φF (c) Es}when φs = F (d) VG when φs=F S:(a) _$ 1015 =ln(NA/n₁）=ln( =11.51 $kT/q$ $\f0}$ F=11.51(kT/q)=(11.51)(0.0259)=0.298V (b) Using Eq.(16.15), 1 1/2 2(11.8)(8.85×10-14)(0.298) 1/2 W= φ}$ = 0.624 μm $qN_}$ (1.6×10-19)(1015) (c) Evaluating Eq.(16.12)atx=0yields E.Thus $qNA}$ (1.6×10-19)(101)(6.24×10-5) &s = $K{sε0$ W= (11.8)(8.85×10-14) 9.56× 103V/cm (d) Substituting into Eq.(16.26) gives $K_{s}$ $V{}$ $ + Ko $x_{0{s$ ...Es evaluated at F (11.8)(10-5)(9.56×103) = 0.298+ = 0.587 V 3.9 Comment: Themanipulationsandresults inthisexercisearefairlyrepresentative. InkT/q units,F|typically ranges between9and18atT=300K.Forthe non- degenerate dopings used in MoS devices,one also expects to be less than one- half a Si band gap expressed in volts (<0.56VatT=300K).As required,the calculated Wis less than WTassociated withthegiven doping (see Fig.16.9).The onlypossiblesurpriseisthe sizeofthesurfaceelectricfield;~104V/cm.Finally, --- Page 607 --- 584 FIELD EFFECTDEVICES the device parameters assumed in this exercise are identical to those used in con- structing Fig.16.10.As must be the case,thecomputed VG agrees with thevalue read from the Fig.16.10 plot. 16.4 CAPACITANCE-VOLTAGECHARACTERISTICS With d.c.currentflowblocked bytheoxide,the majorobservableexhibited by MoS-Cs is capacitance.The capacitance varies as a functionofthe applied gate voltage and the mea- sured capacitance-voltage(C-V)characteristic isofconsiderable practical importance.To the device specialist,the MOS-C C-V characteristic is like a picture window,a window revealing theinternal natureof thestructure.Thecharacteristicservesasa powerful diag- nostic toolfor identifying deviations from the ideal in boththeoxide and the semiconduc- tor.Forthereasonscited,MOs-CC-Vcharacteristicsareroutinelymonitored duringMOS device fabrication. In most laboratories andfabrication facilities,theC-Vmeasurements are performed withautomated equipment such as that previously described in Subsection 7.2.3 andsche- maticallypictured in Fig.7.6.TheMOS-C is positionedon a probing station,normally housed in a light-tight boxto excluderoom light,and is connected by shielded cablesto a C-Vmeter.The meter superimposes a small a.c.signal on topofapreselectedd.c.voltage and detects theresulting a.c.currentflowing through the test structure. The a.c.signal is typically15mVrmsorless,andacommonsignalfrequency is1MHz.Built-inprovisions are made for slowly changing the d.c.voltage to obtain acontinuous (or quasi-continuous) capacitance versus voltage characteristic.Outputfromthemeter is usually displayedon a computer monitorand a hardcopyof the data is produced employing a printer or plotter. Somewhat more sophisticated commercial equipment combining a high-frequency (1MHz)C-Vmeter withlow-frequency (quasistatic)measurementcapabilities is pictured in Fig. 16.11(a). This section is primarily devoted to modeling theobserved formof theMOs-CC-V characteristic intheso-calledlow-frequencyandhigh-frequency limits.These limiting- case designationsoriginallyreferredtothefrequencyofthe a.c.signal used in thecapaci- tance measurement.Althoughthetheoreticaltreatment inthischapter isrestrictedtothe ideal structure,practical measurementconsiderationsare includedattheendofthesection. 16.4.1 Theory and Analysis Qualitative Theory High-and low-frequencyC-V data derivedfrom a representative MOS-capacitor are dis- played in Fig.16.11(b).To explain theobserved formof the C-Vcharacteristics,letus consider how the charge inside an n-type MOS-C responds to the applied a.c.signal as the d.c.bias is systematicallychangedfrom accumulation,throughdepletion,to inversionW --- Page 608 --- MOS FUNDAMENTALS 585 (a) Low-frequency 1.0 0.8 0.6 C 0.4 High-frequency 0.2 -5 -4 -3 -2 1- 0 1 2 VG(volts) (b) Figure16.11 (a) TheKeithley simultaneous high-and low-frequencyC-V system.(b)Sample MOS-Chigh-and low-frequency capacitance-voltage characteristics.The device was fabricated on ND=9.1×1014/cm3(100)Si;=0.119μm.[Part（a)appears intheKeithley1993-1994Test & Measurement Catalog. Photograph courtesy ofKeithley Instruments, Inc. --- Page 609 --- 586 FIELDEFFECTDEVICES begin withaccumulation.Inaccumulation the d.c.state is characterized by the pileupof majority carriers right at theoxide-semiconductor interface.Furthermore,under accumu- lationconditionsthestateofthesystemcan bechanged veryrapidly.Fortypical semicon- ductor dopings,themajoritycarriers,theonlycarriers involved in theoperationofthe accumulated device,canequilibrate witha timeconstantontheorderof10-10to10-13 sec. Consequently,at standard probing frequenciesof1MHz or less it is reasonableto assumethe devicecanfollow theapplieda.c. signalquasistatically,withthe smalla.c. signal adding or subtracting a small ΔQonthetwo sidesof the oxide as shownin Fig.16.12(a).Since the a.c.signal merelyadds or subtracts a charge close to the edgesof an insulator,thechargeconfigurationinsidetheaccumulatedMOS-Cisessentiallythatof anordinary parallel-platecapacitor.Foreither loworhigh probing frequencies wetherefore ΔQ Q Q ΔQ $w_}$ -Q -Q -△Q -△Q M S M 0 S IHT Co CoCs$ (a) (b) △Q ΔQ $W_o}$ $W_{}$ -△Q -△Q (c) (d) Figure16.12 a.c.chargefluctuations inside an n-type MOS-capacitor under d.c.biasing condi- tions corresponding to (a) accumulation,(b)depletion,(c) inversion when ω→0,and (d)inversion whenω→∞.Equivalentcircuitmodels appropriateforaccumulationand depletion biasing arealso shown beneath the block charge diagrams in parts (a)and (b),respectively. --- Page 610 --- MOS FUNDAMENTALS 587 conclude $KOε{AG C(acc) = C0 (16.29) $x_o$ where AG is the area of the MOS-C gate. Under depletion biasing the d.c. state of an n-type MOS structure is characterized by a-Qcharge on the gate and a +Qdepletion layercharge in the semiconductor.The depletionlayercharge is directlyrelated,ofcourse,to thewithdrawalofmajoritycarriers from aneffective width Wadjacent to theoxide-semiconductor interface. Thus,once again,onlymajority carriers are involved in theoperationofthe deviceand thecharge state inside the systemcan bechanged veryrapidly.As picturedin Fig.16.12(b),whenthe a.c. signal places an increased negativecharge ontheMOs-C gate,the depletionlayer inside the semiconductor widens almost instantaneously; that is,the depletion widthquasistati- callyfluctuatesaboutitsd.c.valueinresponsetotheapplieda.c.signal.Ifthe stationary d.c.chargeinFig16.2(b)isconceptuallyliminatdallthatremainsismallfuctuating charge onthe two sidesofa double-layer insulator.Forallprobingfrequencies this situ- ation isclearlyanalogousto two parallel plate capacitors(CoandCs)inseries,where $KO{AG C_o$ = (oxide capacitance) (16.30a) $x_{o$ $K_{{AG$ $C_{s=$ (semiconductor capacitance) (16.30b) W and $C_{_{s$ Co$ C(depl) = (16.31) +0 $K{OW}$ 1+ $K{s}$ Note from Eq.(16.31)that,because Wincreases with increased depletion biasing,C(depl) correspondingly decreases as the d.c. bias is changedfromflatband to the onsetof inversion. Once inversion isachieved we know thatan appreciable numberof minority carriers pile up near theoxide-semiconductor interface in response to the applied d.c.bias.Also, the d.c.widthof the depletionlayertends to maximize at W.The a.c.charge response, however,is not immediatelyobvious.Theinversion layercharge mightconceivablyfluc- tuate in responsetothe a.c.signal as illustrated inFig.16.12(c).Alternatively,the semicon- ductor charge required tobalance ΔQchanges in the gatecharge mightresultfrom small variations in the depletionwidthas pictured inFig.16.12(d).Even a combinationof the two extremes is a logical possibility. Theproblem is to ascertain which alternative de- scribes the actual a.c.charge fluctuation inside an MOS-C.As it turns out,theobserved charge fluctuation dependson thefrequencyof the a.c. signal used inthe capacitance measurement. --- Page 611 --- 588 FIELDEFFECTDEVICES First ofall,ifthe measurementfrequency is verylow (ω→0),minoritycarrierscan be generated or annihilated in response to the applied a.c.signal and the time-varying a.c. state is essentially a successionofd.c.states.Just asin accumulation,one has asituation (Fig.16.12c)where charge is beingaddedor subtracted closeto theedgesofa single-layer insulator.Wetherefore conclude C(inv) ≈Coforω→0 (16.32) If, on the other hand,the measurement frequency is very high (ω →∞),the relatively sluggishgeneration-recombinationprocess willnotbeableto supplyoreliminateminority carriers inresponsetotheapplied a.c.signal.Thenumberofminoritycarriersin theinver- sion layerthereforeremainsfixedatitsd.c.valueandthedepletionwidthsimplyfluctuates aboutthe Wd.c.value.Similartodepletionbiasingthissituation(Fig.16.12d)isequiva- lent to two parallel-plate capacitors in series and $C_{ C_s$ Co$ C(inv)= for ω→∞ (16.33) $C_{o}+ _s}$ 一 $KOW_q}$ 1+ $K_{s_{}$ WT being a constant independent of the d.c. inversion bias makes C(inv)ω→∞= suchthataportionof theinversion layerchargecanbecreated/annihilatedinresponseto the a.c.signal,an inversioncapacitance intermediatebetweenthe high-and low-frequency limits will be observed. An overalltheory can nowbeconstructedbycombining theresultsof the foregoing accumulation,depletion,and inversionconsiderations.Specifically,weexpect the MOS-C capacitance to be approximatelyconstant at Cunderaccumulation biases,to decrease as the d.c.bias progresses through depletion,and to be approximately constant again under inversionbiasesata valueequalto~Cifω→0or C(depl)min ifω→∞.Morever,for ann-type device,accumulating gate voltages (where C≈C)are positive,inverting gate voltages are negative,and the decreasing-capacitance,depletion biasregion ison theordr of a voltor soin width.Quiteobviously,thistheoryforthecapacitance-voltage character- istics is in good agreement with the experimental MoS-CC-VG characteristics presented in Fig.16.11(b). Exercise 16.4 P:Complete the followingtable making useof theideal-structureC-Vcharacteristic and theblockcharge diagrams includedinFigure E16.4.Foreachofthebiasing conditions named inthetable,employletters (a-g)to identifythecorresponding bias point or points on the ideal MOs-CC-V characteristic. Likewise,use a num- ber(1-5)to identify the blockcharge diagram associated witheachof thebiasing conditions. --- Page 612 --- MOS FUNDAMENTALS 589 Bias Condition Capacitance (a-g) Charge Diagram (1-5) Accumulation Depletion Inversion Flat band Depl/inv transition C a b g f c e N-bulk p →_Gq ③ 2 Ionized donors +0 x x x -0 -Q M Tol S Mlo S 4 Holes Ionized donors Holes +Q +Q x x -Q 0- Figure E16.4 transition-d,5. S:Accumulation-g3;depletion-e,1;inversion-a/b/c,4;flatband-f,2；depl/inv --- Page 613 --- 590 FIELDEFFECTDEVICES Delta-Depletion Analysis Building on the development in the previous subsection,it is relatively easy to establish a first-orderquantitativetheorybasedon thedelta-depletionformulation.Specifically,inthe delta-depletionformulationthecharge blocksrepresentingaccumulationand inversion layers in Fig.16.12are formallyreplaced byδ-functionsof chargeright at theoxide- semiconductor interface.Consequently,Cin the delta-depletion solution is precisely equal to Cofor accumulation biases and for inversion biases in the low-frequency limit.On the other hand,the depletion relationship and the high-frequency inversion relationship (Eqs.16.31and16.33,respectively)can be used withoutmodification.Theblockcharge modelingofthedepletionregionsinFig.16.12conformsexactlywiththesimplifiedcharge distributionsassumed in thedepletionapproximation.Withintheframeworkofthedelta- depletion formulation, therefore, Co$ acc (16.34a) C_o}$ depl (16.34b) $K_{OW}$ 1+ $K_{}$ C= $Co$ inv (ω→ 0) (16.34c) C_o$ inv (ω→∞) (16.34d) $K{O0W_}$ 1+ $K_{s}$ Given a set of device parameters,one can compute Cρ and Wfrom previous relation- ships. For the analytical solution to be complete,however,the depletion-bias Win Eq.(16.34b)must be expressed as a functionof VG.Inverting Eq.(16.28)toobtain s(or more precisely,√)asafunctionof VG,andthen substitutingtheresultintoEq.(16.15), yields the required expression. We find $K_s}$ $V_{}$ W= Ko $x_o$ 1+ $V_{s}$ 1 (16.35) V where $K_{s{r}{2}$ $V_{$ 9 NA . .. p-bulk device (16.36) 2 $K{0$ (for n-bulk NA→-D) NotethatifEq.（16.35)issubstitutedintoEq.（16.34b),oneobtainstheverysimpleresult --- Page 614 --- MOSFUNDAMENTALS 591 Co}$ C= (depletion biases) (16.37) $V_{}$ 1+ $\sr{{}$ V A samplesetof low-andhigh-frequencyC-V characteristicsconstructed usingthe results of the delta-depletion analysis is displayed in Fig.16.13. 16.4.2Computations andObservations Exact Computations The delta-depletioncharacteristics,as typifiedbyFig.16.13,are arathercruderepresen- tationofreality. Thefirst-ordertheory doesa credible jobfor gate voltagescomfortably withinagivenbiasingregionbutfailsbadlyintheneighborhoodofthetransitionpoints going from accumulation to depletion and from depletion to inversion.A more accurate modeling oftheobservedcharacteristics isoftenrequired in practicalapplications and is established by working with the exact-charge distribution inside the MOS-capacitor. The resultsoftheexact-chargeanalysis are presented in AppendixC.Althoughthe derivation Low-frequency 1.0 0.8 0.6 C $\oveaeo}$ 0.4 High-frequency 0.2 1 1 1 -2.0 -1.5 -1.0 -0.5 0 0.5 1.0 V_G(volts) Figure16.13 Sample setof low-andhigh-frequencyC-Vcharacteristicsestablishedusingthe delta-depletion theory(x。=0.1m,ND=101/cm,T=300K). --- Page 615 --- 592 FIELDEFFECTDEVICES of the exact-charge relationships is beyondthescopeofthis text,theresults themselves are quitetractable.Highlaccurateideal-structureC-Vcharacteristicscanbeeadil constructed. A numberof sampleC-V characteristics calculated using the exact-charge relation- ships are displayed in Figs.16.14to16.16.These figures,respectively,exhibitthe general effectof varying the doping concentration(Fig.16.14),theoxide thickness (Fig.16.15), andthe device temperature (Fig.16.16).Note inparticularfromFig.16.14the significant increase inthe high-frequency inversion capacitanceand the substantial widening of the depletionbiasregion withincreased doping.In fact,at very high dopings (notshown)the capacitance approaches a constant independentof bias. This should not be an unexpected result,for with increased doping the semiconductor begins to look moreand more like a metal and the MoS-C should be expected to react more and more like a standard capacitor. As illustrated inFig.16.15,anincrease in theoxidethicknessalso widensthedepletion biasregionandaffects the high-frequency inversion capacitance.The increased widthof the depletion bias region with increased xis simply a consequence of a proportionate increase inthe voltage dropacrosstheoxidecomponentof thestructure.Finally,Fig.16.16 nicely displays themoderate sensitivityofthe inversion-bias capacitanceand the near in- sensitivityof the depletion-bias capacitance to changes in temperature. (C) Exercise 16.5 P:(a) Employing theexact-chargerelationships found in Appendix C,writea MATLAB (computer)）program thatcan be usedto calculate and plotlow-frequency CICOversus VG characteristics. Theprogram is to calculate C/Coand the corre- sponding VforU=s/kT/q)steppedinone-unitorlesincrementsfromU = UF-21toUF+21.U=/(kT/q).EmployK=11.8and=3.9;letT= 300K.OnlyNAandaretobeconsideredinput variables.SettingNA=10/cm, use your program to plotout the low-frequency C/Co versus VG curves for = 0.1μm,0.2m,and 0.3μm.Compare your program results with Fig.16.15. (b)Again employing the exact-charge relationships found in AppendixC,follow the same general computational procedures noted in part(a)to generate high-frequency CCOversus VGcharacteristics.Setting=0.1μm,outputhigh-frequencyCCO versusVcurvesforN=10/cm10/cm,and10/cm3.Comparyourro gramresults withFig.16.14.Beadvisedthatthehigh-frequencycalculation isfar more involvedand computation intensive than the low-frequencycalculation) S:(a)/(b) The constructionofa program or programs to generateexact-charge C-V characteristicsis a very informativeexercise.The reader is urged to at leastcomplete part (a)of the exercise before referring to theMOS_CVMATLABfile includedon disk and listed in Appendix M.In any event,characteristics generated bythe reader's program or the MOs_CVprogram shouldcompare favorably with the C-Vcurves plotted inFigs.16.14and16.15. --- Page 616 --- MOS FUNDAMENTALS 593 1.0 n-type 0.8 0.6 $\ora}$ ND=1016 $/cm{3 $\vra{o$ 0.4 /S0=a 0.2 ND=101{4 $/cm^{3 0.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 VG (volts) (a) 1.0 : p-type 0.8 0.6 ……… NA=10^{16/cm^³}$ \r} …… 0.4 NA =10{5/cm^3 0.2 $NA=10^{4}cm^}$ 0.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 VG(volts) (b) Figure 16.14 Doping dependence of the MOS-C high-frequency C-VG characteristic. Sample(a)n-typeand (b)p-typecharacteristicscomputedemploying the exact-charge theory( 0.1μm,T=300 K). --- Page 617 --- 594 FIELD EFFECT DEVICES 1.0 : n-type 0.8 0.6 \ $xo =0.3μm $\vra{o}$ xo =0.2μm 0.4 x_0=0.1μm 0.2 … 0.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 VG (volts) (a) 1.0 : : p-type 0.8 0.6 $\frac{}{0}$ x0=0.3μm $x_{0=0.2μm 0.4 x0=0.1μm 0.2 0.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 VG(volts) (b) Figure16.15 Oxide thickness dependenceof the low-andhigh-frequency MOS-CC-VG charac- teristics. Sample(a) n-typeand(b)p-typecharacteristics computedemploying the exact-charge theory(NAorND=105/cm,T=300K). --- Page 618 --- MOS FUNDAMENTALS 595 1.0 T= 0C T=100℃ 0.8 0.6 C $\Coero}$ 0.4 T= 100℃ 80 60 0 0.2 20 0 0.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 VG(volts) theory，x0=0.1μm,ND=5×104/cm3.) PracticalObservations In the discussion to this point we have more or less sidestepped any clarificationof pre- cisely what was meant by“lowfrequency”and“high frequency”in terms of actual mea- surementfrequencies.Onemightwonder,willa100Hz a.c.signaltypically yield low- frequency C-V characteristics? Perhaps surprisingly,the answer to thequestion is no. Givenmodern-dayMOS-Cswiththeir long carrier lifetimesandlowcarrier generation rates,even probingfrequencies as lowas10Hz,the practical limit in bridge-type measure- ments,will yieldhigh-frequencytypecharacteristics.Ifan MOS-Clow-frequencycharac- teristicisrequired,indirectmeanssuchasthequasistatictechnique]mustbeemplod toconstructthecharacteristic.Inthequasistatictechniqueaslow（typically10-100mV/ sec) linear voltage ramp is appliedtothe MOS-C gate andthe current into the gate is monitored as a functionofthe gate voltage.As is readily confirmed,thequasistaticdis- placementcurrentflowingthroughthedeviceisdirectlyproportionaltothelow-frquncy capacitance;properlycalibrated,themeasured current versus voltage datareplicatesthe desired low-frequency C-Vcharacteristic. On the high-frequency side,one cannot actually let ω→∞ and expectto observe a high-frequencycharacteristic.Measurementfrequencies,infact,seldomexceed1MHz.At higher frequenciesthe resistanceof the semiconductor bulk comes intoplayandlowersthe observed capacitance.Ateven higher frequencies(≥1GHz)one must begin to worry about the response time of the majority carriers. Normally,it is thehigh-frequency characteristicthat is routinelyrecordedandthe standard,almost universal measurementfrequency is1MHz.This is not to say the high- --- Page 619 --- 596 FIELD EFFECTDEVICES 1.0 0.8 0.6 C $\overco$ High-frequency 0.4 R=0 R=0.035 R=0.096 R=0.26 0.2 R=2.6 Total deep depletion Rin V/sec -12 01- -8 -6 -4 -2 0 2 4 VG (volts) Figure16.17 MeasuredC-Vcharacteristics as a functionof the ramprate(R).In inversionthe high-frequencycapacitancewasobtainedbystoppingtherampandallowingthedevicetoequilibrate. frequency characteristics can be recorded at1MHz withoutexercising a certain amountof caution.Suppose,for example,theC-V measurement is performed as described earlier in thischapter,withthe d.c.voltage being rampedfrom accumulation into inversiontoob- tain a continuous capacitance-versus-voltage characteristic.Figure16.17 illustratesthe usual results of such a measurement performed at various ramp rates.Note that at even the slowestramprateonedoesnotproperlyplotouttheinversionportionofthehigh-frequency characteristic.Onemust stop the ramp in inversionandallow thedevicetoequilibrate,or slowly sweep the device backwardfrom inversion toward accumulation,to accuratelyre- cord the high-frequency inversion capacitance. The discussion in the preceding paragraph really serves two purposes,the second being an entry into the important topicof deep depletion.Let us examine the ramped measure- ment (Fig.16.17)ingreater detail.Whentherampvoltage is in accumulationor depletion, onlymajority carriers are involved in theoperation of the device,and the d.c.charge con- figuration inside the structurerapidlyreacts to thechanging gatebias.As the ramppro- gresses fromdepletion into the inversion biasregion,however,a significant numberof minoritycarriers arerequired to attain an equilibriumcharge distribution within the MOS- C.Theminoritycarriers werenot presentpriortoenteringtheinversionbiasregion,cannot enter thesemiconductorfrom the remotebackcontactoracross theoxide,and therefore must becreated in the near-surface regionof thesemiconductor. Thegeneration process, --- Page 620 --- MOS FUNDAMENTALS 597 Missing holes △Q $_}$ Ⅲ $W_$ w→1 $w >W _}$ -△Q (a) (b) Figure16.18 (a)Nonequilibrium charge configuration inside ann-typeMOS-capacitor under deep-depletionconditions.(b)a.c.chargefluctuationsinsidetheMOS-Cwhenthesemicondcoris totally deep depleted. as we have noted several times,israther sluggishand has difficultysupplying the minority carriers needed for the structuretoequilibrate.Thus,as pictured inFig.16.18(a),thesemi- conductor is driven into a nonequilibrium conditionwhere,in balancing thechargeadded to the MOS-C gate,the depletion width becomes greater than WTto offset themissing minority carriers. Thedescribedcondition,the nonequilibriumconditionwherethere is a deficitofminority carriersand a depletion width inexcess oftheequilibrium value,is referred to as deep depletion. The existence ofa W>Wexplains thereduced valuesofCobserved in theramp measurement.Moreover,the decrease incapacitance withincreasedramprate indicates a greaterdeficitofminoritycarriersand awiderdepletionwidth.This is logical,sincethe greaterthe ramprate,the fewerthe numberofminoritycarriers generated prior to arriving at a given inversion bias. The limiting case as faras deep-depletion isconcernedoccurs when the semiconductor is totally devoid of minority carriers—totally deep depleted.Except for a wider depletion width,the total deep depletioncondition shown in Fig.16.18(b)is precisely the same as thesimple depletionconditionpictured in Fig.16.12(b).Consequently,byanalogy,and based on the delta-depletion formulation,the limiting-case capacitance exhibited by the structure under deep depletion conditions should be C_o}$ total deep depletion C= (16.38) (VG>Vp-type;VG<Vn-type) $V_{}$ 1 + $V_{δ}$ Equation(16.38)is inexcellentagreementwithexperimentalobservationsand isessen- tiallyidenticaltotheresultobtainedfromanexact-chargeanalysis.The2.6V/secramp rate curve shown in Fig.16.17 is anexampleofa total-deep-depletioncharacteristic. --- Page 621 --- 598 FIELDEFFECTDEVICES The deeply depletedcondition,we shouldnote,is centralto theoperationofthedy- namic random access memory(DRAM)and the charge-coupled devices(CCDs) men- tioned intheSection15.1GeneralIntroduction.DRAMs use deeply depletedMOS- capacitors as storage elements.InCCDimagers,carriercharge is opticallyproducedand temporarilystored inpartiallydeepdepleted potentialwellsbeneathanarrayofMOS-C gates. Exercise16.6 P:TheexperimentalC-Vcharacteristic shownin Fig.E16.6wasobservedunderthe following conditions:The d.c.bias was changed very slowlyfrompoint（1）to point2).Atpoint(2)theVGsweeprate wasincreasedsubstantially.Uponrriving at point (3)the sweep was stopped andthe capacitance decayed to point (4).Quali- tatively explain the observed characteristic. C 4 3 Figure E16.6 S:In going from point (1)to point （2)the semiconductorclearly equilibrates ateach d.c.bias point because of the very slow speed of the ramp.One thereforeobserves thestandardhigh-frequency C-Vcharacteristic.Withtheramprate increasedat point (2),the semiconductor is no longerable to equilibrateand is consequently driven into deep depletion. W becomes larger than WandC decreases below the minimum high-frequency equilibrium value.When the sweep is stopped and the ca- pacitance increases from point (3)to point(4),equilibrium is systematicallyrestored inside the device through the generationofminority carrier holes in the near-surface region.Astheholesare generatedtheyaddtotheinversion layer,the depletion width correspondingly decreases,andCincreases back to the equilibrium high-frequency value. --- Page 622 --- MOS FUNDAMENTALS 599 16.5 SUMMARYANDCONCLUDINGCOMMENTS This chapter was devoted to introducing basic MOSterminology,concepts,visualization aids,analytical procedures,and the like.We began bydescribingtheMOS-capacitorand clearly defining what was envisioned as the ideal MOS structure. The ideal MOS-C serves not only as a convenienttool forthe introductionof MOSfundamentals,butalso provides a pointofreferenceforunderstanding and analyzing themorecomplexbehaviorofreal MOS structures. Thereference natureof the ideal structure willbecome more apparent in Chapter18 where someof the idealizations will beremoved andtheensuing perturbations on the device characteristics will be carefully examined. TheinternalstateoftheMOS-Cunderstatic-biasing conditions was qualitatively describedusingenergybandandblockcharge diagrams. Withtheaidofthe diagrams the terms accumulation,flatband,depletion,and inversion were given a physicalinter- pretation. Accumulation corresponds to the pileup of majority carriers at the oxide- semiconductor interface;flat band,to no bending of the semiconductor bands,or equiva- lently,to no charge in the semiconductor; depletion,to the repulsionofmajority carriers from the interface leaving behind an uncompensated impurity-ion charge; and inversion,to the pileupof minority carriers at the oxide-semiconductor interface. As a pointof information,itshouldbementioned that,in certainanalyses,it is con- venient and reasonableto dividethe depletionbiasregionas defined herein into two sub- regions.In someMOS publicationsonetherefore findsthetermdepletionisonlyused in referring to band bendings between s=0 and s= F.Weak inversion is used to de- scribe band bendings from s=Ftos=2F.In addition,strong inversion (implying more inversion than weak inversion) replaces inversion as defined herein. Quantitativeexpressionsforthecharge density,electricfield,andelectrostatic poten- tial inside an MOS-C areestablished bysolving Poisson'sequation.Afirst-order solution based on the depletion approximationandδ-functionmodelingof thecarrier charges was presented in the chapterproper. An exact solutionfor theelectrostatic variablescanbe found in Appendix B. The qualitativeand quantitative formalism that hadbeen developed was nextapplied to modeling the MOs-CC-Vcharacteristics in the low-andhigh-frequency limits.Sample ideal-structurecharacteristics were presentedand examined for parametric dependencies. Finally,thepracticalmeaningof"low-frequency”and"high-frequency”wasclarified,and deep depletion was notedto occur when the MOS-C is not allowed to equilibrate under inversion biasing.Deep depletion isanonequilibriumconditionwherethere is a deficitof minority carriers and a depletion width in excess of theequilibrium value. --- Page 623 --- 600 FIELD EFFECT DEVICES PROBLEMS CHAPTER 16 PROBLEMINFORMATIONTABLE Complete Difficulty Suggested Short Problem After Level Point Weighting Description 16.1 16.3.1 2 10 (one each diagram) F,s sets/infer diagrams 16.2 " 2 8(a-2,b-3,c-3) Examine Fig.16.8(c) 16.3 " 2 8 GaAs WT vs.doping plot *16.4 16.3.2 2-3 16 (a::d-2, e-8) Calculate F,W,Es,G 16.5 " 3-4 15(a-6, b-9) Exact VG-s calculation 16.6 16.4.1 2-3 10(a-4,b-6) Derive/use Eq.(16.35) 16.7 " 3 20 (2 each part) Interpret E-band diagram 16.8 " 2 5 (1/2 each entry) CandE-bandvs.bias table 16.9 " 3 16 (2 each part) Given E-band, deduce info 16.10 " 3 8(a-2,b-2, c-4) Intrinsic MOS-C 16.11 16.4.2 3 15 (5 each part) SOS-C 16.12 " 2 8 (4 each part) Given C-V, answer questions 16.13 " 2-3 10 (2 each part) Given C-V,deduce info 16.14 " 2-3 16 (2 each part) Given MOS-C, ans. questions 16.15 " 2-3 12 (2 each part) Given Q-diag., deduce info 16.16 " 3 10 Deep depletion C-V 16.17 " 3 10(a-2, b-3, c-5) xand doping determination In all computations involving an MOs (Si-SiO2) capacitor maintained at T=300 Kemploy kT/q=0.0259 V,n=1010/cm3,Ks=11.8,andK0=3.9. 16.1Forthe F,sparametersetslistedbelowfirstindicatethespecifiedbiasingcondi- tion andthendrawtheenergybanddiagramandblockcharge diagramthatcharacterizethe static state of the system.Assume the MOS structure to be ideal. $φ_s= 3 }$ \}$ φs$ φf$ (a) kT/q = 18, kT/q = 9 (d) kT/q = -15, kT/q (b) -12, $φs=0$ (e) φf}$ =9， φs$ =21 kT/q kT/q $kT/q kTlq f$ $φ_= 24 (c) kTlq = 12, kT/q --- Page 624 --- MOS FUNDAMENTALS 601 16.2Let us examine Fig.16.8,particularlyFig.16.8(c),moreclosely. (a)Draw theblockcharge diagramdescribingthechargesituationinsideanideal -ulk MOS-C biased at the onset of inversion. (b) Is your part (a)diagram in agreement with theplotofρ/qNAversusxin Fig.16.8(c)? Explain why thep/qNAplothas a spike-like nature near=0 and shows a valueof ρ/qNA=-2atx=0. (c) Noting that F/kT/q)=12andT=300K was assumedin constructing Fig.16.8, determineWT.IsthededucedWTconsistentwiththeapproximatechargedistiution shown in Fig. 16.8(c)? 16.3ConstructaWTversusdopingplotsimilartoFig.16.9thatisappropriateforGaAs. AssumeT=300K;=12.85. 16.4 An MOS-Cis maintained atT=300 K,=0.1m,andtheSi dopingis ND= 101/cm^3}Compute: (a)F inkT/q units and in volts; (b)Wwhen φs=2F; (c) Eswhenφs=2F; (d）VG=Vwhen φs=2F.(How is thisresult related to Fig.16.10?) (e) Withthe Si doping（NAor ND),x。,and the depletion-bias s valueof theMOs-C maintained at300K as input variables,write a computer program that automatically calculates F,WEs,and V.Check theoutputof your program against theresults recorded in Exercise16.3.Use theprogram to verifythemanuallygenerated answers to parts (a)-(d) of this problem. 16.5(a)Making useofAppendix B，showthatthe exact-solutionequivalentofEq. (16.28) is kT $U_s}$ $K_{s}$ F(Us,U V$ $+\}$ = q where Us≡φs(kT/q)andUF=F/(kT/q).See Eqs.(B.5),(B.17),and(B.18)in Appendix Bfor the definitionsof LD,(U,U),and U,respectivey. (b) Constructa computerprogramthatcalculates VGas a functionofUs using the part（a） relationship.Only the semiconductor doping and xare to be considered input vari- ables;letT=300Kand stepUinone-unitincrementsfromU=U-21toU= UF+21.Runtheprogramassuming=0.1μmandND=10/cm3.Compareyour numerical results with theexact solution curve inFig.16.10. --- Page 625 --- 602 FIELD EFFECT DEVICES 16.6(a)Following the approach suggested in the text,derive Eq.(16.35). (b）Assuming=0.1μm,ND=105/cm,andT=300K，compute: (i) W_{； (ii) CiCoinversion (ω→∞); (iii)V(delta-depletion theory). (iv)Commenton the comparisonof yourCand Vresults withFig.16.13. 16.7Theenergyband diagramforanideal=0.2μmMOS-CoperatedatT=300Kis sketched inFig.P16.7.Notethattheappliedgatevoltagecausesbandbending inthesemi- conductor suchthatEF=Eat the Si-SiO2interface.Invokethe delta-depletionapproxi- mation as required in answering the questions that follow. EFM Ec EFs 0.29eV Ei 0.56 eV $Ev}$ →x 0 W Figure P16.7 (a) Sketch theelectrostatic potential()inside the semiconductor as a functionof position. (b) Roughlysketchtheelectricfield()insidetheoxideandsemiconductorasafunction of position. (c) Doequilibriumconditionsprevail insidethe semiconductor?Explain. (d) Rouhlskchthelctroconcntionvrsusositioninsidehesemcon (e) What is theelectronconcentrationattheSiSiO2interface? (f) ND = ? (g) $φ _s} =? (h) V_{}==$ (i) What is the voltage drop (Δx)across the oxide? (j)What isthenormalizedsmall-signalcapacitanceCC,oftheMOS-Catthepictured bias point? --- Page 626 --- MOS FUNDAMENTALS 603 16.8Complete thefollowing tablemaking useoftheideal-structureC-Vcharacteristic and energy banddiagrams inFigureP16.8.Foreachof thebiasingconditionsnamed inthe table,employ letters (a-e)to identifythecorresponding bias pointontheideal MOS-C C-Vcharacteristic.Likewise,use a number(1-5)toidentify the band diagramassociated with each of the biasing conditions. Bias Condition Capacitance (a-e) Band Diagram(1-5) Inversion Depletion Flat band $V_{}}$ Accumulation C a b c d →VG 2 3 4 5 Figure P16.8 --- Page 627 --- 604 FIELDEFFECTDEVICES 16.9FigureP16.9isadimensionedenergybanddiagramforanidealMOS-Coperated at T=300K with VG≠0.Notethat EF=Eatthe Si-SiO2interface. Ec k→k 0.56 eV $-Ei$ 0.30eV $EF$ $E} 0.6 eV EFM Figure P16.9 (a) Do equilibrium conditions prevail inside the semiconductor? (b) = ? (c) $\ _s=? (d) V_{G = ? (e) x{0} =$ (f）Drawtheblockcharge diagramcorrespondingtothestatepicturedintheenergy band diagram.Forreference purposes,include the maximum equilibrium depletion width, W, on your diagram. (g) SketchtheenealhapeofthelowrequecCVchaactristictobeexpectedom the given MOS-C.Place an × on the C-Vcharacteristic ata pointthatroughly corre- sponds to the state pictured in the energy band diagram. (h)Forthespecificbiaspointpicturedintheenergybanddiagram,whichofthefollowing is thecorrectexpressionfor thecapacitance exhibited by the structure?Explain. Co$ Co (i)C = (ii)C= $K_{OW}$ $V_s}$ 1+ $K_{s_}$ 1+ $2_{}$ Co$ Co$ $(iii)C= (iv) C= $K{OW_}$ $V_sq}$ 1+ $\sqrk}}$ 1+ $\sr{{}$ --- Page 628 --- MOS FUNDAMENTALS 605 16.10An idealMOScapacitor is constructedon asubstrateof intrinsicsilicon. (a)Sketchtheenergy band diagramforthecapacitorunder flatbandconditions.Include the diagramforallthreecomponentsoftheMOS-CproperlypositiontheFermilevel in the metal and semiconductor,and label the energy levels. (b)Constructblockcharge diagrams representing thecapacitorundertheconditionsof positive and negative gate bias. (c) Invoking the delta-depletion approximation,sketch the low-frequencyC-Vcharacter- isticforthegivenMOS-C.Justifytheformofyoursketch ineachregionofoperation. 16.11Withmodern-day processing it is possible to produce semiconductor-oxide-semi- conductor(SOS)capacitors in which asemiconductorreplacesthe metallicgate inastan- dard MOS-C.Answer the questions posed below assuming an SOS-C composed of two identical-typenondegeneratesiliconlectrodesanidealstructureandbiasingarrag- ment as defined by Fig.P16.11.Includeanycomments which mayhelptoforestall amis- interpretation of the requested pictorial answers. (a) Drawtheenergy band diagramforthe structure when(i)V=0,(i)V>0butsmall, (i)V>0and verylarge,(iv)<0but small,and(v)V<0and verylarge. (b)Draw theblockchargediagramscorrespondingtothefivebiasinconditionsconsid- ered in part (a). (c)S Sketchtheexpected shapeof the high-frequencyC-Vcharacteristic for theSOS-C described inthisproblem.Forreference purposes,also sketchonthe sameplotthe high-frequency C-VGcharacteristic of an MOS-C assumed to have the same semi- conductor doping and oxide thickness as the SOS-C. VG S 0 S Figure P16.11 --- Page 629 --- 606 FIELD EFFECTDEVICES 16.12(a）ConsidertheC-Vcurves shown in Fig.P16.12(a).For whichcurveorcurves is an equilibrium inversion layer present when V>V?Explain. (b) C-Vcurvesderivedfromtwo MOS-Cs with the same gate area(AG)arecompared in Fig.P16.12(b).The MOS-Cexhibiting curve b has (choose one: a thinner,the same,a thicker)oxideand (chooseone: a lower, the same,a higher) doping than the Mos-C C a b C →VG 0 $Vs }$ Figure P16.12(a) C a b VG 0 Figure P16.12(b) --- Page 630 --- MOSFUNDAMENTALS 607 16.13TheCVcharacteristicexhibitedbyanMOS-C（assumedtobeidealisdisplayd in Figure P16.13. C + 100 pF ① 20pF ② V_{q$ 0 Figure P16.13 (a) Is the semiconductorcomponentof the MOS-C doped n-type or p-type?Indicate how you arrived at your answer. (b) Draw theMOS-Cenergybanddiagramcorrespondingtopoint2)ontheC-Vchar- acteristic.(Be sure to includethediagramsforallthree componentsof theMOS-C, show theproper band bending in both theoxide andsemiconductor,andproperly po- sition the Fermi level in the metal and semiconductor.) (c) Draw theblockcharge diagramcorrespondingtopoint1)ontheC-Vcharacterisic. (d) IftheareaoftheMOS-Cis3×10-cm²,whatistheoxidethickness()? (e) Invokingthedelta-depletionapproximationdetermine Wandtheassociatedsemi- conductor doping concentration for the given MOs-C. 16.14AnidealMOS-CisoperatedatT=300K.=0.1μm,ND=2×1015/cman AG=10-3cm^2 (a)Sketchthegeneralshapeofthehigh-requencyC-Vcharacteristictobeexpecdfom the given device. (b) Defining CMAxtobethe maximumhigh-frequencycapacitancedetermineCMA (c) DefiningCtobetheminimumhighfrequenccapacitanceestimateCMIemoy ing the delta-depletion approximation. (d) IfV=V,determine s.(Give both a symbolic and a numerical answer.) --- Page 631 --- 608 FIELD EFFECTDEVICES (e) Compute VT. (f）Supposethegatebiasissuchthat=3F/2.DrawtheMOS-Cenergybanddiagram correspondingto thespecified gate bias.Besuretoinclude the diagramsforallthree componentsof theMOS-C,show the proper band bending in both the oxide and semi- conductor,and properly position the Fermi level in the metal and semiconductor.) (g) Supposethegatebiasissuchtheφs=5/2.Drawtheblockcharge diagramorre- sponding to the specified gate bias. (h)TheC-Vcharacteristicofthedeviceismeasuredasthedc.biasisrapidlysweptrom accumulationinto inversion.Using a dashedline,sketch theexpectedformofthere- sulting C-Vcharacteristiconthesamesetofcoordinatesastheparta)answer. 16.15Thed.c.stateofanidealMOS-capacitor ischaracterizedbytheblockchargedia- gram shown in Fig. P16.15. A $W_s}$ →x M 0 S Figure P16.15 (a)Is the semiconductor n-or p-type? Explain. (b)Isthe device accumulation,depletion,or inversion biased?Explain. (c) Drawtheenergybanddiagramcorrespondingtothechargestatepicturedintheblock charge diagram. (d)Byappropriatelymodifyingtheblockchargediagram,characterizethechargeste inside the MOs-C when a high-frequency a.c.signal is applied to the device. (e） Sketchthegeneralshapeofthehigh-frequencyC-Vcharacteristictobeexpectedfrom the structure.Placean Xon theC-Vcharacteristic at the point whichroughycorre- spondstothechargestatepicturedintheFig.P16.15blockchargediaram. (f)WhilebiasedatthesamegatevoltagegivingrisetotheFig.P16.15diagam,the MOS-Cis somehowtotally deepdepleted.Drawtheblockcharge diagram describing the new state of the system. --- Page 632 --- MOS FUNDAMENTALS 609 16.16Write a MATLAB (computer)program thatcanbeusedtocalculateandplottotal- deep-depletionC/Co versus VG characteristics. To simplifytheprogramming,takethe MOS-Cto be p-type.Let Vbe the gate voltage when s=F.Following the procedure outlined in Exercise16.5,use the low-frequency relationshipsfound in AppendixCto computethe V≤（s≤F）portionofthecharacteristic.CalculateC/C0versus VG for VG>V{employing Eq.（16.38)；stopthecomputationat VG=5V.Combinethe calculatedC/Covaluesfromthe two voltageranges intoa single plot.Setting =0.2μm and NA=7.8×1014/cm,run yourprogram andcompare the plottedoutput withthe total deepdepletioncurve in Fig.16.17.(Itwillbe necessary to mirror your computed p-typecharacteristicabouta verticallinethroughVG=0toobtainthe desired ntype characteristic.) 16.17Theoxidethickness（x）anddoping concentration（NAor ND）requiredincon- structing C-Vcharacteristics and inmodeling MOSdevices areoften deduceddirectly from the measured high-frequency MOS-CC-V data. Let us explore the determination procedure employing the high-frequencyC-Vdata recorded in Fig.16.17. (a）Themeasuredcapacitance withtheMOS-Cheavily accumulatedisusedto determine the oxidethickness.The MOS-C yielding the characteristic shown in Fig.16.17exhib- ited a maximum capacitance（Co）of82pFandhada gate areaof AG=4.75× 10-³cm²2. Determine from the given data. (b) Themeasured high-frequency capacitance withthe MOS-C heavily inverted is used to determinethesemiconductor dopingconcentration.Referring to Fig.16.17,record the minimum high-frequency valueofC/Cobserved when the device is biased far into inversion<4V).PrEq.C.1oundinAppndxC,asciee observedC/Co.Calculate Wef(inv）using thexdetermined inpart(a). Divide Wef（inv)byLD=2.91×10-3cmtoobtain theexperimental valueof W(inv)/LD. (The intrinsic Debye length,LD,is defined in Appendices BandC.The valuequoted here is appropriate for T=300 K.) that[14] $Weff(\inv) ≤2e-|UF|/2{2|UF|-1+ln[1.15(|UF|-1)]}1/2 LD whereUF= F/(kT/q).Using manualor computer-based iterative techniques,deter- minethe valueofUneededtomatchthe part(b)experimental valueof W（inv）/LD. DetermineUFtofour significant figures andcomputethecorresponding NDassuming T=300K.[NOTE:The NDvaluededucedfromFig.16.9correspondingto W=Wef(inv)providesanexcellentfirst guessfor ND,whichin turncan beusedto compute a first-guess value for U.] --- Page 633 --- --- Page 634 --- 17 MOSFETs—TheEssentials MOSFET-based integrated circuits have become the dominant technology in the semicon- ductor industry.Thereare literally hundredsofMOS-transistorcircuits inproductiontoday, ranging fromrather simple logic gates used in digital-signal processing to custom designs withbothlogicand memory functions on the same silicon chip.MOSproducts are found in a mind-boggling number of electronic systems, including the now commonplace per- sonal computer.InitiallytheMOS-transistor was identifiedby severalcompetingacr- nyms,namely,metal-oxide-semiconductortransistor(MOST)insulated gatefieldc transistor(IGFET),andmetal-oxide-semiconductorfieldeffect transistor(MOSFET). (PIGFET andMISFET were even suggested with asmileatonetimeoranother.) Withthe passage oftime,however,the transistor structure hascommonlycome to be known as the MOSFET.In thischapter we areconcerned withdescribingtheoperationof the MOSFET and modelingthe devicecharacteristics. Wecontinue to assume the MOS structuretobe ideal.Moreover,the developmentfocuseson thebasictransistorconfiguration,thelong- channel(orlarge-dimension)enhancement-modeMOSFET.An examinationof small- dimensionefectsandstructural variationsisundertakeninChapter19.Webeginherewith a qualitative discussionofMoSFEToperationand d.c.currentflowinsidethestructure, progress throughaquantitativeanalysisofthed.c.(ID-VD)characteristics,andconclude with an examination of the a.c.response. 17.1QUALITATIVETHEORYOFOPERATION Figure17.1(a)showsacut-awayview,andFig.17.1(b)asimplifiedcross-sectionalviw, ofthe basicMOS-transistorconfiguration.Physically,theMOSFETisessentiallynothing more than anMOS-capacitor with two pn junctions placed immediatelyadjacent to the regionof the semiconductor controlled by theMOS-gate. The Si substrate can beeither p-type (as pictured)or n-type;p+junction islands areofcourse required in n-bulk devices. Also shown in Fig.17.1(b)are the standard terminal and d.c. voltage designations.The draincurrent(ID),whichflowsinresponsetotheappliedterminalvoltages,istheprimary d.c.observable.Consistent withthe naming of the device leads,the current flow is always such thatcarrierslectronsin thepresentcase)enterthestructurethroughthesourc(S leave throughthe drain(D),and are subjectto thecontrolor gating actionof the gate (G). Thevoltageappliedtothegaterelativetogroundis VG,whilethedrainvoltagerelativeo ground is VD.Unless statedotherwise, we willassume that the source and back are grounded.Please notethatundernormaloperationalconditionsthedrainbias isalways 611 --- Page 635 --- 612 FIELDEFFECT DEVICES Phosphosilicate glass n Silicon dioxide Gate Drain Source. p-type substrate (a) S G D VG D(VD≥0 D$ SiO_{2}$ $N+}$ +N $D$ p-Si B (b) Figure17.1 The basic MOSFET structure.(a)Idealized cut-away view of a p-bulk (n-channel) MOSFET.(b)Simplifiedcross-sectional view of the p-bulk (n-channel) structure showing theter- minal designations,carrierandcurrentflow directions,andstandardbiasing conditions.[(a)From Beadle,Tsai,andPlummer[15].Reprinted withpermissionofAT&T.] --- Page 636 --- MOSFETS—THEESSENTIALS 613 such as toreverse bias the drain pn junction(V≥0for the Fig.17.1(b)device).Finally, reflectingthe normalflowofelectrons withinthestructure,thedraincurrentforthe p-bulk device is taken to be positive when flowing from the external circuit into the drain terminal. To determine how thedraincurrent is expected to vary as a functionofthe applid terminalvoltages,letusfirstconceptuallyset V=0andexaminethesituation insidethe structureasafunctionoftheimposedgatevoltage.WhenVGisaccumulationordepletion biased（V≤V,where Visthedepletion-inversiontransition-pointvoltage),thegated region betweenthesourceand drain islandscontainseitheranexcessor deficitofholes and veryfewelectrons.Thus,lookingalong thesurface betweenthen+islands underthecited conditions,oneeffectivelysees anopencircuit.When VGisinversionbiased(V>V), however,an inversion layercontaining mobileelectrons is formedadjacent to theSi sur- face. Now looking along the surface between the n+ islandsone sees,as pictured in Fig.17.2(a),an induced“n-type”region(the inversion layer)or conducting channel con- necting the sourceand drain islands.Naturally,the greaterthe inversion bias,the greater the pileupofelectrons at the Si surface and the greater theconductanceof the inversion layer.An inverting gatebias,therefore,createsorinducesa source-to-drainchannelad determines the maximum conductance of the channel. Turning next to theactionof the drainbias，supposean inversion bias VG>Vis applied to thegateandthedrainvoltage isincreasedinsmallstepsstartingfromVp=0. AtVD=0thesituationinsidethedeviceisaspicturedinFig.17.2a)thermalequilibrium obviouslyprevails,andthedraincurrentisidenticallyzero.WithVsteppedtosmall posi- tive voltages,the surface channelmerelyacts like a simple resistorand a draincurrent proportional to VDbeginstoflow into the drain terminal.The portionof the ID-VDrela- tionshipcorresponding to smallVDbiases is shownas the linefromtheorigin to point A in Fig.17.3.Any V>0bias,it shouldbeinterjected,simultaneously reversebiasesthe drain pn junction,and theresulting reverse bias junctioncurrentflowing into theSi sub- strate doescontribute toID.In well-made devices,however,the junctioncurrent istotally negligiblecompared to the channelcurrent,provided VDis less than the junction break- down voltage. Once VDis increased above a fewtenths of a volt,the device enters a new phase of operation.Specifically,the voltage dropfrom thedraintothe sourceassociated withthe flowofcurrent in thechannelstarts to negatethe inverting effectofthe gate.As pictured in Fig.17.2(b),the depletion region widens in going down thechannelfrom the source to the drain and the numberof inversion layer carriers correspondingly decreases.There- duced numberof carriers decreases the channel conductance, which in turn is reflected as a decrease in the slopeof theobserved ID-VDcharacteristic.Continuing to increasethe drainvoltagecauses aprogressivereductioninthechannelcarrierconcentrationandthe systematic slope-over in the ID-VDcharacteristic noted in Fig.17.3.The greatest decrease in channel carriers occurs near the drain,and eventually the inversion layer completely vanishessurfacedropsbelowNA)inthenear vicinitofthedrain(eFg.17.c).The onsetof surface depletionat thedrainendofthechannel,the specialsituation where the channelcarrierconcentrationattheSi-SiO2interfaceimmediatelyadjacenttothe drain becomesequaltothe bulkdoping concentration,isreferred toas pinch-off.Whenthe --- Page 637 --- 614 FIELDEFFECTDEVICES Inversion layer S D $N^{+}$ $N^{+}$ 1 (a) Depletion region S D $N{}$ $N{}$ (b) S D $N^{+}$ $N{}$ 1 (c) {Dsat DD$ S L D $N^{+}$ $h $N^{+}$ △L 1 (d) Figure 17.2 Visualizationof various phases of VG>VMOSFEToperation.(a）VD=0; (b)channel (inversion layer)narrowing undermoderate VDbiasing;（c)pinch-off;and（d)postpinch- off(VD>VDsat)operation.Notethattheinversion layer widthsdepletion widths,etc.arenotdrawn to scale.) channel pinches off inside the device,the point B is reached on the Fig.17.3 characteristic; that is,the slope of the ID-VDcharacteristic becomes approximately zero. For drainvoltages in excessof thepinch-off voltageVDsat,thepinched-of portonof the channel widens from just a point into a depleted channel section ΔL in extent (see Fig.17.2d).Being a region withfew carriers,and hence lowconductance,the pinched-off ΔL section absorbs mostof the voltage drop in excess of VDsat. Given a long-channel --- Page 638 --- MOSFETS—THEESSENTIALS 615 ID △L~L B △L<<L A VD 0 VDsat Figure17.3General variationofID with VDfora given V>V. device where ∆L≤L,the source to pinch-off regionof the MOSFET willbe essentially identical inshapeand willhave thesameendpoint voltagesforallVD≥VDsat.Whenthe shape of aconductive regionand the potential applied across the region do notchange,the current throughtheregion mustalsoremain invariant. Thus,IDremains approximately constant for drain voltages in excess ofV Dsat provided△L≤L.IfΔLis comparableto L, the same voltagedrop(VDsat)willappear acrossa shorterchannel(L-ΔL)and,as noted in Fig.17.3,the post-pinch-offID in suchdevices will increase somewhat with increasing Thus far we have examined the response of the MOSFET to the separate manipulation of thegateand drainbiases.Toestablishacomplete setofID-VDcharacteristics,itis necessary to combine theresults derivedfrom the separate considerations.Clearly,for VG≤V,thegatebiasdoesnotcreatea surfacechannelandIp≈0foralldrainbiases belowthejunctionbreakdown voltage.ForallV>Vacharacteristicoftheform shown in Fig.17.3 willbeobserved.Since theconductanceof thechannel increases with increas- ingV,itfollowsthattheinitialslopeoftheID-VDcharacteristic willlikewiseincrease with increasing VG.Moreover,the greaterthe numberofinversion layercarriers present wheno increase with increasing VG.From the foregoing arguments one concludes that the varia- tionof ID withVDand VG must beof theform displayed in Fig.17.4. TheID-VDcharacteristicsjustestablishedconfirmthe“transistor”natureof theMOS- FETstructure.Transistoractionresults,ofcourse,when IDflowing in anoutputcircuit is modulated byan input voltageappliedto the gate.Relative to terminology,we shouldnote that the portion of the characteristics where VD> V for a given VG is referred to as the Dsat saturationregionooeratioheprtionofthechrcteristicshereVD<iscalled the linear (or sometimes triode)regionof operation. Also,the MOSFET is known as an n-channel device whenthechannelcarriersare electrons; whenthechannelcarriersare holes the MOSFET is designated a p-channel device. --- Page 639 --- 616 FIELDEFFECTDEVICES

