part17

Appendix A: Elements of Quantum Mechanics

The big question of this appendix. Before we can talk about electrons moving through a silicon crystal, we need to know what an electron does inside a single isolated atom. Newton’s mechanics, which works beautifully for baseballs and planets, gives wrong answers at atomic size. Quantum mechanics is the more accurate theory, and it reduces to Newton’s in the limit of large masses and energies. Think of Newton as the “zoomed-out map” and quantum mechanics as the “street-level survey”.

The appendix has three movements:

1. The quantization concept (A.1): three experimental surprises that broke classical physics.
2. The basic formalism (A.2): five postulates of Schrödinger wave mechanics.
3. Electronic states in atoms (A.3): hydrogen solved exactly, then extrapolated to silicon.

---

A.1 The Quantization Concept

A.1.1 Blackbody radiation

Every warm solid glows. At room temperature or below, the glow is almost entirely infrared, so your eye never notices. An ideal radiator is called a blackbody. Its emitted power versus wavelength (Fig. A.1) has these features:

• Curves for $T=300$, $1000$ and $2000\text{K}$ are humped, with the hump moving to shorter wavelength as $T$ rises (a hotter stove goes from invisible, to dull red, to white).
• The visible window is only $0.4\mu \text{m}\lesssim \lambda \lesssim 0.7\mu \text{m}$.
• A dashed line shows the classical Rayleigh-Jeans prediction for $T=2000\text{K}$. It tracks the data at long wavelengths but shoots upward without bound as $\lambda \to 0$.

The classical story (Rayleigh-Jeans). Heat makes the atoms in a solid vibrate. Model each vibrating atom as a harmonic oscillator with normal-mode frequency $\nu =\omega /2\pi$ and a continuum of allowed energies, distributed by statistics. The emitted light is a sample of this distribution. It agrees at long wavelengths but diverges at short ones. Integrating over all wavelengths gives infinite radiated energy. This is the ultraviolet catastrophe.

Thought experiment. Imagine a vending machine that only accepts whole coins. Low-denomination change is easy to make. But to buy something that costs a “big” coin, you must save up a whole big coin. Planck’s insight is similar. High-frequency oscillators demand a large minimum energy “coin”, and at a given temperature there is rarely enough thermal energy to pay it. So high-frequency modes stay silent and the catastrophe never happens.

Planck’s fix (1901). An oscillator of frequency $\nu$ can only have energies

E_n = n h \nu = n\hbar\omega,\qquad n = 0,1,2,\dots \tag{A.1}

Matching theory to experiment gives

$h=6.63\times 10^{-34}\text{J⋅s},\hslash =\frac{h}{2\pi }.$

(The OCR printed $10^{-4}$; the true exponent is $-34$.) Lesson: at atomic scale energy comes in discrete steps. That is energy quantization.

A.1.2 The Bohr atom

Heated gases emit sharp, discrete spectral lines, not a smooth rainbow. Rutherford (1910) proposed the nuclear atom: electrons of mass $m_0$ and charge $-q$ orbit a nucleus of charge $+Zq$. Two problems follow classically:

• Electrons could orbit at any radius, so the light should be a continuum, not lines.
• An accelerating charge radiates, so the orbiting electron should spiral into the nucleus almost immediately.

Bohr’s postulate (1913). Only orbits with quantized angular momentum are allowed. For hydrogen ($Z=1$), circular orbit:

L_n = m_0 v_n r_n = n\hbar,\qquad n = 1,2,3,\dots \tag{A.2}

Derivation of the orbit radii and energies.

Step 1. Stable orbit means centripetal force equals Coulomb attraction:

\frac{m_0 v_n^2}{r_n} = \frac{q^2}{4\pi\varepsilon_0 r_n^2} \tag{A.3}

Step 2. Solve (A.2) for $v_n$:

$v_n=\frac{n\hslash }{m_0{r}_n}$

Step 3. Substitute into (A.3):

$\frac{m_0}{r_n}\frac{n^2{\hslash }^2}{m_0^2{r}_n^2}=\frac{q^2}{4\pi {\epsilon }_0{r}_n^2}$

Step 4. Cancel and solve for the radius:

r_n = \frac{4\pi\varepsilon_0 (n\hbar)^2}{m_0 q^2} \tag{A.4}

Step 5. Kinetic energy, using (A.3):

\text{K.E.} = \tfrac12 m_0 v_n^2 = \frac12\,\frac{q^2}{4\pi\varepsilon_0 r_n} \tag{A.5a}

Step 6. Potential energy (zero at infinity):

\text{P.E.} = -\frac{q^2}{4\pi\varepsilon_0 r_n} \tag{A.5b}

Step 7. Total energy (note the virial-style factor of one half):

E_n = \text{K.E.}+\text{P.E.} = -\frac12\,\frac{q^2}{4\pi\varepsilon_0 r_n} \tag{A.6}

Step 8. Insert $r_n$ from (A.4):

E_n = -\frac{m_0 q^4}{2(4\pi\varepsilon_0 n\hbar)^2} = -\frac{13.6}{n^2}\ \text{eV} \tag{A.7}

Here $1\text{eV}=1.6\times 10^{-19}\text{J}$.

Spectra. A photon emitted in a jump from level $n$ to level $m<n$ carries energy $E_n-E_m$. These discrete energies agree with experiment. Fig. A.2 shows the hydrogen levels with $E_1=-13.6$, $E_2=-3.40$, $E_3=-1.51$, $E_4=-0.85$, $E_5=-0.54$ eV, and $E_{\infty }=0$ (the OCR lost $E_3$). Transitions ending on $n=1$ form the Lyman series, on $n=2$ the Balmer series, on $n=3$ the Paschen series.

Think of the levels as the rungs of a ladder in a well: you can stand on a rung but never between rungs. Jumping down emits a photon whose color is set by the rung gap.

Bohr’s model failed for helium and beyond, but it cemented two ideas: energy quantization, and the failure of classical mechanics at small scale.

A.1.3 Wave-particle duality

Light as particle:

• Photoelectric effect (Einstein, 1905): illuminated metals eject electrons. Explained by light quanta (photons) with $E=h\nu$.
• Compton effect: x-rays scattered by electrons change frequency exactly as in a billiard-ball collision conserving energy and momentum. With $E=h\nu =m{c}^2$, photon momentum is

$p=mc=\frac{h\nu }{c}=\frac{h}{\lambda }.$

de Broglie (1925) flipped it. If waves can be particles, particles should be waves:

p = \frac{h}{\lambda}\quad\Longleftrightarrow\quad \lambda = \frac{h}{p} \tag{A.8}

Davisson-Germer (1927). A low-energy electron beam hit a nickel crystal. The electron wavelength from (A.8) was comparable to the Ni nearest-neighbor spacing. If electrons were simple bullets they would scatter randomly. Instead the angular intensity showed maxima and minima like light from a diffraction grating, at angles predicted by reflecting waves of wavelength $h/p$ from atomic planes. Protons and neutrons later behaved the same way.

(Footnote: in this appendix only, $p$ denotes momentum. Elsewhere in the book $p$ is hole concentration.)

Conclusion of A.1: observables such as energy and angular momentum are quantized, and all matter has an inherent wave nature.

---

A.2 Basic Formalism

Schrödinger (1926) provided a unified theory, wave mechanics, built on Planck’s quantization and de Broglie’s matter waves. Heisenberg’s matrix mechanics is mathematically different but exactly equivalent. We use Schrödinger’s version.

The five postulates (single-particle system).

1. Wavefunction. There exists $\Psi =\Psi (x,y,z,t)$, possibly complex, from which all dynamic information is obtained. It is the system’s “describing function”.

2. Wave equation. $\Psi$ follows the time-dependent Schrödinger equation:

-\frac{\hbar^2}{2m}\nabla^2\Psi + U\Psi = i\hbar\frac{\partial\Psi}{\partial t} \tag{A.9}

where $m$ is the particle mass, $U$ the potential energy and $i=\sqrt{-1}$.

3. Good behavior. $\Psi$ and $\nabla \Psi$ must be finite, continuous and single-valued everywhere.

4. Probability. ${\Psi }^{\ast }\Psi dV=|\Psi {|}^{2}dV$ is the probability of finding the particle in volume element $dV$. Since the particle must be somewhere,

\int_V \Psi^*\Psi\,dV = 1 \tag{A.10}

over all space (normalization).

5. Operators and expectation values. Each dynamic variable $\alpha$ has an associated operator ${\alpha }_{op}$. Its average (expectation) value is

\langle\alpha\rangle = \int_V \Psi^*\,\alpha_{op}\,\Psi\,dV \tag{A.11}

The operators are fixed by demanding agreement with classical mechanics for large mass and energy.

Table A.1: Dynamic variables and operators

Dynamic variable	Operator
$x,y,z$	$x,y,z$ (multiply)
$f(x,y,z)$	$f(x,y,z)$
$p_x,p_y,p_z$	$\frac{\hslash }{i}\frac{\partial }{\partial x},\frac{\hslash }{i}\frac{\partial }{\partial y},\frac{\hslash }{i}\frac{\partial }{\partial z}$
$E$	$i\hslash \frac{\partial }{\partial t}$

Intuition. $\Psi$ is not a thing you can touch. It is a bookkeeping function whose squared magnitude is a probability density. Postulate 5 says: to ask “what is the average momentum?”, sandwich the momentum operator between ${\Psi }^{\ast }$ and $\Psi$ and integrate.

How problems are solved. In principle: impose the boundary conditions plus postulates 3 and 4, solve (A.9) for $\Psi$, then compute any variable from (A.11). In practice closed-form solutions exist only for idealized or a few practical problems. Often one gets the allowed energies from the constraints alone, or uses expansions, trial wavefunctions or limiting cases.

Status of the equation. Schrödinger’s equation is essentially empirical, like Newton’s laws. Its justification is that every experimental test so far agrees with it, often to extreme precision.

---

A.3 Electronic States in Atoms

A.3.1 The hydrogen atom

A massive fixed nucleus ($+q$) and one electron ($-q$) with fixed total energy $E$. Separating space and time gives

\Psi(x,y,z,t) = \psi(x,y,z)\,e^{-iEt/\hbar} \tag{A.12}

Derivation of the time-independent equation.

Step 1. Substitute (A.12) into (A.9). Time derivative: $i\hslash {\partial }_t\Psi =E\psi e^{-iEt/\hslash }$.

