Main

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 evolution6 of the pseudogap into the superconducting gap, combined with similarities in their gap magnitudes and symmetries5, has led to suggestions that the pseudogap is a precursor superconducting state7,8,9,10 characterized by a broad temperature region over which the superconducting order parameter is finite but the phase fluctuates3,4. This picture of precursor pairing remains controversial however and is challenged by measurements indicating that the pseudogap itself closes at a critical doping concentration11, slightly beyond optimal doping, where superconductivity is most robust12. 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 Hc2 with temperature, as inferred by measurements of the in-plane resistivity ρa b(T,H) in pulsed and d.c. magnetic fields. Previous transport studies in cuprates identified Hc2(T) with the ‘knee’ at or near the top of the ρa b(T) curve or the temperature at which ρa b(H) attained a certain fraction of the normal-state value15. Either technique would lead to a Hc2(T) line with a markedly different shape from that determined by the Nernst effect, for example. In this study, we adopt an alternative approach16,17, using the evolution of the transverse magnetoresistance Δρa b(H)(=ρa b(H)−ρa b(0) with H∥c) with temperature, field and doping, as our probe. In a single-band metal, Δρa b(H) varies as H2 at the lowest fields, this being the lowest-order even response in the longitudinal electrical conductivity to the field-induced Lorentz force18. The same behaviour is also observed in optimally and overdoped cuprates well above the phase-fluctuation regime19. As the lowest order term in the normal-state 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 holes20 leads to a different form of the magnetoresistance at low fields that has been modelled successfully by assuming the co-existence of two types of carriers21.) We therefore identify Hc2 in the optimally and overdoped cuprates as the field at which Δρa 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 H2, rather than Hc2.

Figure 1a shows representative Δρa b(H)/ρa b(0) data at temperatures well above Tc0∼32 K, obtained on a single crystal of La1.79Sr0.21CuO4 (labelled hereafter as LSCO21), which clearly demonstrate the H2 dependence up to μ0H=30 T. Below T∼60 K however, as shown in Fig. 1b, the field profile of ρa b(H) changes and the H2 dependence is only recovered above a T-dependent 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=Tc0, indicating that the Cooper pair amplitude remains large and robust across the superconducting transition.

Figure 1: Fluctuating superconductivity in La1.79Sr0.21CuO4 (LSCO21).
figure 1

a,b, Transverse in-plane magnetoresistance Δρa b(H) (H∥c) plotted versus H2 both above a and below b the onset temperature for fluctuating superconductivity T2. In a, the data are normalized to ρa b(0), the zero-field resistivity value. In b, the superscript ‘+’ indicates that the ρa b(H) have been offset for clarity. The solid lines in a and b highlight the region of H2 magnetoresistance (MR), and the solid arrows and the dashed horizontal error bars in b indicate respectively the upper field H2 and its associated uncertainty (see Error bars section) at which the normal-state (H2) MR is recovered. c, Zero-field resistivity ρa b(T) curve for LSCO21. The superconducting transition temperature Tc0 is defined by the midpoint of the superconducting transition. d, Temperature derivative dρa b/dT of the same resistivity curve shown in c. Here, T2 is indicated by the minimum in dρa b/dT. The dotted line is a guide to the eye. e, Solid circles: temperature dependence of H2 as determined by the restoration of the normal-state MR. Open circles: H2(T) as determined by Nernst effect measurements on LSCO20 (ref. 22). The thick dashed line is a fit to the two-fluid expression H2(T)=H2(0)(1−(T/T2)2) with H2(0)=48.5 T and T2=58.5 K. The vertical dashed line highlights the coincidence of T2 in d and e. The horizontal error bars in c indicate our uncertainty in the two temperature scales Tc0 and T2.

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The evolution of H2(T) for LSCO21 is compared with the temperature dependence of the zero-field resistivity ρa b(T) in Fig. 1c–e. H2(T) is found to drop monotonically to zero at a temperature T2∼60 K (Fig. 1e) that coincides with a well-defined upturn in dρa b/dT (Fig. 1d) marking the onset of superconducting fluctuations. Although the upturn in dρa b/dT is significant (it increases by around 10% on cooling from 60 to 40 K), the absolute change in ρa b(T) from its extrapolated normal-state behaviour is extremely small (of order 1 μΩ cm), despite the fact that over the same temperature interval H2 has increased to around 30 T.

