- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Thermodynamic Evidence for Pressure-Driven Evolution towards Weak-Coupling Superconductivity in Pb

Rustem Khasanov*

  • *Contact author: rustem.khasanov@psi.ch
PDF

Phys. Rev. Lett. 137, 026002 – Published 7 July, 2026

DOI: https://doi.org/10.1103/xqxt-fmh8

Abstract

The thermodynamic critical field Bc provides direct access to the superconducting condensation energy, yet its pressure dependence has been studied much less extensively than that of the transition temperature. Here, muon-spin-rotation relaxation measurements of the thermodynamic critical field Bc of elemental Pb under hydrostatic pressure up to ≃2.3  GPa are reported. From the magnetic-field distribution in the intermediate state, Bc(T) is determined and Bc(0) is extracted at different pressures, thereby providing direct thermodynamic access to the pressure evolution of the condensation-energy scale. It is shown that, within the experimentally accessible pressure range, the pressure dependence of Bc(0) follows that of the superconducting gap Δ(0) more closely than that of the transition temperature Tc. The measured pressure derivative of the coupling ratio α=Δ(0)/(kBTc) is furthermore found to agree with the difference between the pressure derivatives of Bc(0) and Tc, providing a direct thermodynamic confirmation of the relation between these quantities. Together, these results show that pressure suppresses strong-coupling effects in Pb. When viewed together with previously reported high-pressure data for Bc and Tc, the present results indicate that this tendency continues toward a regime in which α becomes nearly pressure independent.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (49)

