Advertisement

Persistent Currents in Mesoscopic Normal Metal Rings

  • Yoseph Imry
Part of the NATO ASI Series book series (NSSB, volume 254)

Abstract

It is well known that persistent currents which “never decay” can flow in superconducting systems. These can be of fundamentally two different varieties: equilibrium currents, such as those shielding the magnetic field in a bulk superconductor or quantizing the magnetic flux in a ring, and extremely long-lived metastable currents (for example those induced in a ring by a time-dependent magnetic flux) which may survive for astronomical times after the flux is appropriately reduced. Basic theoretical considerations for such currents in a ring (we shall use this term for a general doubly-connected geometry, including cylinders etc.) were presented by Byers and Yang [1] and Bloch [2, 3]. There is no reason why the former (equilibrium currents induced by magnetic fields) should not exist in principle in normal (i.e. non-superconducting) systems. In fact, such currents do obviously exist in atoms and molecules when an equilibrium diamagnetic moment is induced in them by a magnetic field. Do such currents exist in finite macroscopic [4] systems? The answer (while still surprising to many) is of course positive. Diamagnetic currents do flow in equilibrium in metals. In fact, whenever the appropriate thermodynamic potential, J, (e.g. E at zero temperatures and F at finite temperatures for canonical systems, FµN for grand canonical ones) depends on the magnetic field, II, the system has an equilibrium magnetization, M, given by \( M =- \frac{{\partial J}}{{\partial H}} \). This magnetization can be regarded as due to some non-zero circulating currents, which for homogeneous systems are easily shown to flow on the surface of the specimen. Let us now consider a general ring geometry with an Aharonov-Bohm (AB) [5] flux Φ through its hole. The above considerations show that if the thermodynamic potential, J, depends on Φ, then an equilibrium current will circulate around the hole, given by
$$ I =- \frac{1}{c}\frac{{\partial J}}{{\partial \Phi }} $$
(1)
.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N. Byers and C.N. Yang, Phys. Rev. Lett. 7, 46 (1961)ADSCrossRefGoogle Scholar
  2. [2]
    F. Bloch, Phys. Rep. 137, A787 (1965)MathSciNetGoogle Scholar
  3. [3]
    F. Bloch, Phys. Rep. B 2, 109 (1970)Google Scholar
  4. [4]
    Y. Imry, Ann. Phys. NY 51, 1 (1969) (in this paper some of the subtleties with taking the “thermodynamic limit” too early have been discussed)ADSCrossRefGoogle Scholar
  5. [5]
    Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (1980)ADSCrossRefGoogle Scholar
  7. [7]
    R.B. Laughlin, Phys. Rev. B 23, 5632 (1981)ADSCrossRefGoogle Scholar
  8. [8]
    Y. Imry, J. Phys. C 16, 3501 (1983)ADSGoogle Scholar
  9. [9]
    M. Büttiker, Y. Imry, and R. Landauer, Phys. Lett. 96 A, 365 (1983)CrossRefGoogle Scholar
  10. [10]
    Y. Imry, in: directions in condensed matter physics, G. Grinstein and G. Mazenko, eds., World Scientific, Singapore (1986)Google Scholar
  11. [11]
    D.J. Thouless, Phys. Rev. Lett. 39, 1167 (1977)ADSCrossRefGoogle Scholar
  12. [12]
    J.T. Edwards and D.J. Thouless, J. Phys. C 5, 807 (1972)ADSGoogle Scholar
  13. [13]
    R.B. Dingle, Proc. Phys. Soc. A 212, 47 (1952)Google Scholar
  14. [14]
