_c3

3.4 Equations of State

Real semiconductors do all the carrier actions at once. To find the state of a semiconductor we must combine the individual actions. This “putting it all together” yields the basic starting equations of device analysis, the equations of state.

3.4.1 Continuity equations

Every carrier action (drift, diffusion, thermal or direct R-G, light generation, …) changes the carrier concentration with time. The total rate of change is the sum of the contributions of the individual processes:

\frac{\partial n}{\partial t} = \left.\frac{\partial n}{\partial t}\right|_{drift} + \left.\frac{\partial n}{\partial t}\right|_{diff} + \left.\frac{\partial n}{\partial t}\right|_{\text{thermal R-G}} + \left.\frac{\partial n}{\partial t}\right|_{\text{other (light, etc.)}} \tag{3.44a}

\frac{\partial p}{\partial t} = \left.\frac{\partial p}{\partial t}\right|_{drift} + \left.\frac{\partial p}{\partial t}\right|_{diff} + \left.\frac{\partial p}{\partial t}\right|_{\text{thermal R-G}} + \left.\frac{\partial p}{\partial t}\right|_{\text{other (light, etc.)}} \tag{3.44b}

This is conservation of carriers: electrons and holes cannot appear or disappear at a point mysteriously; they must arrive by transport or be created there. Hence “continuity” equations.

Compact form of the transport terms. Drift plus diffusion changes the local concentration only if more current flows in than out of a small volume:

\left.\frac{\partial n}{\partial t}\right|_{drift} + \left.\frac{\partial n}{\partial t}\right|_{diff} = \frac{1}{q}\left(\frac{\partial J_{Nx}}{\partial x}+\frac{\partial J_{Ny}}{\partial y}+\frac{\partial J_{Nz}}{\partial z}\right) = \frac{1}{q}\nabla\cdot\mathbf{J}_N \tag{3.45a}

\left.\frac{\partial p}{\partial t}\right|_{drift} + \left.\frac{\partial p}{\partial t}\right|_{diff} = -\frac{1}{q}\nabla\cdot\mathbf{J}_P \tag{3.45b}

(The signs differ because the electron current density ${\mathbf{J}}_N$ counts negative charges flowing; a divergence of electron current means electron charge is leaving, i.e., electrons are added per $\frac{1}{q}\nabla \cdot {\mathbf{J}}_N$. For holes, a positive divergence of ${\mathbf{J}}_P$ means holes are leaving.)

Bank-account analogy. The balance in a small region (the number of carriers) changes by (deposits in) minus (withdrawals out) plus (interest created inside) minus (losses inside): inflow minus outflow of current, plus generation, minus recombination.

Substituting (3.45) into (3.44):

\boxed{\frac{\partial n}{\partial t} = \frac{1}{q}\nabla\cdot\mathbf{J}_N + \left.\frac{\partial n}{\partial t}\right|_{\text{thermal R-G}} + \left.\frac{\partial n}{\partial t}\right|_{\text{other processes}}} \tag{3.46a}

\boxed{\frac{\partial p}{\partial t} = -\frac{1}{q}\nabla\cdot\mathbf{J}_P + \left.\frac{\partial p}{\partial t}\right|_{\text{thermal R-G}} + \left.\frac{\partial p}{\partial t}\right|_{\text{other processes}}} \tag{3.46b}

These are completely general and are the usual starting point of device analysis. In computer simulation they are used directly: the appropriate R-G expression (e.g. (3.34) or the general (3.35)) and any “other process” terms are inserted and $n(x,y,z,t)$, $p(x,y,z,t)$ are solved numerically. For closed-form solutions we use a simplified version, derived next.

3.4.2 Minority carrier diffusion equations

The workhorse equations come from (3.46) under these simplifying assumptions:

1. The system is one-dimensional (everything depends on $x$ only).
2. We restrict attention to minority carriers.
3. $\mathcal{E}\approx 0$ in the region analyzed.
4. The equilibrium minority concentrations do not depend on position: $n_0\ne n_0(x)$, $p_0\ne p_0(x)$.
5. Low-level injection.
6. Indirect thermal R-G dominates the thermal R-G.
7. There are no other processes except possibly photogeneration.

Derivation for electrons (minority carriers in p-type material).

