Fig. 3: The mechanism for interfacial-MSHE.

a Illustration of crystal structure of bulk FGT with symmetry generator C_2yT and M_xT, which prohibit T-odd SHC \({\sigma }_{{yx}}^{{S}_{z}}\). b Illustration that the T-odd SHC \({\sigma }_{{yx}}^{{S}_{z}}\) is allowed by the symmetry-breaking at the interface of FGT/MoTe₂ heterostructure. c Calculated k-resolved spin current dipole of \({\sigma }_{{yx}}^{{S}_{z}}\) in FGT/MoTe₂ heterostructure. The color code represents the magnitude of the spin current dipole. Different from that in bulk FGT, the distribution of k-resolved spin current dipole \({D}_{{yx}}^{{S}_{z}}\) in FGT/MoTe₂ heterostructure results in non-zero integration of spin current dipole and non-zero magnetic spin Hall conductivity. d Calculated temperature dependence of T-odd SHC \({\sigma }_{{yx}}^{{S}_{z}}\), which is in good agreement with our experimental results. The calculated τ in our system is 0.04 ps–0.33 ps. The error bars are determined by the noise level of the measured nonlocal voltage signal. e Schematics of interfacial-MSHE in a ferromagnetic metal/nonmagnetic metal bilayer. M is the magnetization of ferromagnet. \({J}^{S}\) is the spin current over the interface. \({J}^{C}\) is the applied charge current. The gray box represents the nonmagnetic layer and the orange box represents the magnetic layer with perpendicular magnetization in +z direction. f Schematics of the interfacial-MSHE in the same bilayer with the magnetization direction of the magnetic layer reversed. As a time-reversal odd quantity, the direction of the spin current induced by interfacial-MSHE reverses compared to that in e.