_c2

3.2 Diffusion

3.2.1 Definition and visualization

Diffusion is the tendency of particles to spread out, owing to their random thermal motion, migrating (on the macroscopic scale) from regions of high concentration to regions of low concentration. Left alone it ends in a uniform distribution. The diffusing particles need not be charged: it is random thermal motion, not mutual repulsion, that drives the process.

Everyday example: open a perfume bottle in a corner of a room. Even in perfectly still air, random molecular motion spreads the scent everywhere in a short time.

A toy model you can compute by hand (Fig. 3.11). Take a 1-D box with four compartments and $1024$ particles. Every $\tau$ seconds every particle jumps to a neighboring compartment, left or right with equal probability (a jump into a wall is reflected back to the starting compartment). At $t=0$ all particles are in compartment 1.

Time	Compartments 1, 2, 3, 4	How it arises
$0$	1024, 0, 0, 0	start
$\tau$	512, 512, 0, 0	half of 1024 jump right; half try to jump left and are reflected back
$2\tau$	512, 256, 256, 0	from 1: 256 go to 2 and 256 reflect (stay in 1); from 2: 256 go to 3, 256 return to 1
$3\tau$	384, 384, 128, 128	repeating the halve-and-pass rule
$6\tau$	roughly 320, 320, 192, 192 (nearly uniform)	system approaches 256 each

The rule at each step: each compartment hands half its particles to each neighbor. Notice that even though every individual particle moves randomly with no preference, the crowd moves from crowded to empty: randomness plus a gradient produces a systematic flow. The reason is purely statistical: from a crowded compartment many particles jump out and few jump in.

In semiconductors the motion is 3-D and not compartmentalized, but the macroscopic result is the same. Since electrons and holes are charged, their diffusion constitutes a current (Fig. 3.12): both electrons and holes flow from high-concentration to low-concentration regions, but because their charges are opposite, the currents point opposite ways.

Exercise 3.3: “DiffDemo” MATLAB simulation

Problem. (a) Run the program below, which pseudo-animates a 1-D particle system like the one in the text. (b) Re-run with all particles initially in the middle box. (c) Experiment with initially putting particles in several boxes, with the pause time, and with more boxes.

Logic. y holds the number of particles in each of 5 boxes. At each time step, every box sends half of its particles to each neighbor; at the two end boxes, the half that would jump out of the system is reflected and returns to the same box. The new content of box $k$ is therefore half of the left neighbor plus half of the right neighbor, and for the walls the “missing neighbor” is replaced by the box itself. round keeps the counts integer. bar makes the histogram; pause slows the animation.

% Simulation of Diffusion (DiffDemo)
% One-dimensional system, right and left jumps equally probable
close
x = [0.5 1.5 2.5 3.5 4.5];
y = [1.0e6 0 0 0 0];          % initial condition: all particles in box 1
[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;                        % number of jumps; increase for a longer run
 
for ii = 1:N
    % Diffusion step: each box passes half to each neighbour
    bin(1) = round(y(1)/2 + y(2)/2);   % left wall reflects half of box 1
    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);   % right wall reflects half of box 5
    y = bin;
    [xp,yp] = bar(x,y);
    plot(xp,yp); text(0.5,1.1e6,['t=',num2str(ii)]);
    axis([0 5 0 1.2e6]);
    pause(0.5)
end

Part (b): change the initial condition to y = [0 0 1.0e6 0 0];. The pile in the middle box spreads symmetrically left and right, reaching the walls after two jumps, reflecting, and flattening out. Part (c): e.g. y = [5e5 0 0 0 5e5]; gives two piles moving inward and merging in the middle. In every experiment the final state is nearly uniform ($2\times 10^5$ per box for $10^6$ particles in 5 boxes), as in Fig. E3.3 (the $t=15$ panel).

3.2.2 Hot-point probe measurement

Before the diffusion current, a neat application. The hot-point probe quickly tells whether a semiconductor sample is n- or p-type (essential before any resistivity-to-doping conversion).

