Tunable Berry curvature and valley and spin Hall effect in bilayer MoS2

The chirality of electronic Bloch bands is responsible for many intriguing properties of layered two-dimensional materials. We show that in bilayers of transition metal dichalcogenides (TMDCs), unlike in few-layer graphene and monolayer TMDCs, both intralayer and interlayer couplings give important contributions to the Berry curvature in the K and -K valleys of the Brillouin zone. The interlayer contribution leads to the stacking dependence of the Berry curvature and we point out the differences between the commonly available 3R type and 2H type bilayers. Due to the interlayer contribution, the Berry curvature becomes highly tunable in double gated devices. We study the dependence of the valley Hall and spin Hall effects on the stacking type and external electric field. Although the valley and spin Hall conductivities are not quantized, in MoS2 2H bilayers, they may change sign as a function of the external electric field, which is reminiscent of the behavior of lattice Chern insulators.