Quasiclassical theory of charge transport in disordered interacting electron systems
Corresponding Author
P. Schwab
Institut für Physik, Universität Augsburg, 86135 Augsburg, Germany
Search for more papers by this authorCorresponding Author
R. Raimondi
NEST-INFM and Dipartimento di Fisica, Universita di Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
Search for more papers by this authorCorresponding Author
P. Schwab
Institut für Physik, Universität Augsburg, 86135 Augsburg, Germany
Search for more papers by this authorCorresponding Author
R. Raimondi
NEST-INFM and Dipartimento di Fisica, Universita di Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
Search for more papers by this authorAbstract
We consider the corrections to the Boltzmann theory of electrical transport arising from the Coulomb interaction in disordered conductors. In this article the theory is formulated in terms of quasiclassical Green's functions. We demonstrate that the formalism is equivalent to the conventional diagrammatic technique by deriving the well-known Altshuler-Aronov corrections to the conductivity. Compared to the conventional approach, the quasiclassical theory has the advantage of being closer to the Boltzmann theory, and also allows description of interaction effects in the transport across interfaces, as well as non-equilibrium phenomena in the same theoretical framework. As an example, by applying the Zaitsev boundary conditions which were originally developed for superconductors, we obtain the P(E)-theory of the Coulomb blockade in tunnel junctions. Furthermore we summarize recent results obtained for the non-equilibrium transport in thin films, wires and fully coherent conductors.
References
- [1]P. Drude, Ann. Phys. (Leipzig) 1, 6 (1900); Ann. Phys. (Leipzig) 3, 369 (1900).
- [2]N.W. Ashcroft and N.D. Mermin, Solid State Physics (Saunders College Publishing, 1976).
- [3]R. Landauer, Philos. Mag. 21, 863 (1970).
- [4]E.N. Economou and C.M. Soukoulis, Phys. Rev. Lett. 46, 618 (1981).
- [5]D.S. Fisher and P.A. Lee, Phys. Rev. B 23, 6851 (1981).
- [6]M. Büttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B 31, 6207 (1985).
- [7]M. Büttiker, Phys. Rev. Lett. 57, 1761 (1986).
- [8]H.U. Baranger and A.D. Stone, Phys. Rev. B 40, 8169 (1989).
- [9]Y. Meir and N.S. Wingreen, Phys. Rev. Lett. 86, 2512 (1992).
- [10]B.J. van Wees, H. van Houten, C.W.J. Beenakker, J.G. Williamson, L.P. Kouwenhoven, D. van der Marel, and C.T. Foxon, Phys. Rev. Lett. 60, 848 (1988).
- [11]D.A. Wharam, T.J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J.E.F. Frost, D.G. Hasko, D.C. Peacock, D.A. Ritchie, and G.A.C. Jones, J. Phys. C21, L209 (1988).
- [12]E. Scheer, P. Joyez, D. Esteve, C. Urbina and M.H. Devoret, Phys. Rev. Lett. 78, 3535 (1997). E. Scheer, N. Agrait, J.C. Cuevas, A. Levy Yeyati, B. Ludoph, A. Martin-Rodero, G.R. Bollinger, J.M. van Ruitenbeek, and C. Urbina, Nature 394, 154 (1998).
- [13]B.L. Altshuler and A.G. Aronov, in: Electron-Electron Interactions in Disordered Systems, edited by M. Pollak and A. L. Efros (North-Holland, Amsterdam, 1985), p. 1.
- [14]G. Bergmann, Phys. Rep. 107, 1 (1984).
- [15]P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
- [16]S. Chakravarty and A. Schmid, Phys. Rep. 140, 193 (1986).
- [17]B. Kramer and A. MacKinnon, Rep. Prog. Phys. 56, 1469 (1993).
- [18]D. Belitz and T.R. Kirkpatrick, Rev. Mod. Phys. 66, 261 (1994).
- [19]B.L. Altshuler, Sov. Phys.-JETP 48, 670 (1978).
- [20]B.L. Altshuler and A.G. Aronov, Solid State Commun. 36, 115 (1979).
- [21]B.L. Altshuler, A.G. Aronov, and P.A. Lee, Phys. Rev. Lett. 44, 1288 (1980).
- [22]H. Fukuyama, J. Phys. Soc. Jpn. 48, 2169 (1980).
- [23]D.S. Golubev and A.D. Zaikin, Phys. Rev. Lett. 81, 1074 (1998); Phys. Rev. B 59, 9195 (1999); Phys. Rev. B 62, 14061 (2000). D.S. Golubev, A.D. Zaikin, and G. Schön, J. Low Temp. Phys. 126, 1355 (2002).
