Advertisement

Resistance Fluctuations in Small Samples: Be Careful When Playing with Ohm’s Law

  • Sean Washburn
Part of the NATO ASI Series book series (NSSB, volume 254)

Abstract

The discovery of random fluctuations [1,2] in the resistance of metal samples has wrought a small revolution in the understanding of electronic transport in samples with weak disorder (commonly known as metals). The fluctuations arise from quantum mechanical interference among the various trajectories that the electrons “follow” as they migrate from one end of the sample to the other. Obviously then the electrons must retain the coherence of their wavefunctions from one end to the other. By carefully summing up all the possible trajectories and interferences among them, Al’tshuler [3] and Lee et al. [4] have calculated the ensemble averaged variance of the conductance of wires of length L φ to be ((∆G)2) ≡ (G 2) -(G)2 ≡ (e 2/h)2. From an ensemble of classically identical samples (the theorists assume that the samples are carved from some mythical film where, on any scale larger than the mean free path length, the resistivity is utterly homogeneous, and the angular brackets refer to the average over this ensemble), one would find that the conductance fluctuates among the members by an amount characterized by a distribution of width ≃ e 2/h. Owing to a random Aharonov-Bohm (AB) effect, the conductance of a particular sample fluctuates by the same amount as a function of magnetic field [5], and the result of the sensitivity of the conductance to the impurity potential, it also fluctuates by ≃ e 2/h as a function of the Fermi energy [4] or when minute rearrangements occur in the impurity configuration [6]. In principle, this “universal” feature of the conductance of metal wires can be used as a measure of the phase coherence of the carriers and as a exquisitely sensitive probe of changes in the impurity potential [6].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. Blonder, R. Dynes, and A. White, unpublished (1984)Google Scholar
  2. [2]
    C.P. Umbach, S. Washburn, R.B. Laibowitz, and R.A. Webb, Phys. Rev. B 30, 4048 (1984);ADSCrossRefGoogle Scholar
  3. [2a]
    R.A. Webb, S. Washburn, C.P. Umbach, and R.B. Laibowitz, in: ‘Localization, Interaction, and Transport Phenomena’, G. Bergmann, Y. Bruynseraede, and B. Kramer, eds., Springer Ser. in Sol. St. Sc. 61, 121 (1985)CrossRefGoogle Scholar
  4. [3]
    B.L. Al’tshuler, JETP Lett. 41, 648 (1985)ADSGoogle Scholar
  5. [4]
    P. A. Lee, A.D. Stone, and J. Fukuyama, Phys. Rev. B 35, 1039 (1987)ADSCrossRefGoogle Scholar
  6. [5]
    A.D. Stone, Phys. Rev. Lett. 54, 2692 (1985)ADSCrossRefGoogle Scholar
  7. [6]
    B.L. Al’tshuler and B.Z. Spivak, JETP Lett. 42, 447 (1985)ADSGoogle Scholar
  8. [7]
    R. Landauer, Phil. Mag. 21, 863 (1970);ADSCrossRefGoogle Scholar
  9. [7a]
    see also Z. Phys. B 68, 217 (1987), and references cited thereGoogle Scholar
  10. [8]
    P.A. Lee, Physica 140, 169 (1986), and references cited thereCrossRefGoogle Scholar
