Model spin-orbit coupling Hamiltonians for graphene systems

TL;DR

This paper develops comprehensive effective spin-orbit coupling Hamiltonians for various graphene-based systems, considering global and local symmetries, and introduces a previously neglected SOC parameter relevant for graphene on substrates.

Contribution

It provides a systematic derivation of SOC Hamiltonians for different symmetries and adsorption configurations, including a new SOC term for graphene on substrates.

Findings

Identification of symmetry-dependent SOC parameters

Introduction of a third SOC parameter for graphene on substrates

Explicit effective SOC Hamiltonians for various structures

Abstract

We present a detailed theoretical study of effective spin-orbit coupling (SOC) Hamiltonians for graphene based systems, covering global effects such as proximity to substrates and local SOC effects resulting, for example, from dilute adsorbate functionalization. Our approach combines group theory and tight-binding descriptions. We consider structures with global point group symmetries $D_{6h}$, $D_{3d}$, $D_{3h}$, $C_{6v}$, and $C_{3v}$ that represent, for example, pristine graphene, graphene mini-ripple, planar boron-nitride, graphene on a substrate and free standing graphone, respectively. The presence of certain spin-orbit coupling parameters is correlated with the absence of the specific point group symmetries. Especially in the case of $C_{6v}$---graphene on a substrate, or transverse electric field---we point out the presence of a third SOC parameter, besides the conventional intrinsic and Rashba contributions, thus far neglected in literature. For all global structures we provide effective SOC Hamiltonians both in the local atomic and Bloch forms. Dilute adsorbate coverage results in the local point group symmetries $C_{6v}$, $C_{3v}$, and $C_{2v}$ which represent the stable adsorption at hollow, top and bridge positions, respectively. For each configuration we provide effective SOC Hamiltonians in the atomic orbital basis that respect local symmetries. In addition to giving specific analytic expressions for model SOC Hamiltonians, we also present general (no-go) arguments about the absence of certain SOC terms.