---
title: Carrier Transport and Recombination
module: Semiconductors
moduleNumber: 7
lessonNumber: 3
order: 703
summary: >
  Carriers move by drift in a field and by diffusion down a concentration
  gradient, the two tied together by the Einstein relation. This lesson derives
  mobility and its scattering-limited temperature dependence, the drift and
  diffusion currents, the continuity equations, band-to-band and trap-assisted
  recombination, and the minority-carrier lifetime and diffusion length that set
  the length scale of every junction device.
topics: [Semiconductors]
sources:
  - book: Ashcroft & Mermin
    ref: "Ch. 28–29 — Homogeneous and Inhomogeneous Semiconductors"
  - book: Kittel
    ref: "Ch. 8 — Semiconductor Crystals"
  - book: Hook & Hall
    ref: "Ch. 5 — Semiconductors"
---

Knowing [how many carriers](/condensed-matter/semiconductors/intrinsic-and-extrinsic-semiconductors)
a semiconductor holds, the next questions are how fast they move under a field,
how they spread when their concentration is uneven, and how long an excess
population survives before recombining. The three answers — mobility, diffusion,
and lifetime — combine into the transport equations that the p-n junction and
every device built on it obey.

## Drift and mobility

In a field $\vec E$ a carrier of charge $q$ and effective mass $m^\ast$ is
accelerated between collisions and randomized by them after a mean free time
$\tau$. Averaging the equation of motion $m^\ast \dot{\vec v} = q\vec E$ over the
collision time gives a steady **drift velocity** proportional to the field,

$$
\vec v_d = \frac{q\tau}{m^\ast}\vec E \equiv \pm\,\mu\,\vec E, \qquad \mu = \frac{e\tau}{m^\ast},
$$

with the sign following the charge: electrons drift opposite to $\vec E$, holes
along it. The proportionality constant $\mu$ is the **mobility**, the drift speed
per unit field. Both carrier types contribute their drift to the current density,

$$
\vec J_{\text{drift}} = e\,(n\mu_e + p\mu_h)\,\vec E = \sigma \vec E, \qquad \sigma = e(n\mu_e + p\mu_h),
$$

recovering Ohm's law with conductivity $\sigma$. In silicon at room temperature
$\mu_e \approx 1400\ \text{cm}^2/\text{V·s}$ and $\mu_h \approx 450\
\text{cm}^2/\text{V·s}$; electrons are more mobile because their band-edge
effective mass is smaller.

> **Definition (Mobility).** The **mobility** $\mu = e\tau/m^\ast$ of a carrier is
> the magnitude of its drift velocity per unit applied field. It combines the
> scattering time $\tau$ with the band-edge effective mass $m^\ast$, and it sets
> the conductivity $\sigma = e(n\mu_e + p\mu_h)$ jointly with the carrier count.

The linear law holds only at low field. Once the drift energy per free flight
approaches the optical-phonon energy, carriers shed energy to the lattice as fast
as the field feeds it, and the drift velocity **saturates** near $v_{\text{sat}}
\approx 10^{7}\ \text{cm/s}$ for silicon, independent of further increase in field.
Velocity saturation limits the switching current in short-channel transistors.

$$
% caption: Drift velocity versus field. At low field the drift velocity rises
% linearly, its slope equal to the mobility; above a critical field it saturates
% at v_sat as carriers lose energy to optical phonons as fast as the field
% supplies it.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.0,0) node[below] {$E$};
  \draw[->, black] (0,0) -- (0,3.6) node[left] {$v_d$};
  % saturating curve v = vsat * (E/Ec)/(1+E/Ec)
  \draw[acc, very thick, domain=0:6.6, samples=100, variable=\x]
    plot ({\x},{2.7*(\x/1.6)/(1+\x/1.6)});
  \draw[black, dashed] (0,2.7) -- (6.6,2.7);
  \node[anchor=west, font=\scriptsize] at (5.2,2.9) {saturation};
  % low-field tangent (slope = mobility)
  \draw[black, dashed] (0,0) -- (2.2,3.4);
  \node[black, anchor=west, font=\scriptsize] at (1.5,3.35) {slope = mobility};
\end{tikzpicture}
$$

