Skip to main content

Thank you for visiting 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.

Scaling of the strange-metal scattering in unconventional superconductors


Marked evolution of properties with minute changes in the doping level is a hallmark of the complex chemistry that governs copper oxide superconductivity as manifested in the celebrated superconducting domes and quantum criticality taking place at precise compositions1,2,3,4. The strange-metal state, in which the resistivity varies linearly with temperature, has emerged as a central feature in the normal state of copper oxide superconductors5,6,7,8,9. The ubiquity of this behaviour signals an intimate link between the scattering mechanism and superconductivity10,11,12. However, a clear quantitative picture of the correlation has been lacking. Here we report the observation of precise quantitative scaling laws among the superconducting transition temperature (Tc), the linear-in-T scattering coefficient (A1) and the doping level (x) in electron-doped copper oxide La2–xCexCuO4 (LCCO). High-resolution characterization of epitaxial composition-spread films, which encompass the entire overdoped range of LCCO, has enabled us to systematically map its structural and transport properties with unprecedented accuracy and with increments of Δx = 0.0015. We have uncovered the relations Tc ~ (xcx)0.5 ~ (A1)0.5, where xc is the critical doping in which superconductivity disappears and A1 is the coefficient of the linear resistivity per CuO2 plane. The striking similarity of the Tc versus A1 relation among copper oxides, iron-based and organic superconductors may be an indication of a common mechanism of the strange-metal behaviour and unconventional superconductivity in these systems.

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

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Combinatorial synthesis and multiscale structural characterization of LCCO.
Fig. 2: Microregion characterizations of electrical transport properties.
Fig. 3: Quantitative scaling revealed from the systematic spread data and comparison of different unconventional superconductors.

Data availability

The data that support the findings of this study are available in the paper. Additional data are available from the corresponding authors upon reasonable request.


  1. Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).

    ADS  CAS  Google Scholar 

  2. Armitage, N. P., Fournier, P. & Greene, R. L. Progress and perspectives on electron-doped cuprates. Rev. Mod. Phys. 82, 2421–2487 (2010).

    ADS  CAS  Google Scholar 

  3. Scalapino, D. J. A common thread: the pairing interaction for unconventional superconductors. Rev. Mod. Phys. 84, 1383–1417 (2012).

    ADS  CAS  Google Scholar 

  4. Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature 518, 179–186 (2015).

    ADS  CAS  PubMed  Google Scholar 

  5. Gurvitch, M. & Fiory, A. T. Resistivity of La1.825Sr0.175CuO4 and YBa2Cu3O7 to 1100 K: absence of saturation and its implications. Phys. Rev. Lett. 59, 1337–1340 (1987).

    ADS  CAS  PubMed  Google Scholar 

  6. Takagi, H. et al. Systematic evolution of temperature-dependent resistivity in La2–xSrxCuO4. Phys. Rev. Lett. 69, 2975–2978 (1992).

    ADS  CAS  PubMed  Google Scholar 

  7. Cooper, R. A. et al. Anomalous criticality in the electrical resistivity of La2–xSrxCuO4. Science 323, 603–607 (2009).

    ADS  CAS  PubMed  Google Scholar 

  8. Fournier, P. et al. Insulator-metal crossover near optimal doping in Pr2–xCexCuO4: anomalous normal-state low temperature resistivity. Phys. Rev. Lett. 81, 4720–4723 (1998).

    ADS  CAS  Google Scholar 

  9. Jin, K., Butch, N. P., Kirshenbaum, K., Paglione, J. & Greene, R. L. Link between spin fluctuations and electron pairing in copper oxide superconductors. Nature 476, 73–75 (2011).

    CAS  PubMed  Google Scholar 

  10. Taillefer, L. Scattering and pairing in cuprate superconductors. Annu. Rev. Condens. Matter Phys. 1, 51–70 (2010).

    ADS  CAS  Google Scholar 

  11. Greene, R. L., Mandal, P. R., Poniatowski, N. R. & Sarkar, T. The strange metal state of the electron-doped cuprates. Annu. Rev. Condens. Matter Phys. 11, 213–229 (2020).

    CAS  Google Scholar 

  12. Varma, C. M. Colloquium: linear in temperature resistivity and associated mysteries including high temperature superconductivity. Rev. Mod. Phys. 92, 031001 (2020).

    ADS  Google Scholar 

  13. Sarkar, T. et al. Hidden strange metallic state in underdoped electron-doped cuprates. Phys. Rev. B 103, 224501 (2021).

    ADS  CAS  Google Scholar 

  14. Koinuma, H. & Takeuchi, I. Combinatorial solid-state chemistry of inorganic materials. Nat. Mater. 3, 429–438 (2004).

