Honeycomb lattice potentials and Dirac points

Fefferman, Charles; Weinstein, Michael

doi:10.1090/S0894-0347-2012-00745-0

Honeycomb lattice potentials and Dirac points
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by Charles L. Fefferman and Michael I. Weinstein

J. Amer. Math. Soc. 25 (2012), 1169-1220

DOI: https://doi.org/10.1090/S0894-0347-2012-00745-0

Published electronically: June 25, 2012

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Abstract:

We prove that the two-dimensional Schrödinger 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.

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