Axial approximation in the calculation of the valence bands in semiconductor superlattices.

Abstract

We apply the axial approximation, introduced by Altarelli, Ekenberg, and Fasolino [Phys. Rev. B 32, 5138 (1985)] to Si-${\mathrm{Ge}}_{\mathrm{x}}$${\mathrm{Si}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$ superlattices and compare the results to exact calculations. In the axial approximation, the warping of the valence bands is neglected so that a cylindrically symmetric system is obtained. We show that the influence of the warping is roughly the same for every semiconductor. According to our calculations, the axial approximation is satisfactory close to \ensuremath{\Gamma}, but at large values of the transverse crystal momentum the warping becomes so severe that the approximation is quite poor. We have also included the warping by treating it as a perturbation applied to the cylindrically symmetric system. This only leads to an improvement when the warping is small.