  11. [9]
    P.F. Bagwell and T.P. Orlando, Phys. Rev. B 40, 1456 (1989)ADSCrossRefGoogle Scholar
  12. [10]
    N.D. Mermin and N.W. Ashcroft, ‘Solid State Physics’, 345ff, Holt, Rinehart and Winston, New York (1976)Google Scholar
  13. [11]
    See A.G. Aronov and Yu.V. Sharvin, Rev. Mod. Phys. 59, 744 (1987) and references to literature on weak localization and h/2e oscillations from coherent back-scattering cited thereADSCrossRefGoogle Scholar
  14. [12]
    D.J. Thouless, Phys. Rev. Lett. 39, 1167 (1977)ADSCrossRefGoogle Scholar
  15. [13]
    B.L. Al’tshuler and B.I. Shklovskii, JETP 64, 127 (1986)Google Scholar
  16. [14]
    Y. Imry, Europhys. Lett. 1, 249 (1986)ADSCrossRefGoogle Scholar
  17. [15]
    Y. Imry, in: Directions on Condensed Matter Physics, G. Grinstein and E. Mazenko, eds., p. 101, World Publishing, Singapore (1986)CrossRefGoogle Scholar
  18. [16]
    B.L. Al’tshuler and D.E. Khmel’nitskii, JETP Lett. 42, 291 (1985)Google Scholar
  19. [17]
    W.F. Smith et al., unpublishedGoogle Scholar
  20. [18]
    S. Hershfield, Phys. Rev. B 37, 8557 (1988), and unpublished (1989)ADSCrossRefGoogle Scholar
  21. [19]
    F.N.H. Robinson, Phys. Lett. 97 A, 162 (1983); C. Van Haesendonck, unpublished (1985)CrossRefGoogle Scholar
  22. [20]
    S. Feng, P.A. Lee, and A.D. Stone, Phys. Rev. Lett. 56, 1960 (1986)ADSCrossRefGoogle Scholar
  23. [21]
    N.M. Zimmermann and W.W. Webb, Phys. Rev. Lett. 61, 889 (1988);ADSCrossRefGoogle Scholar
  24. [22]
    G.B. Alers, M.B. Weissmann, R.S. Averback, and A. Shyu, Phys. Rev. B 40, 900 (1989);ADSCrossRefGoogle Scholar
  25. [23]
    N.B. Israeloff, M.B. Weissmann, G.J. Nieuwenhuys, and J. Kosiorowska, Phys. Rev. Lett. 63, 794 (1989)ADSCrossRefGoogle Scholar
  26. [22]
    N.O. Birge, B. Golding, and W.H. Iiaemmerle, Phys. Rev. Lett. 63, 195 (1989)ADSCrossRefGoogle Scholar
  27. [23]
    D. Mailly, M. Sanquer, J.-L. Pichard, and P. Pari, Europhys. Lett. 8, 471 (1989)ADSCrossRefGoogle Scholar
  28. [24]
    P. Debray, J.-L. Pichard, J. Vicente, and P.N. Tang, Phys. Rev. Lett. 63, 2264 (1989)ADSCrossRefGoogle Scholar
  29. [25]
    A.D. Stone, Phys. Rev. B 39, 10736 (1989)CrossRefGoogle Scholar
  30. [26]
    J.C. Licini, D.J. Bishop, M.A. Kastner, and J. Melngailis, Phys. Rev. Lett. 55, 2987 (1985)ADSCrossRefGoogle Scholar
  31. [27]
    Y. Aharonov and D. Böhm, Phys. Rev. 115, 485 (1959)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  32. [28]
    E. Merzbacher, Am. J. Phys. 30, 237 (1962)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  33. [29]
    R.G. Chambers,Phys. Rev. Lett. 5, 3 (1960);ADSCrossRefGoogle Scholar
  34. [29a]
    A. Tonomura, N. Osakabe, T. Mat-suda, T. Kawasaki, J. Endo, S. Yano, and H. Yamada, Phys. Rev. Lett. 56, 792 (1986)ADSCrossRefGoogle Scholar
  35. [30]
    B.S. Deaver and W.F. Fairbank, Phys. Rev. Lett. 7, 43 (1961); R. Doll and M. Näbauer, ibid., 51 (1961)ADSCrossRefGoogle Scholar
  36. [31]
    F. London, ‘Supernuids I: Macroscopic Theory of Superconductivity’, p. 152, Wiley, New York (1950);Google Scholar
