Advertisement

What Happens to the Integer Quantum Hall Effect in Three Dimensions?

  • J. T. Chalker
Part of the NATO ASI Series book series (NSSB, volume 370)

Abstract

The integer quantum Hall effect is often advertised as one of the most striking phenomena to occur in the two-dimensional electron gas. It is nevertheless natural and interesting to ask whether it can also take place in a three-dimensional system, and that question is the subject of this article. More specifically, two remarkable aspects of the quantum Hall effect in two dimensions are that the Hall conductance shows plateaus, in which it is constant for a range of Landau level filling factors, and that, when the Hall conductance lies on a plateau, current flows through the sample via edge states. Here we ask, for three-dimensional systems, whether the Hall conductance can also show plateaus, and whether there then exist surface states, analogous to the edge states of a two-dimensional system. Of course, for there to be much to write about, the answers to these questions must in some circumstances be ‘yes’, and our focus will partly be on the conditions necessary to favour the quantum Hall effect in three dimensions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. I. Halperin, Jpn. J. Appl. Phys. 26 Suppl.26-3, 1913 (1987).MathSciNetGoogle Scholar
  2. 2.
    J. T. Chalker and A. Dohmen, Phys. Rev. Lett. 75, 4496 (1995).ADSCrossRefGoogle Scholar
  3. 3.
    L. Balents and M. P. A. Fisher, Phys. Rev. Lett. 76, 2153 (1996).CrossRefGoogle Scholar
  4. 4.
    Y. B. Kim, Phys. Rev. B 53, 16420 (1996).ADSCrossRefGoogle Scholar
  5. 5.
    H. Mathur, Phys. Rev. Lett. 78, 2429 (1997); Y.-K. Yu, cond-mat/9611137.ADSCrossRefGoogle Scholar
  6. 6.
    L. Balents, M. P. A. Fisher, and M. R. Zirnbauer, Nucl. Phys. B 483, 601 (1997); S. Cho, L. Balents, and M. P. A. Fisher, cond-mat/9708038.MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    I. A. Gruzberg, N. Read, and S. Sachdev, Phys. Rev. B 55, 10593 (1997); ibid 56 13218 (1997).ADSCrossRefGoogle Scholar
  8. 8.
    Z. Q. Wang, Phys. Rev. Lett. 78, 126 (1997).ADSCrossRefGoogle Scholar
  9. 9.
    Z. Q. Wang, Phys. Rev. Lett 79, 4002 (1997).ADSCrossRefGoogle Scholar
  10. 10.
    H. L. Stornier, J. P. Eisenstein, A. C. Gossard, W. Wiegmann, and K. Baldwin, Phys. Rev. Lett. 56 85 (1986).ADSCrossRefGoogle Scholar
  11. 11.
    D. Poilblanc, G. Montambaux, M. Heritier, and P. Lederer, Phys. Rev. Lett. 58 270 (1986).ADSCrossRefGoogle Scholar
  12. 12.
    S. Hill, P. S. Sandhu, J. S. Quails, J. S. Brooks, M. Tokumoto, N. Kinoshita, T. Kinoshita, and Y. Tanaka, Phys. Rev. B 55, R4891 (1997).ADSCrossRefGoogle Scholar
  13. 13.
    D. P. Druist, P. J. Turley, E. G. Gwinn, K. Maranowski, and A. C. Gossard, UCSB preprint.Google Scholar
  14. 14.
    For reviews, see The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer, Berlin, 1990); and M. Jansen, O. Viehweger, U. Fastenrath and J. Hajdu Introduction to the Theory of the Integer Quantum Hall Effect (VCH, Weinheim, 1994).Google Scholar
  15. 15.
    D. E. Khmel’nitskii, Pis’ma Zh. Eksp. Teor. Fiz. 38, 454 (1983) [JETP Lett. 38, 552].Google Scholar
  16. 16.
    D. K. K. Lee and J. T. Chalker, Phys.Rev.Lett. 72, 1510 (1994).ADSCrossRefGoogle Scholar
  17. 17.
    Z. Q. Wang, D.-H. Lee, and X. G. Wen, Phys. Rev. Lett. 72, 2454 (1994).ADSCrossRefGoogle Scholar
  18. 18.
    N. Read, unpublished; D.-H. Lee, Phys. Rev. B 50, 10788 (1994); M. R. Zirnbauer, Annalen der Physik 3, 513 (1994).Google Scholar
  19. 19.
    M. A. Martindelgado, R. Shankar, and G. Sierra, Phys. Rev. Lett. 77, 3443 (1996).ADSCrossRefGoogle Scholar
  20. 20.
    J. T. Chalker and P. D. Coddington, J. Phys. C 21, 2665 (1988).ADSCrossRefGoogle Scholar
  21. 21.
    A. MacKinnon and B. Kramer, Phys. Rev. Lett. 47, 1546 (1981); Z. Phys. B 53, 1 (1983).ADSCrossRefGoogle Scholar
  22. 22.
    The directed network model was first introduced, in a different context, in: L. Saul, M. Kardar, and N. Read, Phys. Rev. A 45, 8859 (1992).Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • J. T. Chalker
    • 1
  1. 1.Theoretical PhysicsUniversity of OxfordOxfordUK

Personalised recommendations