# Computation of intrinsic spin Hall conductivities from first principles using maximally-localized Wannier functions

## TL;DR

This paper introduces an efficient first-principles method using Wannier functions to compute intrinsic spin Hall conductivities, significantly reducing computational cost while maintaining accuracy, demonstrated on platinum and doped gallium arsenide.

## Contribution

The authors develop a Wannier-function-based interpolation scheme for calculating spin Hall conductivities from ab initio data, improving efficiency over direct calculations.

## Key findings

- Accurate spin Hall conductivities for platinum and doped gallium arsenide.
- Interpolation scheme reduces computational cost significantly.
- Good agreement with previous first-principles studies.

## Abstract

We present a method to compute the intrinsic spin Hall conductivity from first principles using an interpolation scheme based on maximally-localized Wannier functions. After obtaining the relevant matrix elements among the ab initio Bloch states calculated on a coarse k-point mesh, we Fourier transform them to find the corresponding matrix elements between Wannier states. We then perform an inverse Fourier transform to interpolate the velocity and spin-current matrix elements onto a dense k-point mesh, and use them to evaluate the spin Hall conductivity as a Brillouin-zone integral. This strategy has a much lower computational cost than a direct ab initio calculation, without sacrificing the accuracy. We demonstrate that the spin Hall conductivities of platinum and doped gallium arsenide, computed with our interpolation scheme as a function of the Fermi energy, are in good agreement with those obtained in previous first-principles studies. We also discuss certain approximations that can be made, in the spirit of the tight-binding method, to simplify the calculation of the velocity and spin-current matrix elements in the Wannier representation.