The Light-Emitting Diode (LED) — Question 3

Question 3

Competing lifetimes set the internal quantum efficiency

An injected carrier population N recombines radiatively at rate $N/\tau_r$ and nonradiatively at rate $N/\tau_{nr}$. Let $\tau_r=20.0\,\mathrm{ns}$ and $\tau_{nr}=30.0\,\mathrm{ns}$. Every radiative event emits one photon.

1. Derive and evaluate total lifetime and internal quantum efficiency.

2. If passivation doubles only $\tau_{nr}$, find the new efficiency and total lifetime.

Question 3 - Solution

Competing lifetimes set the internal quantum efficiency

(a) Method and calculation

$$\boxed{\tau^{-1}=\tau_r^{-1}+\tau_{nr}^{-1},\quad\tau=12.0\,\mathrm{ns}}.$$$$\boxed{\eta_{\mathrm{int}}=\frac{1/\tau_r}{1/\tau_r+1/\tau_{nr}}=\frac{\tau_{nr}}{\tau_r+\tau_{nr}}=.600}.$$

(b) Interpretation and further calculation

$$\boxed{\eta_{\mathrm{int,new}}=60/(20+60)=.750}.$$$$\boxed{\tau_{\mathrm{new}}=20(60)/(20+60)=15.0\,\mathrm{ns}}.$$

Physical check

Suppressing nonradiative loss improves efficiency but lengthens the population decay time. The two rates add because they act on the same carrier population.