Disentangling orbital and valley Hall effects in bilayers of transition metal dichalcogenides

It has been recently shown that monolayers of transition metal dichalcogenides (TMDs) in the 2H structural phase exhibit relatively large orbital Hall conductivity plateaus within their energy band gaps, where their spin Hall conductivities vanish. However, since the valley Hall effect (VHE) in these systems also generates a transverse flow of orbital angular momentum it becomes experimentally challenging to distinguish between the two effects in these materials. The VHE requires inversion symmetry breaking to occur, which takes place in the TMD monolayers, but not in the bilayers. We show that a bilayer of 2H-MoS$_2$ is an orbital Hall insulator that exhibits a sizeable OHE in the absence of both spin and valley Hall effects. This phase can be characterised by an orbital Chern number that assumes the value $mathcal{C}_{L}=2$ for the 2H-MoS$_2$ bilayer and $mathcal{C}_{L}=1$ for the monolayer, confirming the topological nature of these orbital-Hall insulator systems. Our results are based on density functional theory (DFT) and low-energy effective model calculations and strongly suggest that bilayers of TMDs are highly suitable platforms for direct observation of the orbital Hall insulating phase in two-dimensional materials. Implications of our findings for attempts to observe the VHE in TMD bilayers are also discussed.