Skip to main content
SpringerLink
Log in

Transport equations for superconductors in the presence of spin interaction

  • Regular Article
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

Quasi-classical theory of superconductivity provides a powerful and yet simple description of the superconductivity phenomenology. In particular, the Eilenberger and Usadel equations provide a neat simplification of the description of the superconducting state in the presence of disorder and electromagnetic interaction. However, the modern aspects of superconductivity require a correct description of the spin interaction as well. Here, we generalize the transport equations of superconductivity in order to take into account space-time dependent electromagnetic and spin interactions on equal footing. Using a gauge-covariant Wigner transformation for the Green-Gor’kov correlation functions, we establish the correspondence between the Dyson-Gor’kov equation and the quasi-classical transport equation in the time-dependent phase-space. We give the expressions for the gauge-covariant current and charge densities (quasi-particle, electric and spin) in the transport formulation. The generalized Eilenberger and Usadel limits of the transport equation are given, too. This study is devoted to the formal derivation of the equations of motion in the electromagnetic plus spin plus particle-hole space. The studies of some specific systems are postponed to future works.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 106, 162 (1957)

    Article  ADS  MathSciNet  Google Scholar 

  2. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108, 1175 (1957)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. J.G. Valatin, Nuovo Cim. 7, 843 (1958)

    Article  MathSciNet  Google Scholar 

  4. N.N. Bogoliubov, Sov. Phys. J. Exp. Theor. Phys. 7, 51 (1958)

    MathSciNet  Google Scholar 

  5. L.P. Gor’kov, Sov. Phys. J. Exp. Theor. Phys. 7, 505 (1958)

    MATH  Google Scholar 

  6. A.A. Abrikosov, L.P. Gor’kov, I.E. Dzyaloshinsky, Methods of Quantum Field Theory in Statistical Physics (Prentice Hall, 1963)

  7. Y. Nambu, Phys. Rev. 117, 648 (1960)

    Article  ADS  MathSciNet  Google Scholar 

  8. Y. Nambu, Phys. Rev. Lett. 4, 380 (1960)

    Article  ADS  Google Scholar 

  9. L.D. Landau, V.L. Ginzburg, Sov. Phys. J. Exp. Theor. Phys. 20, 1064 (1950)

    Google Scholar 

  10. L.P. Gor’kov, Sov. Phys. J. Exp. Theor. Phys. 9, 1364 (1959)

    MATH  MathSciNet  Google Scholar 

  11. A.A. Abrikosov, Sov. Phys. J. Exp. Theor. Phys. 5, 1174 (1957)

    Google Scholar 

  12. B.D. Josephson, Adv. Phys. 14, 419 (1965)

    Article  ADS  Google Scholar 

  13. P.W. Anderson, Phys. Rev. 130, 439 (1963)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. F. Englert, R. Brout, Phys. Rev. Lett. 13, 321 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  15. P. Higgs, Phys. Rev. Lett. 13, 508 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  16. P.-G. De Gennes, Superconductivity of Metals and Alloys (Advanced Book Classics, Westview Press, 1999)

