Band-gap renormalization in semiconductors with multiple inequivalent valleys

The narrowing of direct and indirect band gaps is investigated in highly excited semiconductor materials with multiple inequivalent valleys in the conduction band. The model substance chosen is ${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As close to the crossover from a direct-band-gap to an indirect-band-gap semiconductor. A theoretical model is presented, which is an extension of the universal formula for band-gap renormalization (BGR) to a multivalley scenario. This model accounts explicitly for the electron-exchange contribution to BGR in each conduction-band minimum. Its applicability to the case of high carrier densities is demonstrated by comparison to band-gap energies experimentally determined from time-resolved luminescence spectra. In a scenario of essentially unoccupied higher-energy conduction-band minima, one finds a renormalization of the fundamental gap by full correlation and exchange effects. The higher-energy gaps, on the other hand, narrow mainly due to the interactions between the holes in the valence band, while the electron-exchange effect in the empty valleys is negligible. The case of the electrons being distributed among several minima, such as in direct band-gap ${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As just below the crossover point, is treated with a self-consistent routine for the population and renormalization of each valley. The agreement with experimental data is excellent and the alleged enhancement of the BGR in this scenario is quantitatively explained. The differential renormalization of minima with different populations is utilized to achieve a laser-induced transition from a direct-band-gap to an indirect-band-gap semiconductor in ${\mathrm{Al}}_{0.42}$ ${\mathrm{Ga}}_{0.58}$As. Crossings of various conduction-band minima due to BGR is predicted from the self-consistent multivalley model and demonstrated by the properties of indirect stimulated emission.