Inhomogeneous persistent currents at high temperatures

Solid State Communications

Volumn 100, Issue 3, 1996, Pages 195-199

  Azbel', Mark Ya ^a,b  

^a HARVARD UNIVERSITY   (United States)

^b TEL AVIV UNIVERSITY   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

CALCULATIONS; CYCLOTRONS; DIAMAGNETISM; HIGH TEMPERATURE EFFECTS; MAGNETIC FIELDS; MAGNETIZATION; METALS; PARAMAGNETISM;

CYCLOTRON ORBIT; INHOMOGENEOUS PERSISTENT CURRENTS; MAGNETIC FLUX QUANTA; MESOSCOPIC SCALE; STRONG MAGNETIC FIELD;

ELECTRIC CURRENTS;

EID: 0030269575     PISSN: 00381098     EISSN: None     Source Type: Journal    
DOI: 10.1016/0038-1098(96)00395-X     Document Type: Article

References (52)

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5. PC, related to dHvA, and its Meissner effect, were demonstrated (in the absence of scattering) in [5].
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8. Peierls, R.E., Surprises in Theoretical Physics. Princeton Univ. Press, Princeton, 1979; Teller, E., Z. Phys. 67, 1931, 311. See also Levy, L.P., Reich, D.M., Pfeiffer, L., West, K., Physics B189, 1993, 204; Zhu, J.-X. and Wang, Z.D., Phys. Lett. A203, 1995, 144.
9. Peierls, R.E., Surprises in Theoretical Physics. Princeton Univ. Press, Princeton, 1979; Teller, E., Z. Phys. 67, 1931, 311. See also Levy, L.P., Reich, D.M., Pfeiffer, L., West, K., Physics B189, 1993, 204; Zhu, J.-X. and Wang, Z.D., Phys. Lett. A203, 1995, 144.
10. Peierls, R.E., Surprises in Theoretical Physics. Princeton Univ. Press, Princeton, 1979; Teller, E., Z. Phys. 67, 1931, 311. See also Levy, L.P., Reich, D.M., Pfeiffer, L., West, K., Physics B189, 1993, 204; Zhu, J.-X. and Wang, Z.D., Phys. Lett. A203, 1995, 144.
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12. λλ = 1/2|l|MΩ̃; (b) Eq. (3.982.1).
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17. Cheung, H.-F., Gefen, Y. and Riedel, E.K., IBMJ. Res. Dev. 32, 1988, 359; Phys. Rev. Lett. 62, 1989, 587.
18. Cheung, H.-F., Gefen, Y. and Riedel, E.K., IBMJ. Res. Dev. 32, 1988, 359; Phys. Rev. Lett. 62, 1989, 587.
19. c ≪ R, paramagnetic currents are the Halperin currents: Halperin, B.I., Phys. Rev. B25, 1982, 2185, which flow only at the edges in the direction opposite to the one in the bulk Landau orbits. Even edge states may be broad and slowly changing: see Chklovskii, D.B., Shklovskii, B.I, and Glazman, L.I., Phys. Rev. B46, 1992, 4026; and Knott, R., Dietsche, W., von Klitzing, K., Ebert, K. and Ploog, K., Semicond. Sci. Technol 10, 1995, 117.
20. c ≪ R, paramagnetic currents are the Halperin currents: Halperin, B.I., Phys. Rev. B25, 1982, 2185, which flow only at the edges in the direction opposite to the one in the bulk Landau orbits. Even edge states may be broad and slowly changing: see Chklovskii, D.B., Shklovskii, B.I, and Glazman, L.I., Phys. Rev. B46, 1992, 4026; and Knott, R., Dietsche, W., von Klitzing, K., Ebert, K. and Ploog, K., Semicond. Sci. Technol 10, 1995, 117.