## Scattering and the temperature dependence of mobility

The scattering time $\tau$ is set by whatever deflects a carrier: the thermal
vibration of the lattice and the ionized impurities left behind by doping. Because
the rates add, so do the inverse mobilities — **Matthiessen's rule**,

$$
\frac{1}{\mu} = \frac{1}{\mu_L} + \frac{1}{\mu_I}.
$$

The two mechanisms pull in opposite temperature directions:

- **Lattice (phonon) scattering.** The density of phonons rises with temperature,
  shortening $\tau$; a Boltzmann-averaged cross section gives
  $\mu_L \propto T^{-3/2}$. This dominates at high temperature and in pure crystals.
- **Ionized-impurity scattering.** A carrier is deflected by the Coulomb field of a
  charged dopant. Faster carriers are deflected less, and the thermal speed rises
  with $T$, so $\mu_I \propto T^{3/2}/N_I$ — impurity scattering weakens as the
  crystal warms and strengthens with the ionized-impurity density $N_I$.

The competition makes the mobility rise, peak, and fall as temperature increases,
with the peak shifting to higher $T$ and lower height as doping increases.

$$
% caption: Mobility versus temperature on log axes. Ionized-impurity scattering
% (rising T^{3/2}) dominates at low T and lattice scattering (falling T^{-3/2})
% at high T; their Matthiessen sum peaks between, and heavier doping lowers and
% broadens the peak toward higher temperature.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.0,0) node[below] {$\ln T$};
  \draw[->, black] (0,0) -- (0,4.2) node[left] {mobility};
  % impurity branch rising
  \draw[black, dashed, domain=0.5:4.0, samples=40, variable=\x]
    plot ({\x},{0.6+0.75*\x});
  \node[black, anchor=west, font=\scriptsize] at (0.6,1.7) {impurity};
  % lattice branch falling
  \draw[black, dashed, domain=3.0:6.6, samples=40, variable=\x]
    plot ({\x},{6.9-0.75*\x});
  \node[black, anchor=east, font=\scriptsize] at (6.6,1.7) {lattice};
  % Matthiessen sum (peaked)
  \draw[acc, very thick, domain=0.6:6.4, samples=100, variable=\x]
    plot ({\x},{3.6 - 1.4*((\x-3.5)/2.2)*((\x-3.5)/2.2)});
  \node[acc, anchor=south, font=\scriptsize] at (3.5,3.65) {total};
\end{tikzpicture}
$$

## Diffusion and the Einstein relation

A nonuniform carrier density spreads by **diffusion**, a particle flux down the
gradient. Fick's law gives diffusion current densities

$$
\vec J_n^{\text{diff}} = e D_n \nabla n, \qquad \vec J_p^{\text{diff}} = -e D_p \nabla p,
$$

with $D_n, D_p$ the diffusion coefficients; the sign difference reflects the
opposite charge of the two carriers moving down their respective gradients. The
total electron current is the sum of drift and diffusion,

$$
\vec J_n = e n \mu_e \vec E + e D_n \nabla n,
$$

and similarly for holes. Drift and diffusion are not independent: both arise from
the same random thermal motion interrupted by the same collisions, so their
coefficients are linked. In thermal equilibrium the total current must vanish
everywhere, even where a built-in field and a concentration gradient coexist. In
such a region the carriers follow the Boltzmann distribution
$n(x) \propto e^{e\varphi(x)/k_B T}$ in the electrostatic potential $\varphi$, so
$\nabla n = (e/k_B T)\,n\,\nabla\varphi = -(e/k_B T)\,n\,\vec E$. Setting
$\vec J_n = 0$,

$$
e n \mu_e \vec E + e D_n\!\left(-\frac{e}{k_B T}n\vec E\right) = 0 \;\Longrightarrow\; D_n = \frac{k_B T}{e}\,\mu_e.
$$

> **Theorem (Einstein relation).** The diffusion coefficient and mobility of a
> nondegenerate carrier are proportional through the thermal voltage:
> $D = (k_B T/e)\,\mu$. The same collisions that limit drift limit diffusion, so a
> measurement of one fixes the other. At $300\ \text{K}$ the thermal voltage
> $k_B T/e = 25.9\ \text{mV}$, giving $D_n \approx 36\ \text{cm}^2/\text{s}$ for
> electrons in silicon.

