Abstract
A pivotal step toward understanding unconventional superconductors would be to decipher how superconductivity emerges from the unusual normal state. In the cuprates, traces of superconducting pairing appear above the macroscopic transition temperature T_{c}, yet extensive investigation has led to disparate conclusions. The main difficulty has been to separate superconducting contributions from complex normalstate behaviour. Here we avoid this problem by measuring nonlinear conductivity, an observable that is zero in the normal state. We uncover for several representative cuprates that the nonlinear conductivity vanishes exponentially above T_{c}, both with temperature and magnetic field, and exhibits temperaturescaling characterized by a universal scale Ξ_{0}. Attempts to model the response with standard GinzburgLandau theory are systematically unsuccessful. Instead, our findings are captured by a simple percolation model that also explains other properties of the cuprates. We thus resolve a longstanding conundrum by showing that the superconducting precursor in the cuprates is strongly affected by intrinsic inhomogeneity.
Introduction
Despite tremendous experimental and theoretical efforts over the past three decades, the nature of the superconducting fluctuation regime of the cuprates remains intensely debated^{1}. Experimentally, the problem has been approached using bulk probes, such as conductivity and other transport properties in a wide frequency range^{2,3,4,5,6,7,8,9,10,11,12,13}, magnetic susceptibility^{3,14,15,16}, surface sensitive probes^{17}, local probes, such as muon spin rotation^{18}, and photoemission spectroscopy^{19,20}. Some studies point to the possible persistence of superconducting pairing well above T_{c}, which has been taken as an indication of preformed Cooper pairs related to the appearance of the pseudogap^{13,21}. Other studies indicate that traces of superconductivity emerge at somewhat lower temperatures, and are most prominent at moderate doping^{9,10,14}. High pairing onset temperatures have been related to exotic normalstate physics^{22,23} and to unconventional prepairing^{24,25}, with profound consequences for the mechanism of cuprate superconductivity. However, terahertz and microwave conductivity^{2,5,6,11,12} as well as magnetometry experiments^{15,16} consistently detect superconducting contributions only near T_{c}, irrespective of doping. The resolution of this puzzle would be a crucial step toward understanding the highT_{c} cuprates.
However, in previous experiments it has often been difficult to reliably establish the nonsuperconducting normalstate contribution in order to extract a superconducting signal. Typical approaches involve the extrapolation of hightemperature behavior or the suppression of superconductivity by a magnetic field. The situation is further convoluted due to the complexity of the cuprate phase diagram, which features a dopingdependent pseudogap, as well as universal and compoundspecific ordering tendencies that manifest themselves differently in different experimental observables^{1}. The presence of various kinds of disorder in these complex oxides poses yet another complication^{26}. Data are often discussed assuming preformed Cooper pairs in an extended temperature range^{9,10,14}, or analyzed within the Ginzburg–Landau (GL) framework with possible corrections to the original meanfield theory^{3,4,7,8,12}; yet this has not resulted in a unified picture.
The absence of any discernible signal due to nonsuperconducting contributions renders the nonlinear conductivity technique uniquely suitable to study and model superconductivity emergence. We apply this probe to a number of cuprate families and a variety of experimental conditions. The measurements unambiguously show that the superconducting precursor is limited to a narrow temperature range above T_{c}, which rules out extended fluctuations and prepairing regimes. Importantly, we find that the superconductivity emergence range is not controlled by T_{c}, a crucial qualitative feature of GL theory, but rather by a scale Ξ_{0} that is nearly independent of compound and doping (in the studied doping range p = 0.08–0.19). This robust experimental finding is an important step toward understanding cuprate superconductivity, as it places strong constraints on any theory. We then use a simple model to explain the data: the superconducting gap is known to be spatially inhomogeneous, which results in a distribution of local transition temperatures and naturally leads to percolation. Percolation, and the scalefree fractal structures that emerge from it, is a wellknown and ubiquitous phenomenon: first investigated in the context of polymer growth, it has since been formulated as a mathematical concept and applied to systems as diverse as random resistor networks, organic molecular gels, dilute magnets, the spread of diseases, and the largescale structure of the universe^{27,28}. The basic ingredient in percolation theory is inhomogeneity, and we find that evoking T_{c} inhomogeneity is essential to understand superconductivity emergence in the cuprates. Remarkably, the minimal percolation model that we employ is sufficient to capture the observed unusual exponential temperature and magneticfield dependences of the nonlinear conductivity. We also report complementary linear conductivity measurements and take a fresh look at prior experimental results (torque magnetometry^{15}, resistivity^{7}, Seebeck coefficient^{8}, specific heat^{29}, and tomographic density of states^{19}), to demonstrate that the emergence of superconductivity can be consistently explained with this minimal model. Finally, the universal scale Ξ_{0} is shown to be a direct measure of the superconducting gap distribution width. The underlying inhomogeneity therefore is unrelated to material details, and must be an intrinsic, generic feature of cuprate superconductors.
