Abstract
Despite extraordinary scientific efforts over the past three decades, the cuprate hightemperature superconductors continue to pose formidable challenges. A pivotal problem, essential for understanding both the normal and superconducting states, is to clarify the nature of the superconducting prepairing above the bulk transition temperature T_{c}. Different experimental probes have given conflicting results, in part due to difficulties in discerning the superconducting response from the complex normalstate behavior. Moreover, it has proven challenging to separate common properties of the cuprates from compoundspecific idiosyncrasies. Here we investigate the paraconductivity—the superconducting contribution to the directcurrent (dc) conductivity—of the simpletetragonal model cuprate material HgBa_{2}CuO_{4+δ}. We are able to separate the superconducting and normalstate responses by taking advantage of the Fermiliquid nature of the normal state in underdoped HgBa_{2}CuO_{4+δ}; the robust and simple quadratic temperaturedependence of the normalstate resistivity enables us to extract the paraconductivity above the macroscopic T_{c} with great accuracy. We find that the paraconductivity exhibits unusual exponential temperature dependence, and that it can be quantitatively explained by a simple superconducting percolation model. Consequently, the emergence of superconductivity in this model system is dominated by the underlying intrinsic gap inhomogeneity. Motivated by these insights, we reanalyze published results for two other cuprates and find exponential behavior as well, with nearly the same characteristic temperature scale. The universal intrinsic gap inhomogeneity is not only essential for understanding the supercoducting precursor, but will also have to be taken into account in the analysis of other bulk measurements of the cuprates.
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 T_{c}, there exists no longrange 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 Fermiliquid charge transport^{17,18,19,20,21} and proposed that superconductivity emerges from this state in a universal percolative manner.^{8,9} Directcurrent (dc) conductivity is a highly sensitive probe that can, in principle, provide a unique opportunity to test the consistency of these ideas. Furthermore, the effectivemedium 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 Fermiliquid 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 prepairing regime has been the separation of the superconducting response from the normalstate 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 chargedensitywave 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 chargeorder related signal might be mistaken for superconducting fluctuations,^{34,35} or linear magnetization and conductivity measurements,^{4,12,13,36} where the normalstate behavior is assumed to be linear in temperature, which is not necessarily the case.^{8} Several schemes to systematically subtract the presumed normalstate 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 magnetization^{8} and nonlinear conductivity.^{9} Here we introduce a method to separate the dc paraconductivity from the normalstate contribution, with the resulting highquality data readily comparable to the more involved backgroundfree techniques.
A number of recent experimental investigations consistently point to a simple picture for both the normal state^{17,18,19,20,21} 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 experiments^{21} unequivocally show that the mobile charge carriers behave as a Fermi liquid, even in strongly underdoped compounds. NMR investigations also find an itinerant Fermiliquid component, along with a local, pseudogaprelated signal.^{37} The observations that the magnetoresistivity obeys Kohler scaling with a 1/τ ∝ T^{2} scattering rate^{19} and that the optical scattering rate exhibits conventional scaling with temperature and frequency^{21} are particularly clearcut Fermiliquid signatures. Moreover, magnetization,^{8} highfrequency linear conductivity,^{3,5,6,7} and nonlinear response measurements^{9} indicate that the superconducting emergence regime is limited to a rather narrow temperature range above T_{c} 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 Fermiliquid behavior in the pseudogap regime. We subtract its contribution to the planar resistivity with a reliability approaching the backgroundfree 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 HgBa_{2}CuO_{4+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 CuO_{2} layer per formula unit, and the largest optimal T_{c} (nearly 100 K) of all such singlelayer compounds.^{38} Further evidence for the model nature of Hg1201 comes from the observation of a tiny residual resistivity,^{20,39} of ShubnikovdeHaas oscillations,^{40,41} and of a small density of vortexpinning 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 ∝ T^{2}, the behavior characteristic of a Fermi liquid.^{20}
Results
Paraconductivity measurements
We studied Hg1201 samples with T_{c} ≈ 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 T_{c} in which pure quadraticintemperature resistivity is seen, while being reasonably far away from the doping level (p ≈ 0.09) where weak shortrange 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 T_{c}, T**, and T*. The purely quadratic behavior seen below T** is in agreement with the Fermiliquid character of the mobile holes.^{17,18,19,20} The considerable difference between T** and T_{c} provides for a simple way to assess the superconducting paraconductivity contribution. In order to subtract the normalstate signal and obtain the purely superconducting contribution above T_{c}, we fit ρ(T) = ρ_{0} + a_{2}T^{2} to the data in a temperature range from 100 K to T** ≈ 140 K, where ρ_{0} is the small residual resistivity (the estimated residual resistivity ratio is ~120) and a_{2} a constant. The resultant value of a_{2} = 9.8(1) nΩ cm/K^{2} is consistent with previous measurements^{20} 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) = ρ_{0} + aT^{α} 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 Fermiliquid temperature dependence. We may therefore safely extrapolate the fit to T_{c} to obtain the underlying normalstate contribution.