    L. Günther and Y. Imry, Sol. St. Commun. 7, 1391 (1969); unpublished results (1969–1970)CrossRefGoogle Scholar
  15. [15]
    I.O. Kulik, JETP Lett. 11, 275 (1970)ADSGoogle Scholar
  16. [16]
    N.B. Brandt, E.N. Bogachek, D.V. Gitsu, G.A. Gogadze, I.O. Kulik, A.A. Nikolaeva, and Ya.G. Ponomarev, JETP Lett. 24, 273 (1976)ADSGoogle Scholar
  17. [17]
    M. Schick, Phys. Rev. 166, 401 (1968)ADSCrossRefGoogle Scholar
  18. [18]
    B.L. Altshuler, A.G. Aronov, and B.Z. Spivak, JETP Lett. 33, 94 (1981)ADSGoogle Scholar
  19. [19]
    M. Büttiker, Phys. Rev. B 32, 1846 (1985)CrossRefGoogle Scholar
  20. [20]
    M. Büttiker, Ann. NY Acad. Sci. 480, 194 (1986)ADSCrossRefGoogle Scholar
  21. [21]
    R. Landauer and M. Büttiker, Phys. Rev. Lett. 54, 2049 (1985)ADSCrossRefGoogle Scholar
  22. [22]
    A. Stern, Y. Aharonov, and Y. Imry, Phys. Rev. A 41, 3936, (1990)Google Scholar
  23. [23]
    G. Bergmann, Phys. Rep. 107, 1 (1984)ADSCrossRefGoogle Scholar
  24. [24]
    P.A. Lee and R.V. Ramakrishnan, Revs. Mod. Phys. 57, 287 (1985)ADSCrossRefGoogle Scholar
  25. [25]
    P.W. Anderson, D.J. Thouless, E. Abrahams, and D.S. Fisher, Phys. Rev. B 22, 3519 (1980)ADSCrossRefMathSciNetGoogle Scholar
  26. [26]
    E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Phys. Rev. Lett. A 42, 673 (1979)ADSCrossRefGoogle Scholar
  27. [27]
    R. Landauer, unpublished IBM proposal (1966)Google Scholar
  28. [28]
    R. Landauer, IBM J. Res. Dev. 1, 223 (1957)CrossRefMathSciNetGoogle Scholar
  29. [29]
    Y. Gefen, Y. Imry, and M.Ya. Azbel, Phys. Rev. Lett. 52, 129 (1984)ADSCrossRefGoogle Scholar
  30. [30]
    M. Büttiker, Y. Imry, and M.Ya. Azbel Phys. Rev. A 30, 1982 (1984)CrossRefGoogle Scholar
  31. [31]
    R.A. Webb, S. Washburn, O.P. Umbach, and R.B. Laibowitz, Phys. Rev. Lett. 54, 2696 (1985)ADSCrossRefGoogle Scholar
  32. [32]
    V. Chandrasekhar, M.J. Rooks, S. Wind, and D.E. Prober, Phys. Rev. Lett. 15, 1610 (1988)Google Scholar
  33. [33]
    S. Datta, M. Melloch, S. Bandyopadhyay, R. Noren, M. Vaziri, M. Miller, and R. Reifenberger, Phys. Rev. Lett. 55, 2344 (1986)ADSCrossRefGoogle Scholar
  34. [34]
    Y. Gefen, private communication (March, 1984)Google Scholar
  35. [35]
    D.V. Sharvin and V.V. Sharvin, JETP Lett. 32, 272 (1981)ADSGoogle Scholar
  36. [36]
    B.L. Altshuler, A.G. Aronov, D.E. Khmel’nitskii, A.I. Larkin, in: Quantum Theory of Solids, I.M. Lifschitz, ed., p. 146, Mir Publishers, Moscow (1982)Google Scholar
  37. [37]
    L.P. Levy, G. Dolan, J. Dunsmuir, and H. Bouchiat, Phys. Rev. Lett. 64, 2074 (1990)ADSCrossRefGoogle Scholar
  38. [38]
    O. Entin-Wohlmann and Y. Gefen, Europhys. Lett. 8, 477 (1989)ADSCrossRefGoogle Scholar
  39. [39]
    H.F. Cheung, Y. Gefen, and E.K. Riedel, Phys. Rev. Lett. 62, 587 (1989)ADSCrossRefGoogle Scholar
  40. [40]
    H.F. Cheung, Y. Gefen, E.K. Riedel, and W.H. Shih, Phys. Rev. B 37, 6050 (1988)ADSCrossRefGoogle Scholar
  41. [41]
    H. Bouchiat and G. Montambaux, J. Phys. (Paris) 59, 2695 (1989)CrossRefGoogle Scholar
  42. [42]
    R. Peierls, Quantum Theory of Solids, p. 29, Oxford (1955) (comment attributed to W. Shockley)zbMATHGoogle Scholar
  43. [43]
    U. Sivan and Y. Imry, Phys. Rev. Lett. 61, 1001 (1988)ADSCrossRefGoogle Scholar
  44. [44]
    Y. Imry, Europhys. Lett. 1, 249 (1986)ADSCrossRefGoogle Scholar
  45. [45]
