Skip to main content

Advertisement

SpringerLink
Log in

Condensation Energy in a Superconductor for All Temperatures

  • Published:
Journal of Low Temperature Physics Aims and scope Submit manuscript

Abstract

An exact analytical expression for the condensation energy \(E_{\text{cond}} \left( T \right)\) of a phonon-driven superconductor for all absolute temperatures \(T\) and for any coupling strength is introduced so as to calculate the Helmholtz free energy difference \(F_{s} \left( T \right) - F_{n} \left( T \right)\) between superconducting \(\left( s \right)\) and normal \(\left( n \right)\) states. This is achieved via a boson–fermion ternary gas theory—called the generalized Bose–Einstein condensation (GBEC) theory—which includes two-hole Cooper pairs, two-electron ones as well as single, free/unbound electrons. The GBEC formalism turns out to be quite useful in dealing with nonzero \(T\) values of \(E_{\text{cond}} \left( T \right)\) and reproduces several well-known experimental results. An expression for the condensation energy per atom is also calculated and applied to aluminum and niobium, and both results are compared with experimental data.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. V.V. Tolmachev, Phys. Lett. A 266, 400 (2000)

    ADS  Google Scholar 

  2. S.K. Adhikari, M. de Llano, F.J. Sevilla, M.A. Solís, J.J. Valencia, Physica C 453, 37 (2007)

    ADS  Google Scholar 

  3. M. Grether, M. de Llano, V.V. Tolmachev, Int. J. Quantum Chem. 112, 3018 (2012)

    Google Scholar 

  4. M. de Llano, V.V. Tolmachev, Phys. A 317, 546 (2003)

    Google Scholar 

  5. M. de Llano, V.V. Tolmachev, Ukr. J. Phys. 55, 79 (2010)

    Google Scholar 

  6. L.N. Cooper, Phys. Rev. 104, 1189 (1956)

    ADS  Google Scholar 

  7. A.P. Drozdov, M.I. Eremets, I.A. Troyan, V. Ksenofontov, S.I. Shylin, Nature 525, 73 (2015)

    ADS  Google Scholar 

  8. A.P. Drozdov, P.P. Kong, V.S. Minkov, S.P. Besedin, M.A. Kuzovnikov, S. Mozaffari, L. Balicas, F.F. Balakirev, D.E. Graf, V.B. Prakapenka, E. Greenberg, D.A. Knyazev, M. Tkacz, M.I. Eremets, Nature 569, 528 (2019)

    ADS  Google Scholar 

  9. H. Liu, I.I. Naumov, R. Hoffmann, N.W. Ashcroft, R.J. Hemley, Proc. Natl. Acad. Sci. 114, 6990 (2017)

    ADS  Google Scholar 

  10. M. Somayazulu, M. Ahart, A.K. Mishra, Z.M. Geballe, M. Baldini, Y. Meng, V.V. Struzhkin, and R.J. Hemley. arXiv: 1808.07695 [Cond Mat] (2018)

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

    ADS  MathSciNet  Google Scholar 

  12. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 2005)

    MATH  Google Scholar 

  13. J.F. Annett, Superconductivity, Superfluids, and Condensates (Oxford University Press, Oxford, 2004)

    Google Scholar 

  14. Y. Wada, Phys. Rev. 135, A1481 (1964)

    ADS  Google Scholar 

  15. J. Bardeen, M. Stephen, Phys. Rev. 136, A1485 (1964)

    ADS  Google Scholar 

  16. G.M. Eliashberg, Sov. Phys. JETP 11, 696 (1960)

    Google Scholar 

  17. G.M. Eliashberg, Sov. Phys. JETP 16, 780 (1963)

    ADS  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  19. J. Ranninger, R. Micnas, S. Robaszkiewicz, Ann. Phys. 13, 455 (1988)

