Fig. 4. Angle-dependent spin current and spin Hall effect in monolayer V₂Se₂O.

a Spin-resolved charge conductivity $\sigma$ for electric field along $x$ direction. Inset is the dependency of $\sigma$ on the carrier density n. For light doping, $\sigma$ increases almost linearly with n. b Angle-dependence of the longitudinal (L) and transverse (T) charge conductivity varying with the electric field direction $\theta$, taken at 0.2 eV below the valance band maximum. The transverse charge conductivity is always opposite for spin-up and spin-down electrons, and hence the transverse charge current is always zero and the charge current direction is always $\theta$. c The corresponding angle-dependence of the longitudinal and transverse spin conductivity. d Relation between directions of charge current $\mathbf{J}$ and spin current ${\mathbf{J}}^{\mathrm{S}}$. The spin current always flows along the direction $\pi -\theta$. In details, for $\theta =\pm \pi /2$, the spin current will be along the same direction of the charge current, whereas for $\theta =0$ or $\pi$, the spin current flow along the opposite direction of the charge current. Moreover, for $\theta =\pm \pi /4$ or $\pm 3\pi /4$, there will be large spin Hall effect, where the spin current is perpendicular to the charge current.