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

TL;DR

This paper investigates the orbital Hall effect in bilayer transition metal dichalcogenides using two different theoretical approaches, highlighting the importance of interlayer coupling and non-Abelian orbital magnetic moments in understanding OHE.

Contribution

It compares intra-atomic and Berry-phase based descriptions of orbital angular momentum in bilayer TMDs, revealing qualitative agreement and quantitative differences, and emphasizes the role of interlayer coupling.

Findings

Interlayer coupling significantly influences OHE in bilayer TMDs.

Two approaches to OAM yield qualitatively similar but quantitatively different results.

Non-Abelian orbital magnetic moments are crucial for understanding OHE in coupled bilayers.

Abstract

Using an effective Dirac model, we study the orbital Hall effect (OHE) in bilayers of transition metal dichalcogenides with 2H stacking (2H-TMD). We use first-order perturbation theory in the interlayer coupling of the bilayer system to obtain analytical expressions for the orbital Hall conductivity in the linear response regime. We use two distinct descriptions of the orbital angular momentum (OAM) operator: The first one is the intra-atomic approximation that considers only the intrasite contribution to the OAM [Cysne et al. Phys. Rev. Lett. 126, 056601 (2021)]. The second one uses the Berry-phase formula of the orbital (valley) magnetic moment to describe the OAM operator [Bhowal and Vignale, Phys. Rev. B 103, 195309 (2021)]. This approach includes both intersite and intrasite contributions to the OAM. Our results suggest that the two approaches agree qualitatively in describing the OHE in bilayers of 2H-TMDs, although they present some quantitative differences. We also show that interlayer coupling plays an essential role in understanding the OHE in the unbiased bilayer of 2H-TMD. This coupling causes the Bloch states to become bonding (antibonding) combinations of states of individual layers, demanding the consideration of the non-Abelian structure of the orbital magnetic 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-TMDs based on OHE is very different from the usual picture based on the valley Hall effect, shedding new lights on previous experimental results. We also discuss the effect of the inclusion of a gate-voltage bias in the bilayer system. Our work gives support to recent theoretical predictions on OHE in two-dimensional materials.