▲VG increasing 1 1 1 1 1 1 1 1 $V_V$ $V_{<V}$ 1 T V_{$ Figure17.4General formof theID-VDcharacteristicsexpectedfroma long channel(LL) MOSFET. Exercise17.1 P:Supposetheprecedingsection is to berewritten withthe illustrativeMOSFET changed from an n-channel to a p-channel device.Indicate how the figures in the section must be modified if the MoSFET used for illustrative purposes is a p-channel device. S: The required modifications can be summarized as follows: Figure17.1(a)…p-type substrate→n-type substrate;n+drain/source→p+drain/ source. Figure17.l(b)…holesreplaceelectrons asthechannel carriers; p-Si bulk(sub- strate)→n-Si bulk; VD≥0→VD≤0in the polarity comment near the drain terminal;the arrows are reversed to show positive ID currentflowfromthe source tothedrainandoutofthedraincontact.(IEEEconventionalwaysdefines thecurrentlowingintoadeviceterminaltobepositive.Strictlyadheringtothe convention would make I≤0in p-channel MOSFETs under normal operating conditions. Herein we prefer to deal with positive drain currents.) Figure 17.2…change the source/drain dopings to p+. Figure 17.3…VD→-VDand VDsat→-Vsaalong the -axisof the plot.Alter- natively,theexistingVDlabelingcould be retained andthecharacteristics appro- priatelyredrawnalong the negative portionof the x-axis.(Adhering toIEEE convention,the characteristics are sometimes drawn in an upside-down fashion in the third plot quadrant where both ≤0and VD≤0.)

--- Page 640 --- MOSFETS—THEESSENTIALS 617 Figure17.4..replace“VGincreasing”with“-VGincreasing”and VDwith-VD. Since V<0inap-channeldevice,nocurrentflowsif V>Vandcurrent is observed if V_<V_. In additiontothenotedchanges to thefiguresproperthe inequalitysignsmust be reversed in the Fig.17.2 and 17.3 captions. 17.2QUANTITATIVE-VRELATIONSHIPS During thecourseof MOSFET developmentthere hasevolved a hierarchyof long-channel ID-VDformulations thatprovideprogressively increased accuracy at the expenseof in- creased complexity.We willexaminetwoof the formulations:the”square-law”theoryand the“bulk-charge”theory. Theformerprovides very simplerelationships; the latter is a muchmoreaccuraterepresentationofreality.Interestingly,allbutthefinalderivational steps in the two theories are identical.Comments relative to moreexacting long-channel theories canbefoundat theendof the section. 17.2.1 PreliminaryConsiderations Threshold Voltage From thequalitative descriptionofMOSFEToperation it shouldbeobvious thatthe pa- rameter V plays a prominentrole in determining the precise nature of the devicecharac- teristics.InMOSFETanalyses Vis commonly calledthethresholdorturn-on voltage. Thetransistor starts to carrycurrent (turnson)at theonsetof inversion.Acomputational expression for the importantVparameter is readilyestablished using the resultsof Sub- section16.3.2andthefact that V=Vwhen s=2F.Specifically,given anideal n-channel (or p-bulk) device,simple substitution into Eq.(16.28)yields $K_s$ ..idealn-channel $V=2F+$ (17.1a) Ko V K_{ (p-bulk) devices Analogously, $K_s$ 4qND$.idealp-channelV2$ (-\φ)$(17.1b)Ko$K_{0$ (n-bulk) devices

--- Page 641 --- 618 FIELDEFFECTDEVICES Effective Mobility In deriving quantitative expressions fortheMOSFETd.c.characteristicsone encounters a new parameter knownas the“effective mobility.”Thecarriermobilities,n andμp,were first described inSection3.land were noted tobe a measure of theease of carriermotion within a semiconductor crystal.In the semiconductor bulk,that is,at a pointfarremoved from the semiconductor surface,the carriermobilities are typically determinedbythe amountoflattice scatteringand ionizedimpurityscatteringtakingplaceinsidethematerial. For a given temperature and semiconductor doping,these bulkmobilities( andp)are well-defined and well-documented material constants.Carriermotion in a MOSFET,how- ever,takes place inasurfaceinversionlayer wherethegate-inducedelectricfieldactssoas to acceleratethecarriers toward the surface.Theinversion layercarriers thereforeexperi- ence motion impeding collisions withtheSi surface (seeFig.17.5)inadditionto lattice and ionized impurity scattering. Theadditionalsurfacescattering mechanism lowersthe mobilityofthecarriers,withthecarriersconstrainednearesttheSisurfaceexperiencing the greatestreduction inmobility.Theresultingaveragemobilityofthe inversion layer carriers iscalledtheefectivemobilityand is given thesymboluorp. Seeking toestablisha formal mathematical expression for the effectivemobility,let us consider an n-channel device withthestructureand dimensions specified in Fig.17.6.Let x be the depth into the semiconductor measured from the oxide-semiconductor interface, ythe distance along the channel measuredfrom the source,(y)the channel depth,n(x,y) theelectron concentration ata point (x,y)in thechannel,andμn(x,y)the mobilityof carriers at the (x,y)point in thechannel.Invoking thestandard averaging procedure,the effectivemobilityofcarriersanarbitrarydistancefromthesourceisgivenby $xc(y)\mu n（x,y)n(x,y)dxJo$}=$(17.2)$fxc(y) $n(x,y)ddx$ Jo For futurereferenceitisusefultonotethattheelectroniccharge/cm²inthechannelatan arbitrary pointy is $xc(y)QN(y)=-q$n(x,y)dx$(17.3)SourceDrain$N +}$Inversionlayer$N++}$ Figure17.5Visualizationofsurfacescatteringatthe Si-SiO2interface.

--- Page 642 --- MOSFETS—THEESSENTIALS 619 D G S P-Si y 7 x Z Figure17.6Devicestructuredimensions,andcoordinateorientationsassumedinthequantitative analysis. leading to the alternative expression $c(y)9=“$ μn(x,y)n(x,y)dx$(17.4)$QN（y）o$Ifthedrainvoltageissmall,thechanneldepthandcarrierchargewillbemoreorlessuniformfromsourcetodrainandtheeffectivemobilitywillbeessentiallythesameforally-values.Whenthedrainvoltagebecomeslarge,ontheotherhand,candONvarywithposition,anditisreasonabletoexpectthat\mu likewisevariessomewhatingoingdownthechannelfromthesourcetothedrain.Fortunately,inlong-channeldevicesthecitedy-dependencecantypicallybeneglectedwithoutintroducingasignificanterror.Thus,inthischapterwewillsubsequentlyconsideruntobeadeviceparameterthatisindependentofyandtheapplieddrainvoltageVD.Relativetothedependenceof\bar{u}nontheappliedgatevoltage,increasedinversionbi-asingincreasesthex-directionelectricfieldactingonthecarriersandconfinesthecarriersclosertotheoxide-semiconductorinterface.Surfacescatteringisenhancedanduthere-foredecreaseswithincreasedinversionbiasing—adependencethatcannotbeignored.Theexact\mu nversusVGdependencevariesfromdevicetodevicebutgenerallyfollowstheformdisplayedinFig.17.7.Tofirstorderthedependencecanbemodeledbytheempiricalrelationship$\n{n1V{-}$ μo (17.5)

--- Page 643 --- 620 FIELDEFFECTDEVICES 0.8 □ 0.6 □ □ 回 □ $\}$ 0.4 □ □ μbulk 0.2 0 0 4 8 12 16 VG-VT(volts) Figure17.7Sample variationofunwiththeapplied gate voltage（VD=0).Data（)from Sun and Plummer[16].Equation(17.5) was used to construct the solid-line curve.(μboulk=1340 cm²/V-sec,μ0=847cm²/V-sec,andθ=0.0446/V.) where andθ areconstants.Equation(17.5) was used toconstructthe solid linecurve in Fig.17.7 after determining μ andθfrom a least-squares fit to the experimental data.Fi- nally,notefromFig.17.7thatthesurfacescatteringphenomenoncan berather significant, givingrisetoeffectivemobilitiesconsiderablybelowthebulk. 17.2.2 Square-Law Theory TheMOSFETunderanalysis is takento bea long-channel device withthe structureand dimensionsasspecified inFig.17.6.Thefigure also indicates theassumedcoordinate orientations. For gate voltages above turn-on(VGV),and drain voltages belowpinch-off (0≤V≤VDsat)，the derivationofthe square-lawID-VDrelationship proceeds as follows: In general one can write JN= qμnn+qDn (17.6) Within theconducting channelthecurrentflow is almostexclusively in the y-direction. Moreover,the diffusion componentofthecurrentisoftenfoundto be negligiblewhen dealing withthe more numerous carrier at a given point inside a semiconductor. Thus,

--- Page 644 --- MOSFETS—THE ESSENTIALS 621 basedonprecedentestablishedinsimilarproblems,itisreasonabletoneglectthediffusion componentofthecurrent(qDN∇n)in Eq.(17.6).Implementing the suggested simplifica- tions yields $d\}$ JN ≡JNy=qμnny=-qμ{n (in the conducting channel)) (17.7) dy All ofthequantities in Eq.(17.7)—μn,n,and JNy—are,ofcourse,-and y-position de- pendent.JNy,lke nisexpectedtobequitelarge at=0+andtodropoffrapidlyasone moves into the semiconductor bulk. Sincecurrentflowisrestrictedtothesurfacechannel,thecurrentpassingthroughany cross-sectional plane within the channel must be equal to ID.That is, (17.8a) =(-z4)(-a(,)) (17.8b) Upon examiningEq.(17.8b),note thatthesecondbracketonthe right-handsideofthe equation is justunON(see Eq.17.4).Eq.(17.8b)thus simplifies to dφ $D=-ZnON$ (17.9) dy Next,realizingthatIDis independentof y,wecanrecastEq.(17.9)into amore useful form by integratingIDoverthe lengthof thechannel.Specifically, L VD Ddy=DL=-ZnQNdφ (17.10) or, with n being position independent, Zū} $VD$ID

$QNdφ (17.11) L Jo Ananalytical expressionrelating ON to the channelpotential at an arbitrary point yis obviously required to complete the derivation. Working to establish the requird (a) The minus sign appears in the general formula for IDbecause IDis defined to be positive in the-y-direction. (b) Generally speaking,and dldy are functions of x. The very narrow extentof the inversion layer dictates, however,that is a weak function ofx(≈)in the channel region. Therefore,in writing down the final form ofEq.(17.8),dldy was taken to be constant over the x-width of the channel.

--- Page 645 --- 622 FIELDEFFECTDEVICES expression,werecall that theequilibriuminversion-layercharge inanMOS-Calmost pre- cisely balances the charge addedto the MOS-C gate when VGexceeds V.Inother words charge charge ΔQg -△Qsemi -QN…G≥VT (17.12) gate $c{m}^{}$

$c{m}^{}$ Becausethecharges are addedimmediatelyadjacenttotheedgesoftheoxide,we can also assert charge ΔQgate ≌CΔG=G-） . ..G≥V（17.13) $c{m}^{}$ and therefore QN≡-C（-V）…VG≥V (17.14) where CoK{{$C_{}$ (17.15) I AG}x_o is the oxide capacitance per unit area of the gate. Whereas the entire backside in an MOS-C is grounded,thebottom-side“plate"po- tentialin aMOSFET variesfrom zero at thesource toVpat the drain. As envisionedin Fig.17.8,the MOSFET canbe likenedto a resistive-plate capacitor where theplate-to- plate potential diference is VG at the source, VG-VDat the drain,and V-φat an arbitrary point y.Clearly,the potential dropV-at an arbitrary pointyin the MoSFET functionally replaces the uniform VGpotential drop in an MOS-C.Utilizing Eq.(17.14)we therefore conclude QN(y) -C_(-V-φ) (17.16) _{ V{ ZZ $Si{O}_2$ →y $Si{O}_{}$ Φ V\{e} MOS-C MOSFET Figure17.8 Capacitor-like modelfor determining the charge in the MOSFET channel.

--- Page 646 --- MOSFETS-THE ESSENTIALS 623 An explicitID-VDrelationshipcan nowbeestablished by simply substitutingthe Eq.(17.16)expressionfor QN into Eq.(17.11)and integrating.Theresult is Zūn C_ V_{qr{} $(0\le Vd\le VDsat$Ip$

L (V{-V)D$一2$V{≥V}$(17.17)ItshouldbereemphasizedthattheforegoingdevelopmentandEq.(17.17),inparticu-lar,applyonlybelowpinch-off.Infact,thecomputedIDversusVDforagivenVGactuallybeginstodecreaseifVDvaluesinexcessofDsatareinadvertentlysubstitutedintoEq.(17.17).Aspointedoutinthequalitativediscussion,however,IDisapproximatelyconstantifVDexceedsVDsat.Tofirstorder,thenthepostpinch-offportionofthecharac-teristicscanbemodeledbysimplysettingID|VD>VDsat=D|VD=DSatDt(17.18)or$V{{{}$(17.19)L2TheIDsatrelationshipcanbesimplifiedsomewhatbynotingthatpinch-ofatthedrainendofthechannelimpliesQN(L)\to 0when(L)=VD\to VDsat.ThusfromEq.(17.16)QN(L)=-C(VG-V-VDsat)=0(17.20)orDsat{=}_{-}V(17.21)and$Zu{ _$IDsat(-$ (17.22)

2L Neglecting ns dependenceon VG,Eq.(17.22) predicts a saturation drain current dependence.

--- Page 647 --- 624 FIELD EFFECTDEVICES (C) Exercise 17.2 P:Ifthesquare-lawIDdividedbyZnC/Lisplotted versus VDforselectvaluesof V-V,the resulting normalized characteristics are device independent-the same characteristics are obtainedfor any combination of Z,L,μn,x,and NA.Construct such a“universal”plot specifically showing thecharacteristics correspondingto V-V=1,2,3,and4V. S: The requested plot is presented in Fig.E17.2.The MATLABprogram listedbelow was used to generate the plot. MATLAB program scrit… %“Universal”ID-VDCharacteristics// Square-Law Theory %Initialization close clear %Let VGT=VG-VT; for VGT=4:-1:1, %Primary Computation VD=linspace(0,VGT); ID=VGT.*VD-VD.VD./2; IDsat=VGTVGT/2; VD=[VD,9]; ID=[ID,IDsat]; %Plotting and Labelling ifVGT==4, plot(VD,ID);grid; axis([010010]); xlabel(‘VD(volts)’);ylabel(‘ID/(ZCo/L)’); text(8,IDsat+0.2,‘VG-VT=4V’); hold on else, plot(VD,ID); %The following ‘iflabelsVG-VTcurves<4 ifVGT==3, text(8,IDsat+0.2,VG-VT=3V’); elseifVGT==2, text(8,IDsat+0.2,VG-VT=2V’); else, text(8,IDsat+0.2,VG-VT=1V’);

--- Page 648 --- MOSFETS—THEESSENTIALS 625 end end end hold off 10 9 VG-VT=4V 8 7 ID/(ZCo/L) 6 5 VG-VT=3V 4 3 VG-VT=2V 2 1 VG-VT=1V 0 0 1 2 3 4 5 6 7 8 9 10 VD (volts) Figure E17.2 17.2.3 Bulk-Charge Theory Althoughappearing veryreasonableand soundonfirst inspection,closescrutiny reveals that the square-law theory contains a majorflaw. The capacitor-like modelusedinthe square-law analysis assumed changes in gate charge going downthe MOSFETchannel were balanced solely by changes in QN.This is equivalent to implicitly assuming thede- pletion width at all channel pointsfrom the source to thedrainremains fixedat Weven under VD≠0biasing.Inreality,as pictured in Figs.17.2(b)to (d),the depletion widh widens in progressing from the source to the drain when Vp≠0.This point-to-point varia- tion in the depletion layeror“bulk”charge must be included in any charge balance relationship. With changes in the depletion width,W(y),taken intoaccount,one more accurately deduces QN(y)=-(-V-)+qNA[W(y)-W] (17.23)

--- Page 649 --- 626 FIELD EFFECT DEVICES where, making use of the delta-depletion results in Chapter16, $2K_s$ 71/2 W(y)=$2(2F+\phi )(17.24)$qNA 2K_s{02φ) 1/2 厂 $W_{}$

(17.25) qN{ Thus,combining Eqs.(17.23)to(17.25)and introducing q{A_T Vw (17.26) ℃ one obtains the bulk-charge theory analogue of Eq.(17.16), namely, (17.27) Thepredicted ID -VDrelationship based on the bulk-charge formulation is next readilyobtainedby substituting Eq.(17.27)into Eq.(17.11)andintegrating. Theend result is Zun C_{ Vsfqra}{} 413 $V$ 3/2 $D$

(V{ V)VDDV{}$1+$ L 2 $for0\le V\le Dst(17.28)$andV_{ ≥ V_} Asin the square-lawanalysis,the post pinch-of portionof thecharacteristics are approxi- matelymodeledbysetingIDevaluatedatV>VDsatequaltoDatV=VDat.Likewise, an expressionfor VDsacanbeobtainedbynotingQNyy=→0in Eq.(17.27)when (L) = VD→Dsat.One finds V−+(1+Vk)]+ Dsat = VG-V-Vw{[ V_G 2\} (17.29) Having concluded the mathematical development, let us examine the results and make appropriatecomments.Firstofall,it shouldbe recognized thattheprimary assetof the square-lawtheoryisitssimplicity.Generaltrends,basicinterrelationships,andthelikecan

--- Page 650 --- MOSFETS—THEESSENTIALS 627 Square-law theory 10 Bulk-charge theory(NA=1014/cm) Bulk-charge theory (NA=101/cm3) 8 V_G-VT=4V -1- ID} 6 $Z\bar{u}{n}_o$ L }V_{G-V=3V 4 2 }VG-V}=2V V{G-V}=1V 0 2 4 6 8 10 12 VD(volts) Figure17.9ComparisonoftheVDcharacteristicsderivedfromthesquare-lawandbulkcharge theories.The bulk-chargecurves werecomputed assuming =0.1mandT=30K. be establishedusing the square-lawformulation without an excessive amountof mathe- maticalentanglement.Thebulk-charge theory,ontheotherhand,is ingoodareement withthe experimental characteristics derivedfromlong-channelMoSFETs.Itshouldalso be notedthat,although Eqs.(17.28)and(17.29)are decidedlymorecomplex thantheir square-law analogues,the added terms,the terms not appearing in Eqs.(17.17)and(17.21), respectively,arealwaysnegativeandactprimarilytoreduceIDand VDsatforagivensetof operational conditions.Figure 17.9,which compares the two theories,confirms the fore- going observationandalso illustrates another well-known property—theaccuracyofthe square-lawtheory improves as the substrate doping is decreased.In fact,the bulk-charge theory mathematically reduces to the square-law theory as NA（or ND)→0and →0. Exercise 17.3 P: Suppose the gate and drain of an ideal n-channel MOSFET are tiedtogether as picturedon the following page.。=500Å,NA=101/cm3,Z/L=10,n= 625 cm²/V-sec,andT=300 K.Employing the bulk-charge theory,determine D when (a) V{=VD=1V; (b) V {G=V{}=3V

--- Page 651 --- 628 FIELDEFFECTDEVICES S G,D n{+ $n+$ p-Si S:(a) This is somewhat of a trick question. We note kT\F= $.ln(NA/n）=0.0259ln(1016/1010)=0.358Vqand4qNA\phi$ Ks KO} K{s0 71/2 (11.8)(5×10-6)(4)(1.6×10-19)(1016) =0.716+ (0.358) =1.42V (3.9) (11.8)(8.85×10-14) WithV=1VV<andthereforethetransistoristurnedoff;D=0. (b)ComputingIDby simply substituting VG=Vp=3Vinto Eq.（17.28)givesan incorrectresult.AsnotedinthetextandasilustatedinFig.17.9Dainthebulk- chargetheoryoccursatalower voltagethanthesquare-lawVDsat=VG-V.Since D=V，itfollowsthat VD=>V-V>VDsatWiththegateanddraintied together,the MOSFET is saturated biasedfor all VD=V>V. Respectively employingEq.(17.15),(17.25),and(17.26),we findC=6.90× 10-8F/cm,W=3.06×10-5cm,andVw=0.71V.SubstitutingintoEq（17.29) thenyelsV1.15V.FnllycomuigDusgE17.2wDa =1.15V gives SID=0.382mA. 17.2.4Charge-Sheet and Exact-Charge Theories Boththe square-lawandbulk-charge theories sufferfrom two severe inherent limitations. Forone,the charge in the MOSFET channel(ON in the preceding analyses) was assumed tobeidentically zero for gate voltages atandbelowthethresholdvoltage.Inan actual device,the channel charge becomes small,butdoes notcompletely vanish. As a conse-

--- Page 652 --- MOSFETS—THE ESSENTIALS 629 quence,a residualdrain currentflows betweenthe sourceanddrainatandbelowthe thresholdvoltage. Thisresidualdraincurrentiscalledthe subthresholdcurrentandits precise value isoftenof interest.Second,the square-lawand bulk-charge relationships do not self-saturate—it is necessary toartificiallyconstructtheabovepinch-off portionof the characteristics. Thenotedfailings are removed in thecharge-sheetandexact-chargeformulations; eithertheory can be used to compute the subthreshold current,and boththeories are self- saturating. TheID-VDcomputationalrelationships resultingfrom thecharge-sheetand exact-charge models are reproduced in Appendix D.Theexact-charge resultsareesta- lished by working withthe exact-charge distribution insidethe MOSFET.Althoughnot overly complex,the exact-charge result does involve integrals. Thecharge-sheet model maybe viewed as a simplified versionof theexactformulation.Of thetheories discussed the charge-sheet formulationprovidesthe best trade-off betweenaccuracyandcomplexity SampleID-VDcharacteristicsresultingfromthecited more-exactingtheoriesare shown in Fig.17.10.ComputedandobservedsubthresholdcharacteristicsofID versus VG withVD held constant are presented in Fig.17.11. 3.0 VG}5volts 2.5 2.0 VG} 4 volts 1.5 1.0 VG= 3 volts 0.5 V{G}=2 volts 1 1 1 1 1 1 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 D(volts) Figure17.10 Theoreticalcurrent-voltage characteristicsof an n-channelMOSFET with= 0.05μm,NA=101/cm3,n=550cm²/V-sec,L=7μm,Z=70mandT=23°C.Thesolid- line curves were derivedfrom theexact-charge result while the dashed-line curves were computed using thecharge-sheettheory.ReprintedfromPierretandShields[17©1983,withkindpermissin from Elsevier Science Ltd.)

--- Page 653 --- 630 FIELD EFFECTDEVICES 10-2 10-3 10-4 10-5 NA=10^{4}/}cm3}V} D(A)10-6 NA =10^{15}cm3} 10-7 V} 10-8 10-9 * 10-10 -0.6 -0.4 -0.2 0.0 0.2 0.4 VG(volts) Figure17.11 Subthresholdtransfer characteristicsof n-channel MOSFETs having the samepa- rameters as the Fig.17.10deviceexcept NA=1014/cm3orNA=101/cm3and=0.013μm. The(*)are experimental data. Setting V=1V,the solid-anddashed-linecurves were computed respectively from the exact-charge and charge-sheet relationships found in Appendix D.Theideal- device curves were shifted along the voltage axis to enhance the comparison with the experi- mental data.(Reprintedfrom Pierretand Shields[17],©193,withkind permissionfromElsevier Science Ltd.) 17.3a.c.RESPONSE 17.3.1Small-Signal EquivalentCircuits The a.c.response of the MOSFET,routinely expressed in terms of a small-signal equivalent circuit, is mostconveniently established by considering the two-port network shown in Fig.17.12(a).Initiallywerestrictourconsiderationsto lowoperationalfrequencieswhere capacitiveeffectsmay be neglected.Itshould be noted thatthefollowing developmentand results are very similar to those of the J-FET presentation in Subsection15.2.4.

--- Page 654 --- MOSFETS—THEESSENTIALS 631 G D In MOSFET Out S S (a) 1 id} D + + _gq} $8m_g$ 8d Ud$SS(b)$C_gd$GD++$v_g}C_{s $8m_q$ 8d Ud$ 一 S S (c) Figure17.12(a) TheMOSFET viewed as a two-port network.(b)Low-frequency and (c)high- frequency small-signal equivalentcircuitscharacterizingthea.c.responseoftheMOSFET. We begin by examining the device input.Looking into the input port between the gate and grounded source/substrate,one seesacapacitor.Acapacitor,however,behaves(tofirst order)like anopen circuitat lowfrequencies.Itis standard practice,therefore,to model the low-frequency input to the MoSFET by an open circuit. At the output portthe d.c.draincurrenthas beenestablishedtobea functionofVD and V；thatis,ID=ID(VD,VG).Whena.c.drainand gate potentials，vandvg，are respectivelyaddedtothed.c.drainand gateterminal voltages,VDand VG,thedraincurrent throughthe structureismodifiedtoI(VD,G)+,whereiis thea.c.componentofthe drain current.Providedthe devicecanfollow the a.c.changes in potential,which iscer- tainly the case at lowoperationalfrequencies,one can state +ID(D，）=D（D+,VG+g) (17.30a) and =ID（VD+，VG+g）-D(D，) (17.30b)

--- Page 655 --- 632 FIELD EFFECT DEVICES Expandingthe firsttermontheright-handsideofEq.（17.3Ob)in a Taylor seriesaboutthe d.c.operating point,andkeepingonlyfirst-orderterms inthe expansion(higher-order terms are negligible), one obtains $\partial D$

D（VD+ν，VG+g）=D（VD，VG）+ ${DV$ $v_}+$ (17.31) which when substituted into Eq.(17.30b) gives $\$ $∂}$ $id$ = v_d$ + (17.32) g DimensionallythepartialderivativesinEq.7.32)areconductances.Intodui $∂\D$ gd

…thedrainorchannelconductance (17.33a) VG=constant $D$ 8m _{ … transconductance or mutual conductance (17.33b) VD=constant we can then write =gd+8mg$ (17.34) Equation(17.34)may be viewed asthea.c.-currentnodeequation forthedrainterminal and,by inspection,leads to theoutput portionofthecircuitdisplayed in Fig.17.12(b). Since, as concluded earlier,the gate-to-source or input portionofthe device is simplyan opencircuitFig.17.12(b)thenisthedesiredsmall-signalequivalentcircuitcharacterizing the low-frequency a.c.response of the MOSFET. For field-effect transistors the gmparameter plays aroleanalogoustothe andβin the modelingofbipolar junction transistors. As its name indicates,g may be viewed as eitherthe device outputadmittanceor the a.c.conductance ofthe channelbetween the sourceanddrain.ExplicitgandgmrelationshipsobtainedbydirectdiferentiationofEqs. (17.17),(17.22),and(17.28)usingtheEq.(17.33)definitions are catalogued in Table17.1. At the higheroperational frequencies oftenencountered in practical applications,the Fig.17.12(b)circuitmust bemodifiedtotake intoaccountcapacitivecoupling betweenthe device terminals. The required modification is shown in Fig.17.12(c).A capacitor between the drainand source terminals at theoutput has beenomitted inFig.17.12(c) becausethe draintoourcecacitaceitycllnlibleCwicrovideundeiab- capacitance—thecapacitance resultingfromthe portionof thegate thatoverlapsthedrain

--- Page 656 --- MOSFETS—THE ESSENTIALS 633 Table17.1 MOSFET Small-Signal Parameters. Below pinch-off(VD≤VDsat) Above pinch-off(D> Dsat) Z_C{0(V{-V- Square 0=P8 law 8d

L Bulk Zū{n C_

• V- 8{ = 0 charge 8d = L -Vw(√1+VD/2F-1)] Z_u_{nC{0D$Squarelaw$gm$=L8m=LBulkchargeL8m$ 二 L with VDsat per Eq.(17.29) Etieinttaetebceiioof.).n.).oo unwith VG was neglected in establishing the gmexpressions. island.Theoverlap capacitance is minimizedbyforming a thickeroxide in theoverlap region orpreferably through the useof self-aligned gate procedures.In the self-aligned cessing,usuallpoliliconisdepoitedfir.Aftetheateisdefined,theoucendin islands are subsequentlyformed abutting thegate by diffusionor ion implantation.The remaining capacitor shown in Fig.17.12(c),Cg tance of the MOS gate. 17.3.2 CutoffFrequency Giventhesmall-signalequivalentcircuitofFig.17.12(c),it is possibletoestimatethe maximumoperating frequency or cutof frequencyof anMOS transistor.Let ax bede- fined as the frequency where the MOSFET is no longer amplifying the input signal under optimum conditions—that is,thefrequency where the absolute value of the output current to inputcurrentratio isunity whenthe outputofthetransistoris short-circuited.By inspec- tion,the inputcurrent withtheoutput short-circuited is iin=jω（Cgs+gd)=(2πf)Cog (j=√-1) (17.35) whereCgd is taken tobe small andgs≈.Likewise,the output current is out = 8m^_ (17.36)

--- Page 657 --- 634 FIELDEFFECTDEVICES Thus,settingiouin|=1and solving forf=max，oneobtains {D \max} 8m if V { ≤ Da (17.37) 2πC{o $2\pi L2$ ThelatterformofEq.(17.37)wasestablishedusing thebelowpinch-off gm entryin Table17.1.Theimportantpointtonoteisthatthechannel lengthListhekeyparameterin determining max;increasedMOSFEToperatingfrequencies areachieved by decreasing the channel length. 17.3.3Small-Signal Characteristics Representativeplotsof selectedsmall-signal characteristics thathave received specialat- tentioninthedevice literature are shownin Fig.17.13.g versus VGwithVD=0hasbeen used to obtain a reasonablyaccurateestimateof V.This is accomplished by extrapolating the linear portionof thega-VGcharacteristics into theVG axis andequating the voltage intercept to V.The basis for this procedure can be understood by referring to the below pinch-off gentries in Table17.1.WithVD=0thedrainconductance in both thesquare- law and bulk-charge theories reduces to gd

(V{= 0) (17.38) L To firstorder,then,gis predictedto bea linearfunctionof VG,going to zero whenVG= V.Theexperimental characteristicdoesnotcompletely vanishat VG=Vbecausethere is a smallminority-carrierconcentration in the surface channel at the depletion-inversion transition point. Thisresidualconcentration is neglected inboththesquare-lawandbulk- charge theories.The g versus VG characteristic with VD=0has also been usedto deduce theeffective mobility.Since g is directly proportional tounaccording to Eq.（1.8),u versus VG can be computedreadily from the ga-VG data. This mobility measurement method is accurate provided the device contains a low densityof interfacial traps (see Sub- section18.2.4).Amoderate-to-large density ofinterfacialtraps would spreadoutthe g-VG characteristic and yield a fallaciously low value for un. Thesecond characteristic in Fig.17.13typifies the gate capacitance versus VG depen- dence derived from the MOSFET when the drain is grounded. The MOSFET C-VG (VD=0)characteristic hasbeenusedfor diagnostic purposes in muchthe same manner as the MOS-CC-VG characteristic.The MOSFET characteristic can,in fact,be modeled to first order by the low-frequency MOS-CC-VG theory. Unlike the MOS-C,however, a low- frequency type characteristic isobservedeven whentheMOSFETis probed at frequencies exceeding1MHz.A low-frequencycharacteristic isobtained because thesource anddrain islandssupplytheminoritycarriersrequiredforthestructuretofollowtheac.fluctuaions

--- Page 658 --- MOSFETS—THE ESSENTIALS 635 0.5 0.4 1 0.9 (sw) P8 0.3 $CGmax$ V{}=00$\mathrm{0.80.20.7}V00$ 0.1 0.6 0.5 0 V_s} 4 6 8 10 12 14 -4-3-2-1 0 12 3 45 6 VG(volts) VG(volts) (a) (b) Figure17.13MOSFET small-signal characteristics.(a)8 versus VGwith VD=0;(b)CGversus Vwith V{D= 0. inthe gate potentialwhenthe device is inversion biased.Minority carriersmerely usethe surface channeltoflow laterally into andoutoftheMOS gatearea inresponse to the applied a.c. signal. Exercise 17.4 P:（a) The transconductance（gm）plotted as a functionof VGwith Vheld constant isanothersmall-signal characteristicencounteredquiteoften intheMOSFET litera- ture.Deducetheexpected generalformofthe gm-VGcharacteristic.Specifically, making useof the square-law entries in Table17.1 and assuming V=2V,plot gm/（ZunC/L)versus V(0≤V≤10V)for VD=2,4,and6V.Neglectthe variation of ū with V_. (b)Compare similarexperimental gm-VGcharacteristics with thepart(a)theoretical characteristics.Speculate as to theoriginof any differences in the shapeofthe characteristics. S:(a)Letξ=ZunC/L.As deducedfrom the square-law entries in Table17.1, 8m{T}$ …0≤V-V≤VD

ξ VD …VG-V_≥VD

--- Page 659 --- 636 FIELDEFFECTDEVICES ForallVD,gm/ξ=OifVG<V=2V.ForVG>VbutVG-V<VD，thedevice is saturationbiased and gm/ξincreaseslinearly withV.Once VGreachesthevoltage where VG-=VD,thedevicedropsoutof saturation and gm/ξ=VD=constant. Thegm-VGcharacteristics are thereforeconcluded to beof the form shown below 6+ VD=6V 5 (so) VD=4V 3 VT 2+ VD}=2V 1+ 0 2 4 6 8 10 12 VG(volts) (b) Experimental gm-VG characteristics derivedfrom ann-channel MOSFET are presented below.The data are from PartIIreference[18].The mostnoticeable dif- ference between theory and experiment is the decrease in the measured gm at V>VD+V.Thisfalloffingmisreadilyexplained—itiscausedby thedecrease inμnwith VG that was neglected in drawing the theoretical characteristics. 2.5 VD=6V 2.0 1.5 gm(mS) 1.0 0.5 VD=2V 0 2 4 6 8 10 VG(volts)

--- Page 660 --- MOSFETS—THEESSENTIALS 637 17.4SUMMARY Thischapter was intended to provide an introduction to MOSFETterminology,operation, and analysis.The MOS structure was assumed to be ideal and considerations were limited to the basic transistorconfiguration. We began with a qualitative discussion of MOSFET operationandd.c.currentflowinside thestructure.Whenbiasedintoinversion,theinduced surface inversion layer forms a conducting channelbetween the source and drain contacts. The greatertheapplied gate voltage in excess of turn-on,the largertheconductanceof the internal channel at a given drain voltage.A nonzero drain voltage inturn initiatescurrent flowbetweenthesourceanddrain. Thecurrentflow is proportionalto VDatlowdrain voltages,slopesoverduetochannel narrowing as VDisincreased,andeventuallysaturates once the internal channel vanishes or pinches off near the drain. Thequantitative analysis of theMOSFETd.c.characteristics,considered next,was subjecttotwonotablecomplications.Firstofall,carriers in a surface channel experience motion-impeding collisions withthe Si surface,whichlowerthe mobilityof the cariers and necessitate the introductionofan effectivecarriermobility.Second,the carrierconcen- tration andthereforethecurrentdensity in thesurfacechannelare strong functionsofpo- sition,dropping off rapidly as one proceeds into the semiconductor bulk.Nonetheless,the first-orderresultsfortheMOSFETcurrent-voltagerelationshiparesurprisinglysimple. The resultsof thefirst-ordertheory,referredtohereinas the square-lawtheory, are con- tained in Eqs.(17.17),(17.21),and(17.22).Thebulk-charge theory,a secondformulation culminating in Eqs.(17.28)and(17.29),provides a moreaccuraterepresentationof reality at the expense of complexity. Even more-exacting formulations were discussed briefly with the reader referred to Appendix Dfor computational details. The last section of the chapter was devoted to the a.c.response of the MOSFET.The equivalentcircuitsofFigs.17.12(b)and(c),respectively,specifythesmallsignalresponse at lowandhighfrequencies.WiththeaidoftheFig.17.12(c)circuitit wasestablishedthat a short channel length is necessary to achieve high-frequency,high-speedoperation.It was also pointedoutthatusefulinformationcanoften beextractedfromthe small-signal pa- rameters (g,gm,CG)monitored as a functionof the d.c.terminal voltages. Although designed tobeself-contained,theMOSFETdevelopment in thepresent chapter doescloselyparallelthe J-FETpresentation in Chapter15.Thereadermayfindit a usefulexercise to notesimilaritiesanddifferences in theoperationandanalysisofthe two devices. In Chapter18 we examine the impact of nonidealities on MOSFEToperation. As noted previously,an examinationof small-dimensioneffects and structural variations is undertaken in Chapter 19.

