Calculation of resonant second-order Raman efficiencies for allowed and forbidden scattering

Abstract

The frequency dependence of the cross section for second-order Raman scattering in an insulator is evaluated for incident frequencies below and above the gap. We consider both a deformation potential and the Fr\"ohlich interaction for the electron-one-phonon coupling and take the uncorrelated electron-hole continuum as intermediate states. The following results compare well with recent experiments in GaP: (a) the frequency dependence of the scattering intensity with $\mathrm{TO}(\ensuremath{\Gamma})+\mathrm{LO}(\ensuremath{\Gamma})$ and $2\mathrm{LO}(\ensuremath{\Gamma})$ phonons around the gap; (b) the corresponding selection rules; (c) absolute numbers for the deformation potential and the prefactor of the Fr\"ohlich interaction using measured intensity ratios with $\mathrm{TO}(\ensuremath{\Gamma})$ first-order scattering and a pseudopotential calculation. Calculated line shifts and widths due to phonon dispersion show a strong dependence on the incident frequency around the gap.