Abstract
In underdoped cuprate superconductors, phase stiffness is low and longrange superconducting order is destroyed readily by thermally generated vortices (and antivortices), giving rise to a broad temperature regime above the zeroresistive state in which the superconducting phase is incoherent^{1,2,3,4}. It has often been suggested that these vortexlike excitations are related to the normalstate pseudogap or some interaction between the pseudogap state and the superconducting state^{5,6,7,8,9,10}. However, to elucidate the precise relationship between the pseudogap and superconductivity, it is important to establish whether this broad phasefluctuation regime vanishes, along with the pseudogap^{11}, in the slightly overdoped region of the phase diagram where the superfluid pair density and correlation energy are both maximal^{12}. Here we show, by tracking the restoration of the normalstate magnetoresistance in overdoped La_{2−x}Sr_{x}CuO_{4}, that the phasefluctuation regime remains broad across the entire superconducting composition range. The universal low phase stiffness is shown to be correlated with a low superfluid density^{1}, a characteristic of both underdoped and overdoped cuprates^{12,13,14}. The formation of the pseudogap, by inference, is therefore both independent of and distinct from superconductivity.
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In underdoped cuprates, an energy gap (pseudogap), of as yet unknown origin, appears in the electronic density of states well before superconductivity develops. The continuous evolution^{6} of the pseudogap into the superconducting gap, combined with similarities in their gap magnitudes and symmetries^{5}, has led to suggestions that the pseudogap is a precursor superconducting state^{7,8,9,10} characterized by a broad temperature region over which the superconducting order parameter is finite but the phase fluctuates^{3,4}. This picture of precursor pairing remains controversial however and is challenged by measurements indicating that the pseudogap itself closes at a critical doping concentration^{11}, slightly beyond optimal doping, where superconductivity is most robust^{12}. As it stands, there have been very few studies to date of the fluctuating superconductivity in overdoped, superconducting cuprates to establish the precise relationship between the pseudogap and phase fluctuation regimes.
To address this, we focus here on the evolution of the upper critical field H_{c2} with temperature, as inferred by measurements of the inplane resistivity ρ_{a b}(T,H) in pulsed and d.c. magnetic fields. Previous transport studies in cuprates identified H_{c2}(T) with the ‘knee’ at or near the top of the ρ_{a b}(T) curve or the temperature at which ρ_{a b}(H) attained a certain fraction of the normalstate value^{15}. Either technique would lead to a H_{c2}(T) line with a markedly different shape from that determined by the Nernst effect, for example. In this study, we adopt an alternative approach^{16,17}, using the evolution of the transverse magnetoresistance Δρ_{a b}(H)(=ρ_{a b}(H)−ρ_{a b}(0) with H∥c) with temperature, field and doping, as our probe. In a singleband metal, Δρ_{a b}(H) varies as H^{2} at the lowest fields, this being the lowestorder even response in the longitudinal electrical conductivity to the fieldinduced Lorentz force^{18}. The same behaviour is also observed in optimally and overdoped cuprates well above the phasefluctuation regime^{19}. As the lowest order term in the normalstate magnetoresistance is quadratic, any downward deviation from such quadratic behaviour can only arise from fluctuations associated with some lower resistive state, that is, superconductivity. (In certain underdoped cuprates, the presence of electrons and holes^{20} leads to a different form of the magnetoresistance at low fields that has been modelled successfully by assuming the coexistence of two types of carriers^{21}.) We therefore identify H_{c2} in the optimally and overdoped cuprates as the field at which Δρ_{a b}(H) first recovers its quadratic field dependence. However, as there is no clear thermodynamic phase transition at this particular field scale, in the remainder of this Letter, we shall label it H_{2}, rather than H_{c2}.
Figure 1a shows representative Δρ_{a b}(H)/ρ_{a b}(0) data at temperatures well above T_{c0}∼32 K, obtained on a single crystal of La_{1.79}Sr_{0.21}CuO_{4} (labelled hereafter as LSCO21), which clearly demonstrate the H^{2} dependence up to μ_{0}H=30 T. Below T∼60 K however, as shown in Fig. 1b, the field profile of ρ_{a b}(H) changes and the H^{2} dependence is only recovered above a Tdependent field scale (identified by the vertical arrows) that approaches 50 T at low temperatures. Note that this field scale is already large (∼30 T) at T=T_{c0}, indicating that the Cooper pair amplitude remains large and robust across the superconducting transition.