--- Page 661 --- 638 FIELD EFFECT DEVICES PROBLEMS CHAPTER 17 PROBLEMINFORMATIONTABLE Complete Diffi culty Suggested Short Problem After Level Point Weighting Description 17.1 17.4 1 10 (1 each part) Quick quiz 17.2 ” 2 10(a,c,f-2;b,d, e,g-1) Sample computations 17.3 17.1 2 10(a-3, b-2, c-5) Sketch diagrams 17.4 17.2.1 2 10 Plot VTvs.NAfor selectx 17.5 ” 3 10 VT temperature dependence 17.6 17.2.2 2 5 VDsat from ID maximum 17.7 ” 2-3 10 n→pin Subsection 17.2.2 17.8 ” 1 5 PlotID-VD,nper Eq.(17.5) 17.9 ” 3 10 (5 each part) VD=VG-VBB$ 17.10 ” 3-4 10 ID versus V-V 17.11 ” 3-4 10(a-8,b-2) Circular MOSFET 17.12 ” 3 12(a-2,b-10 discuss-2) ID temperature dependence 17.13 ” 3-4 12 Include R_,RDin theory 17.14 17.2.3 3-4 10 Derive Eq. (17.29) 17.15 ” 3 10 Plot bulk-charge D-VD 17.16 17.2.4 4 25(a-10,b-15) Plot sheet, exact D- 17.17 17.3.1 2 5 Verify Table 17.1 entries 17.18 ” 2 5(a-3, b-2) MOSFET QN,8d 17.19 17.3.2 3 10 Deduce Z,L formatch 17.20 ” 2-3 15(a-3,b:g-2) General MOSFET review 17.21 17.3.3 3 12 (4 each part) Small signal characteristics 17.22 17.4 2 12 Compare FETs 17.1 Quick Quiz Answer the following questions as concisely as possible. (a)Why are thecurrent-carrying contacts in the MOSFETreferred to as thesourceand drain? (b) Precisely what is the channelin MOSFET terminology? (c) Whenrefringto theMOSFETDDcharacteristics,whatexactlyisthesaturation region of operation?

--- Page 662 --- MOSFETS-THEESSENTIALS 609 duced in theMOS-Cdiscussionandthethreshold(turn-on)voltageintroducedinthe MOSFET discussion? (e) Why isthemobility inthesurfacechannelofa MOSFETdifferentfromthecarrier mobility in the semiconductor bulk? (f) Whyis theID-VDtheoreticalformulationofSubsection17.2.2referredtoasthe square-law theory? (g) WhyistheDVDtheoreticalformulationofSubsection17.2.3referredtoasthebulk- charge theory? (h) Whatvariablesareplottedindisplayinthe“subthresholdtransrcharacteisics (i) Whatisthemathematicaldefinitionofthedrainconductance?thetransconductance? (j） Why is theobservedMOSFETC(D=0)curve typicalyalow-frequencychar- acteristic even at a measurementfrequency of1MHz? 17.2 Sample Calculations Simplenumericalcomputationshelptoestablishtheexpected sizeofdevice variables.An idealn-channelMOSFETmaintained atT=300K is characterizedbythefollowing parameters:Z=50μm,L=5μm,。=0.05μm,NA=101/cm,and=800cm²/ V-sec (assumed independent of V). Determine: (a) V； (b） IDsat (square-law theory)if V=2V; (c） IDsat(bulk-charge theory)if =2V; (d)gdifV=2VandVD=0; (e) gm(square-lawtheory)ifV=2Vand VD=2V; (f）gm(bulk-charge theory)ifV=2Vand VD=2V; (g)maxifV=2Vand VD=1V. 17.3Givenan ideal p-channelMOSFETmaintained atroom temperature: (a) Assuming VD=0,sketch the MOSenergyband diagram for the gate region ofthe given transistor at threshold. (b)Assuming VD=0,sketch the MOSblockcharge diagram for the gate regionofthe given transistor at threshold. (c)SketchtheinversionlayeanddepletionregioninsidetheMOSFETatpinch-off.Show and label all parts of the transistor. ●17.4Construct a plotof VT versus NAforidealn-channelMOSFETsoperated atroom temperature.Superimposeon thesame plotthecurvescorrespondingto=0.0,0.02,

--- Page 663 --- 640 FIELDEFFECTDEVICES 0.05,and0.1μm.Let the semiconductor doping varyover therange1014/cm≤NA≤ 1018/cm³；limittheplotted Vto0≤V≤3V. 17.5Explore thetemperature dependenceof the MOSFETthreshold voltage.Considering an ideal n-channel MOSFET,taking x。and NA to be input parameters,and perhaps refer- ring to Exercise 2.4,construct a computer program that calculates and plots VT versus T over the range 200 K≤T≤400K.Record the outputof the program when x=0.1μm and NA=106/cm.Describe thegeneral natureof the VversusTdependence. 17.6IfEq.（17.17)is used tocomputeIDasa functionof VDfora given VG,andifVis allowed toincreaseaboveVDatonefindsIDtobeapeakedfunctionofVDmaximiigt VDat.Theforegoinsuggestscond watoestablistheEq.17.2)rlationhipfr VDat.ilocle points ofafunctioncanbeusedtoderive Eq.(17.21)directlyfromEq.(17.7). 17.7SupposeSubsection17.2.2istoberewritten withtheillustrativeMOSFETchanged fromann-channeltoap-channel device.Indicatehowtheequationsinthesubsectionmust be modified if the MOSFET used for illustrative purposes is a p-channel device. 17.8The unin relationships throughoutthechapter canbereplacedbytheEq.(17.5) expression to approximatelyaccount for thedependenceofnon theapplied gate voltage. Modifythecomputational program inExercise17.2to illustrate theeffectof incorporating the μn-VG dependence.Introducing the new normalizing factor,ZC/L,superimpose plotsoftheID/（Z0C/L）versus VDcharacteristicscorrespondingtoθ=0andθ=0.05/ V.Notethatθ=0shouldyieldcharacteristicsidentical toFig.E17.2,whileθ=0.05/Vis roughlythe value used to fit then-VGdata of Fig.17.7. 17.9Suppose a battery VB≥0is connected between the gate and drainof an idealn- channel MOSFET as pictured in Fig.P17.9.Using the square-law results, (a) SketchDversus VD(D≥0)ifVB=V/2; (b) SketchID versus VD(VD≥0)ifVB=2V. VD {BB + n+n+$ p-Si Figure P17.9

--- Page 664 --- MOSFETS-THEESSENTIALS 641 17.10 The most widelyencountered MOSFET characteristics are a plotof ID versus VD with VGorVVheldconstantatselectvalues.AnalternativeplotofIDversusVGor V-VwithVDheldconstantatselect valuesissometimesuseful.Sketchtheshapeofthe ID versus VG-Vcharacteristicsto be expectedfrom an idealn-channel MOSFET.Specifi- cally show the characteristics corresponding to ,2,3, and 4 V.Explain how you arrived at your sketch. 17.11Alinear devicegeometryandarectangulargateof lengthLbywidthZwere explic- itly assumed in thetext derivationofID-VDrelationships.However,MOSFETs havebeen built with circular geometry as pictured (top view) in Fig.P17.11. S G D FigureP17.11 (a)Ifrandaretheinsideandoutsidediametersofthegatedareashowthatinthe square-law formulation one obtains 2π C0（VG-V)VD- $$V\fra}{} $==$ $n(r{2}) 2 forbelowpinch-offoperationof a MOSFET withcircular geometry. Toderivethe above result,usecylindricalcoordinates(r,θ,z)andappropriatelymodifyEqs.(17.7) through (17.17). (b)Settingr=+LandZ=2πr,showthatthe part （a)resultreducestothelinear geometry result,Eq.(17.17),in the limit where L/r,≤1. 17.12As afollow-uptoProblem17.5,we wishto explorethetemperature dependenceof the MOSFET saturationcurrent.Consider anideal n-channelMOSFETwith=0.1m andNA=101/cm3.AssumeintheMOSFETchannelhasthesametemperaturedepen- dence as μn in the semiconductor bulk. (aEstablishanexpressionforDa()Dat(00K)utilizinthesquare-lawtheory. (b） Setting VG=3V,computeandplotIDsat（T)/IDsat(300K)versusTovertherange 200K≤T≤400 K.Superimpose a plotof μn(T)/μn(300 K)on the same setof

--- Page 665 --- 642 FIELDEFFECTDEVICES coordinates.Repeat the computationandplotting taking VG=10V.Discuss your results. 17.13Aspictured inFig.P17.13,resistances Rs andRDexistbetweenthe soure/drain terminals and thechannel proper. These resistances arisefrom a combination ofthe metal- Si contactresistance andthebulkresistanceof thesource/drain islands. TypicallyRand RD are negligible in long-channel MOSFETs.However,as the dimensionsof MOSFETs are reduced to achieve higheroperating frequencies and higher packing densities, R and RDhave become increasingly important.Working with the square-law theory,show that the sourceand drain resistances are appropriately taken intoaccount byreplacing VD with VD-ID(R+RD)and GwithVG-DRin Eqs.(17.17),(17.21),and(17.22). S G D ID .n-channel.. PW R{s 4 ↑ RD$ y=0 y=L φ(0)= IDS φ(L)=VD-DD Figure P17.13 17.14Derive Eq.(17.29). 17.15Writeacomputerprogramthat canbe used tocalculateand plotID/(ZμnC/L)ver- sus VDcharacteristics based on the bulk-charge theory.Use your program to verify the accuracyofthe bulk-chargecharacteristicspictured in Fig.17.9. 17.16Utilizing therelationships in AppendixD,constructcomputerprogramstocalculate andplottheIp-VDcharacteristics basedon (a)the charge-sheet theory and(b)the exact- charge theory.As a check,run yourprograms to obtain results thatcan becompared with thecharacteristics shown in Fig.17.10and/or Fig.17.11. 17.17Perform themathematical manipulationsto verify the bulk-charge g and gmentries in Table17.1. 17.18Biases V=3Vand VD=0areapplied toanideal n-channelMOSFET withZ= 70μm,L=7μm,=550cm²/V-sec,。=0.05μm,and V=1V.Making useofthe square-law theory, (a)determinetheinversionlayer charge/cm²at themidpoint(=L/2)ofthechannel. (b)determine the drain conductance (gd)at the specified bias point.

--- Page 666 --- MOSFETS—THE ESSENTIALS 643 17.19Acomplementary pairof ideal n-channel and p-channelMOSFETs are tobede- signedsothathedeviceexbtthesamegandwhenequivalntlbiasedandop- erated at T=300 K.The structural parameters of the n-channel device are Z=50 m, L=5μm,=0.05μm,and NA=101/cm3.The p-channel device has thesameoxide thickness and doping concentration,but,becauseofthe lower hole mobility, must have different gate dimensions.Determinethe required ZandLof the p-channel device.Assume theeffectivemobilityofcarriers in both devices isone-halfthebulkmobility. 17.20GeneralMOSFETReview AnID-VDcharacteristic derivedfromanidealMOSFETis pictured in Fig.P17.20.Note that IDsat=10-3Aand VD =5V for the given characteristic.Answer the questions that Dsat follow making useof the square-law theoryand theinformationconveyed in the figure. ID ↑ $Dsat= 10-{3}A ① 1 1 1 1 1 1 1 1 1 1 1 1 1 1 →VD VDsat=5V 10V Figure P17.20 (a)Carefully sketchthe inversionlayerand depletionregion insidetheMOSFETcorre- spondingto point1)onthepicturedcharacteristic.Showandlabelallpartsofthe transistor. (b) Givenaturn-on voltageof V=1V,whatis thegatevoltageone mustapplytothe MOSFET gate to obtain the pictured characteristic? (c)If=0.1m,whatistheinversion-layercharge/cm²atthedrainendofthechannel when the MOSFET is biased at point (2)on the characteristic? (d) Supposethegate voltageisreadjustedsothat VG-=3V.Forthenewcondition, determineIDif V=4V. (e) DeterminegdifthequiescentoperatingpointoftheMOSFETispoint(3)onthepic- tured characteristic. (f）DeterminegifthequiescentoperatingpointoftheMOSFETispoint3）onthe pictured characteristic. (g)IfVD=0(ie.thedrainisshortedtothesourceandback)sketchthegeneralshapeof theCG （gate capacitance) versus VGcharacteristicto beexpectedfromtheMOSFET.

--- Page 667 --- 644 FIELD EFFECTDEVICES 17.21Makingfreeuseofthesquare-lawentriesinTable17.1ignoringthevariationof with V,andemployingonlyone setof coordinates pereach partof the problem,draw (a)g/（ZnC/L)versus VG（0≤VG≤5V)ifV=1Vand VD=0,1,and2V (b）g/（ZunC/L)versus VD（0≤VD≤5V)when VG-V=1,2,and3V. (c）gm/（ZnC/L）versus VD（0≤V≤5V）whenVG-V=1,2,and3V. 17.22Compare theMOSFETand J-FET;concisely describe similaritiesanddifferences in structure, operation, and analysis.

--- Page 668 --- 18 Nonideal IMOS Theideal structure provides a convenient vehiclefor establishing thebasic principlesof MOStheory in a clear and uncomplicatedfashion.RealMOSdevice structures,however, are never perfectly ideal.In this chapter we examine well-documented deviations from the idealthat have beenencountered in MOS device structures. Theeffectof a nonidealityon device characteristicsitsidentified or suspected physical origin,andmethodsimplemented to minimizethe nonidealityare noted.Becauseofease offabricationandfunctionalsim- plicity,the MoS-capacitor has long been the test structure ofchoice forprobing nonideal- ities.Itisunderstandablethereforethatthevastmajorityofnonidealefectsareillustrated using MOS-CC-Vdata.The descriptionherein likewise relies heavilyon thecomparison of real andidealMOS-C C-Vcharacteristics.Nevertheless,any deviationfrom the ideal has a comparable impactontheMOStransistor.To underscore this fact,the chapter con- cludes with a section dealing exclusively with the MOSFET.We discuss hownonidealities can affect the MOSFETthreshold voltage,practical ramifications,and in-use methodsfor adjusting the threshold voltage. 18.1METAL-SEMICONDUCTORWORKFUNCTIONDIFFERENCE Theenergy band diagramsfor theisolatedcomponentsof anA1-SiO2-(p-type) Sisystem are drawn roughly to scale in Fig.18.1(a).Uponexamining this figure,we seethat in areal device theenergydifference between the Fermienergy and the vacuum level is unlikelyto bethe same in the isolated metal and semiconductor componentsof the system; that is,in contrast to theideal structure,Φ ≠Φs =χ+(E-EF)FB.Tocorrectly describereal systems,the idealtheory must be modified to account for this metal-semiconductor work- function difference. In workingtoward therequiredmodification,letusfirstconstructtheequilibrium (VG=0)energybanddiagramappropriateforthe sample systemofFig.18.1(a).Webegin by conceptually connectinga wire between theouterendsof the metal and semiconductor. Thetwo materials arethen brought together in a vacuum untiltheyare a distance xapart. Theconnecting wire facilitates the transferofcharge between the metaland semiconductor and helps maintain the system in an equilibrium state where the respective Fermi levels “line up”asthematerialsarebroughttogether.WiththemetalEandsemiconductorEat the same energy,and ΦM≠x+(E-F)FB,the vacuum levels in the two materials must beat differentenergies.Thus an electricfield,vac,develops betweenthecomponents, with theSi vacuum levelabove the Al vacuumlevel given the situation picturedin 645

--- Page 669 --- 646 FIELDEFFECTDEVICES $xi$ x Φs ΦM$EEEEFAl$SiO_{2$Si(a)11A11111下x\Phi M11$x{} 1 1 Φ'M} 1 1 EE印E 1 1 (b) Figure18.1a)ToscaleenergybanddiagramsfortheisolatedcomponentsoftheAl-SiOSi system.(b)Equilibrium(V=0)energyband diagramtypical ofrealMOSstructures. Fig.18.1(a).Moreover,band bending occurs inside the semiconductor-K&smustequal Evac.vac and the semiconductor band bending increase of course as the components are brought closerandclosertogether.Once the metaland semiconductor are positioned a vac distance x apart,the insulator is next inserted into the empty space between theother two components. Theadditionofthe insulator simplylowerstheeffective surface barriers

--- Page 670 --- NONIDEAL MOS 647 （ΦM→Φ-i=Φandχ→x-i=χ’)andreduces the electricfieldin the region(Ko>1).Theresultingequilibrium energy banddiagram typicalofrealMOS systems is shown in Fig.18.1(b). Thepointto bederivedfromthe preceding argument andFig.18.1(b)is thatthe work- functiondifference modifies therelationship between the semiconductor surface potential and theappliedgatevoltage.Specificallysetting VG=0doesnot giverisetoflatband conditions inside thesemiconductor.Like in a pn junction or MS diode,there is a built-in potential.Theprecise valueofthebuilt-inpotential,bi,canbedeterminedbyequating theenergies from the Fermi levelto thetopof the band diagram as viewedfromthe two sidesof the insulator in Fig.18.1(b).Oneobtains Φ+ q∆Φx=(c-)FB-qs+χ′ (18.1) metal side semiconductor side Thus,takingthemetaltobethezero-potentialreferencepointtheusualprocedureinde- fining built-in potentials), wefind bi=-（φs+Δx)=Ms (18.2) where 1 1 φ\s$一(\Phi M$ $\text{s}=$ 一 −χ′-(-)FB (18.3) q q Perhapstheresulthereshouldhave beenintuitivelyobvious:Thebuilt-in potentialinside a ΦM≠Φ,butotherwise ideal,MOS structure is just the metal-semiconductor work- function difference expressed in volts. In dealing with any nonideality,a majorconcern is the effectof the nonidealityon device characteristics.Generally speaking,one wouldliketo know how the givennon- idealityperturbstheideal-devicecharacteristics.Toillustratethegeneraldetermination procedure and to specifically ascertain theeffectof a Ms≠0,letus suppose Fig.18.1(b) is the energy band diagram foran MOS-C.Also let the broken-line curve in Fig.18.2be the expected formof the high-frequencyC-Vcharacteristic exhibited by an ideal version of this p-bulk MOS-C.Flat band for the ideal device occurs,of course,at a gate biasof zero volts.On theother hand,from a cursory inspectionof Fig.18.1(b)one infers that a negative bias must beappliedtothe nonideal device to achieveflat bandconditions.In fact, a gate voltage V=Ms （where ΦMs<0for the given device)must be appliedtooffset thebuilt-in voltageandachieveas=0.Sincebothdevices willexhibitthesamecapaci- tance underflat-bandconditions,weconcludetheflat-band pointforthe real device will be displaced laterally ΦMs volts along the voltage axis. Asitturns out,we can begintheargument just presented at any pointalong the ideal- device characteristic.There is aone-to-one correspondence between the degreeofand

--- Page 671 --- 648 FIELDEFFECTDEVICES C FB FB ΦMS $φM{s≠0 φMs=0 0 Figure18.2 Effect of aMs≠0on the MOS-Chigh-frequency C-Vcharacteristic. bending or s and theobservedcapacitance.Thus,regardlessofthe referencepointalong the ideal-device C-Vcharacteristic,one mustalways apply anadded Ms volts to the gate of the real device to achieve the same degree of band bending and hence observe the same capacitance.In other words,as pictured in Fig.18.2,the entire real-device C-Vchar- acteristic willbe shifted Ms voltsalong the voltage axisrelativetotheideal-device characteristic. In the preceding discussion the effectof the Ms≠0 nonideality was described in graphicalterms.Alternativelyonecangeneratemathematicalexpressionforthevolae shift,△VG,betweentheidealandrealCVcurves.IfVisthevoltageappliedtothegae of the ideal device to achieve a given capacitance,and VG the real-device gate voltage required to achieve the same capacitance,then,simply converting the C-Vcurve discus- sion into mathematical terms, △Δ{=( {- )mc=φMM (18.4) Itshouldbeinterjectedthatitiscommonpracticeto useVforthegatevoltagewhen referring to the ideal structure.Forpresentation clarity, we have herein limited the useof Vto thischapter where simultaneous reference is made to real and ideal devices. The actual ΔVG=Ms value for a given MOS structure is routinely computedfrom Eq.(18.3)using the Φ-χappropriateforthe systemandthe(E-EF)B deduced from a knowledgeof the doping concentrationinside the semiconductor. The Ms (T=300 K)for the commercially importantn+ poly-Si-gate and Al-gate systems are graphed as a functionof doping in Fig.18.3.The experimentally determined ΦM values for a number of other metal-siliconcombinations are listed in Table18.1.Note from Fig.18.3and the Φ-valueslistedin Table18.1that Ms is moreoftenthan not a negative quantity,especially for p-type devices,and is typicallyquite small-on theorder of one volt or less.

--- Page 672 --- NONIDEAL MOS 649 -0.2 AI(n-Si) -0.4 $n+$ poly (n-Si) -0.6 -0.8 Al(p-Si) -1.0 -1.2 n + poly (p-Si). 1014 1015 1016 1017 1018 NA orND （cm^{³3}) Figure18.3Workfunctiondiferenceasafunctionofthen-and -typedopantconcentationin n+poly-Si-gateandAl-gateSiO-Sistructures.（T=300K.ΦM-χ=-0.18eVforthen+pol- Si-gate structure; ΦM-x=-0.03eV for the Al-gate structure.) Table18.1 Barrier Height Differences in Selected Metal-SiO2-Si Structures. (Data from Kar[19].) MetalGate Material ΦM-x=Φ^M-x^{‘(eV) Ag 0.73 Au 0.82 Cr -0.06 Cu 0.63 Mg -1.05 Sn -0.83 Exercise 18.1 P:ItispossibletobuildanMOS-CwithaMs=0throughtheproperchoiceofgate material and Si doping concentration.Restricting the Si doping to be in the range 1014/cm³≤NAorND≤1018/cm3,andassumingoperationatT=300K,identify thegate-material/doping-concentrationcombinations)thatgivesrisetoMs=0. Employ the ΦM-x’ values given in Table 18.1.

--- Page 673 --- 650 FIELD EFFECT DEVICES S: Noting that (Ec-EF)FB=E-E+(E₁-F)FB ≡EG/2-kTln(ND/n) ..n-type Si ≌EG/2+kT1n(NA/n) .. .p-type Si andemployingkT=0.0259eV,=1.12eV,andn=1010/cm3,onecalculates 0.08 eV≤(Ec-F)FB≤0.32eV V…if1014/cm3≤ND≤1018/cm³ 0.80eV≤(Ec-F)FB≤1.04eV…if1014/cm≤NA≤1018/cm3 Since φMs=(1/q)[Φ−’-(-F)FB],toachieveaMs=0clearlyrequires 0.08eV≤Φ-x’≤0.32eV V.. .or…0.80eV≤ΦM-x’≤1.04eV Examining Table18.1,wefindthattheonly gate materialthatmeets the general requirement is Au with a Φ-χ’=0.82eV. The specific doping of the Au p-Si MOS-Cexhibiting a Ms=0 must be such that (EC-EF)FB=ΦM-χ′=0.82eV or (Ei-EF)FB= 0.26eV and NA=n;e(E-EF)FB/k=1010e0.26/0.0259=2.29×1014/cm³ Thus Au gate;NA=2.29 ×1014/cm³⇒MOS-C with ΦMs=0 18.2 OXIDECHARGES 18.2.1General Information Asmightbeinferredfromthecomments at theendof Section18.1,Ms≠Oisarelatively minornonideality.Thevoltage shiftassociated with Ms≠Oissmall,totallypredictable, and incapableofcausing deviceinstabilities.Oxidecharge,ontheotherhand,can giverise

--- Page 674 --- NONIDEAL MOS 651 Metal K{+ Mobile ion charge Oxide trapped charge Na{+ SiO_2 + + + Fixed charge 一 田 田 + X X X Interface trapped charge !S Figure18.4 Natureandlocationofchargecentersinthermallygrown SiO2-Sistructues. (Adapted from Deal[20l,©1980IEEE.) to farmoresignificanteffects,includinglargevoltageshiftsandinstabilities.Throughex- tensiveresearchanumberofdistinctchargecentershavebeenidentifiedactuallywithinthe oxideorattheSi-SiO2interface.The nature and positionof theoxide charges aresum- marized inFig.18.4. Toestablishthe generaleffectofoxidecharges,letus postulatetheexistenceofa charge distribution,x(),thatvariesinanarbitrarymanneracrossthewidthoftheoxide layer.NotefromtheFig.18.5visualizationofthecharge distributionthat,forconvenience in thisparticularanalysis,theoriginof the x-coordinate has been relocated at the metal- oxide interface.Withtheadditionofthechargecenters,aportionoftheV-sderivation presentedinSubsection16.3.2isnolonger validandmustberevised.Specifically,inplace of Eqs.(16.19) to (16.21),one has,respectively, d $\rho ox(x)ox(18.5)dxKO01$xo =（x)^{0 $\rho ox(x^{{}^{\prime }}dx$ (18.6) dx $KO\epsilon 0$Jx$and1$xo ∆x=x0&0(0)- ρox(x^{)dx'dxx (18.7) $KOε0 Jo Thedouble integral inEq.(18.7)can bereducedto asingle integralemploying integration byparts.Moreover,&x(）=KsEs/KOifa planeof charge [other thanone possibly

--- Page 675 --- 652 FIELD EFFECTDEVICES x M 0 S $x0x=x$ Figure18.5 Arbitrary distribution of oxide charges. includedinxieclueromteoidmonuctorce.i indicated modifications yields K_s} 1 ${}_o$ x_{s xp{ox(x)dxx (18.8)

Ko}

K{0 Jo Since V=+△xforastructure thathaschargecentersintheoxidebutisotherwise ideal,one obtains K_s} 1 _{ V_{ = s+ x_{s xp{ox(x)dxx (18.9) KOK{O$JoHowever,foranidealdevice$K_{s}V{} s+x_{s$(18.10)KO$ Thus ∆V_G(charges) oxide 1 xo $x\rho ox(x)dx$ (18.11) K{ε Jo As emphasized in the development,the voltage translation specified by Eq.(18.11)is valid for anarbitrarycharge distributionand is addedto the Eq.(18.4)voltage translation due to Ms. In the following subsections we systematically review known information about the various types of charge centers and examinetheir specific effecton MOS device characteristics.

--- Page 676 --- NONIDEAL MOS 653 18.2.2 Mobile lons The most perplexing and serious problem encountered in the development of MOS devices canbedescribed asfollows:First,theas-fabricatedearly(c.1960)devicesexhibitedC-V characteristics that were sometimes shifted negatively by tens of volts withrespectto the theoretical characteristics.Second,when subjected tobias-temperature (BT)stressing, a commonreliability-testing procedure where a device is heated underbiasto accelerate device-degrading processes,theMOSstructures displayed asevere instability.The negative shift in the characteristics was increased additionaltens of volts after the device was biased positively and heated up to150C or so.Negative bias-temperature stressing had there- verse effect: The C-Vcurve measured at room temperature after stressing shifted positively ortoward the theoretical curve.Inextremecasesthe instabilitycouldevenbeobservedby simplybiasing the device at room temperature.One might sweeptheC-Vcharacteristics foragiven device,goouttolunchleavingthedevicepositivelybiased,andreturntoreat the C-Vmeasurementonly to find the characteristics had shifted a voltor so toward nega- tivebiases.Notethatthecharacteristicswerealwaysshiftedinthedirectionoppositetothe appliedgate polarityandthattheobservedcurves werealways tothe negative sideofthe theoretical curves. The nature and extentof the problem is nicely summarized inFig.18.6. From apractical standpoint,thenonidealitycausingtheas-fabricatedtranslationand instabilityof theMOSdevicecharacteristicshadto beidentifiedandeliminated.Adevice whose effectiveoperatingpointuncontrollablychangesasafunctionoftime isfairlyuse- less.Itisnowwellestablishedthatthelargeas-fabricatedshiftingandtherelatedinstability can be traced to mobile ions insidethe oxide,principally Na. -△VB =-79V 1.0 0.9 Before bias temperature C \er } 0.8 0.7 After +12V bias After-30Vor0Vbias 100°C 70min 200^{°C60min 0.6 1 1 -140 -120 -100 -80 -60 -40 -20 0 { Figure18.6 Illustrationofthe largeas-fabricatedC-Vcurve shiftingandbias-temperaturein- stabilityobserved withearly MOSdevices.AllC-Vcurves weretaken atroom temperature;= 0.68m.Thearrows adjacenttotheafter+BTcurves indicatethedirectionof thevoltageswep. (From Kerr et al.[2,©1964 byIBMCorporation.Reprinted withpermission.)

--- Page 677 --- 654 FIELD EFFECTDEVICES Ifρion(x) is taken to be the ionic charge distribution with $o$ $Q_M=$ ρion(x)ddx (18.12) Jo beingthetotalionicchargewithintheoxideperunitareaofthegatethenitclearlyolows from Eq.(18.11) that ΔV_{ mobile 1 xo xρion(x)dx (18.13) ions

K{0 Jo Note fromEq.(18.13)that positive ions in theoxide would giverise toa negative shift in the C-Vcharacteristics as observed experimentally,while negative ions would giverise to a positive shift in disagreement with experimentalobservations.Furthermore,because the integrand inEq.(18.13)varies as xρion(x),ΔVGis sensitive totheexactpositionoftheions in the oxide.If,for example,the same QMcharge per unit gate area is positioned (a)near the metal and(b)near the semiconductor as shown inFig.18.7(a),one computes △VG（a）=-（0.05)QM/C。andΔVG（b）=-(0.95）QM/C，whereC=KOε0/Forthe cited example,the shift is predictedto be some19 times larger whenthe ions are located near the oxide-semiconductor interface! Indeed,based on the preceding observations,it is reasonable to speculate that a largeas-fabricated negativeshift in the measuredC-Vchar- acteristicsandtheattendant instability is causedbypositive ions intheoxidethatmove around or redistribute under bias-temperature stressing.The required ion movement, away from themetalfor +BT stressing andtowardthemetal for-BTstressing,is in factcon- sistent withthedirectionofionmotiongrosslyexpectedfromtherepulsive/attractiveaction ofother charges within the structure (see Fig.18.7b). Actualverificationof themobile ionmodelandidentificationoftheculprit (theionic species)rivals someof the bestcourtroom dramas.The suspects were first indicted because of theirpast historyand theiraccessibilityto the sceneofthecrime.Long beforethefab- ricationofthefirstMOSdevice,asfarbackas1888,researchershaddemonstratedthat Na+,Li+,and K+ ions couldmove through quartz，crystalline SiO2，at temperatures below250C.Furthermore,alkali ions,especially sodiumions,wereabundant inchemical reagents,in glass apparatus,onthe handsoflaboratory personnel,and in the tungsten evaporationboatsusedinformingthemetallicgate.Withthesuspectidentified,geatcare was taken to avoid alkali ioncontamination in the formationof theMoSstructure. The net result was devicesthat showedessentially nochange in their C-Vcharacteristics afterthey were subjected to eitherpositive or negative biases for many hours at temperatures up to 200C.Next,other carefully processed devices were purposely contaminated byrinsing the oxidized Si wafers in a dilute solutionof NaCl(orLiCl) prior to metallization.As expected,the purposely contaminateddevices exhibited severe instabilities underbias- temperature stressing.Finally,sodium was positively identified in theoxidesof normally

--- Page 678 --- NONIDEALMOS 655 ρ ion A ρ ion A x -x 0.1x 0.9x 0 x_{ 0 0 x。 Distribution-a Distribution-b (a) M 0 S M 0 S ||||||Ⅲ|||| + + + + + + + + + + + + + + + + + + + + + + (一) biases (+) biases (b) Figure18.7 (a) Two hypothetical ionic charge distributions involving the same total numberof ions situated nearthe metal(distribution-a)andnearthe semiconductor(distribution-b).(b)Expected motion of positive mobile ions within the oxide under(+)and(-)bias-temperature stressing. fabricated devices (no intentionalcontamination)through the useofthe neutronactivation technique;thatistheoxideswerebombarded withasuficientnumberof neutronstocreate aradioactive speciesof sodium. Theanalysisof the resultantradioactivity directly con- firmed the presence of sodium within the oxide. Althoughcaretoliminatealkali-ioncontaminationthroughoutthefabicationprocess did lead to stable MOSdevices,MOSFET manufacturers encountered difficulties in attain- ing andmaintaining therequired degreeofqualitycontrol inproduction-linefacilities. Thus,inaddition to alkali-ionreductionefforts,specialfabrication procedures were de- velopedandimplementedto minimizetheeffectsofresidualalkali-ioncontamination. Two different procedures found widespread usage:phosphorus stabilizationand chlorine neutralization. In phosphorus stabilizationtheoxidized Si wafer is simplyplaced in a phosphorus dif- fusion furnace for a short periodof time.During the diffusion,as ilustrated in Fig.18.8(a, phosphorusenters theouterportionof theSiO2filmand becomes incorporated intothe bonding structure,thereby forming a new thin layer referred to as a phosphosilicate glass. At the diffusion temperature the sodium ions areextremelymobileand invariably wander into thephosphorus-laden regionof theoxide.Once in the phosphosilicate glass the ions

--- Page 679 --- 656 FIELDEFFECTDEVICES Phosphorus Phosphosilicate M glass $Na+$Na+ +N +BN +BN Na{+ Na+ +BN SiO_{2}$Si{O}_2$ $Na+$SiO_{2$SiSi!S(a)ChlorineMMNa+Na+$Na{+ Na+ SiO{}} $Si{O}_2$ Na Na 个 Si Si Chlorine-rich Si material (b) Figure18.8 PictorialdescriptionofMOSstabilizationprocedures:(a)phosphorus stabilization; (b) chlorine neutralization. become trapped and stay trapped when the system is cooled to room temperature.In this positionednear theouter interface where they give rise to the leastamountof as-fabricated C-Vcurve shifting,and are heldfirmly in place during normaloperating conditions.The phosphosilicateglass layer,it shouldbe noted,alsoblocks any subsequentcontamination associated withthegate metallizationorother poststabilization processing steps. Chlorine neutralization involves a totally differentapproach.Asmallamountofchlo- rine inthe form of a chlorine-containing compound is introduced into the furnace ambient during thegrowthof theSiO2layer.As pictured in Fig.18.8(b),thechlorine enters the oxide andreacts toform a new material,believedto be a chlorosiloxane,located atthe oxide-silicon interface.Stabilizationoccurs whenthe ionic sodium migrates into the vi- cinityof theoxide-silicon interface, becomes trapped,and is neutralized.Once neutral- ized,the sodium has no effect on the MOS device characteristics. The shrinking size of MOS device dimensions,calling for gate-oxide thicknesses on the orderof100Å,limitstheuseof thephosphorus stabilizationprocedure inpresent-day structures.Because of a potential polarizationproblem,the phosphosilicate glass can be

--- Page 680 --- NONIDEALMOS 657 onlyasmallfractionoftheoveralloxidethickness.With~100,thegetteringvolume becomes difficulttocontrol.Chlorineneutralizationcontinuesto beemployed,withcareful controlofthechlorine concentrationtoavoid anoxidethickness variationthat can accom- pany the process.Neutralization coupled withimprovements in the purityof fabrication materials (chemicals,gases,etc.) and upgraded processing procedures now permit the rou- tine fabrication of stableMOSdevices.It is commonpractice,nevertheless,to closely monitor furnace tubesandthe processing in general to detect theonsetof ionic contami- nation.As an extensionof the chlorine neutralization procedure,chlorine has come tobe widelyemployed in the pre-oxidationcleaning offurnace tubes.Also,a phosphosilicate glass layer is invariably deposited via chemical vapor techniques to form a protective coating onICs. The layer helps to minimize ionic contamination subsequent to device fabrication. Exercise 18.2 P:Positive bias-temperature(+BT)and negative bias-temperature（-BT)stressing performed fora sufficientamountof timeto respectively cause the mobile ionsto pile up attheO-S andM-Ointerfaces,andthe voltage displacementbetweenthe correspondingCVcurves,are routinelyusedtodeducethetotalmobile-ioncharge cm²(OM)inside MOS-Cs.Suppose an MOS-C with=0.1μmexhibits the post- stressing C-Vcharacteristicspictured below.Assuming themobile ionsare all piled upin aδ-functiondistributionajacenttotheOSinterfaceafterBTsressing,and allpiledup inaδ-functiondistributionadjacenttotheM-Ointerfaceafter-BT stressing,determine QM/q. C After+BT ←10V→ →VG S:After+BTstressing,ion=QMδ().SubstitutingintoEq.（18.13)therefore yields mobile ions 1 _{o x_{Q_}OM}ΔVG$ xQMδ（₀)dx after +BT

K{ε0 二 K{0ε0

C_{ Jo

--- Page 681 --- 658 FIELD EFFECTDEVICES After-BTstressing,ρion=QMδ(0)and mobile ions 1 xo △V_{G$xQM\delta (0)dx=0after-BT二$KOε$JoAssumingallothernonidealitiesareunaffectedbythebias-temperaturestressing,thedisplacementbetweenthetwoC-Vcurvesismerelyduetothedifferencebetweentheabove\Delta VG(+BT)and\Delta VG(-BT);thatis∆VG(+BT)-△VG(-BT)=-10V=-QM/C0Thus$OM}C_o}$ [△（+BT)-△G(-BT)] q q (3.9)(8.85× 10-14) (10)=2.16×101²/cm2 (1.6×10-19)(10-5) 18.2.3 The FixedCharge The gross perturbation associated with mobile ions in the oxidetended to obscure or cover up theeffectsofotherdeviationsfromtheideal.Indeed,withthesuccessfuleliminationof the mobile-ion problem,it became possible to perform a more exacting examinationof the device characteristics.Theresults were rather intriguing.Even in structures free ofmobile ions,and aftercorrecting for Ms≠0,theobserved C-Vcharacteristics were still translated up to a few voltstoward negativebiasesrelative tothetheoreticalcharacteristics.The possibilityof mobile-ion contaminationhad beeneliminated because the structures were stable underbias-temperature stressing.Moreover,for a given setof fabricationconditions the observed ΔVG was completely reproducible.Confirming data were obtained from de- vices independentlyfabricated by a numberof workers at differentlocations.Subsequent testing (by etching theoxide away in small steps and through photo-measurements)re- vealed the unexplained ΔG shift was caused by acharge residing within theoxide very close to theoxide-semiconductor interface. Because this quasi-interfacial charge wasre- produciblyfabricated into thestructureand was fixed in positionunder bias-temperature stressing,the nonideality became known as the“built-in”or”fixed”oxide charge. In modeling the quantitativeeffect of the fixed charge on theC-Vcharacteristics,it is typicallyassumed thecharge is located immediately adjacentto theoxide-semiconductor interface. Under this assumption one can write ρox(x)=QFδ() (18.14)

--- Page 682 --- NONIDEAL MOS 659 whereδ(x)isadelta-functionpositionedattheoxide-semiconductor interfaceandOFis bution”into Eq.(18.11)and simplifying yields V_G(charge) fixed Q} (18.15) $C_o$ From Eq.（18.15)itisobviousthatlikethemobile-ioncharge,thefixedoxidecharge mustbepositivetoaccountforthenegativeΔVG’sobservedexperimentally.Otherrelevant informationaboutthe fixedoxidecharge can be summarized as follows: (1)The fixed charge is independentof theoxide thickness,the semiconductor doping con- centration,and the semiconductor doping type (n or p). (2) Thefixedcharge variesasafunctionoftheSi surfaceorientation；OFislargeston 111}surfaces,smalleston100}surfaces,andtheratioofthefixedchargeonthewo surfaces is approximately 3:1. (3)QFis astrongfunctionoftheoxidationconditionssuchastheoxidizing ambientand furnace temperature.As displayed in Fig.18.9,the fixed charge decreases more or less linearly withincreasingoxidationtemperatures.Itshouldbemphasized,howetha only theterminal oxidationconditions areimportant.If,forexample,aSi wafer isfirst oxidized in watervaporat1000Cfor1h,andthenexposedto a dry O2ambientat 1200Cfor a sufficiently long time toachieve a steady-state condition （~5min),the OF value will reflect only the dry oxidation process at1200C. (4) Annealing (that is,heating)of anoxidized Si wafer in an Aror N2atmosphere fora timesufficient toachievea steady-stateconditionreduces QFto the valueobservedfor dryoxidationsat1200C.Inother words,regardlessoftheoxidationconditions,the fixed chargecanalways bereducedtoaminimum byannealing inan inertatmosphere. Thepreceding experimental factsall providecluesto the physicaloriginofthefixed oxide charge.For one,although doping impurities from the semiconductor diffuse into the oxide during the high-temperatureoxidation process,the fixed charge was noted tobe independent of the semiconductor doping concentrationand doping type. Theexistence of ionized dopingimpurities withintheoxidecan therefore beeliminatedas a possible source ofQF.Second,thecombinationoftheinterfacial positioningofthefixedcharge,theSi- surface orientation dependence,and the sensitivityofQF to the terminaloxidation condi- tions suggests thatthefixedchargeisintimatelyrelated totheoxidizingreactionattheSi SiO2interface.Inthisregard,itshouldbeunderstoodthaduringthethermalformatonof the SiO2layer,theoxidizingspecies diffuses throughtheoxideandreactsattheSi-SiO2 interface to form moreSiO2.Thus,thelastoxideformed,the portionof theoxide con- trolledbytheterminaloxidationconditions,liesclosesttotheSiSiO2interfaceandcon- tains the fixed oxide charge.From considerations such as these,it has been postulatedthat

--- Page 683 --- 660 FIELDEFFECTDEVICES 1.0 $1200°C0.9$1100{ 1000° -Theory 0=0 C 0.8 900° ${\phi }_s==0$ \ova } 0.7 0.6 -10 -8 -6 -4 -2 0 2 G(volts) (a) 10 8 After dry O2oxidation $\text{mfra})$ 6 (111)Si 4 After N{2 anneal 2 0 1 1 1 600 700 800 900 1000 1100 1200 T({C (b) Figure18.9 Effectof theoxidation temperatureand annealing onthe fixed charge in MOSstruc- tures.(a)MeasuredC-VcharacteristicsafterdryOoxidationsat varioustemperatures[=0.2m, ND=1.4×101/cm3,（111)Sisurfaceorientation].(b)Fixedchargeconcentrations-theso-called oxidation trianglespecifying theexpectedOF/q after dryO2oxidationandafter inertambientan- nealing.[(a)From Dealetal.22]Reprintedbypermissionof the publisherTheElectrochemical So- ciety, Inc.]

