Relativistic Tight-Binding Model for Hexagonal Lattice: Application to Graphene

TL;DR

This paper develops a relativistic tight-binding model for graphene, revealing a small spin-orbit induced band gap of 25 μeV at the K points, enhancing understanding of relativistic effects in 2D materials.

Contribution

The study introduces a relativistic tight-binding approach for graphene, incorporating spin-orbit effects into the electronic structure calculations.

Findings

Identification of a 25 μeV spin-orbit gap at K points in graphene.

Application of the relativistic TB method to derive the energy band structure.

Demonstration of relativistic effects on graphene's electronic properties.

Abstract

A non-perturbative relativistic tight-binding (TB) approximation method applicable to crystalline material immersed in a magnetic field was developed in 2015. To apply this method to any material in the magnetic field, the electronic structure of the material in absence of a magnetic field must be calculated. In this study, we present the relativistic TB approximation method for graphene in a zero magnetic field. The Hamiltonian and overlap matrix is constructed considering the nearest neighbouring atomic interactions between the $s$ and $p$ valence orbitals, where the relativistic hopping and overlap integrals are calculated using the relativistic version of the Slater-Koster table. The method of constructing the Hamiltonian and overlap matrix and the resulting energy-band structure of graphene in the first Brillouin zone is presented in this paper. It is found that there is an appearance of a small band-gap at the $\textbf{K}$ points (also known as the spin-orbit gap) due to the relativistic effect, whose magnitude is $25$ $\mu$eV.