Inversion of the experimentally determined resistivity and subtraction of the extrapolated quadratic temperature dependence then gives the superconducting paraconductivity contribution, ∆σ_{dc}, shown in Fig. 2. We present results for the same sample as above, which was chosen from a larger batch of crystals with T_{c} ≈ 80 K due to its welldefined superconducting transitions—for Hg1201, the samplecontacting procedure often induces spurious doping of the sample surface,^{20} which can “short out” the current path at temperatures above the nominal T_{c} 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 T_{c}. We also performed a magnetic susceptibility measurement using vibrating sample magnetometry (VSM) that shows a sharp transition, with a T_{c} value that agrees well with the resistive T_{c} (Fig. 1c). The fieldcool/zerofieldcool susceptibility ratio approaches one and is among the highest observed in the cuprates,^{39} which demonstrates the high quality of the sample.
In order to compare our result for Hg1201 to other cuprates, we use the same normalstate subtraction procedure on published resistivity data for underdoped, detwinnned YBa_{2}Cu_{3}O_{6+δ} (YBCO) and underdoped La_{2−x}Sr_{x}CuO_{4} (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 exponentiallike temperature dependence away from T_{c}, consistent with prior magnetization^{8} and nonlinear response^{9} results. Notably, a similar exponential paraconductivity was previously seen in YBCO at several doping levels, but with a different normalstate subtraction scheme that involves fitting to highmagneticfield data and yields considerably lower sensitivity.^{16}
We emphasize five crucial points: (i) the exponential dependence is qualitatively different from the underlying normalstate powerlaw 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 Fermiliquid normalstate contribution; (iv) the signaltonoise 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 compoundindependent (and nearly dopingindependent) underlying energy scale k_{B}Ξ_{0} 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 T_{c}, the material consists of normal patches with resistance R_{n}, and of superconducting patches with resistance R_{0} (where we will take the limit R_{0} → 0).^{9,46} The fraction of superconducting patches, P, is temperaturedependent: at a critical fraction P_{π} (corresponding to the critical temperature T_{π}), a samplespanning superconducting cluster is formed and hence percolates. In the limit of vanishingly small currents, T_{π} equals T_{c}, but in any experiment, T_{c} is shifted slightly below T_{π} due to the required nonzero currents. The temperaturedependent superconducting fraction originates from an underlying distribution of superconducting gaps, and P is hence directly obtained as the temperature integral of the distribution:
where g(T′) is the normalized local T_{c} distribution function. This is shown schematically in Fig. 3. For concreteness, we use the simplest (Gaussian) distribution with a fullwidthathalfmaximum equal to Ξ_{0}, 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 effectivemedium theory (EMT),^{22} in a form derived specifically for site percolation problems^{23}:
where \(\tilde \sigma _{\rm{dc}}\) is the conductivity normalized to the normalstate value, R_{0} the resistivity of the superconducting patches (which is taken to be vanishingly small), and P_{π} the percolation threshold. While it is known that EMT becomes unreliable in the critical regime close to the percolation threshold^{22} (in our case, about 2 K above T_{c}), we use it for simplicity in the interesting highertemperature regime away from T_{c}. Notably, the validity of EMT has been established also in recent comparison to direct numerical calculations of the resistivity of superconductorresistor 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 < T_{c} + 2 K, is presumably not purely percolative, with critical exponents modified by thermal effects^{47,48}; in order to see a discrepancy between the data and the EMT calculation, a careful powerlaw analysis of the critical regime would need to be undertaken, with more closely spaced measurements around T_{c}. The investigation of criticality is thus not within the scope of the present work.