In Fig. 1e, we also compare our estimates for H2 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 H2, presumably owing to the fact that such estimates rely on a linear extrapolation of a finite Nernst signal ey to zero22. Note that H2(T) obtained in either case follows approximately the simple two-fluid expression, H2(T)=H2(0)(1−(T/T2)2). The phase diagram thus obtained is highly reminiscent of the temperature–field profile expected for a two-dimensional Berezinsky–Kosterlitz–Thouless (BKT) transition23, as is the fluctuation conductivity described in Supplementary Fig. S1. Here Tc0 corresponds to the temperature at which long-range phase coherence is destroyed by thermally generated vortex–antivortex pairs, whereas T2 represents the temperature beyond which the pairing amplitude is zero.

Figure 2a shows H2(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 H2(0) decreases with increasing doping, the onset temperature T2 remains high, and even rises slightly as x increases from 0.21 to 0.24. Beyond x=0.24, both H2(0) and T2 seem to collapse as the edge of the superconducting dome is approached. The evolution of both H2(0) and T2 across the phase diagram are summarized in Fig. 2b and c respectively. In Fig. 2b, values of H2(0) for lower hole concentrations (that is, below x=0.21) are extracted from published Nernst results24 and from low-temperature specific heat25. There is clear agreement between the different experimental techniques. The suppression of H2(0) with overdoping reflects the overall reduction in the superconducting pairing strength as the system is doped further away from the parent Mott insulator.

Figure 2: Evolution of the upper field H2(T) and the phase diagram for overdoped LSCO.
figure 2

a, Temperature dependence of the upper field H2(T) for LSCO21 (solid blue circles), LSCO23 (solid green diamonds), LSCO24 (solid red squares) and LSCO26 (solid black triangles). The dashed lines are fits to the expression H2(T)=H2(0) (1−(T/T2)2). b, The doping dependence of H2 (0) as determined by the Nernst effect (solid diamonds, ref. 24), the coefficient of the H0.5 field-dependence of the low temperature electronic specific heat (solid circles, ref. 25) and our magnetoresistance measurements (solid squares). The solid red curve denotes the doping dependence of the superconducting transition temperature Tc0(x). c, Temperature versus doping phase diagram of LSCO as extracted from the temperature derivative of ρa b(T) (ref. 26). The labels Tc0, T2, T*, Tcoh represent respectively the zero-field superconducting transition temperature, the onset temperature for short-lived vortex excitations, the opening of the pseudogap and the onset of anti-nodal quasiparticle incoherence27. For the purposes of this study, T* and Tcoh are identified with the downturns in dρa b/dT for x<0.19 and x>0.19 respectively26. The solid diamonds correspond to T2 values obtained from Nernst measurements on LSCO (ref. 24). The dashed lines are all guides to the eye. The hatched region reflects current uncertainty in the behaviour of T* and Tcoh around pcrit=0.19. d, Ratio of T2/Tc0 versus hole content p (solid squares) superimposed on a plot of the normalized superfluid density ρs/ρs(pcrit) for LSCO (solid circles, pcrit=0.19±0.01), Ca-doped YBa2Cu3O6+δ (open circles) and Tl0.5−yPb0.5+ySr2Ca1−xYxCu2O7 (open triangles)12, illustrating the anti-correlation between ρs and the extent of the fluctuation regime. For determination of ρs/ρs(pcrit) for LSCO, and its associated error, please refer to Supplementary Table S1. For a description of all the associated errors, please refer to the Error bars section.

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The contrasting doping dependencies of T2 and the pseudogap temperature T* for LSCO are highlighted in Fig. 2c. Here T* is defined as the temperature below which ρa b(T) first deviates from its high temperature T-linear behaviour. For x<0.19, both T*(x) and T2(x) decrease monotonically with increasing x and, indeed, follow an approximate scaling relation with T*∼2T2. 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 T2(x) remains roughly constant between x=0.19 and x=0.24, T* becomes difficult to track as it dips below T2. The precise location of the critical hole doping concentration pcrit 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 well-defined critical concentration around pcrit=0.19±0.01, irrespective of the cuprate system11. Beyond x=0.19, a new temperature scale Tcoh appears in the phase diagram, again associated with the recovery of T-linear resistivity at high temperatures26, but this time corresponding to the loss of quasiparticle coherence, predominantly for states with momenta near the zone boundary27. The crucial distinction between T* and Tcoh in LSCO is the doping dependence; whereas T* decreases with increasing x, Tcoh shows the opposite trend.