21. c ≪ R, paramagnetic currents are the Halperin currents: Halperin, B.I., Phys. Rev. B25, 1982, 2185, which flow only at the edges in the direction opposite to the one in the bulk Landau orbits. Even edge states may be broad and slowly changing: see Chklovskii, D.B., Shklovskii, B.I, and Glazman, L.I., Phys. Rev. B46, 1992, 4026; and Knott, R., Dietsche, W., von Klitzing, K., Ebert, K. and Ploog, K., Semicond. Sci. Technol 10, 1995, 117.
22. Lurye, S. and Kazarinov, S., Phys. Rev. B27, 1983, 1386.
23. Azbel', M.Ya. and Entin-Wohlman, O., Phys. Rev. B32, 1985, 562.
24. l is the localization length, and are vanishingly small for very close energies, which, in general, correspond to regions very remote from each other. The study of linear response magnetization, its average and typical values seen in Shapiro, B., Physica A200, 1993, 498; Raveh, A. and Shapiro, B., Europhys. Lett. 19, 1992, 109; Akkermans, E. and Shapiro, B., Europhys. Lett. 11, 1990, 467. See also [18].
25. l is the localization length, and are vanishingly small for very close energies, which, in general, correspond to regions very remote from each other. The study of linear response magnetization, its average and typical values seen in Shapiro, B., Physica A200, 1993, 498; Raveh, A. and Shapiro, B., Europhys. Lett. 19, 1992, 109; Akkermans, E. and Shapiro, B., Europhys. Lett. 11, 1990, 467. See also [18].
26. l is the localization length, and are vanishingly small for very close energies, which, in general, correspond to regions very remote from each other. The study of linear response magnetization, its average and typical values seen in Shapiro, B., Physica A200, 1993, 498; Raveh, A. and Shapiro, B., Europhys. Lett. 19, 1992, 109; Akkermans, E. and Shapiro, B., Europhys. Lett. 11, 1990, 467. See also [18].
27. l is the localization length, and are vanishingly small for very close energies, which, in general, correspond to regions very remote from each other. The study of linear response magnetization, its average and typical values seen in Shapiro, B., Physica A200, 1993, 498; Raveh, A. and Shapiro, B., Europhys. Lett. 19, 1992, 109; Akkermans, E. and Shapiro, B., Europhys. Lett. 11, 1990, 467. See also [18].
28. Altshuler, B.L., Gefen, Y. and Imry, Y., Phys. Rev. Lett. 66, 1991, 88; Vignale, G., Phys. Rev. B50, 1994, 7668; Altland, A. and Gefen, Y., Phys. Rev. Lett. 71, 1993, 3339; 72, 1994, 2973; Kamenev, A. and Gefen, Y., Phys. Rev. Lett. 70, 1993, 1976; Phys. Rev. B49, 1994, 14474 (and refs. therein).
29. Altshuler, B.L., Gefen, Y. and Imry, Y., Phys. Rev. Lett. 66, 1991, 88; Vignale, G., Phys. Rev. B50, 1994, 7668; Altland, A. and Gefen, Y., Phys. Rev. Lett. 71, 1993, 3339; 72, 1994, 2973; Kamenev, A. and Gefen, Y., Phys. Rev. Lett. 70, 1993, 1976; Phys. Rev. B49, 1994, 14474 (and refs. therein).
30. Altshuler, B.L., Gefen, Y. and Imry, Y., Phys. Rev. Lett. 66, 1991, 88; Vignale, G., Phys. Rev. B50, 1994, 7668; Altland, A. and Gefen, Y., Phys. Rev. Lett. 71, 1993, 3339; 72, 1994, 2973; Kamenev, A. and Gefen, Y., Phys. Rev. Lett. 70, 1993, 1976; Phys. Rev. B49, 1994, 14474 (and refs. therein).
