Valence-band warping in tight-binding models

The spin-orbit nearest-neighbor ${\mathrm{sp}}^3{s}^{*}$ model is widely believed able to reproduce accurately the valence bands of most III-V and group-IV semiconductors, essential for modeling the in-plane dispersion of valence- and inter-band quantum heterostructures. To check this belief, we study valence-band warping in the spin-orbit ${\mathrm{sp}}^3{s}^{*}$ model including interactions up to second-nearest-neighbor, providing exact, analytic formulas for the [110] and [111] light- and heavy-hole masses. Surprisingly, we find that the second-nearest-neighbor model offers significantly more flexibility in fitting these masses independently of those along [001] than does the nearest-neighbor model.