Band-structure topologies of graphene: spin-orbit coupling effects from first principles

TL;DR

This study uses first-principles calculations to analyze how spin-orbit coupling and electric fields influence graphene's band structure, revealing small but significant gaps and splittings relevant for topological and spintronic applications.

Contribution

It provides a detailed first-principles analysis of spin-orbit and electric field effects on graphene's band topology, highlighting the role of higher orbitals and extrinsic splittings.

Findings

Spin-orbit coupling opens a 24 μeV gap at K points.

Electric field induces a 10 μeV Rashba splitting per V/nm.

Structural distortions are minimal and have little effect.

Abstract

The electronic band structure of graphene in the presence of spin-orbit coupling and transverse electric field is investigated from first principles using the linearized augmented plane-wave method. The spin-orbit coupling opens a gap at the $K(K')$-point of the magnitude of 24 $\mu$eV (0.28 K). This intrinsic splitting comes 96% from the usually neglected $d$ and higher orbitals. The electric field induces an additional (extrinsic) Bychkov-Rashba-type splitting of 10 $\mu$eV (0.11 K) per V/nm, coming from the $\sigma$-$\pi$ mixing. A 'mini-ripple' configuration with every other atom is shifted out of the sheet by less than 1% differs little from the intrinsic case.