## Abstract

The perfectly linear temperature dependence of the electrical resistivity observed as *T* → 0 in a variety of metals close to a quantum critical point^{1,2,3,4} is a major puzzle of condensed-matter physics^{5}. Here we show that *T*-linear resistivity as *T* → 0 is a generic property of cuprates, associated with a universal scattering rate. We measured the low-temperature resistivity of the bilayer cuprate Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} and found that it exhibits a *T*-linear dependence with the same slope as in the single-layer cuprates Bi_{2}Sr_{2}CuO_{6+δ} (ref. ^{6}), La_{1.6−x}Nd_{0.4}Sr_{x}CuO_{4} (ref. ^{7}) and La_{2−x}Sr_{x}CuO_{4} (ref. ^{8}), despite their very different Fermi surfaces and structural, superconducting and magnetic properties. We then show that the *T*-linear coefficient (per CuO_{2} plane), *A*_{1}^{□}, is given by the universal relation *A*_{1}^{□}*T*_{F} = *h*/2*e*^{2}, where *e* is the electron charge, *h* is the Planck constant and *T*_{F} is the Fermi temperature. This relation, obtained by assuming that the scattering rate 1/*τ* of charge carriers reaches the Planckian limit^{9,10}, whereby *ħ*/*τ* = *k*_{B}*T*, works not only for hole-doped cuprates^{6,7,8,11,12} but also for electron-doped cuprates^{13,14}, despite the different nature of their quantum critical point and strength of their electron correlations.

## Main

In conventional metals, the electrical resistivity *ρ*(*T*) normally varies as *T*^{2} in the limit *T* → 0, where electron–electron scattering dominates, in accordance with Fermi-liquid theory. However, close to a quantum critical point (QCP) where a phase of antiferromagnetic order ends, *ρ*(*T*) ~ *T*^{n}, with *n* < 2.0. Most striking is the observation of a perfectly linear *T* dependence *ρ*(*T*) = *ρ*_{0} + *A*_{1}*T* as *T* → 0 in several very different materials, when tuned to their magnetic QCP; for example, the quasi-one-dimensional (1D) organic conductor (TMTSF)_{2}PF_{6} (ref. ^{4}), the quasi-2D ruthenate Sr_{3}Ru_{2}O_{7} (ref. ^{3}) and the 3D heavy-fermion metal CeCu_{6} (ref. ^{1}). This *T*-linear resistivity as *T* → 0 has emerged as one of the major puzzles in the physics of metals^{5}, and while several theoretical scenarios have been proposed^{15}, no compelling explanation has been found.

In cuprates, a perfect *T*-linear resistivity as *T* → 0 has been observed (once superconductivity is suppressed by a magnetic field) in two closely related electron-doped materials, Pr_{2−x}Ce_{x}CuO_{4±δ} (PCCO)^{2,16,17} and La_{2−x}Ce_{x}CuO_{4} (LCCO)^{13,14}, and in three hole-doped materials: Bi_{2}Sr_{2}CuO_{6+δ} (ref. ^{6}), La_{2}_{−}_{x}Sr_{x}CuO_{4} (LSCO)^{8} and La_{1.6−x}Nd_{0.4}Sr_{x}CuO_{4} (Nd-LSCO)^{7,11,12}. On the electron-doped side, *T*-linear resistivity is seen just above the QCP^{16} where antiferromagnetic order ends^{18} as a function of *x*, and as such it may not come as a surprise. On the hole-doped side, however, the doping values where *ρ*(*T*) = *ρ*_{0} + *A*_{1}*T* as *T* → 0 are very far from the QCP where long-range antiferromagnetic order ends (*p*_{N} ~ 0.02); for example, at *p* = 0.24 in Nd-LSCO (Fig. 1a) and in the range *p* = 0.21–0.26 in LSCO (Fig. 1b). Instead, these values are close to the critical doping where the pseudogap phase ends (that is, at *p** = 0.23 ± 0.01 in Nd-LSCO (ref. ^{11}) and at *p** ~ 0.18–0.19 in LSCO (ref. ^{8})), where the role of antiferromagnetic spin fluctuations is not clear. In Bi2201, *p** is farther still (see Supplementary Section 10).