The evolution of H_{2}(T) for LSCO21 is compared with the temperature dependence of the zerofield resistivity ρ_{a b}(T) in Fig. 1c–e. H_{2}(T) is found to drop monotonically to zero at a temperature T_{2}∼60 K (Fig. 1e) that coincides with a welldefined upturn in dρ_{a b}/dT (Fig. 1d) marking the onset of superconducting fluctuations. Although the upturn in dρ_{a b}/dT is significant (it increases by around 10% on cooling from 60 to 40 K), the absolute change in ρ_{a b}(T) from its extrapolated normalstate behaviour is extremely small (of order 1 μΩ cm), despite the fact that over the same temperature interval H_{2} has increased to around 30 T.
In Fig. 1e, we also compare our estimates for H_{2} with those determined from earlier Nernst measurements (over a narrower temperature range) on a crystal of similar doping (LSCO20; ref. 22). Reasonable agreement is found between the two approaches, although if anything, it would seem that the Nernst effect tends to underestimate H_{2}, presumably owing to the fact that such estimates rely on a linear extrapolation of a finite Nernst signal e_{y} to zero^{22}. Note that H_{2}(T) obtained in either case follows approximately the simple twofluid expression, H_{2}(T)=H_{2}(0)(1−(T/T_{2})^{2}). The phase diagram thus obtained is highly reminiscent of the temperature–field profile expected for a twodimensional Berezinsky–Kosterlitz–Thouless (BKT) transition^{23}, as is the fluctuation conductivity described in Supplementary Fig. S1. Here T_{c0} corresponds to the temperature at which longrange phase coherence is destroyed by thermally generated vortex–antivortex pairs, whereas T_{2} represents the temperature beyond which the pairing amplitude is zero.
Figure 2a shows H_{2}(T) for a number of overdoped LSCO crystals with Sr concentrations ranging between 0.21 and 0.26. (The analysis presented in Fig. 1 for LSCO21 is repeated in Supplementary Figs S2–S4 for LSCO23, LSCO24 and LSCO26 respectively.) One surprising feature of Fig. 2a is that whereas the absolute value of H_{2}(0) decreases with increasing doping, the onset temperature T_{2} remains high, and even rises slightly as x increases from 0.21 to 0.24. Beyond x=0.24, both H_{2}(0) and T_{2} seem to collapse as the edge of the superconducting dome is approached. The evolution of both H_{2}(0) and T_{2} across the phase diagram are summarized in Fig. 2b and c respectively. In Fig. 2b, values of H_{2}(0) for lower hole concentrations (that is, below x=0.21) are extracted from published Nernst results^{24} and from lowtemperature specific heat^{25}. There is clear agreement between the different experimental techniques. The suppression of H_{2}(0) with overdoping reflects the overall reduction in the superconducting pairing strength as the system is doped further away from the parent Mott insulator.
The contrasting doping dependencies of T_{2} and the pseudogap temperature T* for LSCO are highlighted in Fig. 2c. Here T* is defined as the temperature below which ρ_{a b}(T) first deviates from its high temperature Tlinear behaviour. For x<0.19, both T*(x) and T_{2}(x) decrease monotonically with increasing x and, indeed, follow an approximate scaling relation with T*∼2T_{2}. Such a good correspondence has contributed considerably to the widespread association of the two temperature scales to the same physical phenomenon, namely precursor superconductivity. However, whereas T_{2}(x) remains roughly constant between x=0.19 and x=0.24, T* becomes difficult to track as it dips below T_{2}. The precise location of the critical hole doping concentration p_{crit} at which the cuprate pseudogap closes has long been a controversial issue, although there is now a large body of evidence from bulk physical measurements that T* and the pseudogap energy scale do not extend into the heavily overdoped region of the phase diagram but rather collapse at a welldefined critical concentration around p_{crit}=0.19±0.01, irrespective of the cuprate system^{11}. Beyond x=0.19, a new temperature scale T_{coh} appears in the phase diagram, again associated with the recovery of Tlinear resistivity at high temperatures^{26}, but this time corresponding to the loss of quasiparticle coherence, predominantly for states with momenta near the zone boundary^{27}. The crucial distinction between T* and T_{coh} in LSCO is the doping dependence; whereas T* decreases with increasing x, T_{coh} shows the opposite trend.
