Introduction

After three decades of extensive experimental and theoretical efforts, the nature of the emergence of superconductivity in the cuprates remains controversial.1 At temperatures above the macroscopic superconducting transition temperature Tc, there exists no long-range coherence, yet traces of superconductivity remain observable, and different experimental investigations have led to widely disparate conclusions.2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 It was recently established that the normal state of underdoped cuprates exhibits Fermi-liquid charge transport17,18,19,20,21 and proposed that superconductivity emerges from this state in a universal percolative manner.8,9 Direct-current (dc) conductivity is a highly sensitive probe that can, in principle, provide a unique opportunity to test the consistency of these ideas. Furthermore, the effective-medium approximation required to model such a mixed regime is well established for the dc conductivity response,22,23,24,25,26 whereas calculations for other observables (e.g., magnetic susceptibility) are challenging. Here, we present benchmark dc conductivity data for a pristine cuprate compound, along with modeling and analysis of published results for other cuprates, that confirm the Fermi-liquid nature of the normal state and support the interpretation of percolative superconductivity emergence in a quantitative manner.

The principal problem in many previous investigations of the pre-pairing regime has been the separation of the superconducting response from the normal-state response. Different experimental probes can be sensitive to distinct aspects of the normal state. Moreover, it is well established that the underdoped cuprates also exhibit other electronic ordering tendencies, including charge-density-wave order,27,28,29,30,31,32,33 which has further precluded an unequivocal extraction of superconducting contributions. Prominent examples of such problems include the analysis of the Nernst effect,10,11 where a charge-order related signal might be mistaken for superconducting fluctuations,34,35 or linear magnetization and conductivity measurements,4,12,13,36 where the normal-state behavior is assumed to be linear in temperature, which is not necessarily the case.8 Several schemes to systematically subtract the presumed normal-state contribution have been devised, mainly based on the suppression of superconductivity with external magnetic fields.5,7,16 However, so far, only two experimental techniques are genuinely sensitive to only superconducting signals: nonlinear torque magnetization8 and nonlinear conductivity.9 Here we introduce a method to separate the dc paraconductivity from the normal-state contribution, with the resulting high-quality data readily comparable to the more involved background-free techniques.

A number of recent experimental investigations consistently point to a simple picture for both the normal state17,18,19,20,21 and the superconducting emergence regime.3,6,8,9 Measurements of transport properties, such as the dc resistivity,20 Hall angle,17 magnetoresistivity,19 and optical experiments21 unequivocally show that the mobile charge carriers behave as a Fermi liquid, even in strongly underdoped compounds. NMR investigations also find an itinerant Fermi-liquid component, along with a local, pseudogap-related signal.37 The observations that the magnetoresistivity obeys Kohler scaling with a 1/τ ∝ T2 scattering rate19 and that the optical scattering rate exhibits conventional scaling with temperature and frequency21 are particularly clear-cut Fermi-liquid signatures. Moreover, magnetization,8 high-frequency linear conductivity,3,5,6,7 and nonlinear response measurements9 indicate that the superconducting emergence regime is limited to a rather narrow temperature range above Tc and, importantly, that it can be described with a simple percolation model.9

In the present work, we start from the fact that the normal state displays robust Fermi-liquid behavior in the pseudogap regime. We subtract its contribution to the planar resistivity with a reliability approaching the background-free techniques. With the inherent sensitivity of the dc conductivity to superconducting contributions, this enables us to obtain highly precise insight into the emergence of superconductivity and to test the percolation scenario with unprecedented accuracy. In particular, we report measurements of the dc conductivity for the cuprate HgBa2CuO4+δ (Hg1201) in the underdoped part of the phase diagram. Hg1201 may be viewed as a model compound due to its simple tetragonal structural symmetry, with one CuO2 layer per formula unit, and the largest optimal Tc (nearly 100 K) of all such single-layer compounds.38 Further evidence for the model nature of Hg1201 comes from the observation of a tiny residual resistivity,20,39 of Shubnikov-de-Haas oscillations,40,41 and of a small density of vortex-pinning centers which has enabled the measurement of the triangular magnetic vortex lattice.42 Below the characteristic temperature T** (T** < T*; T* is the pseudogap temperature), the planar resistivity of Hg1201 exhibits quadratic temperature dependence, ρ ∝ T2, the behavior characteristic of a Fermi liquid.20

