Energy spectra for quantum wires and two-dimensional electron gases in magnetic fields with Rashba and Dresselhaus spin-orbit interactions

Abstract

We introduce an analytical approximation scheme to diagonalize parabolically confined two-dimensional (2D) electron systems with both the Rashba and Dresselhaus spin-orbit interactions. The starting point of our perturbative expansion is a zeroth-order Hamiltonian for an electron confined in a quantum wire with an effective spin-orbit induced magnetic field along the wire, obtained by properly rotating the usual spin-orbit Hamiltonian. We find that the spin-orbit-related transverse coupling terms can be recast into two parts $W$ and $V$, which couple crossing and noncrossing adjacent transverse modes, respectively. Interestingly, the zeroth-order Hamiltonian together with $W$ can be solved exactly, as it maps onto the Jaynes-Cummings model of quantum optics. We treat the $V$ coupling by performing a Schrieffer-Wolff transformation. This allows us to obtain an effective Hamiltonian to third order in the coupling strength ${k}_{\text{R}}\ensuremath{\ell}$ of $V$, which can be straightforwardly diagonalized via an additional unitary transformation. We also apply our approach to other types of effective parabolic confinement, e.g., 2D electrons in a perpendicular magnetic field. To demonstrate the usefulness of our approximate eigensolutions, we obtain analytical expressions for the $n\text{th}$ Landau-level ${g}_{n}$ factors in the presence of both Rashba and Dresselhaus couplings. For small values of the bulk $g$ factors, we find that spin-orbit effects cancel out entirely for particular values of the spin-orbit couplings. By solving simple transcendental equations we also obtain the band minima of a Rashba-coupled quantum wire as a function of an external magnetic field. These can be used to describe Shubnikov-de Haas oscillations. This procedure makes it easier to extract the strength of the spin-orbit interaction in these systems via proper fitting of the data.