To make progress, several questions must be answered. Is *T*-linear resistivity as *T* → 0 in hole-doped cuprates limited to single-layer materials with low *T*_{c}, or is it generic? Why is *ρ*(*T*) = *ρ*_{0} + *A*_{1}*T* as *T* → 0 seen in LSCO over an anomalously wide doping range^{8}? Is there a common mechanism linking cuprates to the other metals where *ρ* ~ *T* as *T* → 0?

To establish the universal character of *T*-linear resistivity in cuprates, we have turned to Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} (Bi2212). While Nd-LSCO and LSCO have essentially the same single electron-like diamond-shaped Fermi surface at *p* > *p** (refs ^{19,20}), Bi2212 has a very different Fermi surface, consisting of two sheets, one of which is also diamond-like at *p* > 0.22, but the other is much more circular^{21} (see Supplementary Section 1). Moreover, the structural, magnetic and superconducting properties of Bi2212 are very different to those of Nd-LSCO and LSCO: a stronger 2D character, a larger gap to spin excitations, no spin-density-wave order above *p* ~ 0.1 and a much higher superconducting *T*_{c}.

We measured the resistivity of Bi2212 at *p* = 0.23 by suppressing superconductivity with a magnetic field of up to 58 T. At *p* = 0.23, the system is just above its pseudogap critical point (*p** = 0.22 (ref. ^{22}); see Supplementary Section 2). Our data are shown in Fig. 2. The raw data at *H* *=* 55 T reveal a perfectly linear *T* dependence of *ρ*(*T*) down to the lowest accessible temperature (Fig. 1a). Correcting for the magnetoresistance (see Methods and Supplementary Section 3), as was done for LSCO (ref. ^{8}), we find that the *T*-linear dependence of *ρ*(*T*) seen in Bi2212 at *H* *=* 0 from *T* ~ 120 K down to *T*_{c} simply continues to low temperature, with the same slope *A*_{1} = 0.62 ± 0.06 μΩ cm K^{−1} (Fig. 2b). Measured per CuO_{2} plane, this gives *A*_{1}^{□} ≡ *A*_{1}/*d* *=* 8.0 ± 0.9 Ω K^{−1}, where *d* is the (average) separation between CuO_{2} planes. Remarkably, this is the same value, within error bars, as measured in Nd-LSCO at *p* = 0.24, where *A*_{1}^{□} = 7.4 ± 0.8 Ω K^{−1} (see Table 1).

The observation of *T*-linear resistivity in those two cuprates shows that it is robust against changes in the shape, topology and multiplicity of the Fermi surface. By contrast, the Hall coefficient *R*_{H} is not. In Fig. 2d, we compare *R*_{H}(*T*) in Bi2212 and in Nd-LSCO (and PCCO). We see strong differences, brought about by the different anisotropies in either the inelastic scattering or the Fermi surface, or both^{23}. Nevertheless, *ρ*(*T*) is perfectly linear in both cases. Moreover, the coefficient *A*_{1}^{□} is the same despite the very different spectra of low-energy spin fluctuations, gapped in Bi2212 (ref. ^{24}) and ungapped in Nd-LSCO (ref. ^{25}). We conclude that a *T*-linear resistivity as *T* → 0 is a generic and robust property of cuprates.

Note that *ρ*(*T*) deviates from pure *T*-linearity above a certain temperature, and in this high-*T* regime a generic evolution has also been found in LSCO (ref. ^{26}), with *ρ*(*T*) ~ *A*_{1}*T* + *A*_{2}*T*^{2}. Here we focus strictly on the low-*T* regime of pure *T*-linear resistivity (see Supplementary Section 12). In this regime, and close to the QCP of BaFe_{2}(As_{1−x}P_{x})_{2} (at *x* = 0.31), an empirical scaling relationship between applied magnetic field and temperature has been proposed^{27}, but this scaling does not work very well in Bi2212 (see Supplementary Section 11).