To parameterize the extent of the fluctuation regime, we plot in Fig. 2d the ratio T_{2}/T_{c0} versus hole content p (solid red squares). Despite the nonmonotonic and asymmetric form of T_{2}(x) in Fig. 2c, the ratio T_{2}/T_{c0} turns out to be surprisingly symmetrical, with a minimum at x=p_{crit}. There is no obvious reason for amplitude fluctuations to follow such a trend. According to the original argument of Emery and Kivelson^{1}, phase stiffness is determined predominantly by the superfluid density ρ_{s}; the smaller the ρ_{s}, the weaker the screening and the larger the region over which classical (thermal) phase fluctuations are important. In underdoped cuprates, ρ_{s} is found to decrease monotonically with T_{c0} (ref. 28), indicating that the condensate density drops as p is reduced. Significantly, in overdoped cuprates, ρ_{s} also falls away as T_{c0} decreases^{13,14}.
The solid circles in Fig. 2d correspond to estimates of the normalized superfluid density ρ_{s}/ρ_{s}(p_{crit}) for LSCO (see Supplementary Table S1 for a description of how ρ_{s}/ρ_{s}(p_{crit}) is determined at each p value). Also included in Fig. 2d for comparison are ρ_{s}/ρ_{s}(p_{crit}) values obtained for Cadoped YBa_{2}Cu_{3}O_{6+δ} (open circles) and Tl_{0.5−y}Pb_{0.5+y}Sr_{2}Ca_{1−x}Y_{x}Cu_{2}O_{7} (open triangles)^{12}. The clear anticorrelation between T_{2}/T_{c0} and ρ_{s}(p) supports the Emery–Kivelson picture^{1} and provides firm evidence that the temperature interval T_{c0}≤T≤T_{2} in overdoped cuprates is characterized by the same phase fluctuations that appear on the underdoped side. In underdoped cuprates, the reduction in the superfluid density (that subsequently enhances the phase fluctuation regime) is attributed to the formation of the normalstate pseudogap, which progressively removes spectral weight that is never recovered on entering the full Meissner state. In overdoped cuprates, the strong suppression of ρ_{s} has been attributed both to phase separation^{13} and to pairbreaking^{14}. Whatever its origin, the observed persistence of an extended fluctuation regime beyond x=p_{crit} in LSCO reveals that low phase rigidity in highT_{c} cuprates does not require the presence of a pseudogap in the normalstate excitation spectrum and that the pseudogap itself is not a precondition for the development of an extended region of phasefluctuating superconductivity.
Detailed Nernst data on irradiated samples of underdoped and optimally doped YBa_{2}Cu_{3}O_{7−δ} have shown that although T_{c0} is strongly suppressed with increasing disorder, T_{2} is much less affected^{29}. In accord with the anticorrelation reported here between T_{2}/T_{c0} and ρ_{s}, one can now interpret the expansion of the fluctuation regime in irradiated YBa_{2}Cu_{3}O_{7−δ} as a direct consequence of the strong disorderinduced suppression of the superfluid density^{12}, which ultimately enhances the role of phase fluctuations. The strong disorder inherent in LSCO presumably plays a key role in amplifying T_{2}/T_{c0} here too. The relation between T_{c0} and T_{2} is explored in more detail in the Supplementary Information.
Finally, the observation of an expanded phasefluctuation regime in overdoped cuprates may help to explain why certain spectroscopic probes, such as angleresolved photoemission, tend to advocate scenarios in which T* tracks the superconducting dome on the overdoped side, rather than vanishing inside it. This difference could arise from the difficulty in distinguishing spectroscopically between the loss of states due to the opening of the normalstate pseudogap and the loss of states due to superconducting fluctuations. Note that in LSCO, T* and T_{2} are easily distinguished by inplane resistivity data as they are identified respectively by a downturn and an upturn in dρ_{a b}/dT (ref. 26). The ubiquity of phasefluctuating superconductivity and the coincidence between T_{2}, as determined by dρ_{a b}/dT, and the vanishing of H_{2}(T), as determined by the highfield magnetoresistance, imply that one might now be able to identify the phase fluctuation regime in any cuprate system (at or beyond optimal doping) simply by taking the first (or second) derivative of the zerofield resistivity curve. Indeed, excellent agreement is already noted between the fluctuation onset temperatures determined by the Nernst effect and magnetization in LSCO, Bi_{2}Sr_{2−y}La_{y}CuO_{6} and optimally doped YBa_{2}Cu_{3}O_{7} and the corresponding T_{2} values obtained from d^{2}ρ_{a b}/dT^{2} analysis of Ando and coworkers^{30}. Such agreement suggests that it should be relatively straightforward to generalize these findings to other new or existing cuprate families.
