Comparing the Extrinsic Orbital Hall Effect in Centrosymmetric and Noncentrosymmetric Systems: Insights from Bilayer Transition Metal Dichalcogenides

TL;DR

This paper analyzes the orbital Hall effect (OHE) in bilayer TMDs, focusing on how short-range disorder and inversion symmetry breaking influence extrinsic contributions. Using a quantum kinetic framework, it demonstrates that extrinsic OHE dominates away from band edges in both centrosymmetric and biased bilayers, with inversion symmetry breaking dramatically amplifying the extrinsic component. In centrosymmetric bilayers, OHE arises solely from off-diagonal density-matrix terms, while biasing introduces intraband OAM and enables substantial extrinsic enhancement, potentially exceeding intrinsic OHE by orders of magnitude at high Fermi energies. The results suggest that, in typical doped samples, the OHE is predominantly extrinsic and tunable via gate voltage, offering actionable insights for orbitronics experiments and material design.

Abstract

Both intrinsic and extrinsic orbital Hall effects (OHE) in bilayer transition metal dichalcogenides (TMDs) are investigated in the presence of short-range disorder using quantum kinetic theory. Bilayer TMDs provide an ideal platform to study the effects of inversion symmetry breaking on transport properties due to their unique structural and electronic characteristics. While bilayer TMDs are naturally inversion symmetric, applying a finite gate voltage to create a bias between the layers effectively breaks this symmetry. Our findings reveal that slightly away from the band edges, the extrinsic OHE becomes the dominant contribution in both inversion-symmetric and asymmetric cases, with its prominence increasing significantly as the Fermi energy rises. Furthermore, we demonstrate that breaking inversion symmetry greatly enhances the extrinsic OHE. This enhancement arises from the fundamentally distinct behavior of orbital angular momentum (OAM) in centrosymmetric systems, where intraband components vanish due to symmetry constraints. As a result, in centrosymmetric systems, only the off-diagonal components of the density matrix contribute to the extrinsic OHE. In contrast, in noncentrosymmetric systems, both diagonal and off-diagonal components play a role. Our study suggests that in experimentally relevant, highly doped systems, the OHE is predominantly extrinsic in nature, regardless of whether the system is centrosymmetric or noncentrosymmetric. Importantly, we infer that even a weakly breaking of inversion symmetry can lead to a dramatic enhancement of the OHE, a finding with significant implications for experimental investigations.

Figures (7)

• Figure 1: (Color online) The energy spectrum of the biased bilayer of 2H-MoS$_2$ with energy gap $2m$ near $K$ and $K'$ valley for $V_g=0.3$ eV. The green and the orange lines belong to layer 1 and 2 respectively (Eqs. \ref{['eev']} and \ref{['eec']}) and the red arrows show the orbital magnetic moment in each band. Note that zero energy is defined at the valence band maximum when the gate voltage is absent.
• Figure 2: (Color online) Intrinsic, extrinsic and the total orbital Hall conductivity $\sigma_{\rm{OHE}}$ versus the Fermi energy $\varepsilon_{\rm F}$ for unbiased centrosymmetric bilayer of 2H-MoS$_2$. In both hole-doped and electron-doped systems, the extrinsic conductivity predominates over the intrinsic one as we go away from the band gap, but the extrinsic contribution is minimal close to the gap.
• Figure 3: (Color online) The total orbital Hall conductivity (solid lines) and the intrinsic orbital Hall conductivity (dashed-dotted lines) $\sigma_{\rm{OHE}}$ versus the Fermi energy $\varepsilon_{\rm F}$ for biased bilayer of 2H-MoS$_2$ for $V_g=0.2, 0.4,$ and $0.6$ eV. For higher Fermi energies, the extrinsic term dominates the orbital Hall conductivity for the system with broken inversion symmetry. The overall conductivity rises to a maximum value and then progressively falls until the second band begins to fill and the same process is repeated, while the intrinsic term reduces as it moves further from the gap.
• Figure 4: (Color online) Intrinsic, side-jump, skew and the total orbital Hall conductivity $\sigma_{\rm{OHE}}$ versus the Fermi energy $\varepsilon_{\rm F}$ for biased bilayer of 2H-MoS$_2$ for $V_g=0.4$ eV.
• Figure 5: (Color online) $\sigma_{\rm{OHE}}^{\rm{ext}}/\sigma_{\rm{OHE}}^{\rm{int}}$ versus the Fermi energy $\varepsilon_{\rm F}$ for unbiased and biased ($V_g=0.2$ eV) 2H-MoS$_2$ bilayer in electron-doped region. Notice that because the diagonal elements of the OAM or intraband terms disappear in the centrosymmetric system, the absolute value of $\sigma_{\rm{OHE}}^{\rm{ext}}/\sigma_{\rm{OHE}}^{\rm{int}}$ is lower than in the biased noncentrosymmetric bilayer.