31. Altshuler, B.L., Gefen, Y. and Imry, Y., Phys. Rev. Lett. 66, 1991, 88; Vignale, G., Phys. Rev. B50, 1994, 7668; Altland, A. and Gefen, Y., Phys. Rev. Lett. 71, 1993, 3339; 72, 1994, 2973; Kamenev, A. and Gefen, Y., Phys. Rev. Lett. 70, 1993, 1976; Phys. Rev. B49, 1994, 14474 (and refs. therein).
32. Altshuler, B.L., Gefen, Y. and Imry, Y., Phys. Rev. Lett. 66, 1991, 88; Vignale, G., Phys. Rev. B50, 1994, 7668; Altland, A. and Gefen, Y., Phys. Rev. Lett. 71, 1993, 3339; 72, 1994, 2973; Kamenev, A. and Gefen, Y., Phys. Rev. Lett. 70, 1993, 1976; Phys. Rev. B49, 1994, 14474 (and refs. therein).
33. Altshuler, B.L., Gefen, Y. and Imry, Y., Phys. Rev. Lett. 66, 1991, 88; Vignale, G., Phys. Rev. B50, 1994, 7668; Altland, A. and Gefen, Y., Phys. Rev. Lett. 71, 1993, 3339; 72, 1994, 2973; Kamenev, A. and Gefen, Y., Phys. Rev. Lett. 70, 1993, 1976; Phys. Rev. B49, 1994, 14474 (and refs. therein).
34. 2NA/2πMc) (δω/T). This result holds even for a system of interacting electrons - cf. [20]. However, the response changes sign whenever the Fermi level becomes degenerate, and this can happen at very small fields for a system with many transverse channels.
35. Eckern, V. and Schmid, A., Europhys. Lett. 18, 1992, 457; Kopietz, P., Phys. Rev. Lett. 70, 1993, 3123; Weidenmuller, H.A., Physica A200, 1993, 111; Müller-Groeling, A., Weidenmüller, H.A. and Lewenkopf, C.H., Europhys. Lett. 22, 1993, 193; Smith, R.A. and Ambegaokar, V., Europhys. Lett. 20, 1992, 161; Loss, D., Phys. Rev. Lett. 69, 1992, 343; Abraham, M. and Berkovits, R., Phys. Rev. Lett. 70, 1993, 1509; Bouzerar, G., Poilblanc, D. and Montambeaux, G., Phys. Rev. B49, 1994, 8258; Kato, H. and Yoshioka, D., Phys. Rev. B50, 1994, 4943; Deng, W., Liu, Y. and Gong, C., Phys. Rev. B50, 1994, 7655.
36. Eckern, V. and Schmid, A., Europhys. Lett. 18, 1992, 457; Kopietz, P., Phys. Rev. Lett. 70, 1993, 3123; Weidenmuller, H.A., Physica A200, 1993, 111; Müller-Groeling, A., Weidenmüller, H.A. and Lewenkopf, C.H., Europhys. Lett. 22, 1993, 193; Smith, R.A. and Ambegaokar, V., Europhys. Lett. 20, 1992, 161; Loss, D., Phys. Rev. Lett. 69, 1992, 343; Abraham, M. and Berkovits, R., Phys. Rev. Lett. 70, 1993, 1509; Bouzerar, G., Poilblanc, D. and Montambeaux, G., Phys. Rev. B49, 1994, 8258; Kato, H. and Yoshioka, D., Phys. Rev. B50, 1994, 4943; Deng, W., Liu, Y. and Gong, C., Phys. Rev. B50, 1994, 7655.
37. Eckern, V. and Schmid, A., Europhys. Lett. 18, 1992, 457; Kopietz, P., Phys. Rev. Lett. 70, 1993, 3123; Weidenmuller, H.A., Physica A200, 1993, 111; Müller-Groeling, A., Weidenmüller, H.A. and Lewenkopf, C.H., Europhys. Lett. 22, 1993, 193; Smith, R.A. and Ambegaokar, V., Europhys. Lett. 20, 1992, 161; Loss, D., Phys. Rev. Lett. 69, 1992, 343; Abraham, M. and Berkovits, R., Phys. Rev. Lett. 70, 1993, 1509; Bouzerar, G., Poilblanc, D. and Montambeaux, G., Phys. Rev. B49, 1994, 8258; Kato, H. and Yoshioka, D., Phys. Rev. B50, 1994, 4943; Deng, W., Liu, Y. and Gong, C., Phys. Rev. B50, 1994, 7655.