Results
Nonlinear response
Nonlinear planar response, for current flow along the CuO_{2} planes, is measured with a sensitive contactfree method^{30} (see Methods). The response can be analyzed by decomposing the signal into harmonics,
where J is the response of the sample to an external field K (electric or magnetic), σ_{1} the linear response tensor, and σ_{2}, σ_{3}, etc., the correction nonlinear tensors. Here, we discuss the third harmonic σ_{3}, the lowestorder conventional correction to the linear response (the secondharmonic σ_{2} can only appear if time reversal or inversion symmetry is broken^{31} and is not discussed here). In any alternatingfield experiment, magnetic and electric fields are related, and therefore it is arbitrary if one designates the signal at frequency 3ω as proportional to nonlinear conductivity or (complex) susceptibility. Complementary linear conductivity measurements are performed with a microwave cavity perturbation technique (see Methods).
Temperature dependence
Measurements of the inplane linear and nonlinear response were performed for three representative cuprate families: a nearly optimally doped sample of HgBa_{2}CuO_{4+δ} (Hg1201), a model cuprate system due to its simple structure, high T_{c}, and minimal point disorder effects^{32,33,34,35}; an optimally doped YBa_{2}Cu_{3}O_{7−δ} (YBCO) sample with 3% of Cu substituted by Zn (YBCO–Zn), where Zn dramatically affects the superconducting properties^{36}; and La_{2−x}Sr_{x}CuO_{4} (LSCO), spanning a wide range of doping across the superconducting dome (see Table 1). For all samples, σ_{3} exhibits qualitatively the same temperature dependence (Fig. 1a and Supplementary Figure 1): no signal at high temperatures, a peak at a temperature that we designate as T_{c}, consistent with previous work (see Methods), and a steplike feature below T_{c}. The signal magnitude depends on sample size and shape, and thus is normalized to the peak value. We note that T_{c} as obtained from σ_{3} agrees with the temperature of the peak in the real part of the linear microwave conductivity, which in turn corresponds to the value determined from magnetic susceptibility measurements^{12}.
The measurements clearly show that the nonlinear response decays quickly above T_{c}, which demonstrates the absence of extended fluctuations. Some previous investigations suggested agreement between experiments and GL theory (with various modifications to the theory^{4,7,8,12}) for particular cuprate compounds at particular doping levels; in line with these investigations, we have attempted to analyze our results within the GL framework. Within this framework, we would expect an approximately powerlaw temperature dependence of σ_{3} (see ref. ^{37} and Methods), and a scaling of the data for different compounds with the characteristic scale T_{c}. Figure 1b shows our nonlinear conductivity data in dependence on the GLreduced temperature ln(T/T_{c}) compared to a calculation of σ_{3} using anisotropic GL theory beyond mean field (see Methods), similar to ref. ^{12}. The theory predicts a temperature dependence of σ_{3} that is clearly incompatible with experiment; the agreement cannot be improved by any tuning of the parameters, such as a different definition of T_{c} (see Supplementary Figure 2). Even more importantly, the expected scaling is absent: T_{c} is not the characteristic temperature scale for superconductivity emergence. The scaling argument is valid regardless of the manner in which GL theory is modified. However, the data are remarkably similar on an absolute temperature scale: a simple shift by a sampledependent temperature T_{π} (that is slightly larger than T_{c}) leads to the data collapse shown in Fig. 1c. This implies that a mechanism that gives rise to approximately exponential behavior with a single temperature scale Ξ_{0} underlies the emergence of superconductivity. A similar exponential dependence can also be deduced from linear conductivity (inset in Fig. 1c) and torque magnetometry^{15} experiments, indicating its robustness. Clearly a framework other than GL is needed to explain the data.
Since nanoscale electronic inhomogeneity is well documented in the cuprates, e.g., from nuclear magnetic resonance^{38,39,40} and scanning tunneling microscopy (STM)^{17,41} measurements, we now attempt to gain understanding through a simple percolation model. The basic ingredient of the model is spatial inhomogeneity of local superconducting gaps, with a distribution width that is characterized by the scale k_{B}Ξ_{0}. This distribution corresponds to superconducting patches that proliferate upon cooling, and macroscopic superconductivity then emerges via a percolative process^{42,43}. We calculate the response assuming nearestneighbor site percolation, although the result does not critically depend on the details of the scenario (see Methods). For simplicity, we take the material to be made of perfectly connected square or cubic patches that are either nonsuperconducting, with a normal resistance R_{n}, or superconducting, with a nonlinear resistance R_{s}(j) that depends on the current through the patch j (see Methods). Since we normalize the experimental nonlinear conductivity, we can also normalize the resistances by taking R_{n} = 1. The fraction P of superconducting patches depends on temperature: P → 0 at high temperatures and P → 1 well below T_{c}. At the critical concentration P_{π}, the system percolates—a connected, samplespanning superconducting cluster is formed. P_{π} only depends on the dimensionality of the system^{27} and on the details of the percolation scenario (e.g., site vs. bond percolation), and it corresponds to the temperature T_{π} that can be viewed as the “true” underlying resistive T_{c} in the limit of small currents. In principle, the full temperature dependence of P can be obtained from the underlying gap distribution, but the distribution must be known (or assumed). However, to lowest order, any reasonable distribution yields a linear dependence of P on temperature close to P_{π} (see Fig. 2). We, therefore, approximate P_{π} − P = (T − T_{π})/Ξ_{0}. The temperaturedependent linear and nonlinear responses are then obtained via an effective medium calculation (see Methods), which yields functions that decay nearly exponentially, in very good agreement with the experimental σ_{3} and σ_{1} (Fig. 1c). Within the nearestneighbor sitepercolation model, two values of P_{π} are possible: ≈0.3 for threedimensional (3D) and ≈0.6 for twodimensional (2D) percolation. Better agreement is obtained with P_{π} ≈ 0.3 (see Supplementary Figure 3), which suggests essentially 3D superconductivity emergence^{27} in the samples studied here. We note that we study the inplane response, and thus the only role of interplane coupling in the percolation model is to determine the effective dimensionality, and hence the percolation threshold.