--- Page 684 --- NONIDEAL MOS 661 the fixedoxidecharge is due toexcess ionicsiliconthat has brokenaway from thesilicon properand is waitingtoreactinthevicinityofthe Si-SiO2interface whentheoxidation process is abruptly terminated.Themonolayerofoxideadjacentto the Si surface hasin fact been experimentally determined to bex<2 SiO,which isconsistent withthe excess- Si hypothesis. Annealing in an inertatmosphere,a standardprocedure forminimizing the fixedoxide charge,apparently reduces the excess reaction components and thereby lowers QF. Exercise18.3 P:An MOS-C is characterized by the energy band diagram shown in Fig.E18.3. (a) Roughly sketch the electric field ()inside the oxide and semiconductor as a function of position. (b)Isthereanioniccharge（OM)distributedthroughouttheSiOEplan. (c) Isafixedcharge likelytoexistattheSi-SiO2interface?Explain. $Ec$ EFM Ei EFS $Ev$ Figure E18.3 S:(a)Theelectricfieldisdirectlyproportionaltotheslopeoftheenergybands.The deduced & versus x dependence is sketched below. 8 →x SiO_{2}$ Si

--- Page 685 --- 662 FIELDEFFECTDEVICES (b) Ifρox=0,x=constantandtheoxideenergybandsarea linear-functionof position.If ρx≠0,ifthere ischarge distributed throughouttheoxide,xbecomes a functionof positionandthe oxideenergy bands will in turn exhibitcurvatureSince the oxide energy bands are a linear functionof position in Fig.E18.3 and

ox constant, we conclude [QM≈0]. (c) The normalcomponentof the D-field,where D=Kε,must becontinuous if there is no planeof charge at an interface between two dissimilar materials (see Sub- section16.3.2).Whena planeofcharge doesexist,there isa discontinuity inthe on the oxide side of the interface and D=ε>0on the semiconductor side of theinterface,theremust beaplaneofcharge at theSi-SiO2interfaceinthe device characterized by Fig.E18.3.Moreover, since Qinterface

Ds$ D

KSε0s ox S KεxtheintrcilchagemustbeoitivThefiedchrgecloslppxi matesaplaneofpositivechargeattheSiSiO2interaceand wesuspectOF≠. (In generala D-fielddiscontinuityattheSi-SiO2interface canarisefromother sourcesof interface charge. These include the mobile ioncharge drifted to theO-S interface during +BT stressing and the interfacial trapcharge discussed in the next subsection.) 18.2.4Interfacial Traps Judged intermsoftheirwide-ranginganddegradingeffectontheoperational behaviorof MISdevices,insulator-semiconductor interfacial traps mustbe consideredthemostim- portantnonidealityencountered inMISstructures.Acommonmanifestationofa sigif- cant interfacial trapconcentration withinan MOS-Cis the distortedorspread-out natureof theC-Vcharacteristics.This isnicely illustratedin Fig.18.10,which displays twoC-V curves derivedfrom the same device beforeand after minimizing the numberof Si-SiO2 interfacial traps inside the structure. From prior chapters thereader is familiar with donors,acceptors,andrecombination- generation(R-G)centers,whichintroducelocalizedelectronicstatesinthebulkofasemi conductor.Interfacial traps (alsoreferredtoas surfacestatesor interfacestates)areallowed energy statesin whichelectronsare localizedinthevicinityofamaterialssurface.Allof thebulk centers are found to addlevels to the energy band diagram within the forbidden band gap.Donors,acceptors,andR-Gcentersrespectively introducebulklevels nearE, E,andE.Analogously,asmodeledinFig.18.11,interfacialtraps introduceenergylevels in the forbiddenband gapattheSi-SiO2interface.Note,however,that interface statescan, and normallydo,introduce levels distributedthroughout the bandgap.Interface levelscan alsooccur atenergies greaterthan Eorlessthan E,butsuchlevels are usuallyobscured by the much larger density of conduction or valence band states. Figure18.12provides some insight into thebehaviorandsignificanceof the levels. When ann-bulkMOS-Cis biased into inversionas shown in Fig.18.12(a),the Fermilevel atthe surfaceliesclosetoE.Forthegivensituationessentiallyallofthe interfacialtraps

--- Page 686 --- NONIDEAL MOS 663 n-type (ll)Si No H_{2} anneal --- H_{2 anal Normalized capcitane C/o 1.0 0.8 Before 0.6 After 0.4 0.2 1 1 -25 5-20 -15 -10 -5 0 5 10 15 Gate voltage (V) Figure18.10 C-V characteristics derivedfrom the same MOS-C before(—)and after (----) minimizingthenumberofSi-SiOinterfacialtrapsinsidethestructure.FromRazoukand Deal[23]. Reprinted by permission of the publisher, The Electrochemical Society,Inc.) willbeemptybecause,toafirst-orderapproximation,allenergylevelsaboveEFareempty and allenergylevels belowEF arefilled.Moreover,ifthestates areassumedto bedonor- like in nature(that is,positivelycharged whenemptyand neutral whenfilled withanelec- tron),thenetcharge per unit areaassociated withtheinterfacialtraps,O,willbe positive. Changing the gate bias to achieve depletion conditions (Fig.18.12b)positions the Fermi level somewhere near the middleoftheband gap at the surface.Since the interface levels always remainfixed inenergyrelativetoEandEatthesurface,depletionbiasing Interfacial traps Ec$ Ev 0 S Figure1.1Electiclmodelinofintracialtapsaallowedlctroniclevlslocliedinsace at the oxide-semiconductor interface.

--- Page 687 --- 664 FIELDEFFECTDEVICES ←+一 10++1 EE E{c E{cEF E{Ev} Ev} (a) (b) (c) Figure18.12Fillinoftheinterfacelevelsunder(a)inversion(b)depletionandc)accmlaion biasinginann-typedevice.Thechargestateexhibitedbydonor-likeinterfacialtraps[+”plus)or “O”(neutral)] isnotedtotheleftof therespectivediagrams. obviouslydrawselectronsintothelowerinterfacestate levelsandQTreflectstheadded negativecharge:QI（depletion)<Q(inversion).Finally,withtheMOS-Caccumulation biasedFig.18.2c)electronsfillmostoftheinterfacialtrapsandQapproachesitsmini- mum value.The point isthattheinterfacial trapschargeand discharge asa functionofbias, thereby afecting thecharge distributioninside the device,theV-srelationship,andthe device characteristics in an understandable but somewhat complexmanner. The grosseffectof interfacial traps on theVG-srelationship is actuallyquite easy to establish.SinceO,likeQF,islocatedrightatthe Si-SiO2interface,we can writeby analogy with the fixed charge result △V_{$(interfacial$QI（φs) (18.16) traps C_0o} AsemphasizedinEq.(18.16),theresultherediffersfromthefixed-chargeresultinthat OT varies with s,while OF is aconstant independentof s. Combined with the earlierconsiderations on thefilling of interface levels,Eq.(18.16) helps to explaintheformoftheC-Vcharacteristics presented inFig.18.10.Assuming donor-likeinterfacialtraps,OItakesonitslargest positivevalueunderinversioncondi- tions and givesrise to amoderately large negative shift in theC-Vcharacteristics.In pro- gressing through depletion towardaccumulation,Q decreases,and the translation in he C-Vcurve likewise decreases as observed experimentally.Once in accumulation VG shouldcontinue to decreaseand stillremainnegativeaccording to theFig.18.12model. TheFig.18.10data,on theother hand,exhibits an increasinglypositive shiftin thechar- acteristics with increasedaccumulationbiasing. This discrepancy can be traced to the donor-likeassumption.InactualMOSdevicesthe interfacialtraps in theupper halfofthe

--- Page 688 --- NONIDEALMOS 665 band gapare believedto be acceptor-like in nature(that is,neutral whenempty and negative when filledwithanelectron).Thusuponreaching latband（orroughlyfatband)Q passes through zeroand becomes increasingly negative as moreand more upper bandgap statesarefilledwithlectrons.Qualitativelythen,wecanexplaintheobservedcharcr- istics.Acompletequantitativedescription wouldrequireadetailedknowledgeof theinter- facial trapconcentration versusenergyandadditionaltheoreticalconsiderationstoesta- lish an explicit expression for QI as a function of s. Althoughmodelsthatdetailtheelectricalbehaviorof theinterfacialtrapsexist,the physicaloriginofthetrapshasnotbeentotallyclarified.The weightofexperimentalevi- dence,howeversupportstheviewthattheinteracialtrapsprimarilyarisefromuatisd chemical bonds or so-called”dangling bonds”at the surface of the semiconductor.When the siliconlatticeis abruptlyterminatedalong a givenplanetoform asurface,oneofthe four surface-atombondsis leftdangling as pictured inFig.18.13(a).Logically,thethermal formationoftheSiO2layertiesup some butnotallof theSi-surface bonds.It is there- maining dangling bonds that become the interfacial traps (see Fig.18.13b). To add support to theforegoing physical model, let us perform a simplefeasibillity calculation.Ona(100)surfacethere are6.8×104Siatoms percm².If1/1000ofthese form interfacialtrapsandoneelectroniccharge is associatedwitheachtrap,the structure wouldcontainaQI/q=6.8×101/cm².Choosingan=0.1μmandsubstitutinginto Eq.（18.16),weobtaina△VG（interfacialtraps)=3.15V.Clearly,itonlytakesarelatiely small numberof residual dangling bondsto significantly perturbthe devicecharacteristics and readily account for observed interfacial trap concentrations. Theoverall interfacial trapconcentrationandthedistributionor densityof states as a functionofenergyinthebandgap(giventhesymbolDwithunitsofstates/cm-eVae extremelysensitivetoevenminorfabricationdetailsand varysignificantlyfrom deviceto device.Nevertheless,reproduciblegeneraltrends have been recorded.The interfacial trap density,likethefixedoxidecharge,is greateston{111}Sisurfaces,smalleston{100} surfaces,and the ratioofmidgap stateson the two surfaces is approximately 3:.After oxidationinadryOambienDisrelativelyhigh101to10states/cm-emid- gap,withthe density increasing for increasedoxidation temperatures in a manneralso ambient,however,does notminimize D.Rather,as willbe described shortly,annealing inthepresenceofhydrogenatrelativelylowtemperatures500C)minimizes . atmidgapafteranidealinterface-stateanneal istypically≤10/cm-eVandthedistribu- tion of states asafunctionofenergy isof theform sketched in Fig.18.14.As shown in this figure,the interfacialtrap density is moreor less constantoverthemidgapregionand increases rapidly as one approachestheband edges.Lastly,the states near the two band edges are usuallyaboutequal in numberandopposite in their charging character;that is, states near theconductionand valence bands are believed to be acceptor-like and donor- like in nature, respectively. The very importantannealingofMOSstructurestominimize the interfacialtrapcon- centration isroutinely accomplished inoneoftwo ways,namely,through postmetallization annealingor hydrogen(H）ambientannealing.In thepostmetallizationprocess,which

--- Page 689 --- 666 FIELDEFFECTDEVICES Dangling bond Surface Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si (a) 0 0 Si 0 Si $Si{O}_2$ Interfacial trap Si Si Si Si Si Si Si Si Si Si Si (b) Figure18.13Physicalmodelfortheinterfacialtraps.(a）“Dangling bonds,whichoccur when bonds (relative number greatly exaggerated)that become the interfacial traps.[Part (b)adaptedfrom Deal[24].] requires a chemicallyactive gate materialsuchas Alor Cr,the metallized structureis simplyplaced in a nitrogen ambient at~450C for5to10min.During the formationof MOS structures,minute amounts of water vapor inevitably become adsorbed on the SiO2 surface.Atthe postmetallization annealing temperaturethe active gate materialreacts with the water vapor on the oxide surface to release a hydrogen species thought to be atomic hydrogen. As pictured in Fig.18.15,the hydrogen species subsequently migrates through

--- Page 690 --- NONIDEALMOS 667 E $EcAnnealedAsoxidized$EE$110101011$10^{{1o0{3}D{(states/cm^²-eV) Figure18.14 Energy distributionof interface states within the band gap.General form and ap- proximatemagnitudeoftheinterfacialtrapdensityobservedbeforeandafteraninterfacestate anneal. the SiO2layertothe Si-SiO2interface whereit attaches itself toadangling Sibond,thereby makingthebondelectrically inactive.Thehydrogen-ambientprocessoperatesonasimilar principle,exceptthe hydrogen is supplied directly intheambientand thestructurened not be metallized. Even though weoriginallystated that the interfacial trapproblem wasof paramount importance,it is a challenge to convey the true significance and scopeof the problem. Simplystated,if thethermaloxidedidn'ttie upmostofthedangling Sibonds,andifan annealing process were notavailableforreducing theremaining bondsor interfacialtraps to an acceptable level,MOSdevices would merely bea laboratorycuriosity.Indeed,high interfacial-trapconcentrations have severely stuntedthedevelopmentofother insulator- semiconductor systems.（H_{2}}}（H_{2}}$H$SiO_{2}$ X X !S Figure18.15Modelfortheannihilationof interfacestates during thepostmetallizationannealing process.Hsrepresentthectivehydrogenspciesinvolvedintheprocess;Xsreprsentinterface states.

--- Page 691 --- 668 FIELDEFFECTDEVICES 18.2.5InducedCharges RadiationEffects Radiation damage in solid-state devices has been a major concernof space and military expertssincethelaunchofthefirstTelstarsatellitethroughthevanAllenradiationbeltin theearly1960s.Radiationtemporarily disabledtheTelstar satelliteandcanhavedebilitat- ing effectson most solid-state devices and systems.Radiation in the formof x-rays,ener- geticelectrons,protons,andheavy ionized particlesallhave a similareffecton MOS de- vices.After irradiation,MOSdevicestructures invariablyexhibit bothan increase in the apparentfixedchargewithintheoxideandanincreaseintheinterfacial-trapconcentration. The sequenceof events leading to radiation-induced damage is summarized in Fig. 18.16.Theprimaryeffectdirectlyrelatedtotheionizingradiationisthegenerationof electron-hole pairs throughouttheoxide.A percentageof the generatedelectrons and holes recombine immediately. Theelectricfield intheoxideoperates to separate the surviving carriers,accelerating electrons andholes inopposite directions.Electrons, whichhave a relatively highmobility inSiO2,arerapidly (nsec)sweptoutoftheoxide.Holes,onthe other hand,tendto betrapped neartheirpointoforigin.Over a periodof time theholes migrate totheSi-SiO2interface（assuming ois positiveas showninFig.18.16)where theyeitherrecombine withelectronsfromthesiliconorbecometrappedatdeep-levelsies. Once trappedinthenear vicinityoftheinterface,theholesmimicthefixedchargethereby givingrisetoanapparentincreaseinQF.Theprocessleadingtointerface-statecreationis less well understood.Someof the interfacialtraps are created immediately upon exposure (4) Radiation-induced interface traps within Si bandgap Si + + $Si{O}_2$ + (3) Deep hole trapping near Si-SiO_2 interface Gate 元元不不不 (2) Hopping transport + (1) Electron-hole pairs of holes through generated by ionizing localized states radiation in SiO_2 bulk Figure18.16 Response to ionizingradiationand theresulting damage in MOS structures.(From Srour and McGarrity[25],©1988IEEE.)

--- Page 692 --- NONIDEALMOS 669 to ionizingradiation,whiletheremainderarecreated in proportionto thenumberofholes reachingtheSi-SiO2interface.Ithas beenproposedthattheenergyreleasedinthe deep- leveltrappingofholesbreakstheSi-Hbondsassociated withdeactivatedinterfacialtraps. Over a periodofdays to yearsthetrapped-holechargetendsto beslowlyreducedat room temperature by the capture of electrons injected into the oxide from the metal or the silicon.Naturally,removalofthetrapped-holecharge is greatlyacceleratedbythermal annealing;astandard interfacial-trapanneal totallyremovesboththetrapped holesand the induced interfacial traps.It should be noted,however,thatonce subjected to ionizing radia- tion and recovered through low-temperature annealing,MOSdevices exhibit greatrsen- sitivitytosubsequentradiation.This hasbeenatributedtothecreationofneutraltraps within the oxide in addition to the induced charges. Higher-temperature annealing (T>600C)has been found to be partiallyeffective in removing the neutral traps. Thermal annealing can be readilyperformed to removeradiation damage thatocurs during fabrication.It is alsofeasible to increasetheambienttemperatureofcompleted devices to100Cor sotoacceleratetheremoval of trapped holescaused by ionizingradia- tion.After-the-factrecoveryincompleteddevices,howeverisrelativelylimited.Itisp duced)byemployingempiricallyestablishedoptimum growthconditionssuchasoxidation temperatures below1000C.Otherhardening procedures include A1-shielding,which stops themajorityofenergeticlectronsencounteredinspace,andincreasingthetheshold voltageofMOSFETs so theyare less sensitive to ΔVGchanges caused by radiation.(The general topicofMOSFETthresholdadjustmentis discussed in Section18.3.)Somewhat fortuitously,thereduction inoxidethicknessthataccompaniesreductionsindevicedimen- sions is also leading to harderMOS devices.ΔVGforboth thefixed charge and interfacial traps is proportionalto1/C=/Kε0and thereforeautomatically decreases with de- creasing x.The improvement mayalso be due,in part,to the smallerhole trappingcross sections at the higheroxidefieldsthatexistacross thethinneroxides.Projections even suggestholetrappingandtheassociatedinterfacial-trapproductionmightactuallyvanish as theoxidethickness drops below100A.It is envisionedthatelectrons from themetal and semiconductorcan tunnel intoall partsofa verythinoxideandrapidlyannihilatethe trapped holes. Negative-Bias Instabillity Negative-bias instability is a significant perturbationofMOSdevicecharacteristicsthat occurs as a directresultof stressing the structure witha large negative bias at elevated temperatures.Typicalstressconditions wouldbe anegativegatebias sufficientto produce afieldof2×10V/cmin theoxideandT>250°C.The instability is characterized bya large negative shiftalong thevoltage axis and a distortionof theMOS-CC-Vcurve. Ionimplantation,electron-beamevaporationofmetalsdepositionof special-purposethinfilmsovertheiO layer in ahostileplasmenvironmentsputeringelectron-beamand x-ray lithographyand anumberofother fabrication processes can lead to varying degrees of radiation damage.

--- Page 693 --- 670 FIELDEFFECTDEVICES Similar to ionizing radiation,thestress clearlycauses an increase in theapparent fixed charge withintheoxideandan increaseinthe interfacial-trapconcentration.Anaddedpeak in DInearmidgap isconsidered to bea distinctive signatureof the instability.Note that the C-Vcurve shiftingrelatedtothe negative-bias instability isoppositetothatcaused y alkali-ioncontamination. The exactmechanism causing the negative-bias instability has notbeen established. However,because large holeconcentrations areadjacentto theoxide under theconditions ofthenegativestressithasbeenproposedthattheinstabilitymayresultfromholeinjection fromtheSi into theoxideand thesubsequenttrappingof theholes at deep-level sites near the Si-SiO2interface. The radiation-induced and stress-induced damage would then have arelatedorigin.Indeed,it hasbeenfound thatthe negative-bias instability is enhanced if theMOSstructureisfirstexposedtoionizingradiation.Sensitivitytotheinstabilitycanbe minimized by annealing the device structure in hydrogen at 800C-900C prior to gate deposition. 18.2.6△VSummary In the first two sections of this chapter we cited and summarized fourof the most commonly encountered deviations from the ideal; specifically,the metal-semiconductor workfunction difference,mobile ions in theoxide,fixedoxidecharge,and interfacial traps.It was also noted thationizingradiationand stressing witha large negative biasatelevatedtempera- tures giverisetoadditionaloxidecharges.InbothcasesthereisanincreaseinQandthe apparent QF. Thecombinedeffectof theanalyzed nonidealitieson theV-s relationship is de- scribed by Q} Q_{MM Q{(φ_s) △G=(G-sameφ=M- (18.17) ° C_{oC_{} where xρio}n(x)dx Jo γ{M 三 x0 (18.18) $x_o$ $ρion(x)dx Jo In writing down Eq.(18.17)we recastthemobile ioncontribution(Eq.18.13)toemphasize the similaritybetweenthethreetermsassociated withoxidecharges.γMisaunitless quan- tity representing the centroidof the mobile ion charge in theoxide normalized to the width of theoxidelayer;γM=0if themobile ions areallatthemetal-oxideinterface,while

--- Page 694 --- NONIDEAL MOS 671 γ1ifthemobileionrellileduttheiiO2intraceAsenealul OF,and Ms allleadtoanegativeparalleltranslationoftheCVcharacteristicsalongthe voltageaxis relative to theidealtheory.△Vdueto Q,ontheotherhand,canbeeither positiveor negative,dependson theappliedbias,and tends to distortor spread outthe characteristics. From the discussionof nonidealities,it is safe toconclude thatreal MoSdevicesare not intrinsicallyperfect;thedevicesare in fact intrinsicallyimperfect.Throughanexten- sive research efforthowever,procedures have been developed for minimizing the net effect of nonidealities in MOSstructures. Although constant checks must be run to maintain quality control,manufacturerstodayroutinelyfabricatenear-idealdevices. Weshould mentionthat,whereasthedescribedor similarnonidealitiesapplytoallMIS structures,the specificminimizationproceduresoutlinedhereinapplyonlytothethermallygrownSO- Si system. Exercise18.4 P:Identify the physical causeof the listed oxide charges.Place theappropriate letter in the box preceding the charge. Oxide Charge Physical Cause Fixed charge (a) Phosphorus ions (P+) Mobile ion charge (b) Sodium ions (Na+) (c) Nitrogen ions (N+) Interfacial traps (d) Dangling bonds at the Si surface Apparent fixed charge resulting (e) Trapped electrons from ionizing radiation (f) Ionized Si waiting to be oxidized (g) Ionized oxygen waiting to form SiO2 (h) Trapped holes QF-f,QM-b,QI-d,apparent Q-h. S Exercise18.5 P:AnAl-SiO-SiMOS-Cisto beformedonanNA=10/cm3（100)-orientedSi wafer.Thermal oxidation in dry Oat100Cis followedbyan N2annealsufficient to achievea steady-state condition.The structure is next phosphorus gettered so that

--- Page 695 --- 672 FIELDEFFECTDEVICES distributed intheoxideas shown in Fig.E18.5.The device is postmetallization an- nealed,butthereisaresidualinterfacialtrapdensity.Aconstant D=2×1010/ cm²-eVofacceptor-likestatesisfoundtoexistforallbandgapenergies.Determine the expected flat band voltage of the MOS-CifT=300Kand=0.1m. ρion x 0.1x0 0 xo M 0 S Figure E18.5 S:Underflat-band conditions,s=0andV=0.Itthereforefollows from Eq.(18.17)that $Q$ $OMMM$ B = VG=0=MS $C_o$ C0C_0}$where$KOε$(3.9)(8.85\times 10-14)$C_{0}$

3.45×10-8F/cm2 $x_o$ 10-5 We must systematicallyevaluateeachoftheterms in theVFBrelationship. MS…Given NA=1015/cm3andan Algate,weconcludefrom Fig.18.3that MSs =-0.90 V. OF/C。…The N2annealafterthe dryoxidationminimizesthefixedoxidecharge. Figure18.9(b)indicates anexpectedQF/q=2×101/cm²fora（111)-oriented Si Substrate.It was pointedout in the text discussionthat QF is approximately three times smalleron(100)surfaces. Thus for the given device QF/q =6.67× 1010/cm²and $O$ q(QF/q) (1.6×10-19)(6.67×1010) =0.31V $C_o$ Co$ (3.45× 10-8)

--- Page 696 --- NONIDEAL MOS 673 OMγ/C…ReferringtothespecifiedmobileiondistributioninFig.E18.5,and making use of Eq.(18.18), we conclude 0.10 $0.10$x^{2}xρ{maxdx$20\gamma M0.1x00.$1x0$0.05xo$ρmax$dx$x{xx$JoloandQM\gamma M=q(QM/g）\gamma m(1.6\times 10-19)(2\times 1011)(0.05)0.046V$C_{oC_{o$(3.45\times 10-8)O(0)/C..Acceptor-likecentersare)chargediffilledwithanelectronandneutralifempty.UnderequilibriumconditionstheinterfacialtrapsaremostlyfiledbelowtheFermilevelandmostlyemptyabovetheFermilevel.Thefillingandchargestatesunderflat-bandconditionsarethereforeaspicturedbelow.$Ec$(0)$EFEv}Because D=constant,Q(O)=-qDI△E,where ∆Eis the rangeofenergies in the surface band gapcontaining negativelycharged interfacial traps.Consequently, ∆E=EF-E=EG/2-(Ei-EF)=EG/2-kTln(NA/n₁) =0.56-0.0259ln(1015/1010)=0.26eV and qDI△E (1.6×10-1{9)(2×1010)(0.26) -0.024VC_0} C_{o (3.45 × 10-8) Finally, summing the individual terms in the FB relationship yields VFB=-1.23V.

--- Page 697 --- 674 FIELD EFFECTDEVICES 18.3 MOSFETTHRESHOLDCONSIDERATIONS ThusfarwehavedescribedtheeffectofnonidealitiesintermsofMOS-CC-Vcurve shiftingand distortion.TheVpointon theC-Vcurve,corresponding to the threshold voltageof aMOSFET,isofcourse directlyaffected byanyΔVG displacement.Figure 18.17(a)graphicallyillustrates thegeneraleffectof changes inthethreshold voltageonthe MOSFETID-VDcharacteristics.A hypothetical p-channel MOSFET with an ideal device V=1Vwas assumedinconstructingthefigure.TheΔV leadingto theperturbed $D$ $D$ A V_}=-5V $V_{=-7V{V}_G}=-4V$V_}=-6V △V{=-2V V_{}=-3V VG=-5V V_{}=-2V V_{=-4V →-VD →-VD (a) D D A VG=-5V V=-7V V_G}=-4V △V_1}}=-2V plus QIT distortion V}=-6V VG=-5V V_}=-2V V G}=-4V -VD -VD (b) Figure18.17 General effectof nonidealitieson theMOSFETcurrent-voltage characteristics.The ideal characteristics ofa hypothetical p-channel MOSFET with V=-1V are pictured on the left and the perturbed characteristics are shownontheright.(a)Effectof a simple shift in the threshold voltage caused by ΦMs≠0,the fixed charge,and/ormobile ions.(b)gmdegrading effect of interfacial traps.

--- Page 698 --- NONIDEAL MOS 675 characteristicsinFig.18.17(a)istakentobecaused by Ms,thefixedcharge,and/oralkali ions in theoxide.Note that,analogous to the parallel translation of the MoS-CC-Vchar- acteristics,theormoftheertubedOTchractristicsisunchangedbutlr values are needed to achieve comparable ID current levels.If present in large densities,the interfacialtraps canadditionally decrease thechange in currentresulting from a stepped increaseingatevoltage.Thisefect,equivalenttoareductioninthegmofthetransistor,is pictured in Fig. 18.17(b). While it is truethatboththemobile ionand interfacial trapproblems were minimized early in MOSFETdevelopment,the remaining nonidealities,primarily through their effect on V,have had a very large impact on fabricationtechnologydevice design,and modes of operation.In this section we examine the reasons forthe impactand methods that have evolved for adjusting the MOSFET threshold. 18.3.1 V Relationships For discussion purposes itisdesirabletoestablishanexpressionfortheexpectedthreshold voltage,V,ofarealMOSFET.LetVbethethresholdvoltageofanideal versionofthe given MOSFET.Simply evaluating Eq.(18.17)at s=2F then gives $O$ Q_MγM_ V= V {+ M (18.19) $C0o$ C_0o} $C_o$ Although Eq.(18.19)can beused directly to compute V,it is standard practiceto express the threshold shift intermsof thereal-deviceflat-bandvoltage.Underflat-band conditions s=0,V{=0,andfrom Eq.(18.17), $Q$ $QMM$ Q{(O B =G5=0= (18.20) 一 C_o} C_{ C_{0 IfOIchangeslittleingoingfromφs=0tos=2Fareasonablygoodapproximaion in well-made devices,thenonidalit-rlatedtrms inE.18.19)canbeeplacedby and one obtains V = V{2+BB (18.21)

--- Page 699 --- 676 FIELD EFFECTDEVICES where, repeating Eqs.(17.1), K_s} 4qN_{B (+) forn-channel devices $V$ =2 ± x_o (±φF) (-) for p-channel devices KoK{$ NB= NA or ND as appropriate (18.22) 18.3.2 Threshold,Terminology,and Technology As an entry into the discussion,letus perform a simple thresholdvoltage computation employing relationships developed in the preceding subsection.Suppose the gate material is Al，the Si surfaceorientationis（111),T=300K,=0.1μm,NA=101/cm3，QF/q =2×101/cm²,QM=0,andQ=0.For thegivenn-channel device,one computes φMs=-0.90V,-QF/C。 =-0.93V,FB=-1.83V,V=1.00V,andVT=-0.83V. Observethat whereas Vispositive,asexpected,nonidealitiesofaveryrealisticmaniude cause Vtobe negative.Sincean n-channel deviceturnsonfor gate voltages V>V, the device in questionis already”onat a gate bias of zero volts.Actually,negative biases must beappliedto deplete thesurfacechanneland turnthedeviceoff!For a p-channel device withidentical parameters (except,ofcourse,foran NDdopedsubstrate),oneobtains aV=-1.00V,B=-1.23V,and=-2.23V.Inthep-channelcasetheconsidered nonidealities merely increase the negative voltage required to achieve turn-on. AsillustratedinFig.18.18,a MOSFETiscalledanenhancement modedevice when it is”off”atVG=0V.WhenaMOSFETis”on”atVG=0V,itis calleda depletion mode device.Routinelyfabricated p-channelMOSFETs constructed in thestandardcon- figurationare ideally and practically enhancementmode devices,n-channel MOSFETs are also ideallyenhancementmode devices.However,because nonidealities tendto shiftthe threshold voltagetoward negativebiases inthemannerindicated inour samplecalculation, early n-channel MOStransistors were typicallyof the depletionmode type.Up until a- proximately1977this difference in behaviorledto thetotaldominanceofPMOStech- nology over NMOStechnology; that is,ICs incorporating p-channel MOSFETs dominated the commercial marketplace.Subsequently, as explained under the heading of threshold adjustment,NMOS,which is to be preferred because of the greater mobilityofelectrons compared to holes,benefited from technological innovations widely implemented in the late197Os and is now incorporated in the majority of newly designed ICs. Whileon the topicof the thresholdvoltage in practical devices,it is relevant to note that the inversion thresholdofregionsadjacent to the device proper is alsoofconcern. Consider,for example,the unmetallized region between the two n-channelMoStransistors pictured inFig.18.19(a).Ifthe potential attheunmetallizedouteroxide surfaceisassumed to be zero (normally a fairly reasonableassumption)and if the thresholdvoltage forthe n-channel transistors is negative,thenthe intermediate region betweenthetwo transistors

--- Page 700 --- NONIDEAL MOS 677 n-channelMOSFET ID ↑ S G D D Enhancement mode AllVG0$\leftarrow B$n{}n{} G V{G=0，VG0 No channel S when V_}=00 →VD ID V_{0 S G D D Depletion mode V{} 00$B$n^{}n{}$G$V{<0$ Channel S when V_G=0 →VD Structure Symbol Characteristics Figure18.18 MOSFEToperationalmodes. VG=0channel status,circuitsymbol,andD-D characteristics of n-channel enhancement-mode and depletion-mode MOSFETs. willbe inverted.Inother words,a conducting path,a pseudo-channel,willexist between the transistors. This undesirable condition was another nuisance in earlyNMOStech- nology, where, asalready noted,nonidealitiestended to invert thesurfaceof the semicon- ductor inthe absenceof anapplied gate bias. Unless specialprecautions are taken,un- wanted pseudo-channels between devices can also arise in both n- and p-channel ICs from the potentialappliedto the metal overlays supplying the gate and drainbiases. Toavoid this problem,the oxide in the non-gated portions of theIC,referred to as the field-oxide, is typically muchthickerthan the gate-oxide in the active regionsof thestructure (see Fig.18.19b).The ideabehindthe useof thethickeroxidecan be understoodbyrefering

--- Page 701 --- 678 FIELDEFFECTDEVICES D G S D G S n n + n + n P Intermediate P region (a) D G S Field oxide p + Gate oxide N-Si (b) Figure18.19(a)Visualizationoftheintermediate regionbetweentwoMOSFETs.(b)Identiffica- Thusemployingan（fild-oxid)（gate-oxid)increasesinthefield-oxideas relative to the gated areas in PMOS (and modern NMOS)structures.Inversion of the field- oxideregionsis therebyavoidedat potentialsnormallyrequiredforICoperation. 18.3.3ThresholdAdjustment Several physical factorsaffectthe thresholdvoltage and can therefore beused to varythe VTactuallyexhibited by a givenMOSFET.We have,in fact,alreadycitedthe adjustment of V through a variation of the oxide thickness.Obviously, the substrate doping can also be varied to increase or decrease the threshold voltage.However,although strongly influ- encing the observed VT value,the gate-oxide thickness and substrate doping are predeter- mined in large part by other design restraints. Otherfactors thatplay a significantrole in determining Vare the substrate surface orientationand the materialused informing theMOS gate.As first noted inSubsection 18.2.3,the QF in MOSdevices constructed on(100)surfaces is approximately three times smallerthantheQFin devices constructedon(111)surfaces.The use of(10O)substrates thereforereduces the ΔVG associated withthefixed oxide charge. The use ofa polysilicon

--- Page 702 --- NONIDEALMOS 679 insteadofanAlgate,ontheotherhandchanges Ms.Givenapolysilicongatetheeffec- tive“metal”workfuncton becomes “ΦM”= si +(c-)poly-Si (18.23) and $Ms$ 1 [(EF)polySi(-FB.crystalline-Si] (18.24) q Ifthecalculation inSubsection18.3.2isrevised assuming a(100）surfaceorientation (QF/q=2/3×101/cm²)andap-type polysilicongate where E=E,oneobtains a = pu2-=+= possible in(100)-oriented structures incorporating p-type polysilicon gates. Althoughthe foregoing calculation shows positive threshold voltages are possible,the Vin actual structures may beonly nominallypositive,or the structure impractical—in forming n-channelMOSFETs,the polysilicon is conveniently doped n-type,the same as the drain and source islands,not p-type as assumedabove.Also,a larger threshold voltage may bedesired,orone may desireto modifythethresholdattainablein aPMOS structure, or tailoringofthethresholdforboth n-and p-channel deviceson the sameICchip may be required.For anumberofreasons,it is very desirabletohave aflexiblethresholdadjust- mentprocess where Vcanbecontrolledessentiallyat will.Inmoderndeviceprocessing this is accomplished through the use of ion implantation. The general ion-implantation process was described in Chapter 4.To adjust the thresh- oldvoltage,arelativelysmall,preciselycontrollednumberofeitherboronorphosphorus ions are implanted into the near-surface region of the semiconductor. When the MoS struc- ture is depletion or inversion biased,the implanted dopantadds to the exposed dopant-ion charge neartheoxide-semiconductor interfaceandtherebytranslatestheVexhibitedby the structure.The implantationofboroncauses a positiveshiftin thethresholdvoltage; phosphorusimplantationcauses a negative voltage shift.Forshallowimplantstheproce- dure may be viewed to firstorder as placing an additional“fixed”charge at the oxide- semiconductor interface.If Nis the numberof implanted ions/cm² and Q=±qNis the implant-related donor(+)or acceptor(-)charge/cm² at the oxide-semiconductor inter- face,then, by analogy with the fixed charge analysis implanted O} ΔVG$(18.25)ions$C_{o}$ Assuming,forexample,an N=5×10boronions/cm²and=0.1monecomputes a threshold adjustment of +2.32 V.

--- Page 703 --- 680 FIELDEFFECTDEVICES S G D VGS VDS≥0 $n+$ $n+$ P V_{Bs≤0$B(a)Ec$Ec$下Fp$EF$EF$Ev Ev} q(2φF-Bs) O S FN (b) S (c) Figure18.20 The back-biased MOSFET.(a）Cross-sectionalviewindicating the doublesub- scripted voltage variablesused in the analysis.Also shown are the semiconductorenergy band dia- gramscorrespondingtotheonsetofinversionwhen（b)Bs=0and（cBs0. 18.3.4 Back Biasing Reverse biasing theback contactor bulkof a MOS transistorrelative to the source is an- othermethodthathasbeenemployedto adjustthethresholdpotential.This electrical methodofadjustment, which predates ion implantation,makes use of the so-calledbody effect or substrate-bias effect. To explain the effect, let us consider the n-channel MOSFET shown in Fig.18.20(a). If the back-to-source potential difference(VBs)is zero,inversion occurs of course when the voltage dropacross the semiconductor(s)equals 2as pictured in Fig.18.20(b).If B<0thesemiconductorsltemptstoinertwhenreaches2.Howevri VBs<nirsiolcitdopprtesmcnucrc laterally into the source and drain because these regions are at a lower potential.Not until s=2F-Bsas pictured inFig.18.20(c),willthe surface invertandnormaltransistor action begin.In essence,backbiasing changes the inversion point in the semiconductor from 2F to 2F- VBs.The ideal device threshold potential given by Eq.(18.22) is in

--- Page 704 --- NONIDEAL MOS 681 turn modified to $K_s$ 2q_{BB$VBlatthreshld=2F-BB\pm x0(\pm 2\phi F+BS)$Ko}K{ε0$(+),Bs<0forn-channel(-),Bs>0forp-channel(18.26)K_s$ 2qN{BB$VSlatthreshold=2F\pm xo(\pm 2\Phi F+BS)(18.27)KO$K_{{$VFinally,introducing△V=(GSlathreshold-V),weobtain$V_BsB$ F>0,VBs<0for n-chanel △ΔV{=({-2 1- -1 (18.28) a F<0,Bs>0for p-channel Having established Eq.(18.28),wemake thefollowing observationsconcerningback biasing or thebodyeffect:(1)Backbiasing always increases themagnitudeoftheideal device threshold voltage.Ittherefore makes the p-channel thresholdofactual devices more negative and the n-channel threshold more positiveit cannot be used to reduce the nega- tive thresholdof a p-channel MOSFET.(2) The current-voltage relationships developed in Chapter 17are stll valid when VBs≠0 provided2F→2φF-Bs,G→Gs,VD→ VDs,andVisinterpretedasGSathrehold.3)Caremustbeexercisedindescribinback- biased structures to properly identify voltage differences through the use of double- subscripted voltage variables. 18.3.5 Threshold Summary Allnonidealities shifttheMOSFETthresholdvoltage;interfacial traps inadditionreduce the low-frequency gmof the transistor.Atwo-stepprocedure isfollowed incomputingthe expectedthresholdvoltageofa realMOSFET.Theflat-band voltage is first deducedfrom knowninformionboutdvienidliti.ubquntlBisadedtottshold voltageof an ideal versionofthegivenMOSFET.A transistor that is normally”off”when V=0isreferredto as anenhancementmode device; adepletionmodeMOSFETis”on” or conducting when VG=0.Because ofresidual nonidealities,real n-channel MOSFETs are typically depletion-modedevices. Theattainmentofn-channelenhancementmode MOSFETs,and thresholdadjustment in general,is nowaccomplished through the useof ion implantation.Biasingthebackcontactrelativetothesourcehasalso beenemployedto adjust the threshold voltage.

--- Page 705 --- 682 FIELDEFFECTDEVICES Exercise 18.6 P:TheC-VG(VD=0)characteristicderivedfroman n-channelMOSFETis pi- tured in Fig. E18.6. (a) What is the threshold voltage,V,ofthetransistor?Explainhowyouarrivedat your answer. (b) Is the MOSFET a depletion mode or anenhancement modedevice? Explain. (c) SketchthegeneralformoftheID-VDcharacteristicsexpectedfromthedevice, specificallylabeling thosecharacteristics corresponding to V=-2,-1,0,1, and2V. (d) Given Ms=-1V,Q=0,andthefactthat the device is stable underbia- temperaturestressing,howdoyouexplaintheobservedflatband voltageof-2.2V? (e) Ifthe MOSFETisdopedsuchthatF=0.3V,what substratebias（Bs）mustbe applied to attain a VGSlthreshold =1V? CG Flat band 1 V_G -5 -4-3 -2 -1 0 1 2 Figure E18.6 S: (a) V.InprogressingtowardpoitivebiasestheCVcurveincreases towardC=Catroughlytheinversion-depletiontransition point. VGat theinver- sion-depletion transitionpointis thethreshold voltage in MOSFETs. (b)[ )DepletionMode].With V=-1V,theMOSFETisclearly”on”orconducting when V G }= 0.

--- Page 706 --- NONIDEAL MOS 683 (c) The general formof theID-VDcharacteristicsexpectedfrom the device is sketched below. D VG{}=2V V_G}=1V V_{{}=00 {$VG=-1V,-2V(deviceoff)(d)Sincethedeviceisstableunderbias-temperaturestressing,oneconcludesthatQM=0.WithQM=0,Qr=0,andMs=-1Vunabletoaccountfortheentirefat-bandshiftof-2.2V,theresidualshiftingcanonlybeattributedtothefixedcharge.(e)IfEq.(18.28)issolvedforVBs,oneobtainsVBs=2_2[1-(1+V2NotingGSlattreshol=GS|atresoV=\Delta =2and$V{2=V-VFB=-1.0V+2.2V=1.2V direct substitutionintotheBespressionthenld B = 0.6[1 -(1+ 2 27 =-10.7 V 1.2- 0.6

--- Page 707 --- 684 FIELD EFFECTDEVICES PROBLEMS CHAPTER18 PROBLEM INFORMATION TABLE Complete Diffi culty Suggested Short Problem After Level Point Weighting Description 18.1 18.3.5 1 10 (1 each part) Quick quiz 18.2 18.1 2 6 (2 each part) Polysilicon-gate MOS-C 18.3 ” 2 10 Ms versus doping plot 18.4 18.2.2 2 8 (4 each part) Mobile ion calculation 18.5 18.2.3 3 10(a-1,b-6, c-3) Fixed charge calculation 18.6 ” 2 10 (5 each part) Ms,QF from FB-plot 18.7 18.2.4 2 8 (4 each part) Infer Q-info from E-band 18.8 ’ 3-4 14(a-2, b-6, c-6) Deduce OF fromC-Vdata 18.9 ” 3 10 Sketch pre-annealed C-V 18.10 ” 2-3 8 Dangling bond calculation 18.11 ” 3 10(a-3, b-2,c-3, d-2) Single-level interfacial trap 18.12 18.2.5 3 10 Half-irradiated Mos-C 18.13 18.3.1 2 6 (a-2,b-4) BT-stressed MOSFET Modification to 18.14 18.3.3 2-3 10(2 each) 18.15 ” 2 8(a-4,b-3, c-1) Compute MOSFET B, 18.16 ” 2 8 Compute required N 18.1 Quick Quiz Answer the following questions as concisely as possible. (a) What is the difference between ΦM-χ and Φ-x’? (b)What is involved in subjecting a MOS-C or MOSFETto a bias-temperature stress? (c)Whatis believedtobethephysicaloriginofthefixedoxidechargeinMOSstructures? (d)Theinterfacialtrapdensity()isobservedtodependonthesiliconsurfaceorienta- tion. Describe the observed orientation dependence. (e) What is the neteffectof ionizingradiationonMOSstructures? (f)When performing bias-temperature stressexperiments,how doesone distinguish be- tweenthenegative-biasinstabilityandthevoltageinstabilityarisingfromalkaliions? (g) Under what circumstances would V≠V+B? (h)Explain what is meant by theterm”depletion mode”transistor.

--- Page 708 --- NONIDEAL MOS 685 (iWatistencetenteloi”andoxidin 18.2Considera polysilicon-gate MOS-C where EF-=0.2eVintheheavilydoped gate andEF-E=-0.2eVin the nondegenerately dopedsilicon substrate. Assume the structure to be ideal (other than an obvious Ms ≠0) and x’(polysilicon)= x’(crystalline-Si). (a）Sketchtheenergyband diagramforthepolysilicon-gate MOS-Cunderfat-band conditions. (b)Whatisthe“metal”semiconductorworkfunctiondiferenceforthecitedpolysilicon- gate MOS-C? (c) WillthegivenMOS-Cbeaccumulation,depletion,orinversionbiased whenV=0? Explain. ●18.3 With the Φ-χ’of the MOSsystem as an input parameter, write a computer pro- gram that willoutput a plot similar to Fig.18.3.Use yourprogram toconfirm the Aland n+polyplots in Fig.18.3.Alsooutput a plot appropriate for MOS-Cs with p+polysilicon gates doped such that E=E. 18.4(a)AnMoS-Cis found to possess a uniform distributionof sodium ions in theoxide; that is,ρion（x)=ρ0=constantforallintheoxide.ComputetheΔVGshiftresulting fromthisdistributionofionsifρ/q=101/cm3and。=0.1m. (b)Afterpositivebias-temperature(BT)stressingthesodiumionsofparta)allpilep immediately adjacentto theoxide-semiconductor interface.Determine the ΔVGafter +BT stressing. 18.5Inmodeling thequantitativeeffectof thefixedcharge,it is typicallyassumedthe chargeislocatedimmediatladjacenttotheoxideemiconductorintrace.upposeth chargeisactuallydistributedashortdistanceintotheoxidefromtheSiSiO2interace. (a) Forreferencepurposes,writedown thestandardresultforΔ(fixedcharge). (b) DeterminetheexpectedΔG shiftcaused byanequivalentamountof charge distrib- utedinalinearlyincreasingfashionromeroatadistanceΔxfromtheSiSiO2inter- face to 2Q/Δxright at the Si-SiO2 interface. (c) ComputeΔVG（partb)/ΔVG（parta)assuming∆=10Å=10-cmand=0.1μm 18.6It is possibletodetermine Ms andOF for agivenfabricationprocess and metal SiO-SisystembyfabricatingasetofMOS-Cs withdifferentoxidethicknesses.Theflat-

--- Page 709 --- 686 FIELDEFFECTDEVICES band voltage(B）foreachdeviceinthesetismeasuredandplottedasafunctionof. The fabrication process is assumed to yield ideal devicesother than Ms≠0 and QF≠0. (a) Indicate how Ms and QF can be deducedfrom the VFB versus plot. (b) Given the FB versus data tabulated below,determine Ms and QF/q. _{0(m) VB(V$0.1-\mathrm{0.910.15}-\mathrm{1.040.2}-\mathrm{1.20.25}-\mathrm{1.330.3}-\mathrm{1.5218.7}AnMOS-CischaracterizedbytheenergybanddiagraminFig.P\mathrm{18.7.}Assumethattheinterfacialtrapdensityinthestructureisnegligible.(a)IsQM=0orQM\ne 0intheoxide?Explain.(b)IsQF=0orQF\ne 0attheSi-Si{O}_{2}interface?Explain.$Ec$EFEv$ Metal $Si{O}_2$ Si Figure P18.7 18.8TheC-Vcharacteristicexhibited byan Al-SiO-SicapacitormaintainedatT=300 Kis reproduced in Fig.P18.8.Note thatCMAx=200pF,CMIN=67pF,and the flat-band voltage VFB=-0.71V.Therearenomobileions intheoxide(QM=0),theinterfacial trapdensityisnegligible（Q=0),andAG=2.9×10-3cm².