Results

Paraconductivity measurements

We studied Hg1201 samples with Tc ≈ 80 K (the estimated hole doping level is p ≈ 0.11) that were prepared following established procedures.39,43 This particular doping level was chosen because of a relatively wide temperature range between T** and Tc in which pure quadratic-in-temperature resistivity is seen, while being reasonably far away from the doping level (p ≈ 0.09) where weak short-range CDW correlations are most prominent in Hg1201.32,33,44

Figure 1 shows dc resistivity data for a Hg1201 sample along with the three characteristic temperatures Tc, T**, and T*. The purely quadratic behavior seen below T** is in agreement with the Fermi-liquid character of the mobile holes.17,18,19,20 The considerable difference between T** and Tc provides for a simple way to assess the superconducting paraconductivity contribution. In order to subtract the normal-state signal and obtain the purely superconducting contribution above Tc, we fit ρ(T) = ρ0 + a2T2 to the data in a temperature range from 100 K to T** ≈ 140 K, where ρ0 is the small residual resistivity (the estimated residual resistivity ratio is ~120) and a2 a constant. The resultant value of a2 = 9.8(1) nΩ cm/K2 is consistent with previous measurements20 on Hg1201. A narrowing of the fit range does not change the result of our analysis, which demonstrates the robustness of the procedure (see Fig. 1b inset). Furthermore, if a power law of the form ρ(T) = ρ0 + aTα is fitted in the same temperature range, the exponent is α = 1.98(2), and when the temperature range is varied from 100–140 K to 110–130 K, it stays within 5% of this value (Supplementary Information). The fidelity of the quadratic fit is very high (Fig. 1b), which demonstrates that indeed in this temperature range, the only contribution to the resistivity is the Fermi-liquid temperature dependence. We may therefore safely extrapolate the fit to Tc to obtain the underlying normal-state contribution.

Fig. 1
figure 1

a Temperature dependence of the dc resistivity of a Hg1201 single crystal with characteristic temperatures Tc ≈ 80 K (defined here to correspond to the lowest measurable nonzero resistivity), T** ≈ 140 K (defined as the deviation from low-temperature quadratic behavior), and T* ≈ 260 K (defined as the deviation from high-temperature linear behavior). b DC resistivity plotted versus the square of temperature and fit to ρ = ρ0 + aT2 between 100 K and 140 K (dashed line), which demonstrates Fermi-liquid behavior below T** and a very small residual resistivity in the zero-temperature limit (see text). The inset shows the residuals obtained upon subtracting the fit result from the data for fits between 100 K and 140 K (line) and 110 K and 130 K (symbols). The T2 behavior prevails over a ~40-K range. c Normalized VSM magnetization measurements, obtained with an external field of 15 Oe applied perpendicular to CuO2 planes. Solid circles: zero-field-cooled (ZFC) data; open circles: field-cooled (FC) data. These data demonstrate a well-defined superconducting transition at Tc ≈ 80 K and weak vortex pinning, indicative of a low level of point disorder

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Inversion of the experimentally determined resistivity and subtraction of the extrapolated quadratic temperature dependence then gives the superconducting paraconductivity contribution, ∆σdc, shown in Fig. 2. We present results for the same sample as above, which was chosen from a larger batch of crystals with Tc ≈ 80 K due to its well-defined superconducting transitions—for Hg1201, the sample-contacting procedure often induces spurious doping of the sample surface,20 which can “short out” the current path at temperatures above the nominal Tc and artificially broaden the transition. Such samples are not considered here, although they give similar results, except in a narrow (<1–2 K) temperature range above Tc. We also performed a magnetic susceptibility measurement using vibrating sample magnetometry (VSM) that shows a sharp transition, with a Tc value that agrees well with the resistive Tc (Fig. 1c). The field-cool/zero-field-cool susceptibility ratio approaches one and is among the highest observed in the cuprates,39 which demonstrates the high quality of the sample.

Fig. 2
figure 2

DC paraconductivity for underdoped Hg1201 (Tc ≈ 80 K; circles), obtained by subtracting Fermi-liquid normal-state behavior from the measured resistivity, compared to the paraconductivity of underdoped YBCO (Tc = 47 K; squares) and LSCO (Tc = 28 K, Sr doping level x = 0.1; diamonds) obtained by analyzing the data from ref. 52 following the same procedure as for Hg1201 (see text, Supplementary Information, and Fig. S2). The paraconductivity exhibits strong exponential-like temperature dependence with a universal slope. The full line is the prediction of the superconducting percolation model obtained with effective-medium theory from a direct comparison with the Hg1201 data. The dot dashed line indicates quasi-two-dimensional Gaussian Aslamazov–Larkin (AL) fluctuation paraconductivity (see text)