  17. M. Tinkham, Introduction to Superconductivity, 2nd edn. (Dover Publications Inc., 1996)

  18. G. Eilenberger, Z. Phys. 214, 195 (1968)

    Article  ADS  Google Scholar 

  19. A.I. Larkin, Y.N. Ovchinnikov, Sov. Phys. J. Exp. Theor. Phys. 28, 1200 (1969)

    ADS  Google Scholar 

  20. K.D. Usadel, Phys. Rev. Lett. 25, 507 (1970)

    Article  ADS  Google Scholar 

  21. G. Eliashberg, Sov. Phys. J. Exp. Theor. Phys. 34, 668 (1972)

    ADS  Google Scholar 

  22. L.P. Kadanoff, G. Baym, in Quantum Statistical Mechanics, edited by D. Pines (Westview Press, 1962)

  23. R. Prange, L.P. Kadanoff, Phys. Rev. 134, A566 (1964)

    Article  ADS  Google Scholar 

  24. A. Golubov, M. Kupriyanov, E. Il’ichev, Rev. Mod. Phys. 76, 411 (2004)

    Article  ADS  Google Scholar 

  25. R. Casalbuoni, G. Nardulli, Rev. Mod. Phys. 76, 263 (2004)

    Article  ADS  Google Scholar 

  26. A.I. Buzdin, Rev. Mod. Phys. 77, 935 (2005)

    Article  ADS  Google Scholar 

  27. F. Bergeret, A.F. Volkov, K.B. Efetov, Rev. Mod. Phys. 77, 1321 (2005)

    Article  ADS  Google Scholar 

  28. E. Bauer, M. Sigrist, in Non-Centrosymmetric Superconductors, Lecture Notes in Physics (Springer, Berlin, Heidelberg, 2012), Vol. 847

  29. J. Alicea, Rep. Prog. Phys. 75, 076501 (2012)

    Article  ADS  Google Scholar 

  30. C. Beenakker, Ann. Rev. Condens. Matter Phys. 4, 113 (2013)

    Article  ADS  Google Scholar 

  31. M. Franz, Nat. Nanotechnol. 8, 149 (2013)

    Article  ADS  Google Scholar 

  32. N.H. Lindner, E. Berg, G. Refael, A. Stern, Phys. Rev. X 2, 041002 (2012)

    Google Scholar 

  33. D.J. Clarke, J. Alicea, K. Shtengel, Nat. Commun. 4, 1348 (2013)

    Article  ADS  Google Scholar 

  34. D.J. Clarke, J. Alicea, K. Shtengel, arXiv: 1312.6123 (2013)

  35. P.W. Brouwer, M. Duckheim, A. Romito, F. von Oppen, Phys. Rev. B 84, 144526 (2011)

    Article  ADS  Google Scholar 

  36. F. Pientka, G. Kells, A. Romito, P.W. Brouwer, F. von Oppen, Phys. Rev. Lett. 109, 227006 (2012)

    Article  ADS  Google Scholar 

  37. P. Neven, D. Bagrets, A. Altland, New J. Phys. 15, 055019 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  38. N. Schopohl, J. Low Temp. Phys. 41, 409 (1980)

    Article  ADS  Google Scholar 

  39. N. Hayashi, K. Wakabayashi, P. Frigeri, M. Sigrist, Phys. Rev. B 73, 024504 (2006)

    Article  ADS  Google Scholar 

  40. D. Agterberg, R. Kaur, Phys. Rev. B 75, 064511 (2007)

    Article  ADS  Google Scholar 

  41. J.A. Sauls, M. Eschrig, New J. Phys. 11, 075008 (2009)

    Article  ADS  Google Scholar 

  42. V. Stanev, V. Galitski, arXiv: 1402.5337 (2014)

  43. F.S. Bergeret, I.V. Tokatly, Phys. Rev. B 89, 134517 (2014)

    Article  ADS  Google Scholar 

  44. C. Gorini, P. Schwab, R. Raimondi, A.L. Shelankov, Phys. Rev. B 82, 195316 (2010)

    Article  ADS  Google Scholar 

  45. J. Fröhlich, U. Studer, Rev. Mod. Phys. 65, 733 (1993)

    Article  ADS  Google Scholar 

  46. H.-T. Elze, M. Gyulassy, D. Vasak, Phys. Lett. B 177, 402 (1986)

    Article  ADS  Google Scholar 

  47. U. Eckern, A. Schmid, J. Low Temp. Phys. 45, 137 (1981)

    Article  ADS  Google Scholar 

  48. J. Serene, D. Rainer, Phys. Rep. 101, 221 (1983)

    Article  ADS  Google Scholar 

  49. J. Rammer, H. Smith, Rev. Mod. Phys. 58, 323 (1986)

    Article  ADS  Google Scholar 

  50. F.K. Wilhelm, W. Belzig, C. Bruder, G. Schön, A.D. Zaikin, Superlattices and Microstructures 25, 1251 (1999)

    Article  ADS  Google Scholar 

  51. N.B. Kopnin, Theory of Nonequilibrium Superconductivity (Oxford University Press, 2001)

  52. M. Hillery, R.F. O’Connel, M.O. Scully, E.P. Wigner, Phys. Rep. 106, 121 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  53. A. Polkovnikov, Annal. Phys. 325, 1790 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  54. C. Itzykson, J.B. Zuber, Quantum Field Theory (Dover Publications Inc., 2006)