Equipment (Fig. 3.13a): a hot probe (a soldering iron will do), a cold probe (an ordinary multimeter probe), and a center-zero milliammeter between them. Touch both probes to the sample; the meter needle deflects to one side for n-type and to the other side for p-type.

Explanation (Fig. 3.13b). The hot probe creates a larger population of energetic carriers near its tip (mostly holes in p-type, electrons in n-type, since those are the majority carriers). Because carrier density is higher near the hot tip than elsewhere, diffusion spreads these carriers outward, away from the hot probe. The net effect:

• p-type: holes leave the hot-probe region, leaving net negative charge (ionized acceptors, unbalanced) there; the hot probe becomes negative with respect to the cold one.
• n-type: electrons leave, leaving net positive charge (ionized donors) at the hot probe.

So the current through the meter flows in opposite directions for n and p. It is a gorgeous everyday demonstration that thermal gradient $\to$ carrier gradient $\to$ diffusion current.

3.2.3 Diffusion and total currents

Diffusion current. Diffusion occurs only if more particles sit at one place than at another, i.e., there is a nonzero concentration gradient ($\nabla p\ne 0$ for holes, $\nabla n\ne 0$ for electrons). Quite naturally, the steeper the gradient, the larger the flow. Quantitative analysis confirms this and gives Fick’s law:

\mathbf{F} = -D\,\nabla\eta \tag{3.16}

where $\mathbf{F}$ is the flux (particles per cm${}^2$ per second crossing a plane perpendicular to the flow), $\eta$ is the particle concentration, and $D$ is the (positive) diffusion coefficient. The minus sign says “flow goes down the gradient”. Multiplying the flux by the carrier charge gives current density:

\boxed{\mathbf{J}_{P|diff} = -qD_P\,\nabla p} \tag{3.17a}

\boxed{\mathbf{J}_{N|diff} = +qD_N\,\nabla n} \tag{3.17b}

$D_P$ and $D_N$ (units cm${}^2$/s) are the hole and electron diffusion coefficients. Sign check against Fig. 3.12: for $dp/dx>0$ and $dn/dx>0$ both carriers flow toward $-x$. Holes moving toward $-x$ carry positive current toward $-x$: $J_P<0$ (matches $-q{D}_{P}dp/dx$). Electrons moving toward $-x$ carry current toward $+x$: $J_N>0$ (matches $+q{D}_{N}dn/dx$).

Total currents. Drift and diffusion act simultaneously. Summing (3.4) and (3.17):

\boxed{\mathbf{J}_P = \mathbf{J}_{P|drift} + \mathbf{J}_{P|diff} = q\mu_p\,p\,\boldsymbol{\mathcal{E}} - qD_P\,\nabla p} \tag{3.18a}

\boxed{\mathbf{J}_N = \mathbf{J}_{N|drift} + \mathbf{J}_{N|diff} = q\mu_n\,n\,\boldsymbol{\mathcal{E}} + qD_N\,\nabla n} \tag{3.18b}

\boxed{\mathbf{J} = \mathbf{J}_N + \mathbf{J}_P} \tag{3.19}

These “double-boxed” relations are used (directly or indirectly) in nearly every device analysis.

3.2.4 Relating diffusion coefficients and mobilities

The $D$‘s are the constants of the motion for diffusion, but we need not study them as extensively as the $\mu$‘s, since the two are tied by the Einstein relationship. Its derivation needs two preliminary facts about nonuniformly doped semiconductors at equilibrium.

Fact 1: the Fermi level is constant in equilibrium. Consider a nondegenerate n-type sample whose doping varies with position (Fig. 3.14a). Its equilibrium band diagram is as in Fig. 3.14(b): $E_F$ is closer to $E_C$ where $N_D$ is larger, and bands bend. The key rule:

$\frac{d{E}_F}{dx}=0(\text{also in }y,z)\text{in equilibrium}$

Why? Think of electrons hopping between states of equal energy at neighboring positions. The probability that a state of energy $E$ is filled, $f(E)$, must be the same everywhere; otherwise carriers would preferentially transfer from the better-filled to the emptier states and create a net current, contradicting equilibrium. Since $f(E)=1/[1+e^{(E-E_F)/kT}]$ (Eq. 2.7), a position-independent $f(E)$ requires a position-independent $E_F$. On an equilibrium diagram $E_F$ is a horizontal line.

