Luttinger-like parameter calculations

Abstract

We have calculated the Luttinger-like parameters, used in the $8\ifmmode\times\else\texttimes\fi{}8$ Hamiltonian describing both conduction and valence bands, via a second-order perturbation theory that allows one to make the L\"owdin renormalization valid for a set of nondegenerate energy levels. We show that the formulas given in the paper by Pidgeon and Brown [Phys. Rev. 146, 575 (1966)] are the correct ones, contrary to what is claimed in recent papers. This leads to strong discrepancies when the spin-orbit splitting is not much smaller than the band gap. For example, the ratio between the Luttinger-like parameters ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\gamma}}}_{1},$ calculated in the two cases, is of the order of 1.5 in GaAs, 3 in InAs, and 16 in InSb. The accuracy of the equations used by Pidgeon and Brown is also discussed.