Methods
Single crystals of La_{2−x}Sr_{x}CuO_{4} (LSCO) were grown using a travellingsolventfloatingzone technique. The actual doping x(=p) of each crystal was estimated from its T_{c} using the empirical relation T_{c0}=T_{c}^{opt}(1−82.6(p−0.16)^{2}) (ref. 31). The inplane resistivity ρ_{a b} was measured using a conventional fourprobe a.c. lockin detection technique in a superconducting magnet at the University of Bristol, in steady magnetic fields up to 33 T at the High Magnetic Field Laboratory (HMFL) in Nijmegen, and in pulsed magnetic fields up to 60 T at the LNMCIT in Toulouse. In all reported measurements, the field was applied along the caxis.
Error bars. The error bars in Figs 1e and 2a (and subsequently in Fig. 2b) reflect our uncertainty in the determination of H_{2}(T) from the individual magnetoresistance curves. More explicitly, we fit the highestfield data, well above H_{2}, to a quadratic field dependence and determine the standard deviation σ from the fit. We then identify H_{2} (and its associated uncertainty) with the point at which the data deviate from the extrapolated highfield fit by more than 3σ (±1σ). The error bars in the Nernst and specificheatderived points are as quoted in the corresponding references. In Fig. 2c, the error bars for T*, T_{coh} and T_{2} represent compound errors due to uncertainty in determining the various onset temperatures and the spread in values as measured on samples of the same x. The T*, T_{coh} and T_{2} values quoted are averages of measurements taken on at least three different samples for each doping concentration. In Fig. 2d, the error bars associated with T_{2}/T_{c0} are again compound errors—the large error bars for LSCO26 reflect for the most part our uncertainty in T_{c0}, the superconducting transition of which is invariably broader as it resides at the edge of the superconducting dome where dT_{c0}/dx is the steepest. The error bars for ρ_{s}/ρ_{s}(p_{crit}) represent the standard deviation in the spread of values as listed in Supplementary Table S1.
References
Emery, V. J. & Kivelson, S. A. Importance of phase fluctuations in superconductors with small superfluid density. Nature 374, 434–437 (1995).
Corson, J., Mallozzi, R., Orenstein, J., Eckstein, J. N. & Bozovic, I. Vanishing of phase coherence in underdoped Bi2Sr2CaCu2O8+δ . Nature 398, 221–223 (1999).
Xu, Z. A., Ong, N. P., Wang, Y., Kakeshita, T. & Uchida, S. Vortexlike excitations and the onset of superconducting phase fluctuation in underdoped La2−xSrxCuO4 . Nature 406, 486–488 (2000).
Wang, Y. et al. Fieldenhanced diamagnetism in the pseudogap state of the cuprate Bi2Sr2CaCu2O8+δ superconductor in an intense magnetic field. Phys. Rev. Lett. 95, 247002 (2005).
Ding, H. et al. Spectroscopic evidence for a pseudogap in the normal state of underdoped highTc superconductors. Nature 382, 51–54 (1996).
Loeser, A. G. et al. Temperature and doping dependence of the Bi–Sr–Ca–Cu–O electronic structure and fluctuation effects. Phys. Rev. B 56, 14185–14189 (1997).
Nagaosa, N. & Lee, P. A. Ginzburg–Landau theory of the spinchargeseparated system. Phys. Rev. B 45, 966–970 (1992).
Randeria, M., Trivedi, N., Moreo, A. & Scalettar, R. T. Pairing and spin gap in the normal state of short coherence length superconductors. Phys. Rev. Lett. 69, 2001–2004 (1992).
Perali, A. et al. Twogap model for underdoped cuprate superconductors. Phys. Rev. B 62, R9295–R9298 (2000).
Lee, J. et al. Spectroscopic fingerprint of phaseincoherent superconductivity in the underdoped Bi2Sr2CaCu2O8+δ . Science 325, 1099–1103 (2009).
Tallon, J. L. & Loram, J. W. The doping dependence of T*—what is the real highTc phase diagram? Physica 349C, 53–68 (2001).
Bernhard, C. et al. Anomalous peak in the superconducting condensate density of cuprate highTc superconductors at a unique doping state. Phys. Rev. Lett. 86, 1614–1617 (2001).
Uemura, Y. J. et al. Magneticfield penetration depth in Tl2Ba2CuO6+δ in the overdoped regime. Nature 364, 605–607 (1993).
