Orbital Hall effect from orbital magnetic moments of Bloch states: the role of a new correction term

TL;DR

This paper derives a comprehensive formula for the orbital magnetic moment of Bloch states, including new correction terms that significantly influence the orbital Hall effect in layered materials.

Contribution

It introduces two new gauge-covariant contributions to the orbital magnetic moment formula, enhancing the accuracy of orbital Hall effect predictions in complex materials.

Findings

New correction terms reduce the orbital Hall conductivity in studied bilayers.

The corrections are significant in multilayered van der Waals materials.

The formula applies broadly to non-degenerate Bloch states.

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.