    W. Kohn, Phys. Rev. 133, A161 (1964)ADSCrossRefMathSciNetGoogle Scholar
  46. [46]
    N. Trivedi and D. Brown, Phys. Rev. B 38, 9581 (1988)ADSCrossRefGoogle Scholar
  47. [47]
    F. Wegner, Z. Phys. 25, 327 (1976)Google Scholar
  48. [48]
    F.J. Dyson, J. Math. Phys. 3, 140, 157, 166 (1962)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  49. [49]
    M.L. Mehta, Random Matrices, Academic Press, New York, London (1967)zbMATHGoogle Scholar
  50. [50]
    G. Lopez, P.A. Mello, and T.H. Seligman, Z. Phys. A 392, 351 (1981)ADSCrossRefGoogle Scholar
  51. [51]
    W.L. McMillan, Phys. Rev. B 24, 2739 (1981)ADSCrossRefGoogle Scholar
  52. [52]
    M. Kaveh and N.F. Mott, J. Phys. C 14, 2179 (1981)Google Scholar
  53. [53]
    M.Ya. Azbel, private communication (1980)Google Scholar
  54. [54]
    Y. Imry, Phys. Rev. B 21, 2042 (1980)ADSCrossRefGoogle Scholar
  55. [55]
    B.L. Altshuler, Y. Gefen, and Y. Imry, unpublished results (1990)Google Scholar
  56. [56]
    B.L. Altshuler and B.Z. Spivak, Sov. Phys. JETP 65, 343 (1987)Google Scholar
  57. [57]
    M. Büttiker and T.M. Klapwijk, Phys. Rev. bf 33, 5114 (1986)ADSCrossRefGoogle Scholar
  58. [58]
    A.F. Andreev, JETP 46, 1823 (1964)Google Scholar
  59. [59]
    M.C. Gutzwiller, J. Math. Phys. 12, 343 (1971)ADSCrossRefGoogle Scholar
  60. [60]
    A.I. Larkin and D.E. Khmel’nitskii, Sov. Phys. Uspekhi. 25, 185 (1982)ADSCrossRefGoogle Scholar
  61. [61]
    D.E. Khmel’nitskii, Physica 126 B, 235 (1984)Google Scholar
  62. [62]
    Y. Gefen and Y. Imry, unpublished results (1990)Google Scholar
  63. [63]
    D. Divincenzo and M.P.A. Fisher, unpublished results (1988)Google Scholar
  64. [64]
    A. Stern and Y. Imry, unpublished results (1990)Google Scholar
  65. [65]
    M. Murat, Y. Gefen, and Y. Imry, Phys. Rev. B 34, 659 (1986)ADSCrossRefGoogle Scholar
  66. [66]
    A.D. Stone and Y. Imry, Phys. Rev. Lett. 56, 189 (1985)ADSCrossRefGoogle Scholar
  67. [67]
    Y. Imry and N. Shiren, Phys. Rev. B 33, 7992 (1986)ADSCrossRefGoogle Scholar
  68. [68]
    M. Berry, Proc. Roy. Soc. Lond. A 400, 229 (1985)ADSCrossRefzbMATHGoogle Scholar
  69. [69]
    S. Chakravarty and A. Schmid, Phys. Rep. 140, 195 (1986)ADSCrossRefGoogle Scholar
  70. [70]
    B.L. Altshuler, A.G. Aronov, B.Z. Spivak, D.Yu. Sharvin, and Yu.V. Sharvin, JETP Lett. 35, 5898 (1982)Google Scholar
  71. [71]
    G. Montambaux, H. Bouchiat, D. Szigeti, and R. Friesner, preprint (1990)Google Scholar
  72. [72]
    H. Fukuyama, J. Phys. Soc. Jpn. Lett. 58, 47 (1989)ADSCrossRefGoogle Scholar
  73. [73]
    A. Schmid, unpublished results (1990)Google Scholar
  74. [74]
    Y. Meir, Y. Gefen, and O. Entin-Wohlmann, Phys. Rev. Lett. 63, 798 (1989)ADSCrossRefGoogle Scholar
  75. [75]
    B.L. Altshuler, D.E. Khmel’nitskii, and B.Z. Spivak, Sol. St. Commun. 48, 10 (1983)CrossRefGoogle Scholar
  76. [76]
    V. Ambegaokar and U. Eckern, Phys. Rev. Lett. 65, 381 (1990)ADSCrossRefGoogle Scholar
  77. [77]
    Y. Imry, in: Proc. of the 1969 Stanford Superconductivity Conf., F. Chilton ed., 344, North Holland (1971)Google Scholar
  78. [78]
    M. Heiblum, M.I. Nathan, D.E. Thomas, and CM. Knoedler, Phys. Rev. Lett. 55, 2200 (1985)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Yoseph Imry
    • 1
  1. 1.Department of Nuclear PhysicsWeizmann Institute of ScienceRehovotIsrael

Personalised recommendations