$$
% caption: The two current mechanisms. A field tilts the bands and drives drift
% (carriers slide downhill in potential energy); a concentration gradient drives
% diffusion from the dense region to the dilute one. The Einstein relation ties
% the drift mobility to the diffusion coefficient.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- drift panel ---
  \begin{scope}
    \draw[black, thick] (0,1.6) -- (2.6,0.6);
    \node[black, anchor=south, font=\scriptsize] at (1.3,1.4) {tilted band};
    \fill[acc] (0.6,1.35) circle (2.6pt);
    \draw[acc, ->, very thick] (0.75,1.3) -- (1.7,0.94);
    \node[anchor=north, font=\scriptsize] at (1.3,0.5) {drift};
  \end{scope}
  % --- diffusion panel ---
  \begin{scope}[xshift=4.4cm]
    % dense cluster left, sparse right
    \foreach \p in {(0.2,1.3),(0.45,1.0),(0.2,0.7),(0.55,1.35),(0.35,0.95),(0.6,0.75)}
      \fill[acc] \p circle (2.2pt);
    \fill[acc] (2.1,1.05) circle (2.2pt);
    \draw[acc, ->, very thick] (0.9,1.0) -- (1.8,1.0);
    \node[anchor=north, font=\scriptsize] at (1.3,0.5) {spreading};
    \node[font=\scriptsize, anchor=south] at (0.4,1.5) {dense};
    \node[font=\scriptsize, anchor=south] at (2.1,1.3) {dilute};
  \end{scope}
\end{tikzpicture}
$$

## Generation, recombination, and lifetime

Carriers are continually created (**generation**, rate $G$) and destroyed
(**recombination**, rate $R$) as electrons fall from the conduction band into
empty valence states. In equilibrium generation and recombination balance exactly
at every energy. An excess population — injected by light or by a forward-biased
junction — decays back toward equilibrium at a net rate $U = R - G$.

**Band-to-band (direct) recombination.** An electron and hole meet and recombine,
emitting a photon; the rate is proportional to the product of the populations,
$R = B\,np$. Subtracting the equilibrium generation $G_0 = B\,n_i^2$, the net rate
is

$$
U = B\,(np - n_i^2).
$$

For small excess $\delta p$ of minority holes in n-type material ($n \approx N_d$),
$U \approx B N_d\,\delta p = \delta p/\tau_p$ with the **minority-carrier lifetime**

$$
\tau_p = \frac{1}{B N_d}.
$$

Direct recombination is efficient in **direct-gap** materials like gallium
arsenide, where the band extrema share the same crystal momentum and no phonon is
needed — the basis of the LED and the diode laser.

**Trap-assisted (Shockley-Read-Hall) recombination.** In an **indirect-gap**
material like silicon, band-to-band recombination requires a phonon to conserve
momentum and is slow; recombination instead proceeds through defect levels $E_t$
deep in the gap that capture an electron and then a hole in two steps. The
Shockley-Read-Hall rate through a trap density $N_t$ is

$$
U = \frac{np - n_i^2}{\tau_{p0}(n + n_1) + \tau_{n0}(p + p_1)}, \qquad n_1 p_1 = n_i^2,
$$

where $n_1 = n_i\,e^{(E_t - E_i)/k_B T}$ and the capture times
$\tau_{n0}, \tau_{p0}$ are inversely proportional to $N_t$. The denominator is
smallest — the rate largest — when the trap sits near midgap ($E_t \approx E_i$),
so **deep midgap traps are the most effective recombination centers**. This is why
gold and other deep-level impurities are deliberately added to silicon to speed
switching, and why crystalline defects degrade minority-carrier lifetime.