  37. [32]
    N. Byers and C.N. Yang, Phys. Rev. Lett. 7, 46 (1961)ADSCrossRefGoogle Scholar
  38. [32]
    N.B. Brandt, D.V. Gitsu, V.A. Dolma, and Ya.G. Ponomarev, JETP 92, 515 (1987), and references to earlier work cited thereGoogle Scholar
  39. [33]
    D. Yu. Sharvin and Yu.V. Sharvin, JETP Lett. 34, 272 (1981)ADSGoogle Scholar
  40. [34]
    B.L. Al’tshuler, A.G. Aronov, and B.Z. Spivak, JETP Lett. 33, 101 (1981)ADSGoogle Scholar
  41. [35]
    S. Washburn and R.A. Webb, Adv. Phys. 35, 375 (1986), and a forth-coming review article in Rep. Prog. Phys. (1990)ADSCrossRefGoogle Scholar
  42. [36]
    G. Timp, A.M. Chang, P. Mankiewich, R. Behringer, J.E. Cunningham, T.Y. Chang, and R.E. Howard, Phys. Rev. Lett. 58, 2814 (1987)ADSCrossRefGoogle Scholar
  43. [37]
    B.L. Al’tshuler, V.E. Kravtsov, and I.V. Lerner, JETP Lett. 43, 441 (1986)ADSGoogle Scholar
  44. [37a]
    B.L. Al’tshuler, V.E. Kravtsov, and I.V. Lerner, JETP 64, 1352 (1986)Google Scholar
  45. [38]
    G. Timp et al., Phys. Rev. B 39, 6227 (1989), and references cited thereADSCrossRefGoogle Scholar
  46. [39]
    B. Majlly and M. Sanquer, this volumeGoogle Scholar
  47. [40]
    A.D. Stone and Y. Imry, Phys. Rev. Lett. 56,189 (1986)ADSCrossRefGoogle Scholar
  48. [41]
    Y. Gefen, Y. Imry, and M.Ya. Azbel, Surf. Sc. 142, 203 (1984);ADSCrossRefGoogle Scholar
  49. [41a]
    Y. Gefen, Y. Imry, and M.Ya. Azbel, Phys. Rev. Lett. 52, 129 (1984);ADSCrossRefGoogle Scholar
  50. [41b]
    M. Büttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B 31, 6207 (1985)CrossRefGoogle Scholar
  51. [42]
    D.J. Bishop and G.J. Dolan, Phys. Rev. Lett. 55, 2911 (1985)ADSCrossRefGoogle Scholar
  52. [43]
    R.A. Webb, S. Washburn, C.P. Umbach, and R.B. Laibowitz, Phys. Rev. Lett. 54, 2696 (1985);ADSCrossRefGoogle Scholar
  53. [43a]
    V. Chandrasekhar, M.J. Rooks, S. Wind, and D.E. Prober, Phys. Rev. Lett. 55, 1610 (1985);ADSCrossRefGoogle Scholar
  54. [43b]
    S. Datta, M.R. Melloch, S. Banyopadyay, R. Noren, M. Vaziri, M. Miller, and R. Reifenberger, Phys. Rev. Lett. 55, 2344 (1985)ADSCrossRefGoogle Scholar
  55. [44]
    C.P. Umbach, C. Van Haesendonck, R.B. Laibowitz, S. Washburn, and R.A. Webb, Phys. Rev. Lett. 56, 386 (1986)ADSCrossRefGoogle Scholar
  56. [45]
    Y. Gefen, Y. Imry, and M.Ya. Azbel, Phys. Rev. Lett. 52, 129 (1984);ADSCrossRefGoogle Scholar
  57. [45a]
    Y. Gefen, Y. Imry, and M.Ya. Azbel, Surf. Sc. 142, 203 (1984)ADSCrossRefGoogle Scholar
  58. [46]
    S. Washburn, C.P. Umbach, R.B. Laibowitz, and R.A. Webb, Phys. Rev. B 32, 4879 (1985), and unpublishedCrossRefGoogle Scholar
  59. [47]
    A. Benoit, S. Washburn, R.A. Webb, D. Mailly, and L. Dumoulin, in: Anderson Localization, T. Ando and H. Fukuyama, eds., p. 346, Springer, Heidelberg (1988)CrossRefGoogle Scholar
  60. [48]
    M. Büttiker, Phys. Rev. Lett. 57, 1761 (1986);ADSCrossRefGoogle Scholar
  61. [48a]
    M. Büttiker, IBM J. Res. Devel. 32, 317 (1988), and references cited thereCrossRefGoogle Scholar