Fact 2: equilibrium is a dynamic balance of drift and diffusion. In equilibrium, the total current is zero and the electron and hole systems are decoupled, so $J_N=0$ and $J_P=0$ separately. By (3.18) that does not mean drift and diffusion each vanish; it means they are equal in magnitude and opposite in direction. They vanish individually only for uniform doping: $\nabla n=\nabla p=0$ and $\mathcal{E}=0$.

In Fig. 3.14, $N_D$ increases with $x$, so $n$ has a gradient and an electron diffusion current flows in $+x$ (electrons diffuse toward $-x$). The band bending implies a built-in field pointing toward $-x$, causing electron drift current also in the direction of $\mathcal{E}$, i.e., $-x$. The two components have opposite polarity, and in equilibrium equal magnitude. In fact the built-in field exists in order to oppose the diffusive tendency caused by nonuniform doping.

Take-away: even at equilibrium, nonuniform doping produces carrier-concentration gradients, a built-in electric field, and nonzero (but cancelling) current components. This idea returns in the pn junction.

Derivation of the Einstein relationship. One dimension, nondegenerate, nonuniformly doped, equilibrium. Focus on electrons. Net electron current is zero:

J_N = J_{N|drift} + J_{N|diff} = q\mu_n n\mathcal{E} + qD_N\frac{dn}{dx} = 0 \tag{3.20}

Use the band-bending relation for the field,

\mathcal{E} = \frac{1}{q}\frac{dE_i}{dx} \tag{3.21}

(same as 3.15) and Boltzmann statistics for the electron density,

n = n_i\,e^{(E_F - E_i)/kT} \tag{3.22}

Differentiate (3.22) with respect to $x$, using $d{E}_F/dx=0$ (Fact 1):

$\frac{dn}{dx}=n_i{e}^{(E_F-E_i)/kT}\cdot \left(-\frac{1}{kT}\frac{d{E}_i}{dx}\right)$

\frac{dn}{dx} = -\frac{n}{kT}\frac{dE_i}{dx} \tag{3.23}

Substitute (3.21) and (3.23) into (3.20):

$q{\mu }_{n}n\left(\frac{1}{q}\frac{d{E}_i}{dx}\right)+q{D}_N\left(-\frac{n}{kT}\frac{d{E}_i}{dx}\right)=0$

Collect terms:

\left(n\,\frac{dE_i}{dx}\right)\left[\mu_n - \frac{q}{kT}D_N\right] = 0 \tag{3.24}

Because the doping is nonuniform, $nd{E}_i/dx\ne 0$, so the bracket must vanish:

\boxed{\frac{D_N}{\mu_n} = \frac{kT}{q}} \tag{3.25a}

The same argument for holes gives

\boxed{\frac{D_P}{\mu_p} = \frac{kT}{q}} \tag{3.25b}

Although derived in equilibrium, the relation holds also out of equilibrium (more elaborate arguments show it), as long as the material is nondegenerate (degenerate materials give slightly modified forms).

Numbers. $kT/q$ is a voltage; at 300 K it equals $0.0259$ V. For $N_D=10^{14}$ cm${}^{-3}$ Si at 300 K:

$D_N=\frac{kT}{q}{\mu }_n=(0.0259)(1358)=35.2{\text{cm}}^2/\text{s}$

Memory trick: ”$D$ over $\mu$ equals $kT$ over $q$”; it rhymes, and it rhymes upside down too (”$\mu$ over $D$ equals $q$ over $kT$”).

Physical meaning: mobility measures how readily carriers respond to a push (field), diffusion measures how readily they spread out on their own. Both are governed by the same scattering, hence the tie. And the proportionality constant is the thermal energy $kT$ because diffusion is thermal motion.

