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Using an effective Dirac model, we study the orbital Hall effect (OHE) inbilayers of transition metal dichalcogenides with 2H stacking (2H-TMD). We usefirst-order perturbation theory in the interlayer coupling of the bilayersystem to obtain analytical expressions for the orbital Hall conductivity inthe linear response regime. We use two distinct descriptions of the orbitalangular momentum (OAM) operator: The first one is the intra-atomicapproximation that considers only the intrasite contribution to the OAM [Cysneet al. Phys. Rev. Lett. 126, 056601 (2021)]. The second one uses theBerry-phase formula of the orbital (valley) magnetic moment to describe the OAMoperator [Bhowal and Vignale, Phys. Rev. B 103, 195309 (2021)]. This approachincludes both intersite and intrasite contributions to the OAM. Our resultssuggest that the two approaches agree qualitatively in describing the OHE inbilayers of 2H-TMDs, although they present some quantitative differences. Wealso show that interlayer coupling plays an essential role in understanding theOHE in the unbiased bilayer of 2H-TMD. This coupling causes the Bloch states tobecome bonding (antibonding) combinations of states of individual layers,demanding the consideration of the non-Abelian structure of the orbitalmagnetic moment to the occurrence of OHE. As we discuss throughout the work,the emerging picture of transport of OAM in the unbiased bilayer of 2H-TMDsbased on OHE is very different from the usual picture based on the valley Halleffect, shedding new lights on previous experimental results. We also discussthe effect of the inclusion of a gate-voltage bias in the bilayer system. Ourwork gives support to recent theoretical predictions on OHE in two-dimensionalmaterials.

physical review. b./physical review. b

Orbital Hall effect in bilayer transition metal dichalcogenides: From  the intra-atomic approximation to the Bloch states orbital magnetic moment  approach