Scissors implementation within length-gauge formulations of the frequency-dependent nonlinear optical response of semiconductors

Abstract

We show that previous calculations of the nonlinear optical response of semiconductors using the ``length-gauge'' approach of Aversa and Sipe[Phys. Rev. B 52, 14636 (1995)] have incorrectly implemented the scissors correction. Previous calculations have first modified the expressions such that the crystal momentum position operator interband matrix elements ${r}_{nm}^{a}(\mathbf{k})$ and their generalized $k$-derivatives ${r}_{nm;b}^{a}(\mathbf{k})$ are written in terms of the local density approximation (LDA) momentum matrix elements and energy eigenvalues, which are then adjusted to account for the scissors correction. This method is incorrect as it violates specific sum rules. We give a consistent implementation that respects the sum rules: ${r}_{nm}^{a}(\mathbf{k})$ and ${r}_{nm;b}^{a}(\mathbf{k})$, when written in terms of momentum matrix elements and energy eigenvalues, should be evaluated within Kohn--Sham LDA and not with the scissored one. Calculated spectra via full potential augmented plane wave plus local orbital band structures for the second-harmonic generation tensor ${\ensuremath{\chi}}_{2}^{xyz}(\ensuremath{-}2\ensuremath{\omega};\ensuremath{\omega},\ensuremath{\omega})$ show that the errors in previous calculations are approximately 39% and 28% for GaAs and GaP, respectively.