Large enhancement of the intrinsic spin Hall conductivity in W-N compounds

The ultrathin tungsten (W) layer has been utilized as a spin-generating source for memory application in magnetic heterostructures. We investigate the spin Hall conductivity (SHC, ${\ensuremath{\sigma}}_xy^z$) of W-N compounds, focusing on WN and ${\mathrm{W}}₂\mathrm{N}$, employing first-principles calculations. We evaluate the SHC, formation energy ($\mathrm{\ensuremath{\Delta}}F$), and phonon frequencies for all cases. Three possible structures, including NaCl, hexagonal, and NbO type, are considered for WN. While the SHC of WN in NaCl and a hexagonal structure are substantial, the thermodynamic average of WN is $\ensuremath{-}194(\ensuremath{\hbar}/e)\phantom{\rule{0.28em}{0ex}}\text{S/cm}$, much smaller than pure W. However, ${\mathrm{W}}₂\mathrm{N}$, whose stability is confirmed by phonon calculations, shows a large SHC of ${\ensuremath{\sigma}}_xy^z=\ensuremath{-}966(\ensuremath{\hbar}/e)\phantom{\rule{0.28em}{0ex}}\text{S/cm}$ with a spin Hall angle (${\ensuremath{\theta}}_{\text{SH}}$) of $\ensuremath{-}0.44$, in good agreement with the experiment, enhanced by 18% over $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{W}$. Most of the spin Berry curvature, responsible for such large ${\ensuremath{\sigma}}_xy^z$, stems from the $[1\overline{1}0]$ (${k}_x={k}_y$) plane and is analyzed from a symmetry perspective using representations of group theory.