Fig. 1: Anisotropy of the spin current polarization and the associated SOT in spin source materials with a cubic structure.

a The crystal structure of cubic Mn₃Y (Y= Pt, Ir or Rh) in paramagnetic state. b A schematic of a SOT device using paramagnetic Mn₃Y as the spin source, where an isotropic and y-polarized spin current generated in the bottom Mn₃Y layer enters the adjacent ferromagnetic (FM) layer, exerting an isotropic SOT \(\sim {{{{{\bf{m}}}}}}\times ({{{{{\bf{m}}}}}}\times {{{{{\bf{y}}}}}})\) on the perpendicular magnetization. \({\phi }_{{{{{{\rm{E}}}}}}}\) is the angle between the current and the [100] direction of Mn₃Y. In this case, a sizable external magnetic field is required for a deterministic switching, and the charge current required is large. c The structure of cubic Mn₃Y with noncollinear antiferromagnetism, where the Mn moments form “head-to-head” or “tail-to-tail” noncollinear alignments in \((111)\) kagome planes. d A schematic of a SOT device using noncollinear antiferromagnetic Mn₃Y as the spin source, where an anisotropic spin current generated by Mn₃Y exerts the anisotropic SOT in the adjacent ferromagnetic layer. The presence of the z-polarization in the spin current and the associated unconventional SOC component \(\sim {{{{{\bf{m}}}}}}\times ({{{{{\bf{m}}}}}}\times {{{{{\bf{z}}}}}})\) allows a field-free switching of perpendicular magnetization, which does not require a large charge current. e Theoretical \({\phi }_{{{{{{\rm{E}}}}}}}\) dependence of the SHC \({\sigma }_{{zx}}^{p}\) (\(p=y,\, {x},\, {z}\)) and its decomposition into the contributions from the \({{{{{\mathscr{T}}}}}}\)-even and \({{{{{\mathscr{T}}}}}}\)-odd SHE in Mn₃Y. The parameters used to plot (e) are shown in Supplementary Note 1.