Engineering & Science
SRH, Radiative & Auger Recombination Lifetime + Diffusion Length
Combine defect-mediated Shockley-Read-Hall (τ_SRH), band-to-band radiative (B·Δn), and three-particle Auger (C_a·Δn^2) recombination rates into effective minority-carrier lifetime τ_eff (μs) and diffusion length L = SQRT(D·τ_eff) (μm).
Why solar-cell and LED efficiency droops at high injection: the C·Δn^2 Auger wall!
Minority carriers in crystalline solar cells and LEDs die through three parallel recombination channels whose inverse lifetimes add by Matthiessen’s rate rule: 1 / τ_eff = 1 / τ_SRH + B * Δn + C_a * Δn^2! At low injection, deep mid-gap crystal defects set the Shockley-Read-Hall ceiling τ_SRH; at moderate injection, two-carrier radiative emission B * Δn grows linearly; at concentrator solar or high-current LED injection, three-carrier Auger recombination C_a * Δn^2 explodes quadratically and caps maximum achievable efficiency!
Track float-zone silicon (τ_SRH = 500 μs, radiative B = 4.7 × 10^-15 cm^3/s, ambipolar Auger C_a = 1.66 × 10^-30 cm^6/s, diffusivity D = 30 cm^2/s): at 1-sun injection Δn = 1.0 × 10^15 cm^-3, τ_eff = 498.42 μs; at Δn = 1.0 × 10^16 cm^-3, τ_eff = 451.88 μs with minority diffusion length L = SQRT(D * τ_eff) = 1,164.3 μm (easily crossing a 180 μm wafer!); at concentrator injection Δn = 1.0 × 10^17 cm^-3, Auger slashes τ_eff nearly 10-fold down to 52.44 μs!
| Excess Carrier Injection Δn (cm^-3) | Effective Lifetime τ_eff (μs) | Diffusion Length L (μm) | Limiting Recombination Mechanism |
|---|---|---|---|
| 1-Sun Passivated Cell (1.0 × 10^15) | 498.42 μs | 1,222.8 μm | Defect-Mediated SRH (~99.7%) |
| Moderate Concentrator (1.0 × 10^16) | 451.88 μs | 1,164.3 μm | SRH + Onset of Auger (7.5%) |
| High Injection / Laser (1.0 × 10^17) | 52.44 μs | 396.6 μm | Three-Particle Auger Dominated (87%) |
Combined Effective Carrier Lifetime τ_eff (μs)
Sums inverse SRH (1 / (B2 * 10^-6)), radiative (B3 * B6), and Auger (B4 * B6^2) rates in s^-1 and converts back to microseconds.
=IF(OR(B2 <= 0, B3 < 0, B4 < 0, B6 <= 0), "Invalid Recombination Inputs", ROUND(1000000 / ((1 / (B2 * 0.000001)) + B3 * B6 + B4 * (B6 ^ 2)), 2))Minority-Carrier Diffusion Length L = SQRT(D * τ_eff) (μm)
Computes 10,000 * SQRT(D_cm2s * τ_eff_s) in micrometers to verify whether L exceeds solar wafer thickness W.
=IF(OR(B2 <= 0, B3 < 0, B4 < 0, B5 <= 0, B6 <= 0), "Invalid Recombination Inputs", ROUND(10000 * SQRT(B5 / ((1 / (B2 * 0.000001)) + B3 * B6 + B4 * (B6 ^ 2))), 1))Try the example
Use the original values or change them to test your understanding.