Orbital Magnetism of Bloch Electrons I. General Formula

TL;DR

This paper derives an exact, comprehensive formula for the orbital susceptibility of Bloch electrons, clarifying the physical meaning of each term and analyzing band effects and itinerant features systematically.

Contribution

It presents a new exact formula for orbital susceptibility of Bloch electrons, including four distinct contributions, and clarifies their physical significance and band effects.

Findings

The formula includes four contributions: Landau-Peierls, interband, Fermi surface, and occupied states.

In the atomic limit, only core electron diamagnetism and Van Vleck susceptibility remain.

Band effects are systematically analyzed using a linear combination of atomic orbitals approach.

Abstract

We derive an exact formula of orbital susceptibility expressed in terms of Bloch wave functions, starting from the exact one-line formula by Fukuyama in terms of Green's functions. The obtained formula contains four contributions: (1) Landau-Peierls susceptibility, (2) interband contribution, (3) Fermi surface contribution, and (4) contribution from occupied states. Except for the Landau-Peierls susceptibility, the other three contributions involve the crystal-momentum derivatives of Bloch wave functions. Physical meaning of each term is clarified. The present formula is simplified compared with those obtained previously by Hebborn et al. Based on the formula, it is seen first of all that diamagnetism from core electrons and Van Vleck susceptibility are the only contributions in the atomic limit. The band effects are then studied in terms of linear combination of atomic orbital treating overlap integrals between atomic orbitals as a perturbation and the itinerant feature of Bloch electrons in solids are clarified systematically for the first time.