Calculation of intrinsic spin Hall conductivity by Wannier interpolation

Ab initio calculation of intrinsic spin Hall conductivity (SHC) generally requires a strict convergence criterion and a dense $k$-point mesh to sample the Brillouin zone, making its convergence challenging and time consuming. Here we present a scheme for efficiently and accurately calculating SHC based on the maximally localized Wannier function (MLWF). The quantities needed by the Kubo formula of SHC are derived in the space of the MLWF and it is shown that only the Hamiltonian, the overlap, and the spin operator matrices are required from the initial ab initio calculation. The computation of these matrices and the interpolation of the Kubo formula on a dense $k$-point mesh can be easily achieved. We validate our results by prototypical calculations on fcc Pt and GaAs, which demonstrate that the Wannier interpolation approach is of high accuracy and efficiency. Calculations of $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{Ta}$ and $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{Ta}$ show that SHC of $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{Ta}$ is 2.7 times that of $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{Ta}$, while both have the opposite sign relative to fcc Pt and are an order of magnitude smaller than Pt. The calculated spin Hall angle of $\ensuremath{-}0.156$ is quite consistent with previous experiments on $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{Ta}$, further suggesting intrinsic contribution may dominate in $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{Ta}$. Our approach could facilitate large-scale SHC calculations and may benefit the discovery of materials with high intrinsic SHC.