Exercise 3.4: Using the Exercise 3.2 diagram

Problem. Continue with the semiconductor of Exercise 3.2. (a) The semiconductor is in equilibrium; how do you tell from the diagram? (b) What are $J_N$ and $J_P$ at $x=\pm L/2$? (c) Roughly sketch $n$ and $p$ vs. $x$. (d) Is there an electron diffusion current at $x=\pm L/2$? If yes, in which direction? (e) Using the field found in Exercise 3.2, is there an electron drift current at $x=\pm L/2$? If yes, in which direction? (f) What is the hole diffusion coefficient $D_P$ in the $x>L$ region?

Solution.

(a) $E_F$ is invariant with position (a horizontal line). By Fact 1 this is the signature of equilibrium.

(b) In equilibrium, net currents are identically zero everywhere:

$J_N=0,J_P=0\text{at both }x=\pm L/2$

(c) Use $n=n_i{e}^{(E_F-E_i)/kT}$ and $p=n_i{e}^{(E_i-E_F)/kT}$ with $E_F$ flat. Where $E_i$ lies below $E_F$ (near $x=0$), $n>n_i>p$; where $E_i$ lies above $E_F$ ($|x|\ge L$), $p>n_i>n$. On a log scale, $n(x)$ is a hump centered at $x=0$ with maximum $n=n_i{e}^{0.28/0.0259}\approx 5\times 10^{14}$ cm${}^{-3}$, falling to $\approx n_i^2/N_A\approx 2\times 10^5$ cm${}^{-3}$ for $|x|>L$. $p(x)$ is the mirror image: a dip at $x=0$ and a plateau $\approx 5\times 10^{14}$ for $|x|>L$. The two curves cross at $n=p=n_i$ where $E_i=E_F$.

(d) Yes, because $dn/dx\ne 0$ at both points. From (c): $dn/dx>0$ at $x=-L/2$, so $J_{N|diff}=q{D}_{N}dn/dx>0$, flowing in $+x$. At $x=+L/2$, $dn/dx<0$, so $J_{N|diff}$ flows in $-x$.

(e) Yes. $n$ and $\mathcal{E}$ are both nonzero at $x=\pm L/2$, so $J_{N|drift}=q{\mu }_{n}n\mathcal{E}\ne 0$. It has the direction of $\mathcal{E}$: using the Exercise 3.2(b) field, it flows in $-x$ at $x=-L/2$ and in $+x$ at $x=+L/2$. These exactly cancel the diffusion components, as required by $J_N=0$:

	$x=-L/2$	$x=+L/2$
$J_{N	diff}$	$\to$
$J_{N	drift}$	$\leftarrow$

(f) In Exercise 3.2(d) the $x>L$ region had $N_A\approx 5\times 10^{14}$ cm${}^{-3}$ and ${\mu }_p=459$ cm${}^2$/V s. By the Einstein relationship:

$D_P=\frac{kT}{q}{\mu }_p=(0.0259)(459)=11.9{\text{cm}}^2/\text{s}$

---

3.3 Recombination-Generation

3.3.1 Definition and visualization

When a semiconductor is pushed off equilibrium, an excess or deficit of carriers (relative to the equilibrium values) is created. Recombination-generation (R-G) is nature’s order-restoring mechanism: it stabilizes the excess or deficit (if the push continues) or eliminates it (if the push stops). Since devices work in nonequilibrium conditions, R-G often shapes device characteristics.

• Recombination: a process in which electrons and holes (carriers) are annihilated.
• Generation: a process in which electrons and holes are created.

Unlike drift and diffusion, “R-G” is not one process but a family. The most common members (Fig. 3.15):

(a) Band-to-band recombination. A conduction-band electron falls straight into a valence-band hole and both vanish (“zap!”). The released energy typically becomes a photon (light).

(b) R-G center (indirect) recombination. A third party is involved: an R-G center, a lattice defect or a special impurity (e.g., gold in Si). It provides an allowed level $E_T$ near midgap. The recombination is a two-step process:

1. An electron wanders near the center, is captured by its potential well, loses energy, and is trapped.
2. A hole arrives, is attracted to the trapped electron, loses energy, and annihilates it (equivalently the electron loses energy a second time and drops into a valence-band hole).