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In order to compare our result for Hg1201 to other cuprates, we use the same normal-state subtraction procedure on published resistivity data for underdoped, detwinnned YBa2Cu3O6+δ (YBCO) and underdoped La2−xSrxCuO4 (LSCO) (see Supplementary Information for details). Remarkably, as demonstrated in Fig. 2, we find the same paraconductivity behavior as for Hg1201. The superconducting response clearly exhibits exponential-like temperature dependence away from Tc, consistent with prior magnetization8 and nonlinear response9 results. Notably, a similar exponential paraconductivity was previously seen in YBCO at several doping levels, but with a different normal-state subtraction scheme that involves fitting to high-magnetic-field data and yields considerably lower sensitivity.16

We emphasize five crucial points: (i) the exponential dependence is qualitatively different from the underlying normal-state power-law behavior, and hence a very robust result; (ii) this behavior is observed in three distinct compounds, with the same characteristic temperature scale, which demonstrates a remarkable degree of universality; (iii) the agreement with other experiments, some of which require no background subtraction,8,9 provides additional justification for the validity of our approach to subtract the Fermi-liquid normal-state contribution; (iv) the signal-to-noise ratio of the present Hg1201 data is very high, which enables us to follow the paraconductivity over more than four orders of magnitude; (v) the exponential temperature dependence is incompatible with standard models of superconducting fluctuations, such as Ginzburg–Landau theory.45

Model

A simple superconducting percolation model with a compound-independent (and nearly doping-independent) underlying energy scale kBΞ0 was recently shown to explain nonlinear response data.9 The present dc paraconductivity result provides an ideal testing ground for this model, since the model is naturally formulated in terms of the dc conductivity. In particular, the model assumes that, above Tc, the material consists of normal patches with resistance Rn, and of superconducting patches with resistance R0 (where we will take the limit R0 → 0).9,46 The fraction of superconducting patches, P, is temperature-dependent: at a critical fraction Pπ (corresponding to the critical temperature Tπ), a sample-spanning superconducting cluster is formed and hence percolates. In the limit of vanishingly small currents, Tπ equals Tc, but in any experiment, Tc is shifted slightly below Tπ due to the required nonzero currents. The temperature-dependent superconducting fraction originates from an underlying distribution of superconducting gaps, and P is hence directly obtained as the temperature integral of the distribution:

$$P\left( T \right) = \mathop {\smallint }\nolimits_T^\infty g\left( {T^\prime } \right){\rm{d}}T^\prime$$
(1)

where g(T′) is the normalized local Tc distribution function. This is shown schematically in Fig. 3. For concreteness, we use the simplest (Gaussian) distribution with a full-width-at-half-maximum equal to Ξ0, consistent with previous work.9 Other distributions, such as a skewed Gaussian, the gamma, or the logistic distributions, were also tested, but resulted in no significant differences in the outcome of the calculation—slight discrepancies between calculations with different distributions only appear in the temperature range in which the signal is close to the noise level. This insensitivity to distribution shape is simply the result of the integration over the distribution, rendering the exact shape unimportant. The dc conductivity in dependence on temperature is now obtained using effective-medium theory (EMT),22 in a form derived specifically for site percolation problems23:

$$\frac{{\tilde \sigma _{\rm{dc}} - \left( {1 - P + P/R_0} \right)}}{{1 - P + P/R_0 + \left( {1/P_\pi - 1} \right)\tilde \sigma _{\rm{dc}}}}P + \frac{{\tilde \sigma _{\rm{dc}} - \frac{{1/R_0}}{{P + \left( {1 - P} \right)/R_0}}}}{{P + (1 - P)/R_0 + \left( {1/P_\pi - 1} \right)\tilde \sigma _{\rm{dc}}}}\left( {1 - P} \right) = 0$$
(2)

where \(\tilde \sigma _{\rm{dc}}\) is the conductivity normalized to the normal-state value, R0 the resistivity of the superconducting patches (which is taken to be vanishingly small), and Pπ the percolation threshold. While it is known that EMT becomes unreliable in the critical regime close to the percolation threshold22 (in our case, about 2 K above Tc), we use it for simplicity in the interesting higher-temperature regime away from Tc. Notably, the validity of EMT has been established also in recent comparison to direct numerical calculations of the resistivity of superconductor-resistor networks in two dimensions.24,26 For networks with random (uncorrelated) distribution of superconducting links, EMT is indistinguishable from numerical calculations, while it performs slightly worse in correlated networks. We also note that the narrow critical regime, T < Tc + 2 K, is presumably not purely percolative, with critical exponents modified by thermal effects47,48; in order to see a discrepancy between the data and the EMT calculation, a careful power-law analysis of the critical regime would need to be undertaken, with more closely spaced measurements around Tc. The investigation of criticality is thus not within the scope of the present work.

