Energy Levels of Indirect Excitons in Semiconductors with Degenerate Bands

Abstract

A method previously introduced for direct excitons is extended to investigate indirect excitons in semiconductors with degenerate bands. Even though this method can be applied in the case of a general position in $\overrightarrow{\mathrm{k}}$ space of the conduction-band minima, the present investigation is limited to the most common directions (1, 0, 0) and (1, 1, 1). A splitting of the exciton levels, which is shown to be due to the anisotropy of the conduction-band minima and which is predicted by group-theoretical analysis, is quantitatively given by a simple analytical expression. The symmetry of the phonons which assist indirect optical transitions is also given. Results for the $1s$ and $2s$ exciton states in AlSb, GaP, Si, and Ge are presented. The method is very accurate and agrees well with available experimental data for the first three substances. For Ge its accuracy is not so good, because of the strong anisotropy in the conduction-band minima, but it is expected to be within 10%.