Honeycomb Lattice Potentials and Dirac Points

TL;DR

This paper proves the existence and robustness of Dirac points in the dispersion surfaces of two-dimensional Schrödinger operators with honeycomb lattice potentials, explaining their significance in materials like graphene.

Contribution

It establishes the presence of Dirac points in honeycomb lattice potentials without size restrictions and demonstrates their stability under certain symmetry-breaking perturbations.

Findings

Dirac points occur at vertices of the Brillouin zone.

Dirac points are robust under specific perturbations.

Small generic perturbations eliminate Dirac points.

Abstract

We prove that the two-dimensional Schroedinger operator with a potential having the symmetry of a honeycomb structure has dispersion surfaces with conical singularities (Dirac points) at the vertices of its Brillouin zone. No assumptions are made on the size of the potential. We then prove the robustness of such conical singularities to a restrictive class of perturbations, which break the honeycomb lattice symmetry. General small perturbations of potentials with Dirac points do not have Dirac points; their dispersion surfaces are smooth. The presence of Dirac points in honeycomb structures is associated with many novel electronic and optical properties of materials such as graphene.