We now investigate the strength of the *T*-linear resistivity; that is, the magnitude of *A*_{1}. In Fig. 3b, we plot *A*_{1}^{□} versus *p* for hole-doped cuprates. We see from the LSCO data^{8} that *A*_{1}^{□} increases with decreasing *p* (Fig. 1b), from *A*_{1}^{□} ~ 8 Ω K^{−1} at *p* = 0.26 to *A*_{1}^{□} ~ 15 Ω K^{−1} at *p* = 0.21 (see Supplementary Table 2 in Supplementary Section 13 and Methods). In Nd-LSCO, we see a similar increase (Figs. 1c and 3b), when pressure^{12} is used to suppress the onset of the pseudogap at *p* = 0.22 and *p* = 0.23 (see Supplementary Section 4). In Fig. 1d, we present our data on PCCO at *x* = 0.17 (see also Supplementary Section 5), and compare with previous data on LCCO (ref. ^{14}; Supplementary Section 6). In Fig. 4b, we plot *A*_{1}^{□} versus *x* for electron-doped cuprates, and see that *A*_{1}^{□} also increases with decreasing *x*, from *A*_{1}^{□} ~ 1.5 Ω K^{−1} at *x* = 0.17 to *A*_{1}^{□} ~ 3 Ω K^{−1} at *x* = 0.15 (see Supplementary Table 4 in Supplementary Section 13 and Methods). Note that these values are five times smaller than in hole-doped cuprates.

To summarize: *A*_{1}^{□} increases as the doping is reduced in both hole-doped and electron-doped cuprates; *A*_{1}^{□} is much larger in hole-doped cuprates; *T*-linear resistivity as *T* → 0 is observed over a range of doping, not just at one doping; *T*-linear resistivity does not depend on the nature of the inelastic scattering process (hole-doped versus electron-doped) or on the topology of the Fermi surface (LSCO versus NCCO, Bi2212 versus Nd-LSCO; Supplementary Section 1).

To explain these experimental facts, we consider the empirical observation that the strength of the *T*-linear resistivity for several metals is approximately given by a scattering rate that has a universal value, namely *ħ*/*τ* = *k*_{B}*T* (ref. ^{10}), and test it in cuprates. This observation suggests that a *T*-linear regime will be observed whenever 1/*τ* reaches its Planckian limit, *k*_{B}*T*/*ħ*, irrespective of the underlying mechanism for inelastic scattering^{9}. In the following, we use a standard Fermi-liquid approach to extract effective masses and inelastic scattering rates, as in ref. ^{10}. In the simple case of an isotropic Fermi surface, the connection between *ρ* and *τ* is given by the Drude formula, *ρ* = (*m**/*ne*^{2}) (1/*τ*), where *n* is the carrier density and *m** is the effective mass. Thus, when *ρ*(*T*) = *ρ*_{0} + *A*_{1}*T*, then *A*_{1} *=* (*m**/*n e*^{2})(1/*τ*)(1/*T*) = *α*(*m**/*n*)(*k*_{B}/*e*^{2}*ħ*), with *ħ*/*τ* ≡ *αk*_{B}*T*. In two dimensions, this can be written succinctly as:

where *T*_{F} = (π*ħ*^{2}/*k*_{B})(*nd*/*m**) is the Fermi temperature.

Let us first evaluate *α* in electron-doped cuprates, where the Drude formula is expected to work well, since their single Fermi surface is highly 2D and circular (in the overdoped region^{28}; see Supplementary Section 1). Quantum oscillations in Nd_{2−x}Ce_{x}CuO_{4} (NCCO) provide a direct and precise measurement of *n* and *m** in electron-doped cuprates^{29,30}. The Luttinger rule sets the carrier density to be *n* = (1 − *x*)/(*a*^{2}*d*), given precisely by the oscillation frequency *F* = *nd*(*h*/2*e*), where *x* is the number of doped electrons per Cu atom and *a* is the in-plane lattice constant. In Fig. 4a, we see that *m** increases from 2.3 *m*_{0} at *x* = 0.173 to 3.0 *m*_{0} at *x* = 0.151, where *m*_{0} is the bare electron mass (see Supplementary Table 3 in Supplementary Section 13 and Methods). This increasing value is consistent, within error bars, with specific heat data in PCCO at *x* = 0.15, where *γ* = 5.5 ± 0.4 mJ K^{−}^{2} mol^{−1} (ref. ^{31}), which yields *m** = 3.6 ± 0.3 *m*_{0} (see equation (2) below). We use *n* and *m** to estimate *T*_{F} and then plot, in Fig. 4b, the value of *A*_{1}^{□} predicted by equation (1), for *α* = 1 (solid line in Fig. 4b; Supplementary Table 3 in Supplementary Section 13). Comparison with the measured values of *A*_{1}^{□} in PCCO (red hexagon in Fig. 4b) and in LCCO (blue circles in Fig. 4b), listed in Supplementary Table 4 (see Supplementary Section 13 and Methods), shows that the scattering rate in electron-doped cuprates is given by *ħ*/*τ* = *α**k*_{B}*T*, with *α* = 1.0 ± 0.3; that is, the Planckian limit is observed, within experimental error bars.

