Quantum geometry and quantum capacitance in the orbital magneto-electric effect

Abstract

The orbital magneto-electric effect (OME), the generation of a steady-state orbital angular momentum (OAM) density by an electric field, is a central component of orbitronics. Nevertheless its microscopic interpretation is complicated by the existence of two qualitatively different contributions: the intra-band Edelstein term stemming from the combined effect of the equilibrium OAM and Fermi surface shift, and the inter-band contribution, stemming from Zitterbewegung induced by the electric field. Here we develop a pedagogical density-matrix formulation of the OME within the framework of the modern theory of Bloch-electron OAM, in which the two contributions naturally emerge from distinct parts of the density matrix, and their distinct quantum geometric structures are clearly seen. In the inter-band contribution, in addition to the local- and itinerant-circulation terms, we show that a band-diagonal displacement, or charge dipole, is present, which is linear in the electric field and governed by the normalized quantum metric. For a tilted massive Dirac cone the Zitterbewegung OME and the Fermi-surface orbital response occur for orthogonal electric-field orientations, providing a route to separate them experimentally. The itinerant-circulation contribution is identical to the dipolar one, so in a two-band model the dipole accounts for exactly half of the inter-band response, both in the doped regime and in the insulating gap. Our recent insights identify quantum-metric engineering as a route for enhancing orbital magneto-electric responses and clarify the relation between modern orbital magnetization, polarization, and orbital torque phenomenology.