$$
% caption: Two recombination paths. Direct recombination drops an electron
% straight across the gap emitting a photon; trap-assisted recombination proceeds
% in two steps through a deep level E_t, capturing the electron then the hole,
% and is fastest for a midgap trap.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % band edges
  \draw[acc, thick, fill=acc!8] (0,2.6) rectangle (6.4,3.2);
  \node[anchor=west, font=\scriptsize] at (6.45,2.9) {$E_c$};
  \draw[acc, thick, fill=acc!30] (0,0) rectangle (6.4,0.6);
  \node[anchor=west, font=\scriptsize] at (6.45,0.3) {$E_v$};
  % direct transition
  \fill[acc] (1.4,2.6) circle (2.4pt);
  \draw[acc, ->, very thick] (1.4,2.55) -- (1.4,0.65);
  \node[acc, anchor=west, font=\scriptsize] at (1.5,1.6) {direct};
  % trap
  \draw[black] (3.6,1.5) -- (4.6,1.5);
  \node[anchor=south, font=\scriptsize] at (4.1,1.55) {trap $E_t$};
  \fill[acc] (4.1,2.6) circle (2.4pt);
  \draw[black, ->, very thick] (4.1,2.55) -- (4.1,1.58);
  \draw[black, ->, very thick] (4.1,1.42) -- (4.1,0.65);
  \node[black, anchor=west, font=\scriptsize] at (4.7,2.05) {capture};
\end{tikzpicture}
$$

## The continuity equation and the diffusion length

Conservation of carriers ties together transport, generation, and recombination.
The number of holes in a slab changes by the current flowing in minus the current
out, plus net generation — the **continuity equation**,

$$
\frac{\partial p}{\partial t} = -\frac{1}{e}\nabla\cdot\vec J_p + G - R, \qquad \frac{\partial n}{\partial t} = +\frac{1}{e}\nabla\cdot\vec J_n + G - R.
$$

For minority holes in a field-free n-type region under steady, low-level
injection, drift is negligible, $R - G = \delta p/\tau_p$, and the current is pure
diffusion. The steady continuity equation becomes

$$
D_p\frac{\d^2 \delta p}{\d x^2} - \frac{\delta p}{\tau_p} = 0,
$$

a second-order equation with exponential solutions. Injecting excess holes at
$x = 0$ and requiring decay,

$$
\delta p(x) = \delta p(0)\,e^{-x/L_p}, \qquad L_p = \sqrt{D_p\tau_p}.
$$

> **Definition (Diffusion length).** The **diffusion length** $L_p = \sqrt{D_p
> \tau_p}$ is the mean distance an injected minority carrier travels before
> recombining. It is the geometric mean of the distance set by diffusion and the
> lifetime, and it fixes the length scale of the junction: base widths, absorption
> depths, and the neutral regions of a diode are all measured against $L$.

For silicon with $\tau_p \sim 1\ \mu\text{s}$ and $D_p \approx 12\
\text{cm}^2/\text{s}$, the diffusion length is $L_p \approx 35\ \mu\text{m}$ —
macroscopic, and far larger than the depletion widths of the next lesson, which is
why minority-carrier diffusion, not drift, carries the current of an ideal diode.

$$
% caption: Minority-carrier injection profile. Excess holes injected at x = 0 into
% a neutral n region diffuse inward and recombine, decaying exponentially with the
% diffusion length L_p; the shaded area is the stored excess charge.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.0,0) node[below] {$x$};
  \draw[->, black] (0,0) -- (0,3.6) node[left] {excess $p$};
  % exponential decay
  \draw[acc, very thick, fill=acc!12, domain=0:6.4, samples=80, variable=\x]
    plot ({\x},{3.0*exp(-\x/2.3)}) -- (6.4,0) -- (0,0) -- cycle;
  % mark L_p where it falls to 1/e
  \draw[black, dashed] (2.3,0) -- (2.3,3.0*0.3679);
  \node[anchor=north, font=\scriptsize] at (2.3,-0.1) {$L_p$};
  \draw[black, dashed] (0,3.0*0.3679) -- (2.3,3.0*0.3679);
  \node[anchor=east, font=\scriptsize] at (-0.05,3.0*0.3679) {one $e$-fold};
  \node[acc, anchor=west, font=\scriptsize] at (1.3,2.5) {injected excess};
\end{tikzpicture}
$$

The three transport ingredients — the drift-diffusion current, the continuity
equation, and the recombination lifetime — form a closed system. Applied to a
region of spatially varying doping, they produce the built-in field, the depletion
layer, and the rectifying current of the
[p-n junction](/condensed-matter/semiconductors/the-pn-junction), the subject of
the next lesson.