  55. L.D. Landau, E.M. Lifshitz, Quantum Mechanics (Butterworth-Heinemann, 1974)

  56. L.P. Gor’kov, in Superconductivity (in two volumes): Conventional and unconventional superconductors, edited by K.H. Bennemann, J.B. Ketterson (Springer, Berlin, Heidelberg, 2008), Vol. 1, Chap. 5

  57. C.N. Yang, R. Mills, Phys. Rev. 96, 191 (1954)

    Article  ADS  MathSciNet  Google Scholar 

  58. R. Winkler, in Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems, Springer Tracts in Modern Physics (Springer, Berlin, Heidelberg, 2003), Vol. 191

  59. R.L. Stratonovich, Doklady Akademii Nauk SSSR 109, 72 (1956)

    MathSciNet  Google Scholar 

  60. S. Fujita, Introduction to Nonequilibrium Quantum Statistical Mechanics (Saunders, Philadelphia, 1966)

  61. I. Bialynicki-Birula, Acta Phys. Austriaca Suppl. XVI, 111 (1977)

    Google Scholar 

  62. H.-T. Elze, M. Gyulassy, D. Vasak, Nucl. Phys. B 276, 706 (1986)

    Article  ADS  Google Scholar 

  63. O. Serimaa, J. Javanainen, S. Varró, Phys. Rev. A 33, 2913 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  64. D. Vasak, M. Gyulassy, H.-T. Elze, Annal. Phys. 173, 462 (1987)

    Article  ADS  Google Scholar 

  65. M. Levanda, V. Fleurov, J. Phys.: Condens. Matter 6, 7889 (1994)

    ADS  Google Scholar 

  66. M. Levanda, V. Fleurov, Annal. Phys. 292, 199 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  67. J. Javanainen, S. Varró, O. Serimaa, Phys. Rev. A 35, 2791 (1987)

    Article  ADS  Google Scholar 

  68. J. Winter, J. Phys. Colloques 45, C6.53 (1984)

    Article  Google Scholar 

  69. H.-T. Elze, U. Heinz, Phys. Rep. 183, 81 (1989)

    Article  ADS  Google Scholar 

  70. M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, 1995)

  71. A. Aronov, Y. Gal’perin, V. Gurevich, V. Kozub, Adv. Phys. 30, 539 (1981)

    Article  ADS  Google Scholar 

  72. T. Kita, Phys. Rev. B 64, 054503 (2001)

    Article  ADS  Google Scholar 

  73. D.N. Langenberg, A.I. Larkin, Nonequilibrium superconductivity (North-Holland Pubs., 1986)

  74. B.L. Altshuler, Sov. Phys. J. Exp. Theor. Phys. 48, 670 (1979)

    ADS  Google Scholar 

  75. D. Langreth, Phys. Rev. 148, 707 (1966)

    Article  ADS  Google Scholar 

  76. V.P. Mineev, K. Samokhin, Introduction to Unconventionnal Superconductivity (Gordon and Breach Science Publishers, 1998)

  77. K. Tanaka, M. Eschrig, Supercond. Sci. Technol. 22, 014001 (2009)

    Article  ADS  Google Scholar 

  78. S. Coleman, Aspects of Symmetry: Selected Erice Lectures (Cambridge University Press, 1988)

  79. G. Sarma, J. Phys. Chem. Sol. 24, 1029 (1963)

    Article  ADS  Google Scholar 

  80. J. Zinn-Justin, Quantum field theory and critical phenomena, 4th edn. (Oxford University Press, 2002)

  81. M. Eschrig, J.A. Sauls, New J. Phys. 11, 075009 (2009)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Quantum Information, RWTH Aachen University, 52056, Aachen, Germany

    François Konschelle

Authors
  1. François Konschelle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to François Konschelle.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Konschelle, F. Transport equations for superconductors in the presence of spin interaction. Eur. Phys. J. B 87, 119 (2014). https://doi.org/10.1140/epjb/e2014-50143-0

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjb/e2014-50143-0

Keywords

Search

Navigation