  1. B. Lorenz and C. W. Chu, High pressure effects on superconductivity, in Frontiers in Superconducting Materials, edited by A. V. Narlikar (Springer, Berlin, Heidelberg, 2005), pp. 459–497.
  2. J. J. Hamlin, Superconductivity in the metallic elements at high pressures, Physica (Amsterdam) 514C, 59 (2015).
  3. J. S. Schilling, High-pressure effects, in Handbook of High-Temperature Superconductivity, edited by J. R. Schrieffer and J. S. Brooks (Springer, New York, 2007).
  4. J. S. Schilling and J. J. Hamlin, Recent studies in superconductivity at extreme pressures, J. Phys. Conf. Ser. 121, 052006 (2008).
  5. W. L. McMillan, Transition temperature of strong-coupled superconductors, Phys. Rev. 167, 331 (1968).
  6. P. B. Allen and R. C. Dynes, Transition temperature of strong-coupled superconductors reanalyzed, Phys. Rev. B 12, 905 (1975).
  7. J. P. Carbotte, Properties of boson-exchange superconductors, Rev. Mod. Phys. 62, 1027 (1990).
  8. G. M. Eliashberg, Interactions between electrons and lattice vibrations in a superconductor, Sov. Phys. JETP 11, 696 (1960), https://www.jetp.ras.ru/cgi-bin/e/index/e/11/3/p696?a=list&utm_source=chatgpt.com.
  9. M. Tinkham, Introduction to Superconductivity, 2nd ed. (McGraw-Hill, New York, 1996).
  10. P. G. de Gennes, Superconductivity of Metals and Alloys (CRC Press, Boca Raton, 1999).
  11. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Microscopic theory of superconductivity, Phys. Rev. 106, 162 (1957).
  12. J. R. Schrieffer, Theory of Superconductivity (Addison-Wesley, Reading, Massachusetts, 1964).
  13. I. V. Berman, N. B. Brandt, and N. I. Ginzburg, Investigation of the effect of pressures up to 30 katm on the critical field of tin and indium at temperatures of 0.1–4 K, Sov. Phys. JETP 26, 86 (1968), https://jetp.ras.ru/cgi-bin/e/index/e/26/1/p86?a=list&utm_source=chatgpt.com.
  14. See Supplemental Material at http://link.aps.org/supplemental/10.1103/xqxt-fmh8 for contains details of the sample preparation, the TF-μSR experiments, the α-model description for strong-coupling superconductors, and an estimate of the pressure effect on the Sommerfeld coefficient γe in elemental Pb, which includes Refs. [15–27].
  15. R. Khasanov, Z. Guguchia, A. Maisuradze, D. Andreica, M. Elender, A. Raselli, Z. Shermadini, T. Goko, F. Knecht, E. Morenzoni, and A. Amato, High pressure research using muons at the Paul Scherrer Institute, High Press. Res. 36, 140 (2016).
  16. R. Khasanov, Perspective on muon-spin rotation/relaxation under hydrostatic pressure, J. Appl. Phys. 132, 190903 (2022).
  17. R. Prozorov and V. G. Kogan, Effective demagnetizing factors of diamagnetic samples of various shapes, Phys. Rev. Appl. 10, 014030 (2018).
  18. A. Suter and B. Wojek, Musrfit: A free platform-independent framework for μSR data analysis, Phys. Procedia 30, 69 (2012).
  19. A. Yaouanc and P. Dalmas de Réotier, Muon Spin Rotation, Relaxation and Resonance: Applications to Condensed Matter (Oxford University Press, Oxford, 2011).
  20. A. Amato and E. Morenzoni, Introduction to muon spin spectroscopy, in Lecture Notes in Physics (Springer Cham, 2024).
  21. G. Grimvall, Properties of boson-exchange superconductors, Phys. Kondens. Mater. 9, 283 (1969).
  22. F. Giustino, Electron-phonon interactions from first principles, Rev. Mod. Phys. 89, 015003 (2017).
  23. B. Mühlschlegel, Die thermodynamischen funktionen des supraleiters, Z. Phys. 155, 313 (1959).
  24. R. Khasanov, R. Gupta, D. Das, A. Leithe-Jasper, and E. Svanidze, Single-gap versus two-gap scenario: Specific heat and thermodynamic critical field of the noncentrosymmetric superconductor BeAu, Phys. Rev. B 102, 014514 (2020).
  25. T. Strässle, S. Klotz, K. Kunc, V. Pomjakushin, and J. S. White, Equation of state of lead from high-pressure neutron diffraction up to 8.9 GPa and its implication for the NaCl pressure scale, Phys. Rev. B 90, 014101 (2014).
  26. D. L. Waldorf and G. A. Alers, Low-temperature elastic moduli of lead, J. Appl. Phys. 33, 3266 (1962).
  27. S. K. Xuan, Y. Wang, and Y. Li, The superconducting properties of elemental Pb under pressure, Condens. Matter 10, 49 (2025).
  28. H. Padamsee, J. E. Neighbor, and C. A. Shiffman, Quasiparticle phenomenology for thermodynamics of strong-coupling superconductors, J. Low Temp. Phys. 12, 387 (1973).