Let us now turn to hole-doped cuprates. Here our quantitative estimates will be more approximate, since Fermi surfaces are not circular but diamond-shaped (Supplementary Section 1), but we are looking for a large effect (factor ~5 in *A*_{1}^{□} relative to electron-doped materials) and a qualitative trend (increase in *A*_{1}^{□} as *p* is reduced towards *p**). In the absence of quantum oscillation data for Bi2212, LSCO, Nd-LSCO and Bi2201, we estimate *m** from specific heat data, since in two dimensions the specific heat coefficient *γ* is directly related to *m**:

for a single Fermi surface, where *N*_{A} is Avogadro’s number. This connection between *m** and *γ* was nicely confirmed by quantum oscillations in Tl_{2}Ba_{2}CuO_{6+δ} at *p* ~ 0.3, where *m** = 5.2 ± 0.4 *m*_{0} and *γ* = 7 ± 1 mJ K^{−2} mol^{−1} (ref. ^{32}). In Bi2212, *γ* = 12 ± 2 mJ K^{−2} per mol-Cu at *p* *=* 0.22 = *p** (ref. ^{33}; see Supplementary Section 8), giving *m** = 8.4 ± 1.6 *m*_{0} (equation (2)). Applying equation (1), with *n*(*a*^{2}*d*) = 1 − *p* = 0.77 (for an electron-like Fermi surface; Supplementary Section 1), the Planckian limit predicts *A*_{1}^{□} = 7.4 ± 1.4 Ω K^{−1}, while we measured *A*_{1}^{□} = 8.0 ± 0.9 Ω K^{−1}, so that *α* = 1.1 ± 0.3 (Table 1).

In LSCO, *γ* increases from *γ* = 6.9 ± 1 mJ K^{−}^{2} mol^{−}^{1} at *p* = 0.33 (ref. ^{34}) to *γ* = 14 ± 2 mJ K^{−}^{2} mol^{−}^{1} at *p* *=* 0.26 (ref. ^{35}), showing that *m** increases with reduced doping also in hole-doped cuprates (solid line in Fig. 3a). Applying equation (1) to LSCO data at *p* *=* 0.26, using *n*(*a*^{2}*d*) = 1 − *p* = 0.74 and *m** = 9.8 ± 1.7 *m*_{0} (equation (2); Supplementary Table 1 in Supplementary Section 13), the Planckian limit predicts *A*_{1}^{□} = 8.9 ± 1.8 Ω K^{−1}, while we see *A*_{1}^{□} = 8.2 ± 1.0 Ω K^{−1} (Fig. 1b and Supplementary Table 2 in Supplementary Section 13), so that *α* = 0.9 ± 0.3 (Table 1).

In Nd-LSCO, an increase in *m** has also been observed in recent specific heat measurements^{36}, from *γ* = 5.4 ± 1 mJ K^{−2} mol^{−1} at *p* = 0.40 to *γ* = 11 ± 1 mJ K^{−}^{2} mol^{−1} at *p* *=* 0.27 (Fig. 3a). At *p* = 0.24, the electronic specific heat *C*_{el} varies as *C*_{el}/*T* ~ log(1/*T*), which complicates the estimation of *m**. Taking the mean value between *C*_{el}/*T* = 12 mJ K^{−}^{2} mol^{−1} at 10 K and *C*_{el}/*T* = 22 mJ K^{−}^{2} mol^{−1} at 0.5 K (ref. ^{36}), we get *m** = 12 ± 4 *m*_{0} and hence *α* = 0.7 ± 0.4, consistent with the Planckian limit for a third hole-doped material. See Table 1 for a summary of the numbers.

