Abstract
The nature of the underdoped pseudogap regime of the hightemperature copper oxide superconductors has been a matter of longterm debate^{1,2,3}. On quite general grounds, we expect that, owing to their low superfluid densities and short correlation lengths, superconducting fluctuations will be significant for transport and thermodynamic properties in this part of the phase diagram^{4,5}. Although there is ample experimental evidence for such correlations, there has been disagreement about how high in temperature they may persist, their role in the phenomenology of the pseudogap and their significance for understanding hightemperature superconductivity^{6,7,8,9,10}. Here we use THz timedomain spectroscopy to probe the temporal fluctuations of superconductivity above the critical temperature (T_{c}) in La_{2−x}Sr_{x}CuO_{4} (LSCO) thin films over a doping range that spans almost the entire superconducting dome (x=0.09–0.25). Signatures of the fluctuations persist in the conductivity in a comparatively narrow temperature range, at most 16 K above T_{c}. Our measurements show that superconducting correlations do not make an appreciable contribution to the chargetransport anomalies of the pseudogap in LSCO at temperatures well above T_{c}.
Similar content being viewed by others
Main
In general, continuous phase transitions are typified by fluctuations with correlation length and timescales that diverge near T_{c}. Dynamical measurements such as THz timedomain spectroscopy are sensitive probes of the onset of superconductivity^{11} and measure its temporal correlations on the timescales of interest. In the presence of superconducting vortices such highfrequency measurements are not affected by effects such as vortex pinning, creep and edge barriers, which often complicate interpretation of lowfrequency and d.c. results. In this study, we investigate the fluctuation superconductivity in thin films of LSCO grown by molecular beam epitaxy. This synthesis technique provides exquisite control of the thickness and chemical composition of the films; the intrinsic chemical tunability of LSCO enables us to investigate essentially the entire phase diagram. For details on the films, see Supplementary Information.
In Fig. 1a and b, we show the real (σ_{1}) and imaginary (σ_{2}) parts of the THz conductivity measured at a number of different temperatures for optimally doped LSCO (x=0.16) with T_{c}=41 K. We obtain similar data at other doping levels. The spectra are easily understood in the limiting cases of high and low temperatures. Well above the onset of superconductivity, the real part of the conductivity is almost frequency independent and the imaginary part is small, consistent with the expected behaviour of a metal at frequencies well below the normalstate scattering rate. At the lowest temperature the conductivity is consistent with that expected for a longrange ordered superconductor; σ_{1} is small as most of the lowfrequency spectral weight has condensed into the ω=0 delta function, and the frequency dependence of σ_{2} is very close to 1/ω. Our principal interest, however, is in the interesting transition region around T_{c}, where fluctuations of superconductivity are apparent. Here, both components of the conductivity are enhanced, with σ_{1} first rising and then falling as spectral weight moves to frequencies below the measurement range in the superconducting state.
The fluctuation regime is perhaps most evident in Fig. 1c,d, where we plot the complex conductivity of the same sample as a function of temperature. Above the transition, the real conductivity shows a slow decrease as the temperature is raised, consistent with the increasing normalstate d.c. resistivity. Near T_{c} a ‘loss peak’ occurs owing to the onset of strong superconducting fluctuations; it is exhibited at progressively lower temperatures as the probing frequency is decreased (inset to Fig. 1c). As discussed below, this is a direct consequence of the slowing down of superconducting fluctuations as T is lowered. About 10 K above T_{c}, the imaginary conductivity shows a sharp upturn at low frequency (inset to Fig. 1d), indicating the onset of strong superconducting correlations^{11,12}. Whether such correlations persist well above the 10–15 K range above T_{c} is a more subtle issue, which we discuss below. For all dopings, the region of obvious enhancement is far below the temperature of the diamagnetism and Nernst onset measured in LSCO crystals^{7,10,13}, but is consistent with the onset found in other a.c. conductivity studies^{11,14,15,16,17}.
An important quantity for understanding superconducting fluctuations is the phase stiffness, which is the energy scale for introducing twists in the phase of the complex superconducting order parameter Δe^{i} ϕ(r). As in any continuous elastic medium we can write the energy of a phase deformation in the form of where is a stiffness constant. As a phase gradient is associated with a superfluid velocity, this ‘elastic’ energy is equivalent to the centreofmass kinetic energy. For N particles of mass m, the stiffness is . We can measure this quantity directly through the imaginary part of the fluctuation conductivity σ_{2f}, as k_{B}T_{ϕ}=(ℏω σ_{2f}t)/G_{Q}. Here T_{ϕ} is the twodimensional stiffness of a single CuO_{2} plane given in units of degrees Kelvin, G_{Q}=e^{2}/ℏ is the quantum of conductance and t is the interCuO_{2} plane spacing. The phase stiffness is usually regarded as an equilibrium quantity; the effect of measuring it dynamically at finite frequency ω is to introduce a length scale over which the system is probed. In models with vortex proliferation this is typically proportional to the vortex diffusion length , where D is a vortex diffusion constant.