Transport term. In one dimension

\frac{1}{q}\nabla\cdot\mathbf{J}_N\to\frac{1}{q}\frac{\partial J_N}{\partial x} \tag{3.47}

The electron current is

J_N = q\mu_nn\mathcal{E} + qD_N\frac{\partial n}{\partial x}\ \to\ qD_N\frac{\partial n}{\partial x} \tag{3.48}

(We drop drift because $\mathcal{E}\approx 0$ and the minority concentration $n$ is small, so the product $n\mathcal{E}$ is doubly small. We could not do this for majority carriers.) Since $n=n_0+\Delta n$ and $n_0\ne n_0(x)$:

\frac{\partial n}{\partial x} = \frac{\partial n_0}{\partial x}+\frac{\partial\Delta n}{\partial x} = \frac{\partial\Delta n}{\partial x} \tag{3.49}

Combine (3.47)-(3.49):

\frac{1}{q}\nabla\cdot\mathbf{J}_N\to D_N\frac{\partial^2\Delta n}{\partial x^2} \tag{3.50}

R-G term. Indirect thermal R-G dominance plus low-level injection and minority carriers allow use of (3.34):

\left.\frac{\partial n}{\partial t}\right|_{\text{thermal R-G}} = -\frac{\Delta n}{\tau_n} \tag{3.51}

Other processes. Only light is allowed:

\left.\frac{\partial n}{\partial t}\right|_{\text{other processes}} = G_L \tag{3.52}

($G_L=0$ if the sample is dark.)

Left-hand side. The equilibrium concentration is constant in time, $\partial n_0/\partial t=0$, so

\frac{\partial n}{\partial t} = \frac{\partial n_0}{\partial t} + \frac{\partial\Delta n}{\partial t} = \frac{\partial\Delta n}{\partial t} \tag{3.53}

Putting (3.50)-(3.53) into (3.46a) and treating holes analogously:

\boxed{\frac{\partial\Delta n_p}{\partial t} = D_N\frac{\partial^2\Delta n_p}{\partial x^2} - \frac{\Delta n_p}{\tau_n} + G_L} \tag{3.54a}

\boxed{\frac{\partial\Delta p_n}{\partial t} = D_P\frac{\partial^2\Delta p_n}{\partial x^2} - \frac{\Delta p_n}{\tau_p} + G_L} \tag{3.54b}

The subscripts remind us that these hold only for minority carriers: electrons in p-type, holes in n-type.

Reading the equation as a story (for holes): the excess-hole concentration at a point rises with the curvature of the profile (diffusion refills hollows and flattens peaks), falls because holes recombine (rate $\Delta p/{\tau }_p$), and rises because light makes new ones ($G_L$).

3.4.3 Simplifications and solutions

Further simplifications often reduce the diffusion equation drastically (Table 3.1).

Table 3.1: Common diffusion-equation simplifications

Simplification	Effect
Steady state	$\partial \Delta n_p/\partial t\to 0$ (and $\partial \Delta p_n/\partial t\to 0$)
No concentration gradient (no diffusion current)	$D_N{\partial }^2\Delta n_p/\partial x^2\to 0$ (and $D_P{\partial }^2\Delta p_n/\partial x^2\to 0$)
No drift current or $\mathcal{E}=0$	No further simplification ($\mathcal{E}\approx 0$ was assumed in the derivation)
No thermal R-G	$\Delta n_p/{\tau }_n\to 0$ (and $\Delta p_n/{\tau }_p\to 0$)
No light	$G_L\to 0$

Table 3.2: Common special-case solutions (written for electrons; replace $n\to p$, $N\to P$ for holes)

No.	Given	Simplified equation	Solution
1	Steady state, no light	$0=D_N\frac{d^2\Delta n_p}{d{x}^2}-\frac{\Delta n_p}{{\tau }_n}$	$\Delta n_p(x)=A{e}^{-x/L_N}+B{e}^{x/L_N}$, with $L_N=\sqrt{D_N{\tau }_n}$; $A$, $B$ constants
2	No concentration gradient, no light	$\frac{d\Delta n_p}{dt}=-\frac{\Delta n_p}{{\tau }_n}$	$\Delta n_p(t)=\Delta n_p(0)e^{-t/{\tau }_n}$
3	Steady state, no concentration gradient	$0=-\frac{\Delta n_p}{{\tau }_n}+G_L$	$\Delta n_p=G_L{\tau }_n$
4	Steady state, no R-G, no light	$0=D_N\frac{d^2\Delta n_p}{d{x}^2}$	$\Delta n_p(x)=A+Bx$

(Solution 1 follows by trying $e^{\lambda x}$: ${\lambda }^2=1/(D_N{\tau }_n)$. Solution 2 is separable. Solution 3 is algebraic. Solution 4 integrates twice.)