The energy is typically released as heat (lattice vibrations). The key property of R-G centers is the near-midgap level; that distinguishes them from donors and acceptors (whose levels sit close to a band edge). Fig. 3.16 lists midgap-ish levels from Au, Cu, Mn, Cr, Fe in Si. R-G centers are present even in the purest material, but at concentrations $N_T$ normally far below the dopant concentrations.

Analogy: a stepping-stone. Instead of leaping a wide river (the whole band gap) in one go, an electron hops onto a rock in the middle (the midgap level) and then onto the far bank. Two small hops are far more likely than one huge leap.

(c) Auger recombination. Band-to-band recombination happens simultaneously with a collision between two like carriers. The released energy goes to the surviving carrier, which then “thermalizes”, losing the energy in small steps by lattice collisions (the staircase in Fig. 3.15c). Its rate rises with carrier concentration (more collisions), so it matters in heavily doped regions.

Generation processes (the reverses of the above):

(d) Band-to-band generation: a valence electron is excited to the conduction band either by thermal energy (direct thermal generation) or by absorbing light (photogeneration).

(e) R-G-center generation: thermally assisted creation of carriers through the midgap level.

(f) Impact ionization: the inverse of Auger recombination. A highly energetic carrier collides with the lattice and creates an electron-hole pair. It occurs in high-field regions of devices and will matter for breakdown in pn junctions.

3.3.2 Momentum considerations

All these processes occur all the time, even in equilibrium; the question is the rate. Usually we need only the dominant process. Auger matters only at high doping; impact ionization only in high-field regions; photogeneration only under illumination. So which pair dominates in the low-field, non-degenerately doped, room-temperature, dark case? The candidates are thermal band-to-band R-G and R-G through centers.

The simple energy diagram suggests band-to-band should win, but the energy picture misleads, because crystal momentum must also be conserved in every R-G event. This requires $E$-$k$ diagrams (allowed electron energy vs. crystal momentum $k$, Fig. 3.17):

• Direct semiconductors (e.g., GaAs): the conduction-band minimum and valence-band maximum occur at the same $k$ (here $k=0$).
• Indirect semiconductors (e.g., Si, Ge): the conduction-band minimum is displaced to $k\ne 0$.

Two kinds of transition quanta matter:

• Photons have negligible momentum (but sizable energy $\sim E_G$): a photon-assisted transition is essentially vertical on the $E$-$k$ plot.
• Phonons (lattice-vibration quanta) have small energy (about 10-50 meV) but sizable momentum: a phonon-assisted transition is essentially horizontal.

Electrons and holes sit very near the band-edge extremes (Fig. 3.18), consistent with the carrier distributions studied before.

Hence:

• Direct material: electrons and holes sit at nearly the same $k$, so recombination needs almost no momentum change; emission of a photon conserves both energy and momentum. Band-to-band recombination is efficient. This is the process behind LEDs and junction lasers.
• Indirect material: recombination needs a large $k$ change. A photon alone cannot supply it, so a phonon must be emitted or absorbed simultaneously with the photon: a three-body event of low probability. Band-to-band recombination is therefore negligible compared with R-G-center recombination in Si and Ge.

Why silicon makes lousy LEDs (and GaAs makes good ones): the band gap in Si is indirect. Even in direct materials, the R-G-center route is often still the dominant recombination mechanism. Since R-G centers matter universally and most devices are Si (indirect), the quantitative treatment below focuses on R-G centers (indirect thermal R-G).

3.3.3 R-G statistics

“R-G statistics” is the mathematical description of R-G. Importantly, this is not a current-density relation: an R-G event occurs at a localized position and so does not by itself transport charge. Rather, R-G changes the carrier concentrations, and thereby indirectly affects current. What we need is the time rate of change $\partial n/\partial t$ or $\partial p/\partial t$.

