Spin-orbit interactions in inversion-asymmetric two-dimensional hole systems: A variational analysis
We present an in-depth study of the spin-orbit (SO) interactions occurring in inversion-asymmetric two-dimensional hole gases at semiconductor heterointerfaces. We focus on common semiconductors such as GaAs, InAs, InSb, Ge, and Si. We develop a semianalytical variational method to quantify SO interactions, accounting for both structure inversion asymmetry (SIA) and bulk inversion asymmetry (BIA). Under certain circumstances, using the Schrieffer-Wolff (SW) transformation, the dispersion of the ground state heavy hole subbands can be written as E (k ) =A k2-B k4±C k3 where A ,B , and C are material- and structure-dependent coefficients. We provide a simple method of calculating the parameters A ,B , and C , yet demonstrate that the simple SW approximation leading to a SIA (Rashba) spin splitting ∝k3 frequently breaks down. We determine the parameter regimes at which this happens for the materials above and discuss a convenient semianalytical method to obtain the correct spin splitting, effective masses, Fermi level, and subband occupancy, together with their dependence on the charge density, and dopant type, for both inversion and accumulation layers. Our results are in good agreement with fully numerical calculations as well as with experimental findings. They suggest that a naive application of the simple cubic Rashba model is of limited use in either common heterostructures or quantum dots. Finally, we find that for the single heterojunctions studied here the magnitudes of BIA terms are always much smaller than those of SIA terms.