Empirical tight-binding calculation of dispersion in the second-order nonlinear optical constant for zinc-blende crystals

We calculate both the finite- and zero-frequency values of χ→ ${}_{}{}^{\mathrm{}\left(2\right)\mathrm{}}{}_{}{}^{}$ for the zinc-blende materials GaP, GaAs, GaSb, InAs, and InSb, by employing an empirical tight-binding band-structure technique, used previously to obtain the dispersion in the linear-optical properties. The momentum matrix elements are calculated by three methods: by using k⋅p perturbation theory together with the experimental effective masses, by direct calculation using wave functions from a more fundamental theory, and by fitting ${\epsilon }_1$(0) to experiment. The calculated conduction-band–conduction-band (c-c) momentum matrix elements were found to be approximately an order of magnitude smaller than the experimental matrix elements, while the valence-band–conduction-band (v-c) matrix elements agreed much better. When the experimental c-c matrix elements are used, good agreement is found between theory and experiment for χ→ ${}_{}{}^{\mathrm{}\left(2\right)\mathrm{}}{}_{}{}^{}$ for all materials except InAs and InSb in the low-frequency region. For these latter materials, the discrepancy is accounted for by inadequacies in the empirical tight-binding band structure. It is proposed that Fong and Shen’s [Phys. Rev. B 12, 2325 (1975)] anomalously low results for χ→ ${}_{}{}^{\mathrm{}\left(2\right)\mathrm{}}{}_{}{}^{}$(0) could be due to calculated values for the c-c (or v-c) momentum matrix elements which are too small.