--- Page 710 --- NONIDEAL MOS 687 C ↑ CMAX CMAX=200pF FB CMIN=67pF VFB=-0.71V -CMIN$\to VGVFBB$ T0 Figure P18.8 (a) Adding to a copyof Fig.P18.8 and using a dashed line,sketch theC-Vcurve one would expect from an ideal version of the given MOS-C.Also indicate the flat-band point on the ideal-device characteristic. (b)Ascertain the Sidopingconcentration.Anapproximatedeterminationusingthedelta- depletion C-V formulation is acceptable.) (c) Determine the fixed charge/cm2OF,in the given device. 18.9The C-V characteristics of an MOS-C are measured beforeand after performing a postmetallization anneal to eliminate interfacial traps.The AFTERC-Vcharacteristic ex- hibited by the device is traced in Fig.P18.9.Assuming acceptor-like interfacial traps, sketch theC-V characteristic ofthe device BEFORE annealing onthe same setof coordi- nates. Explain how you arrived at your answer. C →VG Figure P18.9

--- Page 711 --- 688 FIELDEFFECTDEVICES 18.10Ifinterfacial trapsareassociated withresidual“danglingbonds”attheSisurface, and assuming the numberofresidual”dangling bonds”is proportional to the numberofSi surfaceatoms,whichsilicon surface plane,(100)or(110),wouldbe expectedto exhibit the higher density of interfacial traps? Record all work leading to youranswer. 18.11Aratherunusualn-bulkMoS-Cisfound to have interfacial trapsatonlyoneband- gapenergyE,located rightatmidgap(seeFig.P18.11).Answerthefollowingquestions assuming a high-frequencyC-Vmeasurement and anotherwise idealMOS-C.Include a few wordsofexplanation,if necessary,to convey your thought process and to prevent a misinterpretation of the requested sketches. -Surface Ec Interfacial trap level E Figure P18.11 (a)Sketch the generalformof the expectedMOS-CC-V characteristic if the interface statesgivingrisetotheElevelaredonorlikeinnature.Assumethenumberofstas is sufficiently large to perturb the ideal-device characteristic.) (b)Repeat part (a) if the interface states are acceptor-like in nature. (c) Repeatpart(a）assumingtheinterfacestatesaredonor-likebuttheenergylevelis located veryclosetotheconduction band(say-=0.001eV). (d)Repeat part(a)fordonor-likeinterfacialtrapsthatintroduceanenergylevelveryclose to the valence band (say E-E=0.001eV). 18.12Theas-fabricatedC-VcharacteristicderivedfromanMOS-Cis picturedinFig. P18.12.Afterfabrication,precisely one-halfofthe gatearea is exposed to ionizing radia- tion,whichincreases theapparentfixed charge(QF)in theexposed halfof the device.The increase in interfacial trap densitycaused by the irradiation is negligible.Sketchtheex- pected formof theC-Vcharacteristicafter irradiationonthesamesetofcoordinatesasthe pre-irradiationcharacteristic.Also explainhow you arrived at youranswer.HINT: Think of the two halves of the MOS-C as separate capacitors.

--- Page 712 --- NONIDEAL MOS 689 C V_G Figure P18.12 18.13Thebeforebias-temperature stressingID-VD characteristicsofa MOSFETare sketchedinFig.P18.13(a).The g-VG(VD=0)characteristicsofthesame device before and afterbias-temperaturestressing using a positive bias are shown in Fig.P18.13(b). ID8d ↑ ↑ V_{G=- 4V BeforeV{G}=-3V$ +BT After VG}=-2V +BT →-VD 1 →VG -5 -4 -3 -2 -1 0 (a) (b) Figure P18.13 (a)What wastheprobablecauseoftheshiftintheg-characteristicafterpositivebias- temperature stressing? (b)SketchtheID-VDcharacteristicsafterbias-temperaturestressingforgatevoltagesof V=-2,-3,and-4V.Also record yourreasoning. 18.14Ann-channelp-bulk)MOSFETisidealexcept Ms≠0.Indicatehowthefollowing modificationstothestructure,takenindependently,willaffectthethresholdvoltage. Include a few words of explanation in each case. (a)Ionizing radiation causes an apparent QF≠0. (b)Thegate material is changedfrom Al to Cu.

--- Page 713 --- 690 FIELDEFFECTDEVICES (c)There is an increase in the substrate doping. (d)The oxide thickness is decreased. (e) Boron ions are implanted intothe near surface regionof the silicon. 18.15AMOSFETisfabricated with Ms=-0.46V,OF/q=2×101/cm²,QM=0, QI=0,andQ₁/q=-4×10/cm²,=0.05μm，A=10-3cm²,andND=15/ cm.Q1istheionchargeimplantedimmediatelyadjacenttothe Si-SiO2interface. (a) Determine B. (b) Determine VT (c)Is the given MOSFET an enhancement mode or depletion mode device?Explain. 18.16Givenan Al-SiO2-SiMOSFET,T=300K,a Si-substrate dopingof NA=1017/ cm3=100O/q=101/cm2,nointerfacialtrapsandnomobileionsintheoxide. Determinetheboronions/cm²（N)thatmustbeimplantedintothestructuretoachievea V=0.5V.（Assumetheimplantedionscreateanaddednegativecharge rightatthe Si- SiO_2 interface.)

--- Page 714 --- 19 Modern FET Mostly Read Only Structures To achieve higheroperating speeds and increased packing densities,FET device structures havebeen subjectedtogreaterand greaterminiaturization. Thedecrease inFETdevice dimensions,in itself,can lead to major modifications in theobserved devicecharacteristics. Smalldimensioneffects,alsoreferred toasshort-channeleffectsor small geometryeffects, include,forexample,shifts inthethresholdvoltageandanincrease inthesubthreshold current. Thecitedmodifications indevicebehaviorareofmajor importance inpractical applications.Notably,anaccurate predictionofthethreshold voltage is needed to deter- mine logic levels,noisemargins,speed,and nodevoltages,while the subthresholdcurrent affectstheoff-statepowerdissipation,dynamiclogicclockspeeds,andmemoryrefresh times.A large portionof this chapter is devotedto the descriptionand discussionof small dimension effects.It shouldbe understood fromtheoutsetthat smalldimensioneffectsare generallyundesirableandareminimizedoravoided incommercial structures throughthe proper scalingofdevicedimensionsormodifications in devicedesign.Relativetomodifi- cations in device design,the chapterconcludes witha brief surveyofselect implemented and developmental FET structures. 19.1SMALLDIMENSIONEFFECTS 19.1.1Introduction In1965thesmallestMOSFETshadanL~1mil=25 μm.By1990industry-standard versionsof MOSdevice structures hadattained submicron dimensions. The steadypro- gressiontowardsmallerandsmallerFETs ispicturedschematically inFig.19.1.Projec- tions into the near future are summarized in Table19.1. $n$ $n$ P $n+ n+ p 口 1970 1980 1990 2000 Year Figure19.1 The“shrinking”MOSFET.TherelativescalingoftheMOSFETchannellengthin the figure roughly depicts the decrease in minimum feature size of production-line MOSDRAMs. 691

--- Page 715 --- 692 FIELDEFFECTDEVICES Table19.1 SiliconICProjections[26].(DRAMisanacronym for dynamicrandomaccess memory.I/O is an abbreviationfor input/output.Wiring levels refer to the number of metallization levels that interconnect devices on an IC.) YEARof first DRAMshipment 1995 1998 2001 2004 2007 2010 Minimum feature size (m) 0.35 0.25 0.18 0.13 0.10 0.07 ● Microprocessor 250 300 360 430 520 620 Chip size (mm2) DRAM 190 280 420 640 960 1400 Bits/chip in DRAMs 64M 256M 1G 4G 16G 64G Si wafer diameter (mm) 200 200 300 300 400 400 Wiring levels (maximum in on-chip logic) 4-5 5 5-6 6 6-7 7-8 I/O pads per chip (high performance) 900 1350 2000 2600 3600 4800 Desktop 3.3 2.5 1.8 1.5 1.2 0.9 Power supply voltage (V) Portable 2.5 1.8-2.5 0.9-1.8 0.9 0.9 0.9 On-chip 300 450 600 800 1000 1100 Chip frequency (MHz) high performance ●Chip-to-board 150 200 250 300 375 475 The departure from long-channel behavior, which can accompany the noted decrease in device dimensions,is nicely illustrated using observed ID-VDcharacteristics.The onset of short-channel effects is heralded by a significant upward slant inthe post-pinch-off por- tions of the ID-VDcurves.Severe short-channel effects lead to characteristics of the form curves areobservedforgatevoltages belowthreshold.(TheFig.19.2device shouldbe “off”for VG>0V!)Anotherclear manifestationof short-channel effects is provided by the subthresholdtransfercharacteristics.A sample long-channel subthresholdcharacteris- tic was presented inFig.17.11.In long-channelMOSFETs the subthreshold drain current varies exponentially withVG and is independentofVDprovided VD>fewkTlq volts.In short-channel devices,on theother hand,the subthreshold drain current is found to increase systematicallyandsignificantlywithincreasingVD.Shiftsintheobservedthresholdvol- ageconstitutethethirdwidelyquotedindicationofsmall-dimensioneffects.Asisreadily confirmedbyreferringtotherelationshipinSubsection18.3.1thethresholdvoltage in a long-channelMOSFET is independentof thegate lengthand width.Inshort-channel devices,Vbecomes a function of the gate dimensions and the applied biases (see Fig.19.3).

--- Page 716 --- MODERNFETSTRUCTURES 693 4.0 $:=-2V 3.0 -1V () 2.0 0V 1V 2V 3V 1.0 4V 0 S- -10 -15 -20 -25 -30 D(volts) Figure19.2 ID-VDcharacteristicsofa MOSFETexhibiting severe short-channel effects.(Re- printedfromBatemanetal.[27,©1974,withkindpermissionofElsevier ScienceLtd.) 0.6 VD=0.125V 0.4 (s10∧)1 V=4V 0.2 。 1 1 1 0 1 2 3 4 5 6 7 L(m) Figure19.3 Observedthreshold-voltage variation withchannellengthandapplied drainbiasin short-channelMOSFETs.NA=8×101/cm,=0.028μm,andr=1μm.（Adaptedfrom Fichtner and Potzl[28].)

--- Page 717 --- 694 FIELDEFFECTDEVICES Themajorityof small-dimension effects in MOSFETs are associated withthereduc- tion in thechannel lengthL.Itis therefore reasonableto introduceand specify aminimum channellength,Lmin,below whichsignificant short-channel effects are expected tooccur. Crudely speaking,Lminmust be greater thanthe sumof the depletion widths associated withthe source and drain junctions.Values in the 0.l μm to1 μm range are indicated.As suggested by computer simulationsand confirmed by experimentalobservations,a more preciseestimateofLminisgiven by theempiricalrelationship[29] Lmin=0.4[rx（W+W)2]1/} …$x_{i}n\mathring{A}；Lmin，,(19.1)$W_{}，W_Dinm ris the source/drain junction depth,。the oxide thickness,W the depletion widthatthe source junction,and WDthe depletion width at the drain junction.Note from Eq.(19.1) that Lmin can bemade smaller byreducing the depthof the source/drain islands,by reduc- ing the oxide thickness,and/or by increasing the substrate doping,which in turn causes a decrease in W and WD.Allof the above have in fact beenemployed to help assure long- channel operationof MOSFETs with increasingly scaled-down dimensions. The causes underlying departures from long-channelbehavior fall intooneof three generalcategories.Forone,differencesbetweenexperimentandlong-channeltheorymay simply arise from a breakdown of assumptions used in the long-channel analysis.Second, a reduction in device dimensions automatically leads to an enhancementof certain effects that are known to occur but are negligible in long-channel devices.Lastly,some departures from long-channel behaviorarisefrom totally new phenomena. All three categoriesare represented in the following considerationof specific-case effects. 19.1.2 Threshold VoltageModification ShortChannel In enhancement-mode short-channel devices Vis found to monotonically decrease with decreasing channel length L.Qualitatively,the decrease in threshold voltage can be ex- plained asfollows:Beforean inversion layerorchannelforms beneath thegate,thesubgate region must first be depleted (W→W).In a short-channel device the source and drain assist in depleting the region under the gate; that is,a significant portion of the subgate deletion-region charge is balanced bythe charge on theother sideof the sourceand drain pn junctions.Thus les gate charge is required to reach the start of inversionand|V decreases.The smaller L,the greaterthe percentage of charge balanced by the source/drain pn junctions, and the greater the reduction in |V1. Afirst-orderquantitative expressionfor the ΔVassociated withshort-channel effects canbeestablishedusingstraightforwardgeometricarguments.Althoughhighlysimplifed the derivationto be presented is very informative inthat it illustratesthe generalmethodof analysis. The derivation also indicates how parameters such as the source/drain junction depth enter into the specification of short-channel effects.

--- Page 718 --- MODERN FET STRUCTURES 695 As previously established,for an ideal device K_s}$+$Ko}$(19.2)LetObethetotalchare/cminsidethesemiconductor.ApplyinGausslawitiseadilestablishedthat$Q_s=-ε0 (19.3) Combining Eqs.(19.2) and (19.3), we can therefore write x_{oQ{s} Q_s} V{}= s$=\phi s$ (19.4) K{0

WhenV=,s=2FandQs=QB，whereQBisthe bulkordepletion-regioncharge per unit area of the gate. Specialized to the threshold point,Eq.(19.4)thus becomes OBB} (19.5) C_{}$Nextintroducing∆V=V(shortchannel)-(longchannel)(19.6)takingOBLandOBstobethelong-channelandshort-channeldepletion-regioncharges/c{m}^2,respectively,andmakinguseofEq.(19.5),weobtain1OBL{Q}_{B}s$ $△V_T$

C_0} (QBS-QBL)= C_o} 1

QBL (19.7) Tocompletethederivation,itisnecessarytodevelopexpressionsforQBLandOBs in terms of the device parameters. Working towardthisend,considerthe shortn-channel MOSFETpictured inFig.19.4.To simplifytheanalysis,VDistaken tobesmallor zeroso that W≤ Wat all points beneaththecentral portionof the gate.Theshaded areas inthe figure identify thoseportionsof thesubgateregionthatareassumedto becontrolled bythe source and drain pn junctions.In a long-channel device,charge in theentire rectangular region of side-length L is balanced by charges on the gate and qNA(ZLW) $Q_BL= =-qNAW (19.8) ZL

--- Page 719 --- 696 FIELDEFFECTDEVICES L W_ } ri n+n+$sourcedrain{L}^{\prime }WD$W_s$pDepletionboundaryFigure19.4CrosssectionofaMOSFETidentifyingparametersthatenterintotheshort-channelanalysis.Theshadedportionsofthesubgateareaareassumedtobecontrolledbythesourceanddrainpnjunctions.ItisalsoassumedthatVD=0.ZLWisthedepletion-regionvolumeandZLtheareaofthegate.Inashort-channeldevice,thedepletion-regionchargecontrolledbythegateisconfinedtothetrapezoidalregionofsidelengthsLand{L}^{\prime }.Thus-12$qNA$(L+L^{\prime })ZWL+L^{\prime }$QBS=$=-qNAW$ (19.9) 一 ZL 2L Substitutingthe OBLandOBsexpressionsintoEq.(19.7)thengives qNA\_}} L+L{} $\Delta V$ 1- (19.10) C_0o}2L Next,looking at the source region in Fig.19.4 and assuming W= W,one deduces from geometrical considerations 2 (19.11) which can be solved to obtain L{=L-2n[2W{}1+ -1 (19.12) rj Finally,eliminatingLinEq.(19.10)usingEq.19.12),weconclude

--- Page 720 --- MODERN FET STRUCTURES 697 qN_A_} rj 2W_t} ∆V(short channel) = 1+ -1 (19.13) $C_o$ L rj$Althoughafirst-orderresult,the\Delta VgivenbyEq.(19.13)doesexhibitthesameparametricdependenciesinitillynotedintheLindiscussion.Examining\Delta /long-channel),whichistherelevantquantityingaugingtheimportanceofshort-channeleffects,oneagainfindstheeffectsareminimizedbyreducing,reducingr,andincreasingNANarrowWidthThethresholdvoltageisalsoaffectedwhenthelateralwidthZofaMOSFETbecomescomparabletothechanneldepletion-widthWT.Inenhancement-modenarrow-widthde-vices,VisfoundtomonotonicallyincreasewithdecreasingchannelwidthZ.NotethattheZ-dependenceofthethresholdvoltageshiftisoppositetotheL-dependence.Thenarrow-widtheffect,however,isexplainedinmuchthesamemannerastheshort-channeleffect.ReferringtothesideviewofaMOSFETinFig.19.5,notethatthegate-controlleddepletionregionextendstotheside,lyinginpartoutsidetheZ-widthofthegate.Inwide-widthdevices,thegate-controlledchargeinthelateralregionistotallynegligible.Innarrow-widthdevices,ontheotherhand,thelateralchargebecomescomparabletothechargedirectlybeneaththeZ-widthofthegate;thatis,thereisanincreaseintheeffectivecharge/c{m}^{2}beingbalancedbythegatecharge.Thus,addedgatechargeisrequiredtoreachthestartofinversionand|Vincreases.Parallelingtheshort-channelderivationaquantitativeexpressionfor△Vassociatedwiththenarrow-widtheffectisreadilyestablished.Ifthelateralregionsareassumedtobequarter-cylindersofradiusW,thelateralvolumecontrolledbythegateis（\pi /2)WLand$qNA（ ZLW_{ + πra} w\}{L 2 $\text{s}$ QB(narrow width) = -qNAW} 1+ π (19.14) ZL 2 Z Z Field oxide Gate oxide WT W_o} Figure19.5 Side viewofa MOSFETusedtoexplain and analyze the narrow-widtheffect.

--- Page 721 --- 869 FIELD EFFECTDEVICES Replacing QBs in Eq.(19.7) with the narrow-widthQB,one rapidly concludes qNAW_1π}_} ∆V(narrow width)= (19.15) $C_0$ 2Z Thisresultconfirmsourinitialassertionthatthe narrow-widtheffectbecomesimportant when Zis comparable to WT. As a concluding point it should be noted that a combined-effect ΔVmust be estab- lished for MOSFETs that have both a short channel and a narrow width.The short-channel and narrow-width ΔV’s are notsimplyadditive.Thecombined-effect ΔVandmoreex- actingΔVexpressionsfortheindividualeffectscanbefoundinthedeviceliterature.[30] 19.1.3ParasiticBJTAction Containing an oppositely dopedregionbetweenthe sourceandthe drain,theMOSFET bears a striking physicalresemblance to a lateral bipolar junctiontransistor (BJT).Thus, withthedistancebetween the sourceand drain in amodernMOSFETreduced to a value comparable to the base width in a bipolartransistorit is not surprising thatphenomena have been observed that are normally associated with the operation of BJTs. One such phenomenon is source to drain punch-through. When the source and drain are separated by a fewmicrons or less it becomes possible for the pn junction depletion regions around the source and drain to touchor punch-through as pictured in Fig.19.6. When punch-throughoccurs,a significantchange takesplace in theoperationofthe MOSFET.Notably,the gate loses controlof the subgate region exceptfor a small portion of theregionimmediatelyadjacenttothe Si-SiO2interface.Thesource-to-draincurrent is then no longerconstrainedto thesurfacechannel,but begins to flowbeneaththesurface through the touching depletion regions.Analogous to the punch-through current inBJTs, this subsurface”space-charge”current varies asthe squareof the voltage applied etween n{ n{ Depletion-region boundaries Figure19.6 Punch-through and space-charge current in a short-channel MOSFET.

--- Page 722 --- MODERN FET STRUCTURES 669 the sourceand drain.The V>0characteristics presented inFig.19.2are examplesofthe I∞Vcurrent that results from source to drain punch-through. As a practical matter,punch-through in small-dimension MOSFETs is routinely sup- pressed by increasingthe dopingof thesubgate region and therebydecreasing the source drain depletion widths.Thiscan beaccomplished by increasing thesubstrate doping.How- ever,increasing the substrate doping has theadverseeffectofincreasing parasiticcapaci- tances.Consequently,it is common practice to perform a deep-ion implantation to selec- tively increase the doping of the subgate region. ParasiticBJT action involving carriermultiplication and regenerative feedback leads to a second potentiallysignificant perturbationof the MOSFETcharacteristics.There is a certain amountof carriermultiplication in the high-field depletionregion near the drain in allMOSFETs.Inlong-channeldevices themultiplication is negligible.In short-channel devices,however,carriermultiplication coupled withregenerative feedbackcandramati- callyincreasethedraincurrentandplacereducedlimitonthemaximumoperatingV. Catastrophic failure may even occur under extreme conditions. The multiplication and feedback mechanism operating in small-dimension MOSFETs is very similarto that whichlowrs VBR)E inaBJTrelativetoBR)CBoeeSubsection 11.2.4).The basicmechanism is best describedand explained with theaidofFig.19.7. The process is initiatedbychannelcurrententering thehigh-field region in the vicinityof the drain.Uponaccelerationin thehigh-fieldregion,a small percentageof thechannel carriers gainasufficientamountofenergytoproduceelectron-holepairsthroughimpact ionization.For an n-channel device the added electrons drift into the drain,while the added holes are swept into thequasineutralbulk.Because the semiconductor bulk has afinite resistance,thecurrentflowassociated withthe impact-generatedholes givesrise toapo- tential dropbetweenthedepletionregionedgeandthebackcontact.Thispotential dropis V{D$Channel$n+n+$currentCarriermultiplicationAdditionalviaimpactionizationelectroninjection2Holesandcarrierflowintodrainsweptinto+bulk{R}_{BB}$ 3 Potential drop caused by hole current Back-to-source forward biased current enhancement in short-channel MOSFETs.

--- Page 723 --- 700 FIELDEFFECTDEVICES of such a polarity as to foward bias the source pn junction. Forward biasing of the source pn junction in turn leadstoelectron injectionfrom the source pn junction into thequasi- neutral bulkanadditionalelectronflowintothedrain,increasedcarriermultiplicaionand so forth.The process is stableas long asthefractionalincrease in the draincurrentorthe multiplicationfactor is less than1/a,where a is the common-base gainof the parasitic BJT.Athighenhanced currents thereisthepotentialforexcessivecurrentflowthrough the device and device failure. 19.1.4Hot-CarrierEffects OxideCharging Oxide charging,orcharge injectionand trapping in theoxide,is a phenomenonthat occurs in allMOSFETs.In the vicinityofthe drain underoperational conditions,channelcarriers, andcarriersentering the drain depletionregionfromthe substrate,periodically gaina suf- ficientamount ofenergy to surmount the Si-SiO2surface barrier andenter theoxide.Neu- tral centers in theoxide trap a portionof theinjected charge and therebycause a charge buildup within theoxide.In long-channel devices,oxidecharging is thewell-known cause of”walk-out”—a progressive increase in the drain breakdown voltage of MOSFETs op- eratedat large VDbiases.Unfortunately,theeffects ofoxidecharging inshort-channel MOSFETs are decidedlymore serious.This is true because a larger percentage ofthe gated regionisaffctedinthesmllerdvices.Specificllsigificantchanges inndg can result from the oxide-charging phenomenon. Moreover,because oxide charging is cumu- lative over time,the phenomenon tends to limit the useful“life”of a device and must be minimized.A popularapproachfor minimizing hot-carrier effects,theformation ofa lightly doped drain (LDD),is described inSubsection19.2.1. VelocitySaturation Intheconventional analysisof the long-channel MOSFET,there is no theoretical limitation on the velocitythatthecarrierscanattain inthesurfacechannel.Itisimplicitlyassumed thecarriervelocities increaseas neededto supportthecomputedcurrent.Inreality,the 10cm/sec when theaccelerating electric fieldexceeds~3×104V/cm forelectronsand ~10³V/cmforholes(seeFig.3.4inPartI).If,forexample，VD=2VandL=0.5m, there will obviously be points in the MOSFET surfacechannel where theacceleratingelec- tric fieldis greaterthanorequalto4×104V/cm.Limitationof thechannel current dueto velocity saturationisclearlyapossibilityinshort-channeldevices. Velocity saturation has twomaineffectsontheobservedcharacteristics.First,IDsatis significantlyeduced.ThemodifiedItiapproximatlydescribedby IDsat ZCG- )sat (19.16) Second,as canbeinferredfromEq.(19.16)andas illustrated inFig.19.8,thesaturation currentexhibits analmostlinear dependenceonVG-VTas opposedto the conventional square-law dependence.

--- Page 724 --- MODERN FET STRUCTURES 701 V}=5V 4V Current (0.2 mA/div) Current (0.2 mA/div) Current (0.2 mA/div) V{{}5V 4V 3V -4V + 3V i -3V -2V 2V -2V 1V 1V Drain voltage (1V/div) Drain voltage (1 V/div) Drain voltage (1 V/div) (a) (b) (c) Figure19.8IllustrationoftheefctsofvelocitysaturationontheMOSFETI-Vcharacteristics. (a) Experimental characteristics derivedfrom a short-channel MOSFET with L=2.7 m,= 0.05m,=0.4mandN（substrate)10/cm.Comparativetheoreticalcharacterisicscom puted(b)includingvelocitysaturationand（c)ignoringvelocitysaturation.（FromYamaguchi[3, © 1979 IEEE.) VelocityOvershoot／Ballistic Transport Inthe Chapter17modeling ofcarrier drift in the MOSFET surface channel, we implicity assumed thatcarriers experienced numerous scattering events intraveling between the source andthe drain.This is equivalenttoassuming thechannel length(L)ismuchgreater thantheaverage distance (l)between scattering events.Clearly,if theMOSFETchannel length isreduced toa valuecomparabletol,fundamental revisions will berequired in the analyticalformalism.More importantly,however,ifeven smaller-dimension structures couldbebuiltwithL<l,a large percentageofthe carriers wouldthentravelfromthe sourcetothe drainwithoutexperiencing a singlescatteringevent.Theenvisionedprojectile- likemotionofthecarriers isreferred toasballistictransport. ObservableballisticeffectsaretheoreticallypossibleinGaAsstructures whereL≤ 0.3m.Somewhat shorter lengthsare required in Si devices.Experimentally,both Si and GaAs FETs with channel lengths ~0.1 μm have been fabricated in researchlaboratories. Thus,FET structures withLI areattainable withpresent-daytechnology.Moreover, referring to Table19.1,such structures are expectedto become commonplace in the near future. Practicallyspeakingballistictransport isofinterest becauseitcouldlead to super-fast devices.Withreduced scattering,theaverage velocityofcarrierstransversingthe channel can exceed v t.This is referred to as velocity overshoot. Average velocities up to 35% sat larger than the saturation velocity have beenobserved in an L=0.12 μm MOSFET[].

--- Page 725 --- 702 FIELDEFFECTDEVICES Admittedlythereareotherconsiderations,suchastheoperationofthecarrier injecting sourcethatcanlimitballisticdeviceperformance.Nonetheless,ballisticeffectsavebn observed and are likely to play arole in theoperationof future FETs. 19.2SELECTSTRUCTURESURVEY In discussing the essentials of MOSFEToperation, we made use of the basic enhancement- mode structure.Useofthebasicstructureallows oneto focus on the developmentofcon- cepts and the understanding of phenomena.Naturally,given the maturity ofMoSFETtech- nology,there exists a significantnumberof distinct device variations.Modifications to the basic structure have been implemented to solve specific problems or to enhance a specific devicecharacteristic.Itis alsotruethatcloselyrelatedFETdevices fabricated inGaAs and othercompound semiconductors invariably take a somewhat different form than those fab- ricated in Si.Abriefsurveyof selectMOSFETandMOSFET-likestructures has been includedin thissection toprovide some feelforthe variety that existsand thenatureof major modifications.Itshould beemphasized thatthe surveyed devicestructuresconstitute only a sampling,withabias toward structures likely to beencountered in the recentFET literature. 19.2.1MOSFETStructures LDD Transistors As described in the preceding section,reduced dimension devices are more susceptibleto hot-carriereffects.Thefield-aidedinjectionand subsequenttrappingofcarriers in thegate oxide nearthe draincanleadto serious device degradation.Thedegradationeffectsare further worsened by the common practice of using bias voltages that have not been scaled down in proportiontothedevice dimensions. The lightly doped drain(LDD)structure shown in Fig.19.9helps tominimize hot-carriereffects.Thefeatureof note is the lightly doped drainregion between the endof thechannel and the drainproper(the n-region in Fig.19.9).Thereduced doping gradient in going fromthechannel to the drainproper lowers the E-field in the vicinityof the drainand shifts thepositionof the peak E-field toward theendof thechannel.Carrier injectionintotheoxide isthereby reduced andoxide charging correspondingly minimized. Sidewall oxide Poly-Si gate Gate oxide $n$ n n- $n$ source S-d drain Lightly doped drain Figure19.9 Cross sectionofa lightly doped drain (LDD) structure.

--- Page 726 --- MODERN FET STRUCTURES 703 Source Gate Drain Metal Oxide $n+$ $n+$ P p+ Isolation n-Epitaxy p-Substrate (a) Diffusion mask edge →x AR | $Si{O}_2$ $n+$ p $n-$ $n$ 1 i Source i 1 1 1 Channel Drain 1 1 1 |N{-NA 1 1 1 1 n+ P $n-$ $n+$ x (b) Figure19.10(a)Cross sectionofa DMOS structure.(b)Magnifiedcross sectionof thechannel region and lateral doping profile.(From Pocha et al.33,©1974IEEE.) DMOS Adouble-diffused MOSFET(DMOS)structure is pictured inFig.19.10.The structure is distinctive inthatthechannelregionisformed,andthe channel lengthspecified,bythe difference in thelateral extentof twoimpurityprofles.A p-typedopant(e.g.boron)and an n-type dopant (e.g.phosphorus)are admitted and diffused into the Si through the same oxide maskopening.The p-type dopant,which is introduced first, diffuses slightly deeper andfartherto thesidethanthen-type dopant.Theresultis the simultaneousformation of the source and channelregions as clearly shown in the magnifiedcross sectionof Fig.19.1O(b).Themost importantphysicalcharacteristicofDMOSisa shortchannel length(~1μm)that can be established without using small-dimension lithographic masks. The DMOS structure thus boasts high-frequencyoperation,which iscombined with ahigh drain breakdown voltage.Ithas been used in high-frequency analog applications and high- voltage/high-powercircuits.Althoughfirst introduced in theearly197Os,variationsofthe DMOS structure,notably power-DMOS structures,continue to be developed.

--- Page 727 --- 704 FIELD EFFECTDEVICES Oxide Substrate G ND $N$ B S G D 1 1 Gate 1 Channel region 1 1 $n$ N{D $n+$ N{}-N} 1 $N$ $N$ 一 0 1 x B -NA (a) (b) Figure19.11 (a) Cross sectionofa buried-channel MOSFET.(b)Approximate subgate doing profile.(Adaptedfrom Vander Tol andChamberlain[34,©1989IEEE.) Buried-Channel MOSFET Figure19.11shows thecross sectionofa buried-channelMOSFETandtheapproximate subgate-doping profile insidethe transistor.The uniquestructuralfeature,a surface layer beneath the gate with the same doping as the source and drain islands,is typically formed by ion implantation. With a pn junction bottom gate and an MOS top gate,the buried- channel MOSFETis physicallyand functionally a hybridJ-FET/MOSFETstructure.It can be designed to function as a depletion-mode or enhancement-mode device depending on the thickness and doping of the surface layer. The buried-channel MOSFET gets its name from thefact that channelconduction can bemade to take placeaway from theoxide- semiconductor interface. This leads to inherently highercarriermobilities,a reduced inter- facial trap interaction,and a decreased sensitivity to hot-carrier effects. SiGe Devices Withthe developmentof theultrahigh-vacuum chemical-vapor-deposition(UHV/cVD) process[5ithas become possibleto deposithigh-qualitySixGealoyfilmsona production-line basis.Theprocesspermitsatomic-levelcontrolofthefilmthickes,pe- cise control of the film composition,and minimization of background contamination.Films covering the entire range from 0 to 100% Ge content are readily deposited. Reflecting the factthat the Gelattice constant isa= 5.65 Å compared to the smaller a=5.43of siliconSiGealloyspreferablyexhibitalatticeconstantlargernSi. However,ifsufficientlthintypically≤1000,SiGealloyflmsdepositedonaSisu strate are pseudomorphic in nature;that is,theyconform atom-by-atomtothe latice patern

--- Page 728 --- MODERN FET STRUCTURES 705 Gate +poly-Si Source 12 nm SiO_{2 Drain $n+$ Strained-Si $n$ ~6 nm 0.25μm Relaxed Si0.7Ge0.3 2 SiGe Graded buffer 1.5μm 5%→30%Ge Figure19.12 SiGeenhanced-mobilityMOSFET.The n-channelMOSFETincorporates a strained- Si surfacechannel.(From Welseret al.[36],©1994IEEE.) oftheunderlying Si substrate.Siheterojunctionbipolartransistors(HBTs)fabricated with a pseudomorphicSiGe base layer haveachieved the highest operating frequencyof any Si bipolar device. Becauseof theconstrained lattice spacing,pseudomorphic films are subject to a con- siderableamountof internal stress.Growing a SiGealloyfilmbeyond acriticalthickness causes defects to nucleate spontaneously.The defect generation relieves the stressandal- lows thefilm torelax toits preferredlatticeconstant.RelaxedSiGefilmsareof particular interest in the present context because they have beenused infabricating”enhanced mo- bility”MOSFETs as illustrated inFig.19.12. In fabricating theFig.19.12devicea compositionallygradedSiGebufferlayer,are- laxedSi07Ge0.3 layer,andathinpseudomorphicSifilmare sequentiallygrownonaSi substrate.Withstrain-reliefdefectsprimarilyconfinedtothebufferlayer,therelaxed Si0.7Ge03layereffectivelyfunctionsasaqualitysubstrateforthepseudomorphiclayer. ConformingtothelargerlatticeconstantofSi0.7Ge0.3placesthepseudomorphicSifilm undertension.Thestrain intheSilayer inturn gives rise toreduced carrier scatteringand a lowerelectroneffectivemassparalleltotheSiSiOinterface.Thesecombinetoenhance the carriermobilityintheSisurfacechannel.Thepictureddevice,forexample,exhibited a factorof～2enhancement in theobserved low-fieldmobilityover a standard MOSFET. Theenhancedmobility,translating intoincreasedcurrentdriveathigherfields,hasthe potential to extend the performance limits of existing MOStechnology. sOI Structures The termsilicon-on-insulator(SOI)is used to describe structures where devices are fabri- cated insingle-crystalSilayers formedoveraninsulating filmorsubstrate.Thefirstreal- izationofSOIstructuresinvolvedcrystallinefilmsofSidepositedonproperlyoriented sapphire substrates(SoS).Subsequently, laserannealing techniques were used to crystal- lize amorphousSifilmsdepositedoninsulatorssuchasSiO2andSi3N4.Inbothofthese approachestheSifilmqualityhas impededtheirwidespreadutilization.Amorerecent approach,epitaxiallayerovergrowth(ELO),has solvedthefilmqualityproblem.InELO

--- Page 729 --- 706 FIELDEFFECTDEVICES Oxygen Process flow Implant oxygen Silicon Anneal to form SiO2 Silicon Grow CVD epi to $Si{O}_2$ required Si thickness Silicon with permission.) a hole isopened intheinsulator to expose asmallsectionoftheunderlying crystalline-Si substrate.With theexposed substrate acting asaseed,epitaxial Si is then grown up through the holeandsideways over the insulator. Unfortunately,ELO is not considered to be a production-linetechnology.Among all theSOItechnologiesonlySIMOXandBESOI are currentlyconsidered production-linetechnologies that arecapableofyielding Sifilmsof acceptablequality.BothSIMOX and BESOI wafers are available commercially. In BESOI(BondEdSOI)a SiO2layer,whichis to become the insulating layer,is thermally grown on one Si wafer.A second Si wafer is next bonded to the top of the first wafer and annealed.Finally,thetop wafer is ground,polished,andchemicallyetcheduntil the desired surface-layer thickness is attained.In SIMOX(Separation byIMplantationof OXygen)theSOIstructure is formed bycreatinganSiO2layer justbeneaththe surfaceof a bulk-Si waferas illustrated inFig.19.13.TheSIMOXprocess begins withtheion- implantationofoxygen beneaththeSi surface.TheO2implant is normallyperformed em- ploying an accelerating energyof150-200keVandinvolves a total doseof1-2×1018 atoms/cm².Annealingthe structure at anelevatedtemperature(typically1300Cfor roughly6 hours)next allows the implanted O2to react with the Si to form a buriedoxide (BOX)layer. Theannealalsominimizes defects in thesurface Si layer. Additional Si is sometimes deposited by standard epitaxialtechniques to increase thefinal thickness of the Si surface layer.Required thicknessesrange from500Åto2000forCMOSapplications and from 0.3 μm to 10 μm for thick-film bipolar applications. The numerous projected advantages of SOI-based ICs,particularly MOSICs,helpto explain the extensive research effort expended to develop a viableSOItechnology. The dielectricisolationofindividualcircuitelementspossiblewithSOIreducesparasiticcapac-

--- Page 730 --- MODERNFETSTRUCTURES 707 itances,therebygivingrise to higheroperating speeds,andtotally eliminates latch-up. SOI-based MOSFETs are also projected to consume less power,operate at highertem- peratures,provide improved radiation hardness,and exhibit reduced short-channel effects. From a design standpoint,SOIpermits higherpacking densitiesand easier scalingto smaller linewidths.From a fabrication standpoint SOIleads to simpler processing because it permits a reduction in the number of masking (lithographic)steps.Perhaps most impor- tantly,the wafers derivedfrom SIMOXandBESOIaretotallycompatible with production- line equipment and procedures. The mostextensive use of SOI to date is in the production of radiation-hardened SRAMsformilitaryapplications.Expanded use is anticipatedin the areas of very high speed very large scaleCMOSICs,low-power supply-voltage devices, andDRAMS.[38] 19.2.2MODFET(HEMT) ThelastmemberoftheFETfamilyto be surveyed is themodulation dopedfieldeffect transistor(MODFET).AlthoughMODFETappears to bethepreferredacronym forthe device structure,it isoftenalternatively identifiedas the highelectronmobilitytransistor (HEMT).Whenfirstintroduced, the structure was sometimes calledthe selectively dopedheterostructuretransistor(SDHT)orthetwo-dimensionalelectron-gasfieldeffect transistor (TEGFET).A perspective viewof an AlGaAs/GaAsMODFET was presented in Fig.15.7as partofthegeneralintroductiontofield-effectdevices;a simplifiedcross sec- tion of the structure is shown in Fig.19.14(a). Thestructuralsimilarities oftheMODFETandMOSFETareobviousfromFg. 19.14(a)ifone thinksofthe AlGaAslayer as an insulator. The MODFETdoes differ from theMOSFET inthatthe“insulatinglayer,theAlGaAs,is doped.Moreover,theGaAs epilayer is nominallyundoped.Theterm“modulationdoping”arisesfrom themodulation of the dopantsource to selectively dopeonly the AlGaAs layer during the sequential depo- sition of the GaAs and AlGaAs layers.Dopingofthe AlGaAs helps induce asurface chan- nel adjacentto theAlGaAs/GaAs interface. Because the current-carrying GaAs layer is nominallyundoped,thereisonlyminimal scatteringfromresidual（unintentional)dopant impurities in the GaAssurface channel.Very highelectronmobilities are observed atroom temperature,andaneven greatermobilityenhancementrelativetodoped-channel devices liketheMESFET is obtainedat liquidnitrogentemperatures.This is whythestructure is Greaterdetailofthegated regionandtheelectrostaticsituationinsidetheMODFET are pictured in Fig.19.14(b).AlGaAs has a widerband gapthan GaAs and,as previously illustrated inFig.11.19(a),energybandoffsetsariseattheAlGaAs/GaAsinterface.Inthe Latch-up is thetermfor amajorproblemin digitalCMOScircuits where thecircuitrygets stuck inaspeciic like action. Althoughthe HEMTacronymcontinuestobeemployed,itisnowrecognizedthat thelow-fieldmobilityisnot the key parameter in determining the characteristicsof the device under normal operating conditions.Rathe analogoustoMESFEToperationdescribedinSection15.3itisthehigh-fieldvelocitythatbecomesthecritical parameter when large electric fields exist along the FET channel.

--- Page 731 --- 708 FIELDEFFECTDEVICES Schottky $n-GaAs$ S D contact cap layer G n-AlGaAs HH GaAs (nominally undoped) Alloyed Semi-insulating ohmic GaAs contact (a) Schottky Channel contact conduction m , VG n-AlGaAs! p-GaAs 1 1 $N+$ Undoped Inversion layer x_{1 (two-dimensional electron gas) 1 1 1 1 1 1 1 1 一 1 1 1 i 1 1 1 1 1 1 E{ 1 1 Φ 1 M 1 △EC$ Ei -EF Ev (b) Figure19.14 The basic MODFET or HEMT structure.(a) Simplifiedcross section.(b) Section throughthetransistorbeneatthegateandtheassociatedenergybanddiagram.[Partb)rome andLundstrom[39],©1984IEEE.]

--- Page 732 --- MODERNFETSTRUCTURES 709 MODFETtheconduction bandoffset(△E)becomesanelectron containment barrierAs already noted,the AlGaAs is doped to induce an inversion or accumulation layer ofelec- trons at the GaAs surface.(Theconducting electron layer is usuallyreferredto as a two- dimensionalelectron gas in theMODFET literature.) TheAlGaAslayer is alsomadesuf- ficiently thin so that it istotally depleted underequilibriumconditionsby thebuilt-in potentialassociated withtheSchottkygatecontact.Theresultinguncompensateddopant ions in theAlGaAs give rise toan energy band diagram verysimilarto thatof a MOSFET withsodium ions distributed throughouttheoxide (compare Figs.19.14band P18.7).We should note that, because AlGaAs/GaAs is a lattice-matched system, the interface between thetwomaterials isessentiallyfreeoffixedchargesandinterfacialtraps,aprerequsiteof proper device operation. AlthoughexhibitingacceptableelectricalcharacteristicssimilartothoseofotherFETs at lowcurrentlevels,amaterial-related propertyof theAlGaAs/GaAsMODFETleadsto a degradationof device performance at moderateandhigh current levels. Specifically, above a certaincarrierdensity,therelatively smallconductionbanddiscontinuityatthe AlGaAs/GaAs interface permits channelcharge to spillover into the AlGaAs.Conse- quently,thecarrierdensityinthechanneltendstosaturateasafunctionofthegate potn- tial.Device performance istherebydegraded.Seeking asolutiontothe problem has ledto the developmentof two second-generation MODFET structures.Inone structure a pseu- domorphicInGaxAslayer is positionedbetweentheAlGaAsandGaAs.Theadditionof In to GaAsreduces the semiconductor bandgap and increasesthe △Econtainmentbarrier. The larger △Einturn permits highercarrier densitiestobeinduced in theInGaAssurface channel.Thecross sectionof a pseudomorphicMODFETor PHEMTsoldcommercially by Hewlett-Packard isreproduced inFig.19.15.Theothersecond-generationMODFET featuresanaltrntivelicmcedsstemnamlthel0.48I2AI53G47As/ InP system,where InPis thesubstrate material.The ΔEat the AlInAs/InGaAsinterface is approximately two times theconduction band discontinuityof the AlGaAs/GaAs system. Even though theAlInAs/InGaAsMODFETtechnology is relatively immature,it aleady D G S Au $n_{}-}GaAs n-AlGaAs Ohmic contact InGaAs Undoped GaAs Semi-insulating GaAs substrate Figure19.15 Cross section of a commercially available pseudomorphic MODFET or PHEMT. （Fromthe 1993 Hewlett-Packardcatalogueofcommunicationcomponents.[40]Reproducedcourtesy ofHewlettPackard Co., Components Group.)

--- Page 733 --- 710 FIELD EFFECTDEVICES terminal device. Relative to high-frequency operation,notethe mushroom or”T”shape of the gate in Fig.19.15.Attheroutinely submicrongate-lengthsfoundinMoDFETs,theeffectofpara- sitic resistances often masks intrinsic device performance. While maintaining a small “footprint,theTshapeofthegatemetallizationincreasesitscurrent-carryingcross- sectional area and thus reduces its resistance. At present,theMODFETis graduallyreplacing GaAsMESFETs inmanyadvanced commercial and military systemsthat demandultra-high performance.It is an ideal candi- date forapplications involving low-noiseamplification atmicrowave and millimeter wave frequencies.Althoughoriginallyconceived as aultra-high-speed digitallogicdevice,the compound-semiconductor MODFET is now considered unlikely to replace Si devices in the near future. PROBLEMS CHAPTER 19 PROBLEMINFORMATIONTABLE Complete Dificulty Suggested Short Problem After Level Point Weighting Description 19.1 19.2.2 1 15(1 each part) Quick quiz 19.2 19.2.2 1 5 (1/2 each answer) Acronyms and abbreviations 19.3 19.1.2 2-3 12(a-5, b-5,c-2) Lmin,ΔV calculations 19.4 19.2.1 1 10 (2 each part) Identify unique features 19.5 19.2.2 2-3 30 (5 each part) Summarize journal articles 19.1 Quick Quiz Answer the following questions as concisely as possible. (a)Small-dimensioneffects are generallyundesirableand are minimizedor avoided in commercialMOSFET structures through the proper scaling of device dimensionsor modifications in device design.True or false? (b) Namethe three mostcommonly citedindicationsof short-channeleffects. (c) Relativetothreshold-voltagemodification,howdotheshort-channelandnarrow-width effects differ? In what ways are they alike? (d) Forafixedchannellengthandchannel width,nametwootherMOSFETparameters that can beadjustedtominimizesmall-dimensioneffects.Alsoindicatehow thepa- rameters must be adjusted to reduce small-dimension effects. (e) Name two BJT-likephenomena that have beenobserved in short-channel MOSFETs. (f) DescribewhathappensinsideaMOSFET affectedbythehot-carrierphenomenon known as oxide charging.

