The Light-Emitting Diode (LED) — Question 6

Question 6

The ABC recombination model predicts an efficiency maximum

In a supplied uniform LED model with equal electron and hole densities n, the total recombination rate per volume is $R=An+Bn^2+Cn^3$. Only $Bn^2$ is radiative. Assume constant positive coefficients.

1. Derive internal efficiency and the carrier density maximizing it.

2. For $A=10^7\,\mathrm{s^{-1}}$, $B=10^{-16}\,\mathrm{m^3/s}$, and $C=10^{-41}\,\mathrm{m^6/s}$, evaluate the optimum density and efficiency.

Question 6 - Solution

The ABC recombination model predicts an efficiency maximum

(a) Method and calculation

$$\boxed{\eta(n)=\frac{Bn^2}{An+Bn^2+Cn^3}=\frac{Bn}{A+Bn+Cn^2}}.$$$$\frac{d\eta}{dn}=\frac{B(A-Cn^2)}{(A+Bn+Cn^2)^2}=0\Rightarrow\boxed{n_*=\sqrt{A/C}}.$$

(b) Interpretation and further calculation

$$\boxed{n_*=10^{24}\,\mathrm{m^{-3}}}.$$$$\boxed{\eta(n_*)=\frac{B\sqrt{A/C}}{2A+B\sqrt{A/C}}=\frac{10^8}{1.2\times10^8}=.8333}.$$

Physical check

At the optimum, the linear and cubic nonradiative rates are equal. The high-density decrease follows from the supplied cubic term; actual LED droop may involve additional mechanisms.