The phonon assisted absorption of excitons in Cu$_2$O
The basic theoretical foundation for the modelling of phonon assisted absorption spectra in direct bandgap semiconductors, introduced by Elliott 60 years ago using second order perturbation theory, results in a square root shaped dependency close to the absorption edge. A careful analysis of the experiments reveals that for the yellow S excitons in Cu$_2$O the lineshape does not follow that square root dependence. The reexamination of the theory shows that the basic assumptions of constant matrix elements and constant energy denominators is invalid for semiconductors with dominant exciton effects like Cu$_2$O, where the phonon assisted absorption proceeds via intermediate exciton states. The overlap between these and the final exciton states strongly determines the dependence of the absorption on the photon energy. To describe the experimental observed line shape of the indirect absorption of the yellow S exciton states we find it necessary to assume a momentum dependent deformation potential for the optical phonons.