Electrons in certain two-dimensional crystals possess a pseudospin degree of freedom associated with the existence of two inequivalent valleys in the Brillouin zone. If, as in monolayer ${\mathrm{MoS}}_2$, inversion symmetry is broken and time-reversal symmetry is present, equal and opposite amounts of $k$-space Berry curvature accumulate in each of the two valleys. This is conveniently quantified by the integral of the Berry curvature over a single valley—the valley Hall conductivity. We generalize this definition to include contributions from disorder described with the supercell approach, by mapping (“unfolding”) the Berry curvature from the folded Brillouin zone of the disordered supercell onto the normal Brillouin zone of the pristine crystal, and then averaging over several realizations of disorder. We use this scheme to study from first principles the effect of sulfur vacancies on the valley Hall conductivity of monolayer ${\mathrm{MoS}}_2$. In dirty samples the intrinsic valley Hall conductivity receives gating-dependent corrections that are only weakly dependent on the impurity concentration, consistent with side-jump scattering and the unfolded Berry curvature can be interpreted as a $k$-space resolved side jump. At low impurity concentrations skew scattering dominates, leading to a divergent valley Hall conductivity in the clean limit. The implications for the recently observed photoinduced anomalous Hall effect are discussed.

• Received 23 June 2015
• Revised 4 September 2015

DOI:https://doi.org/10.1103/PhysRevB.92.125146