Dirac cones for a mean-field model of graphene

Abstract

In this article, we show that, in the dissociation regime and under a non-degeneracy assumption, the reduced Hartree-Fock theory of graphene presents Dirac points at the vertices of the first Brillouin zone and that the Fermi level is exactly at the coincidence point of the cones. For this purpose, we first consider a general Schrödinger operator $H=-\Delta +V_L$ acting on $L^2({\mathbb{R}}^2)$ with a potential $V_L$ which is assumed to be periodic with respect to some lattice with length scale $L$. Under some assumptions which covers periodic reduced Hartree-Fock theory, we show that, in the limit $L\to \infty$, the low-lying spectral bands of $H_L$ are given to leading order by the tight-binding model. For the hexagonal lattice of graphene, the latter presents singularities at the vertices of the Brillouin zone. In addition, the shape of the Bloch bands is so that the Fermi level is exactly on the cones.

References in corpus (10)

• The electronic properties of graphene
• Dirac materials
• The Rare Two-Dimensional Materials with Dirac Cones
• A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case
• Elliptic operators with honeycomb symmetry: Dirac points, Edge States and Applications to Photonic Graphene
• Coulomb and Riesz gases: The known and the unknown
• Dirac cones for point scatterers on a honeycomb lattice
• Ground state properties of graphene in Hartree-Fock theory
• Anomalous behavior in an effective model of graphene with Coulomb interactions
• The diatomic Hartree model at dissociation