Advertisement

International Journal of Theoretical Physics

, Volume 55, Issue 11, pp 4830–4840 | Cite as

Theory of the Half-integer Quantum Hall Effect in Graphene

  • Shigeji Fujita
  • Akira Suzuki
Article
  • 309 Downloads

Abstract

The unusual quantum Hall effect (QHE) in graphene is described in terms of the composite (c-) bosons, which move with a linear dispersion relation. The “electron” (wave packet) moves easier in the direction [1 1 0 c-axis] ≡ [1 1 0] of the honeycomb lattice than perpendicular to it, while the “hole” moves easier in [0 0 1]. Since “electrons” and “holes” move in different channels, the particle densities can be high especially when the Fermi surface has “necks”. The strong QHE arises from the phonon exchange attraction in the neighborhood of the “neck” surfaces. The plateau observed for the Hall conductivity and the accompanied resistivity drop is due to the superconducting energy gap caused by the Bose-Einstein condensation of the c-bosons, each forming from a pair of one-electron–two-fluxons c-fermions by phonon-exchange attraction. The half-integer quantization rule for the Hall conductivity: (1/2)(2P−1)(4e 2/h), P=1,2,..., is derived.

Keywords

Quantum Hall effect Composite boson (fermion) Superconducting energy gap Phonon exchange attraction 

References

  1. 1.
    Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Katsnelson, M.I., Grigorieva, I.V., Dubonos, S.V., Firsov, A.A.: Nature 438, 197 (2005)Google Scholar
  2. 2.
    Ashcroft, N.W., Mermin, N.D.: Solid State Physics, p 214. Saunders, Philadelphia (1976)Google Scholar
  3. 3.
    Wigner, E., Seitz, F.: Phys. Rev. 43, 804 (1933)Google Scholar
  4. 4.
    Fujita, S., Suzuki, A.: J. Appl. Phys. 107, 013711 (2010)Google Scholar
  5. 5.
    Fujita, S., Jovaini, A., Godoy, S., Suzuki, A.: Phys. Lett. A 376, 2808 (2012)Google Scholar
  6. 6.
    Fujita, S., Takato, Y., Suzuki, A.: Mod. Phys. Lett. B 25, 223 (2011)Google Scholar
  7. 7.
    Zhang, Y., Tan, Y.-W., Stormer, H.L., Kim, P.: Nature 438, 201 (2005)Google Scholar
  8. 8.
    Tsui, D.C., Stormer, H.L., Gossard, A.C.: Phys. Rev. Lett. 48, 1559 (1982)Google Scholar
  9. 9.
    Laughlin, R.B.: Phys. Rev. Lett. 50, 1395 (1983)Google Scholar
  10. 10.
    Girvin, S.M., MacDonald, A.H.: Phys. Rev. Lett. 58, 1252 (1987)Google Scholar
  11. 11.
    Zhang, S.C., Hansson, T.H., Kivelson, S.: Phys. Rev. Lett. 62, 82 (1989)Google Scholar
  12. 12.
    Jain, J.K.: Phys. Rev. Lett. 63, 199 (1989)Google Scholar
  13. 13.
    Jain, J.K.: Phys. Rev. B 40, 8079 (1989)Google Scholar
  14. 14.
    Jain, J.K.: Phys. Rev. B 41, 7653 (1990)Google Scholar
  15. 15.
    Jain, J.K.: Surf. Sci. 263, 65 (1992)Google Scholar
  16. 16.
    Read, N.: Phys. Rev. Lett. 62, 86 (1989)Google Scholar
  17. 17.
    Halperin, B.I., Lee, P.A., Read, H.: 7312 47 (1993)Google Scholar
  18. 18.
    Shankar, R., Murthy, G.: Phys. Rev. Lett. 79, 4437 (1997)Google Scholar
  19. 19.
    Ezawa, Z.F.: Quantum Hall Effects, 2nd ed. World Scientific, Singapore (2008)Google Scholar
  20. 20.
    Fujita, S., Okamura, Y.: Phys. Rev. 369, 155313 (2004)Google Scholar
  21. 21.
    Jain, J.K.: Composite Fermions, Cambridge University Press, Cambridge, UK (2007)Google Scholar
  22. 22.
    Fujita, S., Suzuki, A., Ho, H.C.: arXiv:hep-th/1304.7631v1 [cond-mat.mes-hall]
  23. 23.
    Laughlin, R.B.: Science 242, 525 (1988)Google Scholar
  24. 24.
    Onsager, L.: Phil. Mag. 43, 1006 (1952)Google Scholar
  25. 25.
    Dirac, P.A.M.: Principles of Quantum Mechanics, 4th ed., pp. 248–252, pp. 253–263, p. 267. Oxford Univ. Press, Oxford (1958)Google Scholar
  26. 26.
    Ehrenfest, P., Oppenheimer, J.R.: Phys. Rev. 37, 333 (1931)Google Scholar
  27. 27.
    Bethe, H.A., Jackiw, R.: Intermediate Quantum Mechanics. In: 2nd (ed.) , p 23. Benjamin, New York (1968)Google Scholar
  28. 28.
    Fujita, S., Gau, S.-P., Suzuki, A.: J. Korean Phys. Soc. 38, 1174 (2001)Google Scholar
  29. 29.
    Bardeen, J., Cooper, L.N., Schriefler, J.R.: Phys. Rev. 108, 1175 (1957)Google Scholar
  30. 30.
    Cooper, L.N.: Phys. Rev. 104, 1189 (1956)Google Scholar
  31. 31.
    Fujita, S., Ito, K., Godoy, S.: Quantum Theory of Conducting Matter, Superconductivity, pp. 73–75, pp. 77–79. Springer, New York (2009)Google Scholar
  32. 32.
    Fujita, S., Suzuki, A.: Electrical Conduction in Graphene and Nanotubes, pp. 212–215. Wiley-VCH, Weinheim, Germany (2013)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of PhysicsUniversity at Buffalo, State University of New YorkBuffaloUSA
  2. 2.Department of Physics, Faculty of ScienceTokyo University of ScienceShinjyuku-kuJapan

Personalised recommendations