Fig. 3: spin-Hall Conductivity (SHC) \({\sigma }_{xy}^{z}\) vs energy for Mn₃Sn.

Panel (a) shows the SHC calculated using the conventional spin current formula, while panel (b) shows the SHC calculated using the proper spin current formula. Panels (c) and (d) contain the two torque dipole contributions I₁ and I₂, respectively.