On the effective models of spin-orbit coupling in a two-dimensional electron gas

TL;DR

This paper derives comprehensive effective Hamiltonians for a 2D electron gas with spin-orbit coupling, classifies symmetry-dependent terms, and explores topological insulator models with anisotropic spin interactions.

Contribution

It provides a systematic symmetry-based classification of spin-orbit and magnetic coupling terms in 2D electron gases, including a model for topological insulators.

Findings

Spin-orbit coupling can be cubic in momentum in some bands.

Effective Zeeman interaction shows strong anisotropy.

A simple topological insulator model with spin-orbit coupling is analyzed.

Abstract

We use the method of invariants to derive one- and two-band effective Hamiltonians of a noncentrosymmetric two-dimensional electron gas, in the presence of magnetic field. A complete classification of the antisymmetric spin-orbit and magnetic coupling terms near the $\Gamma$ point is developed for all two-dimensional crystal symmetries. The effective Hamiltonian depends on the symmetry of the Bloch bands at the $\Gamma$ point, which is described by one of the double-valued corepresentations of the two-dimensional magnetic point group. In some bands, the spin-orbit coupling is cubic in the electron momentum and the effective Zeeman interaction is strongly anisotropic. As an example of a two-band effective Hamiltonian, we introduce a simple model of a topological insulator with the intraband and interband spin-orbit coupling and investigate its bulk and boundary properties.