MSCE  Vol.3 No.8 , August 2015
Creation of Bielectron of Dirac Cone: The Tachyon Solution in Magnetic Field
Author(s) Lyubov E. Lokot
Schr&oumldinger equation for pair of two massless Dirac particles when magnetic field is applied in Landau gauge is solved exactly. In this case, the separation of center of mass and relative motion is obtained. Landau quantization ε = ±B/ l for pair of two Majorana fermions coupled via a Coulomb potential from massless chiral Dirac equation in cylindric coordinate is found. The root ambiguity in energy spectrum leads into Landau quantization for bielectron, when the states in which the one simultaneously exists are allowed. The tachyon solution with imaginary energy in Cooper problem (ε 2 < 0) is found. The continuum symmetry of Dirac equation allows perfect pairing between electron Fermi spheres when magnetic field is applied in Landau gauge creating a Cooper pair.

Cite this paper
Lokot, L. (2015) Creation of Bielectron of Dirac Cone: The Tachyon Solution in Magnetic Field. Journal of Materials Science and Chemical Engineering, 3, 71-77. doi: 10.4236/msce.2015.38010.
[1]   Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Katsnelson, M.I., Grigorieva, I.V., Dubonos, S.V. and Firsov, A.A. (2005) Two-Dimensional Gas of Massless Dirac Fermions in Grapheme. Nature, 438, 197-200.

[2]   Vasko, F.T. and Ryzhii, V. (2008) Photoconductivity of Intrinsic Grapheme. Physical Review B, 77, Article ID: 195433.

[3]   Zhang, Y., Tan, Y.-W., Stormer, H.L. and Kim, P. (2005) Experimental Observation of the Quantum Hall Effect and Berry’s Phase in Graphene. Nature, 438, 201-204.

[4]   Wallace, P.R. (1947) The Band Theory of Graphite. Physical Review, 71, 622.

[5]   Semenoff, G. (1984) Condensed-Matter Simulation of a Three-Dimensional Anomaly. Physical Review Letters, 53, 2449.

[6]   Zheng, Y. and Ando, T. (2002) Hall Conductivity of a Two-Dimensional Graphite System. Physical Review B, 65, 245420.

[7]   Gusynin, V.P. and Sharapov, S.G. (2005) Unconventional Integer Quantum Hall Effect in Graphene. Physical Review Letters, 95, 146801.

[8]   Gusynin, V.P. and Sharapov, S.G. (2006) Transport of Dirac Quasiparticles in Graphene: Hall and Optical Conductivities. Physical Review B, 73, 245411.

[9]   Peres, N.M.R., Guinea, F. and Castro Neto, A.H. (2006) Electronic Properties of Disordered Two-Dimensional Carbon. Physical Review B, 73, 125411.

[10]   Berry, M.V. (1984) Quantal Phase Factors Accompanying Adiabatic Changes. Proceedings of the Royal Society of London A, 392, 45.

[11]   Carmier, P. and Ullmo, D. (2008) Berry Phase in Graphene: Semiclassical Perspective. Physical Review B, 77, Article ID: 245413.

[12]   Gusynin, V.P., Sharapov, S.G. and Carbotte, J.P. (2007) Ac Conductivity of Graphene: From Tight-Binding Model to 2 + 1-Dimensional Quantum Electrodynamics. International Journal of Modern Physics B, 21, 4611-4658.

[13]   Alicea, J. and Fisher, M.P.A. (2006) Graphene Integer Quantum Hall Effect in the Ferromagnetic and Paramagnetic Regimes. Physical Review B, 74, Article ID: 075422.

[14]   Novikov, D.S. (2007) Elastic Scattering Theory and Transport in Grapheme. Physical Review B, 76, Article ID: 245435.

[15]   Landau, L.D. and Lifshitz, E.M. (1977) Quantum Mechanics. Non-Relativistic Theory. Pergamon, NewYork.

[16]   Lokot, L.E. (2015) Particle-Hole Pair States of Layered Materials. Physica E, 68, 176-183.

[17]   McClure, J.W. (1956) Diamagnetism of Graphite. Physical Review, 104, 666-671.

[18]   Berestetskii, V.B., Lifshitz, E.M. and Pitaevskii, L.P. (1989) Quantum Electrodynamics. Nauka, Moskow.

[19]   Gamayun, O.V., Gorbar, E.V. and Gusynin, V.P. (2009) Supercritical Coulomb Center and Excitonic Instability in Graphene. Physical Review B, 80, Article ID: 165429.

[20]   Malard, L.M., Guimaraes, M.H.D., Mafra, D.L., Mazzoni, M.S.C. and Jorio, A. (2009) Group-Theory Analysis of Electrons and Phonons in N-Layer Graphene Systems. Physical Review B, 79, Article ID: 125426.

[21]   Nandkishore, R., Levitov, L.S. and Chubukov, A.V. (2012) Chiral Superconductivity from Repulsive Interactions in Doped Graphene. Nature Physics, 8, 158-163.

[22]   Rashba, E.I. and Edelshteyin, B.M. (1969) Magnetic Coulomb Levels Near the Saddle Points. JETP Letters, 9, 475- 480.

[23]   Hartmann, R.R., Shelykh, I.A. and Portnoi, M.E. (2011) Excitons in Narrow-Gap Carbon Nanotubes. Physical Review B, 84, Article ID: 035437.

[24]   Julian, S. (2012) Viewpoint: Pairing with Spin Fluctuations. Physics, 5, 17.

[25]   Lokot, L.E. (2015) Exciton Insulator States for Particle-Hole Pair in ZnO/(Zn,Mg)O Quantum Wells and for Dirac Cone. SSRG International Journal of Material Science and Engineering, 1, 1. arXiv: 1501.03696v2 [cond-mat.mes- hall].

[26]   Stroucken, T., Gronqvist, J.H. and Koch, S.W. (2011) Optical Response and Ground State of Graphene. Physical Review B, 84, Article ID: 205445.

[27]   Gronqvist, J.H., Stroucken, T., Beghauser, G. and Koch, S.W. (2011) Excitons in Graphene and the Influence of the Dielectric Environment. arXiv:1107.5653v1 [cond-mat.mtrl-sci].

[28]   Malic, E., Winzer, T., Bobkin, E. and Knorr, A. (2011) Microscopic Theory of Absorption and Ultrafast Many-Particle Kinetics in Grapheme. Physical Review B, 84, Article ID: 205406.