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# Incoherent transport across the strange-metal regime of overdoped cuprates

## Abstract

Strange metals possess highly unconventional electrical properties, such as a linear-in-temperature resistivity1,2,3,4,5,6, an inverse Hall angle that varies as temperature squared7,8,9 and a linear-in-field magnetoresistance10,11,12,13. Identifying the origin of these collective anomalies has proved fundamentally challenging, even in materials such as the hole-doped cuprates that possess a simple bandstructure. The prevailing consensus is that strange metallicity in the cuprates is tied to a quantum critical point at a doping p* inside the superconducting dome14,15. Here we study the high-field in-plane magnetoresistance of two superconducting cuprate families at doping levels beyond p*. At all dopings, the magnetoresistance exhibits quadrature scaling and becomes linear at high values of the ratio of the field and the temperature, indicating that the strange-metal regime extends well beyond p*. Moreover, the magnitude of the magnetoresistance is found to be much larger than predicted by conventional theory and is insensitive to both impurity scattering and magnetic field orientation. These observations, coupled with analysis of the zero-field and Hall resistivities, suggest that despite having a single band, the cuprate strange-metal region hosts two charge sectors, one containing coherent quasiparticles, the other scale-invariant ‘Planckian’ dissipators.

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## Data availability

The data that support the plots within this paper and other findings of this study are available from the Bristol data repository, data.bris, at https://doi.org/10.5523/bris.150s0lqyd3eh61zsiaj8cj5vqd.

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## Acknowledgements

We thank M. Allan, J. G. Analytis, I. Božović, M. S. Golden, B. Goutéraux, C. Pépin, K. Schalm, H. Stoof and S. Vandoren for insightful discussions during the course of this work. We also thank S. Smit and L. Bawden for initial characterization of some of the Bi2201 single crystals and L. Malone for assistance in the growth of the Tl2201 single crystals. J.A. acknowledges the support of the EPSRC-funded CMP-CDT (ref. EP/L015544/1) and an EPSRC Doctoral Prize Fellowship (ref. EP/T517872/1). A.C. acknowledges support of the EPSRC (ref. EP/R011141/1). We also acknowledge the support of the High Field Magnet Laboratory (HFML) at Radboud University, member of the European Magnetic Field Laboratory (EMFL – also supported by the EPSRC, ref. EP/N01085X/1), and the former Foundation for Fundamental Research on Matter (FOM), which is financially supported by the Netherlands Organisation for Scientific Research (NWO) (grant no. 16METL01, ‘Strange Metals’). Finally, part of this work was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement nos 835279-Catch-22 and 715262-HPSuper).

## Author information

Authors

### Contributions

J.A., M.B., S.F., A.C. and N.E.H. conceived the overall project. J.A., M.B., M.Č., Y.-T.H., C.P. and N.E.H. performed the high-field measurements. Y.H., E.v.H., J.R.C., C.P., T.K. and T.T. grew and characterized the single-crystal samples. J.A. and A.C. performed the SCTIF calculations. J.A., M.B., J.Z. and N.E.H. wrote the manuscript with input from all of the co-authors.

### Corresponding authors

Correspondence to J. Ayres or M. Berben or N. E. Hussey.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Nicholas Breznay and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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 Zero-field resistivities of Tl2201 and Bi2201.

a, b, Zero-field, ambient-pressure resistivity ρab(T) curves for representative Tl2201 (a) and Bi2201 (b) crystals investigated in this study. Note the super-linear T dependence for all samples. The spread in absolute magnitudes of ρab(T) is higher in the Tl2201 crystals owing to the fact that they were mounted for pressure measurements and as such, their absolute resistivities were harder to quantify accurately.

### Extended Data Fig. 2 Quadrature scaling in overdoped Tl2201.

a, ρab(HT) as measured in Tl2201 with Tc = 26.5 K. H2 behaviour cedes to a H-linear resistivity at high fields. b, c, Scaling plots of [ρab(HT) − ρab(0, 0)]/T versus H/T for overdoped Tl2201 (Tc = 26.5 K). As shown in c, there is a clear breakdown of the scaling at low H/T. d, e, Scaling plots of [ρab(HT) − ρab(0, T)]/T versus H/T for the same sample where ρ(0, T) = (T) = ρ0 + AgT + BT2. Note that Ag does not correspond to A, the full T-linear coefficient of the zero-field resistivity, since part of that is contained within the quadrature form. The inclusion of these additional T-dependent terms makes the data collapse over the full range of T. Taking the derivative with respect to H (as done in the main text) provides another means of isolating the quadrature MR from (T). The dashed lines in all panels represent the quadrature expression $$\Delta {\rho }_{ab}(H)=\alpha {k}_{{\rm{B}}}T\sqrt{1+{(\beta {\mu }_{0}H/T)}^{2}}$$ (ρ0 = 15.5 μΩ cm, Ag = 0.14 μΩ cm K−1, B = 0.003 μΩ cm K−2, αkB = 0.04 μΩ cm K−1, γμB = 0.20 μΩ cm T−1). f, The derivatives with respect to magnetic field of the measured curves shown in a. g, When plotted against H/T, the derivatives presented in f collapse onto a universal curve (with the exception of those sections of each field sweep that are in the mixed state).