3.4.4 Problem solving

The book proposes a five-step recipe, illustrated on two sample problems:

1. Review exactly what is given or implied.
2. Characterize the system in equilibrium.
3. Analyze the problem qualitatively (predict the shape before computing).
4. Perform the quantitative analysis: check the simplifying assumptions, write the minority carrier diffusion equation, simplify, solve with boundary conditions.
5. Examine the solution (units, limits, compare with step 3). (“Failing to examine the solution is like growing vegetables and failing to eat the produce.”)

Sample Problem No. 1: light switched on at $t=0$

Problem. A uniformly donor-doped silicon wafer at room temperature is suddenly illuminated at $t=0$. Take $N_D=10^{15}$ cm${}^{-3}$, ${\tau }_p=10^{-6}$ s, and a light-induced creation of $10^{17}$ electron-hole pairs per cm${}^3$ per second throughout the semiconductor. Determine $\Delta p_n(t)$ for $t>0$.

Step 1. Silicon, 300 K, uniform $N_D=10^{15}$ cm${}^{-3}$, $G_L=10^{17}$ cm${}^{-3}$s${}^{-1}$ everywhere. Equilibrium holds for $t<0$.

Step 2 (equilibrium). With $n_i=10^{10}$ cm${}^{-3}$ and $N_D\gg n_i$:

$n_0=N_D=10^{15}{\text{cm}}^{-3},p_0=\frac{n_i^2}{N_D}=\frac{10^{20}}{10^{15}}=10^5{\text{cm}}^{-3}$

Uniform doping means $n_0$ and $p_0$ are the same everywhere.

Step 3 (qualitative). Before $t=0$, $\Delta p_n=0$. After $t=0$, light adds carriers, so $\Delta p_n$ grows. But a bigger $\Delta p_n$ means a faster recombination rate ($\propto \Delta p_n$). Eventually recombination eats exactly what light adds, and steady state results. Expect: $\Delta p_n(t)$ starts at zero, rises at a decreasing rate, saturates. At saturation generation = recombination:

$G_L=\frac{\Delta p_n(\infty )}{{\tau }_p}\Rightarrow \Delta p_{n,max}=G_L{\tau }_p$

(provided low-level injection holds).

Step 4 (quantitative). All seven simplifying assumptions behind the diffusion equation hold: 1-D (really uniform), only minority carriers, $n_0$, $p_0$ position independent, indirect thermal R-G dominant in Si, only photogeneration. Because generation is uniform, the perturbed concentrations are position-independent, so no gradients and $\mathcal{E}=0$. Low-level injection is verified below. So start from

\frac{\partial\Delta p_n}{\partial t} = D_P\frac{\partial^2\Delta p_n}{\partial x^2} - \frac{\Delta p_n}{\tau_p} + G_L \tag{3.55}

with initial condition

\Delta p_n(t)\big|_{t=0} = 0 \tag{3.56}

Since $\Delta p_n$ does not depend on $x$, the curvature term vanishes and (3.55) becomes an ordinary differential equation:

\frac{d\Delta p_n}{dt} + \frac{\Delta p_n}{\tau_p} = G_L \tag{3.57}

Solve it. Particular solution (steady value): $\Delta p_n=G_L{\tau }_p$. Homogeneous solution: $A{e}^{-t/{\tau }_p}$. So

\Delta p_n(t) = G_L\tau_p + A\,e^{-t/\tau_p} \tag{3.58}

Apply $\Delta p_n(0)=0$:

0 = G_L\tau_p + A\ \Rightarrow\ A = -G_L\tau_p \tag{3.59}

\boxed{\Delta p_n(t) = G_L\tau_p\left(1 - e^{-t/\tau_p}\right)} \tag{3.60}

Step 5 (examine). $G_L{\tau }_p$ has the dimensions (cm${}^{-3}$s${}^{-1}$)(s) = cm${}^{-3}$, a concentration, so the answer is dimensionally correct. At $t=0$: 0. As $t\to \infty$: $G_L{\tau }_p$. After one lifetime ($t={\tau }_p$) the excess is $63.2\%$ of its final value; after $3{\tau }_p$ it is $95\%$ (Fig. 3.25). Numbers:

$\Delta p_{n,max}=(10^{17})(10^{-6})=10^{11}{\text{cm}}^{-3}\ll n_0=10^{15}{\text{cm}}^{-3}(\text{low-level injection confirmed})$

so $p_{max}=p_0+10^{11}\approx 10^{11}$ cm${}^{-3}$, i.e., $10^6$ times its equilibrium value.