38. Eckern, V. and Schmid, A., Europhys. Lett. 18, 1992, 457; Kopietz, P., Phys. Rev. Lett. 70, 1993, 3123; Weidenmuller, H.A., Physica A200, 1993, 111; Müller-Groeling, A., Weidenmüller, H.A. and Lewenkopf, C.H., Europhys. Lett. 22, 1993, 193; Smith, R.A. and Ambegaokar, V., Europhys. Lett. 20, 1992, 161; Loss, D., Phys. Rev. Lett. 69, 1992, 343; Abraham, M. and Berkovits, R., Phys. Rev. Lett. 70, 1993, 1509; Bouzerar, G., Poilblanc, D. and Montambeaux, G., Phys. Rev. B49, 1994, 8258; Kato, H. and Yoshioka, D., Phys. Rev. B50, 1994, 4943; Deng, W., Liu, Y. and Gong, C., Phys. Rev. B50, 1994, 7655.
39. Eckern, V. and Schmid, A., Europhys. Lett. 18, 1992, 457; Kopietz, P., Phys. Rev. Lett. 70, 1993, 3123; Weidenmuller, H.A., Physica A200, 1993, 111; Müller-Groeling, A., Weidenmüller, H.A. and Lewenkopf, C.H., Europhys. Lett. 22, 1993, 193; Smith, R.A. and Ambegaokar, V., Europhys. Lett. 20, 1992, 161; Loss, D., Phys. Rev. Lett. 69, 1992, 343; Abraham, M. and Berkovits, R., Phys. Rev. Lett. 70, 1993, 1509; Bouzerar, G., Poilblanc, D. and Montambeaux, G., Phys. Rev. B49, 1994, 8258; Kato, H. and Yoshioka, D., Phys. Rev. B50, 1994, 4943; Deng, W., Liu, Y. and Gong, C., Phys. Rev. B50, 1994, 7655.
40. Eckern, V. and Schmid, A., Europhys. Lett. 18, 1992, 457; Kopietz, P., Phys. Rev. Lett. 70, 1993, 3123; Weidenmuller, H.A., Physica A200, 1993, 111; Müller-Groeling, A., Weidenmüller, H.A. and Lewenkopf, C.H., Europhys. Lett. 22, 1993, 193; Smith, R.A. and Ambegaokar, V., Europhys. Lett. 20, 1992, 161; Loss, D., Phys. Rev. Lett. 69, 1992, 343; Abraham, M. and Berkovits, R., Phys. Rev. Lett. 70, 1993, 1509; Bouzerar, G., Poilblanc, D. and Montambeaux, G., Phys. Rev. B49, 1994, 8258; Kato, H. and Yoshioka, D., Phys. Rev. B50, 1994, 4943; Deng, W., Liu, Y. and Gong, C., Phys. Rev. B50, 1994, 7655.
41. Eckern, V. and Schmid, A., Europhys. Lett. 18, 1992, 457; Kopietz, P., Phys. Rev. Lett. 70, 1993, 3123; Weidenmuller, H.A., Physica A200, 1993, 111; Müller-Groeling, A., Weidenmüller, H.A. and Lewenkopf, C.H., Europhys. Lett. 22, 1993, 193; Smith, R.A. and Ambegaokar, V., Europhys. Lett. 20, 1992, 161; Loss, D., Phys. Rev. Lett. 69, 1992, 343; Abraham, M. and Berkovits, R., Phys. Rev. Lett. 70, 1993, 1509; Bouzerar, G., Poilblanc, D. and Montambeaux, G., Phys. Rev. B49, 1994, 8258; Kato, H. and Yoshioka, D., Phys. Rev. B50, 1994, 4943; Deng, W., Liu, Y. and Gong, C., Phys. Rev. B50, 1994, 7655.
42. Eckern, V. and Schmid, A., Europhys. Lett. 18, 1992, 457; Kopietz, P., Phys. Rev. Lett. 70, 1993, 3123; Weidenmuller, H.A., Physica A200, 1993, 111; Müller-Groeling, A., Weidenmüller, H.A. and Lewenkopf, C.H., Europhys. Lett. 22, 1993, 193; Smith, R.A. and Ambegaokar, V., Europhys. Lett. 20, 1992, 161; Loss, D., Phys. Rev. Lett. 69, 1992, 343; Abraham, M. and Berkovits, R., Phys. Rev. Lett. 70, 1993, 1509; Bouzerar, G., Poilblanc, D. and Montambeaux, G., Phys. Rev. B49, 1994, 8258; Kato, H. and Yoshioka, D., Phys. Rev. B50, 1994, 4943; Deng, W., Liu, Y. and Gong, C., Phys. Rev. B50, 1994, 7655.
