Intrinsic and Rashba Spin-orbit Interactions in Graphene Sheets

TL;DR

This paper derives explicit formulas for intrinsic and Rashba spin-orbit gaps in graphene using a microscopic model and perturbation theory, confirming the form of the effective Hamiltonian with ab initio calculations.

Contribution

It provides explicit expressions for spin-orbit induced gaps in graphene and clarifies their dependence on atomic spin-orbit coupling and external electric fields.

Findings

Rashba interaction is first order in atomic spin-orbit coupling and electric field.

Intrinsic spin-orbit gap is second order in atomic spin-orbit coupling.

Effective Hamiltonian matches recent theoretical proposals.

Abstract

Starting from a microscopic tight-binding model and using second order perturbation theory, we derive explicit expressions for the intrinsic and Rashba spin-orbit interaction induced gaps in the Dirac-like low-energy band structure of an isolated graphene sheet. The Rashba interaction parameter is first order in the atomic carbon spin-orbit coupling strength $\xi$ and first order in the external electric field $E$ perpendicular to the graphene plane, whereas the intrinsic spin-orbit interaction which survives at E=0 is second order in $\xi$. The spin-orbit terms in the low-energy effective Hamiltonian have the form proposed recently by Kane and Mele. \textit{Ab initio} electronic structure calculations were performed as a partial check on the validity of the tight-binding model.