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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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
Published electronically: June 25, 2012
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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
Email: cf@math.princeton.edu

Michael I. Weinstein
Affiliation: Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027
Email: miw2103@columbia.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-2012-00745-0
PII: S 0894-0347(2012)00745-0
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.