Two-photon transitions to exciton states in semiconductors. Application to CuCl

We show that excitons in semiconductors are described by an effective Hamiltonian containing a nonlocal potential. In order to preserve the gauge invariance of the multiphoton transition rates in such systems, we show that a corrective term due to the nonlocal character of the potential has to be added to the standard electron-photon interaction. As an application, we calculate in the velocity and length gauges the two-photon transition rates between the crystal ground state and the $2p$ Wannier exciton in CuCl, and we show analytically and numerically their gauge invariance. We also estimate the contribution to the two-photon absorption coefficient due to transitions to $2p$ excitons in CuCl and we find a general agreement with experiment.