Table 1 Symmetry-based spin Berry curvature-driven intrinsic (TR even) and disorder-driven extrinsic (TR-odd) spin Hall conductivity tensors of a noncollinear AF hexagonal Mn3Sn
SHC tensor | \({{\boldsymbol{\sigma }}}^{x}\) | \({{\boldsymbol{\sigma }}}^{y}\) | \({{\boldsymbol{\sigma }}}^{z}\) |
|---|---|---|---|
Intrinsic TR-even | \(\left(\begin{array}{ccc}0 & 0 & 0\\ 0 & 0 & {\sigma }_{{yz}}^{x}\\ 0 & {\sigma }_{{zy}}^{x} & 0\end{array}\right)\) | \(\left(\begin{array}{ccc}0 & 0 & {\sigma }_{{xz}}^{y}\\ 0 & 0 & 0\\ {\sigma }_{{zx}}^{y} & 0 & 0\end{array}\right)\) | \(\left(\begin{array}{ccc}0 & {\sigma }_{{xy}}^{z} & 0\\ {\sigma }_{{yx}}^{z} & 0 & 0\\ 0 & 0 & 0\end{array}\right)\) |
Extrinsic TR-odd | \(\left(\begin{array}{ccc}0 & {\sigma }_{{xy}}^{x} & 0\\ {\sigma }_{{yx}}^{x} & 0 & 0\\ 0 & 0 & 0\end{array}\right)\) | \(\left(\begin{array}{ccc}{\sigma }_{{xx}}^{y} & 0 & 0\\ 0 & {\sigma }_{{yy}}^{y} & 0\\ 0 & 0 & {\sigma }_{{zz}}^{y}\end{array}\right)\) | \(\left(\begin{array}{ccc}0 & 0 & 0\\ 0 & 0 & {\sigma }_{{yz}}^{z}\\ 0 & {\sigma }_{{zy}}^{z} & 0\end{array}\right)\) |
