Effect of Carrier Scattering on Nonlinear Optical Susceptibility due to Mobile Carriers in InSb, InAs, and GaAs

To estimate the effect of carrier scattering on the low-frequency nonlinear optical response of mobile carriers in semiconductors, the Boltzmann transport equation is solved in a generalized relaxation-time approximation. Two cases, one in which the elastic scattering of carriers by ionized impurities dominates the transport properties and the other in which inelastic scattering by longitudinal optical phonons is the most important scattering mechanism, are considered in detail for InSb, InAs, and GaAs. We find that the inclusion of carrier scattering affects the third-order nonlinear susceptibility in two distinct ways. First, the contribution due to nonparabolicity is modified in a nontrivial fashion. This modification becomes important in the optical-frequency-mixing experiments, when the difference frequency is comparable to the "average" collision frequency. Second, an additional nonlinearity arises because of the energy dependence of carrier collision frequency. This is estimated to be unimportant in most cases for which detailed experimental data are available at present, but is shown to be quite important when the difference frequency becomes smaller. It is found that, at low temperatures, the additional nonlinearity due to energy-dependent scattering can be significant in InSb and InAs samples, with $n\approx {10}^{18}$ ${\mathrm{cm}}^{-3\mathrm{}}$, even for the frequencies used in the presently available experimental investigations.