Abstract
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.
Highlights
Monolayers of MoS2 and related transition-metal dichalcogenides (TMDs) have recently become the subject of intense investigation, due in part to the possibility of manipulating the so-called “valley” degree of freedom [1]
The band structure exhibits a direct gap at the two inequivalent valleys centered at the high-symmetry points K and K = −K in the Brillouin zone, where the topmost valence bands are primarily composed of transition-metal d states [2]
We develop a computational scheme that allows us to include in a realistic manner the effect of impurities in the calculation of the VHC and of the photoinduced anomalous Hall effect (AHE) in TMDs
Summary
Monolayers of MoS2 and related transition-metal dichalcogenides (TMDs) have recently become the subject of intense investigation, due in part to the possibility of manipulating the so-called “valley” degree of freedom [1]. Note that the anomalous Hall response of pristine samples at low temperatures is dominated by skew scattering, with the intrinsic contribution only becoming significant in moderately resistive samples (where it competes with side-jump scattering) This analysis, originally developed for the AHC in ferromagnetic metals, carries over to the VHC and photoinduced AHC in TMDs. It is well established that sulfur vacancies constitute the main source of disorder in MoS2 [13,14,15,16,17,18]. Recall that the definition of the VHC in Eq (5) requires identifying the individual valley domains in the BZ where the Berry curvature is to be integrated It is not clear a priori how to do so in the context of a supercell calculation, since the electronic states cannot be labeled by wave vectors in the normal BZ of Fig. 1. IV, and the Appendixes present the details of the ab initio calculations, the BZ unfolding method, and the effective-Hamiltonian methodology
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