Epilogue: link to the photoconductive decay measurement. The stroboscope output can be modeled as a pulse train (Fig. 3.26a). (3.60) describes the carrier build-up during a pulse. However the flash lasts $\sim 1\mu$s while ${\tau }_p\lesssim 150\mu$s, so $t_{on}/{\tau }_p\ll 1$ and we see only the very first part of the rising exponential before the light goes off. For the light-off transient: set $G_L=0$ and use $\Delta p_n(0)=$ (value at the end of the light-on period), which is Solution No. 2 of Table 3.2:

\Delta p_n(t) = \Delta p_n(0)\,e^{-t/\tau_p} \tag{3.61}

(with $t=0$ reset to the instant the light turns off), which is exactly Eq. (3.37). The combined on-off response is sketched in Fig. 3.26(b).

Sample Problem No. 2: non-penetrating light on a semi-infinite bar

Problem. The $x=0$ end of a uniformly doped semi-infinite Si bar with $N_D=10^{15}$ cm${}^{-3}$ is illuminated so as to create $\Delta p_{n0}=10^{10}$ cm${}^{-3}$ excess holes at $x=0$ (Fig. 3.27a). The wavelength is such that no light penetrates into the interior ($x>0$). Determine $\Delta p_n(x)$.

Step 1 and 2. Silicon, steady state (we are asked for $\Delta p_n(x)$ not $\Delta p_n(x,t)$), $N_D=10^{15}$, so $n_0=10^{15}$, $p_0=10^5$ cm${}^{-3}$ uniform. Boundary conditions: $\Delta p_n(0)=\Delta p_{n0}=10^{10}$ cm${}^{-3}$ and $\Delta p_n\to 0$ as $x\to \infty$ (semi-infinite bar: a surface perturbation cannot reach infinity). Non-penetrating light means $G_L=0$ for $x>0$. The temperature is not stated, so assume the usual 300 K.

Step 3 (qualitative). Light creates carriers right at $x=0$. With more carriers at one point than elsewhere, diffusion spreads the excess into the bar. As the holes diffuse, they also recombine (rate $\propto \Delta p_n$) and each lives only ${\tau }_p$ on average, so fewer and fewer survive deeper into the bar. Expect a profile decreasing monotonically from $\Delta p_{n0}$ at $x=0$ to zero as $x\to \infty$.

Step 4 (quantitative). Check the assumptions: 1-D; minority holes only; $p_0$, $n_0$ position independent; indirect thermal R-G dominates; no other process for $x>0$; low-level injection since $10^{10}\ll 10^{15}$. The only question is whether $\mathcal{E}\approx 0$: the hole pile-up makes a small net positive charge, but it is tiny ($\Delta p_{n0}\ll n_0$) and, in practice, the majority electrons redistribute to cancel much of it. Experience shows $\mathcal{E}\approx 0$ to be reasonable.

Steady state with $G_L=0$ for $x>0$:

D_P\frac{d^2\Delta p_n}{dx^2} - \frac{\Delta p_n}{\tau_p} = 0\qquad x>0 \tag{3.62}

Boundary conditions:

\Delta p_n\big|_{x=0^+} = \Delta p_n\big|_{x=0} = \Delta p_{n0} \tag{3.63}

\Delta p_n\big|_{x\to\infty} = 0 \tag{3.64}

This is Solution No. 1 of Table 3.2:

\Delta p_n(x) = Ae^{-x/L_P}+Be^{x/L_P} \tag{3.65}

L_P = \sqrt{D_P\tau_p} \tag{3.66}

Since $e^{x/L_P}\to \infty$ as $x\to \infty$, (3.64) can be satisfied only if $B=0$. Then (3.63) gives

A = \Delta p_{n0} \tag{3.67}

\boxed{\Delta p_n(x) = \Delta p_{n0}\,e^{-x/L_P}} \tag{3.68}

Step 5. Matches the qualitative picture: a monotone exponential decay from $\Delta p_{n0}$ to zero, with a characteristic decay length $L_P$ (Fig. 3.27b).

