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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The experimental data presented in this paper are available at http://wrap.warwick.ac.uk/152398/. The results of the conductivity simulations are available from the corresponding authors upon reasonable request.
The code used to compute the conductivity is available from the corresponding authors upon reasonable request.
Martin, S., Fiory, A. T., Fleming, R. M., Schneemeyer, L. F. & Waszczak, J. V. Normal-state transport properties of Bi2+xSr2−yCuO6+δ crystals. Phys. Rev. B 41, 846–849 (1990).
Löhneysen, H. V. et al. Non-Fermi-liquid behavior in a heavy-fermion alloy at a magnetic instability. Phys. Rev. Lett. 72, 3262–3265 (1994).
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).
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).
Legros, A. et al. Universal T-linear resistivity and Planckian dissipation in overdoped cuprates. Nat. Phys. 15, 142–147 (2019).
Daou, R. et al. Linear temperature dependence of resistivity and change in the Fermi surface at the pseudogap critical point of a high-Tc superconductor. Nat. Phys. 5, 31–34 (2009).
Matt C. E. et al. Electron scattering, charge order, and pseudogap physics in La1.6−xNd0.4SrxCuO4: an angle-resolved photoemission spectroscopy study. Phys. Rev. B 92, 134524 (2015).
Hlubina, R. & Rice, T. M. Resistivity as a function of temperature for models with hot spots on the Fermi surface. Phys. Rev. B 51, 9253–9260 (1995).
Stojković, B. P. & Pines, D. Theory of the longitudinal and Hall conductivities of the cuprate superconductors. Phys. Rev. B 55, 8576–8595 (1997).
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).
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).
Cao, Y. et al. Strange metal in magic-angle graphene with near Planckian dissipation. Phys. Rev. Lett. 124, 076801 (2020).
Parcollet, O. & Georges, A. Non-Fermi-liquid regime of a doped Mott insulator. Phys. Rev. B 59, 5341–5360 (1999).
Davison, R. A., Schalm, K. & Zaanen, J. Holographic duality and the resistivity of strange metals. Phys. Rev. B 89, 245116 (2014).
Hartnoll, S. A. Theory of universal incoherent metallic transport. Nat. Phys. 11, 54–61 (2015).
Patel, A. A. & Sachdev, S. Theory of a Planckian metal. Phys. Rev. Lett. 123, 066601 (2019).
Cha, P., Wentzell, N., Parcollet, O., Georges, A. & Kim, E.-A. Linear resistivity and Sachdev–Ye–Kitaev (SYK) spin liquid behavior in a quantum critical metal with spin-1/2 fermions. Proc. Natl Acad. Sci. USA 117, 18341–18346 (2020).
Corson, J., Orenstein, J., Oh, S., O’Donnell, J. & Eckstein, J. N. Nodal quasiparticle lifetime in the superconducting state of Bi2Sr2CaCu2O8+δ. Phys. Rev. Lett. 85, 2569 (2000).
Kaminski, A. et al. Momentum anisotropy of the scattering rate in cuprate superconductors. Phys. Rev. B 71, 014517 (2005).
Collignon, C. et al. Fermi-surface transformation across the pseudogap critical point of the cuprate superconductor La1.6−xNd0.4SrxCuO4. Phys. Rev. B 95, 224517 (2017).
Chambers, R. G. The kinetic formulation of conduction problems. Proc. Phys. Soc. A 65, 458–459 (1952).
Prange, R. E. & Kadanoff, L. P. Transport theory for electron–phonon interactions in metals. Phys. Rev. 134, A566–A580 (1964).
Abrahams, E. & Varma, C. M. Hall effect in the marginal Fermi liquid regime of high-Tc superconductors. Phys. Rev. B 68, 094502 (2003).
Horio, M. et al. Three-dimensional Fermi surface of overdoped La-based cuprates. Phys. Rev. Lett. 121, 077004 (2018).
Michon, B. et al. Thermodynamic signatures of quantum criticality in cuprate superconductors. Nature 567, 218–222 (2019).
Narduzzo, A. et al. Violation of the isotropic mean free path approximation for overdoped La2−xSrxCuO4. Phys. Rev. B 77, 220502 (2008).
Abrahams, E. & Varma, C. M. What angle-resolved photoemission experiments tell about the microscopic theory for high-temperature superconductors. Proc. Natl Acad. Sci. USA 97, 5714–5716 (2000).
Chang, J. et al. Anisotropic breakdown of Fermi liquid quasiparticle excitations in overdoped La2−xSrxCuO4. Nat. Commun. 4, 2559 (2013).
Pelc, D. et al. Resistivity phase diagram of cuprates revisited. Phys. Rev. B 102, 075114 (2020).
Kovtun, P. K., Son, D. T. & Starinets, A. O. Viscosity in strongly interacting quantum field theories from black hole physics. Phys. Rev. Lett. 94, 111601 (2005).
Giraldo-Gallo, P. et al. Scale-invariant magnetoresistance in a cuprate superconductor. Science 361, 479–481 (2018).
Peierls, R. On the theory of the magnetic change in resistance. Ann. Phys. 10, 97–110 (1931).
Hayes, I. M. et al. Scaling between magnetic field and temperature in the high-temperature superconductor BaFe2(As1−xPx)2. Nat. Phys. 12, 916–919 (2016).
Nakamae, S. et al. Electronic ground state of heavily overdoped nonsuperconducting La2−xSrxCuO4. Phys. Rev. B 68, 100502 (2003).
Cooper, R. A. et al. Anomalous criticality in the electrical resistivity of La2−xSrxCuO4. Science 323, 603–607 (2009).
Abdel-Jawad, M. et al. Anisotropic scattering and anomalous normal-state transport in a high-temperature superconductor. Nat. Phys. 2, 821–825 (2006).
Proust, C., Boaknin, E., Hill. R. W., Taillefer, L. & Mackenzie, A. P. Heat transport in a strongly overdoped cuprate: Fermi liquid and a pure d-wave BCS superconductor. Phys. Rev. Lett. 89, 147003 (2002).
Ramshaw, B. J. et al. Broken rotational symmetry on the Fermi surface of a high-Tc superconductor. npj Quantum Mater. 2, 8 (2017).
Newville, M., Stensitzki, T., Allen, D. B. & Ingargiola, A. LMFIT: non-linear least-square minimization and curve-fitting for python. Zenodo https://zenodo.org/record/11813#.YNRimOhKhPY (2014).
Chakravarty, S., Sudbø, A., Anderson, P. W. & Strong, S. Interlayer tunneling and gap anisotropy in high-temperature superconductors. Science 261, 337–340 (1993).
Helm, T. Electronic Properties of Electron-Doped Cuprate Superconductors Probed by High-Field Magnetotransport. PhD thesis, Technical Univ. Munich (2013).
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).
Bangura, A. F. et al. Fermi surface and electronic homogeneity of the overdoped cuprate superconductor Tl2Ba2CuO6+δ as revealed by quantum oscillations. Phys. Rev. B 82, 140501 (2010).
Analytis, J. G., Abdel-Jawad, M., Balicas, L., French, M. M. J. & Hussey, N. E. Angle-dependent magnetoresistance measurements in Tl2Ba2CuO6+δ and the need for anisotropic scattering. Phys. Rev. B 760, 104523 (2007).
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.
The authors declare no competing interests.
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
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.
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.
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.
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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). https://doi.org/10.1038/s41586-021-03697-8