- [24]B.L. Altshuler, M.E. Gershenson, and I.L. Aleiner, Physica E 3, 58 (1998).
- [25]I.L. Aleiner, B.L. Altshuler, and M.E. Gershenson, Phys. Rev. Lett. 82, 3190 (1999).
- [26]I.L. Aleiner, B.L. Altshuler, and M.G. Vavilov, J. Low Temp. Phys. 126, 1377 (2002).
- [27]R. Raimondi, P. Schwab, and C. Castellani, Phys. Rev. B 60, 5818 (1999).
- [28]D. Cohen and Y. Imry, Phys. Rev. B 59, 11143 (1999).
- [29]T.R. Kirkpatrick and D. Belitz, Phys. Rev. B 65, 195123 (2002).
- [30]A. Völker and P. Kopietz, Phys. Rev. B 65, 045112 (2002).
- [31]D.N. Langeberg and A.I. Larkin (eds.), Non-equilibrium Superconductivity (North-Holland, Amsterdam, 1986).
- [32]C.J. Lambert and R. Raimondi, J. Phys., Condens. Matter. 10, 901 (1998).
- [33]L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys.-JETP 20, 1018 (1964)].
- [34]J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986).
- [35]U. Eckern and A. Schmid, J. Low Temp. Phys. 45, 137 (1981).
- [36]G. Eilenberger, Z. Phys. 214, 195 (1968).
- [37]A.L. Shelankov, J. Low Temp. Phys. 60, 29 (1985).
- [38]R.E. Prange and L.P. Kadanoff, Phys. Rev. A 134, 566 (1964).
- [39]M.Y. Kuprianov and V.F. Lukichev, Zh. Eksp. Teor. Fiz. 94, 139 (1988); Sov. Phys.-JETP 67, 1163 (1988).
- [40]B. Laikhtman and S. Luryi, Phys. Rev. B 49, 17177 (1994).
- [41]C.J. Lambert, R. Raimondi, V. Sweeney, and A.F. Volkov, Phys. Rev. B 55, 6015 (1997).
- [42]L.P. Gorkov, A.I. Larkin, and D.E. Khmelnitskii, Pis'ma Zh. Eksp. Teor. Fiz. 30, 248 (1979) [JETP Lett. 30, 228 (1979)].
- [43]E. Abrahams, P.W. Anderson, D.C. Licciardello, and T.V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
- [44]S. Hershfield and V. Ambegaokar, Phys. Rev. B 34, 2147 (1986).
- [45]A.A. Abrikosov, L.P. Gorkov, and I.Y. Dzyaloshinskii, Quantum Field Theoretical Methods in Statistical Physics (Pergamon Press, Oxford, 1965).
- [46]A.M. Finkelstein, Zh. Eksp. Teor. Fiz. 84, 168 (1983) [Sov. Phys.-JETP 57, 97 (1983)].
- [47]C. Castellani, C. Di Castro, P.A. Lee, and M. Ma, Phys. Rev. B 30, 527 (1984).
- [48]G. Bergmann, Phys. Rev. B 35, 4205 (1987).
- [49]A.M. Rudin, I.L. Aleiner, and L.I. Glazman, Phys. Rev. B 55, 9322 (1997).
- [50]G. Strinati, C. Castellani, C. Di Castro, and G. Kotliar, Phys. Rev. B 44, 6078 (1991).
- [51]A. Kamenev and A. Andreev, Phys. Rev. B 60, 2218 (1999).
- [52]C. Chamon, A.W.W. Ludwig, and C. Nayak, Phys. Rev. B 60, 2239 (1999).
- [53]G. Zala, B.N. Narozhny, and I.L. Aleiner, Phys. Rev. B 64, 201201 (2001); Phys. Rev. B 64, 214204 (2001); Phys. Rev. B 65, 020201 (2002).
- [54]P. Schwab and R. Raimondi, Eur. Phys. J. B 24, 525 (2001).
- [55]M. Leadbeater, R. Raimondi, P. Schwab, and C. Castellani, Eur. Phys. J. B 15, 277 (2000).
- [56]K.E. Nagaev, Phys. Lett. A 189, 134 (1994).
- [57]G.-L. Ingold and Yu.V. Nazarov, in: Single Charge Tunneling, NATO ASI Series B, Vol. 294, edited by H. Grabert and M. Devoret (Plenum Press, New York, 1992), Chap. 2.
- [58]G.J. Dolan and D.D. Osheroff, Phys. Rev. Lett. 43, 721 (1979).
- [59]P.W. Anderson, E. Abrahams, and T.V. Ramakrishnan, Phys. Rev. Lett. 43, 718 (1979).
- [60]T. Tsuzuki, Physica B & C 107, 679 (1981).