Finally, a stringent test of whether the Planckian limit operates in cuprates is provided by Bi2201, since in this particular cuprate the pseudogap critical point that controls *T*-linear scattering occurs at a much higher doping than in other cuprates, namely *p** ~ 0.4 (see Supplementary Section 10). Despite this doubling of *p** and the very different volume of the Fermi surface relative to Bi2212, LSCO and Nd-LSCO, we find that *α* = 1.0 ± 0.4 in Bi2201 (Table 1 and Supplementary Section 10).

In summary, our estimations reveal that the scattering rate responsible for the *T*-linear resistivity in PCCO, LCCO, Bi2212, LSCO, Nd-LSCO and Bi2201 tends to the same universal value, namely *ħ*/*τ* = *αk*_{B}*T*, with *α* = 1.0 (Table 1). A constant value of *α* in equation (1) implies that *A*_{1}^{□} ~ 1/*T*_{F}, so that, in essence, *A*_{1}^{□} ~ *m**. This explains why the slope of the *T*-linear resistivity is much larger in hole-doped than in electron-doped cuprates, since the effective mass is much higher in the former (Fig. 3a versus Fig. 4a). It also explains why *A*_{1}^{□} increases in LSCO when going from *p* = 0.26 to *p* = 0.21 (Fig. 1b) and in Nd-LSCO (under pressure) when going from *p* = 0.24 to *p* = 0.22 (Fig. 1c). Indeed, as shown in Fig. 3, *A*_{1}^{□} (Fig. 3b) and *m** (Fig. 3a) in LSCO and Nd-LSCO are seen to rise in tandem with decreasing *p* (we make the natural assumption that *m** continues to rise until *p* reaches *p** and that the pressure does not change the specific heat significantly above *p*^{*} in Nd-LSCO). Moreover, a Planckian limit on scattering provides an explanation for the ‘anomalous’ range in doping over which *ρ* ~ *A*_{1}*T* is observed in LSCO (ref. ^{8}). As doping decreases below *p* ~ 0.33, scattering increases steadily until *p** ~ 0.18–19, but the inelastic scattering rate 1/*τ* cannot exceed the Planckian limit, reached at *p* ~ 0.26. Thus, between *p* ~ 0.26 and *p* = *p**, *ρ*(*T*) is linear and 1/*τ* saturates. The continuous increase of *A*_{1} below *p* ~ 0.26 can be understood if we assume that *m** continues to increase in the range *p** < *p* < 0.26 (ref. ^{36}), since *A*_{1} ~ *m**(1/*τ*) ~ *m**. If one could lower *p**, the range of *T*-linear resistivity would expand further. This is indeed what happens in Nd-LSCO when *p** is lowered by applying pressure^{12} (Fig. 3b).

The fact that *α* ~ 1.0 in cuprates has far-reaching implications since other metals with *T*-linear resistivity as *T* → 0 also appear to have *α* ~ 1.0 (ref. ^{10}). The case is particularly clear in the organic conductor (TMTSF)_{2}PF_{6}, a well-characterized single-band metal whose resistivity is perfectly *T*-linear as *T* → 0 (ref. ^{4}), where *α* = 1.0 ± 0.3 (Table 1 and Supplementary Section 9). For such dramatically different metals as the quasi-1D organics and the cuprates—not to mention the heavy-fermion metals and the pnictides^{10}—to all have quantitatively the same scattering rate in their respective *T*-linear regimes, there must be a fundamental and universal principle at play. Our findings support the idea^{9,10} that *T*-linear resistivity is achieved when the scattering rate hits the Planckian limit, given by *ħ*/*τ* = *k*_{B}*T*, whatever the scattering process, whether by antiferromagnetic spin fluctuations or not. If Planckian dissipation is the fundamental principle, new theoretical approaches are needed to understand how it works^{37,38,39}.