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3.5 Supplemental Concepts

3.5.1 Diffusion lengths

The situation of Sample Problem 2 (excess minority carriers appear on a plane, diffuse away, and decay exponentially with a characteristic length) is so common that the characteristic length has a name:

\boxed{L_P \equiv \sqrt{D_P\tau_p}\quad\text{(holes in n-type)}} \tag{3.69a}

\boxed{L_N \equiv \sqrt{D_N\tau_n}\quad\text{(electrons in p-type)}} \tag{3.69b}

called the minority carrier diffusion lengths. They represent the average distance a minority carrier diffuses into a sea of majority carriers before being annihilated. Check with Sample Problem 2: the mean position of the excess holes is

\langle x\rangle = \frac{\int_0^\infty x\,\Delta p_n(x)\,dx}{\int_0^\infty\Delta p_n(x)\,dx} = \frac{\int_0^\infty x\,e^{-x/L_P}dx}{\int_0^\infty e^{-x/L_P}dx} = \frac{L_P^2}{L_P} = L_P \tag{3.70}

(using ${\int }_0^{\infty }x{e}^{-x/L}dx=L^2$ and ${\int }_0^{\infty }{e}^{-x/L}dx=L$).

Piranha analogy. A small group of animals tries to cross a piranha-infested stretch of the Amazon. Each animal is “eaten” at random with a fixed chance per meter of swimming. The average distance covered before being eaten is the diffusion length. Longer lifetime (fewer piranhas) and faster diffusion (stronger swimmers) both lengthen the distance, hence $L=\sqrt{D\tau }$.

Size. For $T=300$ K, $N_D=10^{15}$ cm${}^{-3}$ Si (${\mu }_p=458$ cm${}^2$/V s) and ${\tau }_p=10^{-6}$ s:

$L_P=\sqrt{D_P{\tau }_p}=\sqrt{\frac{kT}{q}{\mu }_p{\tau }_p}={\left[(0.0259)(458)(10^{-6})\right]}^{1/2}=3.44\times 10^{-3}\text{cm}$

(about $34\mu$m). Because lifetimes vary over orders of magnitude, diffusion lengths do too.

3.5.2 Quasi-Fermi levels

Why we need them. In equilibrium the Fermi level $E_F$ nails down both carrier concentrations:

n_0 = n_i\,e^{(E_F-E_i)/kT} \tag{3.71a}

p_0 = n_i\,e^{(E_i-E_F)/kT} \tag{3.71b}

so $E_F$ and ($n_0,p_0$) carry the same information, and the band diagram lets us read off the concentrations at a glance. But in a nonequilibrium state the Fermi level is not even defined, and one number cannot describe two independently changed concentrations.

Example (Sample Problem 1 at $t\gg {\tau }_p$). Before the light: $n_0=N_D=10^{15}$, $p_0=10^5$ cm${}^{-3}$ (equilibrium diagram, Fig. 3.28a). After the steady state is reached: $\Delta n=\Delta p=G_L{\tau }_p=10^{11}$ cm${}^{-3}$, $p=p_0+\Delta p\approx 10^{11}$ cm${}^{-3}$ and $n\approx n_0$. The electron concentration is essentially unperturbed, but $p$ is up by a factor $10^6$. The equilibrium diagram no longer describes this.

Definition. Introduce two energies: $F_N$ (electron quasi-Fermi level) and $F_P$ (hole quasi-Fermi level), defined to relate to the nonequilibrium concentrations the same way $E_F$ relates to the equilibrium ones (nondegenerate case):

n \equiv n_i\,e^{(F_N-E_i)/kT}\qquad\text{or}\qquad F_N = E_i + kT\ln\!\left(\frac{n}{n_i}\right) \tag{3.72a}

p \equiv n_i\,e^{(E_i-F_P)/kT}\qquad\text{or}\qquad F_P = E_i - kT\ln\!\left(\frac{p}{n_i}\right) \tag{3.72b}

$F_N$ and $F_P$ are conceptual constructs: their values are determined entirely from knowledge of $n$ and $p$. They are defined so that when a perturbed system relaxes to equilibrium, $F_N\to E_F$, $F_P\to E_F$, and (3.72) $\to$ (3.71).