In order to obtain the limit of zero R_{0}, we use different small values in the numerical calculation, until no significant changes in the output are seen (typically for R_{0} on the order of 10^{−5} R_{n}). For simplicity, we take R_{n} to be constant in the temperature interval of interest—this is appropriate, because its relative change (due to the T^{2} dependence) is about 25% over a 10K interval, whereas the paraconductivity changes by a factor of about 10^{2} 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,
where d is the distance between CuO_{2} 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 work^{4,16,36} to analyze the paraconductivity close to T_{c}. 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 T_{c} is not sufficient to perform a reliable critical exponent analysis. Yet away from T_{c} (T > T_{c} + 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 Ξ_{0} of the T_{c} distribution. Other parameters that enter the calculation are constrained: R_{n} is simply the normalstate resistivity, T_{π} is slightly larger than T_{c} (in the present calculation it was taken to be T_{c} + 1 K, but we note that our definition of T_{c} 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 P_{π} was taken to be 0.3, consistent with the prior nonlinear conductivity analysis.^{9} The critical concentration P_{π} is not arbitrary; it is determined by the details of the percolation model^{49}—site or bond percolation, percolation with or without fartherneighbor corrections, etc.—and by the dimensionality of the percolation process. The model yields virtually the same temperature dependence for different values of P_{π}, with a corresponding change in Ξ_{0}: a smaller P_{π} implies a larger Ξ_{0}, and vice versa. We therefore cannot distinguish among specific percolation scenarios, such as twodimensional versus threedimensional percolation. Prior comparison between linear and nonlinear response indicated that a threedimensional site percolation model with P_{π} ≈ 0.3 is appropriate,^{9} leading us to use the same value here. Remarkably, the value Ξ_{0} = 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 simpletetragonal 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 Fermiliquid 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 T_{c} and the underlying Fermiliquid T^{2} 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 bismuthbased cuprates or twinned YBCO^{17,19}). Notably, below optimal doping, LSCO exhibits lowtemperature resistivity upturns^{50} (typically revealed in the absence of superconductivity upon applying a high caxis 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 normalstate subtraction procedure is not possible for optimally doped compounds, where T** becomes comparable to, or smaller than T_{c} 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 (temperaturedependent) delocalization of charge^{53} explains the apparently Tlinear 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 zerothorder 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 realspace superconducting gap disorder, such as scanning tunneling microscopy,^{55,56} which have observed superconducting gap distributions with a width comparable to k_{B}Ξ_{0}. It is also qualitatively consistent with NMR results that demonstrate a considerable distribution of local electric field gradients,^{57,58} and with Xray experiments that find percolative structures^{59,60} in oxygendoped La_{2}CuO_{4+δ} and YBCO.
The clear confirmation of the superconducting percolation scenario in the present work suggests that, fundamentally, both the normalstate 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 nonFermiliquid 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 T_{c}. 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 prepairing 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 Fermiliquid 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/m^{2}, chosen to be small enough to lead to negligible Joule heating, but large enough for a good signaltonoise ratio for the paraconductivity. Temperature was stabilized before each datum was measured.
Data availability
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.
References
 1.
Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to hightemperature superconductivity. Nature 518, 179–186 (2015).
 2.
Corson, R., Mallozzi, L., Orenstein, J., Eckstein, J. N. & Božović, I. Vanishing of phase coherence in underdoped Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}. Nature 398, 221–223 (1999).
 3.
Orenstein, J., Corson, J., Oh, S. & Eckstein, J. N. Superconducting fluctuations in Bi_{2}Sr_{2}Ca_{1x}Dy_{x}Cu_{2}O_{8+δ} as seen by terahertz spectroscopy. Ann. Phys. 15, 596–605 (2006).
 4.
Silva, E., Sarti, S., Fastampa, R. & Giura, M. Excess conductivity of overdoped Bi_{2}Sr_{2}CaCu_{2}O_{8+x} crystals well above T _{c}. Phys. Rev. B 64, 144508 (2001).
 5.
Grbić, M. S. et al. Microwave measurements of the inplane and caxis conductivity in HgBa_{2}CuO_{4+δ}: discriminating between superconducting fluctuations and pseudogap effects. Phys. Rev. B 80, 094511 (2009).
 6.
Bilbro, L. S. et al. Temporal correlations of superconductivity above the transition temperature in La_{2x}Sr_{x}CuO_{4} probed by terahertz spectroscopy. Nat. Phys. 7, 298–302 (2011).
