Fig. 3

Detecting spin-orbit coupling induced valley Hall effects(SVHEs). a Total valley Hall conductivity \(\sigma _{{{xy}}}^{\mathrm{V}}\) as a function of chemical potential μ for gated MoS₂ (red curve) and pristine sample (black curve). The blue dashed line indicates the location where \(\mu = 2|\beta _{{\mathrm{so}}}^{\mathrm{c}}|\). b Polar Kerr effect measurements to detect SVHEs in Mo-based transition-metal dichalcogenides(TMDs). For Mo-based TMDs, SVHEs strongly enhance \(\sigma _{{{xy}}}^{\mathrm{V}}\) in the regime \(\mu \sim 2|\beta _{{\mathrm{so}}}^{\mathrm{c}}|\) comparing to the intrinsic value. This creates a significant valley imbalance n_V and valley magnetization at the sample boundaries, which can be signified by a large Kerr angle θ_K. c Total \(\sigma _{{{xy}}}^{\mathrm{V}}\) versus chemical potential μ for polar TMD WSeTe (red curve) and pristine WSe₂ (black curve). Clearly, in the regime \(\mu \ < \ 2|\beta _{{\mathrm{so}}}^{\mathrm{c}}|\) the sign of \(\sigma _{{{xy}}}^{\mathrm{V}}\) in WSeTe is reversed due to SVHEs. d Schematics for polar Kerr experiments to detect SVHEs in tungsten-based polar TMD WSeTe. The reversed valley current is signified by the sign reversal of θ_K