--- Page 734 --- MODERN FET STRUCTURES 711 (g) Whyisoxidechargingassociatedwithhot-carriereffectsmoreimportantinshort- channel devices? (h)HowcanyoutllwhethervlocitsaturationisffectingtheDVDcharactristicsof a MOSFET? (i) Indicate what is meant byballistic transport. (j) Indicate what is meant by velocity overshoot. (k) Describeapopularapproachforminimizingtheoxidechargingproblem. (1) What exactly is a pseudomorphic film? (m)Cite two examples of pseudomorphicfilms in device structures. (n)Briefly summarize the SIMOX process. (o)What is thedifference between a MODFET and a HEMT? 19.2 Acronyms and Abbreviations Indicate what thefollowingacronyms andabbreviationsstandfor:LDD,DMOS,SO SOS, ELO,SIMOX,BESOI, BOX,MODFET, andPHEMT. 19.3)Maknuoftheameruplieintefiuecationcomputetheexpcted fortheFig.19.3devicewhenV=0.125V.Assumen+-pdrainandsourcestep junctionsandthatthesourceandbackaregrounded.Commentonyourcomputational result. (b) Utilizing Eq.(19.13),whatis the ΔVexpectedfortheFig. 19.3device when L=1μmand VD=0.125V? VD=4Vdata? Explain. 19.4Brieflindicatetheuniquephysicalatureoftheollowintransistorstructures: (a) LDD transistors (b) DMOS (c) Buried-channel MOSFET (d) Enhanced-mobility (strained-Si) MOSFET (e) SOI structures 19.5Readthefollowingjournal articlesandprepareaone-totwo-page summaryofeach article. (a) J.J.Sanchez,K.K.Hsueh,and T.A.DeMassa,“Drain-EngineeredHot-Electron Re- Sistant DeviceStructures: AReview, IEEE Trans. on Electron Devices,36,1125 (June 1989). (b)M.J.VanderTolandS.G.Chamberlain,“Potential andElectron DistributionModl fo the Buried-Channel MOSFET,“IEEETrans.on Electron Devices,36,670(April1989).

--- Page 735 --- 712 FIELDEFFECTDEVICES (c)B.S.Meyerson,“UHV/CVDGrowthofSi and Si:Ge Alloys:Chemistry,Physics,and Device Applications,”Proc.IEEE,80,1592(Oct.1992). (d)B.S.Meyerson,“Ultrahigh-VacuumCVDProcessMakesSiGeDevices,“SolidState Technology,37, 53 (Feb.1994). (e)L.Peters,“SOITakesOver WhereSilicon LeavesOff”SemiconductorInternational, 16, 48 (March 1993). (f） L.D.Nguyen,L.E.Larson,andU.K.Mishra,“Ultra-High-Speed Modulation-Doped Field-Effect Transistors:A Tutorial Review”Proc.IEEE,80,494(April1992).

--- Page 736 --- R3 Part SUPPLEMENTANDREVIEW ALTERNATIVE/SUPPLEMENTALREADINGLIST Type Relevant Author(s) Level (A-Alt., S-Supp.) Chapters General References Neamen A Undergraduate 11-13 Singh A Undergraduate 8,9 Yang A/S Undergraduate 8-13 For Selected Topics Advanced Undergrad 5, 6 (MESFETs) Ladbrooke S to Professional 7 (MODFETs) Pulfrey and Tarr A/S Undergraduate 7 (MOSFETs) Undergraduate to Sah S 6 (MOSFETs) Graduate (variable) 3,4 (CCDs) Schroder S Graduate 6(Modern FET) (1) P.H.Ladbrooke, MMIC Design: GaAs FETs and HEMTs,Artech House, Boston, © 1989. (2) D.A.Neamen, Semiconductor Physics and Devices,Basic Principles,Irwin, Home- wood, IL,© 1992. (3)D.L.PulfreyandN.G.TarrIntroductiontoMicroelectronic Devices,PrenticeHall, EnglewoodCliffs,NJ,©1989. R3 (4)C.T.Sah，Fundamentalsof Solid-StateElectronics,World Scientific，Singapore, © 1991. 713

--- Page 737 --- 714 FIELD EFFECT DEVICES (5)D.K.Schroder,AdvancedMOSDevices,VolumeVII intheModular Serieson Solid State Devices,editedbyG.W.NeudeckandR.F.Pierre,Addison-Wesley,Reading,MA, © 1987. (6)J.Singh,Semiconductor Devices,anIntroduction,McGraw-Hill,NewYork,©1994. (7) E.S.Yang,Microelectronic Devices,McGraw Hill,NewYork,©198. FIGURESOURCES/CITEDREFERENCES (1)W.Shockley”AUnipolarField-EfectTransistor”ProceedingsIRE,40,1365 (Nov. 1952). (2)G.C.enIossUpolfctTaocig 970 (Aug. 1953). (3)G.C.DaceyandI.M.Ross,“TheField-EffectTransistor,BellSystem Technical Journal, 34,1149 (Nov. 1955). (4)D.Kahng and M.M.Atalla,“Silicon-Silicon DioxideFieldInduced Surface Devices,” presentedattheIRE-AIEESolid-State DeviceResearchConference,CarnegieInstituteof Technology,Pittsburgh, PA, 1960. letin,FairchildSemiconductor,Nov.1964. (6) R.H.Dennard,“Field-Efect TransistorMemoryU.S.Patent3387286,application filed July14,1967,granted June 4,1968. (7) P.Wyns and R.L.Anderson,“Low-TemperatureOperationofSiliconDynamic Random-AccessMemories,”IEEETransactionsonElectronDevices,36,1423(August 1989). (8)C.T.Sah,“EvolutionoftheMOSTransistor—FromConceptiontoVLSIProceed- ingsIEEE,76,1280(Oct.1988). (9)M.F.Tompsett,G.F.Amelio,andG.E.Smith,“ChargeCoupled8-Bit Shift Register,” AppliedPhysicsLetters,17,111(1970).C.H.SequinandM.F.Tompsett,Charge Trans- fer Devices,Advances in Electronics andElectronPhysics,Supplement 8,Academic Press, New York, © 1975. SemiconductorTriodes,in Technical DigestofIEEE1963Int.Solid-StateCircuitConff, pp.32-33,Feb.20,1963.F.M.Wanlass,“Low Stand-By Power Complementary Feld- EffectCircuitry,U.S.Patent3356858filedJune18,193,issuedDec.5,97. R3 (11)C.G.Fonstad,Microelectronic Devices andCircuits,McGraw-Hill,NewYork, © 1994.

--- Page 738 --- SUPPLEMENT AND REVIEW 715 (12) T.J.Drummond,W.T.Masselink,and H.Morkoç,“Modulation-DopedGaAs/ (Al,Ga)AsHeterojunctionField-EffectTransistors:MODFETs,”ProceedingsIEEE,74, 773 (June 1986). (13)M.Kuhn,“AQuasi-Static Techniquefor MOSC-Vand SurfaceStateMeasure- ments,“Solid-State Electronics,13,873(970). (14) A. Berman and D.R. Kerr,“InversionCharge Redistribution ModelofHigh- FrequencyMOSCapacitance,“Solid-StateElectronics,17,735(July1974). (15)W.E.Beadle,J.C.C.Tsai,andR.D.Plummer,Quick ReferenceManualforSilicon Integrated Circuit Technology, Wiley, NewYork, ©1985. (16)S.C.Sunand J.D.Plummer”ElectronMobility inInversionandAccumulation LayersonThermallOxidiediliconurceIEEETranctionsonElcroDcs, ED-27,1497(August 1980). State Electronics,26,143 (1983). (18)“Practical Applicationsofthe4145A Semiconductor Parameter Analyzer:DCPa- March 1982. （19）S.Kar,“DeterminationofSi-MetalWorkFunctionDiferences byMOSCapacitance Technique,“Solid-State Electronics,18,169(1975). (20)B.E.DStanrdiedTrmilogyforOxieChaesAsoiaedwhm Oxidizedilicon,IEEETransactionsonElectronDevices,ED-27,606(Marc10. (21)D.R.Kerr,J.S.Logan,P.J.Burkhardt,andW.A.Pliskin,“StabilizationofSiO2 PassivationLayerswithPO,BMJournalofResearch&Development8376(1964). (22)B.E.Deal,M.Sklar,A.S.Grove,andE.H.Snow,“Characteristicsofthe Surface- StateCharge(Qss)ofThermallyOxidizedSilicon”JournaloftheElectrochemicalSoci- ety, 114,266(1967). (23)R.R.RazoukandB.E.Deal,“DependenceofInterface State DensityonSillicon ThermalOxidationProcessVariablesJournaloftheElectrochemicalSociety12, 1573 (1979). (24)B.E.Deal,“TheCurrentUnderstandingofCharges intheThermallyOxidizedSili- con Structure,JournaloftheElectrochemicalSociety,121,198C(174). (25) J.R.SrourandJ.M.McGarrity,“RadiationEffectsonMicroelectronics inSace” ProceedingsIEEE,76,1443(1988). (26) (a) P. Singer,“1995: Looking Down the Road to Quarter-MicronProduction,” SemiconductorInternational,18,46(Jan.1995).(b)ProcessesoftheFuture:Updated R3 RoadmapIdentifieTechnicaStrategicChallenesSolidSateTechnolo2 (Feb. 1995).

--- Page 739 --- 716 FIELD EFFECTDEVICES (27)I.M.Bateman,G.A.Armstrong,and J.A.Magowan,“DrainVoltage Limitationsof MOS Transistors,“Solid-State Electronics,17,539(June1974). (28)W.Fichtnerand H.W.Potzl,“MOSModellingbyAnalyticalApproximations. I.SubthresholdCurrentandThresholdVoltage”InternationalJournalofElectronics,46, 33(1979). (29)J.R.Brews,W.Fichtner,E.H.Nicollian,andS.M.Sze,“GeneralizedGuidefor MOSFETMinimization,”IEEEElectron Device Letters,EDL-1,2(1980). (30)T.A.DeMassaand H.S.Chien,“ThresholdVoltageofSmall-Geometry SiMOS- FETs,“Solid-State Electronics,29,409(1986). (31)K.Yamaguchi,“Field-DependentMobilityModelfor Two-DimensionalNumerical AnalysisofOT,“IEETaaiononlconvcsD-668ul79). (32) F.Assaderaghi,P.K.Ko,andC.Hu,“ObservationofVelocityOvershootinSilicon Inversion Layers,“IEEEElectron DeviceLetters,14,484(Oct.1993). (33）M.D.Pocha,A.G.Gonzalez,andR.W.Dutton,ThresholdVoltageControllaility in Double-Diffused-MOSTransistors”IEEETransactionsonElectronDevices,ED-21, 778(1974). (34)M.J.VanderTolandS.G.Chamberlain,PotentialandElectron DistributionModel for theBuried-ChannelMOSFET,”IEEE Transactions on ElectronDevices,36,670 (April 1989). (35) B.S.Meyerson,“UHV/CVDGrowthof Si and Si:Ge Alloys:Chemistry,Physics, andDeviceAplictionsroceegIEE0,Oct.92). (36)J.Welser,J.L.Hoyt,andJ.F.Gibbons,“ElectronMobilityEnhancement inStrained- Si N-TypeMetal-Oxide-SemiconductorField-Effect Transistors,IEEEElectronDevice Letters,15,100(March1994). (37)H.H.Hosack,T.W.Houston,andG.P.Pollack,“SIMOXSilicon-on-Insulator: Ma- terialsandDevicesolidtateTechnology33,61Dec.990). (38)L.Peters,“SOITakesOverWhereiliconLeavesOffSemiconductorInternational, 16,48 (March 193). (39)R.F.Pierret andM.S.Lundstrom,“Correspondence BetweenMOS andModulation- DopedStructures,”IEEETransactionsonElectron Devices,ED-31,383(March1984). (40) Hewlett-PackardCommunications Components,GaAs andSiliconProductsDe- signer’s Catalog, © 1993. R3

--- Page 740 --- SUPPLEMENTANDREVIEW 717 REVIEWLISTOFTERMS Definingthefollowingtermsusingyourown wordsprovidesarapidreviewofthPart material. (1) field effect (41) n-channel, p-channel (2) J-FET (42) saturation (of MOSFET D-D) (3)DRAM (43) linear (triode) region (4) CCD (44) threshold voltage (5) MOSFET (45) effective mobillity (6) CMOS (46) bulk charge (7) MODFET (47) subthreshold transfer characteristics (8) MESFET (48) overlap capacitance (9) MOS-C (49) self-aligned gate (10) source (50) flat-band voltage (11) drain (51) mobile ions (12) gate (52) fixed charge (13) channel (53) interfacial traps (14) pinch-off (54) trapped charge (15) D-VD saturation (55) bias-temperature (BT) stressing (16) gradual channel approximation (56) phosphorus stabilization (17) square-law relationship (57) chlorine neutralization (18) drain conductance (58) oxidation triangle (19) transconductance (59) donor-like trap (20) MMIC (60) acceptor-like trap (21) D-MESFET (61) dangling bonds (22) E-MESFET (62) postmetallization annealing (23) long channel (63) hydrogen-ambient annealing (24) short channel (64) “hardened” oxides (25) MIS (65) negative-bias instability (26) bulk (66) enhancement mode MOSFET (27) block charge diagram (67) depletion mode MOSFET (28) accumulation (68) gate-oxide (29) flat band (MOS) (69) field-oxide (30) depletion (MOS) (70) body effect (substrate-bias effect） (31) inversion (71) oxide charging (32) weak inversion (72) ballistic transort (33) strong inversion (73) velocity overshoot (34) surface potential (74) LDD (35) delta-depletion approximation (75) DMOS (36) oxide capacitance (76) pseudomorphic (37) semiconductor capacitance (77) SOI R3 (38) quasistatic technique (78) SIMOX (39) deep depletion (79) modulation-doped (40) total deep depletion (80) PHEMT

--- Page 741 --- 718 FIELDEFFECTDEVICES PARTIII—REVIEWPROBLEMSETSANDANSWERS The following problem sets were designed assuming a knowledge—at times an integrated knowledge—of the subjectmatter inChapters16-18ofPartIII.The sets could serveas a review or as a means of evaluating the reader’s mastery of the subject.Problem Set A was adaptedfrom aone-hour“open-book”examination;ProblemSet B was adaptedfrom a one-hour “closed-book”examination. Ananswer key is included at the end ofthe problem sets. ProblemSet A Problem A1 Theenergyband diagramforap-Si/SiO2/n-Si(SOS)capacitorunderflat-bandconditions is given below.To achieve the pictured state,theremustofcourse be a non-zero voltage appliedtotheSOS-Cgate.TheSOS-Cisidealexcept foranon-ero workfunctiondifer- ence.T=300K,NA（p-side)=101/cm3，ND（n-side）=1015/cm3,n=100/cm3,= 5×$10-6cm,and{A}_{=}$10-3cm^2. VG S-d $Si{O}_2$ n-Si x^} x^} EFN EFP (a)Whatisthevoltagebeingappliedtothep-Si/SiO/n-SiSOS-Ctoachievethepictured fAlat-band condition?Give both the polarity and magnitudeof (b)Sketch theenergy banddiagram and theassociated blockchargediagramfortheSOS-C tive words to your sketches as necessary toforestallamisinterpretationofyouranswer. (c)Sketch the energy band diagram and the associated block charge diagram for the Sos-C when a large negative gate voltage is applied to the device. R3 (d) Make a sketchof thehigh-frequency C-VG characteristicto be expectedfromthe sOS-C described in this problem.Explain how you arrived at your C-VG sketch. (e) Invoking the delta-depletion approximation,determine the minimum capacitance ex- hibited by the device?Give both a symbolic and numerical answer.

--- Page 742 --- SUPPLEMENTANDREVIEW 719 Problem A2 RefertoFig.17.10in Subsection17.2.4.Forthepurposes ofthisproblem,consider thecharacteristicsplotted inthefiguretobeexperimentalcharacteristicsofaMOSFET under test. (a)Completingthefigurebelow,carefullysketchtheinversionlayerand depletionregion inside thegiven MOSFETcorrespondingto the biasing pointof V=V=5V.Also insert the doping type of the source and drain islands. S D (b)Invoking the square-law relationships,and making use onlyof the plottedcharacteris- tics (ignore the figure caption information),roughly determine theMoSFETthreshold voltage (V).Explain how you arrived at your answer. (c) Givenonlytheparametricvalues inthe figurecaption(ignoring thecharacteristics themselves), what is the expected MOSFET threshold voltage(V)? (d)Making useonlyoftheplottedcharacteristics,estimategifthequiescentoperating pointof the MOSFETis VG=5V,VD=0.Indicatehow youarrived at your answer. (e) DeterminegmifthequiescentoperatingpointoftheMOSFETisV=VD=5V. Indicate how you arrived at your answer. Problem A3 Thethree parts ofthis problem are similar. A pairofMOS-Cs are taken to beidentical except forone physical difference.The physical difference causes a voltage displacement of the C-Vcharacteristicsderivedfromthetwo devicesas illustratedbelow. Defining δVG=FB1-FB2，whereFB1andFB2arethefat-bandgatevoltagesofdevices#1and 2respectively,your job in each case will be to determine δVG. C ↑ VFB2 δVG VFB1 Device #2 Device #1 →VG R3 (a) Two MOS-Cs are identical in every way except device l has an aluminum gate while device #2 has a gate madeof gold. Determine the expected value ofδVG.Record your work.

--- Page 743 --- 720 FIELD EFFECTDEVICES (b) TwoMOS-Cs are identical in every way except device l is fabricatedon a (100)- oriented Si surfaceand device #2 is fabricatedon a(111)-oriented Si surface.Inboth devicesthe interfacialtrapcharge is negligible(Q=0).Also,bothdevices were annealed in N2aftera dry O2oxidation to minimizethe fixed charge.Assuming C= 3×10-8farads/cmdetermine （approximately)theexpected valueofδV.Record your work. (c) TwoMOS-Csareidenticalinevery wayexcepttheNa+ionsindevice#lare allpiled uprightatthemetal-oxide interface,while indevice 2the Na+ions are all piled up right at the oxide-semiconductor interface. Taking C=3×10-farads/cm²and QM/q=5×101/cm²,determine theexpected valueofδV.Record your work. NOTE:You are to assumeδ-function type ionic distributions in working thisproblem. Problem Set B I. MOSFundamentals A totallydimensionedenergybanddiagramforanMO”S-Crecentlyfabricatedin are- search laboratory is shownbelow.(The”Ois actually ZnSeandthe semiconductoris GaAs.)The device is maintained atT=300K,kT/q=0.0259 V,n=2.25×106/cm, OF=0,andQ=0.Usethecitedenergyband diagramandthegiven information in answering Problems 1-10. 0.76eV 0.37eV 0.14eV $Ec$ EF$0.71eV$—E_iEv}$ 1.08eV x R3 0 a

--- Page 744 --- SUPPLEMENTANDREVIEW 721 tion. (Let =0 in the semiconductor bulk.) →x 0 a (2)Roughly sketchtheelectricfield()inside the semiconductor as a functionof position. 8 →x 0 a (3) Do equilibrium conditions prevail inside the semiconductor? (a) Yes (b) No (c) Can’t be determined (4) { }== (a) 4.03×10²{0/cm^³ (b) 8.13×101}/cm³ (c) 1.00×101}/cm³ (d) 5.01×108}/cm³ (5) {} ==$ (a)-0.57V (b)-0.39V (c) 0V (d) 0.39V R3 (e) 0.57V

--- Page 745 --- 722 FIELD EFFECTDEVICES (6) For the pictured condition the M”O”S-Cis (a) Accumulated. (b) Depleted. (c) Inverted. (d)Biasedatthedepltioninversiontranstioon. (7) What is the metal-semiconductor workfunction difference (Ms)? (a) -0.39V (b) -0.25V (c) 0V (d) 0.25V (e) 0.39V (8) What voltage must be applied to thegate to achieve flat-band conditions? (a) -0.39V (b) -0.25V (c) 0V (d) 0.25V (e) 0.39V (9) Invokingthedlta-deltionapproximationdtermiethenormlied low-frequency small-signal capacitance,C/Co,at the pictured bias point. (a) 0.25 (b) 0.41 (c) 0.56 (d) 0.83 (10)Asnoted in theenergy band figure,ais the distancefromthe”oxide”-semiconductor interface to thequasineutral semiconductor bulk.Determine the lengthofa at the pictured bias point. (a) 0.112μm (b) 0.205 μm (c) 0.428 μm (d) 0.813 μm R3

--- Page 746 --- SUPPLEMENT AND REVIEW 723 I. MOSFET 1010/cm²，x=500Å，A=10-3cm²,andNA=10/cm3.AssumeT=300K. (11) Determine the flat-band gate voltage, FB. (a) -2.05V (b) -1.01V (c) -0.89V (d) 0V (12) Determine the gate voltage at the onset of inversion,V. (a)-1.01V (b) -0.21V (c) 0.80V (d) 1.81V (13) The given MOSFET is (a) An enhancement-mode MOSFET. (b) A depletion-mode MOSFET. (c) Abuilt-in channel MOSFET. （14)IftheinternalconditioninsidetheMOSFETisasshownbelowtothe left,identify the correspondingoperational pointon theID-Dcharacteristicat the right. o} S D $N^{+}$ $N$ D E 1 C B →V{D A (15) AtV-V=3Vand VD=1Vthe MOSFETexhibits a drain currentofID= 2.5×10-4amp.Using the square-lawformulation,determine the draincurent if V{-V=3VandVD=4V. (a) 3.5× 10-4amp R3 (b) 4.0×$10-4 amp (c) 4.5× 10-4amp (d) 1.0× 10-3} amp

--- Page 747 --- 724 FIELDEFFECTDEVICES .1 True orFalse (16)The“fieldeffect”is thephenomenon wherecarriersareacceleratedby anelectric field impressed parallel to the surface of the semiconductor. (a) True (b) False (17) Theelectronaffinityofasemiconductoristhediferenceinenergbetweenthe vacuum level and Eat the surface of the semiconductor. (a) True (b) False C-V characteristics. (a) True (b) False (19) The nonequilibrium condition where there is a deficitofminoritycarriers and ade- pletion width in excess ofthe equilibrium value is referred to as”deep inversion.” (a) True (b) False (20)Thevoltage shiftduetomobileionsintheoxideisataminimumwhentheionsare located midway between the gate and semiconductor. (a) True (b) False (21)TheinterfacialtrapchargeO)istypicallyafunctionoftheappliedgatevoltage. (a) True (b) False (22)The“bulk-charge”theoryforthed.c.characteristicsofaMOSFETderivesitsname depletion-region charge beneath the MOSFETchannel. (a) True (b) False (23)Let gd bethe drain or channel conductanceofa MOSFET.By definition,at lowfre- quenciesgd=IDGVD=constant R3 (a) True (b) False

--- Page 748 --- SUPPLEMENTANDREVIEW 725 ebulkmobilityofthesamecarriersbecauseoftheaddedscatteringassciated e depletion region charge. 1e se lern-day MOSstructures the“M”in MOSisoftenheavilydopedpolycrystal- 1e ise -Set A 1 1 kT EFN-P)=-[(FN-E₁)+(E-FP)]= [ln(ND/n₁）+ln(NA/n₁)] q q 0.0259)ln（1015/1010)）=0.596V Accumulation layer of holes EFn →x Accumulation EFp layer of electrons p-Si O n-Si Holes (inversion layer) Ionized Fp W_s} donors →x Ionized 7 W_sT} acceptors EFn Electrons (inversion layer) p-si O n-Si R3

--- Page 749 --- 726 FIELDEFFECTDEVICES (d) When V>0,both semiconductor components are accumulated. Thus C approaches CO at large positive gate biases. When VG<0,the two semiconductors components first depleteandtheninvert.Inversionoccursatthe samebiasvoltageforthetwosides thereforeexpctedtosmoothldcreasetoinatlageneativebiasesAssk below,the deduced characteristic shouldlook very similar to a standard n-bulk high- frequency MOS-C C-Vcurve. C Co+$\to VG0VBB$ (e) Refectingontheanswerstopreviouspartsoftheproblem,particularlypart(c),weare led to model the SOS-C as threecapacitors in series.In the equivalentcircuit forthe SOS-C shownbelow,CSp andCSn arerespectively the p-andn-sidesemiconductor capacitance. Csp Co Csn$BecauseNA(p-side)=D(n-side),CSp=n=Csand111112C$\oa{o}$ + Cse + Csn

C_o} + or $C_{{}_s}$ Co$Co$ C= + 2Co2K_OW$1+1+$K_{s}$ TheminimumcapacitanceoccursunderinversionbiasingwhereW=W.Performing the indicated computations gives R3

--- Page 750 --- SUPPLEMENTANDREVIEW 727 $kT$ φ$ln(NA/n）=(0.0259)ln（1015/1010)=0.298Vq$2K_{$1/22(11.8)(8.85\times 10-14)(0.596)71/2$ = (2φ$=8.82\times 10-5cm二(1.6\times 10-19)(1015)$K{O{AG (3.9)(8.85× 10-14)(10-3) C_o$69.0pF二$x{o }$(5\times 10-6)andCo69.0Cminn$ 5.45 pF 2K{OW_ 2(3.9)(8.82× 10-5) 1+ 1+ K_{s_} (11.8)(5× 10-6) Problem A2 (a) When V= VD=5Vthe MOSFET is biased above pinch-off. Pinched-off region Inversion layer S D $n$ $n$ Depletion region outline (b) Inthe square-lawtheory,VDsat=V-Vor V=-Dsat.Fromaninspection ofthe VG=5Vcharacteristic in Fig.17.10,weestimate VDsat4V.Thus V= 5-4≈1V. (c) SinceV=Vforthegivendevice（thecharacteristics weregeneratedassumingn ideal device), kT f$

q R3

--- Page 751 --- 728 FIELDEFFECTDEVICES and $K_s$ 4qNAV_{}= 2+$\phi f$ Ko$V$K{sε(11.8)(5× 10-6) (4)(1.6×10-19)(10{15) = 2(0.298)+ (0.298)= 0.80 V (3.9) V (11.8)(8.85×10-14) (d)SincegDDvitfollowsthatgcanbededucedfromtheslopeoftheappro- priate VG=constantD-DcharacteristicevaluatedattheVDoperatingpoint.Specifi- cally，theslopeoftheV=5VcharacteristicinFig.17.10evaluatedatV=0yields g=(3.0×10-3A)/(2.3 V)=1.30×10-3S (e) Inspecting the devicecharacteristics,we find the MOSFETissaturation biased when VG=V=5V.Whenbiasedabovepinch-of，gm=（ZnC/L）VDsatusingeitherthe square-laworbulk-chargeentryinTable17.1.VDsat≈4VwhenV=5Vasnotedin answering part (b). AlsoK{O (3.9)(8.85 × 10-14) C_{ 6.90×10-8F/cm^2

= 二 $x_o$ 5× 10-6 Thus $Zu{n}_{}$ (70)(550)(6.9×10-8)(4) $8m$

VDsat 1.52× 10-3S L 7 Problem A3 As can beinferredfromEq.(18.17),oras Eq.(18.20)statesexplicitly, O} $Q_{MM}$ Q(O) 一 C_{ $C_{}$ This basic relationship is employed in all three parts of the problem. (a) The materialusedtoformthegateonlyaffects Ms.Consequently, δVG=FB1-B2=MS|A1-ΦMS|AU or, since(E-F)FBis the samefor both devices, R3 b =-0.03V-0.82V=-0.85V

--- Page 752 --- SUPPLEMENTANDREVIEW 729 The Φ-x’for Al was noted in the Fig.18.3caption, while the ΦM-χ’for Auis listed in Table 18.1. (b) In general the siliconsurfaceorientationaffectsbothOFandQ.However,theprob- lem statement indicates Q=0.Thus here O} O} δG= VFB{1-B2

C0$+$C_{}$(100)(111)AsreadfromFig.18.9(b),thereisaresidualQF/q=2\times 101/c{m}^{2}intheMOS-Cfabricatedona(111)-orientedSisurface.Moreover,itwaspointedoutinthetextdis-cussionthatQFisapproximatelythreetimessmalleron(100)surfaces.(Similarobser-vationsweremadeinExercise\mathrm{18.5.})Wecanthereforewrite2QF(2)(1.6\times 10-19)(2\times 1011)\delta V_{G}W$}}$一=0.71V(3)(3\times 10-8)(111)(c)Beingidenticalexceptforthesodiumiondistribution,theflat-bandvoltagesofthetwoMOS-Cswillexhibitthevoltagedisplacement$Q_{{MM}Q_MM}$ δG=VFB1- VFB2

C_{o}$+$C_0}#1 #2 As discussed in Subsection18.2.6,γM=0if the ions are piled up adjacent to the metal-oxideinterface,whereasγm=1iftheionsare piled upadjacenttotheoxide- semiconductor interface. With M=0 and γM2=1,QM}$(1.6\times 10-19)(5\times 101)$δV_{=$=2.67V$C_{}$ (3× 10-8) Answers—Set B (1)…has the same shape as the “upside down”of the bands. φ →x 0 a R3

--- Page 753 --- 730 FIELDEFFECTDEVICES (2) …is proportional to the slope of the bands. 8 →x 0 a (3) a…Thedeviceis zero biased;thusthe semiconductor mustbeinequilibrium.Eis indeed shown energy independent inside the semiconductor. (4)b…ND=ne(F-E)/k=(2.25×106)e0.57/0.0259=8.13×1015/cm³. (5) c…Per Eq.(16.1),V=-(1/q)[(metal)−F(semi)]=0. (6) b (7)d. ..ΦMs=（1/q)[ΦM-χ’-(E-F)FB]=0.76-0.37-0.14=0.25V (8)d. ..SinceQM=QF=Q=0,△V=(G-V|same φs=MS Under flat-bandconditions V=0and VG=B=ΦMs=0.25V (9) c…Thedelta-depletionanalysis gave(Eq.16.37), C 1 V1+V{c{ It should bepointedout that VG,not V,was actually used intext Eq.(16.37).How- ever,an ideal structure wasassumed throughoutChapter16(all theVGinChapter16 arein reality V).Asnoted in theanswerto Problem8,△=(-lsame φs= Ms.SinceV=0forthepicturedbiaspointhecorresponding=Ms. V_δ}

9 K_{sx{fr2} N{= -0.117V (2)(9)²(8.85×10-14) C 1 =0.56 0.25 1+ 0.117 (10) a…Thequantity aisofcourse just the depletion width W.SinceC/Co was deter- R3 mined in Problem 9, Wcan be computed using Eq.(16.34b).

--- Page 754 --- SUPPLEMENTANDREVIEW 731 C 一 Co_ov } (_{0 1+ K_{s{} 2(i.61) 0012 qm $K_s$ $$}$ (12.85)(10-5) W

Ko x_{o C 1 (9) x_{\GQ (11) b. . .VFB= φ{Ms $C_{}$

$\phi MsM$ 一 q K{Oε0q (1.6×10-19)(5×10-6)(5×1010) =-0.89 一 (3.9)(8.85 × 10-14) = -1.01V (12) b. . .V=V{+FB 4gN_AK_{s}V{}2+$x0Ko$ $Ks\epsilon 0$ (N} \015$kT$φ$ ln =0.0259 ln = 0.298 V

\f0} q n_i 1/2 (11.8)(5× 10-6) (4)(1.6×10-19)(1015)(0.298) V{2}=}2(0.298)+ (3.9) (11.8)(8.85×10-14) =0.80V VT=0.80-1.01=-0.21V (13)b…SincetheMOSFETconductsfor G>V,thedeviceis”on”at VG=0andis therefore a depletion-mode MOFET. (14)D…TheMOSFETchannelisshownjust beingpinchedoff.Thiscorrespondstothe start of saturation, point D. (15) c. ..IfV-V=3Vand V=1V,theMOSFETis biasedbelowpinch-ofand, in the square-law formulation, Zu{n_} $D==$ L or Zu{C_} ID} 2.5 ×10-4 10-4amps/volt² R3

3 - 0.5 L (V_{-V)VD-V{/22$

--- Page 755 --- 732 FIELD EFFECT DEVICES When VG-V=3Vand VD=4V,thedeviceis saturationbiased,and Z\n _} (10-4)(3){^}$ D = Dsaa

2L 2 = 4.5 × 10-4 amps (16) b (17) a (18) a (19)b (20)b (21) a (22) a (23) b (24) b (25) a R3

--- Page 756 --- Appendix A ELEMENTS OF QUANTUMMECHANICS Beforeprogressingtothemodelingofcarriers inacrystalonefirst mustbeableto describe the electronic situation inside an isolated semiconductor atom.Unfortunately, the”every- day”descriptive formalismknown as classical (Newtonian)mechanics yields inaccurate results whenappliedto the electrons in semiconductor atoms or,more generally, when appliedto any system with atomic dimensions. The mathematical formalism known as Quantum Mechanics must be employed in treating atomic dimension systems.Quantum mechanics is a more precise description of nature thatreduces to classical mechanics inthe limit where the masses and energies of the particles involved are large. The first sectionofthis appendixcontains a discussionofkeyobservations andasso- ciated analyses leading to the developmentofquantum mechanics. This is followed bya brief survey of the basic quantum mechanical formalism.Thefinal section contains a sum- mary of the quantum mechanical solution for the electronic states in atoms-the informa- tion needed for the eventual modeling of carriers in a crystal. A.1THEQUANTIZATIONCONCEPT A.1.1BlackbodyRadiation Itis a well-known fact that a solidobject willglowor giveoff light if it is heated to a sufficientlyhightemperatureActuallyolidbodiesinequilibriumwiththeirsurrondings emit a spectrumofradiationatalltimes.Whenthetemperatureofthebody isator below room temperature,however,the radiation is almost exclusively in the infrared and therefore not detectable bythe humaneye.Foran ideal radiator,called a blackbody,the spectrumor wavelength dependence of the emitted radiation is as graphed in Fig.A.1. Various attempts to explain theobserved blackbody spectrum were made in the latter halfof the nineteenth century. Themost successfulof the arguments,allof which were based on classical mechanics,was proposed by Rayleigh and Jeans.Heat energy absorbed by a material was known to cause a vibration of the atoms within the solid.The vibrating atoms were modeled as harmonic oscillators with a spectrum of normal mode frequencies, ν=ω/2π,and a continuumof allowed energies distributed in accordance with statistical considerations. Theemitted radiation was in essence equated to a sampling of the energy distributioninsidethesolid.TheRayleigh-Jeans”law”resultingfrom this analysis is 733

--- Page 757 --- 734 APPENDIX A 10^4 103 Rayleigh-Jeans Law (T= 2000 K) 10^{² T=2000K 10 100 10-1 T=1000K 10-2 T=300K 10-3 1 一 0.1 1.0 10 100 λ(μm) Figure A.1 Wavelength dependence of the radiation emitted by a blackbody heated to 300K, 1000K,and2000K.Notethatthe visibleportionofthespectrumisconfinedto wavelengths.4m ≤λ≤0.7 μm. The dashed line is the predicted dependence for T=2000 K based on classical considerations. shown as a dashedline in Fig.A.1.As is evidentfrom Fig.A.1,theclassical theory was in reasonably good agreement with experimentalobservations at the longer wavelengths. Over the short-wavelength portion of the spectrum, however, there was total divergence between experimentandtheory.This came to beknown as the”ultravioletcatastrophe,” since integrationoverall wavelengthstheoreticallypredictedaninfiniteamountofradiated energy. In1901Max Planck provided a detailed theoretical fit to the observed blackbody spec- trum.Theexplanation was basedonthe then-startling hypothesisthatthe vibrating atoms

--- Page 758 --- ELEMENTS OFQUANTUM MECHANICS 735 in a materialcouldonlyradiateorabsorbenergy in discretepackets.Specifically,fora givenatomicoscillatorvibratingatafrequencyνPlanckpostulatedthattheenergyofhe oscillator was restricted to the quantized values En=nhν=nhω n = 0,1,2,… (A.1) Anh value of6.63×10-4 joule-sec (h=h/2π) was obtained by matching theory to experiment and has subsequently come to be known as Planck’s constant. The point to be learnedfrom the blackbody discussion is that,foratomic dimension systems,theclassical view,whichalways allows acontinuumofenergies,is demonstrably incorrect.Extremelysmalldiscretestepsinenergy,orenergyquantization,canoccurand is a central feature of quantum mechanics. A.1.2TheBohrAtom Anotherexperimentalobservationthat puzzled scientistsofthe nineteenth century wasthe sharp,discrete spectrallinesemitted by heated gases.The first steptoward unravelingthis puzzle was provided by Rutherford,who advanced the nuclear model for the atom in1910. Atoms were viewed as being composedofelectrons with a small rest mass m and charge -q orbiting a massive nucleus with charge +Zq,where Z was an integer equal to the numberoforbiting electrons.Lightemission from heatedatomscouldthen beassociated with the energy lostbyelectrons in goingfrom ahigher-energytoa lower-energyorbit. Classically,however,theelectronscouldassume acontinuumofenergies and theoutput spectrum should likewise be continuous—not sharp,discrete spectral lines. The nuclear model itself posed somewhat of a dilemma. According to classical theory, whenever a charged particle isaccelerated,the particlewillradiateenergy. Thus,basedonclassical arguments,theangularlyacceleratedlectronsinanatomshouldcontinuouslyloseenergy and spiral into the nucleus in a relatively short period of time. In1913Niels Bohr proposed a model that both resolved the Rutherford atom dilemma and explained the discrete natureof thespectra emittedby heated gases.Buildingon Planck’shypothesis,Bohrsuggestedthattheelectronsinanatom wererestrictedtocertain well-definedorbits,or,equivalently,assumedthattheorbitingelectronscouldtakeononly certain (quantized) values of angular momentum L. Forthe simple hydrogenatom with Z=1and acircularelectronorbit,theBohrpos- tulate can be expressed mathematically in the following manner: Ln=morn=nh n=1,2,3,… (A.2) where mis theelectronrest mass,vis the linearelectron velocityandris the radiusof the orbitfor a given valueof n.Sincetheelectronorbits are assumed to bestable,the centripedalforceontheelectronmov²/)mustpreciselybalancethecoulombicatraction

--- Page 759 --- 736 APPENDIX A (q²14πεorinrationalizedMKSunits）betweenthenucleusandtheorbitingelectron. Therefore,one can also write m_{^{} $q^{}$ (A.3) 4π{^} r{$where\epsilon isthepermittivityoffreespace.CombiningEqs.(A.2)and（A.3),oneobtains4\pi \epsilon 0(nh{)}^{}$ rn}$

m_{o^{} (A.4) Next,byexamining thekineticenergy（K.E.)andpotentialenergy(P.E.)componentsof the total electron energy ()in the various orbits,we find 1 一1 K.E.=

m_{0^{}= (q^{²14πε0n (A.5a) 2 2 and P.E.=-q^{²214πε0_n (P.E.set = 0 at r=∞) (A.5b) Thus $En=K.E.+P.E.=1(q^{²}14\pi \epsilon 0n$ (A.6) 2 or, making use of Eq.(A.4), m_{^{} 13.6 $E_n$ eV (A.7) 2(4πε0nh)² ^{}$Theelectronvolt(eV)introducedinEq.(A.7)isanon-MKSunitofenergyequalto1.6\times$10-19joules. With theelectronenergies in the hydrogen atom restrictedto the values specifiedby Eq.(A.7),the lightenergies thatcan beemittedby theatomuponheating are now discrete in nature and equal to E-E,n>n.As summarized in Fig.A.2,the allowedeergy transitions are found to be in excellent agreement with the observed photo-energies. Although the Bohrmodel was immensely successful in explaining the hydrogen spec- tra,numerous attemptstoextendthe“semi-classical”Bohranalysis tomore complexat- oms such as helium proved to be futile. Success along these lines had to awaitfurther development of the quantum mechanical formalism.Nevertheless,the Bohr analysis rein- forced the conceptof energy quantizationand the attendantfailureofclassical mechanics

--- Page 760 --- ELEMENTSOFQUANTUM MECHANICS 737 E 4 (eV) E0o 0 $E_s$ -0.54 $E_4$ -0.85 $E3$ 11- Paschen series E_{2 -3.40 Balmer series E_{1 -13.6 Lyman series FigureA.2HydrogenatomenergylevelsaspredictedbytheBohrtheoryandthetransitionscor- responding to prominent,experimentally observed,spectral lines. in dealing with systems on an atomic scale.Moreover,the quantizationof angular momen- tum inthe Bohrmodelclearlyextendedthe quantumconcept,seeminglysuggestinga gen- eral quantization of atomic-scale observables. A.1.3Wave-Particle Duality Aninterplay between lightand matter was clearlyevident in the blackbodyand Bohratom discussions.Thosetopics can be treated,however,without disturbing the classicalview- pointthatelectromagneticradiation(light,x-rays,etc.)is totallywave-like in natureand

--- Page 761 --- 738 APPENDIXA matter (anatom,an electron)is totallyparticle-like innature.Adifferentsituationarises in treatingthephotoelectriceffect-theemissionofelectronsfromtheilluminatedsurfaceof a material.Toexplainthephotoelectric effect,as arguedbyEinstein in1905,onemust view theimpinging lightto becomposedofparticle-likequanta(photons)withanenergy E=hv.The particle-like propertiesofelectromagnetic radiation were later solidifiedin the explanationof the Comptoneffect.The deflected portionof an x-ray beam directedat solids was foundto undergo a change in frequency. Theobserved change in frequency was precisely whatone wouldexpectfrom a“billiardball”typecollisionbetweenthex-ray quantaandelectrons in thesolid.Insuchacollisionbothenergyand momentummustbe conserved.Noting thatE=hv=mc²,where m is the“massof the photon andcthe velocity of light,the momentum of the photon was taken to bep= mc=hvlc=hλ λbeing the wavelength of the electromagnetic radiation. Bythemid-192osthe wave-particledualityofelectromagneticradiation wasanestab- lished fact.Noting this factand the generalreciprocity of physicallaws,Louis de Broglie in1925made a rather interesting conjecture.He suggested that since electromagnetic ra- diation exhibited particle-likeproperties,particles shouldbeexpectedto exhibit wave-like properties.De Brogliefurther hypothesized that,paralleling the photon momentumcal- culation,the wavelength characteristicof a given particle with momentumpcouldbe computed from† p=h/λ … de Broglie hypothesis (A.8) Although pure conjectureat the time,the de Broglie hypothesis was quickly substan- tiated.Evidenceofthe wave-like properties ofmatter was firstobtained by Davissonand Germer from an experiment performed in1927.In their experiment,a low-energy beamof electrons was directed perpendicularly atthe surface ofa nickel crystal.The energyofthe electrons was chosen suchthatthe wavelengthof theelectrons as computedfromthe de Broglierelationship was comparable to the nearest-neighbordistance betweennickel atoms.If theelectrons behaved as simple particles,one would expect the electrons to scatter moreor less randomly inalldirections from thesurfaceof the nickelcrystal (assumedto be roughon anatomic scale).Theangular distribution actuallyobserved was quitesimilar to theinterferencepatternproducedby light diffractedfrom agrating.Infacttheangular positionsof maxima andminimaofelectron intensitycouldbepredicted accurately using the de Broglie wavelengthandassuming wave-likereectionfromatomicplanesinsidethe nickelcrystal.Later experiments performed byotherresearchers likewise confirmedthe inherent wave-like properties of heavier particles such as protons andneutrons. In summary, then,based on experimental evidence—a portion of which has been dis- +UseofthesymboltorepresentthemomentumofaparticleisconfinedtothisappendixThroughoutthe remainderofthetextisdefinedastheholeconcentrationinitiallyintroducedinSubsection2.

--- Page 762 --- ELEMENTSOFQUANTUM MECHANICS 739 cussed herein under theheadingsofblackbodyradiation,theBohratom,and the wave- particle duality—one is led to conclude thatclassical mechanics does not accuratelyde- scribe theactionof particleson anatomic scale.Experiments pointto a quantizationof observables (energy, angular momentum,etc.)and to the inherent wave-like nature of all matter. A.2 BASICFORMALISM Theaccumulationofexperimental dataand physical explanations intheearly twentieth century that were atodds with the classical laws of physics emphasized the need fora revisedformulationofmechanics.In1926Schrodinger notonlyprovidedtherequired revision,but established a unified scheme validfor describing both the microscopic and macroscopic universes.The formulation,called wave mechanics,incorporated the physical notionsof quantizationfirstadvancedbyPlanckand the wave-like natureofmatter hy- pothesizedbydeBroglie.Itshouldbementionedthatatalmostthesametimeanalternative formulationcalled matrixmechanics was advanced by Heisenberg.Although very different in theirmathematicalorientations,thetwo formulations were later shown to be precisely equivalent and were merged under the general heading of quantum mechanics. Herein we will restrictourselvestothe Schrodinger wave mechanical description,whichis somewhat simpler mathematically andmore readilyrelated to the physics ofa particular problem.Our general approach will be to presentthe five basic postulatesof wave mechanics and to subsequently discuss the postulates to provide some insight into the formulation. For a single-particlesystem,the five basic postulates of wave mechanics are as follows: (1) There exists a wavefunction,Ψ=Ψ(x,y,z,t),from whichone can ascertain the dy- namic behaviorof the system and all desired system variables.Ψmightbe calledthe “describing function”for the system.Mathematically,Ψ is permitted to be a complex quantity(withrealandimaginaryparts)and willingeneral,beafunctionofthespace coordinates (x,y,z) and time t. (2) TheΨforagivensystemandspecifiedsystemconstraintsisdeterminedbysolvingthe equation, $h^{}$ mey =(2xx）n+ΦΔ (A.9) 2m 1e where m is the mass of the particle,U is thepotentialenergyof the system,and i=√-1.Eq.(A.9) is referred to as the time-dependent Schrodinger equation,or simply, the wave equation. (3)Ψand Ψmustbe finite,continuous,andsingle-valuedfor all valuesofx,y,andt.