  29. D. C. Johnston, Elaboration of the α− model derived from the BCS theory of superconductivity, Supercond. Sci. Technol. 26, 115011 (2013).
  30. N. B. Brandt, I. V. Berman, and Yu. P. Kurkin, Investigation of the critical field curves of lead at pressures up to 130 kbar and temperatures down to 0.1 K, Zh. Eksp. Teor. Fiz. 42, 869 (1975) [Sov. Phys. JETP69, 1710 (1975)], https://jetp.ras.ru/cgi-bin/e/index/e/42/5/p869?a=list&utm_source=chatgpt.com.
  31. C. P. Poole, Jr., H. A. Farach, R. J. Creswick, and R. Prozorov, Superconductivity, 3rd ed. (Academic Press, Amsterdam, 2014).
  32. C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, New York, 1996).
  33. R. Prozorov, Equilibrium topology of the intermediate state in type-I superconductors of different shapes, Phys. Rev. Lett. 98, 257001 (2007).
  34. R. Prozorov, A. F. Fidler, J. R. Hoberg, and P. C. Canfield, Suprafroth in type-I superconductors, Nat. Phys. 4, 327 (2008).
  35. R. Khasanov, M. M. Radonjić, H. Luetkens, E. Morenzoni, G. Simutis, S. Schönecker, W. H. Appelt, A. Östlin, L. Chioncel, and A. Amato, Superconducting nature of the Bi-II phase of elemental bismuth, Phys. Rev. B 99, 174506 (2019).
  36. R. Karl, F. Burri, A. Amato, M. Donegà, S. Gvasaliya, H. Luetkens, E. Morenzoni, and R. Khasanov, Muon spin rotation study of type-I superconductivity: Elemental β−Sn, Phys. Rev. B 99, 184515 (2019).
  37. R. Khasanov, H. Luetkens, A. Amato, and E. Morenzoni, Structural phases of elemental Ga: Universal relations in conventional superconductors, Phys. Rev. B 101, 054504 (2020).
  38. R. Khasanov, R. Gupta, D. Das, A. Amon, A. Leithe-Jasper, and E. Svanidze, Multiple-gap response of type-I noncentrosymmetric BeAu superconductor, Phys. Rev. Res. 2, 023142 (2020).
  39. R. Khasanov and G. A. Ummarino, Pressure weakens coupling strength in In and Sn elemental superconductors, Phys. Rev. B 110, 214515 (2024).
  40. R. Khasanov, R. Vocaturo, O. Janson, A. Koitzsch, R. Gupta, D. Das, N. P. M. Casati, M. G. Vergniory, J. van den Brink, and E. Svanidze, Influence of pressure on the properties of the multigap type-I superconductor BeAu, Phys. Rev. B 111, 104507 (2025).
  41. V. S. Egorov, G. Solt, C. Baines, D. Herlach, and U. Zimmermann, Superconducting intermediate state of white tin studied by muon-spin-rotation spectroscopy, Phys. Rev. B 64, 024524 (2001).
  42. H. Leng, J.-C. Orain, A. Amato, Y. K. Huang, and A. de Visser, Type-I superconductivity in the Dirac semimetal PdTe2 probed by μSR, Phys. Rev. B 100, 224501 (2019).
  43. M. Gladisch, D. Herlach, H. Metz, H. Orth, G. zu Putlitz, A. Seeger, H. Teichler, W. Wahl, and W. Wigand, Muon spin rotation in superconductors, Hyperfine Interact. 6, 109 (1979).
  44. V. G. Grebinnik, I. I. Gurevich, V. A. Zhukov, A. I. Klimov, L. A. Levina, V. N. Maiorov, A. P. Manych, E. V. Mel’nikov, B. A. Nikol’skii, A. V. Pirogov, A. N. Ponomarev, V. S. Roganov, V. I. Selivanov, and V. A. Suetin, Investigation of superconductors by the muon technique, Zh. Eksp. Teor. Fiz. 79, 518 (1980) [Sov. Phys. JETP52, 261 (1980)], https://jetp.ras.ru/cgi-bin/e/index/e/52/2/p261?a=list&utm_source=chatgpt.com.
  45. J. Beare, M. Nugent, M. N. Wilson, Y. Cai, T. J. S. Munsie, A. Amon, A. Leithe-Jasper, Z. Gong, S. L. Guo, Z. Guguchia, Y. Grin, Y. J. Uemura, E. Svanidze, and G. M. Luke, μSR and magnetometry study of the type-I superconductor BeAu, Phys. Rev. B 99, 134510 (2019).
  46. V. Kozhevnikov, A. Suter, T. Prokscha, and C. Van Haesendonck, Experimental study of the magnetic field distribution and shape of domains near the surface of a type-I superconductor in the intermediate state, J. Supercond. Novel Magn. 33, 3361 (2020).
  47. R. Khasanov, D. Das, D. Jakub Gawryluk, R. Gupta, and C. Mielke, Isotropic single-gap superconductivity of elemental Pb, Phys. Rev. B 104, L100508 (2021).
  48. W. L. McMillan and J. M. Rowell, Lead phonon spectrum calculated from superconducting density of states, Phys. Rev. Lett. 14, 108 (1965).
  49. A. Chainani, T. Yokoya, T. Kiss, and S. Shin, Photoemission spectroscopy of the strong-coupling superconducting transitions in lead and niobium, Phys. Rev. Lett. 85, 1966 (2000).

Outline

Information

Phys. Rev. Lett. 137, 026002– Published 7 July, 2026
  • Received 23 March 2026
  • Accepted 10 June 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation