Description of the orbital Hall effect from orbital magnetic moments of Bloch states: the role of a new correction term in bilayer systems

TL;DR

The paper derives a gauge-covariant orbital magnetic moment (OMM) operator for Bloch electrons by incorporating the Berry connection both in the k-derivative of Bloch states and in the position operator, revealing two new corrections $g^{\rm I}_{nn'{\bf k}}$ and $g^{\rm II}_{nn'{\bf k}}$ that extend beyond the conventional semiclassical picture. The full OMM is shown to be $m_{nn'{\bf k}}=m^{\rm SR}_{nn'{\bf k}}+g^{\rm I}_{nn'{\bf k}}+g^{\rm II}_{nn'{\bf k}}$, with $g^{\rm I}$ restoring gauge covariance for nondegenerate bands and $g^{\rm II}$ providing additional, potentially large corrections. Applying this complete formalism to bilayer 2H-TMD and biased bilayer graphene, the authors demonstrate that the new $g^{\rm II}$ term reduces the orbital Hall conductivity plateau compared to calculations neglecting it, highlighting heightened sensitivity of multilayer van der Waals materials to OMM corrections. This work advances the formal understanding of orbital magnetization transport and bolsters the theoretical foundations of orbitronics by providing a rigorously gauge-covariant framework for OMM in non-degenerate Bloch bands.

Abstract

We present a rigorous derivation of the matrix elements of the orbital magnetic moment (OMM) of Bloch states. Our calculations include the Berry connection term in the k-derivatives of Bloch states, which was omitted in previous works. The resulting formula for the OMM matrix elements applies to any non-degenerate Bloch states within Hilbert space. We identify two new contributions: the first restores gauge covariance for non-degenerate states, while the second, being itself gauge covariant, can provide significant quantitative corrections depending on the system under study. We examine their impact on the orbital Hall effect in two bilayer systems: a 2H transition metal dichalcogenide bilayer and a biased bilayer graphene. In both cases, these new terms reduce the orbital Hall conductivity plateau compared with results that neglect them, suggesting that multi-layered van der Waals materials may be particularly susceptible to the derived OMM corrections. Our findings may contribute to the formal understanding of electronic OMM transport and to the conceptual foundations of the emerging field of orbitronics.

Figures (2)

• Figure 1: (a) The Fermi momenta of the bilayer 2H-MoS$_2$ bands as a function of Fermi energy. (b) Orbital Hall conductivity of a bilayer 2H-MoS$_2$ as a function of Fermi energy, with the OAM operator described by different approaches: The intra-atomic approximation is shown by the gray curve (see Ref. Cysne-PhysRevLett.126.056601). The Bloch state OMM approach, considering only the $m^{\rm SR}_{nn'{\bf k}}+g^{\rm I}_{nn'{\bf k}}$ [Eqs.(\ref{['mSR']}) and (\ref{['gauge1']})] (see Ref. Cysne-Vignale-Tatiana-PhysRevB.105.195421), is shown by the blue curve. The red curve shows the result including the contribution from Eq. (\ref{['gauge2']}) in the OMM operator, $m^{\rm SR}_{nn'{\bf k}}+g^{\rm I}_{nn'{\bf k}}+g^{\rm II}_{nn'{\bf k}}$. The shaded rectangle highlights the energy bandgap in the bilayer 2H-MoS$_2$ band structure.
• Figure 2: Orbital Hall conductivity of bilayer graphene subjected to an external perpendicular electric field [Eq.(\ref{['HBGB']})]. Here, we consider two values for the external electric field parameter: $\Delta=100 \text{meV}$ (solid lines) and $20 \text{meV}$ (dashed lines). The blue curve represents the results when only $m^{\rm SR}_{nn'{\bf k}}+g^{\rm I}_{nn'{\bf k}}$ are included in the description of the orbital magnetic moment. Red curves represent the results when the complete description $m^{\rm SR}_{nn'{\bf k}}+g^{\rm I}_{nn'{\bf k}}+g^{\rm II}_{nn'{\bf k}}$ is used. Here, we set the momentum relaxation time parameter to $\eta=4$ meV.