--- Page 763 --- 740 APPENDIXA (4) If Ψ* is thecomplex conjugateof Ψ,ΨΨdV=平²dVis to be identifiedas the probabilitythattheparticle willbefoundin thespatial volumeelementdV.Henceby implication, I=APΦ (A.10) where v indicates an integration over all space. (5) Onecan associateaunique mathematicaloperator witheachdynamic system variable such as position or momentum.The value—or, more precisely,the expectation value— Specificall,antobtestmriof ieta the associated mathe- matical operator,the desired expectation value,(α),is computedfrom

(A.11) The unique mathematical operator associated with a given system variable has been established by requiring the wave mechanical expectation value to approach thecorre- spondingvaluederivedfromclassicalmechanicsinthelarg-mass/high-energlimit. An abbreviated listing of dynamic variables and associated operators is presented in Table A.1. The solutionof problems using wave mechanics is in principle quite straightforward. Subject tothe constraints (boundaryconditions)inherent in a problem andthe additional Table A.1 Dynamic Variable/Operator Correspondence. Dynamic Mathematical Variable (α) Operator (αop) Expectation Value-(α) x，y，z ← x，y，z f(x，y,z) ← f(x,y,z) h∂h∂hα $px=$P_x}，}P_y}，}P{}$\leftarrow \to APi\partial xi\partial yi\partial zi\partial x$ h∂ E ←→

--- Page 764 --- ELEMENTSOFQUANTUM MECHANICS 741 constraintsimposed bypostulates3and4,onesolvesSchrdingersequationforthesystem wavefunction Ψ.Once Ψis known,system variables of interest can be deducedfrom Eq.(A.11)per the postulate5recipe.The straightforwardapproach,however,isoften dif- ficult to implement.Except for simpleproblems of an idealized nature and a very select numberof practical problems,it is usually impossible toobtain a closed-form solution to Schrodinger’s equation.Nevertheless,inmanyproblems the constraints imposed onthe solution can beused to deduce informationaboutthe system variables,notablytheallowed system energies, withoutactually solving for the system wavefunction.Another common approach is to useexpansions,trial (approximate) wavefunctions,or limiting-casesolu- tions to deduce information of interest. Finally,acommentis inorderconcerning the“derivation”ofSchrodingers equation and theoriginoftheother basic postulates.Althoughexcellenttheoretical argumentscan be presented to justify the form of the equation,Schrodinger’s equation is essentiallyan empiricalrlationship.LikeNewtonslaws,Schrodingersequationandtheotherbasicpos- tulates of quantum mechanics constitute a generalized mathematical description of the physical worldextrapolatedfrom specific empiricalobservations.Relativeto the validity oftheformulationitcanonlybestatedthatwheneversubjecttotest byexperimenhe predictionsof thequantum mechanical formulation have beenfoundto be in agreement withobservationsto withinthe limitofexperimentaluncertainty,whichinmanycasesas been extremely small. A.3 ELECTRONICSTATESINATOMS We examine here the applicationof thequantummechanical formalism to the hydrogen atom and thesolutionresultsforatoms in general.Itshouldbereiteratedthattheoverall goal of the appendix is to provide informationabout the electronicstates in isolated semi- conductor atoms as a prelude to the eventual modeling of carriers in a semiconductor crystal. The hydrogenatom is the logical place to begin the quantum mechanical analysis because it is the simplestofatoms and because results can be compared with the semi- classicalBohr solution. Although the hydrogen atom analysis yields a completeclosed- formsolution,thetreatmentandsolutionare hardlytrivial.Wewillonlyindicatethe solution procedureandreviewkeyresults.Informationabouttheelectronicstates in multi- electron atoms is extrapolated from the hydrogen atom results. A.3.1The Hydrogen Atom The hydrogen atom consists ofa relativelymassive nucleus with charge +q surrounded by an electron with charge-q. With littleerror,the nucleus can be considered fixed in space and theproblemreducedtoa single particlesystem(theelectron)that is assumed to have afixed total energy E.Inotherwords,the hydrogenatom is taken to be isolated in space and notsubjecttoanyperturbationsthatcouldlead toachange in the total eergy.

--- Page 765 --- 742 APPENDIX A Forany single-particlesystem with afixedtotalenergyEthepositionandtimecoor- dinates can be separated yielding a general solutionofthe form Ψ(x,y,z,t)=ψ(x,y,z)e-iEt/h (A.12) Direct substitutionof Eq.(A.12)into Eq.(A.9),andthe subsequentsimplificationand rearrangement of the result, gives 2m +{2 [E-U(x,y,z）]ψ=0 (A.13) ra{{2} Eq.(A.13),which must be solved to obtain ψ(x,y,z),is known as the time-independent Schrodinger equation. In the hydrogenatom m=mand the-qelectron iselectrostatically attractedtothe +nucleusattheoriginofcoordinates.AnotedintheBohranalysisthepotentialenergy associated with theelectrostatic attraction is $q^2$ U=(A.14) 4πε{ wherer=√+y+isthedistancefromthenucleus.Thustheequationtobesolved takes on the specific form 2moq^{}{24+$E+=0(A.15)$h^{2}$4\pi \epsilon 0rInprincipleonecouldseekasolutiontoEq.(A.15)employingCartesian(x,y,z)coordi-nates.However,giventhesphericallysymmetricnatureofthepotentialenergyitismoreconvenienttoemployspherical(r,\theta ,)coordinates.Insphericalcoordinatesthedesiredwavefunctionsolutionbecomes\psi (r,\theta ,)and1\partial 1\partial 1$a^{24}∇{24= r2 + sinθ + (A.16) $r^{22}$ 1e ∂r $r^{2}sin\theta \theta ee$r^{2si}n²}θ aφ^{}$ Equation (A.15)can be solved using the separation-of-variables technique whereone assumes the wavefunction can be written as the product of three functions separately de- pendent onr,θ,and .The procedure yields an ordered set of bound-state (E<0) wave- function solutions. Arising from the separation constants, and associated with each solu- tion, there is a unique group of three quantum numbers. The standard symbols,allowed values, and full names of the three parameters are as follows:

--- Page 766 --- ELEMENTSOFQUANTUM MECHANICS 743 n = 1,2,3. …principal quantum number l=0, 1, 2,…n -1 … azimuthal quantum number m=-ltol … magnetic orbital quantum number The ψn,m（r,θ)solutionscorrespondington=1andn=2arepresented in TableA.2 forillustrativeandreferencepurposes.The aappearinginthesolutionsistheBohrradius and is numericallyequalto the ground state Bohrorbit；that is,a=4πεh²/m0q²as deduced from Eq.(A.4). Letusexamineandcommentontheresults.Supposefirstof allthat the 1,00solution is substituted into Eq.(A.15) and theresulting expression solved for E.Oneobtains $h^{}$ 1 m_{0q 4 E_{1}0,0$

$2mora2$

2(4πε0{)^{$(A.17)NotethatE100isidenticaltoEoftheBohranalysis.Similarlyifthen=2wavefunctionsinTableA.2aresubstitutedintoEq.(A.15)andtheresultingexpressionssolvedforEoneobtains1$m_{q^{4 E20,0=E₂1-1=2₁,0=1,1= (A.18) 2(4πε{0h^{ TableA.2 TheHydrogen Atom ψ,mSolutionsCorresponding ton=1andn=2. a0=4πεh²/m₀q²=Bohrradius.(J.L.Powelland B.Crasemann, Quantum Mechanics, Addison-Wesley Publishing Co., Reading, MA, ©1961.) 1 _{1,0.0= $2\ e-rlao 1-r/2a0 {2}0.0 e-r/2a0

2\2a{{2} r/2a0 ${}_{2}1，-1=$ 4\s^} e-r12ao e-iφ sinθ r/2a0 $21，0==$ e-r12ao cosθ r/2a0 ${2}11 e-r/2ao eiφ sinθ

4\r}a^{32}$ e

--- Page 767 --- 744 APPENDIXA Then=2 states areallassociated withthe same total energy,and theenergy is identical to E2of the Bohranalysis.Thegeneral pointto bemade is thatthequantumanalysisyilds the same predicted energy levels as the Bohranalysis.Moreover,knowledge of theprinci- palquantumnumber,n,completelyspecifiesthetotalenergyofanelectron ina particular state.Clearlcorrespondstotheroundstatewhilewavefunctionsassociatedwith larger n-values correspond to excited states. Whenthere is more thanone allowed state at a givenenergy,thestates are saidtobe degenerate.TheIandmof degenerate states come intoplay if,for example,thehydrogen atom were perturbed by amagneticfield.Becauseofthediferentspatial distributionofthe wavefunctions,theinteraction withthemagneticfieldwouldcauseasplittingoftheenergy levels and thereby remove the degeneracy. While onthe topicof degeneratestates,it is convenient to pointoutthat a fourth quantum number is actually requiredto completely specifya quantum state.More precise analysesindicatelectronsandothersubatomicparticlesexhibitpropertycallspin,which becomes importantinparticle-particleinteractions.Theelectron is visualized as spinning about an axis through its center ineitheraclockwiseor counterclockwise direction.This givesrise to two spin statesoftenreferred to as spin-upand spin-down.Theassociated spin quantum number,s,can takeon the valuesofs=+ands=-.Spin causes atwo-fold degeneracy to be associated with each of the states in Table A.2. Acomment is also inorderconcerning the spatial distributionof theallowed states. As noted in the section on basic formalism,Ψ*ΨdV represents the probability thata par- ticle willbefound in a spatial volume element dV.To provide a specific example,the probabilityof finding anelectron in the ground state at a distance betweenr andr+dr fromthenucleusisequalto400dr.Aplotof40versusaisdisple in Fig.A.3(a).Whereastheprobabilityoffinding theground-state electronincreasestoa maximum at the Bohrradius,and the peak probabilityprogressivelymoves to largerrasn isincreaedtreisigicantprobablitoffinintecoovrageofitaces fromthe nucleus. This is in totalcontrastto the Bohrmodelwheretheelectron isassumed to be in anorbit at an r= constant distancefrom the nucleus.Infact,the electronis sometimesconceivedasacharge“cloud”distributedinproportiontothe||dVproba- bilityasilustratedinFg.A.(b).The100wavefunctionuedinconstructingFg.A.) isofcourse sphericallysymmetric.Wavefunctions with l≠O would exhibitcharge clouds with an angular dependence. A.3.2 Multi-Electron Atoms Thewavefunctionsolutions,energy levelsandprobabilitydistributionsestablishedore hydrogen atom are specific to the hydrogen atomand cannotbe applied withoutmodifica- tion to more complex atoms.However,theallowedelectronic states in multi-electron atoms are uniquely characterized employing the same set of four quantum numbers (n,l,m, and s) introduced in the hydrogen atom analysis.The same general energyorderalso applies; n=1is associated with the lowestenergy state,n=2with the next lowest energy state, and so on. The foregoing,coupled withrestrictions placed on multi-electron systems,per-

--- Page 768 --- ELEMENTSOFQUANTUM MECHANICS 745 0.6 0.5 0.4 4πr²1410.00 0.3 (1/a0units) 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 rlao (a) (b) FigureA.3Forahydrogenatominthe1ogroundstate:aprobabilityoffindingthelron at a distancerfromthe nucleus;(b)cloud-likerepresentationof theelectroniccharge.

--- Page 769 --- 746 APPENDIX A mitsonetoinferinformationabouttheelectronicstructureofmorecomplexatoms wthout actually solving for the electronic wavefunctions. One of therestrictions referred to above goes by the nameof thePauli Exclusion Principle.The PauliExclusionPrinciple dictates that no twoelectrons in a system canbe characterized by the same set of quantum numbers.For example,one and onlyoneelectron in a multi-electron atom can have n=1,1=0,m=0,ands=.A second implicit restriction isthattheelectronicconfigurationbesuchastominimizethe systemenergy when a multi-electronatom is in its ground state. As a generalrule this means electrons populate states with the lowest possible n-values. Electronic informationpertinentto thefirst fourteen elements (up toSi)inthePeriodic TableoftheElements is presented inTableA.3.The top portionofTableA.3lists thesets of four quantum numbers corresponding to the lowest energy states.The bottom-line entry is the spectroscopic designationforthe state specified by the quantum number set.The number inthe bottom-lineentrygivesthen-value whiletheletter identifiesthel-value according to the scheme = 0, 1, 2, 3 (4,5 …) ↑↑↑↑ s p d f (g,h…) Theratheroddleteringofthefirstfourl-valuesstemsfromearlyspectroscopicworkwhere thetransitionsbetweenstates wereassociated with spectral linesnamed sharp,principal, diffuse,and fundamental.Generally speaking,in multi-electronatoms the s-states have a slightly lowerenergythanp-statesandthereforeappearfirstinthe listingofstatesp-levels corresponding to a given n-value have the same energy. The bottom portionof Table A.3 shows the ground-stateelectronic configuration in elements up to Si. The readershould verifythatthearrangementsareconsistentwithpreviouslycitedfacts and restrictions.The spectroscopic shorthand notationfortheelectronconfigurationis given inthefar-right column. The superscript on a letter in the shorthand notation indicates the number ofelec- trons with the same nl combination. Table A.3 is veryuseful for inferring major features of theelectronic configuration in an isolated Si atom.Siis of particular interestofcourse because it is presentlythe preemi- nent semiconductor material.From thetable we see thatthecompletefilling ofallowed states with a givenn-valueleads to anextremely stable,tightly bound,electronicconfigu- ration;namely,it leads to the inert gases Helium and Neon.Understandably,as envisioned in Fig.A.4,the twon=1and eightn=2electrons in Si likewise populate deep-lying energy levels tightlyboundto the nucleusof theatom. The binding is so strong,in fat, that these tenelectrons remain essentially unperturbed during chemical reactions or normal atom-atom interactions,withtheten-electron-plus-nucleus combinationoftenbeingr- ferred to as the coreofthe atom. Theremainingfour electrons are an“add-on”tothe stableNeon configurationandare expectedto berather weakly bound. They are collec- tively called valenceelectrons becauseof theirstrong participation in chemicalreactions

--- Page 770 --- ELEMENTS OFQUANTUM MECHANICS 747 TableA.3 Energy States and the Electronic Configuration in Elements1-14.Atoms are assumed to be in t the ground state. n 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 Quantum 1 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 Numbers m 0 0 0 0 -1 -1 0 0 1 1 0 0 -1 -1 0 0 1 1 S 12 $-\text{fra}12$ $\frac{}{2}$ $-\text{fra}12$ \fra } {2 } $-\frac{}{2}$ $\frac{}{2}$ $-\frac{1}{2}$ $\frac{}{2}$ $-\frac{}{2}$ $\text{fra}12$ $-\frac{}{2}$ \fra} {2 } $-\frac{}{}$ $\frac{}{2}$ $-\frac{}{2}$ $\frac{}{2}$ $-\frac{}{2}$ State $1s1s2s2s2p2p2p2p2p2p3s3s3p3p3p3p3p3pFilledStatesAtomicElectronicNumberElement1s1s2s2s2p2p2p2p2p2p3s3s3p3pConfiguration1H$1s 2 He $1{\text{s}}^{}$ 3 Li $1s22$ 4 Be 1s{2s^{} 5 B 1s{2s{22 6 C 1s{2s{2} 7 N 1s{2s{23} 8 0 $1s2s2$ 6 F 1s{2s{25 10 Ne 1s{2s{26 1 Na 1s²2{s{²2p63s 12 Mg $1s^{2}2s^{2}2p63s^{2}13A1$1s²2s{²2p63s{2 14 Si $1s²2s²}2p63s²}3{

--- Page 771 --- 748 APPENDIX A n=2 8 Electrons Six allowed levels at same energy Two allowed levels at same energy sp n=1 n= 3 2 Electrons FigureA.4Schematicrepresentationoftheelectronicconfigurationinanisolated,unperurbed Si atom. and atom-atom interactions. Reflecting the information in Table A.3,and as emphasized pictorially in Fig. A.4,the valence electrons occupy the two 3s states and two of the six available 3p states.Finally,weshouldmention thattheelectronicconfiguration in the32 electron Ge-atom(germaniumbeing theotherelemental semiconductor) is essentially identical to the Si-atom configuration except the Ge-core contains 28electrons.

--- Page 772 --- Appendix E B MOSSEMICONDUCTOR ELECTROSTATICS—EXACTSOLUTION Definition of Parameters To streamlinethemathematical presentation,it is customary in the exactformulationto introduce the normalized potentials φ(x) E(bulk)-E₁(x) (x)n

(B.1) $kT/q$ kT φsE(bulk)-E(surface) U_s} (B.2) 二 $kT/qkTand$ E(bulk)-EF U_} (B.3) 二 $kT/q$ kT (x), s,and F were formally defined in Chapter16(also see Fig.16.7).U(x) is clearly the electrostatic potential normalizedtokT/q and is usuallyreferred to as”the potential” is simply called the doping parameter.xis of course the depth into the semiconductor as measured from the oxide-semiconductor interface. Because the electricfield is assumed to vanish in the semiconductor bulk (idealization 5,Section16.1),it is permissible to treat the semiconductoras ifitextendedfromx=0tox=∞.NotethatU(x→∞)=0inagreement with the choice of =0in the semiconductor bulk. In addition to the normalized potentials,quantitative expressions for the band bending inside of a semiconductor are normally formulated in terms of a special length parameter known as theintrinsic Debye length. The Debye length is a characteristic lengththat was originally introduced in the studyof plasmas.(Aplasma is a highlyionizedgascontaining an equal numberof positive gas ions and negative electrons.)Whenever a plasma is per- turbed byplacingacharge inor near itthemobile species alwaysrearrangeso asto shield the plasma properfrom the perturbing charge. The Debye length is the shielding distance, 749

--- Page 773 --- 750 APPENDIX B or roughlythe distance wheretheelectric field emanating from the perturbing charge falls offby a factorof1le.In the bulk,or everywhereunderfat-bandconditions,thesemicon- ductor can be viewed as a type of plasma with its equal numberof ionized impurity sites andmobile electronsorholes. The placementofcharge near thesemiconductor,onthe MOS-C gate for example,thencauses the mobile species inside the semiconductortore- arrange so as to shieldthe semiconductor properfrom theperturbing charge.The shielding distanceor band-bending region is againon theorderofa Debye length,the bulkorextrin- sic Debye length LB,where 1/2 $KSε0kT L_B (B.4)

[q^²2({nblk+ pbulk) AlthoughthebulkDebye lengthcharacterizationappliesonlytosmall deviations fromfat band,it is convenientto employ the Debye lengthappropriate for an intrinsic material as a normalizing factor in theoretical expressions. The intrinsic Debye length,LD,is obtained fromthemoregeneralLBrelationshipbysettingbulk=bulk=n;that is, 1/2 $K_{\epsilon }0T$ LD (B.5)

2q^{2_}} Exact Solution Expressionsfor thecharge density,electricfield,andpotentialasafunctionof position inside thesemiconductor areobtainedby solving Poisson’s equation. Since theMOS-Cis assumed to be aone-dimensional structure(idealization 7,Section16.1),Poisson’sequa- tion simplifies to d8 9 (p-n+ ND-NA) (B.6) dx $K\epsilon 0$K_ Maneuvering to recast the equation in a form more amenable to solution, we note 1 dE_{i}(x$ kTdU 8= (B.7) q dx xpb Thefirstequality in Eq.(B.7)isarestatementofEq.(3.15)in PartI.Thesecondequality follows from the Eq.(B.1)definitionofUandthe factthatdE(bulk）/dx=0.In a similar vein we can write p= n;e[Ei(x)−EF]/kT= n;eUF-U(x) (B.8a) n=n;e[EF-E(x)]/k=neU(xU (B.8b)

--- Page 774 --- MOSSEMICONDUCTOR ELECTROSTATICS—EXACT SOLUTION 751 Moreover,sinceρ=0 andU=0in the semiconductor bulk, O=Pbulk-nblk+ND-NA=n;eUF-n₁e-UF+ND-A (B.9) IO ND-NA=n₁(e-UF-e) (B.10) Substituting the foregoingE, p,n,and ND-NA expressions into Eq.(B.6) yields ρ=qn(eF-U-eU-UF+e-UF-e) (B.11) and $d^2$ q^{2 (an-a-aa+n—n-n (B.12)

$2xp$Kε0T or,in terms of the intrinsic Debye length, $d^{}u$ 1 d_{^{}

$2ra$ $(a—a∂+n-a-a-) (B.13) We turn next to themain task at hand.Poisson’s equation,Eq.(B.13),is to be solved subject to the boundary conditions: dU 8=0 or

0 atx=∞ (B.14a) dx and U= U{s$atx=0(B.14b)MultiplyingbothsidesofEq.(B.13)bydU/dx,integratingfrom=\infty toanarbitrarypointx,andmakinguseoftheEq.(B.14a)boundarycondition,wequicklyobtain(kT/q2$8^{2}= [(I—n)-∂+(I-+n-∂)] (B.15) LD Equation (B.15)isofthe form y²=a²,which has tworoots,y=aandy=-a.Ascan be deduced by inspection using the energy band diagram, we must have>0when U>0 and<0whenU<0.Since the right-hand side of Eq.(B.15)is always positive(a≥0), theproperpolarityfor theelectricfield isobtained by choosing the positive root when U>0 and the negative root whenU<0.We can therefore write

--- Page 775 --- 752 APPENDIXB kTdU kT F(U，UF) 8= \_s} (B.16) q dx q LD where [(I—)an-a+(I-+n-∂)]=（d’n） (B.17) and +1 if U_{s}> 0 (B.18) ifU_{s<0 To complete the solution,one separates theU andx variables in Eq.(B.16)and,making use of the Eq.(B.14b)boundary condition,integrates from x=0 to an arbitrary point x. The end result is Eq. (B.19), Us dU’ x (B.19) F(U{'i) 二 L{v}$ Althoughnot inatotallyexplicitform,Eqs.(B.11),(B.16),and(B.19)collectiely constitute an exact solutionfor theelectrostatic variables.For a givenU,numericaltech- niques can be used to computeUas a function ofx from Eq.(B.19).OnceU as a function ofx is established,direct substitutioninto Eqs.(B.11)and(B.16)yields ρ and as a functionofx. The Fig.16.8plotsofU=/(kT/q)versus x andρ versus x wereconstructed following the cited procedure.

--- Page 776 --- Appendix C MOS C-VSUPPLEMENT An analysis based on the exact charge distribution inside an ideal MOs-Cyields thefollow- ing capacitance-voltage relationships:† Co$C=(C.1)1+$K_{s_{}$2F(U_s，UF)\mathrm{7...}acc(C.2a)eUF(1-e-us)+e-U(es-1)]${rDW_ff=$...flatband(C.2b)$(a-a+） _{sLD 2F(Us，UF) ..depl/inv (C.2c) [（+I）/（I-s)n-a+(sn-a-1） where 0 ...low frequency limit (C.3a) Δ= (eUs-Us-1)/F(U,UF) Us eUF(1- e-)(e{-U-1) … high frequency limit (C.3b) dU (p-type MOS-C) Jo{+}2F^{3}}U，U) $[(I-n-)n-+(I-+n-∂)]=(-n）d (C.4) Except for Eqs.(C.2c)and(C.3b),the relationships are validforeithern-or p-type devices. The required modificationofEqs.（C.2c)and（C.3b)forn-typedevicesisnotedinthetext.Foraderivationofthelow-frequency relationship see A.S.Grove,B.E.Deal,E.H.Snow,andC.T.Sah,“Investigation of Thermally Oxidised Sllion SurfacesUsing Metal-Oxide-Semiconductor Structures”Solid-StateElectronics,8,145(1965).Thehigh- frequencyresult,which includes the so-calledrearrangementcapacitanceor capacitancecontribution fromthe movement of inversion-layer carriers,is adapted from J.R.Brews,“An Improved High-Frequency MOSCapaci- tance Formula,” J.Appl.Phys.,45,1276(1974). 753

--- Page 777 --- 754 APPENDIXC (U,U)=F(U=U,UF) (C.5) 1/2 L{D (C.6)

2q^{2_{} φ_} (C.7)

$kT/q\phi s$ $U_s$ (C.8)

$kT/q$0_s+1 if U_{ > (C.9) if U_{s< 0$and{V}_{G}kT$U_{s+Us}K_{s}$ F(U,UF) (C.10)

KO}LDD q IshouldbenotedthatF(UULDU,U,andcomeromtheexactolutionr semiconductorelectrostatics.Foradditionalinformationaboutthecitedquantities,see Appendix B. Unlikethe delta-depletionresult,Ccannot beexpressedexplicitlyas a functionofVG in the exact charge formulation.Both variables,however,have been relatedto Us and the capacitance expected from the structure for a given applied gate voltage can be computed numerically.The low-frequency computation is simple enough that it can be performedon a hand calculator. The usual andmosteficientcomputational procedure is tocalculateC and thecorrespondingVGfora setof assumedUvalues.Typically,a sufficient setof (C,VG)pointstoconstructtheC-VGcharacteristic willbegeneratedifUisstepped by whole-number units （-5,-4,…)over the normaloperating range ofU values (UF-21≤U≤UF+21atroomtemperature).Itshouldbe noted that care must be exercised if U=0is included as one of the computational points.AtU=0the Eq.(C.2b)expressionfor Wfmust be employed;the accumulationanddepletion/inversion relationships become indeterminate(0/0)ifUis setequal to zero.Also,Eqs.(C.2c)and u-(n-)dxdx←-）dxdxpue[()dx-(d-)dx Eq.(C.2c),while[exp(U）-U-1]→[exp(-U）+U-1] and exp(U){[1- exp(-U)]·[exp(U)-U-1]}→exp(-U){[exp(U)-1][exp(-U)+U-1]}in Eq.C.b.Alterntivlyitispossibletoobtinan-typechaactristicbysimpluning thecalculationsforanequivalentlydoped p-typedeviceandthenchangingthe sinofall computed VGvalues.The latter procedure works becauseofthe voltage symmetry etween ideal n-and p-type devices.

--- Page 778 --- Appendix D MOSI I-VSUPPLEMENT An analysis basedon the exact charge distribution inside an ideal n-channel (p-bulk) MOSFETyieldsthefollowingcurrent-voltagerelationship: Zu{n_} exact $I$ charge) 二 L Zun_} $K_s$ 2 + L $KOLDD$ Jo (D.1) where F(U,UF,ξ)=[eUF(e-U+U-1)+e-UF(eU-5-U-e-ξ)]12 (D.2) The corresponding charge-sheet relationship is charge Zun_} $ID$

sheet L (D.3) where ( 2 K_{s{} NA} V\{fra}{} (D.4) q KOL{D√ n_{$ Note:The formof the relationships quoted herein are from R.F.Pierretand J.A.Shields,“Simplified Long- Channel MOSFETTheory,”Solid-State Electronics,26,143(1983).See H.C.Pao andC.T.Sah,Solid-State Electronics,9,927(1966)fortheoriginal exact-chargeanalysis,andJ.R.Brews,Solid-StateElectronics,21,45 (1978) for the original charge-sheet analysis. 755

