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 ~ (xc – x)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.
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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.
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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.
The authors declare no competing interests.
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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.
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.
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.
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.
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.
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Yuan, J., Chen, Q., Jiang, K. et al. Scaling of the strange-metal scattering in unconventional superconductors. Nature 602, 431–436 (2022). https://doi.org/10.1038/s41586-021-04305-5
Frontiers of Physics (2022)