Variational analysis of the Rashba splitting in III–V semiconductor inversion layers

Abstract

A spin-dependent variational theory is used to analyze the Rashba spin-orbit splitting in two-dimensional electron gases formed in III--V semiconductor inversion layers. The spin split conduction subbands in CdTe/InSb, insulator/InAs, InP/InGaAs, InAlAs/InGaAs, and AlGaAs/GaAs heterojunctions are calculated. The theory, presented here in detail, is based on the 8 $\ifmmode\times\else\texttimes\fi{}$ 8 $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ Kane model and on the introduction of simple and convenient spin-dependent Fang-Howard trial functions, and leads to analytical expressions for the split subbands, as well as allows for a detailed knowledge of the Rashba spin-orbit coupling, including its explicit dependence on structure parameters and its decomposition into separate contributions. The Rashba coupling parameter and the population difference in the spin-split subbands, as experimentally determined from the beating pattern of the Shubnikov-de Haas (SdH) oscillations, are obtained as a function of the electron density (${n}_{s}$). The separate contributions to the particularly large Rashba splitting in CdTe/InSb heterojunctions are also computed and discussed. It is shown, for example, that due to the spin-dependent boundary conditions, the direct Rashba spin-orbit coupling term in the effective Hamiltonian dominates the splitting only for ${n}_{s}>{10}^{10}$ cm${}^{\ensuremath{-}2}$ while it is the barrier penetration kinetic energy term that gives the largest contribution to the Rashba effect at lower densities.