Step 2. Cancel the common factor $e^{-iEt/\hslash }$:

$-\frac{{\hslash }^2}{2m}{\nabla }^2\psi +U\psi =E\psi$

Step 3. Rearrange:

\nabla^2\psi + \frac{2m}{\hbar^2}\left[E - U(x,y,z)\right]\psi = 0 \tag{A.13}

For hydrogen $m=m_0$ and

U = -\frac{q^2}{4\pi\varepsilon_0 r},\quad r=\sqrt{x^2+y^2+z^2} \tag{A.14}

\nabla^2\psi + \frac{2m_0}{\hbar^2}\left[E + \frac{q^2}{4\pi\varepsilon_0 r}\right]\psi = 0 \tag{A.15}

Because $U$ depends only on $r$, use spherical coordinates:

\nabla^2 = \frac{1}{r^2}\frac{\partial}{\partial r}\!\left(r^2\frac{\partial}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\!\left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial\phi^2} \tag{A.16}

Separation of variables, $\psi =R(r)\Theta (\theta )\Phi (\varphi )$, yields bound states ($E<0$). Each is labeled by three quantum numbers arising from the separation constants:

Symbol	Allowed values	Name
$n$	$1,2,3,\dots$	principal quantum number
$l$	$0,1,2,\dots ,n-1$	azimuthal quantum number
$m$ (or $m_l$)	$-l,\dots ,0,\dots ,+l$	magnetic orbital quantum number

Table A.2: Hydrogen wavefunctions ${\psi }_{n,l,m}$ for $n=1,2$ ($a_0=4\pi {\epsilon }_0{\hslash }^2/m_0{q}^2$, the Bohr radius, equal to the ground-state Bohr orbit from (A.4)):

${\psi }_{1,0,0}=\frac{1}{\sqrt{\pi a_0^3}}{e}^{-r/a_0}$

${\psi }_{2,0,0}=\frac{1}{4\sqrt{2\pi a_0^3}}\left(2-\frac{r}{a_0}\right)e^{-r/2a_0}$

${\psi }_{2,1,-1}=\frac{1}{8\sqrt{\pi a_0^3}}\frac{r}{a_0}{e}^{-r/2a_0}{e}^{-i\varphi }\sin \theta$

${\psi }_{2,1,0}=\frac{1}{4\sqrt{2\pi a_0^3}}\frac{r}{a_0}{e}^{-r/2a_0}\cos \theta$

${\psi }_{2,1,1}=\frac{1}{8\sqrt{\pi a_0^3}}\frac{r}{a_0}{e}^{-r/2a_0}{e}^{i\varphi }\sin \theta$

(The OCR garbled the prefactors. The normalization constants above are the standard values satisfying (A.10).)

Energies. Substituting ${\psi }_{1,0,0}$ into (A.15) and solving for $E$:

E_{1,0,0} = -\frac{\hbar^2}{2m_0a_0^2} = -\frac{m_0q^4}{2(4\pi\varepsilon_0\hbar)^2} \tag{A.17}

This is identical to Bohr’s $E_1$. Substituting any $n=2$ function gives

E_{2,0,0}=E_{2,1,-1}=E_{2,1,0}=E_{2,1,1} = -\frac{m_0q^4}{2(4\pi\varepsilon_0\hbar)^2}\cdot\frac14 \tag{A.18}

equal to Bohr’s $E_2$. So quantum mechanics reproduces Bohr’s levels, but without ad hoc assumptions. Knowing $n$ alone fixes the energy. $n=1$ is the ground state, larger $n$ are excited states.

Degeneracy. States sharing one energy are degenerate. The four $n=2$ states are degenerate. A magnetic field would split them because their spatial shapes interact differently with the field. This is where $l$ and $m$ matter.

Spin. A fourth quantum number is needed. Electrons have spin, pictured as spinning clockwise or counterclockwise, with $s=+\frac{1}{2}$ or $s=-\frac{1}{2}$. Every state of Table A.2 is thus two-fold degenerate. (The spin picture is only a mental image.)

Where is the electron? The probability of finding the ground-state electron between $r$ and $r+dr$ is $4\pi r^2|{\psi }_{1,0,0}{|}^{2}dr$. Plotted versus $r/a_0$ (Fig. A.3a) it rises to a maximum at $r=a_0$ and decays. For larger $n$ the peak shifts outward, and there is significant probability over a range of distances. This contrasts sharply with Bohr’s electron on a rail at fixed $r$. Picture instead a charge cloud with density $\propto |\psi {|}^2$ (Fig. A.3b), spherically symmetric for ${\psi }_{1,0,0}$; states with $l\ne 0$ give clouds with angular dependence.

A.3.2 Multi-electron atoms

Hydrogen’s wavefunctions do not transfer directly, but the same four quantum numbers $(n,l,m,s)$ label states in any atom, and the energy ordering follows $n$. Two rules let us deduce structure without solving anything:

1. Pauli exclusion principle: no two electrons may share the same set of quantum numbers. (Like theater seats: each seat is a quantum state, one person per seat.)
2. Minimum energy: in the ground state electrons fill the lowest available states.

Spectroscopic letters for $l=0,1,2,3,4,5,\dots$ are $s,p,d,f,g,h,\dots$ (from sharp, principal, diffuse, fundamental). In multi-electron atoms $s$ states lie slightly below $p$ states of the same $n$.

Counting. Level $n$ holds $2n^2$ electrons: $n=1$ has 2, $n=2$ has 8 (2 in $2s$, 6 in $2p$), $n=3$ begins with $3s$ (2) and $3p$ (6).

Table A.3: Quantum numbers and ground-state configurations, elements 1 to 14

States in filling order (each line is one electron slot):

$n$	$l$	$m$	$s$	state
1	0	0	$-\frac{1}{2}$	1s
1	0	0	$+\frac{1}{2}$	1s
2	0	0	$-\frac{1}{2}$	2s
2	0	0	$+\frac{1}{2}$	2s
2	1	$-1$	$-\frac{1}{2}$	2p
2	1	$-1$	$+\frac{1}{2}$	2p
2	1	0	$-\frac{1}{2}$	2p
2	1	0	$+\frac{1}{2}$	2p
2	1	1	$-\frac{1}{2}$	2p
2	1	1	$+\frac{1}{2}$	2p
3	0	0	$-\frac{1}{2}$	3s
3	0	0	$+\frac{1}{2}$	3s
3	1	$-1$	$-\frac{1}{2}$	3p
3	1	$-1$	$+\frac{1}{2}$	3p
3	1	0	$-\frac{1}{2}$	3p
3	1	0	$+\frac{1}{2}$	3p
3	1	1	$-\frac{1}{2}$	3p
3	1	1	$+\frac{1}{2}$	3p

Z	Element	Configuration
1	H	$1s^1$
2	He	$1s^2$
3	Li	$1s^{2}2s^1$
4	Be	$1s^{2}2s^2$
5	B	$1s^{2}2s^{2}2p^1$
6	C	$1s^{2}2s^{2}2p^2$
7	N	$1s^{2}2s^{2}2p^3$
8	O	$1s^{2}2s^{2}2p^4$
9	F	$1s^{2}2s^{2}2p^5$
10	Ne	$1s^{2}2s^{2}2p^6$
11	Na	$1s^{2}2s^{2}2p^{6}3s^1$
12	Mg	$1s^{2}2s^{2}2p^{6}3s^2$
13	Al	$1s^{2}2s^{2}2p^{6}3s^{2}3p^1$
14	Si	$1s^{2}2s^{2}2p^{6}3s^{2}3p^2$

The superscript counts electrons sharing an $nl$ label.

Silicon. A completely filled $n$ shell gives a very stable, inert configuration (He, Ne). The two $n=1$ and eight $n=2$ electrons of Si sit deep and tightly bound. These ten electrons plus the nucleus form the core, which stays essentially untouched in chemistry. The remaining four electrons are the valence electrons. They occupy the two $3s$ states and two of the six $3p$ states (Fig. A.4: shells $n=1$ with 2 electrons, $n=2$ with 8 electrons and 8 slots, $n=3$ with 4 electrons in a level with 2 $s$ and 6 $p$ slots). These weakly bound four electrons do the bonding, and, as the rest of the book shows, form the carriers in the crystal. Germanium ($Z=32$) is the same, except its core has 28 electrons.

Analogy. The core is a locked vault; the valence electrons are the coins in your pocket. Chemistry and conduction happen with the pocket coins.

---

Appendix B: MOS Semiconductor Electrostatics, Exact Solution

The story. Chapter 16 solved the MOS capacitor with the depletion approximation, treating the band-bending region as a box with a sharp edge. Here we throw away that shortcut and solve Poisson’s equation with every mobile carrier included. The payoff is a result valid from accumulation through depletion to strong inversion, and it needs one trick: multiplying Poisson’s equation by $dU/dx$ so that it integrates exactly, like using the energy integral for a pendulum.

B.1 Definition of parameters

Three normalized quantities keep the algebra light. Everything is measured in units of the thermal voltage $kT/q$ (about 25.9 mV at 300 K):

U(x) \equiv \frac{\phi(x)}{kT/q} = \frac{E_i(\text{bulk}) - E_i(x)}{kT} \tag{B.1}

U_s \equiv \frac{\phi_s}{kT/q} = \frac{E_i(\text{bulk}) - E_i(\text{surface})}{kT} \tag{B.2}

U_F \equiv \frac{\phi_F}{kT/q} = \frac{E_i(\text{bulk}) - E_F}{kT} \tag{B.3}

• $U(x)$ is the electrostatic potential normalized to $kT/q$ (“the potential”).
• $U_F$ is the doping parameter: positive for $p$-type, negative for $n$-type.
• $x$ is the depth into the semiconductor from the oxide-semiconductor interface.

Because the field vanishes in the bulk (idealization 5 of Section 16.1), the semiconductor can be treated as running from $x=0$ to $x=\infty$. Then $U(x\to \infty )=0$, consistent with $\varphi =0$ in the bulk.

The Debye length

Drop a charged bead into a plasma (a gas of free positive ions and electrons). The mobile charges crowd around and screen it. The screening distance, over which the field falls to about $1/e$, is the Debye length.

