Magnetic moment of electrons in systems with spin-orbit coupling

TL;DR

This work advances the relativistic treatment of electron magnetism in systems with spin–orbit coupling by showing that the magnetic moment operator fails to coincide with the naive $-\partial H/\partial\boldsymbol B$ once relativity is accounted for, defining the abnormal magnetic moment as the commutator-based deviation. It develops a general framework—via Dirac→Pauli reductions and Kane-model decoupling—for computing SOC corrections to the magnetic moment in vacuum and in solids, including explicit expressions for conduction-band moments and their spin/orbital decompositions, which become basis-dependent and violate conventional spin algebra. The authors also formulate a branch-resolved Kubo theory for the linear kinetic magnetoelectric effect, revealing an intrinsic interbranch contribution tied to noncommuting position and $\partial/\partial\boldsymbol B$ operators, related to Berry-curvature concepts. Collectively, the paper provides a consistent methodology to treat relativistic magnetic phenomena, clarifies limitations of the modern theory of orbital magnetization in SOC systems, and offers concrete expressions for Kane-model corrections relevant to semiconductors and spintronic materials.

Abstract

Magnetic effects originating from spin-orbit coupling (SOC) have been attracting major attention. However, SOC contributions to the electron magnetic moment operator are conventionally disregarded. In this work, we analyze relativistic contributions to the latter operator, including those of the SOC-type: in vacuum, for the semiconductor 8 band Kane model, and for an arbitrary system with two spectral branches. In this endeavor, we introduce a notion of relativistic corrections to the operation $\partial/\partial\boldsymbol B$, where $\boldsymbol B$ is an external magnetic field. We highlight the difference between the magnetic moment and $-\partial H/\partial\boldsymbol B$, where $H$ is the system Hamiltonian. We suggest to call this difference the abnormal magnetic moment. We demonstrate that the conventional decomposition of the total magnetic moment into the spin and orbital parts becomes ambiguous when relativistic corrections are taken into account. The latter also jeopardize the "modern theory of orbital magnetization" in its standard formulation. We derive a linear response Kubo formula for the kinetic magnetoelectric effect projected to individual branches of a two branch system. This allows us, in particular, to identify a source of this effect that stems from noncommutation of the position and $\partial/\partial\boldsymbol B$ operators' components. This is an analog of the contribution to the Hall conductivity from noncommuting components of the position operator. We comment on the relation between such contributions and the Berry curvature theory. We also report several additional observations related to the electron magnetic moment operator in systems with SOC and other relativistic corrections.