Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

A universal scaling relation in high-temperature superconductors

Abstract

Since the discovery of superconductivity at elevated temperatures in the copper oxide materials1 there has been a considerable effort to find universal trends and correlations amongst physical quantities, as a clue to the origin of the superconductivity. One of the earliest patterns that emerged was the linear scaling of the superfluid density (ρs) with the superconducting transition temperature (Tc), which marks the onset of phase coherence. This is referred to as the Uemura relation2, and it works reasonably well for the underdoped materials. It does not, however, describe optimally doped (where Tc is a maximum) or overdoped materials3. Similarly, an attempt to scale the superfluid density with the d.c. conductivity (σdc) was only partially successful4. Here we report a simple scaling relation (ρsσdcTc, with σdc measured at approximately Tc) that holds for all tested high-Tc materials. It holds regardless of doping level, nature of dopant (electrons versus holes), crystal structure and type of disorder5, and direction (parallel or perpendicular to the copper–oxygen planes).

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Figure 1: Plot of the superfluid density (ρs) versus the product of the d.c. conductivity (σdc) and the superconducting transition temperature (Tc) for a variety of copper oxides and some simple metals.
Figure 2: As Fig. 1 but for copper oxides only, and including data for the poorly conducting c axis.

References

  1. Bednorz, J. G. & Müller, K. A. Possible high Tc superconductivity in the Ba-La-Cu-O system. Z. Phys. B 64, 189–193 (1986)

    Article  ADS  CAS  Google Scholar 

  2. Uemura, Y. J. et al. Universal correlations between Tc and ns/m* (carrier density over effective mass) in high-Tc cuprate superconductors. Phys. Rev. Lett. 62, 2317–2320 (1989)

    Article  ADS  CAS  Google Scholar 

  3. Niedermayer, C. et al. Muon spin rotation study of the correlation between Tc and ns/m* in overdoped Tl2Ba2CuO6+δ . Phys. Rev. Lett. 71, 1764–1767 (1993)

    Article  ADS  CAS  Google Scholar 

  4. Pimenov, A. et al. Universal relationship between the penetration depth and the normal-state conductivity in YBCO. Europhys. Lett. 48, 73–78 (1999)

    Article  ADS  CAS  Google Scholar 

  5. Eisaki, H. et al. Effect of chemical inhomogeneity in bismuth-based copper oxide superconductors. Phys. Rev. B 69, 064512 (2004)

    Article  ADS  Google Scholar 

  6. Basov, D. N. et al. In-plane anisotropy of the penetration depth in YBa2Cu3O7-x and YBa2Cu4O8 superconductors. Phys. Rev. Lett. 74, 598–601 (1995)

    Article  ADS  CAS  Google Scholar 

  7. Homes, C. C. et al. Effect of Ni impurities on the optical properties of YBa2Cu3O6+x . Phys. Rev. B 60, 9782–9792 (1999)

    Article  ADS  CAS  Google Scholar 

  8. Liu, H. L. et al. Doping-induced change of optical properties in underdoped cuprate superconductors. J. Phys. Condens. Matter 11, 239–264 (1999)

    Article  ADS  Google Scholar 

  9. Puchkov, A. V., Timusk, T., Doyle, S. & Herman, A. M. ab-plane optical properties of Tl2Ba2CuO6+δ . Phys. Rev. B 51, 3312–3315 (1995)

    Article  ADS  CAS  Google Scholar 

  10. Homes, C. C., Clayman, B. P., Peng, J. L. & Greene, R. L. Optical properties of Nd1.85Ce0.15CuO4 . Phys. Rev. B 56, 5525–5534 (1997)

    Article  ADS  CAS  Google Scholar 

  11. Singley, E. J., Basov, D. N., Kurahashi, K., Uefuji, T. & Yamada, K. Electron dynamics in Nd1.85Ce0.15CuO4+δ: Evidence for the pseudogap state and unconventional c-axis response. Phys. Rev. B 64, 224503 (2001)

    Article  ADS  Google Scholar 

  12. Startseva, T. et al. Temperature evolution of the pseudogap state in the infrared response of underdoped La2-xSrxCuO4 . Phys. Rev. B 59, 7184–7190 (1999)

    Article  ADS  CAS  Google Scholar 

  13. Pronin, A. V. et al. Direct observation of the superconducting energy gap developing in the conductivity spectra of niobium. Phys. Rev. B 57, 14416–14421 (1998)

    Article  ADS  CAS  Google Scholar 

  14. Klein, O., Nicol, E. J., Holczer, K. & Grüner, G. Conductivity coherence factors in the conventional superconductors Nb and Pb. Phys. Rev. B 50, 6307–6316 (1994)

    Article  ADS  CAS  Google Scholar 

  15. Ando, Y. et al. Metallic in-plane and divergent out-of-plane resistivity of a high-Tc cuprate in the zero temperature limit. Phys. Rev. Lett. 77, 2065–2068 (1996)

