Third-order optical nonlinearities in semiconductors: The two-band model

We calculate the coherent electronic contributions to the third-order optical response ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$(-ω;ω,Ω,-Ω) of bulk semiconductors in the independent-particle approximation using a simple two-band model. The formalism used to derive this response coefficient naturally accounts for all relevant contributions and, in contrast to existing results in the literature, leads to physically realistic, nondivergent expressions in the limits ω,Ω→0. Such well behaved infrared limits imply that the imaginary part of our ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$ correctly describes the dispersion of nondegenerate absorption; indeed for Ω=0 our results are consistent with predictions from Franz-Keldysh theory. Complementing these results, we can now also unambiguously extract from the real part of ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$ the below band gap, two-band model predictions for the nonlinear refractive index, the dc Kerr effect, and the virtual photoconductivity; all of these predict a finite, real ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$(0;0,0,0) as physically expected for clean, cold semiconductors. Finally, our specific results help expose more general consequences of the gauge choice when employing common approximate band-structure models.