Fig. 1. Schematics of C-paired spin-valley locking (SVL) and its properties.

a In T-paired SVL, the splitting between spin-up (red) and spin-down (blue) bands is from strong spin-orbit coupling (SOC) when inversion symmetry is broken, and the two valleys are related by the time-reversal (T) symmetry. b However, for C-paired SVL, the spin splitting is owing to exchange couplings between itinerant electrons and local magnetic moments, and different valleys are related by a crystal symmetry (a mirror symmetry $M_{\varphi }$ as an example). c A strong valley polarization can be easily induced by a strain, and the piezomagnetism will simultaneously happen upon finite doping. M first increase linearly with the strain and finally gets saturated for large strain with M equal to n. d Contribution to the charge current from different valleys are in general different, which gives rise to a non-zero spin current ${\mathbf{J}}^{\mathrm{S}}={\mathbf{J}}_{\mathbf{K}}-{\mathbf{J}}_{{\mathbf{K}}^{\prime }}$. One typical result ($\varphi =\frac{\pi }{6}$) is in the bottom panel. Both longitudinal and transverse spin currents depend on the electric field E direction $\theta$ with a period of $\pi$. When $\theta =\varphi$ or $\varphi +\pi$, where E does not break the mirror symmetry, the longitudinal charge currents from different valleys are the same, whereas transverse charge currents are exactly opposite, which result in pure spin currents perpendicular to the charge currents, i.e., spin Hall effect.