Niedermayer, Ch. et al. Muon spin rotation study of the correlation between Tc and ns/m* in overdoped Tl2Ba2CuO6+δ . Phys. Rev. Lett. 71, 1764–1767 (1993).
Ando, Y. et al. Resistive upper critical fields and irreversibility lines of optimally doped highTc cuprates. Phys. Rev. B 60, 12475–12479 (1999).
RullierAlbenque, F. et al. Total suppression of superconductivity by high magnetic fields in YBa2Cu3O6.6 . Phys. Rev. Lett. 99, 027003 (2007).
Cooper, R. A. et al. Anomalous criticality in the electrical resistivity of La2−xSrxCuO4 . Science 323, 603–607 (2009).
Pippard, A. B. Magnetoresistance in Metals (Cambridge Univ. Press, 1989).
Kimura, T. et al. Inplane and outofplane magnetoresistance in La2−xSrxCuO4 single crystals. Phys. Rev. B 53, 8733–8742 (1996).
LeBoeuf, D. et al. Electron pockets in the Fermi surface of holedoped highTc superconductors. Nature 450, 533–536 (2007).
Rourke, P. M. C. et al. Fermisurface reconstruction and twocarrier model for the Hall effect in YBa2Cu4O8 . Phys. Rev. B 82, 020514(R) (2010).
Wang, Y. et al. High field phase diagram of cuprates derived from the Nernst effect. Phys. Rev. Lett. 88, 257003 (2002).
Doniach, S. & Huberman, B. A. Topological excitations in twodimensional superconductors. Phys. Rev. Lett. 42, 1169–1172 (1979).
Wang, Y., Li, L. & Ong, N. P. Nernst effect in highTc superconductors. Phys. Rev. B 73, 024510 (2006).
Wang, Y. & Wen, H. H. Doping dependence of the upper critical field in La2−xSrxCuO4 from specific heat. Europhys. Lett. 81, 57007 (2008).
Hussey, N. E. et al. Dichotomy in the Tlinear resistivity in highTc cuprates. Preprint at http://arxiv.org/abs/0912.2001v1.
Kaminski, A. et al. Crossover from coherent to incoherent electronic excitations in the normal state of Bi2Sr2CaCu2O8+δ . Phys. Rev. Lett. 90, 207003 (2003).
Uemura, Y. J. et al. Universal correlations between Tc and ns/m* (carrier density over effective mass) in highTc cuprate superconductors. Phys. Rev. Lett. 62, 2317–2320 (1989).
RullierAlbenque, F. et al. Nernst effect and disorder in the normal state of highTc cuprates. Phys. Rev. Lett. 96, 067002 (2006).
Ando, Y., Komiya, S., Segawa, K., Ono, S. & Kurita, Y. Electronic phase diagram of highTc cuprate superconductors from a mapping of the inplane resistivity curvature. Phys. Rev. Lett. 93, 267001 (2004).
Tallon, J. L. et al. Generic superconducting phase behaviour in highTc cuprates: Tc variation with hole concentration in YBa2Cu3O7−δ . Phys. Rev. B 51, 12911–12914 (1995).
Acknowledgements
The authors would like to acknowledge R. A. Cooper for experimental assistance, S. M. Hayden and O. J. Lipscombe for providing us with the LSCO23 crystals, and J. P. Annett, A. Carrington, B. Gyorffy, R. H. McKenzie, T. Senthil, N. Shannon, T. Timusk, Y. J. Uemura and J. A. Wilson for fruitful discussions. This work was supported by EPSRC (UK), MEXTCT2006039047, EURYI, the National Research Foundation, Singapore and EuroMagNET under EU contract 228043. N.E.H. acknowledges a Royal Society Wolfson Research Merit Award.
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All authors made critical comments on the manuscript. Y.T., T.A. and Y.K. synthesized the samples. P.M.C.R., I.M., X.X., Y.W., B.V., C.P., E.V.K., U.Z. and N.E.H. carried out the transport measurements. P.M.C.R., I.M. and N.E.H. analysed and interpreted the transport data.
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Rourke, P., Mouzopoulou, I., Xu, X. et al. Phasefluctuating superconductivity in overdoped La_{2−x}Sr_{x}CuO_{4}. Nature Phys 7, 455–458 (2011). https://doi.org/10.1038/nphys1945
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DOI: https://doi.org/10.1038/nphys1945
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