For P_{π} ≈ 0.3, the resultant characteristic scale Ξ_{0} lies in a narrow range for all investigated samples (see Table 1), Ξ_{0} = 27 ± 2 K, and hence is de facto universal (the stated uncertainty is 1 s.d. from the mean of the data in Table 1). If we assume 2D percolation and P_{π} ≈ 0.6, the agreement between σ_{3} and σ_{1} is not as good, and the corresponding Ξ_{0} is smaller by about a factor of two. We emphasize that the calculated σ_{3} is effectively insensitive to model details such as the parameters of the patch nonlinear response R_{s}, rendering Ξ_{0} the sole parameter (within a given percolation model). This insensitivity to specifics is a consequence of percolation physics, where model details are unimportant close to the threshold and the response of the largest clusters dominates.
An important feature can be inferred from the comparison of linear and nonlinear response. Within the effective medium calculation, the linear conductivity determines the net current through the sample, given an applied electric field. Yet the nonlinear resistance of the superconducting patches, R_{s}, is current dependent. The thirdharmonic signal therefore depends on the third power of the current (to lowest order), which implies that σ_{3} α σ_{1}^{3}. This is indeed borne out by experiment, as seen in Fig. 1d. In contrast, in GL theory both responses are determined by the electric field, and their ratio has a more complex temperature dependence (see ref. 37 and Methods). The apparent characteristic temperature scales for σ_{3} and σ_{1}, therefore differ because of the nonlinear nature of σ_{3}, but the underlying scale Ξ_{0}, which determines the range of superconducting pairing emergence, is the same for both responses. This also implies that the superconducting contribution to the linear response should be discernable up to significantly higher temperatures than the nonlinear part, if the experimental signaltonoise ratios are similar. Measurements throughout the phase diagram of LSCO consistently confirm this trend (Fig. 1d), which strongly supports the percolation model.
Magneticfield effect
As a further test of the model, we investigate the influence of an external magnetic field on the emergence regime. Although the quantitative effects of a magnetic field are difficult to determine within our simple effective medium approach, we can make qualitative predictions. One would expect the field to greatly influence the superconducting percolation process. Both the critical current of a superconducting patch and the number of patches decrease with increasing field. Above a characteristic field H_{0}, the critical currents of all patches are small, except for the samplespanning cluster at temperatures below T_{π}. At fields significantly above H_{0}, the nonlinear response should therefore only exhibit a steplike feature close to T_{π}. Furthermore, H_{0} is related to the macroscopic critical field H_{c2}, as both fields are determined by the underlying superfluid stiffness: H_{0} describes the properties of the finitesized clusters below and above T_{π}, whereas H_{c2} pertains to the samplespanning cluster below T_{π}. In agreement with these expectations, we find that an external magnetic field strongly suppresses the nonlinear response, rendering it stepshaped far above H_{0} (Fig. 3a). Once the highfield steplike response is subtracted (see Supplementary Note 4 and Supplementary Figure 5), the data for all samples exhibit universal scaling (Fig. 3b). We apply the same effective medium calculation as for the temperature dependence, assuming a phenomenological powerlaw dependence of the effective patch critical current on H/H_{0} (see Supplementary Note 4 and Supplementary Figure 4), and find good agreement with experiment (Fig. 3b). For H >> H_{0}, only large superconducting clusters survive. Since the clustersize distribution in any percolation model is generally exponential for the largest clusters^{27}, this leads to an exponentiallike field dependence of the highfield response, as also observed in prior torque measurements^{15}.
H_{0} is about two orders of magnitude smaller than H_{c2}, consistent with the percolation scenario, since H_{0} is a property of the average (small) cluster. As seen from Fig. 3b, the doping dependencies of the two characteristic fields are remarkably similar, including a minimum close to the “1/8 anomaly” of LSCO and YBCO^{8,44,45}. The substitution of 3% Cu with Zn in YBCO causes a dramatic decrease of H_{0}, in agreement with established effects of Zn on superconductivity in cuprates^{36}.
