Generation, Recombination & Carrier Lifetime

How excess carriers appear and disappear. Learn thermal and optical generation, excess carriers and low-level injection, recombination mechanisms — radiative (band-to-band), Shockley-Read-Hall through traps, Auger, and surface recombination — minority-carrier lifetime, diffusion length L = √(Dτ), steady-state and transient behaviour, photoconductivity, and why lifetime matters for diodes, BJTs, solar cells, LEDs and switching speed (lifetime control with Au/Pt or irradiation) — with Python computing SRH lifetimes, diffusion lengths, decay transients and Auger limits.

In equilibrium, generation and recombination balance. Light, injection through a junction, or high fields create excess carriers; recombination removes them with a characteristic lifetime τ. How far minority carriers diffuse before recombining — the diffusion length — shapes nearly every bipolar device.

Long lifetime is good for solar cells and BJT gain; short lifetime is good for fast-switching diodes. Engineers control it deliberately.

An analogy

Excess carriers are like visitors released into a city: they wander (diffuse) until they check into a hotel (recombination centre). Many hotels (traps) mean short stays and short journeys; few hotels let them wander far — the diffusion length.

Generation

Thermal generation (constantly, balanced by recombination in equilibrium), optical generation (photons with hν ≥ E_g), impact ionisation (high fields, avalanche) and band-to-band tunnelling (very high fields). In depleted regions recombination is suppressed, so net thermal generation there produces the reverse leakage current of junctions.

Recombination mechanisms

Three main paths:

Diffusion length and device design

L = √(Dτ) is the average distance a minority carrier travels before recombining. For BJT gain, base width must be ≪ L_n; for solar cells, L must exceed the absorber thickness; for fast diodes, lifetime is deliberately reduced (gold/platinum doping, electron or proton irradiation) to shrink stored charge and reverse-recovery time — at the cost of higher forward voltage and leakage.

Measurement

Lifetime is measured by photoconductance decay (µ-PCD, QSSPC), time-resolved photoluminescence, open-circuit voltage decay and reverse-recovery methods. Surface recombination velocity is separated by varying wafer thickness or passivation.

Formulas

Worked example: Diffusion length

Given: D_n 30 cm²/s, τ 1 µs

Find: L_n.

Answer: 55 µm

Worked example: Steady-state excess

Given: G 10²¹ cm⁻³s⁻¹, τ 1 µs

Find: Δn.

Answer: 10¹⁵ cm⁻³

Worked example: Decay

Given: τ 5 µs

Find: Time to fall to 1%.

Answer: 23 µs

Recombination mechanisms

  E_c ───●────────────●──────────────●●──────
          │ radiative   │ SRH via trap    │ Auger: energy given to a 3rd carrier
          │ (photon hν) ▼                 ▼
  E_t          ────────■──── (deep trap near mid-gap)
          ▼             ▼
  E_v ───○────────────○──────────────○───────
  Direct gap → radiative dominates; Si → SRH (traps) and Auger (heavy doping)

Radiative recombination emits light and dominates in direct-gap LEDs/lasers. In silicon, recombination occurs mainly through defect traps (SRH) — most effective when traps lie near mid-gap — and via Auger processes at high carrier densities. Surfaces add their own recombination unless passivated.

Lifetimes, diffusion lengths and transients

import math
# Recombination: SRH lifetime, diffusion length, excess-carrier decay and photoconductivity
vth, sigma = 1e7, 1e-15                         # thermal velocity cm/s, capture cross-section cm^2
for Nt in (1e11, 1e13, 1e15):
    tau = 1 / (sigma * vth * Nt)
    L = math.sqrt(12.0 * tau)                     # Dp ~ 12 cm^2/s (holes in n-type Si)
    print(f"trap density {Nt:.0e}: tau = {tau*1e6:9.3f} us, diffusion length L = {L*1e4:8.1f} um")
tau = 10e-6
for t in (0, 10e-6, 30e-6, 50e-6):
    print(f"t = {t*1e6:4.0f} us after the light pulse: excess carriers {1e14*math.exp(-t/tau):.2e} cm^-3")
G = 1e20                                          # generation rate cm^-3 s^-1
dn = G * tau
q = 1.602e-19
print(f"steady state under light: dn = G*tau = {dn:.1e} cm^-3 -> conductivity rise {q*dn*(1350+480):.3f} S/cm")
Cn = 2.8e-31                                       # Auger coefficient cm^6/s
print(f"Auger lifetime at n = 1e19: {1/(Cn*1e19**2)*1e9:.1f} ns (dominates in heavy doping)")
# trap density 1e+11: tau =  1000.000 us, diffusion length L =   1095.4 um
# trap density 1e+13: tau =    10.000 us, diffusion length L =    109.5 um
# trap density 1e+15: tau =     0.100 us, diffusion length L =     11.0 um
# t =    0 us after the light pulse: excess carriers 1.00e+14 cm^-3
# t =   10 us after the light pulse: excess carriers 3.68e+13 cm^-3
# t =   30 us after the light pulse: excess carriers 4.98e+12 cm^-3
# t =   50 us after the light pulse: excess carriers 6.74e+11 cm^-3
# steady state under light: dn = G*tau = 1.0e+15 cm^-3 -> conductivity rise 0.293 S/cm
# Auger lifetime at n = 1e19: 35.7 ns (dominates in heavy doping)

Trap density controls lifetime directly: 10¹¹ traps give 1 ms and a millimetre-long diffusion length (solar-grade), 10¹⁵ traps give 100 ns and 11 µm (fast-recovery diodes). After a light pulse excess carriers decay exponentially with τ. Under continuous illumination Δn = Gτ raises conductivity — the basis of photoconductors. In heavily doped emitters, Auger limits lifetime to nanoseconds.

Common mistakes

Applications

Key takeaways

Going further