 7.
Grbić, M. S. et al. Temperature range of superconducting fluctuations above T _{c} in YBa_{2}Cu_{3}O_{7δ} single crystals. Phys. Rev. B 83, 144508 (2011).
 8.
Yu, G. et al. Universal superconducting precursor in the cuprates. Preprint at https://arxiv.org/abs/1710.10957 (2017).
 9.
Pelc, D. et al. Emergence of superconductivity in the cuprates via a universal percolation process. Preprint at https://arxiv.org/abs/1710.10219 (2017).
 10.
Xu, Z. A., Ong, N. P., Wang, Y., Kakeshita, T. & Uchida, S. Vortexlike excitations and the onset of superconducting phase fluctuation in underdoped La_{2x}Sr_{x}CuO_{4}. Nature 406, 486–488 (2000).
 11.
Wang, Y., Li, L. & Ong, N. P. Nernst effect in highT _{c} superconductors. Phys. Rev. B 73, 024510 (2005).
 12.
Wang, Y. et al. Fieldenhanced diamagnetism in the pseudogap state of the cuprate Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} superconductor in an intense magnetic field. Phys. Rev. Lett. 95, 247002 (2005).
 13.
Li, L. et al. Diamagnetism and Cooper pairing above T _{c} in cuprates. Phys. Rev. B 81, 054510 (2010).
 14.
Dubroka, A. et al. Evidence of a precursor superconducting phase at temperatures as high as 180K in RBa_{2}Cu_{3}O_{7−δ} (R=Y, Gd, Eu) superconducting crystals from infrared spectroscopy. Phys. Rev. Lett. 106, 047006 (2011).
 15.
Uykur, E., Tanaka, K., Masui, T., Miyasaka, S. & Tajima, S. Persistence of the superconducting condensate far above the critical temperature of YBa_{2}(Cu,Zn)_{3}O_{y} revealed by caxis optical conductivity measurements for several Zn concentrations and carrier doping levels. Phys. Rev. Lett. 112, 127003 (2014).
 16.
RullierAlbenque, F., Alloul, H. & Rikken, G. Highfield studies of superconducting fluctuations in highT _{c} cuprates: Evidence for a small gap distinct from the large pseudogap. Phys. Rev. B 84, 014522 (2011).
 17.
Barišić, N. et al. Evidence for a universal Fermiliquid scattering rate throughout the phase diagram of the copperoxide superconductors. Preprint at https://arxiv.org/abs/arxiv:1507.07885 (2015).
 18.
Li, Y., Tabis, W., Yu, G., Barišić, N. & Greven, M. Hidden Fermiliquid charge transport in the antiferromagnetic phase of the electrondoped cuprate superconductors. Phys. Rev. Lett. 117, 197001 (2016).
 19.
Chan, M. K. et al. Inplane magnetoresistance obeys Kohler’s rule in the pseudogap phase of cuprate superconductors. Phys. Rev. Lett. 113, 177005 (2014).
 20.
Barišić, N. et al. Universal sheet resistance and revised phase diagram of the cuprate hightemperature superconductors. Proc. Natl Acad. Sci. USA 110, 12235–12240 (2013).
 21.
Mirzaei, S.I. et al. Spectroscopic evidence for Fermi liquidlike energy and temperature dependence of the relaxation rate in the pseudogap phase of the cuprates. Proc. Natl Acad. Sci. USA 110, 5774–5778 (2013).
 22.
Kirkpatrick, S. Percolation and conduction. Rev. Mod. Phys. 45, 574–588 (1973).
 23.
Nakamura, M. Conductivity for the sitepercolation problem by an improved effectivemedium theory. Phys. Rev. B 29, 3691–3693 (1984).
 24.
Caprara, S., Grilli, M., Benfatto, L. & Castellani, C. Effective medium theory for superconducting layers: a systematic analysis including space correlation effects. Phys. Rev. B 84, 014514 (2011).
 25.
Bucheli, D., Caprara, S., Castellani, C. & Grilli, M. Metalsuperconductor transition in lowdimensional superconducting clusters embedded in twodimensional electron systems. New J. Phys. 15, 023014 (2013).
 26.
Caprara, S. et al. Inhomogeneous multi carrier superconductivity at LaXO_{3}/SrTiO_{3} (X = Al or Ti) oxide interfaces. Supercond. Sci. Technol. 28, 045004 (2015).
