Phonon-Assisted Two-Photon Exciton Transitions in Semiconductors

Abstract

The theory of phonon-assisted two-photon exeiton transitions is developed from third-order time-dependent perturbation theory. Three different band models are introduced:a) a four-band model corresponding to Loudon’s three-hand model in the case of direct twophoton transitions;b) a two-band model corresponding to that of Mahan for the two-photon transitions case;c) an intermediate (three-band) model which seems to be more convenient for this type of problem. Selection rules are different for the different models. For example, in modela) only δ-excitons are allowed, whilst in modelb)s andd excitons are permitted. In modelc) we find that the selection rules allowp-exciton states only. The symmetry of the phonons which contribute to the transitions for each model, and a group-theoretical analysis of the final exciton symmetry, are given for GaP, Ge and Si. We find that the threeband model is the most favourable for indirect two-photon processes. Numerical calculations, considering the different energy denominators which appear in the expression of the transition rates, show that the three-band model has the largest absorption coefficient, and the twoband model the smallest. This is due to the different matrix elements and energy denominators used for each model.