## Methods

### Samples

#### Bi2212

Our thin film of Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} (Bi2212) was grown epitaxially at 740 °C on a SrTiO_{3} substrate by radiofrequency magnetron sputtering with O_{2}/Ar gas and fully oxygen overdoped after deposition^{40}. The film thickness was measured by deposition rate calibration, giving *t* = 240 ± 15 nm. The film was patterned by mechanical scribing (avoiding the need for a lithography resist) into the shape of a Hall bar consisting of two large pads (for current) connected by a narrow bridge (275 µm wide) between 2 couples of voltage pads distant by 1.15 mm for longitudinal and transverse resistance measurements. Six gold contacts were deposited by sputtering on the different pads and gold wires were attached with silver paint.

The superconducting transition temperature *T*_{c} = 50 K was determined as the temperature below which the zero-field resistance *R* = 0. The hole doping *p* is obtained from *T*_{c}, using the usual convention^{21,22}, according to which our overdoped sample has a nominal doping *p* = 0.23. This means that its doping is just slightly above the end of the pseudogap phase^{22} (see Supplementary Section 2). It is also just above the Lifshitz transition where its anti-bonding band crosses the Fermi level to produce an electron-like diamond-shaped Fermi surface^{21} (see Supplementary Section 1).

#### PCCO

Our thin films of Pr_{2−x}Ce_{x}CuO_{4±δ} (PCCO) were grown by pulsed laser deposition on LSAT substrates under 200 mTorr of N_{2}O using targets including an excess of Cu to suppress the growth of parasitic phases^{41}. Films were then annealed for 4 min in vacuum. The film thickness was measured via the width of X-ray diffraction peaks, giving *t* = 230 ± 30 nm. A very small amount of parasitic phase was detected in the X-ray diffraction spectra. However, its impact on the cross-section of the films should be much smaller than the uncertainty coming from the thickness measurement. Six indium–silver contacts were applied in the standard geometry.

The superconducting transition temperature *T*_{c} = 13 K was determined as the temperature below which the zero-field resistance *R* = 0. The electron concentration is taken to be the cerium content, *x* = 0.17, with an error bar ±0.005. This means that our samples have a concentration slightly above the quantum critical point where the Fermi surface of PCCO is known to undergo a reconstruction by antiferromagnetic ordering^{16}. The Fermi surface of NCCO at that doping could not be simpler: it is a single circular cylinder^{28} (see Supplementary Section 1).

### Measurement of the longitudinal and transverse resistances

The longitudinal resistance *R*_{xx} and transverse (Hall) resistance *R*_{xy} of our Bi2212 film were measured in Toulouse in pulsed fields up to 58 T. The measurements were performed using a conventional 6-point configuration with a current excitation of 0.5 mA at a frequency of ~10 kHz. A high-speed acquisition system was used to digitize the reference signal (current) and the voltage drop across the sample at a frequency of 500 kHz. The data were post-analysed with software to perform the phase comparison. Data for the rise and fall of the field pulse were in good agreement, thus excluding any heating due to eddy currents. Tests at different frequencies showed excellent reproducibility.

*R*_{xx} and *R*_{xy} of our Bi2212 film were also measured in Orsay, at *H* *=* 0 and *H* *=* 9 T, respectively.

The longitudinal resistance *R*_{xx} values of our three PCCO films were measured in Sherbrooke in a zero field and in a steady field of 16 T.

### Values of *m** and *A*
_{1}

#### Hole-doped cuprates

The values of *p* and *m** used in Fig. 3a are listed in Supplementary Table 1 (see Supplementary Section 13). For Nd-LSCO, the value of *p* with its error bar is taken from ref. ^{11}. For LSCO, the value of *p* is taken from refs ^{34,35}, and we assume the same error bar as for Nd-LSCO. The value of *m** is obtained from the measured specific heat *γ*, via equation (2). For Nd-LSCO, the value of *γ* with its error bar is taken from ref. ^{36}, except for *p* = 0.24, where we take the average between the electronic specific heat *C*_{e}/*T* at *T* = 10 K (12 mJ K^{−2} mol^{−1}) and at *T* = 0.5 K (22 mJ K^{−2} mol^{−1}), given that *C*_{e}/*T* is not constant at low *T* (ref. ^{36}). For Bi2212 and LSCO, we estimate *γ* and its error bar from the data published in ref. ^{33} and in refs ^{34,35}, respectively (see Supplementary Section 8). With these values of *m**, we calculate *T*_{F} = (π*ħ*^{2}/*k*_{B})(*nd*/*m**), using *n* = (1 − *p*)/(*a*^{2}*d*) since the Fermi surface of Bi2212, LSCO and Nd-LSCO is electron-like at the dopings considered here (see Supplementary Section 1). We then obtain the Planckian limit on the resistivity slope, namely *A*_{1}^{□} = *h*/(2*e*^{2}*T*_{F}), whose values are listed in the last column of Supplementary Table 2 (see Supplementary Section 13) and plotted in Fig. 3b (solid black line). For Bi2201, the values of *n*, *m** and *A*_{1} are given in Supplementary Section 10, with associated error bars and references.