A semiconductor in the bulk, or under flat bands, is a plasma of ionized impurities and mobile carriers. A charged MOS gate is the “bead”, and the band-bending region is the screening cloud, about one Debye length thick. The extrinsic (bulk) Debye length is

L_B = \left[\frac{K_s\varepsilon_0 kT}{q^2\,(n_{bulk}+p_{bulk})}\right]^{1/2} \tag{B.4}

It is exact only for small departures from flat band, but it is convenient to normalize with the intrinsic Debye length, obtained by setting $n_{bulk}=p_{bulk}=n_i$:

L_D = \left[\frac{K_s\varepsilon_0 kT}{2q^2 n_i}\right]^{1/2} \tag{B.5}

B.2 Exact solution

Step 1: Poisson’s equation in one dimension

The MOS-C is one-dimensional (idealization 7), so

\frac{d\mathcal{E}}{dx} = \frac{\rho}{K_s\varepsilon_0} = \frac{q}{K_s\varepsilon_0}\,(p - n + N_D - N_A) \tag{B.6}

Step 2: Express field and carrier densities through $U$

The field is the slope of the bands:

\mathcal{E} = \frac{1}{q}\frac{dE_i(x)}{dx} = -\frac{kT}{q}\frac{dU}{dx} \tag{B.7}

(First equality: field is the gradient of the electron potential energy divided by $q$. Second: definition (B.1) and $d{E}_i(\text{bulk})/dx=0$.)

With Boltzmann statistics,

p = n_i e^{[E_i(x)-E_F]/kT} = n_i\,e^{U_F-U(x)} \tag{B.8a}

n = n_i e^{[E_F-E_i(x)]/kT} = n_i\,e^{U(x)-U_F} \tag{B.8b}

Step 3: Eliminate the doping with bulk charge neutrality

In the bulk $\rho =0$ and $U=0$:

0 = p_{bulk}-n_{bulk}+N_D-N_A = n_i e^{U_F}-n_i e^{-U_F}+N_D-N_A \tag{B.9}

N_D - N_A = n_i\left(e^{-U_F}-e^{U_F}\right) \tag{B.10}

Step 4: The charge density

\rho = q\,n_i\left(e^{U_F-U}-e^{U-U_F}+e^{-U_F}-e^{U_F}\right) \tag{B.11}

Step 5: Poisson’s equation in normalized form

Put (B.7) in (B.6) and divide by $kT/q$:

\frac{d^2U}{dx^2} = -\frac{q^2 n_i}{K_s\varepsilon_0 kT}\left(e^{U_F-U}-e^{U-U_F}+e^{-U_F}-e^{U_F}\right) \tag{B.12}

From (B.5), $q^2{n}_i/(K_s{\epsilon }_{0}kT)=1/(2L_D^2)$. Factor the bracket as $e^{U_F}(e^{-U}-1)-e^{-U_F}(e^U-1)$:

\frac{d^2U}{dx^2} = -\frac{1}{2L_D^2}\left[e^{U_F}\left(e^{-U}-1\right)-e^{-U_F}\left(e^{U}-1\right)\right] \tag{B.13}

(The OCR loses the factor of one half and some signs; this is the form consistent with (B.11) and (B.5).)

Step 6: Boundary conditions

\mathcal{E}=0\ \ \text{or}\ \ \frac{dU}{dx}=0 \quad\text{at } x=\infty \tag{B.14a}

U = U_s \quad\text{at } x=0 \tag{B.14b}

Step 7: First integral (the energy trick)

Multiply (B.13) by $dU/dx$. The left side becomes $\frac{1}{2}d[(dU/dx{)}^2]/dx$. Integrate from $\infty$ to $x$ and use (B.14a):

${\left(\frac{dU}{dx}\right)}^2=-\frac{1}{L_D^2}{\int }_0^U\left[e^{U_F}(e^{-u}-1)-e^{-U_F}(e^u-1)\right]du$

The integrals are

${\int }_0^U(e^{-u}-1)du=-(e^{-U}-1+U),{\int }_0^U(e^u-1)du=e^U-1-U$

so

${\left(\frac{dU}{dx}\right)}^2=\frac{1}{L_D^2}\left[e^{U_F}\left(e^{-U}+U-1\right)+e^{-U_F}\left(e^U-U-1\right)\right]$

and in terms of the field

\mathcal{E}^2 = \left(\frac{kT/q}{L_D}\right)^2\left[e^{U_F}(e^{-U}+U-1)+e^{-U_F}(e^{U}-U-1)\right] \tag{B.15}

Step 8: Choose the sign of the square root

Equation (B.15) has the form $y^2=a^2$ with roots $\pm a$. From the band diagram, $\mathcal{E}>0$ when $U>0$ and $\mathcal{E}<0$ when $U<0$. The right side is always nonnegative, so take the positive root for $U>0$ and the negative root for $U<0$:

\mathcal{E} = -\frac{kT}{q}\frac{dU}{dx} = \frac{kT}{q}\,\frac{\varsigma_s\,F(U,U_F)}{L_D} \tag{B.16}

F(U,U_F) = \left[e^{U_F}\left(e^{-U}+U-1\right)+e^{-U_F}\left(e^{U}-U-1\right)\right]^{1/2}\ \ge 0 \tag{B.17}

\varsigma_s = \begin{cases} +1 & U_s>0\\ -1 & U_s<0\end{cases} \tag{B.18}

(The book’s List of Symbols calls this “sign ($\pm$) of $U_s$”. I write it ${\varsigma }_s$ to avoid a clash with permittivity.)

Step 9: Separate variables and integrate

From (B.16), $dU/dx=-{\varsigma }_{s}F/L_D$, so $dU/F=-{\varsigma }_{s}dx/L_D$. Integrate from the surface ($x=0$, $U=U_s$) to the point $x$ with potential $U$, using ${\varsigma }_s^2=1$:

\int_{U}^{U_s}\frac{dU'}{F(U',U_F)} = \varsigma_s\,\frac{x}{L_D} \tag{B.19}

What this buys us. Equations (B.11), (B.16) and (B.19) together are the exact solution. For a given $U_s$, evaluate the integral numerically to get $x$ versus $U$, invert to $U(x)$, then substitute into (B.11) for $\rho (x)$ and (B.16) for $\mathcal{E}(x)$. The Fig. 16.8 plots of $U(x)$ and $\rho (x)$ were made this way.

Sanity check. For small $U$, $F\approx |U|\sqrt{(e^{U_F}+e^{-U_F})/2}$, so (B.19) gives $U\approx U_s{e}^{-x/L_B}$, an exponential decay with the extrinsic Debye length, exactly as the plasma picture predicts.

---

Appendix C: MOS C-V Supplement

Why a supplement? The delta-depletion C-V curve of Chapter 16 is explicit: $C$ versus $V_G$ in closed form. The exact-charge version cannot be written that way. Instead both $C$ and $V_G$ are written as functions of the surface potential $U_s$, and you sweep $U_s$ to trace the curve, like a parametric curve $(x(t),y(t))$.

C.1 The capacitance relationships

For an ideal MOS-C with exact charge distribution:

\frac{C}{C_o} = \frac{1}{1+\dfrac{K_o\,W_{\text{eff}}}{K_s\,x_o}} \tag{C.1}

This is two capacitors in series, $C={\left(C_o^{-1}+C_s^{-1}\right)}^{-1}$, with $C_o=K_o{\epsilon }_0/x_o$ and $C_s\equiv K_s{\epsilon }_0/W_{\text{eff}}$. The “effective width” depends on the bias region.

Accumulation ($p$-type, $U_s<0$):

W_{\text{eff}} = \frac{U_s}{|U_s|}\,L_D\,\frac{2F(U_s,U_F)}{e^{U_F}\left(1-e^{-U_s}\right)+e^{-U_F}\left(e^{U_s}-1\right)} \tag{C.2a}

Flat band ($U_s=0$):

W_{\text{eff}}=W_{fb} = L_D\left[\frac{2}{e^{U_F}+e^{-U_F}}\right]^{1/2} \tag{C.2b}

Depletion/inversion ($p$-type, $U_s>0$):

W_{\text{eff}} = \frac{U_s}{|U_s|}\,L_D\,\frac{2F(U_s,U_F)}{e^{U_F}\left(1-e^{-U_s}\right)+e^{-U_F}\left(e^{U_s}-1\right)\left(1+\Delta\right)} \tag{C.2c}

where $\Delta$ distinguishes the measurement frequency:

\Delta = 0\qquad\text{(low-frequency limit)} \tag{C.3a}

\Delta = \frac{e^{U_s}-U_s-1}{F(U_s,U_F)}\times\big[\text{prefactor, Eq. (C.3b)}\big]\times\int_{0}^{U_s}\frac{\left(1-e^{-U}\right)\left(e^{U}-U-1\right)}{2F^3(U,U_F)}\,dU\qquad\text{(high-frequency limit)} \tag{C.3b}

The integrand is the function

\frac{\left(1-e^{-U}\right)\left(e^{U}-U-1\right)}{2F^3(U,U_F)} \tag{C.4}

Reading note on (C.3b). In the scan the factor between $(e^{U_s}-U_s-1)/F$ and the integral is garbled beyond recovery (fragments of $e^{U_F}$, $(1-e^{-U_s})$ and $(e^{-U}-1)$ survive). The structure shown and the integrand (C.4) are legible. This correction is Brews’s “rearrangement capacitance”, the contribution of inversion-layer carriers shifting position (not number) under the small-signal probe.

Why $\Delta =0$ at low frequency? Minority carriers can be generated and recombined fast enough to follow the signal, so the semiconductor responds fully and the charge change is just $d{Q}_s/d{\varphi }_s$. At high frequency the minority carriers cannot be exchanged with the bulk, so the capacitance stays near its minimum in inversion.

Footnotes: all relations except (C.2c) and (C.3b) hold for both $n$- and $p$-type devices. The low-frequency relation traces to Grove, Deal, Snow and Sah (Solid-State Electronics, 8, 145, 1965); the high-frequency formula is adapted from J. R. Brews, J. Appl. Phys. 45, 1276 (1974).