Photogeneration

Light incident on a semiconductor (Fig. 3.19) is partly reflected and partly transmitted. For monochromatic light of wavelength $\lambda$ and frequency $\nu$, if the photon energy exceeds the gap, $h\nu >E_G$, it is absorbed and creates electron-hole pairs along its path. The intensity decays as

I = I_0\,e^{-\alpha x} \tag{3.26}

where $I_0$ is the intensity just inside the surface ($x=0^{+}$) and $\alpha$ is the absorption coefficient, a material- and wavelength-dependent quantity (Fig. 3.20). Since each absorbed photon makes one electron-hole pair (Fig. 3.15d), the generation rate has the same $e^{-\alpha x}$ dependence, and electrons and holes are created in equal numbers:

\left.\frac{\partial n}{\partial t}\right|_{light} = \left.\frac{\partial p}{\partial t}\right|_{light} = G_L(x,\lambda) \tag{3.27}

G_L(x,\lambda) = G_{L0}\,e^{-\alpha x} \tag{3.28}

$G_L$ is the photogeneration rate (number per cm${}^3$ per second). The subscript “light” is needed because $n$ and $p$ can be changed by several processes at once.

Two common simplifying assumptions: (i) uniform illumination throughout the sample; (ii) all the light absorbed in an infinitesimally thin surface layer. The characteristic depth is $1/\alpha$ (the average penetration depth).

• When $h\nu \approx E_G$, $\alpha$ is small and $1/\alpha$ is huge (about 1 cm in Si), so $e^{-\alpha x}\approx 1$ over any practical depth: $G_L\approx$ constant, uniform absorption.
• At $\lambda =0.4\mu$m in Si, $\alpha \approx 10^5$ cm${}^{-1}$ (from Fig. 3.20), so the penetration depth is $1/\alpha \sim 10^{-5}$ cm $=0.1\mu$m, a few thousand angstroms at most: surface absorption.

Indirect thermal recombination-generation

The general R-G-center statistics are beyond our scope, but we can motivate the special-case result most problems use. Define:

Symbol	Meaning
$n_0,p_0$	carrier concentrations in equilibrium
$n,p$	carrier concentrations under arbitrary conditions
$\Delta n=n-n_0,\Delta p=p-p_0$	deviations from equilibrium (positive = excess, negative = deficit)
$N_T$	number of R-G centers per cm${}^3$

Low-level injection is required for the special case. It means the perturbation is small relative to the majority carrier density:

$\Delta p\ll n_0,n\approx n_0\text{(n-type)}$

$\Delta n\ll p_0,p\approx p_0\text{(p-type)}$

Example: $N_D=10^{15}$ cm${}^{-3}$ Si at room temperature, perturbation $\Delta p=\Delta n=10^{10}$ cm${}^{-3}$. Then $n_0=N_D=10^{15}$ and $p_0\approx n_i^2/N_D=10^5$ cm${}^{-3}$. So $n=n_0+\Delta n\approx n_0$ and $\Delta p=10^{10}\ll n_0$: low-level injection. But look at the minority carriers: $p=p_0+\Delta p\approx 10^{10}\gg p_0=10^5$! Under low-level injection the majority concentration is essentially unperturbed, while the minority concentration can change by many orders of magnitude.

Heuristic derivation of the rate. Consider (Fig. 3.21) an n-type sample with an excess of holes $\Delta p\ll n_0$ relaxing toward equilibrium via R-G centers. What controls the recombination rate $\partial p/\partial t{|}_R$?

• To destroy a hole, it must go from the valence band to an electron-filled R-G center. More filled centers $\Rightarrow$ faster recombination. In equilibrium $E_F$ is comfortably above $E_T$, so essentially all centers are filled; with $\Delta p\ll n_0$, electrons vastly outnumber holes and instantly refill any vacated center. So the number of filled centers stays $\approx N_T$: the rate is proportional to $N_T$.
• More holes available means more holes annihilated per second: the rate is proportional to $p$.