 27.
Tranquada, J. et al. Evidence for stripe correlations of spins and holes in copper oxide superconductors. Nature 375, 561–563 (1995).
 28.
Fauqué, B. et al. Magnetic order in the pseudogap phase of highT _{c} superconductors. Phys. Rev. Lett. 96, 197001 (2006).
 29.
Li, Y. et al. Magnetic order in the pseudogap phase of HgBa_{2}CuO_{4+δ} studied by spinpolarized neutron diffraction. Phys. Rev. B 84, 224508 (2011).
 30.
Chang, J. et al. Direct observation of competition between superconductivity and charge density wave order in YBa_{2}Cu_{3}O_{6.67}. Nat. Phys. 8, 871–876 (2012).
 31.
Ghiringhelli, G. et al. Longrange incommensurate charge fluctuations in (Y,Nd)Ba_{2}Cu_{3}O_{6+x}. Science 337, 821–825 (2012).
 32.
Tabis, W. et al. Charge order and its connection with Fermiliquid charge transport in a pristine highT _{c} cuprate. Nat. Commun. 5, 5875 (2014).
 33.
Tabis, W. et al. Synchrotron xray scattering study of chargedensitywave order in HgBa_{2}CuO_{4+δ}. Phys. Rev. B 96, 134510 (2017).
 34.
CyrChroiniere, O. et al. Enhancement of the Nernst effect by stripe order in a highT _{c} superconductor. Nature 458, 743–745 (2009).
 35.
Laliberte, F. et al. Fermisurface reconstruction by stripe order in cuprate superconductors. Nat. Commun. 2, 432 (2011).
 36.
Caprara, S., Grilli, M., Leridon, B. & Lesueur, J. Extended paraconductivity regime in underdoped cuprates. Phys. Rev. B 72, 104509 (2005).
 37.
Haase, J. et al. Twocomponent uniform spin susceptibility of superconducting HgBa_{2}CuO_{4+δ} single crystals measured using ^{63}Cu and ^{199}Hg nuclear magnetic resonance. Phys. Rev. B 85, 104517 (2012).
 38.
Eisaki, H. et al. Effect of chemical inhomogeneity in bismuthbased copper oxide superconductors. Phys. Rev. B 69, 064512 (2004).
 39.
Barišić, N. et al. Demonstrating the model nature of the hightemperature superconductor HgBa_{2}CuO_{4+δ}. Phys. Rev. B 78, 054518 (2008).
 40.
Chan, M. K. et al. Single reconstructed Fermi surface pocket in an underdoped singlelayer cuprate superconductor. Nat. Commun. 7, 12244 (2016).
 41.
Barišić, N. et al. Universal quantum oscillations in the underdoped cuprate superconductors. Nat. Phys. 9, 761–764 (2013).
 42.
Li, Y., Egetenmeyer, N., Gavilano, J. L., Barišić, N. & Greven, M. Magnetic vortex lattice in HgBa_{2}CuO_{4+δ} observed by smallangle neutron scattering. Phys. Rev. B 83, 054507 (2011).
 43.
Zhao, X. et al. Crystal growth and characterization of the model hightemperature superconductor HgBa_{2}CuO_{4+δ}. Adv. Mater. 18, 3243–3247 (2006).
 44.
Hinton, J. P. et al. The rate of quasiparticle recombination probes the onset of coherence in cuprate superconductors. Sci. Rep. 6, 23610 (2016).
 45.
Tinkham, M. Introduction to Superconductivity (McGrawHill, New York, 1996).
 46.
Muniz, R. A. & Martin, I. Method for detecting superconducting stripes in hightemperature superconductors based on nonlinear resistivity measurements. Phys. Rev. Lett. 107, 127001 (2011).
 47.
Cardy, J. Scaling and Renormalization in Statistical Physics (Cambridge University Press, Cambridge, 1996).
 48.
Bergman, D. J., Aharony, A. & Imry, Y. Percolation mechanism for long range magnetic order in disordered systems. J. Mag. Mag. Mater. 7, 217–219 (1978).
 49.
Stauffer, D. & Aharony, A. Introduction to Percolation Theory (Taylor & Francis, London, 1994).
 50.
