Phonon-assisted two-photon absorption in the presence of a dc-field: the nonlinear Franz-Keldysh effect in indirect gap semiconductors

The two-photon absorption coefficient of an indirect gap semiconductor (phonon-assisted two-photon absorption) in the presence of a strong dc-electric field applied perpendicular to the direction of propagation of the optical field is calculated using the formalism developed elsewhere (Aspnes 1996 Phys. Rev. B. 147 554). We show that depending on the type of transition (i.e., allowed-allowed, allowed-forbidden or forbidden-forbidden), the absorption coefficient followed different dispersion relations. In the limit of a weak electric field, we recovered results previously calculated using perturbation theory. In the strong dc-field regime, we found that below the rescaled energy gap given by Eg/N, where N is the number of photons, the tunnelling effect is present, but to our surprise, above the rescaled gap, the Franz-Keldysh oscillations are present only for the allowed-allowed transition. This absence of the oscillations in the allowed-forbidden and forbidden-forbidden transitions is possible due to the weak coupling of the tails of the electron and hole wavefunctions