Ab initio calculation of intrinsic spin Hall conductivity of Pd and Au

An {\it ab initio} relativistic band structure calculation of spin Hall conductivity (SHC) ($\sigma_{xy}^z$) in Pd and Au metals has been performed. It is found that at low temperatures, intrinsic SHCs for Pd and Au are, respectively, $\sim 1400 (\hbar/e)(\Omega {\rm cm})^{-1}$ and $\sim 400 (\hbar/e)(\Omega {\rm cm})^{-1}$. The large SHC in Pd comes from the resonant contribution from the spin-orbit splitting of the doubly degenerated 4$d$ bands near the Fermi level at symmetry $\Gamma$ and X points, and the smaller SHC in Au is due to the broad free electron like 6$s6p$-bands. However, as the temperature increases, the SHC in Pd decreases monotonically and reduces to $\sim 330 (\hbar/e)(\Omega {\rm cm})^{-1}$ at 300 K, while the SHC in Au increases steadily and reaches $\sim 750 (\hbar/e)(\Omega {\rm cm})^{-1}$ at room temperature. This indicates that the gigantic spin Hall effect [$\sigma_{xy}^z \approx 10^5 (\hbar/e)(\Omega {\rm cm})^{-1}$] observed recently in the Au/FePt system [T. Seki, {\it et al.}, Nature Materials {\bf 7}, 125 (2008)] is due to the extrinsic mechanisms such as the skew scattering by the impurities in Au.