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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
MathSciNet review: 2947949
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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.

References [Enhancements On Off] (What's this?)

  • M.J. Ablowitz and Y. Zhu, Nonlinear waves in shallow honeycomb lattices, SIAM J. Appl. Math., 72 (2012).
  • J. E. Avron and B. Simon, Analytic properties of band functions, Ann. Physics 110 (1978), no. 1, 85–101. MR 475384, DOI
  • O. Bahat-Treidel, O. Peleg, and M. Segev, Symmetry breaking in honeycomb photonic lattices, Optics Letters, 33 (2008).
  • M.V. Berry and M.R. Jeffrey, Conical Diffraction: Hamilton’s diabolical point at the heart of crystal optics, Progress in Optics, 2007.
  • M.S. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh, 1973.
  • V. V. Grushin, Application of the multiparameter theory of perturbations of Fredholm operators to Bloch functions, Mat. Zametki 86 (2009), no. 6, 819–828 (Russian, with Russian summary); English transl., Math. Notes 86 (2009), no. 5-6, 767–774. MR 2643450, DOI
  • F.D.M. Haldane and S. Raghu, Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry, Phys. Rev. Lett., 100 (2008), p. 013904.
  • I. N. Herstein, Topics in algebra, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. MR 0171801
  • R. Jost and A. Pais, On the scattering of a particle by a static potential, Phys. Rev. (2) 82 (1951), 840–851. MR 44404
  • C. Kittel, Introduction to Solid State Physics, 7th Edition, Wiley, 1995.
  • Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625
  • P. Kuchment, The Mathematics of Photonic Crystals, in “Mathematical Modeling in Optical Science”, Frontiers in Applied Mathematics, 22 (2001).
  • Peter Kuchment and Olaf Post, On the spectra of carbon nano-structures, Comm. Math. Phys. 275 (2007), no. 3, 805–826. MR 2336365, DOI
  • A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim, The electronic properties of graphene, Reviews of Modern Physics, 81 (2009), pp. 109–162.
  • Roger G. Newton, Relation between the three-dimensional Fredholm determinant and the Jost functions, J. Mathematical Phys. 13 (1972), 880–883. MR 299123, DOI
  • K. S. Novoselov, Nobel lecture: Graphene: Materials in the flatland, Reviews of Modern Physics, 837–849 (2011).
  • O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D.N. Christodoulides, Conical diffraction and gap solitons in honeycomb photonic lattices, Phys. Rev. Lett., 98 (2007), p. 103901.
  • Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
  • Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153
  • J. C. Slonczewski and P. R. Weiss, Band structure of graphite, Phys. Rev., 109 (1958), pp. 272–279.
  • P.R. Wallace, The band theory of graphite, Phys. Rev., 71 (1947), p. 622.
  • Z. Wang, Y.D. Chong, J.D. Joannopoulos, and M. Soljacic, Reflection-free one-way edge modes in a gyromagnetic photonic crystal, Phys. Rev. Lett., 100 (2008), p. 013905.
  • H.-S. Philip Wong and D. Akinwande, Carbon Nanotube and Graphene Device Physics, Cambridge University Press, 2010.

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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

Michael I. Weinstein
Affiliation: Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027
MR Author ID: 181490

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.