Discussion
Previous reports suggest that percolation processes might play a role in understanding the properties of the cuprates^{42} (see also Supplementary Note 3). However, our work demonstrates for the first time that a universal percolation process can describe the prepairing regime. The percolation picture is in excellent agreement with the temperature and magnetic field dependencies of σ_{1} and σ_{3}, and one would expect it to provide an explanation of other experimental results as well. Qualitatively, several previous studies indicate that the superconducting precursor appears within a roughly constant temperature range above T_{c}, similar to Fig. 1d; this is visible, e.g., in highfrequency conductivity measurements^{6,11,12}, specific heat results^{46}, and resistivity curvature plots^{47}. More quantitatively, Fig. 4 demonstrates the similarity of superconducting precursor in several observables. An exponential tail is observed in the dc conductivity^{7} of YBCO at various hole doping levels, with a universal slope (Fig. 4b). Notably, YBCO in particular is structurally complex, with alternating CuO_{2} planes and CuO chains whose filling depends on oxygen concentration; the exponential behaviour, however, is robust and does not depend on the arrangement of the chains. The Nernst effect^{8} in EuLSCO shows an exponential dependence as well (Fig. 4c). Although this measurement can be described by 2D Gaussian theory close to T_{c}, where corrections to the simple percolation picture are expected, once the data are plotted on an absolute temperature scale, the exponential tail is apparent and reveals the same underlying temperature/energy scale Ξ_{0}. Torque magnetometry measurements on several cuprate families, including underdoped LSCO, bismuth cuprates and Hg1201^{15}, as well as YBCO^{16}, exhibit both an exponential signal decrease above T_{c} (Fig. 4d) and a universal temperature scale T_{d}. The exponential dependences at temperatures well above T_{c} are a consequence of the tail of the superconducting gap distribution, and for σ_{1} and σ_{3} the effective medium calculation smoothly continues this dependence down to T_{c}. Finally, roughly exponential tails are observed above T_{c} in specific heat studies^{26,46,48}; it is possible to calculate this within the percolation model by convoluting the meanfield specific heat step at T_{c} with the gap distribution function (see Methods). This procedure yields agreement with experiment and reveals a scale Ξ_{0} similar to that obtained from conductivity (Fig. 4e). Notably, critical fluctuations are observed in the specific heat close to the macroscopic T_{c}, which could be also important for other observables in a small temperature range around T_{c}.
We note that the percolation model discussed here is somewhat different from the standard textbook case, in that both the normal and percolating (superconducting) patches have nonzero conductivity. Therefore, instead of a discontinuity at T_{π} and powerlaw behavior above the percolation temperature (that is predicted if one of the phases is insulating^{27}), the calculation yields smooth exponentiallike behavior. Yet the underlying distribution of superconducting cluster sizes should still be scalefree (i.e., follow a power law) close to the percolation threshold. A signature of this might be observed with other experimental probes, e.g., recent optical pumpprobe experiments^{49} uncovered hitherto unexplained powerlaw superconducting correlations above T_{c}.
We emphasize that the heterogeneity that gives rise to superconducting percolation is qualitatively different from the disorder discussed previously in the context of “dirty” and granular superconductors^{50,51,52,53}, and from inhomogeneity induced by doping. In alloys^{50} and films^{51,52}, the electronic meanfree path is extremely shortened by scatterers, while in granular materials differing Josephson couplings between granules cause superconducting percolation^{50}. Yet here we find that nanoscale gap inhomogeneity is crucial: the superconducting gap, and hence the local T_{c} displays spatial variations and causes the percolation we observe. Related gap disorder (on scales much larger than the superconducting coherence length) has been employed previously in modeling the magnetization of select cuprate and other superconductors^{53}, but not applied universally or used to calculate transport properties. Inhomogeneity and a residual zerotemperature component of uncondensed carriers has been shown to be essential to understand lowtemperature superfluid density and optical response of several cuprates^{54}. Spatial gap inhomogeneity also naturally explains the gap filling recently observed in a tomographic densityofstates photoemission experiment^{19}. As demonstrated in Fig. 4f, g, a quantitative description of this result can be obtained simply by positing that the measured density of states is an average over spatial regions with inhomogeneous gaps, again with a distribution width of k_{B}Ξ_{0} ~ 3 meV, which further supports the percolation scenario (see Methods for details).
Perhaps the most unexpected result of our study, which covers the doping range from the very underdoped (p = 0.08) to the overdoped (p = 0.19) part of the phase diagram, is the existence of a (nearly) doping and sampleindependent percolation scale Ξ_{0}, which implies a common intrinsic origin of the gap inhomogeneity in all cuprates, irrespective of material details. Doping does not significantly alter this scale, but affects the macroscopic T_{c} or, equivalently, the critical percolation temperature. Several distinct types of disorder are generally present in the cuprates: the lamellar structure is intrinsically frustrated, which causes structural inhomogeneity; the hole doping process introduces defects into the crystal structure; and doping a strongly correlated electronic system may induce electronic frustration and inhomogeneity. These different kinds of disorder are typically compound and doping dependent^{26,55}, and various experimental techniques have been used to study them. Residual resistivity, a measure of point disorder, is compounddependent, and can be very small in cuprates such as Hg1201^{33,34}. Furthermore, quantum oscillation experiments point to a high degree of doping (hole concentration) homogeneity in oxygendoped compounds such as Hg1201^{35}, thallium cuprates^{56}, and YBCO^{57}. However, this does not preclude nanoscale electronic inhomogeneity unrelated to point disorder. This reasoning is supported by the fact that we find consistent results for distinctly different cuprates^{26}: Hg1201, where dopingrelated point disorder resides relatively far away from the CuO_{2} planes; LSCO, which exhibits considerable (La/Sr) point disorder in close proximity to the CuO_{2} planes; and YBCO–Zn, where Zn directly introduces point disorder within the CuO_{2} plane.