--- Page 779 --- 756 APPENDIX D In both theories $kT$

$U}$ (D.5) q $kT}$ VsO$ Uso$ (9.a) q kT V_sL U_sL (D.7) q and $kT$ V{=$ U{D$ (D.8) q Finally,the normalized surface potentials at the source(U0)and drain(UL)arerespec- tively computed from kT $K_{s}$ $V_{}=$ $UsO+}$ F(U0,U,0) ..(UsO> 0) (D.9a) $K_{0}D$ q and kT $K{s{}$ V_G +7s}$ F(USL,UF,UD) ...(USL > 0) (D.9b) = $KO}LDD$ q To generate a set of ID-Vcharacteristics, VG and VDare systematically stepped over the desired rangeofoperation.Foreach VG and VDcombination,Eqs.(D.9a)and (D.9b) are iterated to determine U0 and UL at the specifiedoperating point.Once U0 andUsL are known,ID can then be computed using either Eq.(D.1)or Eq.(D.3).The process is repeatedforeach VG-VDcombination. Thecharacteristics for a p-channel device canbe establishedbyrunning thecalculations for an equivalentlydopedandbiased n-channel device.Naturally,thebiasing-polaritiesmust be reversed in plotting the p-channelcharac- teristics.Foradditionalinformation about the exact-charge formalism,the reader isre- ferred to Appendixes B and C. --- Page 780 --- Appendix E LIST OF SYMBOLS A anode A area; arbitrary constant a latticeconstant;gradingconstant;half-widthofthechannelregionin a J-FET; widthof the channel region in a MESFET $a{$ Bohr radius A Richardson's consstant (120 amps/cm²-K2) A* modified Richardson's constant (see Eq.14.19) AG gate area B base C capacitance C collector C speed of light CD$ diffusion capacitance $a emitter-to-base capacitance in the high-frequency Hybrid-Pi model CG MOSFET gate capacitance $C_d$ cuit for the J-FET and MOSFET $s}$ circuit for the J-FET and MOSFET 5 junction or depletion region capacitance $cn$ electron capture coefficient Co$ oxide capacitance (pF) 757 --- Page 781 --- 758 APPENDIXE oxide capacitance per unit area (pF/cm2) $d_}$ hole capture coefficient $C_s$ semiconductor capacitance D drain DB$ minority-carrier diffusioncoefficient in the BJT base DC minority-carrierdiffusioncoefficient in theBJTcollector DE minority-carrier diffusion coefficient in theBJTemitter D{T$ densityofinterfacial traps (states/cm-eV) DN electrondifusioncoeficient(cm²/se) $Doxv}$ dielectric displacement in the oxide D{$ hole diffusion coeffi cient (cm²/sec) D\emi dielectricdisplacement inthesemiconductor E emitter E energy 88 electric field electric field in the oxide 8s surfaceelectricfield,electricfieldinthe semiconductorattheoxide- semiconductor interface 8y y-direction component of the electric field $E{$ vacuum level,minimumenergy anelectron must possesstocompletely free itself from a material $EA acceptor energy level $E{B$ binding energy at dopant (donor,acceptor) sites $Ec$ minimum conduction band energy $ED donor energy level $EF$ Fermienergyor Fermilevel $EFM Fermilevel in themetal $EFn$ Fermi level on the n-side of a pn junction $EFp Fermi levelon thep-sideof a pn junction EFS Fermilevel in the semiconductor --- Page 782 --- LIST OF SYMBOLS 759 $EG band gap or forbidden gap energy $EH electron binding energywithin thehydrogenatom $E{$ intrinsic Fermi level $E_{n$ $Eph photon energy (hv) $E{}$ trap or R-G center energy level $Ev}$ maximum valence band energy F force f frequency (Hz) f(E) Fermi function (nn)d field function (see Eq. B.17) $F_{1}2}$ Fermi-Dirac integral oforder1/2 FF fill factor $f\m{xx$ $F{N quasi-Fermilevel(orenergy)forelectrons $F{ quasi-Fermi level (or energy)for holes $f\oq }$ unity beta frequency of a BJT D conductance G gate $G0$ lowfrequencyconductanceofapnjunctiondiode;channelconductancein a J-FET if there were no depletion regions $gc( ) density of conduction band states GD diffusion conductance P8 drain or channel conductance photogenerationratenumberofelectron-holepairscreatedpercmsec gm transconductance $8v(E) density of valence band states h Planck's constant h/2π 一 current; light intensity --- Page 783 --- 760 APPENDIX E ~ a.c. current; √-1in Appendix A $\o$ saturationcurrentinanidealdiode;lightintensityatx=0 IAK anode to cathode current $IBB}$ d.c. base current $BB$ total (a.c.+ d.c.) base current $i_{b}$ a.c. base current $IC$ d.c.collector current ic total (a.c.+ d.c.) collector current $ic$ a.c. collector current $ICBO$ collector to base current when E =0 ICEO collector toemitter current when B=0 $ICn$ d.c.collector current due to electrons $IC_ d.c. collector current due to holes $ID$ d.c. drain current in a field-effect transistor $i_$ small-signal drain current $IDO$ V=0saturationdraincurrent inaJ-FET Idark dark current IDIFF diffusion current (same as the ideal diode current) idiff a.c.component of the diffusion current IDsat saturation drain current $IE$ d.c. emitter current IEn d.c.emitter current due to electrons $IEp d.c.emitter current due to joles $If$ steady-stateforward-bias current $IOF$ effectivediodeforwardsaturationcurrent(Ebers-Mollmodel) $IG gate current in an SCR IL current due to light $IMo→s$ currentduetolectronsdriftinromthemetaltothesemiconducorin MS (n-type) diode $I}$ hole current --- Page 784 --- LIST OF SYMBOLS 761 IPldrift hole current due to drift $I}$ steady-state reverse-bias current $IR-G$ recombination-generation current $IROR$ effective diode reverse saturation current (Ebers-Moll model) $Is$ reverse-bias saturation current in an MSdiode $sc$ short-circuit current in a solar cell $\so→M}$ currentduetoelectronsdriftingfromthesemiconductortothemetalinan MS (n-type) diode √-\$ 一 J,J current density (amps/cm2) Jdrift total current density due to drift JN,JN electron current density JNx,JNy，JNz x,y, and z direction components oftheelectroncurrent density JNldiff electron current densitydue to diffusion JNldrift electron current density due to drift J_， J_$ hole current density J_x，JPy，_$ x, y, and z direction components of the hole current density JPldiff hole current density due to diffusion JPldrift hole current density due to drrift K cathode k Boltzmann constant (8.617×10-5eV/K) K wavenumber(parameter proportional to theelectroncrystal momentum) K.E. kinetic energy KO oxide dielectric constant $K_s$ semiconductor(usually Si)dielectric constant L length of the J-FETor MOSFET channel azimuthal quantum number L' reduced channel length defined in Fig.19.4 $LB minoritycarrierdiffusionlengthintheBJTbase;extrinsicDebyelength LC minoritycarrierdiffusion length in theBJTcollector --- Page 785 --- 762 APPENDIXE LD intrinsic Debye length LE minoritycarrierdiffusionlengthintheBJTemitter Lmin minimum MOSFETchannel lengthyielding long-channel behavior N electronminoritycarrier diffusion length $Ln angular momentum corresponding to the n quantum number holeminority-carrier diffusionlength M carrier multiplication factor m particle mass m magnetic orbital quantum number $m_{0$ electron rest mass m * electron effective mass m * hole effective mass n energy quantum number n electron carrier concentration(numberofelectrons/cm³) $n {}$ heavily doped n-type material $n_{o$ equilibrium electron concentration $n_{}$ defined electron concentration (see Eq.3.36a) NA total number of acceptor atoms/cm³ $No{}$ number of ionized(negativelycharged) acceptors/cm $N{B$ bulk semiconductordoping(NAorNDas appropriate);dopingconcentra- tion in the BJT base nbulk electron concentration in the semiconductor bulk Nc effectivedensitofconductionbandsttedopinconcenttoie BJT collector $nco$ equilibriumelectronconcentrationinthecollectorofapnpBT $NDD$ total number of donor atoms/cm³ $N\r{}$ NE doping concentration in the BJT emitter $nEO$ equilibriumelectronconcentrationintheemitterofapnpBJT $N_}$ number of implanted ions/cm2 --- Page 786 --- LIST OF SYMBOLS 763 $n$ intrinsic carrier concentration $N number of R-G centers/cm³ $Nv}$ effective densityof valence band states hole concentration(numberofholes/cm³);momentum in Appendix A $p^+$ heavily doped p-type material P.E. potential energy Po equilibrium hole concentration $P_{$ defined hole concentration (see Eq. 3.36b) $PBO$ equilibriumholeconcentration inthebaseof a pnp BJT Pbulk hole concentration in the semiconductor bulk $P_{s$ hole concentration at the semiconductor surface (number/cm³) Q general designation for a charge 9 magnitudeoftheelectronic charge(1.60×10-19coul) Q_B excess minoritycarrierchargein thequasineutralbase;bulkordeletion- region charge per unit area of the MOSFET gate QBL QB in a long-channel MOSFET QBS QB in a short-channel MOSFET QF fixedoxidecharge perunitarea attheoxide-semiconductorinterface $Q$ implant-relatedcharge/cmlocatedattheoxide-semiconductorinterface QI QM totalmobileioncharge withintheoxideperunitareaoftheMOSgate QN totalelectroniccharge/cm2intheMOSFETchannel(n-channeldevice) Qo-s chargeperunitarea locatedattheoxide-semiconductorinterface QP excess hole charge Q_s$ total charge in thesemiconductor per unitareaofthegate R ramp rate (see Fig. 16.17) $r{B{，b}$ base resistance $rc{,$ collector resistance RD channel-to-drain resistance in a J-FETor MOSFET $rE,e$ emitter resistance --- Page 787 --- 764 APPENDIX E rj depth of thesource and drain islands in a MOSFET load resistor $^n$ radius of the Bohrorbit corresponding to the n quantum number ro output resistance in the BJT Hybrid-Pi model RP projected range in ion implantation Rs series resistance; sample resistance；source-to-channel resistance in a J-FETorMOSFET rμu feedthrough resistance in the BJT Hybrid-Pi model $rπ$ input resistance in the BJT Hybrid-Pi model S source S probe-to-probe spacing in a four-point probe S spin quantum number T temperature time t tON triggering time in an SCR TR tuning ratio t$ recovery time (pn diode); rise time (BJT) t reverse recovery time (pn diode) $t_s$ storage delay time (pn diode) tsd storage delay time (BJT) n potentialenergyin AppendixA;electrostaticpotentialnormalizedtokT/q in Appendices B-D UD$ drain voltage normalized tokT/q semiconductor doping parameter U{s$ normalized surfacepotential,Uevaluated at the oxide-semiconductor interface $\s}$ sign(±) of Us Uso$ normalized surfacepotential at x=0 in a MOSFET USL normalized surface potential atx=Lin a MOSFET V voltage, electrostatic potential --- Page 788 --- LIST OF SYMBOLS 765 V volume velocity $V{}$ applied d.c. voltage $_{}$ applied a.c. voltage VAK anode-to-cathode voltage $_{B$ defined voltage (see Eq.D.4) beb a.c. base-to-emitter voltage VBBF forward-bias blocking voltage in PNPN devices $bib$ "built-in" junction voltage V{BR reverse-biaspnjunctionbreakdownvoltagereverse-biasblockingvoltge in PNPN devices $V{BSB$ back-to-source voltage $VCBB d.c. collector-to-base voltage VCBO collector-to-base breakdown voltage when E=0 $vce$ a.c.collector-to-emitter voltage VCEO collector-to-emitter breakdown voltage when B=0 $V{D$ d.c.drain voltage $P_{$ drift velocity vector $P_{a$ drift velocity;a.c.drain voltage VDs drain-to-source voltage VDsat saturation drain voltage $v_sat saturation drift velocity VEB d.c. emitter-to-base voltage VEC d.c. emitter-to-collector voltage $VBB$ flat-band voltage VG d.c. gate voltage $g$ a.c. gate voltage $V { d.c.gate voltage applied to an ideal device VGS gate-to-source voltage --- Page 789 --- 766 APPENDIXE $V {$ gate-to-source voltage being appliedto an ideal device $V_}$ junction voltage V_{oc open circuit voltageof a solarcell $V_$ pinch-off gate voltage in a J-FET $V_s$ d.c. source voltage $_s}$ pulsed source voltage V_SO$ surface potential at x=0 in a MOSFET V{SL surface potential at x=Lin a MOSFET $VT$ turn-on voltage $V {}$ ideal device inversion-depletion transition point gate voltage $V{ defined voltage (see Eq. 17.24) $V_s}$ defined voltage (see Eq. 16.36) W depletion width; quasineutral width of the BJT base $W _{}$ quasineutral widthof the N2base in an SCR $W _{}$ quasineutral width of the P3 base in an SCR $W_{BB}$ total widthof the base in a BJT $WD$ drain pn junction depletion width in a MOSFET $^{}M}$ effective depletion width inaMOSFET(see Appendix C) $WN2$ widthofthe N2base in an SCR $W_{3}$ widthof the P3base in anSCR $W_{s}$ source pn junction depletion width in a MOSFET $W_{T}$ MOSdepletion width whenthe semiconductor is biased at the inversion- depletion transition point $x_c$ width of the base in a narrow-base pn junction diode；depth of the MOSFET channel $x _}$ n-side widthof the pn junction depletion region $x_o$ oxide thickness $x{$ p-side widthof the pnjunctiondepletionregion Y admittance --- Page 790 --- LIST OF SYMBOLS 767 $D diff usion admittance Z widthof the J-FETor MOSFET channel α absorption coefficient $αdc common base d.c.current gain $αF$ αF=αdc,forward gain (Ebers-Moll model) reverse gain (Ebers-Moll model) $α base transport factor βdc common emitter d.c.current gain X semiconductor electron affinity x^' ectivemconductorctronafiitinnMrct Xi insulator (oxide) electron affinity Xsi silicon electron affi nity Δ frequency parameterintheexact-chargeCVtheory(seeE.C.) $∆c$ conduction band offsetenergy in a heterojunction $∆E valence band offset energy in a heterojunction Δ_ox$ voltage drop across the oxide $∆semi$ voltage drop across the semiconductor ΔL decrease inthechannllength underabove-pinch-ofconditions Δn ∆n=nndevionintelcroncocntionromitsqulim value ∆nc excesselectronconcentration inthecollectorofa pnpBJT ∆nE excesselectronconcentration in theemitterof a pnp BT $Δn{$ ∆n in p-type material Δp value $∆P{B$ excess holeconcentration in thebaseof a pnp BJT $∆p{n$ ∆p in n-type matrial ΔQ general designation for achange in charge △Q gate change in the gate charge/cm2 ΔQsemi change in the charge/cm2insidethe semiconductor --- Page 791 --- 768 APPENDIX E $∆_{$ straggle in ion implantation $ΔVG difference between the actual device and ideal devicegate voltagerequired to achieve a given semiconductor surface potential $ΔV{}$ change in the threshold voltage due to small-dimension effects $ΔV{s}$ change inthethresholdvoltageduetobackbiasing(specificallyappliedto ideal devices) $△Φ{B$ changeintheMSbarrierheightcausedbySchottkybarrierlowerig ε permittivity permittivityoffree space(8.85×10-14farad/cm) ε0 electrostatic potentialinsidethe semiconductorcomponentofanMOS device Φ_B}$ surface potential-energybarrierheight in an MSdiode ΦBO B barrier height when=0 at the MS interface referencevoltagerelatedtothesemiconductordopingconcentraion $ΦMM}$ metal workfunction $Φ{}$ Φ=Φ−,effective metal workfunctioninanMOSsructre φMsM$ metal-semiconductorworkfunctiondifrenceexpressedinvolts \ox}$ voltage inside the oxide Φs$ semiconductor workfunction φs semiconductor surface potential r four-point probe correction factor γ emitter efficiency γ{M$ normalizedcentroidofmobileion charge in theoxide η=(E-E)/kT;powerconversion efficiency of a solarcell;external efficiency of an LED n_{c ηc=（EF-E)/kT $η_v ηv=（E-EF）/kT λ wavelength of light λG wavelengthof light corresponding to the semiconductor band gap μ0 low-fieldelectronorholemobility;mobilityfit parameter bulk carrier mobility in the bulk of a semiconductor --- Page 792 --- LIST OF SYMBOLS 769 $μn$ electron mobility $p$ hole mobility $\ }$ effective electron mobility $\$ effective hole mobility frequency of light resistivity（ohm-cm);chargedensity（coul/cm) ρion ionic charge density inside the oxide Pox$ charge density in the oxide σ conductivity To defined carrier lifetime(see Eq.6.44) B minority-carrier lifetime in the BJT base TC minority-carrier lifetime intheBJTcollector E minority-carrier lifetime inthe BJTemitter n$ electron minority-carrier lifetime T hole minority-carrier lifetime τ{1$ base transit time time-dependent wavefunction time-independent wavefunction ω angular frequency (radians) --- Page 793 --- --- Page 794 --- Appendix M MATLAB Program Script Exercise 10.2 (BJT_Eband) % BJT Equilibrium Energy Band Diagram Generator % This program plots out the BJT equilibrium energy band diagram %Original version authored by AaronLuft as a course projectforProf.GerryNeudeck % Major revisions by R.F.Pierret WB=1.0e-4;%Total base width in cm;1.0e-4cm=1micrometer close %Constants T=300; % Temperature in Kelvin k=8.617e-5; % Boltzmann constant eV/K e0=8.85e-14; % permittivity of free space (f/cm) q=1.602e-19; % charge on an electron (coul) KS=11.8; %Dielectric constantof Si at 300K ni=1.0e10; % intrinsic conc.of Silicon at 300K EG=1.12; % Silicon band gap (eV) %end constants %General Computations and Manipulations NE=DOPING(1); % Emitter doping and type NB = DOPING(2); % Base doping and type NC=DOPING(3); % Collector doping and type sE = sign (NE); sB = sign (NB); sC = sign (NC); NE=abs(NE); % Emitter doping NB=abs(NB); % Base doping NC=abs(NC); % Collector doping Ei_emitter=[(sE*k*T*log(NE/ni)). (-sB*k*T*log（NB/ni））]; 771 --- Page 795 --- 772 APPENDIX M （（/）**））= (-sC*k*T*log（NC/ni)）]; Vbi=[(sum(Ei_emitter))(sum(Ei_collector))]; svbi = sign (Vbi); Vbi =abs (Vbi); % Depletion widthon emitter sideof EB junction （（N+N)*N)）A*N*/0*S*= % Depletion width on base side of EB junction xBeb=sqrt(2*KS*eO/q*NE*Vbi(1)/(NB*(NE+NB))); % Depletion width on base side ofCB junction xBcb=sqrt(2*KS*eO/q*NC*Vbi(2)/(NB*(NC+NB))); % Depletion widthon collector sideof EB junction xC=sqrt(2*KS*eO/q*NB*Vbi(2)/(NC*(NB+NC))); W=WB-xBeb-xBcb; ifW<0 error('For the given DOPING and WB,the base is totally depleted.') end if(xC>xE) % Adjust the x-axis foroptimum looking plot HIGH_X=1.5; LOW_X=xC/xE; else HIGH_X=xE/xC; LOW_X=1.5; end VMAX=3; % Maximum Plot Voltage plot（[-LOW_X*xE HIGH_X*xC+WB],[0VMAX],); hold on; % EB JUNCTION xlft=-LOW_X*xE; % Leftmost x position xrght=xBeb +W/2; % Rightmost x position x=linspace(xlft,xrght, 200); sVx=-svbi(1)*sE*sB; （0=<x)*（x=>x)*（0*SX*）/（x-x)*N*b-（））*XA=X （0>X2X-=<X)*（O*SX）/2v(x+x）*N*b*O*XA=2×A Vx=Vx1+Vx2; % V as a function of x EF=Vx(1)+VMAX/2-sE*k*T*log(NE/ni); % Fermi level Ec=-Vx+EG/2+VMAX/2; Ev=-Vx-EG/2+VMAX/2; Ei=-Vx+VMAX/2; --- Page 796 --- MATLAB PROGRAMSCRIPT 773 LEc=Ec(1); LEv=Ev(1); LEi=Ei(1); % Plot Vvs x plot（x,Ec); %Ec plot(x,Ev); %Ev plot(x,Ei,'w:'); %Ei plot（[xlft0],[EFEF]'w'); % EF on left :（M[-X[）d % Junction center % CBJUNCTION xlft=-xBcb-W/2; % Leftmost x position xrght= HIGH_X*xC; % Rightmost x position x=linspace(xlft,xrght,200); sVx=-svbi(2)*sC*sB; Vx1=sVx*（Vbi(2)-q*NC.*(xC-x).^2/(2*KS*eO).*(x<=xC)).*(x>=0); （0x2x-=<x)*（*(x+x)*N*b*0*= Vx=Vx1+Vx2;%Vas afunctionofx OFFSET=(Ec(200))-(-Vx(1)+EG/2+VMAX/2); Ec=(-Vx+EG/2+VMAX/2)+OFFSET; Ev=(-Vx-EG/2+VMAX/2)+OFFSET; Ei=(-Vx+VMAX/2)+OFFSET; X=x+WB; % Plot Vvs x plot (x,Ec); %Ec plot(x, Ev); %Ev plot(x,Ei,'w:'); %Ei plot（[O xrght+WB],[EF EF],'w'); % EF on right plot([WBWB],[0.15VMAX-0.15],'w-); % Junction center if(sC==-1) RIGHT='N'; else RIGHT='P'; end if(sB==-1) MIDL='N'; else MIDL='P'; end if(sE==-1) LEFT='N'; --- Page 797 --- 774 APPENDIX M else LEFT='P'; end A=-LOW_X*xE/2; B=WB+((HIGH_X*xC+WB)-WB)/2; text(A,2.5,LEFT); text(WB/2,2.5,MIDL); text(B,2.5,RIGHT); text(x(200),Ec (200),'Ec'); text(x(200),Ei(200),'Ei'); text(x(200),Ev(200),'Ev'); text(x(200),EF,'EF'); REG=[LEFT(1)MIDL(1)RIGHT(1)]; TITLE=[（'Energybanddiagramforthe)(REG)(device')]; title (TITLE); Exercise 11.7 (BJT) and Exercise 11.10(BJT plus) NOTE:Theitalicizedlines intheBJT/BJTplus listingareaddedtotheBJTproramto form theBJTplus program. Subprogram BJT0 is arun-timerequirementof both BJT and BJTplus; subprogram BJTmod must also be present when BJTplus is executed.Constants, material parameters,and the W=W Ebers-Moll parameters are specified or computed in B the BJT0 subprogram.Computations relatedto base-widthmodulation are performed in the BJTmod subprogram. BJT/BJTplus %BJT Common Base/Emitter Input/Output Characteristics %Modified versionofBJTincluding Base-WidthModulationand %Carrier Multiplication %Input Ebers-Moll Parameters BJTO %Limiting Voltages used in Calculation VbiE=kT*log(NE*NB/ni^2); VbiC=kT*log(NC*NB/ni^2); VCBO=50;VCE0=50; VCBO=60*(NC/1.0e16)^(-3/4); m=6;VCEO=VCBO*(1-aF)^(1/m); %ChoiceofCharacteristic and Special Calculations format compact echo on --- Page 798 --- MATLAB PROGRAM SCRIPT 775 %THISPROGRAMCOMPUTESBJT INPUTANDOUTPUTCHARACTERISTICS % Subprograms BJT0 and BJTmod are run-time requirements. % Modify entries in BJT0to change device/material parameters. % Modify axis commands to change plot min/max values. echo off close c=menu(SpecifythedesiredcharacteristicCommonBaseInput' 'CommonBaseOutputCommonEmitterInputCommonEmitterOutput); j=input('Specify number of curves per plot.'); ifc~=2, bw=input('Include base-width modulation?1-Yes,2-No.'); else end ii=2; ifc==4&bw==1, ii=input('Include impact ionization?1-Yes,2-No.'); else end %Calculation Proper for i=1:j, %Common-Base Input Characteristics ifc==1, VCB=-(i-1)*10; VEB=0:0.005:VbiE; jj=length(VEB); ifbw==1, BJTmod %Base-Width Modulation subprogram else end IE=(IFO.*(exp(VEB/kT)-1)- aR.*IR0.*(exp(VCB/kT)-1))*1.0e3; %1.0e3 inthepreceeding equation changesIE units tomA ifi==1, plot(VEB,IE);axis([0.350.8505]); grid;xlabel(VEB(volts)')；ylabel('IE(mA)"); else plot(VEB,IE); end %Common-Base Output Characteristics elseif c==2, IE=(j-i)*1.0e-3; VCB1=2:-0.01:0; VCB2=0:-VCB0/200:-VCB0; VCB=[VCB1,VCB2]; --- Page 799 --- 776 APPENDIX M jj=length(VCB); IC=(aF*IE-(1-aF*aR)*IR0*(exp(VCB/kT)-1))*(1.0e3); ifi==1, plot(-VCB,IC);axis([-VCB0/10VCB001.3e3*IE]); grid;xlabel（-VCB(volts));ylabel(C(mA)'); text(5,1.1e3*IE,IEstep=1mA'); else plot(-VCB,IC); end else end %Common-Emitter Input Characteristics ifc==3, VEC=(i-1)*5; VEB=0:0.005:VbiE; jj=length(VEB); ifbw==1, VCB=VEB-VEC; BJTmod else end IBO=(1-aF).*IFO+(1-aR).*IRO; IB1=(1-aF).*IFO+(1-aR).*IR0.*exp(-VEC/kT); IB=(IB1.*exp(VEB/kT)-IBO)*(1.0e6); ifi==1, plot(VEB,IB);axis([.35.85-520]; grid;xlabel(VEB(volts))；ylabel('IB(μA)'); else plot(VEB,IB); end %Common-Emitter Output Characteristics elseif c==4, IB=(j-i)*2.5e-6; VECA=0:0.01:VCE0/50; VECB=VCE0/50:VCE0/200:VCE0; VEC=[VECA,VECB]; jj=length(VEC); ifbw==1, VEB=0;%Neglect xnEB variation with bias VCB=VEB-VEC; BJTmod else end ifi==1, --- Page 800 --- MATLAB PROGRAM SCRIPT 777 M=1.0./(1-(-VCB/VCBO).^m）; aF=M.*aF; else end IBO=(1-aF).*IFO+(1-aR).*IRO; IB1=(1-aF).*IFO+(1-aR).*IRO.*exp(-VEC/kT); IC=((aF.*IFO-IRO.*exp(-VEC/kT)).*(IB+IBO)./IB1+IR0-aF.*IRO)*(1.0e3); ifi==1, jA=length(VECA); plot(VEC,IC);axis([0VCE002.5*IC(jA)]); grid;xlabelVEC(volts));ylabel（'IC(mA)); text(5,2*IC(jA),IBstep=2.5uA'); else plot(VEC,IC); end else end hold on end hold off BJTO %BJT Constants and Ebers-Moll Parameters (subprogram BJT0) %Universal Constants q=1.602e-19; k=8.617e-5; e0=8.85e-14; %Device/Miscellaneous Parameters A=1.0e-4; %Ain cm2 WB=2.5e-4; %WBincm T=300;kT=k*T; %Material Parameters ni=1.0e10; KS=11.8; NE=1.0e18; NB=1.5el6; NC=1.5el5; %Mobility Fit Parameters NDref=1.3e17;NAref=2.35e17; μnmin=92;μpmin=54.3; μn0=1268;μp0=406.9; an=0.91;ap=0.88; --- Page 801 --- 778 APPENDIX M μE=μnmin+μn0./(1+(NE/NDref).^an); μB=μpmin+μp0./(1+(NB/NAref).^ap); μC=μnmin+μnO./(1+(NC/NDref).^an); TauE=1.0e-7; TauB=1.0e-6; TauC=1.0e-6; DE=kT*μE; DB=kT*μB; DC=kT*μC; LE=sqrt(DE*TauE); LB=sqrt(DB*TauB); LC=sqrt(DC*TauC); nEO=ni^2/NE; pBO=ni^2/NB; nCO=ni^2/NC; %Ebers-Moll ParameterComputation(W=WB) W=WB; fB=(DB/LB)*pBO*(cosh(W/LB)/sinh(W/LB)); IFO=q*A*((DE/LE)*nEO+fB); IRO=q*A*((DC/LC)*nCO+fB); aF=q*A*(DB/LB)*(pBO/sinh(W/LB))/IF0; aR=q*A*(DB/LB)*(pBO/sinh(W/LB))/IR0; BJTmod %Base-widthmodulation-includedcalculationofEbers-Moll parameters %Subprogram BJTmod xnEB=sqrt((2*KS*eO0/q)*(NE/(NB*(NE+NB)))*(VbiE-VEB)); xnCB=sqrt((2*KS*eO/q)*(NC/(NB*(NC+NB)))*(VbiC-VCB)); W=WB-xnEB-xnCB; fB=(DB/LB)*pBO*(cosh(W/LB)./sinh(W/LB)); IFO=q*A.*((DE/LE)*nEO+fB); IRO=q*A.*((DC/LC)*nCO+fB); aF=q*A*(DB/LB)*(pBO./sinh（W/LB))/FO; aR=q*A*(DB/LB)*(pB0./sinh(W/LB))./IR0; Exercise 16.5 (MOS_CV) %LOW and/or HIGH-frequency p-type MOS-C C-V CHARACTERISTICS %Subprogram CVintgrd is a run-time requirement. %Initialization and Input format compact close clear --- Page 802 --- MATLAB PROGRAM SCRIPT 779 S=menu('ChoosethedesiredplotLow-fC-V'High-fC-VBoth); NA=input('Please input the bulk doping in/cm3,NA='); xo=input('Please input the oxide thickness in cm, xo='); xmin=input('Specify VGmin(volts）,VGmin='); xmax=input('Specify VGmax(volts),VGmax='); global UF %Constants and Parameters e0=8.85e-14; q=1.6e-19; k=8.617e-5; KS=11.8; KO=3.9; ni=1.0e10; T=300; kT=k*T; %Computed Constans UF=log(NA/ni); LD=sqrt((kT*KS*e0)/(2*q*ni)); %Gate Voltage Computation US=UF-21:0.5:UF+21; F=sqrt(exp(UF).*(exp(-US)+US-1)+exp(-UF).*(exp(US)-US-1)); （*（*O）/（Ox*S)*(Sn)Sq/Sn)+S∩)*L=A %Low-frequency Capacitance Computation DENOML=exp(UF).*(1-exp(-US))+exp(-UF).*(exp(US)-1); WL=(US./abs(US)).*LD.*(2*F)./DENOML; :（（0X*S）/（TM*O）+I）/=T %High-frequency Capacitance Computation if s\=1, jj=length(US); nn=0; for ii=1:jj, if US(ii)<3, elseif nn==0, INTG=QUAD('CVintgrd3,US(ii),0.001); nn=1; else INTG=INTG+QUAD(CVintgrd,US(ii-1),US(i),0.001); end ifUS(ii)<3, cH(ii)=cL(ii); else （DLNI*（）dx*（）)/(-)Sn-(())x）=P --- Page 803 --- 780 APPENDIXM DENOMH=exp(UF).*(1-exp(-US(i)))+exp(-UF).*((exp(US(i))-1)(1+d)); WH=LD.*(2*F(ii))./DENOMH; （（OX*S）/（HM*O）+I)/OI=（）H end end else end %Plotting the Result ifs==1, plot(VG,cL); elseif s==2, plot(VG,cH); else plot(VG,cL,--',VG,cH); text(0.8*xmin,.17,---Low-f'color,,); text(0.8*xmin,.12,__High-f'color',[1,0,]); end axis([xmin,xmax,0,1]); text(0.8*xmin,.27,['NA=',num2str(NA)'/cm3']); text(0.8*xmin,.22,['xo=',num2str(xo),'cm']); xlabel('VG(volts));ylabel('C/CO');grid CVintgrd function [y]= cvintegrand(U) global UF F=sqrt(exp(UF).*(exp(-U)+U-1)+exp(-UF).*(exp(U)-U-1)); y=(1-exp(-U)).*(exp(U)-U-1)./(2*F^3); --- Page 804 --- INDEX A binding energy, 37 Abrupt junction, 309 donor in silicon, 35 Absorption coefficient,110,111,112 Ar annealing, 659 Acceptor-like interfacial traps, 665 Arsenic (As) Acceptors,35-40,58 binding energy, 37 binding energy, 37 donor in silicon, 35 definition, 40 Asymmetrical junction,250,251 ionized,36,58 Auger recombination,106,107,108 Accumulation,568,569,570,571,576-577, Avalanche breakdown,264-268,414 599 Avalanche multiplication,265,266,414–417, Admittance, pn junction small signal,301- 419 326 Avalanche photodiodes,355-356 Aluminum (Al) Avalanching,264,268 acceptor in silicon, 35 binding energy, 37 B contact metal,499 Backbiasing,680-681 energy band diagram,646 Backcontact,564,680 gate material, 563 Ballistic transport,701-702 interconnect metal, 500 Band alignment, 429,430 postmetallization annealing,666 Band bending, 89-93 shielding, 669 Band gap,27-28,31-32,354 spiking, 499,500 Band gap wavelength,348,349,362 workfunction difference,649 Bands. See Energy band diagram AIAs,348,362 Barrier heights.See Electron affinity,Metal AlGaAs workfunction alloy semiconductor, 3, 4,5 Base,371 band alignment, 430 Base transit time,454,473 in HBTs,430-432 Base transport factor,383,395,399,400 inLEDs,363,364,365-366 Base width modulation,410-412,470 inMODFETs,529,707-709 bcc unit cell,8,9 AlGaN,366 BESOI,70,707 AlInAs,430,709 Bias-temperature(BT) stressing,653,654, AlInGaP,363,64 655,658,669 Alkali ions. See Sodium BiCMOS,426,427 Alloys. See Semiconductors, alloys Bilateral diode,44 Aluminum gallium arsenide. See AlGaAs Bilateral SCR,474 Amorphous, 6 Bipolar junctiontransistor.See BJT Amphoteric dopant, 39 BJT Angular momentum, 24 active biasing mode,374,375 Annealing,659,660,665-667,669,670 amplification by,381,382 Anode (in SCR),463 base,371,380 Antimony (Sb) base carrier distribution,394,398-399 781 --- Page 805 --- 782 INDEX base transit time, 454 small-signal equivalent circuit,443-449 base transport factor,383,395,399,400 static characteristics,389-433 base width modulation,410-412 storage delay time, 457 buried layer, 378 transconductance, 446 carrier multiplication and feedback, 415, transient (switching)response,449–458 416,419 turn-off transient,456-457,458,459 charge control relationships,452–454 turn-on transient, 454-456,458,459 circuit symbol, 372 VCBO,408,410 collector,37,380 VCEO,41010$ collector current,396,404 Blackbody radiation,733-735 comb structure, 422 Block charge diagrams,566-567 common base,373 a.c. probed MOS-C,586 common base d.c.current gain,383-384, deeply depleted MOS-C,597 396,399,400,42,46 static biasing states,569,570,575 common base input characteristics, 408 Blocking voltage, 464 common base output characteristics, 408 Body effect. See Back biasing common collector,373 Bohratommodel,24-25,735-737 common emitter, 373 Bohrradius,735,743,744 common emitter d.c.current gain,384,396, Bonding model,26,27,29-30,35-36 399,400 Boron (B) common emitter input characteristics,409 acceptor in silicon,35 common emitter output characteristics,409 binding energy, 37 current crowding,420,41–422 and threshold adjustment, 679 cutoff biasing mode,374,375 Breakdown,260,263-270 Early effect, 410 avalanche,264-268 Ebers-Mollequations,403-407 critical electric field,266 Ebers-Moll model,404 curvature effect,267,356 electrostatics,377,378 edge,267,356 emitter,371,380 in BJTs,414 emitter current,39,404 in Schottky diodes,490 emitterefficiency,382,395,399,400,422 in SCRs,466,467,470 energy band diagram,377,378-380 Zener, 268-270 fabrication,374-376,378 Breakdown voltage,263 fall time,457 Buffer layer,355,705 forward gain, 403,404 Built-in charge. See Fixed charge graded base,423-424 Built-in voltage,200,201,203-206 Gummel plot,424-425 Bulk,564,573,49 hybrid-pimodel,446-449 Bulk charge, 625 ICBO,383,405-406,422 Bulk charge theory,625-628 ICE0,384,422 Buried-channelMOSFET,704 intrinsic, 421 Buried layer, 377 inverted biasing mode,374,375 notation,371-372,390 C performance parameters,382-384,395- Capacitance 396,398-400 diffusion, pn diode,315–323 polysilicon emitter,426-429 junction,BJT,447,448,49 punch-through,412-414 junction, pn diode, 301–308 quasineutral base width, 378 MOS-C,584-598 reverse gain, 403,404 oxide,587,622 R-G current, 422 Schottky diode,493-496 rise time, 457 semiconductor, 587 saturation biasing mode, 374,375 Capacitance-voltage characteristics.See C-V Schottky diode clamped,458,497 characteristics series resistance, 421 Capture coefficients, 114 --- Page 806 --- INDEX 783 Carrier properties, 32–40 process flow, 166–173 Carriers,23,29-31 sample die, 173 charge, 32 Crystal growth,16-18 concentration calculation,59-61 silicon purification, 17 concentration temperature dependence, 60, single crystal formation,17–18 65-66 Crystal lattices,8-11.See also Unit cell distribution plots, 47 Crystalline,6 effective mass,32–34 CuInSe,360 electrons,29,30 Current crowding,420,421-422 equilibrium distribution, 46–48 Current density holes,30,31 carrier,99,134 majority,40 definition, 77 minority, 40 diffusion,98-99 Cathode (in SCR),463 drift,77-79 CCD,527,58,598 relationships summary, 137 CdS photoconductor, 147 total steady state, 99 CdTe,360 Cutoff frequency, 633-634 Channel,52,613 C-Vcharacteristics,MOS-C,584-598 Channel conductance. See Drain conductance deep depletion,596-598 Chargecontrol approach,282-284,333, delta-depletionanalysis,590-591 452-454 doping dependence,52-593 Charge coupled device. See CCD exact charge relationships,753–754 Charge density, 198 experimental,585,596 Charge neutrality relationship,57–58 high frequency,584,585,588,590,591, Chemical vapor deposition(CVD),164,165 595,596,753,754 Chlorine neutralization,656,657 low frequency,584,585,588,590,591,595, Chromatic dispersion,353 753,754 Chrome(Cr),649,66 measurement of,584,595,596 CMOS,528,529 oxide thickness dependence,592,594 Collector,371 qualitative theory, 584–588 Compensated, 60 quasistatic measurement technique, 595 Compositionally graded,363,430-431, temperature dependence, 592, 595 705 theoretical,591,593–595 Compound semiconductors, 3, 4, 5 C-Vcharacteristics,pn diode,305-313 Compton effect, 738 CVD (chemical vapor deposition),164,165 Concentration formulas Czochralskimethod,17-18 compensated semiconductor, 60 doped semiconductor, 60 D electrons,51,52,53,59,60 D-MESFET,551,552 holes,51,52,53,59,60 Danglingbonds,665,66,667 intrinsic semiconductor,34,60 de Broglie hypothesis,738 Conductance Debye length drain (in J-FET),548,549 extrinsic,750 drain (in MOSFET), 632 intrinsic,750,751 diffusion,315–321 Deep depletion,596-598,599 reverse bias diode,313-315 Defect, 26,27,105 Conductionband,27,28 Degenerate quantum states, 744 Conductivity,85,117-118 Degenerate semiconductor, 52 Conductivity modulation,281 Delta-depletion Continuity equations,121-122 approximation, 577,578 Core, atomic, 25 C-Vanalysis,590-591 Core electrons, 25 electrostatics solution,576-579 Covalent bonding,26 Delta-function,577,578,581,590,658,659 CPU (central processing unit) Density of states,41-42,47 --- Page 807 --- 784 INDEX Depletion,568,569,570,599 Drain conductance Depletionapproximation,206-209 J-FET,548,549 Depletion-layer capacitance. See Junction MOSFET,632 capacitance DRAM,526,527,598,692 Depletion mode MOSFET,676 Depletion region, 200 current,76-79 Depletion width,578 definition-visualization,75-76 and deep depletion,597 velocity,76,77,78 MOS-C,579 Drive-in,153,154,155 MOS-Ceffective,753 Dry etch, 169 pn junction, 214, 216 Dual-gate SCR,474 DIAC,474,475 Dynamic random access memory. See DRAM Diamond, band gap,31 Diamond lattice, 9-10 E Die,166,173 Early effect,410 Dielectricconstant,37,198 Ebers-Moll equations,403-407,446 Dielectric displacement,580,581,662 Ebers-Mollmodel, 404 Diffusion E-k plots,108,109,361 current,98-99 E-MESFET,551,552 of carriers,94-104 Eddy current, 88,89 masks,152 Effective density of states,51 solid state, 152–155 Effective mass,32-34 Diffusionadmittance,315-323 Effective mobility,618-620,634,635 Diffusion capacitance,317,320 mathematical expression,618 Diffusion coeffi cients,98,99,101-103 measurement of,634,635 Diffusion conductance,317,320 position dependence, 619 Diffusion current. See alsoIdealdiode voltage dependencies, 619,620 equation Einstein relationship,11-103 in pn junction diode, 273 Electric field,89,90,91,92 in Schottky diode,491,492 Electron Diffusion lengths,131-132 capture coefficient,114, 115 Dipole layer, 213 carrier, 29,30 Direct semiconductor,108,354,361,362 charge,32 DMOS,703 concentration formulas,51,52,53,9,60 Donor-like interfacial traps663,64,665 continuity equation, 121 Donors,35-40,58 crystal momentum,108,109 binding energy, 36–37 diffusion current,95,98-99 definition, 40 diffusion equation,122-123 energy band model, 38 diffusion length, 131 ionized,35,58 drift current,78,79 Dopant effective mass,32–34 binding energies, 37 Einstein relationship,102 definition, 40 equilibrium distribution,46–48 in GaAs, 39 intrinsicconcentration,34,53-57 in Si, 35 lifetime,115,116 Doping, 35-40 mobility,78,79–84 Doping parameter, 749 quasi-Fermi level, 133 Doping profile,196,197 rest mass,32 Doping potential() Electron affinity,478,565 definition,572,573 Electron crystal momentum,361 properties, 573, 574 Electronic quantum states Dose, 156 hydrogen atom,24,741-744,45 Double-diffused MOSFET.SeeDMOS multi-electronatoms,744,745-748 Drain,526,5,611 silicon,25,747,748 --- Page 808 --- INDEX 785 Electron volt (eV),24,736 Fermi energy. See Fermi level Electrostatic potential Fermi function,42-46 basic relationships, 90, 91 plots,43,46 inside MOSFET,572, 573 temperature dependence, 45–46 normalized,749 Fermi level surface,572,573,574 determination of position, 61-64 Electrostatics,195.See also specific device. equilibrium position independence, 100 Emitter efficiency,382,395,399,00, in Fermi function,42-45 422,432 intrinsic, 48,49,62 Emitter,371 quasi-,132-136 Energy band diagram (ormodel) Fermi-level pinning, 498 band bending, 90,100 Fick's law,98 basic model,26-29 Field effect, 525 BJT,377,378-380 Field emission, 499 different materials, 31 Fieldoxide,677,678 donor level included,38 Fillfactor,356,357 dopant action visualized,38 Fixed charge,658-661 Fermi levelsadded,49 Flat,12 HBT,431 Flatband,478,570,599,647 ideal MOS structure, 566 Flat-band voltage,672,675 insulator (surface-included),5,646 Forbidden gap. See Band gap metal (surface included),478,565,646 Four-point probe, 87-89 p-i-n photodiode, 353 Freeze-out, 65,66 quasi-Fermi levels added,132,135 real MOS structure, 646 G R-G visualized, 106 Gallium (Ga) semiconductor (surface included), 478, acceptor in silicon, 35 565,646 binding energy, 37 Enhancement mode MOSFET,676 Gallium arsenide (GaAs) Epilayer,164,375 and LEDs,362 Epitaxial layer overgrowth(ELO),705,706 and HBTs,430-432 Epitaxy, 164-165 andMESFET,550-552,556 Equations of state, 120-123 andMODFET,529,707-710 summary, 137 and solar cells,357,358,60 Equilibrium, 23 ballistic transport in, 701 Equivalent circuit band gap energy, 31,32 BJT,444,447 band gap wavelength, 348 Ebers-Moll, 404 carrier effective masses, 34 hybrid-pi,447 compound semiconductor, 3, 5 J-FET,547 doping of, 39 MOS-C,586 intrinsic carrier concentration,34,54 MOSFET,631 lattice constant,10 pn diode,302,316 MS contacts, 498 two-transistor, 467 semi-insulating(S.I.), 430 Evaporation, 162,163 temperature considerations,66, 67 Excess charge. See Stored charge unit cell,10,11 Exciton,362 GaAsP,362-365 Extrinsic Debye length,750 Gain Extrinsic semiconductor, 40 common base d.c.current,383-384,36, Extrinsic temperature region,65,66 399,400,425,426 common emitter d.c.current,384,36, F 399,400 fcc unit cell, 8,9 GaN, 363 Fermi-Dirac integral, 50 GaP,348,362,363,34,365 --- Page 809 --- 786 INDEX Gate,463,526,51,611 Hot-point probe, 97-98 Gate oxide,677,678 Hybrid-pimodel,446-449 Gate voltage relationship, ideal device,580- Hydrogen anneal,665-667,670 582 Hydrogen atom,24,744-745 Generation. See Recombination-generation Hyperabrupt junction,305,306 Germanium (Ge) Hyperabrupt varactor diode,307,308 avalanche photodiode, 356 band alignment with GaAs,430 I band gap energy, 31,32 Ideal diode equation,249,235-260 carrier effective masses, 34 boundary conditions,24–246 elemental semiconductor,3 law of the junction, 245 indirect semiconductor,362 qualitative derivation,235-241 intrinsic carrier concentration,34 quantitative derivation,247–249 isolated atom,25,748 saturationcurrent(),250-254,258-259 lattice constant,20 solution strategy, 241–246 solar cell substrate,362 Ideality factor,276,277,321 unit cell,9 IEEE current convention,371 Gettering,116,656 IGFET.See MOSFET Gold (Au),105,116 Impact ionization, 264 Grading constant, 223 Impurity atoms. See Dopant Gradual channel approximation,539,552,554 Indirect semiconductor,108,354,361,362 Grain boundaries,427 Indium (In) Grains,427 acceptor in silicon, 35 GTO SCR,471 binding energy, 37 Gummel plot,424,425 Induced charges,668-670 InGaAs,354,355,360,430,709 H InGaAsP,355,360 Hardened oxides,69 InGaN,366 HBT,429-432 Ingot, 17,18 HEMT.See MODFET InP,348,356,360,430 Heterojunction, 429 Instability,653,654,669-670 Heterojunctionbipolartransistor.See HBT Insulator. See also Silicon dioxide High-level injection,279-281 energy band diagram,31,565,646 Holes perfect, 564 capture coefficient,113,114,115 Interfacial traps,62-667 carrier,29-31 acceptor-like, 665 charge, 32 density of,665,667 concentration formulas,51,52,53,59,60 donor-like,663,664,665 continuity equation, 121 effect on C-Vcharacteristics,662,663, diffusion current,95,98–99 664,665 diffusion equation, 122-123 effect onMOSFETcharacteristics,674,675 diffusion length,131,132 electricalmodel,662,663 drift current,77–78 physical model,65,666 effective mass,33,34 Intrinsic carrierconcentration,34-35,53-57 Einstein relationship,102 Intrinsic Debye length,750,51 equilibrium distribution, 46–48 Intrinsic Fermilevel,48,49,62 intrinsic concentration,34,53–57 Intrinsic semiconductor,34, 40,60 lifetime,115,116 Intrinsic temperature region,65,66 lifetime measurement,116-120 Intrinsic transistor, 421 mobility,78,79-84 Inversion,569,570,599 quasi-Fermi level, 133 Inversion layer,578,613 Hot carrier diode,497 Ion implantation,155-159,679 Hot-carrier effects,700-702 Ionization. See Acceptors andlor Donors --- Page 810 --- INDEX 787 Ionizing radiation, 668-669 Lightlydoped drain. See LDD transistors Isoelectronic trap, 364,365 Linearly graded junction,197 Lithium (Li),alkali ion,654 J Lithography,159-162 Junction capacitance,301-308 Low-level injection, 112 Junctionfieldeffect transistor (J-FET),525, LPCVD,164 526,530-550 a.c. response,54–550 M basic device structure,526,530 Mask,159,161 channel,532 Mass of a carrier. See Effective mass channel narrowing,532,533 Matrix mechanics, 739 drain conductance,548,549 MBE,429,529 equivalent circuit, 547 Mean free time, 79 modern device structure,531 MESFET,550-556 pinch-off,532,533,534 Metal-insulator-semiconductor(MIS), pinch-off gate voltage,536 563,671 qualitative theory ofoperation,531–536 Metallurigical junction,195 quantitative ID-VDrelationships,537-547 Metal-oxide-semiconductor capacitor. See saturation,534,535 MOS-C small-signal equivalent circuit,547 Metal-oxide-semiconductor field effect tran- small-signal parameters (table of),549 sistor. See MOSFET source/drain resistance,541,543,545-547 Metals,31-32 square-law relationship,541, 545 Metal-semiconductor contact. See MS contact transconductance,548,549 Metal-semiconductor field effect transistor. waterfall analogy,55,536 See MESFET Metal-semiconductor workfunction difference, K 645-650 Kinetic energy Metal workfunction,478,565 of carriers,89,90,92,93 Millerindices,12-16 electron in H-atom, 736 directions, 15 equivalent planes, 14 L indexing procedure, 13, 14 Latch-up,707 notation summary, 15 Lattice6-11.See alsoUnitcell overbar, 14 Latticeconstant,8,9,354 Minority carrier diffusion equations,122-123 Lattice-matched system, 429 Minoritycarrierdiffusion lengths,131-132 Law of the junction, 245 Minoritycarrier lifetimes,115,116 LDD transistors, 700,702 MIS structures,563,671 LED,347,361-368 MMIC,551 AlGaAs,365-366 Mobileions,653-658,670,671 AlInGaP,366 effectonMOSFETcharacteristics,653 basic structure and operation,361-362 model,655 GaAs0.6P0.4,362-364 normalized charge centroid,60,671 GaAsP:N,364-365 Mobility,78,79–84 GaN,366 doping dependence,80,81 GaP:N,364-365 relationship to scattering, 79 GaP:Zn-O,365 temperature dependence,81–83 packaging,366-367 units,79 photon extraction, 367-368 MODFET,528-530,707-710 SiC,366 Modulation doped, 707 Light-activated SCR (LASCR),471 Modulation doped field effect transistor. See Light-activated switch (LAS),471 MODFET Light emitting diode. See LED Molecular beam epitaxy. See MBE --- Page 811 --- 788 INDEX Momentum,107-109 N MOS-C Narrow-base diode,284-288 biasing of, 567-571 N2 annealing, 659,660 C-Vcharacteristics,584-598,753-754 Nearest neighbors,10,11,26 energy band diagram,56,646 Negative bias instability,669-670 ideal,564 Neutron activation, 655 physical device,53,564 NMOS,676,677,678,679 MOSFET,526,527,611-637,674-684, Nondegenerate semiconductor, 52,63 691-707 Notch,12 a.c.response,630-636 np product, 55 action ofdrainbias,613-615 n-type material,40,48–49 action of gate bias,613 n+-material,63,166 basic structure,611,612 bulk-charge theory,625-628 0 buried-channel,704 Off-state,327,443,450 channel,613 Ohmic contact,209,480-483,498–500, charge-sheet theory,628-630,755-756 564 cutoff frequency, 633–634 On-state,327,443,450 C-Vcharacteristic,634-635 Overlap capacitance. See MOSFET, gate-to- depletion mode, 676 drain capacitance effect of nonidealities,674-675 Oxidation,149-151,152,153 enhanced mobility structure, 705 dry, 150 enhancement mode,676 wet, 150 equivalent circuit, 631 growth curves, 152,153 exact-charge theory,628-630,755-756 pyrogenic,151 flat-band voltage,675 Oxidation triangle,660 gate-to-drain capacitance,631,632 Oxide capacitance,587,622 gate-to-source capacitance,631,633 Oxide dielectric constant, 581 linear region of operation,615 Oxide charging,700,702 narrow width, 697-698 Oxygen(O),149,150,151,365 pinch-off,613 qualitative theoryof operation,611-616 P saturation, 615 P. See Phosphorus short-channel,694-697 Parallel plate capacitor, 303 small-signal characteristics,634-636 Pauli exclusion principle,27,746 small-signal equivalent circuits,630- PECVD,164 633 Periodic Table (abbreviated）,5 small-signal parameters (table of),633 Perturbation,112,113 square-law theory,620-625,626,627 PHEMT,709 subthreshold,629,630,692 Phonon,109 threshold voltage,617,674–681 Phosphorus (P) transconductance, 632 binding energy, 37 triode region of operation, 615 donor in silicon, 35 MOST.See MOSFET inMOSFET,655,656,679 MS contact. See also Schottky diode Phosphorus stabilization,655-656,657 energy band diagram,479,480,482 Phosphosilicate glass,655,66,657 Fermi-level pinning, 498 Photo-BJT.See Phototransistor ideal,477-483 Photoconductor,116,147 ohmic,480-483,498-500 Photocurrent, 350 rectifying,480-483,497–498 Photodetector, 347 MS diode. See Schottky diode Photodiode,349-356 Multiplication factor,265,266,267-268, avalanche,355-356 414,416,469,470 I-Vcharacteristics,350 Mutual conductance. See Transconductance p-i-n,352,355 --- Page 812 --- INDEX 789 pn junction,349–352 turn-on transient,38–341 spectral response,350-352 Zener process, 28–270 Photogeneration,106,107,110-112 pn junction photodiodes,34–352 Photon, 109 PNPN diode,463,472 Photoresist,159,161,162 Point defect,26,27 Phototransistor,416,417 Poisson's equation,197–198 Pinch-off,532,533,534,613 Polycrystalline, 6 Pinch-off gate voltage,536 Polycrystalline Si (polysiliconpoly) p-i-n photodiodes,352–355 polyemitter BJT,426-429 Planes, crystal. See Miller indices deposition of, 164 PMOS,676,678,679 MOS gate material,168,563,649,650,679 pn junction Si crystal production,17 abrupt, 309 PolysiliconemitterBJT,426-429 asymmetrical,250,251 Polysilicon MOS gate,563,649,650,679 built-in voltage,200,201,203–206 Postmetallizationannealing,665-667 depletion approximation,206-209 Potassium(K),alkali ion,654 depletion region, 200 Potential. See Electrostatic potential depletion width,214,216 Potential energy electrostatics qualitative solution,198-202 of carriers,89,0,91 energy band diagram, 199,211 electron in H-atom,736,742 hyperabrupt, 05, 06 in Schrodinger equation, 739 linearlygraded electrostatics,22-226 Predeposition,153,154,155 space charge region, 200 Primitive unit cell,8 step junction electrostatics,209–218 Principleofdetailed balance,114 pn junction diode Profiling,312,313 avalanching,264-268 Programmable unijunction transistor,475,476 breakdown,260,263–270 Projected range, 158 breakdown voltage, 23 Pseudo-channel, 677 carrier concentrations,255-256,257 Pseudomorphic,429,704,705,709 carrier currents,254–255 p-type material, 40,48-49 conductance,reverse bias,313-315 p+-material, 63 C-V,303,305-313 Punch-through,287,288,412-414,698,699 depletion width,214,216 PUT,475,476 diffusionadmittance,315–323 diffusion capacitance,315–323 Q diffusion conductance,315-321 Quantizationconcept,23-25,733-739 high-level injection,279–281 Quantum mechanics,25,33,733–748 idealityfactor,276,277,321 basic postulates, 739–740 1-Vcharacteristics,experimental,261,262 degenerate states, 744 junction capacitance,301–308 expectation values, 740 junction voltage,278,279 quantum numbers,742,743,746,47 narrow-base diode, 284–288 spin, 744 punch-through,287,288 Quantum numbers,742,743,74,747 recovery time, 327,328 Quasi-Fermi levels,132-136 reverse recovery time,37,328 Quasineutral base width, 378 reverse injection, 330 Quasineutral regions, 242 R-Gcurrent,270-277 Quasistaticassumption,305,340,341,444, saturation current(0),250-254,258-259 548,631 series resistance,278-279,301,302 Quasistatic,definition, 305 small-signal admittance,301-326 Quasistatic technique, 595 small-signal equivalent circuit,302,316 storage delay time,327,328,33–338 R stored charge,255,256,282,315,329,330 Radiation damage,668,669 turn-off transient,327-338 Radiation effects,668-669 --- Page 813 --- 790 INDEX Radiation hardening,669 circuit symbol, 465 Rayleigh-Jeans law,733,734 di/dt burnout, 472 Recombination-generation,105-120 du/dt effect, 473 Auger,106,107 gate turn-off, 471 band-to-band, 105 I-Vcharacteristics,465 definition-visualization,105-107 light activated, 471 R-Gcenter (indirect),105,112-115 operational theory,465–470 statistics,110-115 shorted cathode,41–472 Recovery time,37,328 structure,463–465 Rectifying contact.See MScontact triggering time, 473 Refractive index,367,368 two transistor model, 467 Refractory metals, 499 Seed crystal, 17 Regenerative process,415,467,699–700 Self-aligned gate, 633 Relative eye response, 348 Semiconductors Resistivity, 85-89 alloys,3,4,5,354 Reverse recovery time,327,328 composition, 3–5 R-G. See Recombination-generation compounds,3, 4, 5, 354 R-Gcenter,105,10,107 degenerate, 52 R-G current direct, 108 inBJT,422 elemental, 3, 4 in MS diode,491,492 extrinsic, 40 in pn diode, 270–277 indirect, 108 Richardson's constant, 489 intrinsic,34,40,60 list of, 4 S material classification,31–32 Saturation nondegenerate, 52 in BJT,374,375 purity, 5 in J-FET,534,535 structure, 6 in MOSFET,615 uniformly doped, 57 in pn junction diode,250-254,258–259 Semiconductor workfunction,478,564,565 in Schottky diode, 490 Sheet resistivity,89 Saturation(drift)velocity,78,553,554-555, Shockley diode,463 700 Shockley equation. See Ideal diode equation Scattering,75,79,81,83 Short-channel Schottky barrier lowering,490,492 MESFET,552-556 Schottky diode MOSFET,691-697 a.c.response,493–496 saturated velocitymodel,554-555 built-in voltage, 483 two region model,555-556 capacitance,493–496 variablemobility model,553-554 depletion width, 486 Shorted cathode SCRconfiguration,471-472 electrostatics,483–487 SiC,67,348,32,63 I-Vcharacteristics,487-493 SiGe,429,704-705 real (nonideal),497–498 Silicides,499 saturation current, 490 Silicon (Si) Schottky barrier lowering,490,492 band gap energy, 31, 32 thermionic emission current, 488 carrier effective masses, 34 transient response,496-497 carrier mobility, 80,82 SchottkydiodeclampedBJT,458,497 crystal growth,16-18 Schrödinger wave equation dopant site binding energy,37 time-dependent, 739 dopants, 35 time independent, 742 elemental semiconductor, 3, 4 SCR ingot, 17,18 blocking voltage, 464 intrinsiccarrierconcentration,34,54,55-57 circuit operation,470-471 isolated atom,25,748 --- Page 814 --- INDEX 791 lattice constant, 10 Step junction, 197 minority carrier lifetimes,116 Step recovery diode,338 MScontacts,498,499,500 Storage delay time,327,328,333-338,457 polycrystalline,17, 164 Storage time,327 polysiliconemitter BJT,426-429 Storedcharge,255,256,282,315,329,330,455 polysilicon gate,168,563,649,60,679 Straggle, 158 resistivity, 86 Strong inversion, 59 unit cell,9,10 Substrate-bias effect. See Backbiasing wafer, 12 Subthreshold transfer characteristics,629, Siliconcontrolledrectifier.SeeSCR 630,692 Silicon controlled switch(SCS),474 Surface potential,572,573,574 Silicon dioxide (SiO_2) Surface scattering,618,619,620 band gap, 31 Surface states. See Interfacial traps energy band diagram,31,646 growth of and the fixed charge,659 T insulator in MOS structures,563,564 Temperature dependence mobile ion contamination, 654 acceptor/donor action, 38–39 radiation hardened, 669 carrier concentrations, 65-67 Si-SiO2interfacial traps,651,663,666 Fermi function,45-46 stabilization of,655-657 intrinsic carrier concentration,54 thermal oxidation product,149,150 mobility, 81-83 SIMOX,706,707 Thermal motion,76 Small-dimension effects,691-702 Thermionic emission,488,499 Small-signal equivalent circuit.See Equivalent Threshold voltage circuit adjustment,678-681 Snap back diode, 338 definition, 617 Sodium (Na),653,654,655,656 measurement of, 634 SOI(silicon-on-insulator),159,705-707 narrow width modification,697-698 Solar cell,347,356-361 relationships,617,675,676 amorphous-Si, 360 short-channel modification,694-697 antireflection coating, 359 Thyristor, 463 concentrator,360,361 Ti,499,500 fill factor,356,357 TiN,500 fingers,358 Titaniumsilicide（TiSi2),499,500 1-Vcharacteristics,357 Total internal reflection,368 light trapping, 359 Transconductance open circuit voltage,35,357 BJT,446 power conversion efficiency,357–360 J-FET,548,549 shadowing, 359 MOSFET,632 short circuit current,356,357 Transit time, 454,473 textured surface, 359 Trapped charge,651,668 thin film,360,361 TRIAC,474,475 Solar spectral irradiance,358 Tuning ratio,307 SOS (Silicon-on-sapphire), 705 Tunneling,268,269,498,499,669 Source,526,531,611 Turn-onvoltage.See Threshold voltage Space charge region, 200 Two-dimensional electron gas, 709 Spiking, 499,500 Spin, 744 U Sputtering,162-164 Ultraviolet castastrophe,734 Square-law theory,620-625,626,627 Unijunction transistor (UJT), 45,476 Square-law relationship Unipolar transistor, 531 J-FET,641 Unit cell,7–11 MOSFET,623 bcc,8,9 Steady state, 124 diamond, 9, 10 --- Page 815 --- 792 INDEX fcc,8,9 Walk-out,700 simple cubic, 8 Wavefunction,739 zincblende,10 Wave mechanics, 739 Wave-particle duality,737-739 V Weak inversion, 599 Vacuum level,477,478,565 Wet etch,169 Valence band,27,28 Wide-base diode,244 Valence electrons,25,746 Workfunction,478,565 Varactor,307 Velocityovershoot,701-702 Z Velocity saturation,554-556,700-701 Zener process, 268-270 Zincblende unit cell,10,11 W Zn,365 W(tungsten), 499,500 ZnSe,3,5,348,362 Wafer,12 --- Page 816 --- Document generated by Anna’s Archive around 2023-2024 as part of the DuXiu collection (https://annas-blog.org/duxiu-exclusive.html). 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