  62. [49]
    S. Maekawa, Y. Isawa, H. Ebisawa, J. Phys. Soc. Jpn. 56, 25 (1987);CrossRefGoogle Scholar
  63. [49a]
    Y. Isawa, H. Ebisawa, and S. Maekawa, in: Anderson Localization, T. Ando and H. Fukuyama, eds., p. 329, Springer, Heidelberg (1988)CrossRefGoogle Scholar
  64. [50]
    M. Büttiker, Phys. Rev. M. B 35, 4123 (1987)ADSCrossRefGoogle Scholar
  65. [51]
    A. Yu. Zyuzin and B.Z. Spivak, ZhETF 93, 994 (1987) [JETP 93, 560 (1987)]Google Scholar
  66. [51a]
    A. Yu. Zyuzin and B.Z. Spivak, Sol. St. Commun. 67, 75 (1988)ADSCrossRefGoogle Scholar
  67. [52]
    CL. Kane, R.A. Serota, and P.A. Lee, Phys. Rev. B 37, 6701 (1988)ADSCrossRefGoogle Scholar
  68. [53]
    C.L. Kane, P.A. Lee, D.P. DiVinzenzo, Phys. Rev. B 38, 2995 (1988)ADSCrossRefGoogle Scholar
  69. [54]
    S. Hershfield and V. Ambegaokar, Phys. Rev. B 38, 7909 (1988);ADSCrossRefGoogle Scholar
  70. [54a]
    S. Hershfield,Ann. Phys. 196, 12 (1989)ADSCrossRefGoogle Scholar
  71. [55]
    A. Benoit, S. Washburn, C.P. Umbach, R.B. Laibowitz, and R.A. Webb, Phys. Rev. Lett. 57, 1765 (1986)ADSCrossRefGoogle Scholar
  72. [56]
    A.D. Benoit, C.P. Umbach, R.B. Laibowitz, and R.A. Webb, ibid 58, 2343 (1987);ADSCrossRefGoogle Scholar
  73. [56a]
    W.J. Skocpol, P.M. Mankiewich, R.E. Howard, L.D. Jacket, D.M. Tennant, and A.D. Stone, ibid 58, 2347 (1987)ADSCrossRefGoogle Scholar
  74. [57]
    H. Haucke, S. Washburn, A. Benoit, C.P. Umbach, and R.A. Webb, subm. to Phys. Rev. B (1990)Google Scholar
  75. [58]
    C.P. Umbach, P. Santhanam, C. Van Haesendonck, R.A. Webb, Appl. Phys. Lett. 50, 1289 (1987)ADSCrossRefGoogle Scholar
  76. [59]
    V. Chandrasekhar, P. Santhanam, and D.E. Prober, Phys. Rev. Lett. 61, 2253 (1988)ADSCrossRefGoogle Scholar
  77. [60]
    C.W.J. Beenakker and H. van Houten, Phys. Rev. B 37, 6544 (1988)ADSCrossRefGoogle Scholar
  78. [61]
    V. Chandrasekhar, P. Santhanam, and D.E. Prober, subm. to Phys. Rev. B (1990)Google Scholar
  79. [62]
    D.P. DiVincenzo and C.L. Kane, Phys. Rev. B 38, 3006 (1988)ADSCrossRefGoogle Scholar
  80. [63]
    H. Baranger, A.D. Stone, D.P. DiVincenzo, Phys. Rev. B 37, 6521 (1988)ADSCrossRefGoogle Scholar
  81. [64]
    D.E. Khmel’nitskii and A.I. Larkin, Phys. Scripta T 14, 4 (1986)CrossRefGoogle Scholar
  82. [65]
    R.A. Webb, S. Washburn, and C.P. Umbach, Phys. Rev. B 37, 8544 (1988), and unpublishedCrossRefGoogle Scholar
  83. [66]
    P. deVegvar, G. Timp, P.M. Mankiewich, R. Behringer, J.C. Cunningham, Phys. Rev. B 40, 3491 (1988). Some remarks here about the relations among energy scales in the metal experiments [65] are erroneousCrossRefGoogle Scholar
  84. [67]
    Y. Imry and S. Washburn, unpublished (1989)Google Scholar
  85. [68]
    A.I. Larkin and K.A. Matveev, JETP 66, 580 (1987);Google Scholar
  86. [68a]
    L.I. Glazman and K.A. Matveev, JETP 67, 1276 (1987)Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Sean Washburn
    • 1
  1. 1.IBM Thomas J. Watson Research CenterYorktown HeightsUSA

Personalised recommendations