Ando, Y., Boebinger, G. S., Passner, A., Kimura, T. & Kishio, K. Logarithmic divergence of both inplane and outofplane normalstate resistivities of superconducting La_{2x}Sr_{x}CuO_{4} in the zerotemperature limit. Phys. Rev. Lett. 75, 4662–4665 (1995).
 51.
RullierAlbenque, F., Alloul, H., Balakirev, F. & Proust, C. Disorder, metalinsulator crossover and phase diagram in highT_{c} cuprates. Europhys. Lett. 81, 1–6 (2008).
 52.
Ando, Y. et al. Electronic phase diagram of highT _{c} cuprate superconductors from a mapping of the inplane resistivity curvature. Phys. Rev. Lett. 93, 267001 (2004).
 53.
Pelc, D. et al. Unusual behavior of cuprates explained by heterogeneous charge localization Preprint at https://arxiv.org/abs/1710.10221 (2017).
 54.
Char, K. & Kapitulnik, A. Fluctuation conductivity in inhomogeneous superconductors. Z. Phys. B 72, 253–259 (1988).
 55.
Boyer, M. C. et al. Imaging the two gaps of the hightemperature superconductor Bi_{2}Sr_{2}CuO_{6+x}. Nat. Phys. 3, 802–806 (2007).
 56.
Lv, Y.F. et al. Mapping the electronic structure of each ingredient oxide layer of highT _{c} cuprate superconductor Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}. Phys. Rev. Lett. 115, 237002 (2015).
 57.
Singer, P. W., Hunt, A. W. & Imai, T. ^{63}Cu NQR evidence for spatial variation of hole concentration in La_{2−x}Sr_{x}CuO_{4}. Phys. Rev. Lett. 88, 047602 (2002).
 58.
Rybicki, D. et al. Spatial inhomogeneities in singlecrystal HgBa_{2}CuO_{4+δ} from ^{63}Cu NMR spin and quadrupole shifts. J. Supercond. Nov. Magn. 22, 179–183 (2009).
 59.
Poccia, N. et al. Percolative superconductivity in La_{2}CuO_{4.06} by lattice granularity patterns with scanning micro xray absorption near edge structure. Appl. Phys. Lett. 104, 221903 (2014).
 60.
Campi, G., Ricci, A., Poccia, N. & Bianconi, A. Imaging spatial ordering of the oxygen chains in YBa_{2}Cu_{3}O_{6+y} at the insulatortometal transition. J. Supercond. Nov. Magn. 27, 987–990 (2014).
 61.
Reber, T. J. et al. Prepairing and the “filling” gap in the cuprates from the tomographic density of states. Phys. Rev. B 87, 060506(R) (2013).
Acknowledgements
P.P. acknowledges funding by the Croatian Science Foundation Project No. IP2016067258. D.P. and M.P. acknowledge funding by the Croatian Science Foundation under Grant No. IP1120132729. The work at the TU Wien was supported by FWF project P27980N36 and the European Research Council (ERC Consolidator Grant No. 725521). 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.
Author information
Affiliations
Contributions
P.P. and D.P. contributed equally to this work. P.P. performed the conductivity measurements. D.P. performed the model calculations. Y.T., K.V., Z.A., V.N., and G.Y. grew and characterized the samples. K.V. performed the magnetization measurements. D.P., N.B., and M.G. initiated the percolation analysis. D.P., N.B., and M.G. wrote the manuscript, with input from all authors. M.P., N.B., and M.G. supervised the project.
Corresponding authors
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Popčević, P., Pelc, D., Tang, Y. et al. Percolative nature of the directcurrent paraconductivity in cuprate superconductors. npj Quant Mater 3, 42 (2018). https://doi.org/10.1038/s4153501801152
Received:
Revised:
Accepted:
Published:
Further reading

Dopingdependent phonon anomaly and chargeorder phenomena in the HgBa2CuO4+δ and HgBa2CaCu2O6+δ superconductors
Physical Review B (2020)

Resistivity phase diagram of cuprates revisited
Physical Review B (2020)

Phonon spectrum of underdoped HgBa2CuO4+δ investigated by neutron scattering
Physical Review B (2020)

Unusual Dynamic Charge Correlations in SimpleTetragonal HgBa2CuO4+δ
Physical Review X (2020)

Superconducting fluctuations probed by the Higgs mode in Bi2Sr2CaCu2O8+x thin films
Physical Review B (2020)