Past exam of the laser science and quantum information processing course of the University of Oxford / 2021 / 2 / b / Solution

A quasi-Fermi level is the chemical potential parameter of an approximately thermalized carrier population within one energy band. Rapid intraband scattering can establish a Fermi-Dirac distribution in each energy band before slower interband recombination equilibrates their populations. Thus separate $E_{\mathrm{Fc}}$ and $E_{\mathrm{Fv}}$ describe a pumped population inversion even though the entire semiconductor is far from thermal equilibrium.

At a common positive temperature, the Fermi-Dirac distribution is strictly decreasing in $E-E_F$. ConsequentlyThe joint density of states is nonzero only above the band gap, soIn particular the quasi-Fermi-level splitting must exceed $E_g$. The upper boundary is transparency, where stimulated emission balances absorption; the lower boundary has zero joint density of states in this ideal three-dimensional parabolic model. The strict occupation comparison uses $T>0$; at exactly zero temperature occupations become step functions, and their supports should be checked explicitly, as in the next part.