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

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 light 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.

Figure 1

(a) Side view of a bilayer of TMD with 2H stacking. Each monolayer of TMD in the H structural phase is composed of an atomic plane of transition metal (black spheres) sandwiched by two atomic planes of chalcogens (yellow spheres). The top and the bottom layers have a relative rotation angle of 180 degrees, making the bilayer system centrosymmetric. $\mathcal{I}$ represents a point of inversion symmetry of the bilayer. (b) Schematic representation of OHE, i.e., the transverse flux of orbital-current induced by a longitudinal ($\widehat{x}$ direction) electric field. In the picture, the orbital-current flows in the $-\widehat{y}$ direction and has OAM polarized in the $\widehat{z}$ direction. (c) The energy spectrum of the bilayer of 2H-TMD near valleys $K$ (left) and $K^{\prime }$ (right) of BZ. The solid black curve shows the spectrum obtained by numerical diagonalization of the Hamiltonian of Eq. (1) without considering the spin-orbit coupling ($\lambda \to 0$). The dashed orange curve shows the energy spectra obtained by first-order perturbation theory expansion in interlayer hopping $t_{\perp }$ [Eqs. (10), (11), (12)].

Figure 2

(a) OHC as a function of the Fermi energy for unbiased bilayer of 2H-${\mathrm{MoS}}_2$. The two curves show the orbital conductivities calculated using intra-atomic approximation [Eqs. (19, 20, 21)] and Bloch state orbital magnetic moment approach [Eqs. (31)–(33)] for the OAM operator. (b) Fermi-momentum [Eq. (22)], as a function of Fermi energy, for the valence and conduction band states in the unbiased bilayer of 2H-${\mathrm{MoS}}_2$. The vertical continuous black lines in both panels delimit the insulating gap of the unbiased bilayer 2H-${\mathrm{MoS}}_2$ [Fig. 1]. The horizontal dashed black line in (a) signals the quantized orbital Hall conductivity in the intra-atomic approximation when Fermi-energy lies in the insulating gap $[{\overline{\sigma }}_{\mathrm{OH}}^{\text{Intra}}=4(e/2\pi )$, Eq. (23)]. Inl (b), $a=3.160\text{Å}$ and $t=1.137\text{eV}$ (see Sec. 2).

Figure 3

The effect of inversion symmetry-breaking gate potential in OHC computed using intra-atomic approximation [Eqs. (40, 41, 42)] (a) and Bloch state orbital magnetic moment description of OAM [Eqs. (43)–(45)] (b). (c) Fermi momentum [Eqs. (38) and (39)], as a function of Fermi energy, for the valence and conduction band states in bilayer of 2H-${\mathrm{MoS}}_2$ with an gate voltage bias $U=0.2\text{eV}$. In (c), $a=3.160\text{Å}$ and $t=1.137\text{eV}$ [see Sec. 2].