Fig. 1

Spin-orbit coupling induced valley Hall effects. a Schematics for the Ising spin-orbit coupling (SOC) (orange arrows), the Rashba SOC (golden arrows) and the spin-type Berry curvatures \({\mathrm{\Omega }}_{{\mathrm{spin}}}^{{\mathrm{c}}, - }\) (red/blue arrows) in the lower spin bands represented by red/blue pockets above K/−K-points. b Valley Hall effects due to \({\mathrm{\Omega }}_{{\mathrm{spin}}}^{\mathrm{c}}\). White arrows indicate out-of-plane gating fields/electric polarization labeled by E_G. c Magnitudes of spin-type Berry curvature \(|{\mathrm{\Omega }}_{{\mathrm{spin}}}^{{\mathrm{c}}, \pm }|\) near the conduction band edge (red solid curve) and orbital-type Berry curvature \(|{\mathrm{\Omega }}_{{\mathrm{orb}}}^{\mathrm{c}}|\) (black solid curve) near K-points. Rashba coupling strength is set to be \(\alpha _{{\mathrm{so}}}^{\mathrm{c}} = 21.4\,{\mathrm{meV}} \cdot\) Å according to \(\alpha _{{\mathrm{so}}}^{\mathrm{c}}k_{\mathrm{F}} \approx 3\,{\mathrm{meV}}\)[33], comparable to \(2|\beta _{{\mathrm{so}}}^{\mathrm{c}}| = 3\,{\mathrm{meV}}\) ^31,32. Parameters for \(|{\mathrm{\Omega }}_{{\mathrm{orb}}}^{\mathrm{c}}|\) are set to be: Δ = 0.83 eV, V_F = 3.51 eV ⋅ Å⁸. Clearly, \(|{\mathrm{\Omega }}_{{\mathrm{spin}}}^{{\mathrm{c}}, \pm }|\) is nearly ten times of \(|{\mathrm{\Omega }}_{{\mathrm{orb}}}^{\mathrm{c}}|\). d \(|{\mathrm{\Omega }}_{{\mathrm{spin}}}^{{\mathrm{c}}, \pm }|\) as a function of \(\alpha _{{\mathrm{so}}}^{\mathrm{c}}\) at the K-points. Evidently, \(|{\mathrm{\Omega }}_{{\mathrm{spin}}}^{{\mathrm{c}}, \pm }|\) scales quadratically with \(\alpha _{{\mathrm{so}}}^{\mathrm{c}}\)