Second-order optical response in semiconductors

We present a new general formalism for investigating the second-order optical response of solids, and illustrate it by deriving expressions for the second-order susceptibility tensor ${\ensuremath{\chi}}_{2}(\ensuremath{-}{\ensuremath{\omega}}_{\ensuremath{\Sigma}};{\ensuremath{\omega}}_{\ensuremath{\beta}},{\ensuremath{\omega}}_{\ensuremath{\gamma}}),$ where ${\ensuremath{\omega}}_{\ensuremath{\Sigma}}={\ensuremath{\omega}}_{\ensuremath{\beta}}+{\ensuremath{\omega}}_{\ensuremath{\gamma}},$ for clean, cold semiconductors in the independent particle approximation. Based on the identification of a polarization operator $\mathbf{P}$ that would be valid even in a more complicated many-body treatment, the approach avoids apparent, unphysical divergences of the nonlinear optical response at zero frequency that sometimes plague such calculations. As a result, it allows for a careful examination of $\mathit{actual}$ divergences associated with physical phenomena that have been studied before, but not in the context of nonlinear optics. These are (i) a coherent current control effect called ``injection current,'' or ``circular photocurrent,'' and (ii) photocurrent due to the shift of the center of electron charge in noncentrosymmetric materials in the process of optical excitation, called ``shift current.'' The expressions we present are amenable for numerical calculations, and we demonstrate this by performing a full band-structure calculation of the shift current coefficient for GaAs.