Application to Sample Problem 1 at $t\gg {\tau }_p$. Since $n\approx n_0$: $F_N\approx E_F$. For holes, with $p=10^{11}$ cm${}^{-3}$, $n_i=10^{10}$, $kT=0.0259$ eV:

$F_P=E_i-kT\ln \frac{p}{n_i}=E_i-0.0259\ln (10)=E_i-0.060\text{eV}$

Drawing $F_N$ and $F_P$ on the diagram in place of $E_F$ (Fig. 3.28b) shows at a glance (i) the system is not in equilibrium ($F_N\ne F_P$), and (ii) it is in low-level injection: $F_N$ barely moved from the equilibrium $E_F$ while $F_P$ moved from far below $E_i$ (about $0.30$ eV below $E_F$… i.e., $E_F-E_i=0.298$ eV, so $E_F$ is $0.30$ eV above $E_i$ and $F_P$ is $0.06$ eV below $E_i$) up toward the mid-gap.

Compact current expressions. Start from the hole current:

J_P = q\mu_pp\,\mathcal{E} - qD_P\nabla p \tag{3.73}

(same as 3.18a). Differentiate (3.72b):

\nabla p = \frac{n_i}{kT}\,e^{(E_i-F_P)/kT}\left(\nabla E_i - \nabla F_P\right) \tag{3.74a}

= \frac{p}{kT}\left(q\boldsymbol{\mathcal{E}} - \nabla F_P\right) \tag{3.74b}

where we used $\mathcal{E}=\nabla E_i/q$ (3-D version of (3.15)). Eliminate $\nabla p$ from (3.73) with (3.74b):

$J_P=q{\mu }_{p}p\mathcal{E}-q{D}_P\frac{p}{kT}\left(q\mathcal{E}-\nabla F_P\right)$

Collect the terms in $\mathcal{E}$ and $\nabla F_P$:

J_P = q p\left(\mu_p - \frac{qD_P}{kT}\right)\mathcal{E} + \mu_p\!\left(\frac{qD_P}{kT\mu_p}\right)p\,\nabla F_P\ ... \tag{3.75}

More cleanly: $J_P=qp\mathcal{E}\left({\mu }_p-\frac{q{D}_P}{kT}\right)+\frac{q{D}_{P}p}{kT}\nabla F_P$. Now invoke the Einstein relationship $q{D}_P/kT={\mu }_p$: the first bracket vanishes and the coefficient of $\nabla F_P$ is ${\mu }_{p}p$. (Sign check: for holes, $\nabla F_P>0$ should drive current down the hole-energy hill, i.e. in the sense given below.) The result:

\boxed{\mathbf{J}_P = \mu_p\,p\,\nabla F_P} \tag{3.76a}

and similarly

\boxed{\mathbf{J}_N = \mu_n\,n\,\nabla F_N} \tag{3.76b}

(The book’s expressions are in this form. In the standard sign convention, with energies measured for electrons, ${\mathbf{J}}_N={\mu }_{n}n\nabla F_N$ and ${\mathbf{J}}_P={\mu }_{p}p\nabla F_P$ both point toward increasing quasi-Fermi energy; in a detailed check, one verifies these reduce to the drift-diffusion forms using the Einstein relation. The statement of consequence is what we need.)

The big interpretation. $J_P\propto \nabla F_P$ and $J_N\propto \nabla F_N$. Therefore a quasi-Fermi level that varies with position ($d{F}_P/dx\ne 0$ or $d{F}_N/dx\ne 0$) shows at a glance that a current is flowing, while a flat quasi-Fermi level means no current of that carrier type. Drift and diffusion unify into a single idea: carriers flow because their (quasi-)Fermi level is not flat, exactly like water flowing because the water level is not flat.

Analogy: two reservoirs. $F_N$ and $F_P$ are the “water levels” of the electron and hole reservoirs. In equilibrium both reservoirs sit at the same level $E_F$ and nothing flows. When we perturb the system, the two levels separate (here $F_N-F_P\ne 0$), and slopes in these levels drive flows.

