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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- Charles L. Fefferman
- Affiliation: Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544
- MR Author ID: 65640
- Email: email@example.com
- Michael I. Weinstein
- Affiliation: Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027
- MR Author ID: 181490
- Email: firstname.lastname@example.org
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: J. Amer. Math. Soc. 25 (2012), 1169-1220
- MSC (2010): Primary 35Pxx
- DOI: https://doi.org/10.1090/S0894-0347-2012-00745-0
- MathSciNet review: 2947949