Fig. 3
figure 3

Schematic representation of the superconducting site percolation model, as a two-dimensional cross section of the full three-dimensional model (upper row). Dark red patches are superconducting with vanishing resistance, whereas light gray patches have nonzero normal-state resistance. As per Eq. (1), the fraction of superconducting patches is simply obtained by integrating the local gap (i.e., Tc) distribution function, taken to be a Gaussian for simplicity (lower row). Note that for a typical three-dimensional percolation model the critical fraction is ~0.3

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In order to obtain the limit of zero R0, we use different small values in the numerical calculation, until no significant changes in the output are seen (typically for R0 on the order of 10−5 Rn). For simplicity, we take Rn to be constant in the temperature interval of interest—this is appropriate, because its relative change (due to the T2 dependence) is about 25% over a 10-K interval, whereas the paraconductivity changes by a factor of about 102 in the same interval. The calculated temperature dependence shown in Fig. 2 closely matches the experimental findings over the entire range of about four orders of magnitude in ∆σ. For comparison, we also plot in Fig. 2 the standard 2D Aslamazov–Larkin (AL) paraconductivity,

$$\Delta \sigma _{\rm{dc}}^{\rm{AL}} = \frac{{e^2}}{{16\hbar {\rm{d}}{\it{\epsilon }}}}$$
(3)

where d is the distance between CuO2 planes (taken to be 9.5 Å) and \({\it{\epsilon }} = \ln T/T_{\rm{c}}\). This and modified fluctuation expressions have been used in previous work4,16,36 to analyze the paraconductivity close to Tc. We emphasize that this regime is not the focus of our work. In addition, as noted above, the density of our data points close to Tc is not sufficient to perform a reliable critical exponent analysis. Yet away from Tc (T > Tc + 2 K), the measured exponential dependence is clearly captured by the EMT calculation, while the AL paraconductivity decays much more slowly. Physically, the reason for this dramatic difference is that in the percolation scenario, the effective spatial range of superconducting correlations is severely limited (by the mean superconducting patch size, that is in turn determined by P).

Effectively, the percolation calculation of the paraconductivity only has one free parameter: the width Ξ0 of the Tc distribution. Other parameters that enter the calculation are constrained: Rn is simply the normal-state resistivity, Tπ is slightly larger than Tc (in the present calculation it was taken to be Tc + 1 K, but we note that our definition of Tc as the lowest temperature with nonzero resistivity is somewhat arbitrary; different definitions, such as the midpoint of the transition measured by susceptibility, easily lead to a 1 K difference), and the critical concentration Pπ was taken to be 0.3, consistent with the prior nonlinear conductivity analysis.9 The critical concentration Pπ is not arbitrary; it is determined by the details of the percolation model49—site or bond percolation, percolation with or without farther-neighbor corrections, etc.—and by the dimensionality of the percolation process. The model yields virtually the same temperature dependence for different values of Pπ, with a corresponding change in Ξ0: a smaller Pπ implies a larger Ξ0, and vice versa. We therefore cannot distinguish among specific percolation scenarios, such as two-dimensional versus three-dimensional percolation. Prior comparison between linear and nonlinear response indicated that a three-dimensional site percolation model with Pπ ≈ 0.3 is appropriate,9 leading us to use the same value here. Remarkably, the value Ξ0 = 26(1) K that yields the best agreement with the data in Fig. 2 is in excellent agreement with nonlinear conductivity and microwave linear response for a number of cuprate compounds (Hg1201, LSCO, and YBCO) and for a range of doping levels.9 Furthermore, a similar universal scale is found in torque magnetization measurements on four different cuprate families, including Hg1201.8

