Why Dirac points in graphene are where they are?

TL;DR

This paper explains why Dirac points occur in graphene and similar lattices using group theory, linking band merging and linear spectrum behavior to geometric properties of conical points.

Contribution

It provides a simple group theory-based explanation for the location of Dirac points in graphene and related lattices, applicable to arbitrary tight-binding models.

Findings

Dirac points occur at Brillouin zone corners in graphene.

Linear spectrum near Dirac points is explained by geometric conical surface properties.

Applicable to monolayer, bilayer graphene, and Kagome lattices.

Abstract

We present a simple group theory explanation of the fact that the energy bands merge in the corners of the Brillouin zone for graphene and for two particular cases of Kagome lattice for arbitrary tight--binding Hamiltonian. We connect the linearity of the spectrum in the vicinity of these points for monolayer graphene, bilayer graphene for AA stacking and Kagome lattice with the properties of the conical points of the surface, known from geometry.