    ADS  CAS  PubMed  Google Scholar 

  15. Yuan, J., Stanev, V., Gao, C., Takeuchi, I. & Jin, K. Recent advances in high-throughput superconductivity research. Supercond. Sci. Technol. 32, 123001 (2019).

    ADS  CAS  Google Scholar 

  16. Sarkar, T. et al. Ferromagnetic order beyond the superconducting dome in a cuprate superconductor. Science 368, 532–534 (2020).

    ADS  CAS  PubMed  Google Scholar 

  17. Schneider, T. & Singer, J. M. Phase Transition Approach to High Temperature Superconductivity (Imperial College Press, 2000).

  18. Legros, A. et al. Universal T-linear resistivity and Planckian dissipation in overdoped cuprates. Nat. Phys. 15, 142–147 (2019).

    CAS  Google Scholar 

  19. Ayres, J. et al. Incoherent transport across the strange-metal regime of overdoped cuprates. Nature 595, 661–666 (2021).

    ADS  CAS  PubMed  Google Scholar 

  20. Božović, I., He, X., Wu, J. & Bollinger, A. T. Dependence of the critical temperature in overdoped copper oxides on superfluid density. Nature 536, 309–311 (2016).

    ADS  PubMed  Google Scholar 

  21. Anderson, P. W. The resonating valence bond state in La2CuO4 and superconductivity. Sci. New Ser. 235, 1196–1198 (1987).

    CAS  Google Scholar 

  22. Varma, C. M., Littlewood, P. B., Schmitt-Rink, S., Abrahams, E. & Ruckenstein, A. E. Phenomenology of the normal state of Cu-O high-temperature superconductors. Phys. Rev. Lett. 63, 1996–1999 (1989).

    ADS  CAS  PubMed  Google Scholar 

  23. Sachdev, S. Quantum phase transitions. Phys. World 12, 33–38 (1999).

    CAS  Google Scholar 

  24. Zaanen, J. Why the temperature is high. Nature 430, 512–513 (2004).

    ADS  CAS  PubMed  Google Scholar 

  25. Bruin, J. a. N., Sakai, H., Perry, R. S. & Mackenzie, A. P. Similarity of scattering rates in metals showing T-linear resistivity. Science 339, 804–807 (2013).

    ADS  CAS  PubMed  Google Scholar 

  26. Licciardello, S. et al. Electrical resistivity across a nematic quantum critical point. Nature 567, 213–217 (2019).

    ADS  CAS  PubMed  Google Scholar 

  27. Moriya, T. & Ueda, K. Spin fluctuations and high temperature superconductivity. Adv. Phys. 49, 555–606 (2000).

    ADS  CAS  Google Scholar 

  28. Monthoux, P. & Pines, D. Spin-fluctuation-induced superconductivity and normal-state properties of YBa2Cu3O7. Phys. Rev. B 49, 4261–4278 (1994).

    ADS  CAS  Google Scholar 

  29. Bourbonnais, C. & Sedeki, A. Link between antiferromagnetism and superconductivity probed by nuclear spin relaxation in organic conductors. Phys. Rev. B 80, 085105 (2009).

    ADS  Google Scholar 

  30. Doiron-Leyraud, N. et al. Correlation between linear resistivity and Tc in the Bechgaard salts and the pnictide superconductor Ba(Fe1–xCox)2As2. Phys. Rev. B 80, 214531 (2009).

    ADS  Google Scholar 

  31. Sedeki, A., Bergeron, D. & Bourbonnais, C. Extended quantum criticality of low-dimensional superconductors near a spin-density-wave instability. Phys. Rev. B 85, 165129 (2012).

    ADS  Google Scholar 

  32. Fang, L. et al. Roles of multiband effects and electron-hole asymmetry in the superconductivity and normal-state properties of Ba(Fe1–xCox)2As2. Phys. Rev. B 80, 140508 (2009).

    ADS  Google Scholar 

  33. Fradkin, E., Kivelson, S. A. & Tranquada, J. M. Colloquium: theory of intertwined orders in high temperature superconductors. Rev. Mod. Phys. 87, 457–482 (2015).

    ADS  CAS  Google Scholar 

  34. Rullier-Albenque, F., Alloul, H. & Tourbot, R. Influence of pair breaking and phase fluctuations on disordered high Tc cuprate superconductors. Phys. Rev. Lett. 91, 047001 (2003).

    ADS  CAS  PubMed  Google Scholar 

  35. Emery, V. J. & Kivelson, S. A. Superconductivity in bad metals. Phys. Rev. Lett. 74, 3253–3256 (1995).

    ADS  CAS  PubMed  Google Scholar 

  36. Hussey, N. E., Gordon-Moys, H., Kokalj, J. & McKenzie, R. H. Generic strange-metal behaviour of overdoped cuprates. J. Phys. Conf. Ser. 449, 012004 (2013).

