On the zero-field orbital magnetic susceptibility of Bloch electrons in graphene-like solids: Some rigorous results

TL;DR

This paper rigorously derives a comprehensive formula for the zero-field orbital susceptibility of Bloch electrons in graphene-like solids, applicable to multiple bands and suitable for numerical computation, with explicit results for two-band models.

Contribution

It provides a rigorous, explicit, and numerically feasible formula for orbital susceptibility in graphene-like materials, including degenerate and gapped multi-band cases.

Findings

Derived a complete susceptibility formula valid at zero temperature and fixed density.

Explicit expressions for two-band gapped models are provided.

Discussed the singular behavior at the Dirac point in gapless models.

Abstract

Starting with a nearest-neighbors tight-binding model, we rigorously investigate the bulk zero-field orbital susceptibility of a non-interacting Bloch electrons gas in graphene-like solids at fixed temperature and density of particles. In the zero-temperature limit and in the semiconducting situation, we derive a complete expression which holds for an arbitrary number of bands with possible degeneracies. In the particular case of a two-bands gapped model, all involved quantities are explicitly written down. Besides the formula that we obtain have the special feature to be suitable for numerical computations since it only involves the eigenvalues and associated eigenfunctions of the Bloch Hamiltonian, together with the derivatives (up to the second order) w.r.t. the quasi-momentum of the matrix-elements of the Bloch Hamiltonian. Finally we give a simple application for the two-bands gapped model by considering the case of a dispersion law which is linear w.r.t. the quasi-momentum in the gapless limit. Through this instance, the origin of the singularity, which expresses as a Dirac delta function of the Fermi energy, implied by the McClure's formula in purely monolayer graphene is discussed.