The cuprates also appear to exhibit inherent structural inhomogeneity, an elegant demonstration of which comes from conductivity and hydrostatic relaxation experiments that show stretched exponential behavior characteristic of glassy materials^{55}. Moreover, Xray experiments find complex fractal interstitialoxygendopant structures linked to percolative superconductivity^{58}. Local electrostatic disorder has been studied via nuclear quadrupole resonance and revealed that LSCO and Bibased compounds exhibit higher levels of such disorder^{38,59} than oxygendoped cuprates such as Hg1201 and YBCO, where the dopant atoms reside far from the CuO_{2} planes^{39,40}. Importantly, however, none of these experiments directly detect superconducting gap disorder, making it difficult to establish a relationship between electrostatic/doping inhomogeneities and superconducting gap distributions. STM does probe local gap distributions on the sample surface, but has been applied only to a select number of cuprates, and it is not trivial to separate the superconducting gap from the more inhomogeneous higherenergy (pseudo)gap^{17}. We emphasize that the gap distribution (with width k_{B}Ξ_{0}) relevant for our model likely is not precisely the same as the gap distribution seen by STM, but rather a coarsegrained distribution of mean local gaps (averaged over the local superconducting coherence lengths). Moreover, we expect the distribution to be effectively narrower below T_{c} because of proximity effects, i.e., largegap superconducting regions may induce a gap in neighboring regions. Nevertheless, STM clearly reveals disorder structures in both underdoped and overdoped^{41} Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}, and extended analysis shows a correlation between the presence of inhomogeneous highenergy gaps and superconductivity^{60}. A recent phenomenological model^{61} based on inhomogeneous, temperature and doping dependent (de)localization of one hole per primitive cell can explain the main features of the cuprate phase diagram and superconductivity. It is conceivable that a universal scale k_{B}Ξ_{0} emerges via a complex renormalization of these highenergy localization gaps^{61} (see also Supplementary Note 3). In this case, the gap disorder would not necessarily be related to any local doping inhomogeneity: the material may be homogeneously doped, yet possess an underlying gap distribution.
The existence of exponential behavior with a universal scale Ξ_{0} also shows that expected GL superconducting fluctuations are considerably weaker than inhomogeneity effects. Conversely, if GL fluctuations were important, the simple percolation model with a single Ξ_{0} would not describe the measurements. This furthermore points to 3D percolation—except perhaps in the special case of strong stripe correlations in Labased cuprates^{8,62} such as La_{1.875}Ba_{0.125}CuO_{4} and La_{1.8−x}Eu_{0.2}Sr_{x}CuO_{4} (see Supplementary Note 2)—since the strong vortex–antivortex fluctuations expected in the 2D case should significantly broaden the onset of superconductivity.
To conclude, we have employed a new approach to the cuprate prepairing problem by studying nonlinear conductivity. The unexpected scaling of nonlinear and linear conductivity for widely different cuprates constitutes a benchmark result for any theory of superconducting prepairing in these materials. Taking into account the wellestablished fact that significant gap inhomogeneity is present in the cuprates, we have provided a simple framework in which inhomogeneity plays a pivotal role. Our results thus show that intrinsic and universal superconducting gap inhomogeneity is highly relevant to understanding the superconducting properties of the cuprates.
Methods
Samples
The Hg1201 and LSCO samples are single crystals of wellestablished high quality used in previous work^{63,64} with volumes of about 1 mm^{3}. Hg1201 is grown using an encapsulation method, while LSCO crystals are grown in a traveling floating zone furnace. YBCO–Zn is an oriented powder sample with 3% Cu substituted with Zn, which enables us to discern effects of intentionally introduced CuO_{2} plane disorder. This sample was prepared using a standard solidstate reaction, used in prior Zn nuclear quadrupole resonance experiments and characterized in detail^{36}. See Table 1 for additional sample information.
Linear and nonlinear conductivity
Nonlinear response measurements typically require relatively large applied fields in order to detect the small signals. Hence, the most serious problem that plagues nonlinear measurements of conductive systems is Joule heating, i.e., the variation of the conductivity/susceptibility with temperature induced by resistive heating of the sample. If a constant or slowly varying electric field is used to detect nonlinear response, the large current will heat the sample during measurement, and spurious nonlinear contributions will appear if the resistance depends on temperature. Conventionally, millisecond field pulses are used to alleviate the heating problem, but heating still plays a role for highly conducting samples and needs to be disentangled from other possible contributions^{65,66}. A pivotal step in our experiment is the use of a highfrequency excitation field—if the frequency is high enough, the timedependent temperature change of bulk samples cannot follow the rapidly changing field, and no heatinginduced nonlinear signal is observed. An average, timeindependent heating is still present, but this does not influence the measurement of the nonlinear response. In principle, timeindependent heating may cause a small shift of the sample temperature, but this was determined to be negligible in our case from a comparison of T_{c} (peak positions) in linear and nonlinear conductivity throughout the phase diagram of LSCO. The nonlinear conductivity experiments are performed with a contactless radiofrequency twocoil setup, with excitation frequency ω/2π = 17 MHz and phase sensitive detection at 3ω/2π using a Stanford Research Systems SR844 RF lockin amplifier. The coil system is kept at the constant temperature of the liquid helium bath, while the sample temperature is varied independently. We use a nonresonant circuit (a coil with silver paint serving as a distributed capacitance) for excitation, and a tuned resonant LC circuit for detection. A thinwalled glass tube separates the vacuum of the sample space from the liquid helium bath and introduces no distortions to the signal. The sample is mounted on a sapphire holder with temperature control sensitivity better than 1 mK. The setup was previously tested under various conditions^{30,67}. Notably, a similar methodology was used in the past to study the nonlinear Meissner effect at low temperatures^{68}. The electric fields with the samples may be estimated using Maxwell’s equations, which gives an amplitude E ~ BωL, where B is the magnetic field amplitude, ω ~ 2π·20 MHz the oscillation frequency, and L ~ 2 mm the typical linear sample dimension. The amplitude of the magnetic field was deliberately kept small, and estimated to be about 0.1 G from the characteristics of the excitation circuit and coil. The electric field amplitude is then E ~ 0.02 V/cm.