Discussion

For simple-tetragonal Hg1201, the paraconductivity is a very sensitive probe of the emergence of superconductivity, and it is accurately described by the superconducting percolation scenario, with the same universal characteristic temperature scale observed for other observables in several different cuprate families.8,9 We demonstrate that the superconducting contribution can be simply obtained by assuming a Fermi-liquid normal state below the characteristic temperature T**. Furthermore, we show that this procedure can be readily applied to other underdoped cuprates for which T** is sufficiently large compared to Tc and the underlying Fermi-liquid T2 resistive behavior is not masked by effects due to low structural symmetry and/or point disorder (such nonuniversal effects are prevalent, e.g., in the bismuth-based cuprates or twinned YBCO17,19). Notably, below optimal doping, LSCO exhibits low-temperature resistivity upturns50 (typically revealed in the absence of superconductivity upon applying a high c-axis magnetic field), but the doping level of the sample analyzed here is sufficiently high for the upturn to be negligible in the temperature range of interest. For YBCO, such nonuniversal upturns generally appear at lower doping levels in the absence of intentionally introduced point disorder.51 The remarkable similarity of the paraconductivity for Hg1201, YBCO and LSCO constitutes a strong confirmation both for the validity of the subtraction scheme and of our main conclusions regarding a universal percolative superconductivity emergence in the cuprates.

The normal-state subtraction procedure is not possible for optimally doped compounds, where T** becomes comparable to, or smaller than Tc and the resistivity no longer exhibits quadratic temperature dependence.20,52 Yet, we note that a recently proposed model for the normal state of the cuprates that considers the gradual (temperature-dependent) delocalization of charge53 explains the apparently T-linear resistivity and provides highly accurate predictions for the resistivity also near optimal doping. In principle, this insight could thus be used to extract the paraconductivity for compounds for which T** is low.

The present work does not provide microscopic insight into the gap inhomogeneity and its origin, and in this respect the percolation model is phenomenological. It should furthermore be viewed as a zeroth-order approximation, since it treats the superconducting patches as essentially static, and does not take into account correlations and an explicit Josephson coupling between the patches. A more refined treatment should also consider thermal fluctuations and pair breaking in the patches, along the lines explored in the context of disordered superconducting films.54 Nevertheless, our basic percolation model provides a remarkable description of the conductivity and is highly consistent with experiments sensitive to real-space superconducting gap disorder, such as scanning tunneling microscopy,55,56 which have observed superconducting gap distributions with a width comparable to kBΞ0. It is also qualitatively consistent with NMR results that demonstrate a considerable distribution of local electric field gradients,57,58 and with X-ray experiments that find percolative structures59,60 in oxygen-doped La2CuO4+δ and YBCO.

The clear confirmation of the superconducting percolation scenario in the present work suggests that, fundamentally, both the normal-state carriers and superconducting emergence are rather conventional in the underdoped cuprates, once the underlying gap disorder is taken into account. Our result excludes the possibility of extended fluctuations usually associated with non-Fermi-liquid models.1,10,11,12,13 It also shows that it is difficult to observe the usual Ginzburg–Landau fluctuation regime in the conductivity, because inhomogeneity effects dominate—the percolation description holds down to temperatures very close to Tc. Along with magnetometry as well as linear and nonlinear conductivity data, the basic percolation model naturally explains other seemingly unconventional features such as the “gap filling” seen in photoemission data,61 and thus provides a unifying understanding of superconducting pre-pairing in the cuprates.9 The dc conductivity measurements presented here have put the scenario to a stringent quantitative test, and hence constitute a crucial, independent confirmation in a model cuprate system. Furthermore, the universal gap distribution that gives rise to the observed paraconductivity is clearly intrinsic and unrelated to point disorder, since the three cuprates considered here exhibit considerably different point disorder characteristics.38 The robustness of our result therefore mandates a paradigm change in the field of cuprate superconductivity, namely that the itinerant carriers are well described by Fermi-liquid concepts, whereas the emergence of superconductivity is dominated by the gap inhomogeneity inherent to these lamellar oxides.

Methods

Hg1201 samples were grown and characterized using established methods,39,43 annealed in flowing nitrogen at 465 °C for 1 month, and quenched to room temperature. The samples were cleaved and contacted by sputtering gold electrodes, which were connected to wires using silver paint resulting in a low contact resistance, as described in detail in previous work.39

The magnetization measurements were obtained using the VSM measurement system for the Quantum Design, Inc., Physical Property Measurement System (PPMS). Measurements were performed in the temperature range from 20 to 110 K, enabling precise determination of the magnetization properties of the sample near the superconducting transition.

Electrical conductivity was measured using a true dc technique with current reversal, utilizing Quantum Design, Inc., PPMS. The current density was 0.02 A/m2, chosen to be small enough to lead to negligible Joule heating, but large enough for a good signal-to-noise ratio for the paraconductivity. Temperature was stabilized before each datum was measured.