Skip to main content

Thank you for visiting nature.com. 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.

Linear-in temperature resistivity from an isotropic Planckian scattering rate

Abstract

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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

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 http://wrap.warwick.ac.uk/152398/. 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.

References

  1. 1.

    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).

    ADS  CAS  Google Scholar 

  2. 2.

    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).

    ADS  PubMed  PubMed Central  Google Scholar 

  3. 3.

    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 

  4. 4.

    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  PubMed Central  Google Scholar 

  5. 5.

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

    CAS  Google Scholar 

  6. 6.

    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).

    CAS  Google Scholar 

  7. 7.

    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).

    ADS  Google Scholar 

  8. 8.

    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).

    ADS  CAS  Google Scholar 

  9. 9.

    Stojković, B. P. & Pines, D. Theory of the longitudinal and Hall conductivities of the cuprate superconductors. Phys. Rev. B 55, 8576–8595 (1997).

    ADS  Google Scholar 

  10. 10.

    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  PubMed Central  Google Scholar 

  11. 11.

    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  PubMed Central  Google Scholar 

  12. 12.

    Cao, Y. et al. Strange metal in magic-angle graphene with near Planckian dissipation. Phys. Rev. Lett. 124, 076801 (2020).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  13. 13.

    Parcollet, O. & Georges, A. Non-Fermi-liquid regime of a doped Mott insulator. Phys. Rev. B 59, 5341–5360 (1999).

    ADS  CAS  Google Scholar 

  14. 14.

    Davison, R. A., Schalm, K. & Zaanen, J. Holographic duality and the resistivity of strange metals. Phys. Rev. B 89, 245116 (2014).

    ADS  Google Scholar 

  15. 15.

    Hartnoll, S. A. Theory of universal incoherent metallic transport. Nat. Phys. 11, 54–61 (2015).

    CAS  Google Scholar 

  16. 16.

    Patel, A. A. & Sachdev, S. Theory of a Planckian metal. Phys. Rev. Lett. 123, 066601 (2019).

    ADS  MathSciNet  CAS  PubMed  PubMed Central  Google Scholar 

  17. 17.

    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).

    MathSciNet  CAS  PubMed  PubMed Central  Google Scholar 

  18. 18.

    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).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  19. 19.

    Kaminski, A. et al. Momentum anisotropy of the scattering rate in cuprate superconductors. Phys. Rev. B 71, 014517 (2005).

    ADS  Google Scholar 

  20. 20.

    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).

    ADS  Google Scholar 

  21. 21.

    Chambers, R. G. The kinetic formulation of conduction problems. Proc. Phys. Soc. A 65, 458–459 (1952).

    ADS  MATH  Google Scholar 

  22. 22.

    Prange, R. E. & Kadanoff, L. P. Transport theory for electron–phonon interactions in metals. Phys. Rev. 134, A566–A580 (1964).

    ADS  MATH  Google Scholar 

  23. 23.

    Abrahams, E. & Varma, C. M. Hall effect in the marginal Fermi liquid regime of high-Tc superconductors. Phys. Rev. B 68, 094502 (2003).

    ADS  Google Scholar 

  24. 24.

    Horio, M. et al. Three-dimensional Fermi surface of overdoped La-based cuprates. Phys. Rev. Lett. 121, 077004 (2018).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  25. 25.

    Michon, B. et al. Thermodynamic signatures of quantum criticality in cuprate superconductors. Nature 567, 218–222 (2019).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  26. 26.

    Narduzzo, A. et al. Violation of the isotropic mean free path approximation for overdoped La2−xSrxCuO4. Phys. Rev. B 77, 220502 (2008).

    ADS  Google Scholar 

  27. 27.

    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).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  28. 28.

    Chang, J. et al. Anisotropic breakdown of Fermi liquid quasiparticle excitations in overdoped La2−xSrxCuO4. Nat. Commun. 4, 2559 (2013).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  29. 29.

    Pelc, D. et al. Resistivity phase diagram of cuprates revisited. Phys. Rev. B 102, 075114 (2020).

    ADS  CAS  Google Scholar 

  30. 30.

    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).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  31. 31.

    Giraldo-Gallo, P. et al. Scale-invariant magnetoresistance in a cuprate superconductor. Science 361, 479–481 (2018).

    ADS  CAS  Google Scholar 

  32. 32.

    Peierls, R. On the theory of the magnetic change in resistance. Ann. Phys. 10, 97–110 (1931).

    Google Scholar 

  33. 33.

    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).

    Google Scholar 

  34. 34.

    Nakamae, S. et al. Electronic ground state of heavily overdoped nonsuperconducting La2−xSrxCuO4. Phys. Rev. B 68, 100502 (2003).

    ADS  Google Scholar 

  35. 35.

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

    ADS  CAS  Google Scholar 

  36. 36.

    Abdel-Jawad, M. et al. Anisotropic scattering and anomalous normal-state transport in a high-temperature superconductor. Nat. Phys. 2, 821–825 (2006).

    CAS  Google Scholar 

  37. 37.

    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).

    ADS  PubMed  PubMed Central  Google Scholar 

  38. 38.

    Ramshaw, B. J. et al. Broken rotational symmetry on the Fermi surface of a high-Tc superconductor. npj Quantum Mater. 2, 8 (2017).

    ADS  Google Scholar 

  39. 39.

    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).

  40. 40.

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

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  41. 41.

    Helm, T. Electronic Properties of Electron-Doped Cuprate Superconductors Probed by High-Field Magnetotransport. PhD thesis, Technical Univ. Munich (2013).

  42. 42.

    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 

  43. 43.

    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).

    ADS  Google Scholar 

  44. 44.

    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).

    ADS  Google Scholar 

Download references

Acknowledgements

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

Affiliations

Authors

Contributions

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.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

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

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 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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Comments

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.

Search

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