Nonlinear optical susceptibilities of semiconductors: Results with a length-gauge analysis

We present a simple prescription for the derivation of electronic contributions to the nonlinear optical response of crystals in the independent particle approximation. Semiconductor Bloch equations are found that include previously neglected effects of intraband motion. Applying perturbation theory to clean, cold semiconductors we find expressions for the susceptibilities lacking the unphysical divergences at zero frequency that have plagued other calculations. For these materials we present well-behaved, general expressions for ${\mathrm{\chi }}^{\mathrm{}\left(2\right)\mathrm{}}$ and ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$ for arbitrary frequency mixing and give an explicit demonstration of the finite zero-frequency value of ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$. We further show how second-order photogalvanic effects are contained in certain physical zero-frequency divergences of ${\mathrm{\chi }}^{\mathrm{}\left(2\right)\mathrm{}}$, and consider the corresponding physical zero-frequency divergences of ${\mathrm{\chi }}^{\mathrm{}\left(3\right)\mathrm{}}$.