We performed complementary microwave (linear) conductivity experiments with a resonant cavity perturbation technique^{69} extensively used to study cuprate superconductors^{5,12}. The sample was mounted in an evacuated elliptical microwave cavity made of copper and immersed in a liquid helium bath. The complex conductivity of the sample was obtained by measuring the temperature dependence of the Qfactor and resonant frequency of the cavity by recording the cavity resonance curve using microwave frequency modulation and employing a demodulator. The signal from the demodulator was fed into a SR830 lockin amplifier and the Qfactor determined from measurements of the higher harmonic components of the modulation frequency. The microwave cavity resonance was close to 10 GHz, while the modulation frequency was 990 Hz. Similar to the nonlinear conductivity experiment, the cavity was kept at constant temperature, while the sample temperature was varied in a wide range. We obtained the superconducting response above T_{c} by subtracting the conductivity measured with an external magnetic field of 16 T (perpendicular to the CuO_{2} planes) from the zerofield conductivity. No appreciable difference in conductivity was observed between 12 and 16 T in the relevant temperature range.
GL theory
Classic GL superconducting fluctuations have been extensively investigated in the cuprates using linear response in a wide frequency range^{3,4,5,6,7,11,12,70,71,72}. Nonlinear response is a better probe of fluctuation contributions, since in linear response one must always attempt to determine and subtract a normalstate contribution, a complication that is absent in the third harmonic. At a quantitative level, the linear and nonlinear GLfluctuation response has been calculated beyond mean field^{37} and with included anisotropy^{72,71,73} (only linear conductivity). For an isotropic typeII superconductor, the nonlinear conductivity in both the Gaussian and critical fluctuation regimes is shown to be proportional to the linear conductivity, as follows. In general, one can define a fielddependent conductivity, σ(E) = σ(E = 0)Σ(E). The scaling function Σ(E) has different forms in different fluctuation regimes and for small/large electric fields, but in the smallfield approximation the leading term is always 1 + A(E/E_{0})^{2}, where A is a numerical constant and E_{0} a reference electric field^{37}. The field E_{0} depends on temperature through the meanfield correlation length (ξ), as E_{0} ~ ξ^{−3}. Therefore, σ_{3} = σ_{1}A/E_{0}^{2}~ σ_{1}ξ^{6}. Since, in GL theory, the linear and nonlinear responses are due to the same fluctuation physics, such a scaling relationship between the two should hold regardless whether abplane/caxis anisotropy and shortwavelength cutoffs^{70,71,72,73} are included. Thus one can directly compare the temperature dependence of the linear/nonlinear response to the predictions of linear GLfluctuation theory. Due to the low frequency of our experiment, the linear response simply corresponds to the inplane dc linear conductivity, \(\sigma _{{\mathrm{ab}}}^{{\mathrm{DC}}}\). The dc conductivity is given by^{71}
where ξ is the superconducting coherence length, the indices ab and c correspond to inplane and caxis quantities, respectively, z is the dynamical exponent, and f (Q_{ab}, Q_{c}) a function of the temperaturedependent anisotropic fluctuation cutoffs Q_{ab} and Q_{c} in reciprocal space. The cutoffs are \(Q_{{\mathrm{ab}},{\mathrm{c}}} = \sqrt 3 {\mathrm{\Lambda }}_{{\mathrm{ab}},{\mathrm{c}}}\xi _{{\mathrm{ab}},{\mathrm{c}}}(T)/\xi _{0{\mathrm{ab}},{\mathrm{c}}}\), where Λ_{ab,c} are temperatureindependent cutoff scales. Since the electric fields applied in our measurements are small, they are significantly below E_{0} (except perhaps in the closest vicinity of T_{c}, where E_{0} rapidly goes to zero).
The relevant dimensionless temperature variable for ξ(T) in GL theory is ln(T/T_{c}), and data for several cuprates are plotted vs. this GL reduced temperature in Figs. 1b and S2. The theoretical prediction obtained from Eq. (1) using the realistic parameters^{12} Λ_{ab} = 0.1 and Λ_{c} = 0.02 is shown in Fig. 1b. The theoretical prediction clearly decays much faster than the data for all investigated samples. We note that the choice of a different value for T_{c} cannot improve the agreement between data and theory. This is demonstrated in Supplementary Figure 2 for the case of LSCO with x = 0.15 (measured T_{c} = 37.2 K). Better agreement can be obtained if the reduced temperature variable is multiplied by materialdependent constants for different samples, but in the case of GL fluctuations these would be additional arbitrary nonuniversal free parameters without obvious physical meaning. Even more importantly, the shape of the temperature dependence cannot be satisfactorily reproduced by GL theory, whatever scaling one employs on the temperature axis.
