Tunable Berry curvature, valley and spin Hall effect in Bilayer MoS2

We show that in bilayers of transition metal dichalcogenides (TMDCs) both intra-layer and inter-layer couplings give important contributions to the Berry curvature in the K and K valleys of the Brillouin zone. Because of the inter-layer contribution the Berry curvature is stacking dependent and the commonly available 3R type and 2H type bilayers have different and highly tunable Berry curvature properties. The Berry curvature leads to valley Hall and spin Hall effects and we study them in 2H stacked bilayer MoS2. Interestingly, the Hall conductivities may change sign as a function of the external electric field in this system which is reminiscent of the properties of lattice Chern insulators.