43. Eckern, V. and Schmid, A., Europhys. Lett. 18, 1992, 457; Kopietz, P., Phys. Rev. Lett. 70, 1993, 3123; Weidenmuller, H.A., Physica A200, 1993, 111; Müller-Groeling, A., Weidenmüller, H.A. and Lewenkopf, C.H., Europhys. Lett. 22, 1993, 193; Smith, R.A. and Ambegaokar, V., Europhys. Lett. 20, 1992, 161; Loss, D., Phys. Rev. Lett. 69, 1992, 343; Abraham, M. and Berkovits, R., Phys. Rev. Lett. 70, 1993, 1509; Bouzerar, G., Poilblanc, D. and Montambeaux, G., Phys. Rev. B49, 1994, 8258; Kato, H. and Yoshioka, D., Phys. Rev. B50, 1994, 4943; Deng, W., Liu, Y. and Gong, C., Phys. Rev. B50, 1994, 7655.
44. Eckern, V. and Schmid, A., Europhys. Lett. 18, 1992, 457; Kopietz, P., Phys. Rev. Lett. 70, 1993, 3123; Weidenmuller, H.A., Physica A200, 1993, 111; Müller-Groeling, A., Weidenmüller, H.A. and Lewenkopf, C.H., Europhys. Lett. 22, 1993, 193; Smith, R.A. and Ambegaokar, V., Europhys. Lett. 20, 1992, 161; Loss, D., Phys. Rev. Lett. 69, 1992, 343; Abraham, M. and Berkovits, R., Phys. Rev. Lett. 70, 1993, 1509; Bouzerar, G., Poilblanc, D. and Montambeaux, G., Phys. Rev. B49, 1994, 8258; Kato, H. and Yoshioka, D., Phys. Rev. B50, 1994, 4943; Deng, W., Liu, Y. and Gong, C., Phys. Rev. B50, 1994, 7655.
45. F/ξ. So, at high temperatures equation (15) may be valid in a wider region of magnetic fields.
46. Sinai, I., Topics in Ergodic Theory. Princeton Univ. Press, 1994; and Nakamura, Quantum Chaos, Cambridge; Chirikov, B., Quantum Chaos (Edited by G. Casati), Cambridge Univ. Press, 1995. Note that the fluctuations in the number of states in a given energy interval, which are crucial for dHvA and ABE, may be different for chaotic and random spectrum. In the former case they are logarithmic, in the latter case they are "conventional" square root ones, in these numbers of states.
47. Sinai, I., Topics in Ergodic Theory. Princeton Univ. Press, 1994; and Nakamura, Quantum Chaos, Cambridge; Chirikov, B., Quantum Chaos (Edited by G. Casati), Cambridge Univ. Press, 1995. Note that the fluctuations in the number of states in a given energy interval, which are crucial for dHvA and ABE, may be different for chaotic and random spectrum. In the former case they are logarithmic, in the latter case they are "conventional" square root ones, in these numbers of states.
48. Sinai, I., Topics in Ergodic Theory. Princeton Univ. Press, 1994; and Nakamura, Quantum Chaos, Cambridge; Chirikov, B., Quantum Chaos (Edited by G. Casati), Cambridge Univ. Press, 1995. Note that the fluctuations in the number of states in a given energy interval, which are crucial for dHvA and ABE, may be different for chaotic and random spectrum. In the former case they are logarithmic, in the latter case they are "conventional" square root ones, in these numbers of states.
49. For an example of the level spacing and the Thouless energy effects on the total PC see, e.g. Altland, A. et al., Europhys. Lett. 20, 1992, 155; Ann. Phys. (N.Y.) 219, 1992, 148.
50. For an example of the level spacing and the Thouless energy effects on the total PC see, e.g. Altland, A. et al., Europhys. Lett. 20, 1992, 155; Ann. Phys. (N.Y.) 219, 1992, 148.
51. -1 practically always allows for semiclassical approach.
52. It was presented in an invited talk at the APS 1996 March meeting - see Bull. Am. Phys. Soc. 41, 1996, 730.