    CAS  Google Scholar 

  37. Maier, T. A., Karakuzu, S. & Scalapino, D. J. Overdoped end of the cuprate phase diagram. Phys. Rev. Res. 2, 033132 (2020).

    CAS  Google Scholar 

  38. Zhang, X. et al. Quantum criticality tuned by magnetic field in optimally electron-doped cuprate thin films. Phys. Rev. B 103, 014517 (2021).

    ADS  CAS  Google Scholar 

  39. Jin, K., Zhu, B. Y., Wu, B. X., Gao, L. J. & Zhao, B. R. Low-temperature Hall effect in electron-doped superconducting La2–xCexCuO4 thin films. Phys. Rev. B 78, 174521 (2008).

    ADS  Google Scholar 

  40. Jin, K., Zhang, X. H., Bach, P. & Greene, R. L. Evidence for antiferromagnetic order in La2–xCexCuO4 from angular magnetoresistance measurements. Phys. Rev. B 80, 012501 (2009).

    ADS  Google Scholar 

  41. Sarkar, T. et al. Fermi surface reconstruction and anomalous low-temperature resistivity in electron-doped La2–xCexCuO4. Phys. Rev. B 96, 155449 (2017).

    ADS  Google Scholar 

  42. Ando, Y., Komiya, S., Segawa, K., Ono, S. & Kurita, Y. Electronic phase diagram of high-Tc cuprate superconductors from a mapping of the in-plane resistivity curvature. Phys. Rev. Lett. 93, 267001 (2004).

    ADS  PubMed  Google Scholar 

Download references


We thank R. Greene for many fruitful discussions and inspiring comments during the preparation of the manuscript. We thank T. Schneider for discussions surrounding the scaling relations. We thank G. Zhang, Z. Weng, Y. Li, Y. Zhou and E. Liu for fruitful discussions. We thank S. Tu and Y. Zou for experimental assistance. This work was supported by the National Key Basic Research Program of China (grant nos. 2017YFA0302900, 2017YFA0303003 and 2018YFB0704102), the National Natural Science Foundation of China (grant nos. 11888101, 11927808, 11834016 and 11961141008), Beijing Natural Science Foundation (grant no. Z190008), the Strategic Priority Research Program (B) of Chinese Academy of Sciences (grant nos. XDB25000000 and XDB33000000), the CAS Interdisciplinary Innovation Team, Key-Area Research and Development Program of Guangdong Province (grant no. 2020B0101340002). The work at the University of Maryland was supported by grant nos. AFOSR FA9550-14-10332, AFOSR FA9550-22-10023, ONR N00014-13-1-0635 and ONR N00014-15-2-222, and NIST grant no. 60NANB19D027. This research used BL12.3.2 of the Advanced Light Source, a US Department of Energy Office of Science User Facility under contract no. DE-AC02-05CH11231.

Author information

Authors and Affiliations



K. Jin conceived the project. K. Jin, J.Y. and Q.C. supervised the experiments. Z.F., Z.L. and H.Y. synthesized the composition-spread films. G.H., J.Z., X.Z. and M.Q. performed the transport measurements. N.T. and I.T. carried out the synchrotron XRD measurements. X.J., Y.S., Y.Z., Z.G.C., I.T., J.Y., Q.C., K. Jin and Z.Z. analysed the experimental data. K. Jiang, Y.-f.Y., T.X. and J.H. provided theoretical understanding. Q.C, J.Y., I.T. and K. Jin wrote the manuscript with input from all authors.

Corresponding authors

Correspondence to Jiangping Hu, Ichiro Takeuchi or Kui Jin.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review information

Peer review information

Nature thanks the anonymous reviewers for their contribution to the peer review of this work.

Additional information

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

Extended data figures and tables

Extended Data Fig. 1 Temperature dependence of resistivity for different bridges (100-μm width) across a combinatorial La2–xCexCuO4 (LCCO) film.

a, The doping varies from 0.123 to 0.185 from the top to the bottom, as indicated by the arrow. These curves provide the raw data for extracting the A1 vs. x, and A1 vs. Tc dependences shown in Figs. 3a and 3b of the main text. b, The low-temperature range of the colored lines in panel a. The doping for each curve is (normal-state resistivity from top to bottom): 0.123, 0.129, 0.138, 0.144, 0.148, 0.152, 0.156, 0.160, 0.162, 0.165, 0.167, 0.168, 0.169, and 0.175.

Extended Data Fig. 2 Definition of Tc in the main text.

The solid red curve is a typical ρ(T) curve of a 100-μm bridge from the combinatorial La2–xCexCuO4 (LCCO) film. The black dashed line is a linear extension of the normal-state resistivity to lower temperatures. The arrow indicates where the resistivity starts to deviate from the linear extension, which is defined as Tc in this work.

Extended Data Fig. 3 Comparison of different Tc definitions.

a, Tc(cross) is defined as the temperature where the linear extrapolation crosses extrapolation of the bulk of the resistance drop due to superconducting transition. Tc(90%RN) is defined as the temperature where the resistivity is 90% of the normal-state resistivity. The lower dashed line is obtained by maintaining the slope of the linear extrapolation of the normal-state resistivity, while multiplying the intercept by 0.9. b, Tc as a function of doping. The solid line shows the fit with the formula: Tc (xc - x)0.5. c, Comparison of the scaling relation between (A1)0.5 and different definitions of Tc. d, Tc0 is extracted by extrapolating the bulk of the resistance drop to zero resistivity. e, Tc as a function of doping. The solid line shows the fit with the formula: Tc (xc - x)0.5. f, Comparison of the scaling relation between (A1)0.5 vs. Tc(onset) and (A1)0.5 vs. Tc0. Although there are a few kelvins difference between different definitions of Tc, the parabolic relation between Tc and x and linear scaling relation between (A1)0.5 and Tc are valid in all three.

Extended Data Fig. 4 Linear-in-T resistivity in superconducting LCCO.

a–b, Temperature dependence of resistivity ρ(T) in zero field (diamonds), fitted by ρ(T) = ρ0 + A1T (red line), for x ≈ 0.123 (a) and 0.146 (b). c, ρ(T) of x ≈ 0.160 at B = 0 (grey diamonds) and B = 5 T (blue circles). The red line is the linear fit to the 5 T data. The insets show the fitting quality presented as Δρ / ρ0 vs. T, where Δρ = ρ - (ρ0 + A1T).

Extended Data Fig. 5 Temperature dependence of resistivity for the LCCO film around optimal doping (x = 0.10).

Magnetic fields are applied along the c-axis direction of the LCCO film: 0 T (black squares), 15 T (green circles) and 55 T (orange diamonds). The linear-in-T resistivity is gradually recovered at high magnetic fields. Dashed lines mark the linear-in-T resistivity region at B = 0 and 55 T. Data adapted from ref. 38.

Extended Data Fig. 6 Evolution of Hall signal as a function of doping measured on a combinatorial film.

a–e, The Hall resistivity (ρxy) as a function of magnetic field (applied perpendicular to the ab-plane of the LCCO film) at different temperatures, for Ce doping x ≈ 0.117 (a), 0.132 (b), 0.145 (c), 0.157 (d), and 0.174 (e). f, ρxy vs. B at T = 2 K for different Ce concentrations from 0.117 to 0.174. g, Temperature dependence of the Hall coefficient (RH) for different Ce concentrations from 0.117 to 0.174, measured at a magnetic field of 14 T. h, RH at T = 2 K and B = 14 T as a function of Ce doping. i, The corresponding Ce doping dependence of Hall number nH = V/eRH.

Extended Data Fig. 7 Tc versus x plots for different unconventional superconductors.

Symbols are data extracted from literature and solid curves are fits with the formula Tc (xc - x)n, with n being a fitting parameter. a, Data for Ba(Fe1–xCox)2As2 extracted from ref. 10. b, Data for La2–xSrxCuO4 (black squares: ref. 17; green dots: ref. 7; blue triangles: ref. 42). c, Data for La2–xSrxCuO4 extracted from the ρ(T) curves of ref. 20. The exponent n for different materials fall in the range of 0.4–0.6. For Ba(Fe1–xCox)2As2, the data is scarce and the fitting uncertainty is relatively large. La2–xSrxCuO4 is one of the most intensively studied hole-doped copper oxide, thus there are more data in literature. However, data from different studies (panel b) are quite scattered thus the fit is poor. In panel c, the data are extracted from a relatively comprehensive study of La2–xSrxCuO4 (ref. 20), which is in a relatively good fit to the formula Tc (xc - x)n with n = 0.51 ± 0.07, consistent with the value reported in this work, and the uncertainty is smaller compared to the other two panels.

Extended Data Fig. 8 Temperature dependence of the resistivity for a bridge with Tc ~ 10 K (x = 0.16) at B = 0 T and B = 5 T.

The red straight line indicates the extrapolation of the low-temperature linear fit to higher temperatures.

Extended Data Table 1 Doping, c-axis, Tc and linear-in-T coefficients for the data shown in Fig. 3 of the main text

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yuan, J., Chen, Q., Jiang, K. et al. Scaling of the strange-metal scattering in unconventional superconductors. Nature 602, 431–436 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


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.


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