Honeycomb lattice potentials and Dirac points

Authors:
Charles L. Fefferman and Michael I. Weinstein

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

MSC (2010):
Primary 35Pxx

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

Published electronically:
June 25, 2012

MathSciNet review:
2947949

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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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Additional Information

**Charles L. Fefferman**

Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544

MR Author ID:
65640

Email:
cf@math.princeton.edu

**Michael I. Weinstein**

Affiliation:
Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027

MR Author ID:
181490

Email:
miw2103@columbia.edu

Keywords:
Honeycomb lattice potential,
graphene,
Floquet-Bloch theory,
dispersion relation

Received by editor(s):
February 16, 2012

Received by editor(s) in revised form:
May 24, 2012

Published electronically:
June 25, 2012

Additional Notes:
The first author was supported in part by US-NSF Grant DMS-09-01040

The second author was supported in part by US-NSF Grant DMS-10-08855

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.