An implementation of spin-orbit coupling for band structure calculations with Gaussian basis sets: Two-dimensional topological crystals of Sb and Bi.

Sahar Pakdel,J J Palacios,Mahdi Pourfath

doi:10.3762/bjnano.9.94

Abstract

We present an implementation of spin–orbit coupling (SOC) for density functional theory band structure calculations that makes use of Gaussian basis sets. It is based on the explicit evaluation of SOC matrix elements, both the radial and angular parts. For all-electron basis sets, where the full nodal structure is present in the basis elements, the results are in good agreement with well-established implementations such as VASP. For more practical pseudopotential basis sets, which lack nodal structure, an ad-hoc increase of the effective nuclear potential helps to capture all relevant band structure variations induced by SOC. In this work, the non-relativistic or scalar-relativistic Kohn–Sham Hamiltonian is obtained from the CRYSTAL code and the SOC term is added a posteriori. As an example, we apply this method to the Bi(111) monolayer, a paradigmatic 2D topological insulator, and to mono- and multilayer Sb(111) (also known as antimonene), the former being a trivial semiconductor and the latter a topological semimetal featuring topologically protected surface states.

Highlights

The topological character of topological materials in most relevant cases originates from relativistic corrections that cannot be neglected in the Hamiltonian of heavy elements, from spin–orbit coupling (SOC)
We have presented an implementation of SOC suitable for density functional theory (DFT) band structure calculations based on contracted Gaussian-type orbitals (CGTOs) basis sets
We evaluate both angular and radial part of the SOC relativistic correction to the Hamiltonian, considering the spherical harmonics and CGTOs as the angular and radial part of the basis functions, respectively

Summary

Introduction

The topological character of topological materials (mostly insulators and non-insulators) in most relevant cases originates from relativistic corrections that cannot be neglected in the Hamiltonian of heavy elements, from spin–orbit coupling (SOC). Such materials are usually characterized by non-zero topological invariants that can be either. As the SOC is increased from zero towards its nominal value, it pushes up the valence band while bringing down the conduction band of the imaginary SOC-free material In this process, the gap closes and reopens again, giving rise to the non-zero topological invariant

Methods

Results

Conclusion