- [61]M. Kaveh, M.J. Uren, R.A. Davies and M. Pepper, J. Phys. C 14, 413 (1981).
- [62]B.L. Altshuler and A.G. Aronov, Pis'ma Zh. Eksp. Teor. Fiz. 30, 514 (1979) [JETP Lett. 30, 482 (1979)].
- [63]G. Bergmann, Z. Phys. B 49, 133 (1982).
- [64]G. Bergmann, W. Wei, Y. Zou, and R.M. Mueller, Phys. Rev. B 41, 7386 (1990).
- [65]B.L. Altshuler, A.G. Aronov, and D.E. Khmelnitskii, J. Phys. C 15, 7367 (1982).
- [66]H. Fukuyama and E. Abrahams, Phys. Rev. B 27, 5976 (1983).
- [67]W. Eiler, J. Low Temp. Phys. 56, 481 (1984).
- [68]A. Stern, Y. Aharonov, and Y. Imry, Phys. Rev. A 41, 3436 (1990).
- [69]J.J. Lin and L.B. Bird, J. Phys.: Condens. Matter 14, R501 (2002).
- [70]D. Hoadley, P. McConville, and N.O. Birge, Phys. Rev. B 60, 5617 (1999).
- [71]I.L. Aleiner and Ya.M. Blanter, Phys. Rev. B 65, 115317 (2002).
- [72]P. Mohanty, E.M.Q. Jariwala, and R.A. Webb, Phys. Rev. Lett. 78, 3366 (1997).
- [73]P. Mohanty, E.M.Q. Jariwala, and R.A. Webb, Fortschr. Phys. 46, 779 (1998).
- [74]A. Zawadowski, J. von Delft, and D.C. Ralph, Phys. Rev. Lett. 83, 2632 (1999).
- [75]Y. Imry, H. Fukuyama, and P. Schwab, Europhys. Lett. 47, 608 (1999).
- [76]A.B. Gougam, F. Pierre, H. Pothier, D. Esteve, and N.O. Birge, J. Low Temp. Phys. 118, 447 (2000).
- [77]T.W. Jing, N.P. Ong, T.V. Ramakrishnan, J.M. Tarascon, and K. Remschnig, Phys. Rev. Lett. 67, 761 (1991).
- [78]S.J. Hagen, X.Q. Xu, W. Jiang, J.L. Peng, Z.Y. Li, and R.L. Greene, Phys. Rev. B 45, 515 (1992).
- [79]P. Fournier, J. Higgins, H. Balci, E. Maiser, C.J. Lobb, and R.L. Greene, Phys. Rev. B 62, R11993 (2000).
- [80]C. Castellani, C. Di Castro, G. Kotliar, and P.A. Lee, Phys. Rev. Lett. 56, 1179 (1986).
- [81]A. Schmid, Z. Phys. 271, 251 (1974).
- [82]E. Abrahams, P.W. Anderson, P.A. Lee, and T.V. Ramakrishnan, Phys. Rev. B 24, 6783 (1981).
- [83]B.L. Altshuler, D. Khmelnitzkii, A.I. Larkin, and P.A. Lee, Phys. Rev. B 22, 5141 (1980).
- [84]Ya.M. Blanter, Phys. Rev. B 54, 12807 (1996).
- [85]S. Levitov and S.V. Shytov, Pis'ma Zh. Eksp. Teor. Fiz. 66, 200 (1997) [JETP Lett. 66, 214 (1997)].
- [86]P. Kopietz, Phys. Rev. Lett. 81, 2120 (1998).
- [87]V. Ambegaokar, U. Eckern, and G. Schön, Phys. Rev. Lett. 48, 1745 (1982).
- [88]E. Ben-Jacob, E. Mottola, and G. Schön, Phys. Rev. B 51, 2064 (1983).
- [89]T.-L. Ho, Phys. Rev. Lett. 51, 2060 (1983).
- [90]G. Schön and A.D. Zaikin, Phys. Rep. 198, 237 (1990).
- [91]H. Grabert and M. Devoret (eds.), Single Charge Tunneling, NATO ASI Series B, Vol. 294, (Plenum Press, New York, 1992).
- [92]Yu.V. Nazarov, Zh. Eksp. Teor. Fiz. 95, 975 (1989) [JETP Lett. 66, 214 (1989)].
- [93]Yu.V. Nazarov, Fiz. Tverd. Tela 31, 188 (1989) [Sov. Phys.-Solid State 31, 1581 (1989)].
- [94]Yu.V. Nazarov, Pis'ma Zh. Eksp. Teor. Fiz. 49, 105 (1989) [JETP Lett. 49, 126 (1989)].