The minimal percolation model
The main idea of the model is that nanoscale superconducting patches form and proliferate in the material (Fig. 2b), and that macroscopic superconductivity then emerges via a percolation process. We assume perfectly connected square or cubic patches (2D or 3D nearestneighbor site percolation) that are either nonsuperconducting, each with a normal resistance R_{n}, or superconducting, each with a nonlinear resistance^{43}
where j is the current through the superconducting patch, R_{0} its residual resistivity (due to the finite size of the patch, and R_{0} << R_{n}), and J_{c} the patch critical current; j and J_{c} are dimensionless currents. We assume the patches to be static (which is probably not a good approximation well above T_{c}) and neglect Josephson couplings and proximity effects (not a good approximation very close to T_{c}). The fraction of superconducting patches is taken to be P, with P → 0 at high temperatures and P → 1 well below T_{c}. The critical concentration at which the system percolates, P_{π}, depends on the dimensionality of the system and the chosen percolation model. In the nearestneighbor sitepercolation scenario used here, we have^{27} P_{π} ≈ 0.3 (3D) and P_{π} ≈ 0.6 (2D). Site percolation is physically realistic in the case of superconducting patches with different local T_{c} values, as seen in STM^{17}, but the particular choice of the percolation model does not critically affect the modeling, as we show below. In order to make a quantitative comparison to experiment, a dependence of P on temperature must be assumed. The simplest possibility is a linear dependence, P_{π} − P = (T − T_{π})/Ξ_{0}, where Ξ_{0} is the universal temperature/energy scale that connects P and T. Physically, the linear dependence is equivalent to taking the distribution of local superconducting gaps to be a simple boxcar function of width Ξ_{0}. Yet the linear term is the leading term for any realistic distribution, and thus this approximation is always valid not too far from T_{c}. Our goal, in the spirit of the minimal model, is to avoid any assumptions related to the gap distribution. This approach in the case of linear and nonlinear conductivity gives good results. We illustrate the difference between our assumption of a linear dependence of P on T and a more realistic Gaussian distribution of local gaps in Fig. 2a. At high temperatures, the Gaussian distribution results in better asymptotic behavior, which eliminates the artificial cutoff present in the linear approximation and gives rise to exponential tails of conductivity, magnetization, etc. Yet in the temperature range where linear and nonlinear conductivity is measurable, the differences are minimal.
The linear and nonlinear responses are calculated via effective medium theory^{74}, using the form most appropriate for site percolation^{75}. Since the experimental nonlinear response is normalized, we also normalize the calculated response by R_{n} (i.e., take that R_{n} = 1). In order to calculate the thirdorder nonlinear conductivity, the dependence of the voltage on current was determined, and σ_{3} was obtained through an expansion in powers of voltage. Due to the percolative nature of the system, σ_{3} is insensitive to the values of R_{0} and J_{c} in the region of interest close to T_{π} (as long as the current j is much smaller than J_{c}). Thus the only parameters entering the calculation of σ_{3} are Ξ_{0} and the percolation threshold concentration P_{π} of superconducting patches (which depends on the number of spatial dimensions, on site vs. bond percolation, etc.). R_{0} is used in the linear response calculation, and was determined to be 0.005R_{n}, which is realistic for nanoscale patches at a finite excitation frequency^{43,50}.
In order to obtain Ξ_{0} and to determine if 3D (with P_{π} ≈ 0.3) or 2D (with P_{π} ≈ 0.6) site percolation is more appropriate, we simultaneously calculate the linear and nonlinear conductivity and compare to the measurements (Fig. 1c). Although the results do not critically depend on P_{π}, a 3D sitepercolation model with P_{π} = 0.31 yields the best agreement with the data. For example, it enables the linear and nonlinear response in LSCO to be described with a single Ξ_{0} = 28.0 ± 0.4 K, whereas in the 2D model the discrepancy between Ξ_{0} obtained from linear and nonlinear conductivities differs at least by 25% (Figure S3). With P_{π} = 0.31 fixed, individual fits to only the nonlinear response of all investigated compounds (Table 1) gives the overall estimate Ξ_{0} = 27 ± 2 K, whereas the simultaneous calculation of both σ_{1} and σ_{3} for LSCO gives the higher precision above. We emphasize that the parameter R_{0} does not influence the determination of Ξ_{0}: R_{0} influences the shape of the linear conductivity curve, whereas Ξ_{0} sets the range of the superconducting contribution. T_{π} is calculated separately in a modelfree way to obtain the best data scaling, with typical uncertainties smaller than 0.05 K. The LSCO0.15 data are taken as a reference since they exhibit the best signaltonoise ratio. Notably, the determination of Ξ_{0} is independent of T_{π}, since the exponential decay rate constant of σ_{3} is simply inversely proportional to Ξ_{0} (i.e., the rate is 42.6 K/Ξ_{0}). The calculated curves depart from measurements close to the macroscopic T_{c}, which is expected—once a significant volume fraction of the sample is superconducting, Josephson couplings can no longer be neglected, macroscopic phase coherence sets in, and the simple percolation picture needs corrections.
