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

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 performed, in the spirit of the tight-binding method, to simplify the calculation of the velocity and spin-current matrix elements in the Wannier representation.