- [95]A. Kamenev and Y. Gefen, Phys. Rev. B 965428 (1996).
- [96]E.V. Sukhorukov and A.V. Khaetskii, Phys. Rev. B 56, 1456 (1997).
- [97]G.M. Minkov, A.V. Germanenko, S.A. Negachev, O.E. Rut, and E.V. Sukhorukov, Phys. Rev. B 59, 13139 (1999).
- [98]F. Pierre, H. Pothier, P. Joyez, N.O. Birge, D. Esteve, and M.H. Devoret, Phys. Rev. Lett. 86, 1590 (2001).
- [99]I.S. Beloborodov, K.B. Efetov, A. Altland, and F.W.J. Hekking, Phys. Rev. B 63, 115109 (2001).
- [100]J. Rollbühler and H. Grabert, Phys. Rev. Lett. 87, 126804 (2001).
- [101]A.V. Zaitsev, Zh. Eksp. Teor. Fiz. 86, 1742 (1984) [Sov. Phys.-JETP 59, 1015 (1984)].
- [102]A.L. Shelankov, Fiz. Tverd. Tela 26, 1615 (1984) [Sov. Phys.-Solid State 26, 981 (1984)]
- [103]B.L. Altshuler, A.G. Aronov, and A.Yu. Zyuzin, Zh. Eksp. Teor. Fiz. 86, 709 (1984) [Sov. Phys.-JETP 59, 415 (1984)].
- [104]J.T. Masden and N. Giordano, Phys. Rev. Lett. 49, 819 (1982); Phys. Rev. B 27, 6522 (1983); Phys. Rev. B 36, 4197 (1987).
- [105]P. Santhanam, Phys. Rev. Lett. 35, 8737 (1987).
- [106]D.B. Gutman and Y. Gefen, Phys. Rev. B 64, 205317 (2001).
- [107]K.E. Nagaev, Phys. Lett. A 169, 103 (1992).
- [108]K.E. Nagaev, Phys. Rev. B 52, 4740 (1995).
- [109]V.I. Kozub and A.M. Rudin, Phys. Rev. B 52, 7853 (1995).
- [110]Y. Naveh, D.V. Averin, and K.K. Likharev, Phys. Rev. B 58, 15371 (1998); Y. Naveh, in: XVIII Rencontres de Moriond: Quantum Physics at Mesoscopic Scale, edited by D. C. Glattli and M. Sanquer (Editions Frontiérs, France, 1999).
- [111]H. Pothier, S. Gueron, N.O. Birge, D. Esteve, and M.H. Devoret, Phys. Rev. Lett. 79, 3490 (1997).
- [112]P. Schwab and R. Raimondi, Eur. Phys. J. B 30, 5 (2002).
- [113]H.B. Weber, R. Häussler, H. v. Löhneysen, and J. Kroha, Phys. Rev. B 63, 165426 (2001).
- [114]Y.V. Nazarov, Phys. Rev. Lett. 82, 1245 (1999).
- [115]A. Kamenev, Phys. Rev. Lett. 85, 4160 (2000).
- [116]M.V. Feigelman, A. Kamenev, A.I. Larkin, and M.A. Skvortsov, Phys. Rev. B 66, 054502 (2002).
- [117]D. Golubev and A. Zaikin, Phys. Rev. Lett. 86, 4887 (2001); cond-mat/0307161.
- [118]A. Levy Yeyati, A. Martin-Rodero, D. Esteve, and C. Urbina, Phys. Rev. Lett. 87, 046802 (2001).
- [119]L.H. Yu and D. Natelson, Appl. Phys. Lett. 82, 2332 (2003); cond-mat/0307576.
- [120]V.E. Kravtsov and V.I. Yudson, Phys. Rev. Lett. 70, 210 (1993).
- [121]S.A. Vitkalov, G.M. Gusev, Z.D. Kvon, G.I. Leviev, and V.I. Fal'ko, Zh. Eksp. Teor. Fiz. 94, 376 (1988) [Sov. Phys.-JETP 67, 1080 (1988)].
- [122]J. Liu and N. Giordano, Phys. Rev. B 43, 1385 (1991).
- [123]Ya.M. Blanter and M. Büttiker, Phys. Rep. 336, 1 (2000).
- [124]W. Belzig and Yu.V. Nazarov, Phys. Rev. Lett. 87, 067006 (2001); Phys. Rev. Lett. 87, 197006 (2001). Yu.V. Nazarov and D.A. Bagrets, Phys. Rev. Lett. 88, 196801 (2002).
- [125]M.V. Feigelman, A.I. Larkin, and M.A. Skvortsov, Phys. Rev. B 61, 12361 (2000).