    Google Scholar 

  20. R. Friedberg, T.D. Lee, Phys. Rev. B 40, 6745 (1989)

    ADS  Google Scholar 

  21. Y. Bar-Yam, Phys. Rev. B 43, 359 (1991)

    ADS  Google Scholar 

  22. R. Friedberg, T.D. Lee, H.C. Ren, Phys. Rev. B 45, 10732 (1992)

    ADS  Google Scholar 

  23. J. Ranninger, J.M. Robin, M. Eschrig, Phys. Rev. Lett. 74, 4027 (1995)

    ADS  Google Scholar 

  24. V.B. Geshkenbein, L.B. Ioffe, A.I. Larkin, Phys. Rev. B 55, 3173 (1997)

    ADS  Google Scholar 

  25. T. Domanski, J. Ranninger, Phys. Rev. B 63, 134505 (2001)

    ADS  Google Scholar 

  26. I. Chávez, M. Grether, M. de Llano, J. Supercond. Nov. Magn. 28, 309 (2015)

    Google Scholar 

  27. M. Girardeau, J. Math. Phys. 1, 516 (1960)

    ADS  MathSciNet  Google Scholar 

  28. W.J. Mullin, A.R. Sakhel, J. Low Temp. Phys. 166, 125 (2012)

    ADS  Google Scholar 

  29. M.V.D. Berg, J.T. Lewis, J.V. Pulé, Helv. Phys. Acta 59, 1271 (1986)

    MathSciNet  Google Scholar 

  30. V.A. Zagrebnov, J.B. Bru, Phys. Rep. 350, 291 (2001)

    ADS  MathSciNet  Google Scholar 

  31. T.A. Mamedov, M. de Llano, J. Phys. Soc. Jpn. 79, 044706 (2010)

    ADS  Google Scholar 

  32. T.A. Mamedov, M. de Llano, J. Phys. Soc. Jpn. 80, 074718 (2011)

    ADS  Google Scholar 

  33. T.A. Mamedov, M. de Llano, Philos. Mag. 93, 2896 (2013)

    ADS  Google Scholar 

  34. T.A. Mamedov, M. de Llano, Philos. Mag. 94, 4102 (2014)

    ADS  Google Scholar 

  35. R.M. Carter, M. Casas, J.M. Getino, M. de Llano, A. Puente, H. Rubio, D.M. van der Walt, Phys. Rev. B 52, 16149 (1995)

    ADS  Google Scholar 

  36. M. Casas, J.M. Getino, M. de Llano, A. Puente, R.M. Quick, H. Rubio, D.M. van der Walt, Phys. Rev. B 50, 15945 (1994)

    ADS  Google Scholar 

  37. N.N. Bogoliubov, J. Phys. URSS 11, 23 (1947)

    Google Scholar 

  38. N.N. Bogoliubov, Il Nuovo Cim. 7, 794 (1958)

    ADS  Google Scholar 

  39. A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971)

    Google Scholar 

  40. C.P. Poole, Superconductivity (Academic Press, Amsterdam, 2007)

    Google Scholar 

  41. F.J. Blatt, Modern Physics (McGraw-Hill, New York, 1992)

    Google Scholar 

  42. V.Z. Kresin, S.A. Wolf, Rev. Mod. Phys. 81, 481 (2009)

    ADS  Google Scholar 

  43. M.L. Kulić, Phys. Rep. 338, 1 (2000)

    ADS  Google Scholar 

  44. A.P. Drozdov, V.S. Minkov, S.P. Besedin, P.P. Kong, M.A. Kuzovnikov, D.A. Knyazev, and M.I. Eremets, arXiv: 180807039 [Cond-Mat] (2018)

Download references

Acknowledgements

We thank T.A. Mamedov for bringing Ref. [14] to our attention and acknowledge the support of UNAM-DGAPA-PAPIIT (Mexico) Grant IN102417. FZ acknowledges funding from CONACYT Grant No. 358174/291001.

Author information

Authors and Affiliations

  1. Posgrado en Ciencia e Ingeniería de Materiales, Universidad Nacional Autónoma de México, 04510, Mexico City, Mexico

    J. Ortega, F. Zúñiga & M. de Llano

  2. SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, UK

    F. Zúñiga

Authors
  1. J. Ortega
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Zúñiga
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. de Llano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Zúñiga.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ortega, J., Zúñiga, F. & de Llano, M. Condensation Energy in a Superconductor for All Temperatures. J Low Temp Phys 201, 489–499 (2020). https://doi.org/10.1007/s10909-020-02514-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10909-020-02514-2

Keywords

Search

Navigation