Journal of Superconductivity

, Volume 16, Issue 6, pp 913–917 | Cite as

Quantum Phase in Nanoscopic Superconductors

  • Z. Gedik


Using the pseudospin representation and the SU(2) phase operators we introduce a complex parameter to characterize both infinite and finite superconducting systems. While in the bulk limit the parameter becomes identical to the conventional order parameter, in the nanoscopic limit its modulus reduces to the number parity effect parameter and its phase takes discrete values. We evaluate the Josephson coupling energy and show that in bulk superconductor it reproduces the conventional expression and in the nanoscopic limit it leads to quantized Josephson effect. Finally, we study the phase flow or dual Josephson effect in a superconductor with fixed number of electrons.

mesoscopic superconductivity order parameter Josephson effect quantum phase 


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  1. 1.
    M. T. Tuominen,J. M. Hergenrother,T. S. Tighe,andM. Tinkham,Phys. Rev. Lett. 69,1997(1992).Google Scholar
  2. 2.
    P. Lafarge, P. Joyez, D. Esteve, C. Urbina, and M. H. Devoret, Phys. Rev. Lett. 70, 994(1993).Google Scholar
  3. 3.
    D. C. Ralph, C. T. Black, and M. Tinkham, Phys. Rev. Lett. 74,3241(1995);76, 688 (1996); 78, 4087 (1997).Google Scholar
  4. 4.
    J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175(1957).Google Scholar
  5. 5.
    N. N. Bogolubov, Sov. Phys. JETP 7, 41(1958).Google Scholar
  6. 6.
    K. A. Matveev and A. I. Larkin, Phys. Rev. Lett. 78, 3749(1997).Google Scholar
  7. 7.
    H. Flocard, in Atomic Clusters and Nanoparticles, Les Houches Lectures Session LXXIII, C. Guet, P. Hobza, F. Spiegelman, and F. David, eds. (Springer, Berlin, 2001), p. 221.Google Scholar
  8. 8.
    P. W. Anderson, Phys. Rev. 110, 985(1958).Google Scholar
  9. 9.
    N. N. Bogolubov, Sov. Phys. JETP 34, 73(1958).Google Scholar
  10. 10.
    A. Vourdas, Phys. Rev. A 41, 1653(1990).Google Scholar
  11. 11.
    J. von Delft, A. D. Zaikin, D. S. Golubev, and W. Tichy, Phys. Rev. Lett. 77, 3189(1996).Google Scholar
  12. 12.
    R. W. Richardson, Phys. Lett. 3, 277(1963).Google Scholar
  13. 13.
    A. Mastellone, G. Falci, and R. Fazio, Phys. Rev. Lett. 80, 4542(1998).Google Scholar
  14. 14.
    S. D. Berger and B. I. Halperin, Phys. Rev. B 58, 5213(1998).Google Scholar
  15. 15.
    F. Braun and J. von Delft, Phys. Rev. Lett. 81, 4712(1998).Google Scholar
  16. 16.
    J. Dukelsky and G. Sierra, Phys. Rev. Lett. 83, 172(1999).Google Scholar
  17. 17.
    I. O. Kulik, H. Boyaci, and Z. Gedik, Physica C 352, 46(2001).Google Scholar
  18. 18.
    B. Jankó, A. Smith, and V. Ambegaokar, Phys. Rev. B 50, 1152(1994).Google Scholar
  19. 19.
    D. S. Golubev and A. D. Zaikin, Phys. Lett. A 195, 380(1994).Google Scholar
  20. 20.
    A. S. Shumovsky, in Modern Nonlinear Optics, Advances in Chemical Physics, 2nd edn., Vol. 119, Part 1,M. W. Evans, ed. (Wiley, New York, 2001), p.395.Google Scholar
  21. 21.
    R. Rossignoli, N. Canosa, and P. Ring, Ann. Phys. (N.Y.) 275, 1(1999).Google Scholar
  22. 22.
    G. Rickayzen, Theory of Superconductivity (Interscience, New York, 1965).Google Scholar
  23. 23.
    For a review, see J. von Delft, Ann. Phys. (Leipzig) 10, 219(2001).Google Scholar
  24. 24.
    G.-S. Tian, L.-H. Tang, and Q.-H. Chen, Europhys. Lett. 50, 361(2000).Google Scholar
  25. 25.
    O. Penrose and L. Onsager, Phys. Rev. 104, 576(1956).Google Scholar
  26. 26.
    C. N. Yang, Rev. Mod. Phys. 34, 694(1962).Google Scholar
  27. 27.
    L. Amico and A. Osterloh, Phys. Rev. Lett. 88, 127003(2002).Google Scholar
  28. 28.
    H.-Q. Zhou, J. Links, R.H. McKenzie, and M.D. Gould, Phys. Rev. B 65, 060502(R), 2002.Google Scholar
  29. 29.
    R. W. Richardson, J. Math. Phys. (N.Y.) 6,1034(1965).Google Scholar
  30. 30.
    R. W. Richardson, Phys. Rev. Lett. 14, 325(1965).Google Scholar
  31. 31.
    P. W. Anderson, Phys. Rev. 112, 1900(1958).Google Scholar
  32. 32.
    W. Thirring, Commun. Math. Phys. 7, 181(1968).Google Scholar
  33. 33.
    M. Wagner, in Unitary Transformations in Solid State Physics (North Holland, Amsterdam, 1986), p. 11.Google Scholar
  34. 34.
    B. D. Josephson, Phys. Rev. Lett. 1, 251(1962).Google Scholar
  35. 35.
    B. D. Josephson, in Superconductivity, R. D. Parks, ed. (Dekker, New York, 1969), p. 423.Google Scholar
  36. 36.
    I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, San Diego, CA, 1980), p. 992.Google Scholar
  37. 37.
    I. O. Kulik and I. K. Yanson,The Josephson Effect in Superconductive Tunneling Junctions, P. Gluck, trans. (Israel Program for Scientific Translations, Jerusalem, Israel, 1972).Google Scholar
  38. 38.
    A. Barone and G. Paterno, Physics and Applications of the Josephson Effect (Wiley, New York, 1982).Google Scholar
  39. 39.
    V. M. K. Bagci, O. Gülseren, T. Yildirim, Z. Gedik, and S. Ciraci, Phys. Rev. B 66, 045409(2002).Google Scholar
  40. 40.
    Z. Gedik, Solid State Commun. 124, 473(2002).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • Z. Gedik
    • 1
  1. 1.Faculty of Engineering and Natural SciencesSabanci UniversityIstanbulTurkey

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