The values of *p* and *A*_{1} used in Fig. 3b are listed in Supplementary Table 2 (see Supplementary Section 13). For Nd-LSCO, the value of *p* with its error bar is taken from ref. ^{11}. For LSCO, the value of *p* is taken from refs ^{8,42}, and we assume the same error bar as for Nd-LSCO. For Nd-LSCO, the value of *A*_{1} is obtained from a linear fit to the raw data in Fig. 1a (*p* = 0.24, at *H* = 16 T) and in Fig. 1c (*p* = 0.22 and 0.23, at *H* = 33 T and *P* = 2 GPa). Note that the magnetoresistance is very weak in Nd-LSCO. For example, at *p* = 0.24, *A*_{1} = 0.47 μΩ cm K^{−1} at *H* = 33 T (Fig. 1c) versus *A*_{1} = 0.49 μΩ cm K^{−1} at *H* = 16 T (Fig. 1a). For LSCO, the value of *A*_{1} is obtained from a linear fit to the raw data in Fig. 1b (*p* = 0.26, at *H* = 18 T) and to the magnetoresistance-corrected data in Supplementary Section 7 (*p* = 0.21 and 0.23). Note that the magnetoresistance in LSCO does not significantly change the slope *A*_{1} (Fig. 1b versus Supplementary Fig. 7). For Bi2212, the value of *A*_{1} is obtained from a linear fit to magnetoresistance-corrected data (Fig. 2b and Supplementary Section 3). The error bar on *A*_{1} is in all cases taken to be ±10%, the estimated uncertainty in measuring the geometric factor of small samples. The values of *A*_{1} listed in Supplementary Table 2 (see Supplementary Section 13) are used to obtain the experimental values of *A*_{1}^{□} = *A*_{1}/*d* that are plotted in Fig. 3b.

#### Electron-doped cuprates

The values of *x* and *m** used in Fig. 4a are listed in Supplementary Table 3 (see Supplementary Section 13). For NCCO, the value of *x* is obtained from the frequency *F* of quantum oscillations, measured precisely^{29,30} (see Supplementary Section 1), via *x* = 1 − (2*eFa*^{2}/*h*). The value of *m** is obtained directly from the high-frequency quantum oscillations, as reported (with error bar) in refs ^{29,30}. At the lowest doping (for example, *x* = 0.15), the Fermi surface of NCCO is reconstructed by an antiferromagnetic order leading to small electron and hole pockets. The small gap between the pockets decreases as the doping increases and, at *x* = 0.17, angle-resolved photoemission spectroscopy data show that the large hole-like Fermi surface is recovered^{28}. At doping levels *x* = 0.151 to 0.163, we assume that the effective mass of the high quantum oscillation frequency due to magnetic breakdown is representative of the large hole-like Fermi surface, as expected in semiclassical theory^{43}. The continuous trend of decreasing effective mass in the doping range *x* = 0.151 to 0.163 is confirmed by the smaller (and smoothly connected) value at *x* = 0.173.

For PCCO, *x* is taken to be the cerium content, with an error bar ±0.005. Here *m** is obtained from the measured specific heat *γ*, via equation (2), and the value of *γ* (with its error bar) is taken from ref. ^{31}. With these values of *m**, we calculate *T*_{F} = (π*ħ*^{2}/*k*_{B})(*nd*/*m**), using *n* = (1 − *x*)/(*a*^{2}*d*) since the Fermi surface of NCCO and PCCO is hole-like (see Supplementary Section 1). The good agreement between the specific heat data in PCCO at *x* = 0.15 and the values of the effective mass in NCCO (Fig. 4a) further confirms the validity of our approach regarding the extraction of *m**.