### Extended Data Fig. 3 Success of ADMR-derived modelling of the in-plane transport of overdoped Tl2201.

a, The c-axis ADMR of Tl2201 with Tc = 15 K measured at 50 K and at various (labelled) azimuthal angles taken from ref. 48. b, Projection of the in-plane Fermi surface derived from the ADMR fitting. c, Schematic showing the isotropic T2 component (black solid line) and anisotropic T + T2 component (red solid line) of the scattering rate as deduced from the ADMR fitting. d, Black dots: ρab(T) data for overdoped Tl2201 (Tc = 15 K) in which superconductivity has been suppressed by a magnetic field (H c)46 and corresponding simulation based on the ADMR fitting48. The difference in the residual resistivities is probably because different samples have been used in the two studies46,48. e, Corresponding simulation for RH(T)48. f, Simulation of RH(H) = ρxy(H)/H at various temperatures as indicated. g, Same simulation data plotted versus H/ρ(0) where here, ρ(0) is the zero-field resistivity at each temperature. h, RH(H) versus H/ρ(0) data taken from ref. 25. For overdoped Tl2201 (Tc = 25 K) for comparison with the simulation in g. The larger absolute values of RH in h relative to g are due to the fact that the high-field data in h are taken on a sample with a higher Tc value where the anisotropy in τ−1(ϕ) is expected to be larger.

### Extended Data Fig. 4 Failure of ADMR-derived modelling to reproduce quadrature scaling.

a, The field dependence of the longitudinal resistivity (ρ(T)) determined with the SCTIF using parameters derived from the ADMR parameterization for overdoped Tl2201. b, Δρ—the change in ρxx with field—at selected temperatures. c, Corresponding derivative plots of ρ(H) showing distinctly non-quadrature behaviour. d, As a consequence, the data fail to collapse when plotted against μ0H/T.

### Extended Data Fig. 5 Failure of the SCTIF to reproduce both the MR and Hall response.

Simulations of the MR and Hall responses within the SCTIF given different parameterizations of vF(ϕ) and τ−1(ϕT). In each simulation, the experimentally determined Fermi surface (kF(ϕ)) has been used. Note that the SCTIF is slow to converge at low fields and so the simulations do not extend all the way to H = 0. Simulation 1, ADMR-derived parameterization of overdoped Tl2201 albeit with no anisotropy at T = 0 and an anisotropic term that increases strictly linearly with T. Simulation 2, A scenario incorporating the vF anisotropy derived from tight-binding modelling of ARPES measurements37. Simulation 3, A scenario in which the anisotropy ratio of τ−1(ϕ) is strictly T-independent in order to generate an MR with a maximum slope that is also independent of temperature (reminiscent of quadrature scaling). Simulation 4, A scenario in which a similar τ−1(ϕ) parameterization to that used to model Nd-LSCO53 is applied to overdoped Tl2201. Simulation 5, Simulation for overdoped Bi2201 with an enhanced anisotropy in vF and τ−1(T = 0) consistent with ARPES54. Column 1, The Fermi surface parameterizations kF(ϕ) and vF(ϕ). Column 2, τ−1(ϕT). Column 3, dρ/d(μ0H) versus H. Column 4, dρ/d(μ0H) versus H/T. Column 5, RH(HT).

### Extended Data Fig. 6 Kohler versus quadrature scaling in Bi2201.

a, Δρab/ρab(0) plotted versus (H/ρab(0))2 for a Bi2201 sample with Tc = 13 K. In a system that shows Kohler scaling, these curves would collapse. Clearly, that is not the case here. b, Δρab plotted versus H2 for the same Bi2201 sample. The dotted lines are fits to the function f(x) = A(μ0H)2 in the regions where the MR is strictly quadratic. Note that the quadrature form of the MR is only purely quadratic in the zero-field limit whereas fits to the data are taken at finite field ranges. Our simulations have shown that fitting up to μ0Hmax = βT (with β as given in Fig. 2 of the main text) agrees with the zero-field limit within a few percent and falls within our experimental error. c, T dependence of ab(0) (with A taken from the fits in b) compared to the square of the Hall coefficient $${R}_{{\rm{H}}}^{2}$$. d, Temperature derivative of ρab(T) for the same sample. Note that the onset of superconducting fluctuations appears only below 30 K. e, The product AT plotted over the full temperature range. The dotted line is a guide to the eye.

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Ayres, J., Berben, M., Čulo, M. et al. Incoherent transport across the strange-metal regime of overdoped cuprates. Nature 595, 661–666 (2021). https://doi.org/10.1038/s41586-021-03622-z

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