Gauge-covariant theory of the orbital Hall effect: role of itinerant orbital currents and a non-Kubo contribution

Abstract

The orbital Hall effect is central to the emerging field of orbitronics, yet its first-principles description relies on an atom-centered approximation (ACA) of the orbital angular momentum (OAM) that discards itinerant contributions. Attempts to go beyond it have used off-diagonal elements of the ground-state orbital magnetization formula, which are not gauge covariant and treat the OAM as a fixed operator. Here we construct a gauge-covariant OAM operator in a Wannier basis that is consistent with the modern theory of orbital magnetization for all band pairs. Because this operator depends on the geometry of the Bloch states, an applied electric field modifies the operator itself, giving rise to a non-Kubo contribution to the orbital Hall conductivity (OHC). First-principles calculations for transition metals and monolayer MoS${}_2$, which are frequently used in spin-orbitronics experiments, show that OHC contributions originating from the itinerant-circulation of the Wannier functions are often comparable to or larger than those originating from their local-circulation, which is especially evident in Pt and Fe. Separating the OHC instead into local and non-local terms, we find that the former closely resemble the ACA results, while the latter often dominate and reverse the sign predicted by the ACA. Our results call for a reassessment of how orbital currents are identified in experiments.

arXiv comment21 pages, 3 figures, 2 tables