Carrier Transport and Recombination

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

v_{d} = \frac{q τ}{m ^{*}} E \equiv \pm μ E, μ = \frac{e τ}{m ^{*}},

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

J_{drift} = e (n μ_{e} + p μ_{h}) E = σ E, σ = e (n μ_{e} + p μ_{h}),

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

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 $v_{sat} \approx 1 0^{7} cm/s$ for silicon, independent of further increase in field. Velocity saturation limits the switching current in short-channel transistors.

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.

Scattering and the temperature dependence of mobility

The scattering time $τ$ 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}{μ} = \frac{1}{μ _{L}} + \frac{1}{μ _{I}} .

The two mechanisms pull in opposite temperature directions:

Lattice (phonon) scattering. The density of phonons rises with temperature, shortening $τ$ ; a Boltzmann-averaged cross section gives $μ_{L} \propto T^{- 3/2}$ . 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 $μ_{I} \propto T^{3/2} / N_{I}$ — 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.

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.

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

J_{n}^{diff} = e D_{n} \nabla n, J_{p}^{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,

J_{n} = e n μ_{e} 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 $n (x) \propto e^{e φ (x) / k_{B} T}$ in the electrostatic potential $φ$ , so $\nabla n = (e / k_{B} T) n \nabla φ = - (e / k_{B} T) n E$ . Setting $J_{n} = 0$ ,

e n μ_{e} E + e D_{n} (- \frac{e}{k _{B} T} n E) = 0 ⟹ D_{n} = \frac{k _{B} T}{e} μ_{e} .

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.

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, $R = B n p$ . Subtracting the equilibrium generation $G_{0} = B n_{i}^{2}$ , the net rate is

U = B (n p - n_{i}^{2}) .

For small excess $δ p$ of minority holes in n-type material ( $n \approx N_{d}$ ), $U \approx B N_{d} δ p = δ p / τ_{p}$ with the minority-carrier lifetime

τ_{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{n p - n _{i}^{2}}{τ _{p 0} ( n + n _{1} ) + τ _{n 0} ( p + p _{1} )}, n_{1} p_{1} = n_{i}^{2},

where $n_{1} = n_{i} e^{(E_{t} - E_{i}) / k_{B} T}$ and the capture times $τ_{n 0}, τ_{p 0}$ 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.

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.

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 J_{p} + G - R, \frac{\partial n}{\partial t} = + \frac{1}{e} \nabla \cdot J_{n} + G - R .

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

D_{p} \frac{d ^{2} δ p}{d x ^{2}} - \frac{δ p}{τ _{p}} = 0,

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

δ p (x) = δ p (0) e^{- x / L_{p}}, L_{p} = D_{p} τ_{p} .

For silicon with $τ_{p} \sim 1 μ s$ and $D_{p} \approx 12 cm^{2} / s$ , the diffusion length is $L_{p} \approx 35 μ m$ — 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.

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.

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, the subject of the next lesson.