With a positive proportionality constant $c_p$ (and a minus sign since $p$ decreases):

\left.\frac{\partial p}{\partial t}\right|_R = -c_p\,N_T\,p \tag{3.29}

Generation depends only on the number of empty R-G centers, a small number held at its equilibrium value, so the generation rate is a constant. In equilibrium generation and recombination must balance exactly (the Principle of Detailed Balance: each process and its inverse balance independently of other processes):

\left.\frac{\partial p}{\partial t}\right|_G = +c_p\,N_T\,p_0 \tag{3.30}

Net rate:

\left.\frac{\partial p}{\partial t}\right|_{R-G} = \left.\frac{\partial p}{\partial t}\right|_R + \left.\frac{\partial p}{\partial t}\right|_G = -c_pN_T\,(p - p_0) \tag{3.31}

or

\left.\frac{\partial p}{\partial t}\right|_{\text{thermal R-G}} = -c_pN_T\,\Delta p\qquad\text{(holes in n-type)} \tag{3.32a}

and analogously

\left.\frac{\partial n}{\partial t}\right|_{\text{thermal R-G}} = -c_nN_T\,\Delta n\qquad\text{(electrons in p-type)} \tag{3.32b}

with $c_n$ and $c_p$ the electron and hole capture coefficients.

Compact form. The products $c_p{N}_T$ and $c_n{N}_T$ have units of 1/time (since $\partial p/\partial t$ is concentration/time and $\Delta p$ is concentration). Define

\tau_p \equiv \frac{1}{c_pN_T} \tag{3.33a}

\tau_n \equiv \frac{1}{c_nN_T} \tag{3.33b}

so

\boxed{\left.\frac{\partial p_n}{\partial t}\right|_{\text{thermal R-G}} = -\frac{\Delta p_n}{\tau_p}\qquad\text{(holes in n-type)}} \tag{3.34a}

\boxed{\left.\frac{\partial n_p}{\partial t}\right|_{\text{thermal R-G}} = -\frac{\Delta n_p}{\tau_n}\qquad\text{(electrons in p-type)}} \tag{3.34b}

(the subscripts $n$ or $p$ on the carrier symbol remind us we are dealing with the minority carrier in that region).

Notes:

• Derived for steady or slowly varying conditions, but works well for nearly all transients.
• $\Delta p<0$ is allowed and gives $\partial p/\partial t>0$: a carrier deficit makes generation outpace recombination, restoring equilibrium. The sign convention thus describes the net of recombination and generation.
• Equations (3.34) apply only to minority carriers under low-level injection.

General steady-state result (valid for any injection level, both carriers, nondegenerate semiconductor):

\left.\frac{\partial p}{\partial t}\right|_{\text{thermal R-G}} = \left.\frac{\partial n}{\partial t}\right|_{\text{thermal R-G}} = -\frac{np - n_i^2}{\tau_p(n+n_1)+\tau_n(p+p_1)} \tag{3.35}

with

n_1 \equiv n_i\,e^{(E_T-E_i)/kT} \tag{3.36a}

p_1 \equiv n_i\,e^{(E_i-E_T)/kT} \tag{3.36b}

(Problem 3.15 asks you to show that (3.35) reduces to (3.34); a complete derivation is given there.)

3.3.4 Minority carrier lifetimes

General information. The constants ${\tau }_n$ and ${\tau }_p$ are the “constants of the motion” for R-G. Return to Fig. 3.21: the excess holes do not all disappear simultaneously; some live briefly, some for a long time. If thermal R-G is the only relaxation process, the average excess-hole lifetime works out to $⟨t⟩={\tau }_p$ (or ${\tau }_n$ for electrons). So ${\tau }_n$ and ${\tau }_p$ are the minority carrier lifetimes: the average time an excess minority carrier survives in a sea of majority carriers.

Unlike the $\mu$‘s and $D$‘s there are no handy plots of $\tau$ vs. doping, because $\tau$ depends on the poorly controlled R-G center concentration $N_T$ (3.33), not on the carefully controlled dopant concentration. Even within one sample, the nature and concentration of dominant R-G centers change during fabrication.

• Gettering (a processing step that sweeps out unwanted impurities) can reduce $N_T$ to a very low level and give ${\tau }_{n(p)}\sim 1$ ms in Si.
• Intentional gold doping raises $N_T$ and gives ${\tau }_{n(p)}\sim 1$ ns.
• Lifetimes in finished Si devices lie roughly midway between those extremes.

A lifetime measurement: photoconductivity decay. Most lifetime measurements use a device structure and fit theory to the device characteristics. The one described here uses a bar of semiconductor with ohmic contacts at both ends (a photoconductor). A light pulse creates excess carriers (a situation like Fig. 3.21) and a measurable conductivity increase; after the light stops, recombination makes the conductivity decay back, and the decay rate gives the lifetime.