C.2 Definitions used

F(U,U_F) = \left[e^{U_F}(e^{-U}+U-1)+e^{-U_F}(e^{U}-U-1)\right]^{1/2} \tag{C.5}

L_D = \left[\frac{K_s\varepsilon_0 kT}{2q^2 n_i}\right]^{1/2} \tag{C.6}

U_F = \frac{\phi_F}{kT/q} \tag{C.7}

U_s = \frac{\phi_s}{kT/q} \tag{C.8}

\varsigma_s = \begin{cases}+1 & U_s>0\\-1 & U_s<0\end{cases} \tag{C.9}

C.3 The gate voltage

V_G = \frac{kT}{q}\left[U_s + \varsigma_s\,\frac{K_s\,x_o}{K_o\,L_D}\,F(U_s,U_F)\right] \tag{C.10}

Derivation.

1. For an ideal device $V_G={\varphi }_s+V_{ox}$.
2. Gauss: $Q_s=-K_s{\epsilon }_0{\mathcal{E}}_s$, with ${\mathcal{E}}_s={\varsigma }_s(kT/q)F(U_s,U_F)/L_D$ from (B.16).
3. Oxide drop: $V_{ox}=-Q_s/C_o={\varsigma }_s\frac{kT}{q}\frac{K_s{x}_o}{K_o{L}_D}F$.
4. Add ${\varphi }_s=(kT/q)U_s$ to get (C.10). (The OCR drops the $x_o$ in the printed fraction.)

C.4 How to compute a C-V curve

$C$ cannot be written explicitly in $V_G$, but both are known functions of $U_s$. The efficient recipe:

1. Choose $U_s$ values, stepped by whole numbers ($\dots ,-5,-4,\dots$) or half units, over the normal range $U_F-21\le U_s\le U_F+21$ at room temperature (the MATLAB program uses steps of 0.5).
2. For each $U_s$ compute $F$, then $V_G$ from (C.10), then $W_{\text{eff}}$ from (C.2), then $C/C_o$ from (C.1).
3. Plot the $(V_G,C/C_o)$ pairs.

Cautions:

• At $U_s=0$ the accumulation and depletion/inversion expressions become $0/0$. Use the flat-band expression (C.2b) there.
• For $n$-type devices the text gives modified forms of (C.2c) and (C.3b) (the exponentials are mirrored). The simplest alternative: compute the equivalently doped $p$-type device and change the sign of every computed $V_G$. This works because ideal $n$- and $p$-type devices are voltage mirror images.
• The low-frequency calculation is simple enough for a hand calculator.

---

Appendix D: MOSFET I-V Supplement

The picture. The square-law MOSFET equation treats channel charge crudely. The exact-charge analysis (Pao and Sah) uses the true carrier distribution in the semiconductor. The charge-sheet analysis (Brews) squeezes the inversion layer into a zero-thickness sheet, which is nearly as accurate and far easier. Both rest on the electrostatics of Appendix B.

D.1 The two current relationships

For an ideal $n$-channel (p-bulk) MOSFET the exact-charge relationship is Eq. (D.1) and the charge-sheet relationship is Eq. (D.3). The exact-charge integrand uses the generalized field function

F(U,U_F,\xi) = \left[e^{U_F}\left(e^{-U}+U-1\right)+e^{-U_F}\left(e^{U-\xi}-U-e^{-\xi}\right)\right]^{1/2} \tag{D.2}

where $\xi$ is the normalized channel quasi-Fermi potential, running from 0 at the source to $U_D$ at the drain. With $\xi =0$, (D.2) collapses to the equilibrium $F(U,U_F)$ of (B.17), as it must.

Reading note. In the scan the typeset bodies of (D.1), (D.3) and the definition (D.4) are damaged beyond reliable recovery. What is legible: an overall prefactor $Z{\mu }_n/L$ (width, mobility, length); in (D.1) a second term proportional to $K_s/(K_o{L}_D)$; integrals between the source-end surface potential $U_{s0}$ and drain-end surface potential $U_{sL}$; and in the charge-sheet case a voltage $V_B$ defined in (D.4) from $K_s$, $N_A$, $K_o$, $L_D$ and $n_i$. The physics common to both is the drift-diffusion channel current

$I_D=\frac{Z{\mu }_n}{L}{\int }_{\text{source}}^{\text{drain}}\left|Q_N\right|d{V}_c,$

where $Q_N$ is the electron charge per unit area in the channel and $V_c$ the channel quasi-Fermi potential. $Q_N$ comes from the exact electrostatics (exact charge) or from gate-charge balance with the bulk depletion charge subtracted (charge sheet).

Sources: R. F. Pierret and J. A. Shields, “Simplified Long-Channel MOSFET Theory,” Solid-State Electronics, 26, 143 (1983); H. C. Pao and C. T. Sah, Solid-State Electronics, 9, 927 (1966) for the exact-charge analysis; J. R. Brews, Solid-State Electronics, 21, 345 (1978) for charge-sheet.

D.2 Normalized voltages (both theories)

U_G = \frac{V_G}{kT/q} \tag{D.5}

U_{s0} = \frac{V_{s0}}{kT/q} \tag{D.6}

U_{sL} = \frac{V_{sL}}{kT/q} \tag{D.7}

U_D = \frac{V_D}{kT/q} \tag{D.8}

$V_{s0}$ and $V_{sL}$ are the surface potentials at the source end ($y=0$) and the drain end ($y=L$). (Only the first symbol of (D.5) is clear in the scan; $U_G=V_G/(kT/q)$ is the quantity that fits.)

D.3 Surface potentials at the channel ends

From the gate-voltage relationship of Appendix C, now with the channel quasi-Fermi level:

V_G = \frac{kT}{q}\left[U_{s0} + \frac{K_s x_o}{K_oL_D}\,F(U_{s0},U_F,0)\right]\quad (U_{s0}>0) \tag{D.9a}

V_G = \frac{kT}{q}\left[U_{sL} + \frac{K_s x_o}{K_oL_D}\,F(U_{sL},U_F,U_D)\right]\quad (U_{sL}>0) \tag{D.9b}

At the source the channel quasi-Fermi potential is 0, at the drain it is $U_D$.

D.4 Procedure to generate $I_D$-$V_D$ characteristics

1. Step $V_G$ and $V_D$ systematically over the desired range.
2. For each $(V_G,V_D)$ pair, iterate (D.9a) and (D.9b) to find $U_{s0}$ and $U_{sL}$. (Since $V_G$ rises monotonically with $U_s$ for $U_s>0$, bisection or Newton iteration works.)
3. With $U_{s0}$ and $U_{sL}$ known, compute $I_D$ from (D.1) or (D.3).
4. Repeat for each $V_G$-$V_D$ combination.
5. For a $p$-channel device, run the calculation for an equivalently doped and biased $n$-channel device and reverse the bias polarities when plotting.

---

Appendix E: List of Symbols

A field guide to the notation. Many symbols are reused: a letter that means “hole concentration” in the main text means “momentum” in Appendix A, and $U$ is potential energy in Appendix A but normalized electrostatic potential in Appendices B to D. Subscripts carry the meaning (B, E, C for BJT regions; n, p for carrier type; 0 for equilibrium or oxide). Entries below are grouped alphabetically (Latin first, then Greek), repairing OCR damage from context.

