Symmetry Classification of Energy Bands in Graphene and Silicene

DOI: 10.4236/graphene.2013.22011   PDF   HTML   XML   7,464 Downloads   13,244 Views   Citations


We present the results of the symmetry classification of the electron energy bands in graphene and silicene using group theory algebra and the tight-binding approximation. The analysis is performed both in the absence and in the presence of the spin-orbit coupling. We also discuss the bands merging in the Brillouin zone symmetry points and the conditions for the latter to become Dirac points.

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E. Kogan, "Symmetry Classification of Energy Bands in Graphene and Silicene," Graphene, Vol. 2 No. 2, 2013, pp. 74-80. doi: 10.4236/graphene.2013.22011.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films”, Science, Vol. 306, No. 5696, 2004, pp. 666-669. doi:10.1126/science.1102896
[2] W. M. Lomer, “The Valence Bands in Two-Dimensional Graphite,” Proceedings of the Royal Society A, Vol. 227, No. 1170, 1955, pp. 330-349. doi:10.1098/rspa.1955.0014
[3] J. C. Slonczewski and P. R. Weiss, “Band Structure of Graphite,” Physical Review, Vol. 109, No. 2, 1958, pp. 272-279. doi:10.1103/PhysRev.109.272
[4] G. Dresselhaus and M. S. Dresselhaus, “Spin-Orbit Interaction in Graphite,” Physical Review, Vol. 140, No. 1A, 1965, pp. A401-A412. doi:10.1103/PhysRev.140.A401
[5] F. Bassani and G. P. Parravicini, “Band Structure and Op tical Properties of Graphite and of the Layer Compounds GaS and GaSe,” Nuovo Cimento B, Vol. 50, No. 1, 1967, pp. 95-128. doi:10.1007/BF02710685
[6] L. M. Malard, M. H. D. Guimaraes, D. L. Mafra, M. S. C. Mazzoni and A. Jorio, “Group-Theory Analysis of Electrons and Phonons in N-Layer Graphene Systems,” Physical Review B, Vol. 79, No. 12, 2009, pp. 125426-125433. doi:10.1103/PhysRevB.79.125426
[7] J. L. Manes, “Existence of Bulk Chiral Fermions and Crystal Symmetry,” Physical Review B, Vol. 85, No. 15, 2012, pp. 155118-115125. doi:10.1103/PhysRevB.85.155118
[8] E. Kogan and V. U. Nazarov, “Symmetry Classification of Energy Bands in Graphene,” Physical Review B, Vol. 85, No. 11, 2012, pp. 115418-115423. doi:10.1103/PhysRevB.85.115418
[9] C.-C. Liu, H. Jiang and Y. G. Yao, “Low-Energy Effective Hamiltonian Involving Spin-Orbit Coupling in Silicene and Two-Dimensional Germanium and Tin,” Physical Review B, Vol. 84, No. 19, 2011, pp. 195430- 195440. doi:10.1103/PhysRevB.84.195430
[10] N. D. Drummond, V. Zolyomi and V. I. Falko, “Electrically Tunable Band Gap in Silicone,” Physical Review B, Vol. 85, 2012, pp. 075423-1-075423-7.
[11] C. Kittel, “Quantum Theory of Solids,” John Wiley & and Sons, Inc., New York, London, 1963.
[12] W. A. Harrison, “Solid State Theory,” McGraw Hill Book Company, New York, London, Toronto, 1970.
[13] L. D. Landau and E. M. Lifshitz, “Quantum Mechanics,” Pergamon Press, Oxford, 1991.
[14] R. S. Knox and S. Gold, “Symmmetry in the Solid State”, W. A. Benjamin, Inc., New York, Amsterdam, 1964.
[15] C. Thomsen, S. Reich and J. Maultzsch, “Carbon Nanotubes: Basic Concepts and Physical Properties,” Wiley Online Library, Wiley-VCH Verlag GmbH, 2004.
[16] M. I. Petrashen, E. D. Trifonov and J. L. Trifonov, “Applications of Group Theory in Quantum Mechanics,” Dover Publications, New York, 2009.
[17] G. F. Koster, J. O. Dimmock, R. G. Wheeler and H. Statz, “Properties of the Thirty-Two Point Groups,” MIT Press, Cambridge, 1964.
[18] F. Hund, “über den Zusammenhang Zwischen der Symmetrie eines Kristallgitters und den Zust?nden seiner Elek- tronen,” Zeitschrift für Physik, Vol. 99, No. 1-2, 1936, pp. 119-136. doi:10.1007/BF01847819
[19] C. Herring, “Accidental Degeneracy in the Energy Bands of Crystals,” Physical Review B, Vol. 52, No. 4, 1937, pp. 365-373. doi:10.1103/PhysRev.52.365

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