One can perform a similar effective medium calculation for 3D nearestneighbor bond (P_{π} = 0.25) rather than site percolation, or for any other percolation model with a similar critical concentration, and fit to the nonlinear data. This in itself poses no problems and will increase Ξ_{0} (by about 20%). However, the linear conductivity provides a constraint—similar to the 2D case, it cannot be simultaneously obtained with the same Ξ_{0} (the difference being about 10%, larger than the uncertainties). Also, the corrections due to P vs. T nonlinearity may become important. In any case, the difference between P_{π} = 0.31 and 0.25 is not very significant in view of the crudeness of the modeling, but the data do support 3D percolation. The cuprate superconductors are known to be strongly anisotropic; in the sitepercolation model, this translates to anisotropy within the patches (i.e., they are elongated in the cdirection), but this does not change the percolation threshold. Since we measure inplane response, the threshold is the only important parameter. A possible exception would be systems with effectively decoupled layers (such as Eu–LSCO and LBCO close to doping 1/8, as discussed).
Modeling of specific heat
Specific heat measurements in several cuprates^{29,48} show hightemperature tails above T_{c}. Here, we show that the tails can be modeled in a quantitative fashion by simply convoluting the standard meanfield step in specific heat at the (local) T_{c} with the gap distribution. We model the meanfield superconducting contribution to the specific heat coefficient by a simple linear dependence below the local T_{c}, Δγ_{loc} = a(T/T_{c} − 1/2), and take it to be zero above T_{c}. Such a form is not correct at low temperatures, but is appropriate close to T_{c}. The coefficient in a system with a distribution of T_{c} is then just
where g is the distribution function. The calculated Δγ is shown in Fig. 4e in comparison with data on Y_{0.8}Ca_{0.2}Ba_{2}Cu_{3}O_{6.75} from ref. 29, with a Gaussian distribution of gaps centered at 75 K (the macroscopic T_{c} is some 6 K larger, in accordance with Fig. 3a). The best agreement with the data above T_{c} is obtained with a distribution width of 35 K, slightly larger than the values of Ξ_{0} obtained from linear and nonlinear conductivity in the main text. Close to the macroscopic T_{c}, the measurements also show a fluctuationinduced peak, which is not included in the simple meanfield summation that we have performed. Despite the simplicity, our approach is in acceptable quantitative agreement with the tail in the specific heat coefficient, which is an important confirmation of our model using a bulk thermodynamic probe.
Percolation interpretation of the tomographic density of states
Along with transport properties, the percolation model can be used to explain other seemingly unconventional results for the cuprates. One example is the tomographic density of states (TDOS) obtained in recent photoemission measurements^{19}. The effective gap obtained in these experiments does not close at the macroscopic T_{c}, but a “filling” of the density of states is observed to extend to temperatures ~1.2T_{c} (Fig. 3f). The gap filling was attributed to an increased superconducting pairbreaking rate, and the response above T_{c} to preformed pairs. However, as we now show, both effects arise naturally if one assumes a spatial gap distribution. In ref. 19, the density of states was fitted to the standard expression
where ω is the frequency relative to the Fermi level, Γ the pairbreaking rate, and ∆ the superconducting gap. To describe the data with this formula, the pairbreaking rate must increase to Γ ~ ∆ close to T_{c}, which signals that the description is no longer physically valid. We find that the experimental result can be quantitatively reproduced by employing a temperatureindependent Γ and by considering that the experiment measures the average density of states in a system with a realspace gap distribution. We then simply convolute the density of states with a gap distribution function, and employ the standard BCS temperature dependence for the gaps. Notably, a similar procedure was recently used to model ARPES data^{76} in Bi_{2}Sr_{2}CaCu_{2}O_{8+y.} A Gaussian gap distribution with mean ∆_{m} = 9.6 meV and full width at half maximum ∆_{0} = 3.2 meV (in line with ref. 17 and with our nonlinear conductivity measurements) yields rather good agreement with the TDOS experiment at all temperatures (Fig. 3g). This constitutes a strong, independent confirmation of the percolation/gap disorder scenario.
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All data needed to evaluate the conclusions in the paper are present in the paper and/or Supplementary information. Additional data related to this paper may be requested from the authors.
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Acknowledgments
We thank A.V. Chubukov and J.M. Tranquada for comments on the manuscript. D.P., M.V., M.S.G., and M.P. acknowledge funding by the Croatian Science Foundation under grant no. IP1120132729. The work at the University of Minnesota was funded by the Department of Energy through the University of Minnesota Center for Quantum Materials under DESC0016371. The work at the TU Wien was supported by FWF project P27980N36 and the European Research Council (ERC Consolidator Grant no. 725521). We acknowledge M.K. Chan for contributing to Hg1201 sample preparation and characterization.
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D.P. and M.V. built the nonlinear conductivity setup, performed measurements, and analyzed the data. M.S.G. and M.P. built the microwave conductivity setup and performed linear conductivity measurements. M.P. supervised all conductivity experiments. M.P. and N.B. initiated the paraconductivity studies. D.P., G.Y., M.G., and N.B. conceived the idea to pursue the percolationbased data analysis. D.P. performed the percolation calculations. T.S. prepared the LSCO samples. D.P. prepared the YBCOZn sample. G.Y. and N.B. prepared the Hg1201 sample. D.P., M.G., and N.B. wrote the paper with input from all authors.
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Pelc, D., Vučković, M., Grbić, M.S. et al. Emergence of superconductivity in the cuprates via a universal percolation process. Nat Commun 9, 4327 (2018). https://doi.org/10.1038/s4146701806707y
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