E.1 Latin symbols

Symbol	Meaning
$A$	anode; area; arbitrary constant
$a$	lattice constant; grading constant; half-width of the channel region in a J-FET; width of the channel region in a MESFET
$a_0$	Bohr radius
$A^{\ast \ast }$	Richardson’s constant (120 A/cm${}^2$-K${}^2$)
$A^{\ast }$	modified Richardson’s constant (see Eq. 14.19)
$A_G$	gate area
$B$	base
$C$	capacitance; collector
$c$	speed of light
$C_D$	diffusion capacitance
$C_{\pi }$	emitter-to-base capacitance in the high-frequency hybrid-pi model
$C_G$	MOSFET gate capacitance
$C_{gd},C_{gs}$	capacitances in the small-signal circuit for the J-FET and MOSFET (gate-drain, gate-source)
$C_J$	junction or depletion region capacitance
$c_n$	electron capture coefficient
$C_o$	oxide capacitance (pF)
$C_o^{\prime }$	oxide capacitance per unit area (pF/cm${}^2$)
$c_p$	hole capture coefficient
$C_s$	semiconductor capacitance
$D$	drain
$D_B,D_C,D_E$	minority-carrier diffusion coefficient in the BJT base, collector, emitter
$D_T$	density of interfacial traps (states/cm${}^2$-eV)
$D_N,D_P$	electron, hole diffusion coefficient (cm${}^2$/s)
$D_{ox}$	dielectric displacement in the oxide
$D_{semi}$	dielectric displacement in the semiconductor
$E$	emitter; energy
$\mathcal{E}$	electric field
${\mathcal{E}}_{ox}$	electric field in the oxide
${\mathcal{E}}_s$	surface electric field, field in the semiconductor at the oxide-semiconductor interface
${\mathcal{E}}_y$	$y$-direction component of the electric field
$E_0$	vacuum level: minimum energy an electron needs to free itself completely from a material
$E_A$	acceptor energy level
$E_B$	binding energy at dopant (donor, acceptor) sites
$E_C$	minimum conduction band energy
$E_D$	donor energy level
$E_F$	Fermi energy or Fermi level
$E_{FM}$	Fermi level in the metal
$E_{Fn},E_{Fp}$	Fermi level on the $n$-side, $p$-side of a pn junction
$E_{FS}$	Fermi level in the semiconductor
$E_G$	band gap or forbidden-gap energy
$E_H$	electron binding energy in the hydrogen atom
$E_i$	intrinsic Fermi level
$E_n$	energy of the $n$th level (Appendix A)
$E_{ph}$	photon energy ($h\nu$)
$E_t$	trap or R-G center energy level
$E_V$	maximum valence band energy
$F$	force
$f$	frequency (Hz)
$f(E)$	Fermi function
$F(U,U_F)$	field function (see Eq. B.17)
$F_{1/2}$	Fermi-Dirac integral of order 1/2
$FF$	fill factor
$F_N,F_P$	quasi-Fermi level (or energy) for electrons, holes
$f_T$	unity-beta (unity-gain) frequency of a BJT
$G$	conductance; gate
$G_0$	low-frequency conductance of a pn junction diode; channel conductance in a J-FET if there were no depletion regions
$g_C(E)$	density of conduction band states
$G_D$	diffusion conductance
$g_d$	drain or channel conductance
$G_L$	photogeneration rate (number of electron-hole pairs created per cm${}^3$-s)
$g_m$	transconductance
$g_V(E)$	density of valence band states
$h$	Planck’s constant
$\hslash$	$h/2\pi$
$I$	current; light intensity
$i$	a.c. current; $\sqrt{-1}$ in Appendix A
$I_0$	saturation current in an ideal diode; light intensity at $x=0$
$I_{AK}$	anode-to-cathode current
$I_B$	d.c. base current
$i_B$	total (a.c. + d.c.) base current
$i_b$	a.c. base current
$I_C$	d.c. collector current
$i_C$	total (a.c. + d.c.) collector current
$i_c$	a.c. collector current
$I_{CBO}$	collector-to-base current when $I_E=0$
$I_{CEO}$	collector-to-emitter current when $I_B=0$
$I_{Cn},I_{Cp}$	d.c. collector current due to electrons, holes
$I_D$	d.c. drain current in a field-effect transistor
$i_d$	small-signal drain current
$I_{DO}$	$V_G=0$ saturation drain current in a J-FET
$I_{dark}$	dark current
$I_{DIFF}$	diffusion current (same as the ideal diode current)
$i_{diff}$	a.c. component of the diffusion current
$I_{Dsat}$	saturation drain current
$I_E$	d.c. emitter current
$I_{En},I_{Ep}$	d.c. emitter current due to electrons, holes
$I_F$	steady-state forward-bias current
$I_{F0}$	effective diode forward saturation current (Ebers-Moll model)
$I_G$	gate current in an SCR
$I_L$	current due to light
$I_{M\to S}$	current due to electrons drifting from the metal to the semiconductor in an MS ($n$-type) diode
$I_P$	hole current
$I_{P	drift}$
$I_R$	steady-state reverse-bias current
$I_{R\text{-}G}$	recombination-generation current
$I_{R0}$	effective diode reverse saturation current (Ebers-Moll model)
$I_S$	reverse-bias saturation current in an MS diode
$I_{SC}$	short-circuit current in a solar cell
$I_{S\to M}$	current due to electrons drifting from the semiconductor to the metal in an MS ($n$-type) diode
$J,\mathbf{J}$	current density (A/cm${}^2$)
$J_{drift}$	total current density due to drift
$J_N,{\mathbf{J}}_N$	electron current density
$J_{Nx},J_{Ny},J_{Nz}$	$x,y,z$ components of the electron current density
$J_{N	diff},,J_{N
$J_P,{\mathbf{J}}_P$	hole current density
$J_{Px},J_{Py},J_{Pz}$	$x,y,z$ components of the hole current density
$J_{P	diff},,J_{P
$K$	cathode
$k$	Boltzmann constant ($8.617\times 10^{-5}$ eV/K)
$\mathbf{K}$	wavenumber (parameter proportional to the electron crystal momentum)
K.E.	kinetic energy
$K_O$	oxide dielectric constant
$K_S$	semiconductor (usually Si) dielectric constant
$L$	length of the J-FET or MOSFET channel
$l$	azimuthal quantum number
$L^{\prime }$	reduced channel length defined in Fig. 19.4
$L_B$	minority-carrier diffusion length in the BJT base; extrinsic Debye length
$L_C$	minority-carrier diffusion length in the BJT collector
$L_D$	intrinsic Debye length
$L_E$	minority-carrier diffusion length in the BJT emitter
$L_{min}$	minimum MOSFET channel length yielding long-channel behavior
$L_n$	electron minority-carrier diffusion length; angular momentum for the $n$ quantum number (Appendix A)
$L_p$	hole minority-carrier diffusion length
$M$	carrier multiplication factor
$m$	particle mass; magnetic orbital quantum number
$m_0$	electron rest mass
$m_n^{\ast },m_p^{\ast }$	electron, hole effective mass
$n$	energy quantum number; electron carrier concentration (electrons/cm${}^3$)
$n^{+}$	heavily doped $n$-type material
$n_0$	equilibrium electron concentration
$n_P,n_N$	defined electron concentration (see Eq. 3.36a)
$N_A$	total number of acceptor atoms/cm${}^3$
$N_A^{-}$	number of ionized (negatively charged) acceptors/cm${}^3$
$N_B$	bulk semiconductor doping ($N_A$ or $N_D$ as appropriate); doping concentration in the BJT base
$n_{bulk}$	electron concentration in the semiconductor bulk
$N_C$	effective density of conduction band states; doping concentration in the BJT collector
$n_{C0}$	equilibrium electron concentration in the collector of a pnp BJT
$N_D$	total number of donor atoms/cm${}^3$
$N_D^{+}$	number of ionized (positively charged) donors/cm${}^3$
$N_E$	doping concentration in the BJT emitter
$n_{E0}$	equilibrium electron concentration in the emitter of a pnp BJT
$N_I$	number of implanted ions/cm${}^2$
$n_i$	intrinsic carrier concentration
$N_T$	number of R-G centers/cm${}^3$
$N_V$	effective density of valence band states
$p$	hole concentration (holes/cm${}^3$); momentum in Appendix A
$p^{+}$	heavily doped $p$-type material
P.E.	potential energy
$p_0$	equilibrium hole concentration
$p_N,p_P$	defined hole concentration (see Eq. 3.36b)
$p_{B0}$	equilibrium hole concentration in the base of a pnp BJT
$p_{bulk}$	hole concentration in the semiconductor bulk
$p_s$	hole concentration at the semiconductor surface (number/cm${}^3$)
$Q$	general designation for a charge
$q$	magnitude of the electronic charge ($1.60\times 10^{-19}$ C)
$Q_B$	excess minority-carrier charge in the quasineutral base; bulk or depletion-region charge per unit area of the MOSFET gate
$Q_{BL}$	$Q_B$ in a long-channel MOSFET
$Q_{BS}$	$Q_B$ in a short-channel MOSFET
$Q_F$	fixed oxide charge per unit area at the oxide-semiconductor interface
$Q_I$	implant-related charge/cm${}^2$ located at the oxide-semiconductor interface
$Q_M$	total mobile ion charge within the oxide per unit area of the MOS gate
$Q_N$	total electronic charge/cm${}^2$ in the MOSFET channel ($n$-channel device)
$Q_{o\text{-}s}$	charge per unit area located at the oxide-semiconductor interface
$Q_P$	excess hole charge
$Q_s$	total charge in the semiconductor per unit area of the gate
$R$	ramp rate (see Fig. 16.17)
$r_B,r_b$	base resistance
$r_C,r_c$	collector resistance
$R_D$	channel-to-drain resistance in a J-FET or MOSFET
$r_E,r_e$	emitter resistance
$r_j$	depth of the source and drain islands in a MOSFET
$R_L$	load resistor
$r_n$	radius of the Bohr orbit corresponding to the $n$ quantum number
$r_o$	output resistance in the BJT hybrid-pi model
$R_P$	projected range in ion implantation
$R_S$	series resistance; sample resistance; source-to-channel resistance in a J-FET or MOSFET
$r_{\mu }$	feedthrough resistance in the BJT hybrid-pi model
$r_{\pi }$	input resistance in the BJT hybrid-pi model
$S$	source; probe-to-probe spacing in a four-point probe
$s$	spin quantum number
$T$	temperature
$t$	time
$t_{ON}$	triggering time in an SCR
$TR$	tuning ratio
$t_r$	recovery time (pn diode); rise time (BJT)
$t_{rr}$	reverse recovery time (pn diode)
$t_s$	storage delay time (pn diode)
$t_{sd}$	storage delay time (BJT)
$U$	potential energy in Appendix A; electrostatic potential normalized to $kT/q$ in Appendices B to D
$U_D$	drain voltage normalized to $kT/q$
$U_F$	semiconductor doping parameter
$U_s$	normalized surface potential, $U$ evaluated at the oxide-semiconductor interface
${\varsigma }_s$	sign ($\pm$) of $U_s$
$U_{s0}$	normalized surface potential at $x=0$ (source end) in a MOSFET
$U_{sL}$	normalized surface potential at $x=L$ (drain end) in a MOSFET
$V$	voltage, electrostatic potential; volume; velocity
$V_A$	applied d.c. voltage
$v_a$	applied a.c. voltage
$V_{AK}$	anode-to-cathode voltage
$V_B$	defined voltage (see Eq. D.4)
$v_{be}$	a.c. base-to-emitter voltage
$V_{BBF}$	forward-bias blocking voltage in PNPN devices
$V_{bi}$	”built-in” junction voltage
$V_{BR}$	reverse-bias pn junction breakdown voltage; reverse-bias blocking voltage in PNPN devices
$V_{BS}$	back-to-source voltage
$V_{CB}$	d.c. collector-to-base voltage
$V_{CBO}$	collector-to-base breakdown voltage when $I_E=0$
$v_{ce}$	a.c. collector-to-emitter voltage
$V_{CEO}$	collector-to-emitter breakdown voltage when $I_B=0$
$V_D$	d.c. drain voltage
${\mathbf{v}}_d$	drift velocity vector
$v_d$	drift velocity; a.c. drain voltage
$V_{DS}$	drain-to-source voltage
$V_{Dsat}$	saturation drain voltage
$v_{sat}$	saturation drift velocity
$V_{EB}$	d.c. emitter-to-base voltage
$V_{EC}$	d.c. emitter-to-collector voltage
$V_{FB}$	flat-band voltage
$V_G$	d.c. gate voltage
$v_g$	a.c. gate voltage
$V_G^{\prime }$	d.c. gate voltage applied to an ideal device
$V_{GS}$	gate-to-source voltage
$V_{GS}^{\prime }$	gate-to-source voltage applied to an ideal device
$V_J$	junction voltage
$V_{OC}$	open-circuit voltage of a solar cell
$V_P$	pinch-off gate voltage in a J-FET
$V_S$	d.c. source voltage
$v_s$	pulsed source voltage
$V_{S0}$	surface potential at $x=0$ in a MOSFET
$V_{SL}$	surface potential at $x=L$ in a MOSFET
$V_T$	turn-on (threshold) voltage
$V_T^{\prime }$	ideal-device inversion-depletion transition point gate voltage
$V_x$	defined voltage (see Eq. 17.24)
$V_y$	defined voltage (see Eq. 16.36)
$W$	depletion width; quasineutral width of the BJT base
$W_{N2}^{\prime }$	quasineutral width of the N2 base in an SCR
$W_{P3}^{\prime }$	quasineutral width of the P3 base in an SCR
$W_{BB}$	total width of the base in a BJT
$W_D$	drain pn junction depletion width in a MOSFET
$W_{eff}$	effective depletion width in a MOS capacitor (see Appendix C)
$W_{N2}$	width of the N2 base in an SCR
$W_{P3}$	width of the P3 base in an SCR
$W_S$	source pn junction depletion width in a MOSFET
$W_T$	MOS depletion width when the 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
$x_n$	$n$-side width of the pn junction depletion region
$x_o$	oxide thickness
$x_p$	$p$-side width of the pn junction depletion region
$Y$	admittance
$Y_D$	diffusion admittance
$Z$	width of the J-FET or MOSFET channel