    Article  ADS  CAS  Google Scholar 

  16. Dordevic, S. V. et al. Global trends in the interplane penetration depth of layered superconductors. Phys. Rev. B 65, 134511 (2002)

    Article  ADS  Google Scholar 

  17. Basov, D. N., Timusk, T., Dabrowski, B. & Jorgensen, J. D. c-axis response of YBa2Cu4O8: A pseudogap and possibility of Josephson coupling of CuO2 planes. Phys. Rev. B 50, 3511–3514 (1994)

    Article  ADS  CAS  Google Scholar 

  18. Homes, C. C., Timusk, T., Bonn, D. A., Liang, R. & Hardy, W. N. Optical properties along the c axis of YBa2Cu3O6+x, for x = 0.50 → 0.95: Evolution of the pseudogap. Physica C 254, 265–280 (1995)

    Article  ADS  CAS  Google Scholar 

  19. Schützmann, J., Tajima, S., Miyamoto, S. & Tanaka, S. c-Axis optical response of fully oxygenated YBa2Cu3O7-δ: Observation of dirty-limit-like superconductivity and residual unpaired carriers. Phys. Rev. Lett. 73, 174–177 (1994)

    Article  ADS  Google Scholar 

  20. Basov, D. N. et al. Sum rules and interlayer conductivity of high-Tc cuprates. Science 283, 49–52 (1999)

    Article  ADS  CAS  Google Scholar 

  21. Tanner, D. B. et al. Superfluid and normal-fluid densities in high-Tc superconductors. Physica B 244, 1–8 (1998)

    Article  ADS  CAS  Google Scholar 

  22. Orenstein, J. et al. Frequency- and temperature-dependent conductivity in YBa2Cu3O6+x crystals. Phys. Rev. B 42, 6342–6362 (1990)

    Article  ADS  CAS  Google Scholar 

  23. Shibauchi, T. et al. Anisotropic penetration depth in La2-xSrxCuO4 . Phys. Rev. Lett. 72, 2263–2266 (1994)

    Article  ADS  CAS  Google Scholar 

  24. Lawrence, W. E. & Doniach, S. in Proc. 12th Int. Conf. Low Temperature Physics (ed. Kando, E.) 361 (Academic, Kyoto, 1971)

    Google Scholar 

  25. Bulaevskii, L. N. Magnetic properties of lamellar superconductors with weak interaction between the layers. Sov. Phys. JETP 37, 1133–1139 (1973)

    ADS  Google Scholar 

  26. Ambegaokar, V. & Baratoff, A. Tunneling between superconductors. Phys. Rev. Lett. 10, 486–489 (1963)

    Article  ADS  Google Scholar 

  27. Hardy, W. N., Bonn, D. A., Morgan, D. C., Liang, R. & Zhang, K. Precision measurements of the temperature dependence of λ in YBa2Cu3O6.95: Strong evidence for nodes in the gap function. Phys. Rev. Lett. 70, 3999–4002 (1993)

    Article  ADS  CAS  Google Scholar 

  28. Shen, Z.-X. et al. Anomalously large gap anisotropy in the a-b plane of Bi2Sr2CaCu2O8+δ . Phys. Rev. Lett. 70, 1553–1556 (1993)

    Article  ADS  CAS  Google Scholar 

  29. Chakravarty, S., Sudbo, A., Anderson, P. W. & Strong, S. Interlayer tunneling and gap anisotropy in high-temperature superconductors. Science 261, 337–340 (1993)

    Article  ADS  CAS  Google Scholar 

Download references

Acknowledgements

We thank A. Chubukov, P. D. Johnson, S. A. Kivelson, P. A. Lee, D. B. Tanner, J. J. Tu, Y. Uemura and T. Valla for discussions. Work in Canada was supported by the Natural Sciences and Engineering Research Council of Canada, and the Canadian Institute for Advanced Research. The HgBa2CuO4+δ crystal growth work at Stanford University was supported by the Department of Energy's Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. Work at the University of California at San Diego was supported by the National Science Foundation and the Department of Energy. Work at Brookhaven was supported by the Department of Energy.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to C. C. Homes.

Ethics declarations

Competing interests

The authors declare that they have no competing financial interests.

Supplementary information

Supplementary Tables 1 and 2

Supplementary Table 1: The values for a variety of high-Tc cuprate superconductors for the critical temperature, dc conductivity close to the critical temperature, superfluid density and the penetration depth, in the a-b planes. Supplementary Table 2: The same quantities described in Supplementary Table 1, except along the c axis. (DOC 36 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Homes, C., Dordevic, S., Strongin, M. et al. A universal scaling relation in high-temperature superconductors. Nature 430, 539–541 (2004). https://doi.org/10.1038/nature02673

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature02673

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing