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Linear-in temperature resistivity from an isotropic Planckian scattering rate


A variety of ‘strange metals’ exhibit resistivity that decreases linearly with temperature as the temperature decreases to zero1,2,3, in contrast to conventional metals where resistivity decreases quadratically with temperature. This linear-in-temperature resistivity has been attributed to charge carriers scattering at a rate given by ħ/τ = αkBT, where α is a constant of order unity, ħ is the Planck constant and kB is the Boltzmann constant. This simple relationship between the scattering rate and temperature is observed across a wide variety of materials, suggesting a fundamental upper limit on scattering—the ‘Planckian limit’4,5—but little is known about the underlying origins of this limit. Here we report a measurement of the angle-dependent magnetoresistance of La1.6−xNd0.4SrxCuO4—a hole-doped cuprate that shows linear-in-temperature resistivity down to the lowest measured temperatures6. The angle-dependent magnetoresistance shows a well defined Fermi surface that agrees quantitatively with angle-resolved photoemission spectroscopy measurements7 and reveals a linear-in-temperature scattering rate that saturates at the Planckian limit, namely α = 1.2 ± 0.4. Remarkably, we find that this Planckian scattering rate is isotropic, that is, it is independent of direction, in contrast to expectations from ‘hotspot’ models8,9. Our findings suggest that linear-in-temperature resistivity in strange metals emerges from a momentum-independent inelastic scattering rate that reaches the Planckian limit.

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Fig. 1: T-linear resistivity and the angle-dependent magnetoresistance technique.
Fig. 2: ADMR and quasiparticle scattering rate of Nd-LSCO at p = 0.24.
Fig. 3: Transport coefficients of Nd-LSCO at p = 0.24.
Fig. 4: Comparison of two overdoped cuprates: Nd-LSCO and Tl2201.

Data availability

The experimental data presented in this paper are available at The results of the conductivity simulations are available from the corresponding authors upon reasonable request.

Code availability

The code used to compute the conductivity is available from the corresponding authors upon reasonable request.


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We acknowledge helpful discussions with J. Analytis, D. Chowdhury, N. Doiron-Leyraud, N. Hussey, M. Kartsovnik, S. Kivelson, D.-H. Lee, P. A. Lee, S. Lewin, A. Maharaj, K. Modic, C. Murthy, S. Musser, C. Proust, S. Sachdev, A. Shekhter, S. Todadri, A.-M. Tremblay and C. Varma. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by the National Science Foundation Cooperative Agreement No. DMR-1644779 and the State of Florida. P.A.G. acknowledges that this project is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 681260). J.-S.Z. was supported by an NSF grant (MRSEC DMR-1720595). L.T. acknowledges support from the Canadian Institute for Advanced Research (CIFAR) as a Fellow and funding from the Natural Sciences and Engineering Research Council of Canada (NSERC; PIN: 123817), the Fonds de recherche du Québec - Nature et Technologies (FRQNT), the Canada Foundation for Innovation (CFI) and a Canada Research Chair. This research was undertaken thanks in part to funding from the Canada First Research Excellence Fund. Part of this work was funded by the Gordon and Betty Moore Foundation’s EPiQS Initiative (Grant GBMF5306 to L.T.) B.J.R. and Y.F. acknowledge funding from the National Science Foundation under grant no. DMR-1752784.

Author information




A.L., P.A.G., L.T. and B.J.R. conceived the experiment. J.Z. grew the sample. A.L., F.L. and C.C. performed the sample preparation and characterization. G.G., Y.F., A.L., D.G., P.A.G. and B.J.R. performed the ADMR measurements at the National High Magnetic Field Laboratory in Tallahassee. G.G., Y.F., S.V. and B.J.R. performed the data analysis and simulations. G.G., L.T. and B.J.R. wrote the manuscript with input from all other co-authors. L.T. and B.J.R. supervised the project.

Corresponding authors

Correspondence to Louis Taillefer or B. J. Ramshaw.

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The authors declare no competing interests.

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Peer review information Nature thanks Antoine Georges and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 ADMR experimental set up.

a, A photograph of the sample on the rotator. The two samples here are mounted on a G-10 wedge to provide an azimuthal angle ϕ of 30°. Additional wedges provided angles of ϕ = 15° and ϕ = 45°. b, ADMR as a function of θ angle from −15° to 110° and ϕ = 0 at T = 20 K for Nd-LSCO p = 0.24, showing the symmetry of the data about these two angles.

Extended Data Fig. 2 Calculated and measured Sommerfeld coefficients of Nd-LSCO.

a, The Sommerfeld coefficient γ for Nd-LSCO as a function of doping. The measured values (red circles) are obtained from measurements of the electronic specific heat Cel/T at T = 10 K (ref. 25). For the calculated γ (black dashed, dotted and solid lines), we use the tight-binding parameters from our ADMR analysis for three different values of t, as indicated. The grey band represents the region of consistency between the calculations and the data. b, Electronic specific heat Cel/T as a function of temperature for Nd-LSCO p = 0.24, 0.27, 0.36 and 0.40 (ref. 25). The data are the solid lines and the dashed lines represent extrapolations.

Extended Data Fig. 3 Fit of the Nd-LSCO p = 0.24 data with different scattering-rate models.

a, ADMR data on Nd-LSCO p = 0.24 at T = 25 K and B = 45 T. b, c, e, f, Best fits for the ADMR data in a using the Fermi surface in Fig. 1d and an isotropic scattering-rate model (b), and three different anisotropic scattering-rate models: cosine (c), tanh (e) and polynomial (f). d, The three different anisotropic scattering rates as a function of the azimuthal angle ϕ at T = 25 K.

Extended Data Fig. 4 ADMR and quasiparticle scattering rate of Nd-LSCO at p = 0.24 for the tanh model.

This figure is the same as Figs. 2a, 3a, b, except that the ADMR has been fitted using the tanh model instead of the cosine model (Extended Data Fig. 3).

Extended Data Fig. 5 ADMR and quasiparticle scattering rate of Nd-LSCO at p = 0.24 for B = 35 T.

a, b, This figure is the same as Fig. 2a, c except that the ADMR data are taken at B = 35 T (a). The fit has been carried out using the cosine model. b shows that scattering-rate values are identical to within a percent of those obtained from the fit to the data at B = 45 T, shown in Fig. 2c.

Extended Data Table 1 Tight-binding parameters from the fit to the ADMR data at p = 0.24
Extended Data Table 2 Results of the fit of the Nd-LSCO p = 0.24 data with the cosine scattering-rate model

Supplementary information

Supplementary Information

This file contains Supplementary Notes and Data, Supplementary Figures 1–2 and Supplementary Refernces.

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Grissonnanche, G., Fang, Y., Legros, A. et al. Linear-in temperature resistivity from an isotropic Planckian scattering rate. Nature 595, 667–672 (2021).

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