A universal scaling relation in high-temperature superconductors


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).

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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.


  1. 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)

  2. 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)

  3. 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)

  4. 4

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

  5. 5

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

  6. 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)

  7. 7

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

  8. 8

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

  9. 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)

  10. 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)

  11. 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)

  12. 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)

  13. 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)

  14. 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)

  15. 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)

  16. 16

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

  17. 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)

  18. 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)

  19. 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)

  20. 20

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

  21. 21

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

  22. 22

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

  23. 23

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

  24. 24

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

  25. 25

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

  26. 26

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

  27. 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)

  28. 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)

  29. 29

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

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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.

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Correspondence to C. C. Homes.

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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)

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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

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