E.2 Greek symbols

Symbol	Meaning
$\alpha$	absorption coefficient
${\alpha }_{dc}$	common-base d.c. current gain
${\alpha }_F$	${\alpha }_F={\alpha }_{dc}$, forward gain (Ebers-Moll model)
${\alpha }_R$	reverse gain (Ebers-Moll model)
${\alpha }_T$	base transport factor
${\beta }_{dc}$	common-emitter d.c. current gain
$\chi$	semiconductor electron affinity
${\chi }^{\prime }$	effective semiconductor electron affinity in an MOS structure
${\chi }_i$	insulator (oxide) electron affinity
${\chi }_{Si}$	silicon electron affinity
$\Delta$	frequency parameter in the exact-charge C-V theory (see Appendix C)
$\Delta E_C$	conduction band offset energy in a heterojunction
$\Delta E_V$	valence band offset energy in a heterojunction
${\Delta }_{ox}$	voltage drop across the oxide
${\Delta }_{semi}$	voltage drop across the semiconductor
$\Delta L$	decrease in the channel length under above-pinch-off conditions
$\Delta n$	$\Delta n=n-n_0$, deviation in electron concentration from its equilibrium value
$\Delta n_C$	excess electron concentration in the collector of a pnp BJT
$\Delta n_E$	excess electron concentration in the emitter of a pnp BJT
$\Delta n_p$	$\Delta n$ in $p$-type material
$\Delta p$	$\Delta p=p-p_0$, deviation in hole concentration from its equilibrium value
$\Delta p_B$	excess hole concentration in the base of a pnp BJT
$\Delta p_n$	$\Delta p$ in $n$-type material
$\Delta Q$	general designation for a change in charge
$\Delta Q_G$	change in the gate charge/cm${}^2$
$\Delta Q_{semi}$	change in the charge/cm${}^2$ inside the semiconductor
$\Delta R_P$	straggle in ion implantation
$\Delta V_G$	difference between the actual device and ideal device gate voltage required to achieve a given semiconductor surface potential
$\Delta V_T$	change in the threshold voltage due to small-dimension effects
$\Delta V_T^{\prime }$	change in the threshold voltage due to back biasing (specifically applied to ideal devices)
$\Delta {\Phi }_B$	change in the MS barrier height caused by Schottky barrier lowering
$\epsilon$	permittivity
${\epsilon }_0$	permittivity of free space ($8.85\times 10^{-14}$ F/cm)
$\varphi$	electrostatic potential inside the semiconductor component of an MOS device
${\Phi }_B$	surface potential-energy barrier height in an MS diode
${\Phi }_{B0}$	${\Phi }_B$ when $\mathcal{E}=0$ at the MS interface
${\varphi }_F$	reference voltage related to the semiconductor doping concentration
${\Phi }_M$	metal work function
${\Phi }_M^{\prime }$	${\Phi }_M^{\prime }={\Phi }_M-{\chi }_i$ (effective metal work function in an MOS structure)
${\varphi }_{MS}$	metal-semiconductor work-function difference expressed in volts
${\varphi }_{ox}$	voltage inside the oxide
${\Phi }_S$	semiconductor work function
${\varphi }_s$	semiconductor surface potential
$\Gamma$	four-point probe correction factor
$\gamma$	emitter efficiency
${\gamma }_M$	normalized centroid of mobile ion charge in the oxide
$\eta$	$\eta =(E-E_F)/kT$; power conversion efficiency of a solar cell; external efficiency of an LED
${\eta }_c$	${\eta }_c=(E_F-E_C)/kT$
${\eta }_v$	${\eta }_v=(E_V-E_F)/kT$
$\lambda$	wavelength of light
${\lambda }_G$	wavelength of light corresponding to the semiconductor band gap
${\mu }_0$	low-field electron or hole mobility; mobility fit parameter
${\mu }_{bulk}$	carrier mobility in the bulk of a semiconductor
${\mu }_n$	electron mobility
${\mu }_p$	hole mobility
${\mu }_{neff}$	effective electron mobility
${\mu }_{peff}$	effective hole mobility
$\nu$	frequency of light
$\rho$	resistivity ($\Omega$-cm); charge density (C/cm${}^3$)
${\rho }_{ion}$	ionic charge density inside the oxide
${\rho }_{ox}$	charge density in the oxide
$\sigma$	conductivity
${\tau }_0$	defined carrier lifetime (see Eq. 6.44)
${\tau }_B$	minority-carrier lifetime in the BJT base
${\tau }_C$	minority-carrier lifetime in the BJT collector
${\tau }_E$	minority-carrier lifetime in the BJT emitter
${\tau }_n$	electron minority-carrier lifetime
${\tau }_p$	hole minority-carrier lifetime
${\tau }_t$	base transit time
$\Psi$	time-dependent wavefunction
$\psi$	time-independent wavefunction
$\omega$	angular frequency (radians)

Where the OCR corrupted a subscript in the source (for example several entries for $C_d$, $C_s$, $g_d$, $V_s$), the symbol was repaired by its description and the surrounding alphabetical order. Entries whose scan lost both symbol and subscript (for example a repeated "$E_n$" row and an "$a$" row after “emitter-to-base capacitance”) are folded into the nearest legible entry.

---

Appendix M: MATLAB Program Scripts

What these are. Four programs, each tied to an end-of-chapter exercise, that turn the book’s equations into pictures: the equilibrium band diagram of a BJT (Exercise 10.2), the BJT input/output characteristics with and without non-ideal effects (Exercises 11.7 and 11.10), and the exact-charge MOS C-V curve (Exercise 16.5). Treat each one as a lab bench: change a doping, a width, a thickness, and watch the physics respond.

General notes on the cleaned code:

• The scan lost some characters (decimal exponents such as 1.5e16, the digit 0 in names like IF0, IR0, VCB0, VCE0, nE0, pB0, nC0, parentheses, quotes, and Greek letters in names). They are restored below. Greek-letter variable names (mu) are written in ASCII (mun_min, mup_min, mun0, mup0) because MATLAB identifiers cannot contain Greek letters.
• A few lines in the scan were printed upside-down or reversed; they are rebuilt from the corresponding physics equation, and noted where this happened.
• Units are CGS-flavored: cm, cm${}^{-3}$, cm${}^2$/V-s, F/cm, with energies in eV.

---

M.1 Exercise 10.2: BJT_Eband (equilibrium energy band diagram of a BJT)

Purpose. Plots $E_C$, $E_V$, $E_i$ and $E_F$ versus position through an emitter-base-collector structure at equilibrium, for any combination of $n$ and $p$ doping (npn, pnp, or even odd combinations). The user supplies a vector DOPING = [NE NB NC] where the sign encodes the type (positive = $p$-type, negative = $n$-type) and the magnitude is the concentration in cm${}^{-3}$. Authored originally by Aaron Luft as a course project for Prof. Gerry Neudeck; major revisions by R. F. Pierret.

The physics inside.

1. Built-in potentials of each junction from the Fermi-level offsets:

$V_{bi}={\left(E_i-E_F\right)}_{\text{side 1}}-{\left(E_i-E_F\right)}_{\text{side 2}},(E_i-E_F)=\pm kT\ln \frac{N}{n_i}$

2. Depletion widths of an abrupt junction between regions with concentrations $N_1,N_2$:

$x_1=\sqrt{\frac{2K_S{\epsilon }_0}{q}\frac{N_2{V}_{bi}}{N_1(N_1+N_2)}}$

3. Quasineutral base width $W=W_B-x_{B,EB}-x_{B,CB}$ (error if the base is totally depleted).
4. Parabolic band bending in each depletion region, plotted as $E_C=-V(x)+E_G/2+V_{max}/2$ and $E_V=E_C-E_G$ with $E_i$ at midgap.
5. The two junction plots are stitched together with an offset so the bands are continuous across the base.

% BJT Equilibrium Energy Band Diagram Generator
% This program plots out the BJT equilibrium energy band diagram.
% Original version authored by Aaron Luft as a course project for Prof. Gerry Neudeck.
% Major revisions by R.F. Pierret.
% DOPING = [NE NB NC] must be defined before running (sign: +p-type, -n-type).
 
WB = 1.0e-4;      % Total base width in cm; 1.0e-4 cm = 1 micrometer
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 constant of Si at 300 K
ni = 1.0e10;      % intrinsic conc. of silicon at 300 K
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))];
Ei_collector = [(sB*k*T*log(NB/ni)) (-sC*k*T*log(NC/ni))];
Vbi  = [sum(Ei_emitter) sum(Ei_collector)];
svbi = sign(Vbi);
Vbi  = abs(Vbi);
 
% Depletion width on emitter side of EB junction
xE   = sqrt(2*KS*e0/q*NB*Vbi(1)/(NE*(NE+NB)));
% Depletion width on base side of EB junction
xBeb = sqrt(2*KS*e0/q*NE*Vbi(1)/(NB*(NE+NB)));
% Depletion width on base side of CB junction
xBcb = sqrt(2*KS*e0/q*NC*Vbi(2)/(NB*(NC+NB)));
% Depletion width on collector side of CB junction
xC   = sqrt(2*KS*e0/q*NB*Vbi(2)/(NC*(NB+NC)));
W = WB - xBeb - xBcb;
if W < 0
    error('For the given DOPING and WB, the base is totally depleted.')
end
 
if (xC > xE)      % Adjust the x-axis for optimum 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], [0 VMAX], 'w');
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;
Vx1 = sVx*(Vbi(1) - q*NB.*(xBeb-x).^2/(2*KS*e0).*(x<=xBeb)).*(x>=0);
Vx2 = sVx*(q*NE.*(x+xE).^2/(2*KS*e0).*(x<0).*(x>=-xE));
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;
LEc = Ec(1);
LEv = Ev(1);
LEi = Ei(1);
% Plot V vs x
plot(x, Ec);            % Ec
plot(x, Ev);            % Ev
plot(x, Ei, 'w:');      % Ei
plot([xlft 0], [EF EF], 'w');                 % EF on left
plot([0 0], [0.15 VMAX-0.15], 'w-');          % Junction center
 
% CB JUNCTION
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*e0).*(x<=xC)).*(x>=0);
Vx2 = sVx*(q*NB.*(x+xBcb).^2/(2*KS*e0).*(x<0).*(x>=-xBcb));
Vx  = Vx1 + Vx2;        % V as a function of x
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 V vs x
plot(x, Ec);            % Ec
plot(x, Ev);            % Ev
plot(x, Ei, 'w:');      % Ei
plot([0 xrght+WB], [EF EF], 'w');             % EF on right
plot([WB WB], [0.15 VMAX-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';
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 = ['Energy band diagram for the ', REG, ' device'];
title(TITLE);

Reconstruction notes. In the scan, the lines defining xE, Vx1 and Vx2 (EB and CB) were printed reversed or garbled. They were rebuilt from the abrupt-junction depletion formulas above: on the lightly doped side the potential rises as $V_{bi}-qN(x_{edge}-x{)}^2/2K_S{\epsilon }_0$ and on the heavily doped side as $qN(x+x_{edge}{)}^2/2K_S{\epsilon }_0$, which match each other at $x=0$ and give total drop $V_{bi}$. The statement x = x + WB shifts the CB junction plot to the right by the base width.

---

M.2 Exercises 11.7 and 11.10: BJT and BJTplus (with BJT0 and BJTmod)

Purpose. Computes and plots the four classic BJT characteristics from the Ebers-Moll model: common-base input ($I_E$ vs $V_{EB}$), common-base output ($I_C$ vs $V_{CB}$), common-emitter input ($I_B$ vs $V_{EB}$) and common-emitter output ($I_C$ vs $V_{EC}$). BJT (Exercise 11.7) is the ideal version. BJTplus (Exercise 11.10) adds base-width modulation (Early effect) and carrier multiplication (avalanche in the collector junction). The book’s note: the italicized lines in the BJT listing are the additions that form BJTplus. Subprogram BJT0 is a run-time requirement of both; subprogram BJTmod must also be present for BJTplus.

Division of labor.

• BJT0: constants, device geometry, doping, mobility model, lifetimes, diffusion coefficients and lengths, equilibrium minority concentrations, and the Ebers-Moll parameters at $W=W_B$.
• BJTmod: recomputes the Ebers-Moll parameters with the voltage-dependent quasineutral base width $W=W_B-x_{nEB}-x_{nCB}$.
• BJT/BJTplus: menu-driven main program.

Equations implemented.

Ebers-Moll in terms of ${\alpha }_F,{\alpha }_R$ and the saturation currents $I_{F0}$, $I_{R0}$:

$I_E=I_{F0}\left(e^{V_{EB}/kT}-1\right)-{\alpha }_R{I}_{R0}\left(e^{V_{CB}/kT}-1\right)$

$I_C={\alpha }_F{I}_E-\left(1-{\alpha }_F{\alpha }_R\right)I_{R0}\left(e^{V_{CB}/kT}-1\right)$

(the second form is used for the common-base output).

With a finite base,

$f_B=\frac{D_B}{L_B}{p}_{B0}\coth \frac{W}{L_B},I_{F0}=qA\left[\frac{D_E}{L_E}{n}_{E0}+f_B\right],I_{R0}=qA\left[\frac{D_C}{L_C}{n}_{C0}+f_B\right]$

${\alpha }_F=\frac{qA(D_B/L_B)p_{B0}/\sinh (W/L_B)}{I_{F0}},{\alpha }_R=\frac{qA(D_B/L_B)p_{B0}/\sinh (W/L_B)}{I_{R0}}$

Breakdown models (empirical): $V_{CB0}=60(N_C/10^{16}{)}^{-3/4}$ volts, $V_{CE0}=V_{CB0}(1-{\alpha }_F{)}^{1/m}$ with $m=6$. Multiplication factor $M=1/[1-(|V_{CB}|/V_{CB0}{)}^m]$, applied as ${\alpha }_F\to M{\alpha }_F$.

BJT / BJTplus (main program)

% BJT Common Base/Emitter Input/Output Characteristics
% Modified version of BJT including Base-Width Modulation and
% Carrier Multiplication (BJTplus)
 
% Input Ebers-Moll Parameters
BJT0
 
% Limiting Voltages used in Calculation
VbiE = kT*log(NE*NB/ni^2);
VbiC = kT*log(NC*NB/ni^2);
VCB0 = 50; VCE0 = 50;
VCB0 = 60*(NC/1.0e16)^(-3/4);
m = 6;
VCE0 = VCB0*(1-aF)^(1/m);
 
% Choice of Characteristic and Special Calculations
format compact
echo on
% THIS PROGRAM COMPUTES BJT INPUT AND OUTPUT CHARACTERISTICS
% Subprograms BJT0 and BJTmod are run-time requirements.
% Modify entries in BJT0 to change device/material parameters.
% Modify axis commands to change plot min/max values.
echo off
close
c = menu('Specify the desired characteristic', 'Common Base Input', ...
    'Common Base Output', 'Common Emitter Input', 'Common Emitter Output');
j = input('Specify number of curves per plot.');
if c ~= 2,
    bw = input('Include base-width modulation? 1-Yes, 2-No.');
else
end
ii = 2;
if c == 4 & bw == 1,
    ii = input('Include impact ionization? 1-Yes, 2-No.');
else
end
 
% Calculation Proper
for i = 1:j,
 
    % Common-Base Input Characteristics
    if c == 1,
        VCB = -(i-1)*10;
        VEB = 0:0.005:VbiE;
        jj = length(VEB);
        if bw == 1,
            BJTmod        % Base-Width Modulation subprogram
        else
        end
        IE = (IF0.*(exp(VEB/kT)-1) - aR.*IR0.*(exp(VCB/kT)-1))*1.0e3;
        % 1.0e3 in the preceding equation changes IE units to mA
        if i == 1,
            plot(VEB, IE); axis([0.35 0.85 0 5]);
            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];
        jj = length(VCB);
        IC = (aF*IE - (1-aF*aR)*IR0*(exp(VCB/kT)-1))*(1.0e3);
        if i == 1,
            plot(-VCB, IC); axis([-VCB0/10 VCB0 0 1.3e3*IE]);
            grid; xlabel('-VCB (volts)'); ylabel('IC (mA)');
            text(5, 1.1e3*IE, 'IE step = 1 mA');
        else plot(-VCB, IC);
        end
    else
    end
 
    % Common-Emitter Input Characteristics
    if c == 3,
        VEC = (i-1)*5;
        VEB = 0:0.005:VbiE;
        jj = length(VEB);
        if bw == 1,
            VCB = VEB - VEC;
            BJTmod
        else
        end
        IB0 = (1-aF).*IF0 + (1-aR).*IR0;
        IB1 = (1-aF).*IF0 + (1-aR).*IR0.*exp(-VEC/kT);
        IB = (IB1.*exp(VEB/kT) - IB0)*(1.0e6);
        if i == 1,
            plot(VEB, IB); axis([.35 .85 -5 20]);
            grid; xlabel('VEB (volts)'); ylabel('IB (uA)');
        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);
        if bw == 1,
            VEB = 0;          % Neglect xnEB variation with bias
            VCB = VEB - VEC;
            BJTmod
        else
        end
        if ii == 1,
            M = 1.0./(1 - (-VCB/VCB0).^m);
            aF = M.*aF;
        else
        end
        IB0 = (1-aF).*IF0 + (1-aR).*IR0;
        IB1 = (1-aF).*IF0 + (1-aR).*IR0.*exp(-VEC/kT);
        IC = ((aF.*IF0 - IR0.*exp(-VEC/kT)).*(IB+IB0)./IB1 + IR0 - aF.*IR0)*(1.0e3);
        if i == 1,
            jA = length(VECA);
            plot(VEC, IC); axis([0 VCE0 0 2.5*IC(jA)]);
            grid; xlabel('VEC (volts)'); ylabel('IC (mA)');
            text(5, 2*IC(jA), 'IB step = 2.5 uA');
        else plot(VEC, IC);
        end
    else
    end
    hold on
end
hold off

What each branch does.

Menu choice	Sweeps	Curve parameter	Output
1. Common-base input	$V_{EB}$ from 0 to $V_{biE}$ in 5 mV steps	$V_{CB}=0,-10,-20,\dots$ V	$I_E$ (mA) vs $V_{EB}$; shows how base-width modulation shifts the curves
2. Common-base output	$V_{CB}$ from +2 V down to $-V_{CB0}$	$I_E$ stepped in 1 mA	$I_C$ (mA) vs $-V_{CB}$; nearly flat lines (high ${\alpha }_F$) and slightly negative-$V_{CB}$ saturation region
3. Common-emitter input	$V_{EB}$	$V_{EC}=0,5,10,\dots$ V	$I_B$ ($\mu$A) vs $V_{EB}$
4. Common-emitter output	$V_{EC}$ from 0 to $V_{CE0}$	$I_B$ stepped in 2.5 $\mu$A	$I_C$ (mA) vs $V_{EC}$; with the options on, shows Early-effect slope and avalanche upturn

Reconstruction notes. In the scan the comparison ii == 1 shows as i == 1 in the multiplication step; physics and the earlier prompt show it must test ii (the impact-ionization flag). The digit zero and letter O were confused in IF0, IR0, VCB0, VCE0, IB0; the zeros are restored. Axis command arguments, run together in the scan, were separated by position (for example [.35 .85 -5 20]).

BJT0 (parameters and Ebers-Moll constants)

% 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;      % A in cm^2
WB = 2.5e-4;      % WB in cm
T  = 300; kT = k*T;
% Material Parameters
ni = 1.0e10;
KS = 11.8;
NE = 1.0e18;
NB = 1.5e16;
NC = 1.5e15;
% Mobility Fit Parameters
NDref = 1.3e17; NAref = 2.35e17;
mun_min = 92;   mup_min = 54.3;
mun0 = 1268;    mup0 = 406.9;
an = 0.91;      ap = 0.88;
muE = mun_min + mun0./(1 + (NE/NDref).^an);
muB = mup_min + mup0./(1 + (NB/NAref).^ap);
muC = mun_min + mun0./(1 + (NC/NDref).^an);
TauE = 1.0e-7;
TauB = 1.0e-6;
TauC = 1.0e-6;
DE = kT*muE;
DB = kT*muB;
DC = kT*muC;
LE = sqrt(DE*TauE);
LB = sqrt(DB*TauB);
LC = sqrt(DC*TauC);
nE0 = ni^2/NE;
pB0 = ni^2/NB;
nC0 = ni^2/NC;
% Ebers-Moll Parameter Computation (W = WB)
W  = WB;
fB = (DB/LB)*pB0*(cosh(W/LB)/sinh(W/LB));
IF0 = q*A*((DE/LE)*nE0 + fB);
IR0 = q*A*((DC/LC)*nC0 + fB);
aF = q*A*(DB/LB)*(pB0/sinh(W/LB))/IF0;
aR = q*A*(DB/LB)*(pB0/sinh(W/LB))/IR0;

Explanation. With $N_E=10^{18}$, $N_B=1.5\times 10^{16}$, $N_C=1.5\times 10^{15}$ (cm${}^{-3}$), the default is a heavily doped emitter, moderately doped base and lightly doped collector (minority concentrations $n_{E0}$, $p_{B0}$, $n_{C0}$ and the quantities named in the List of Symbols). Mobility follows the Caughey-Thomas style fit

$\mu ={\mu }_{min}+\frac{{\mu }_0}{1+(N/N_{ref}{)}^{\alpha }}$

with the electron and hole constants listed. $D=\mu kT/q$ (with $kT$ in eV, so $D=kT\mu$), $L=\sqrt{D\tau }$.

BJTmod (base-width-modulation subprogram)

% Base-width modulation: recalculation of Ebers-Moll parameters
% Subprogram BJTmod
xnEB = sqrt((2*KS*e0/q)*(NE/(NB*(NE+NB)))*(VbiE-VEB));
xnCB = sqrt((2*KS*e0/q)*(NC/(NB*(NC+NB)))*(VbiC-VCB));
W  = WB - xnEB - xnCB;
fB = (DB/LB)*pB0*(cosh(W/LB)./sinh(W/LB));
IF0 = q*A.*((DE/LE)*nE0 + fB);
IR0 = q*A.*((DC/LC)*nC0 + fB);
aF = q*A*(DB/LB)*(pB0./sinh(W/LB))./IF0;
aR = q*A*(DB/LB)*(pB0./sinh(W/LB))./IR0;

Explanation. xnEB and xnCB are the depletion widths penetrating into the base from the emitter and collector junctions at the present bias, $x=\sqrt{\frac{2K_S{\epsilon }_0}{q}\frac{N_{\text{other}}}{N_B(N_{\text{other}}+N_B)}(V_{bi}-V)}$. Subtracting them from $W_B$ gives the voltage-dependent quasineutral base width $W$, and the element-wise operators (./, .*) let all voltage points be processed as arrays. Increasing reverse collector bias shrinks $W$, raises ${\alpha }_F$ and gives the upward slope of the output curves (Early effect).

---

M.3 Exercise 16.5: MOS_CV (exact-charge MOS capacitor C-V curves)

Purpose. Computes and plots the low-frequency and/or high-frequency C-V characteristics of a $p$-type MOS-C using the exact-charge relationships of Appendix C. The user chooses the plot type (low-f, high-f or both), enters $N_A$, $x_o$, and the gate-voltage window. Subprogram CVintgrd provides the integrand of Eq. (C.4) and is a run-time requirement.

Algorithm.

1. $U_F=\ln (N_A/n_i)$, $L_D=\sqrt{kT{K}_S{\epsilon }_0/(2q{n}_i)}$ (with $kT$ in eV).
2. Create the vector $U_s$ from $U_F-21$ to $U_F+21$ in steps of 0.5.
3. Evaluate $F(U_s,U_F)$ and the gate voltage $V_G=kT\left[U_s+\mathrm{sgn}(U_s)\frac{K_S{x}_o}{K_O{L}_D}F\right]$ (C.10).
4. Low-frequency: $W_L=\mathrm{sgn}(U_s)L_{D}2F/\text{DENOML}$ with DENOML $=e^{U_F}(1-e^{-U_s})+e^{-U_F}(e^{U_s}-1)$, then $C_L/C_o=1/[1+K_O{W}_L/(K_S{x}_o)]$.
5. High-frequency (only for $U_s\ge 3$): accumulate the integral ${\int }_0^{U_s}$(C.4) with numerical quadrature, form $\Delta$ (C.3b), $W_H$ from (C.2c), then $C_H/C_o$. Below $U_s=3$ the two curves coincide.
6. Plot, label, and annotate with $N_A$ and $x_o$.

% 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
s = menu('Choose the desired plot', 'Low-f C-V', 'High-f C-V', 'Both');
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 Constants
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));
VG = kT*(US + (US./abs(US)).*(KS*xo)/(KO*LD).*F);
 
% Low-frequency Capacitance Computation
DENOML = exp(UF).*(1-exp(-US)) + exp(-UF).*(exp(US)-1);
WL = (US./abs(US)).*LD.*(2*F)./DENOML;
cL = 1./(1 + (KO*WL)/(KS*xo));
 
% 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('CVintgrd', 0.001, US(ii), 0.001);
            nn = 1;
        else
            INTG = INTG + quad('CVintgrd', US(ii-1), US(ii), 0.001);
        end
        if US(ii) < 3,
            cH(ii) = cL(ii);
        else
            % d implements Delta of Eq. (C.3b): (exp(Us)-Us-1)/F times the
            % prefactor of (C.3b) times the integral INTG
            d = (exp(US(ii))-US(ii)-1)/F(ii) * INTG;   % prefactor omitted (illegible in scan)
            DENOMH = exp(UF).*(1-exp(-US(ii))) + exp(-UF).*((exp(US(ii))-1)*(1+d));
            WH = LD.*(2*F(ii))./DENOMH;
            cH(ii) = 1/(1 + (KO*WH)/(KS*xo));
        end
    end
else
end
 
% Plotting the Result
if s == 1,
    plot(VG, cL);
elseif s == 2,
    plot(VG, cH);
else
    plot(VG, cL, '--', VG, cH);
    text(0.8*xmin, .17, '--- Low-f', 'color', [0 0 1]);
    text(0.8*xmin, .12, '___ High-f', 'color', [1 0 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

function [y] = CVintgrd(U)
% Integrand of Eq. (C.4): (1-exp(-U))(exp(U)-U-1)/(2 F^3)
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);

Reconstruction notes. In the scan the lines for VG, cL, d and cH(ii) were printed reversed and are rebuilt from (C.10), (C.1) and (C.3b). The decisive caveat is the line for d: the scan’s version is unreadable in its prefactor, so the line above shows the legible factors of (C.3b) only and is annotated accordingly. Everything else (flow control, quadrature accumulation, plot annotations) is intact. The first quadrature call in the scan has swapped limits; the version above integrates from $0.001$ (avoiding the $0/0$ at $U=0$) up to US(ii) and then accumulates segment by segment. If $U_s$ ever equals exactly 0 in the vector US, the sign factor US./abs(US) yields NaN; use the flat-band formula (C.2b) or shift the grid slightly.

---

End Matter: Index and Document Notes

Index (book pages 781 to 792, scan pages 804 to 815)

The subject index is an alphabetical locator rather than new technical content. It is organized from A (abrupt junction, absorption coefficient, accumulation, acceptors, admittance, aluminum, amorphous, angular momentum, annealing, avalanche) to Z (Zener process, zincblende unit cell). Its largest entries are the book’s main device topics, each with sub-entries and page ranges:

Heavily indexed topic	Where it lives (book pages)
BJT (biasing modes, gain, Ebers-Moll, hybrid-pi, switching)	371 to 459
pn junction and pn diode (electrostatics, ideal diode, breakdown, C-V, small signal, transients)	195 to 340
MOS-C (biasing, C-V, deep depletion, exact solution)	563 to 600, 749 to 754
MOSFET (square law, bulk-charge, charge-sheet, small-signal, short-channel)	611 to 637, 674 to 707, 755 to 756
J-FET, MESFET	525 to 556
Schottky diode and MS contacts	477 to 500
Photodiode, solar cell, LED	347 to 368
SCR and other PNPN devices	463 to 476
Carrier properties (mobility, effective mass, Fermi function, concentrations)	32 to 67, 75 to 137
Quantum mechanics and Bohr atom	733 to 748
Debye length	749 to 751

Useful navigation tips: C-V topics include exact-charge relationships at pages 753 to 754; hydrogen atom quantum states at 741 to 745; Debye length at 750 to 751; the exact-charge and charge-sheet MOSFET relationships at 755 to 756.

The last page

The final scan page (816) is not textbook content. It is a provenance notice from the digitization source (a DuXiu-collection scan archived around 2023 to 2024, with embedded page images and file metadata such as 792 main pages and 815 total pages). It carries no technical information and can be ignored.

---

Study Notes for This Part

• Appendix A is where the core ideas begin: Planck’s quantized energies, Bohr’s $E_n=-13.6/n^2$ eV, de Broglie’s $\lambda =h/p$, Schrödinger’s equation and its five postulates, hydrogen quantum numbers $(n,l,m,s)$, the Pauli principle, and the Si configuration $1s^{2}2s^{2}2p^{6}3s^{2}3p^2$ (ten-electron core plus four valence electrons).
• Appendix B is one beautiful trick: $d^{2}U/d{x}^2=f(U)$ is integrated once by multiplying by $dU/dx$, giving $\mathcal{E}={\varsigma }_s(kT/q)F/L_D$ and a quadrature for $x(U)$.
• Appendix C turns B into a parametric C-V curve through $W_{\text{eff}}$ and $V_G(U_s)$, with the low- and high-frequency difference carried by $\Delta$.
• Appendix D uses the same $F$ with a channel quasi-Fermi shift $\xi$ to obtain the drain current, evaluated between $U_{s0}$ and $U_{sL}$.
• Appendix E is a reference table; remember that symbols are overloaded between appendices.
• Appendix M converts the equations into four MATLAB experiments; changing a single parameter in BJT0 or the inputs of MOS_CV is the best way to build intuition.