To parameterize the extent of the fluctuation regime, we plot in Fig. 2d the ratio T2/Tc0 versus hole content p (solid red squares). Despite the non-monotonic and asymmetric form of T2(x) in Fig. 2c, the ratio T2/Tc0 turns out to be surprisingly symmetrical, with a minimum at x=pcrit. There is no obvious reason for amplitude fluctuations to follow such a trend. According to the original argument of Emery and Kivelson1, phase stiffness is determined predominantly by the superfluid density ρs; the smaller the ρs, the weaker the screening and the larger the region over which classical (thermal) phase fluctuations are important. In underdoped cuprates, ρs is found to decrease monotonically with Tc0 (ref. 28), indicating that the condensate density drops as p is reduced. Significantly, in overdoped cuprates, ρs also falls away as Tc0 decreases13,14.

The solid circles in Fig. 2d correspond to estimates of the normalized superfluid density ρs/ρs(pcrit) for LSCO (see Supplementary Table S1 for a description of how ρs/ρs(pcrit) is determined at each p value). Also included in Fig. 2d for comparison are ρs/ρs(pcrit) values obtained for Ca-doped YBa2Cu3O6+δ (open circles) and Tl0.5−yPb0.5+ySr2Ca1−xYxCu2O7 (open triangles)12. The clear anti-correlation between T2/Tc0 and ρs(p) supports the Emery–Kivelson picture1 and provides firm evidence that the temperature interval Tc0≤T≤T2 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 normal-state pseudogap, which progressively removes spectral weight that is never recovered on entering the full Meissner state. In overdoped cuprates, the strong suppression of ρs has been attributed both to phase separation13 and to pair-breaking14. Whatever its origin, the observed persistence of an extended fluctuation regime beyond x=pcrit in LSCO reveals that low phase rigidity in high-Tc cuprates does not require the presence of a pseudogap in the normal-state excitation spectrum and that the pseudogap itself is not a precondition for the development of an extended region of phase-fluctuating superconductivity.

Detailed Nernst data on irradiated samples of underdoped and optimally doped YBa2Cu3O7−δ have shown that although Tc0 is strongly suppressed with increasing disorder, T2 is much less affected29. In accord with the anti-correlation reported here between T2/Tc0 and ρs, one can now interpret the expansion of the fluctuation regime in irradiated YBa2Cu3O7−δ as a direct consequence of the strong disorder-induced suppression of the superfluid density12, which ultimately enhances the role of phase fluctuations. The strong disorder inherent in LSCO presumably plays a key role in amplifying T2/Tc0 here too. The relation between Tc0 and T2 is explored in more detail in the Supplementary Information.

Finally, the observation of an expanded phase-fluctuation regime in overdoped cuprates may help to explain why certain spectroscopic probes, such as angle-resolved 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 normal-state pseudogap and the loss of states due to superconducting fluctuations. Note that in LSCO, T* and T2 are easily distinguished by in-plane resistivity data as they are identified respectively by a downturn and an upturn in dρa b/dT (ref. 26). The ubiquity of phase-fluctuating superconductivity and the coincidence between T2, as determined by dρa b/dT, and the vanishing of H2(T), as determined by the high-field 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 zero-field resistivity curve. Indeed, excellent agreement is already noted between the fluctuation onset temperatures determined by the Nernst effect and magnetization in LSCO, Bi2Sr2−yLayCuO6 and optimally doped YBa2Cu3O7 and the corresponding T2 values obtained from d2ρa b/dT2 analysis of Ando and co-workers30. Such agreement suggests that it should be relatively straightforward to generalize these findings to other new or existing cuprate families.

Methods

Single crystals of La2−xSrxCuO4 (LSCO) were grown using a travelling-solvent-floating-zone technique. The actual doping x(=p) of each crystal was estimated from its Tc using the empirical relation Tc0=Tcopt(1−82.6(p−0.16)2) (ref. 31). The in-plane resistivity ρa b was measured using a conventional four-probe a.c. lock-in 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 LNMCI-T in Toulouse. In all reported measurements, the field was applied along the c-axis.

Error bars. The error bars in Figs 1e and 2a (and subsequently in Fig. 2b) reflect our uncertainty in the determination of H2(T) from the individual magnetoresistance curves. More explicitly, we fit the highest-field data, well above H2, to a quadratic field dependence and determine the standard deviation σ from the fit. We then identify H2 (and its associated uncertainty) with the point at which the data deviate from the extrapolated high-field fit by more than 3σ (±1σ). The error bars in the Nernst- and specific-heat-derived points are as quoted in the corresponding references. In Fig. 2c, the error bars for T*, Tcoh and T2 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*, Tcoh and T2 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 T2/Tc0 are again compound errors—the large error bars for LSCO26 reflect for the most part our uncertainty in Tc0, the superconducting transition of which is invariably broader as it resides at the edge of the superconducting dome where dTc0/dx is the steepest. The error bars for ρs/ρs(pcrit) represent the standard deviation in the spread of values as listed in Supplementary Table S1.