Under conditions of intense optical pumping or electrical injection it is possible to establish a temperature of excited carriers, $T_e$, larger than the temperature of the lattice, $T_L$, for periods of time sufficient for many effects to be observed. It is well known that semiconductor band gaps are a function of temperature, but the variation with the two temperatures, $T_e$ and $T_L$, when these are different seems not to have been discussed previously. Simple thermodynamic arguments may be applied when it is recognized that a band gap is a chemical potential. The simple formula, ${\Delta E}_{\mathrm{cv}}(T_e,T_L)={\Delta H}_{\mathrm{cv}}\left(T_L\right)-T_e{\Delta S}_{\mathrm{cv}}\left(T_L\right)$, is deduced. Physically this formula states that the vibronic degeneracy of the electronic states (valence and conduction band or bonding and antibonding) among which the carriers are distributed with characteristic temperature $T_e$ is determined by the lattice temperature, $T_L$. Thus when $T_e>>T_L$, anamalously large variations in the gap will occur. It is found that under certain conditions loss of energy from the carrier system to the lattice will cause the density of excited carriers to increase, rather than decrease.

• Received 31 July 1980

DOI:https://doi.org/10.1103/PhysRevB.23.5543