Large intrinsic spin Hall conductivity in orthorhombic tungsten

In this paper, we explored metastable phases showing large intrinsic spin Hall conductivity (SHC) in bulk tungsten (W) using a crystal structure search scheme based on an evolutionary algorithm (EA) and first-principles calculations. W has been known to have the most stable phase with the body-centered-cubic (bcc) structure $\left(\alpha \text{−}\mathrm{W}\right)$ and the metastable phase with a cubic $Pm\overline{3}n$ structure $\left(\beta \text{−}\mathrm{W}\right)$, the latter of which shows larger SHC in $\beta \text{−}\mathrm{W}$. Through the EA search, we obtained 15 metastable structures in addition to $\alpha \text{−}\mathrm{W}$ and $\beta \text{−}\mathrm{W}$ and found that orthorhombic $Fddd$ and $Cmcm$ structures show larger SHC at the Fermi level. The $Fddd$ structure is formed by adding a fourfold helical distortion along the [110] direction to bcc and the $Cmcm$ structure is obtained by further distortion of $Fddd$. Although the energies are higher by about 300 meV/atom than that of bcc, the SHC values at the Fermi level are 1710 (${\hslash /e)\left(\mathrm{\Omega }\mathrm{cm}\right)}^{-1}$ for $Fddd$ and 1573 (${\hslash /e)\left(\mathrm{\Omega }\mathrm{cm}\right)}^{-1}$ for $Cmcm$, which are almost twice as large as that of $\alpha \text{−}\mathrm{W}$ and exceed 1455 (${\hslash /e)\left(\mathrm{\Omega }\mathrm{cm}\right)}^{-1}$ calculated for $\beta \text{−}\mathrm{W}$.