Circuit (Fig. 3.22): sample resistance $R_s$ in series with load resistor $R_L$ and a dc supply $V_A$; the output $v_L$ is measured by a scope. $R_s$ varies with time, so $v_L$ tracks the conductivity changes.

For an n-type sample with uniform photogeneration, the excess minority holes decay after the light is switched off as (derived later, Section 3.5, and in Sample Problem 1)

\Delta p(t) = \Delta p_0\,e^{-t/\tau_p} \tag{3.37}

($\Delta p_0$ = excess at $t=0$ when light is switched off). The conductivity is

\sigma = \frac{1}{\rho} = q(\mu_n n + \mu_p p) \tag{3.38a}

= q\left[\mu_n(n_0+\Delta n) + \mu_p(p_0+\Delta p)\right] \tag{3.38b}

With $\Delta n=\Delta p$ and, in n-type material, $n_0\approx N_D\gg p_0$:

\sigma = \underbrace{q\mu_nN_D}_{\sigma_{dark}} + \underbrace{q(\mu_n+\mu_p)\,\Delta p}_{\Delta\sigma(t)} \tag{3.39}

where ${\sigma }_{dark}$ is the dark conductivity and $\Delta \sigma (t)\propto \Delta p(t)$ is the time-decaying part. To link $v_L$ to $\Delta \sigma$, use the voltage divider:

v_L = i\,R_L = V_A\,\frac{R_L}{R_s + R_L} \tag{3.40}

with the sample resistance (length $\ell$, area $A$)

R_s = \frac{\rho\ell}{A} = \frac{\ell}{\sigma A} = \frac{\ell}{(\sigma_{dark}+\Delta\sigma)A} = \frac{R_{sd}}{1+\Delta\sigma/\sigma_{dark}} \tag{3.41}

where $R_{sd}=\ell /({\sigma }_{dark}A)$ is the dark resistance. Substituting (3.41) into (3.40) and rearranging:

v_L = \frac{V_A}{1 + \dfrac{R_{sd}}{R_L}\cdot\dfrac{1}{1+\Delta\sigma/\sigma_{dark}}} \tag{3.42}

If the load is matched, $R_L=R_{sd}$, and the light is weak so $\Delta \sigma /{\sigma }_{dark}\ll 1$ (equivalent to low-level injection), expand:

v_L\approx \frac{V_A}{2} + \frac{V_A}{4}\,\frac{\Delta\sigma}{\sigma_{dark}} = \underbrace{\frac{V_A}{2}}_{\text{d.c.}} + \underbrace{v_0\,e^{-t/\tau_p}}_{\text{decay transient}} \tag{3.43}

(The step: with $R_L=R_{sd}$, $v_L=V_A/[1+1/(1+x)]=V_A(1+x)/(2+x)\approx \frac{V_A}{2}(1+x)(1-x/2)\approx \frac{V_A}{2}(1+x/2)=\frac{V_A}{2}+\frac{V_A}{4}x$ with $x=\Delta \sigma /{\sigma }_{dark}$.) An exponential decay with time constant ${\tau }_p$ is observed.

Implementation (Fig. 3.23). A dc supply provides $V_A$; a digitizing oscilloscope (Tektronix 11401) captures the decay; four bar-shaped Si samples sit in a box with the $R_L=R_{sd}$ load resistor. A stroboscope supplies $\sim 1\mu$s, high-intensity flashes spaced from 2.5 ms to 0.5 s apart. Light is first passed through a small Si wafer, which acts as a filter passing only near-bandgap radiation (weakly absorbed, so photogeneration is nearly uniform through the bar).

Result (Fig. 3.24). The transient component of $v_L$ (mV) vs. time (ms): the linear plot shows the exponential shape, the semilog plot shows a straight line (at least for $0.1\text{ms}\le t\le 0.6\text{ms}$), whose slope gives $-1/{\tau }_p$. For the probed sample, ${\tau }_p\approx 150\mu$s.

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