We then obtain the Planckian limit on the resistivity slope, namely *A*_{1}^{□} = *h*/(2*e*^{2}*T*_{F}), whose values are listed in the last column of Supplementary Table 3 (see Supplementary Section 13) and plotted in Fig. 4b (solid black line).

The values of *x* and *A*_{1} used in Fig. 4b are listed in Supplementary Table 4 (see Supplementary Section 13). In all cases, *x* is taken to be the cerium content, with an error bar ±0.005. For PCCO at *x* = 0.17, the value of *A*_{1} is obtained from a linear fit to the raw data in Supplementary Section 5. Within error bars, the same value is measured in all three PCCO films, whether at *H* = 0 or at *H* = 16 T. For LCCO, the value of *A*_{1} is obtained from a linear fit to the raw data in Fig. 1d. Also listed in Supplementary Table 4 (see Supplementary Section 13) are the values of *A*_{1} obtained from a linear fit to the raw zero-field data in LCCO (see Supplementary Section 6). The error bar on *A*_{1} is ±15% for our PCCO film, the uncertainty in measuring the film thickness. We apply the same error bar for LCCO. The values of *A*_{1} listed in Supplementary Table 4 (see Supplementary Section 13), both in zero field and in finite field, are used to obtain the experimental values of *A*_{1}^{□} = *A*_{1}/*d* that are plotted in Fig. 4b (as filled and open circles, respectively).

## Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors (L.T. or C.P.) upon reasonable request.

## Additional information

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## Acknowledgements

The authors would like to thank K. Behnia, C. Bourbonnais, R. Greene, S. Hartnoll, N. Hussey, M.-H. Julien, D. Maslov, J. Paglione, S. Sachdev, A.-M. Tremblay and J. Zaanen for fruitful discussions. A portion of this work was performed at the LNCMI, a member of the European Magnetic Field Laboratory (EMFL). C.P. acknowledges funding from the French ANR SUPERFIELD, and the LABEX NEXT. P.F. and L.T. acknowledge support from the Canadian Institute for Advanced Research (CIFAR) and funding from the Natural Sciences and Engineering Research Council of Canada (NSERC), the Fonds de recherche du Québec - Nature et Technologies (FRQNT) and the Canada Foundation for Innovation (CFI). L.T. acknowledges support from 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.).

## Author information

### Affiliations

#### Institut Quantique, Département de Physique & RQMP, Université de Sherbrooke, Sherbrooke, Québec, Canada

- A. Legros
- , F. Laliberté
- , M. Dion
- , M. Lizaire
- , N. Doiron-Leyraud
- , P. Fournier
- & L. Taillefer

#### SPEC, CEA, CNRS-UMR 3680, Université Paris-Saclay, Gif sur Yvette Cedex, France

- A. Legros
- & D. Colson

#### Laboratoire National des Champs Magnétiques Intenses (CNRS, EMFL, INSA, UJF, UPS), Toulouse, France

- S. Benhabib
- , W. Tabis
- , B. Vignolle
- , D. Vignolles
- & C. Proust

#### AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Krakow, Poland

- W. Tabis

#### Laboratoire de Physique des Solides, Université Paris-Sud, Université Paris-Saclay, CNRS UMR 8502, Orsay, France

- H. Raffy
- , Z. Z. Li
- & P. Auban-Senzier

#### Canadian Institute for Advanced Research, Toronto, Ontario, Canada

- P. Fournier
- , L. Taillefer
- & C. Proust

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### Contributions

A.L., S.B., W.T., B.V., D.V. and C.P. performed the transport measurements at the LNCMI. A.L., F.L., M.L. and N.D.-L. performed the transport measurements at Sherbrooke. H.R., P.A.-S. and Z.Z.L. prepared the Bi2212 film, which was then characterized by A.L., H.R., P.A.-S., Z.Z.L. and D.C. M.D. and P.F. prepared and characterized the PCCO films. A.L., F.L., L.T. and C.P. wrote the manuscript, in consultation with all authors. L.T. and C.P. co-supervised the project.

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to L. Taillefer or C. Proust.

## Supplementary information

### Supplementary Information

10 Figures, 4 Tables, 11 References

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