Exercise 3.5: Quasi-Fermi levels in the illuminated bar

Problem. In Sample Problem 2, non-penetrating illumination of a semiconductor bar produced a steady-state excess-hole concentration $\Delta p_n(x)=\Delta p_{n0}\exp (-x/L_P)$. With low-level injection and $p=p_0+\Delta p$: $n\approx n_0$, $p=p_0+\Delta p_{n0}{e}^{-x/L_P}$. (a) Use (3.72) to establish relationships for $F_N$ and $F_P$ in the illuminated bar. (b) Simplify $F_P$ in the region where $\Delta p_n\gg p_0$. (c) Using (a), (b), sketch the energy band diagrams of the bar for equilibrium and the illuminated steady state (assume $\mathcal{E}=0$). (d) Is there a hole current in the illuminated bar in steady state? (e) Is there an electron current? ($N_D=10^{15}$, $\Delta p_{n0}=10^{10}$ cm${}^{-3}$, Si, 300 K.)

Solution.

(a) Since $n\approx n_0$, from (3.72a) $F_N\approx E_F$. For holes, substitute $p$ into (3.72b):

$F_P=E_i-kT\ln \frac{p}{n_i}=E_i-kT\ln \left[\frac{p_0}{n_i}+\frac{\Delta p_{n0}}{n_i}{e}^{-x/L_P}\right]$

(b) If $\Delta p_n\gg p_0$, then $(\Delta p_{n0}/n_i)e^{-x/L_P}\gg p_0/n_i$ and

$F_P\approx E_i-kT\ln \left[\frac{\Delta p_{n0}}{n_i}{e}^{-x/L_P}\right]$

Split the logarithm of a product:

$F_P=E_i-kT\ln \frac{\Delta p_{n0}}{n_i}+kT\frac{x}{L_P}$

so $F_P$ is a linear function of $x$ with slope $kT/L_P$.

(c) Numbers: $\Delta p_{n0}=10^{10}$, $n_i=10^{10}$, $p_0=n_i^2/N_D=10^5$ cm${}^{-3}$.

• (i) Near $x=0$, $\Delta p_n\gg p_0$ and $F_P$ is linear in $x$.
• (ii) At $x=0$: $\Delta p_{n0}=n_i$, so $\ln (\Delta p_{n0}/n_i)=0$ and $F_P(0)=E_i$.
• (iii) For large $x$, $F_P\to E_F$ (back to equilibrium).
• (iv) $F_N-E_i\approx E_F-E_i=kT\ln (N_D/n_i)=0.0259\ln (10^5)=0.30$ eV.

Sketch: $E_C$, $E_V$, $E_i$ flat (since $\mathcal{E}=0$). $F_N$ is a horizontal line $0.30$ eV above $E_i$ (coinciding with $E_F$ of the equilibrium diagram). $F_P$ starts at $E_i$ at $x=0$, rises linearly with slope $kT/L_P$ (an excess-hole pile gives $F_P$ near mid-gap), then bends over and merges into $F_N=E_F$ at large $x$ (roughly where $\Delta p_n$ falls to $p_0$, $x\approx L_P\ln (10^5)\approx 11.5L_P$).

(d) Yes. By (3.76a), $J_P={\mu }_{p}p\nabla F_P\ne 0$ wherever $d{F}_P/dx\ne 0$, which is the case near $x=0$ (and everywhere that $F_P$ is sloping).

(e) Appearances deceive: yes, $J_N\ne 0$ near $x=0$, even though $F_N$ looks flat. In steady state with no current leaving the bar at $x=0$ and no net electron or hole pile-up, the total current is the same at every point and must vanish at $x=0$ (no current leaves the bar there). Hence $J_N(x)=-J_P(x)$, and since $J_P\ne 0$ near $x=0$, $J_N\ne 0$. The apparent discrepancy is resolved by (3.76b): $J_N={\mu }_{n}n\nabla F_N$ is proportional to both $n$ and $\nabla F_N$. The electron concentration (majority) is enormously larger than the minority hole concentration, so $d{F}_N/dx$ must be correspondingly smaller than $d{F}_P/dx$ to carry the same current. The slope of $F_N$ is simply too tiny to be seen on a band diagram.

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3.6 Summary and Concluding Comments

Most of this chapter examined the three primary carrier actions:

• Drift: charged-particle motion in response to an electric field. Current: $J_{N|drift}=q{\mu }_{n}n\mathcal{E}$, $J_{P|drift}=q{\mu }_{p}p\mathcal{E}$ (Eqs. 3.4).
• Diffusion: migration from regions of high concentration to regions of low concentration due to random thermal motion. Current: Eqs. 3.17-3.19.
• Recombination-generation: annihilation and creation of carriers; changes the concentrations in time (Eqs. 3.34).

Each has its constant of the motion: mobilities $\mu$ (drift), diffusion coefficients $D$ (diffusion), minority-carrier lifetimes $\tau$ (R-G). Mobilities are well characterized versus doping and temperature (Figs. 3.5 and 3.7, and the fit in Exercise 3.1). Diffusion coefficients in nondegenerate semiconductors follow from the mobilities via the Einstein relationship (3.25). Lifetimes must be measured for the sample at hand.

Mathematically combining the actions gives the continuity equations (3.46), which simplify to the minority carrier diffusion equations (3.54) under common conditions. Table 3.1 gives additional simplifications; Table 3.2 gives the most-used solutions; Table 3.3 collects the key working relationships:

Table 3.3: Carrier action equation summary

Equations of state

$\frac{\partial n}{\partial t}=\frac{1}{q}\nabla \cdot {\mathbf{J}}_N+{\frac{\partial n}{\partial t}|}_{\text{thermal R-G}}+{\frac{\partial n}{\partial t}|}_{\text{other}}$

$\frac{\partial p}{\partial t}=-\frac{1}{q}\nabla \cdot {\mathbf{J}}_P+{\frac{\partial p}{\partial t}|}_{\text{thermal R-G}}+{\frac{\partial p}{\partial t}|}_{\text{other}}$

$\frac{\partial \Delta n_p}{\partial t}=D_N\frac{{\partial }^2\Delta n_p}{\partial x^2}-\frac{\Delta n_p}{{\tau }_n}+G_L,\frac{\partial \Delta p_n}{\partial t}=D_P\frac{{\partial }^2\Delta p_n}{\partial x^2}-\frac{\Delta p_n}{{\tau }_p}+G_L$

Current and R-G relationships

${\mathbf{J}}_N={\mathbf{J}}_{N|drift}+{\mathbf{J}}_{N|diff}=q{\mu }_{n}n\mathcal{E}+q{D}_N\nabla n,{\frac{\partial n}{\partial t}|}_{\text{i-thermal R-G}}=-\frac{\Delta n}{{\tau }_n}$

${\mathbf{J}}_P={\mathbf{J}}_{P|drift}+{\mathbf{J}}_{P|diff}=q{\mu }_{p}p\mathcal{E}-q{D}_P\nabla p,{\frac{\partial p}{\partial t}|}_{\text{i-thermal R-G}}=-\frac{\Delta p}{{\tau }_p}$

$\mathbf{J}={\mathbf{J}}_N+{\mathbf{J}}_P$

Key parametric relationships

$\frac{D_N}{{\mu }_n}=\frac{kT}{q},L_N\equiv \sqrt{D_N{\tau }_n},{\tau }_n=\frac{1}{c_n{N}_T}$

$\frac{D_P}{{\mu }_p}=\frac{kT}{q},L_P\equiv \sqrt{D_P{\tau }_p},{\tau }_p=\frac{1}{c_p{N}_T}$

Resistivity and electrostatic relationships

$\rho =\frac{1}{q({\mu }_{n}n+{\mu }_{p}p)},\rho \approx \frac{1}{q{\mu }_n{N}_D}(\text{n-type}),\rho \approx \frac{1}{q{\mu }_p{N}_A}(\text{p-type})$

$\mathcal{E}=\frac{1}{q}\frac{d{E}_C}{dx}=\frac{1}{q}\frac{d{E}_V}{dx}=\frac{1}{q}\frac{d{E}_i}{dx},V=-\frac{1}{q}(E_C-E_{ref})$

Quasi-Fermi level relationships

${\mathbf{J}}_N={\mu }_{n}n\nabla F_N,{\mathbf{J}}_P={\mu }_{p}p\nabla F_P$

Auxiliary topics met along the way: resistivity and its measurement (four-point probe, eddy current), the hot-point probe, constancy of the equilibrium Fermi level, nonuniform doping and the built-in field, $E$-$k$ diagrams (direct and indirect semiconductors), lifetime measurement by photoconductive decay, and diffusion lengths. The energy band diagram itself was developed further: (i) an electric field causes band bending, and the shape of $V(x)$ and $\mathcal{E}(x)$ can be read from the diagram; (ii) R-G centers add a near-midgap level that dominates thermal communication between the bands; (iii) quasi-Fermi levels describe nonequilibrium conditions.

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