In Fig. 2, we plot the quantity ω σ_{2} as a function of frequency and temperature in a and b, respectively, for an underdoped sample (x=0.09) with T_{c}=22 K. This quantity is proportional to the phase stiffness T_{ϕ} when the superconducting signal dominates over the normalstate background. Owing to uncertainties in its possible form, we have abstained from subtracting a background from the plotted signal. We find, however, that different choices for backgrounds have little effect on our conclusions. Deep in the superconducting state, there is essentially no frequency dependence to ω σ_{2}, which is consistent with the fact that the system’s phase is ‘stiff ’ on all length scales. At higher temperatures the curves in Fig. 2b spread as a frequency dependence is acquired. Interestingly, the temperature where the spreading first becomes significant is very close to the temperature where the Kosterlitz–Thouless–Berenzinskii (KTB) theory (black dashed line) would predict a discontinuous jump in this quantity for an isolated CuO_{2} plane^{18}. LSCO is of course only a quasitwodimensional system, but the fact that we observe a spreading at the KTB prediction shows that at some temperature T_{KTB}^{eff}<T_{c} there begin to be significant fluctuations of a twodimensional character even below the transition. T_{c} itself does not occur until higher temperatures, as it is controlled by threedimensional couplings. All of our underdoped samples, as well as previous measurements of BSCCO films^{11}, show this behaviour. As shown by the frequency dependence of ω σ_{2} above T_{KTB}^{eff}, the fluctuations first degrade the stiffness on long length and timescales, that is, low frequencies. To compare across the phase diagram, in Fig. 2c we plot ω σ_{2} for many dopings at a frequency 0.8 THz. Qualitatively, we regard the onset temperature T_{o} as the temperature where the quantity ω σ_{2} presents a substantial deviation from the trend of the highertemperature normal state. As a quantitative measure, we find that all samples exhibit a very sharp deviation out of the smooth hightemperature signal in this range in plots of the second derivative (Fig. 2d) versus temperature. We take this deviation as the quantitative measure of T_{o} (with error bars). For all dopings, T_{o} is no more than 16 K above T_{c}. It is also interesting to note that the difference between T_{o} and T_{c} is relatively constant over the experimental range.
As mentioned above, the data in Figs 1 and 2 are consistent with superconducting correlations that are typified by a slowing down of the characteristic fluctuation rates as temperature is decreased. In general, the diverging length and timescales near a continuous phase transition lend themselves to a scaling analysis in which response functions can be written in terms of these diverging scales. Close to a superconducting transition we expect that the relation
holds for the portion of the conductivity due to superconducting fluctuations σ_{f}. Here T_{ϕ}^{0} is a temperaturedependent prefactor and Ω is the characteristic fluctuation rate. Temperature dependencies enter only through the quantities Ω and T_{ϕ}^{0}. This scaling function is similar to the one proposed by Fisher, Fisher and Huse^{19} and is identical to the one used in previous THz measurements on underdoped BSCCO (ref. 11). Note that equation (1) is a very general form that does not assume any particular dimensionality of the system or character (vortex, Gaussian and so on) of the fluctuations or functional dependencies on temperature of Ω and T_{ϕ}^{0}. Also note that although the fluctuation conductivity σ_{f}=σ_{f}e^{i} φ is a complex quantity all prefactors to the scaling function are real and therefore the phase of is equal to the phase of σ_{f}. is expected to exhibit singleparameter scaling and thus a collapse of φ measured at different temperatures as a function of the reduced frequency ω/Ω yields the temperaturedependent Ω. In Fig. 3a, we show the collapsed phase φ=tan^{−1}σ_{2}/σ_{1} of the x=0.16 sample as a function of the reduced frequency ω/Ω for 45 different temperatures in the 30–55 K range. As expected, the phase is an increasing function of ω/Ω, with the metallic limit φ=0 reached at ω/Ω0 and φ becoming large (but bounded by π/2) as .
We were able to carry out the scaling analysis and obtain similarly good data collapse for all eight samples in the range x=0.09–0.25. In Fig. 3b we plot their extracted fluctuation rates Ω as a function of temperature. In almost all samples the fluctuation rate increases very quickly (almost exponentially) within 10 K of T_{c}. Note that the characteristic fluctuation frequency Ω is actually smooth through T_{c}, in a manner quite different than expected from the Fisher–Fisher–Huse scaling, and it seems to extrapolate to zero near the effective T_{KTB}^{eff}, where twodimensionallike fluctuations become strong. At a temperature we denote as T_{Q}, the extracted fluctuation rate crosses over to a regime where it grows at a much smaller rate, which is proportional to temperature as α k_{B}T/ℏ. As the resistivity itself is linear in T (see Supplementary Information), T_{Q} is an alternative measure of the temperature scale where we cannot distinguish superconducting fluctuations from normalstate transport. Interestingly, we continue to obtain good scaling and data collapse as we continue the analysis for another 5–10 K above T_{Q}. This behaviour may be related to the ω/T scaling that has been seen in the normal state^{20}.
In Fig. 4 we construct a phase diagram of the fluctuation regime summarizing our results. We compare T_{c} of films and bulk crystals, the diamagnetism onset from ref. 10, the region of THz conductivity onset T_{o} (as defined by the sudden change in curvature of ω σ_{2} from Fig. 2d) and T_{Q}, where Ω begins to grow linearly with temperature. Unsurprisingly, to within experimental uncertainty T_{Q} and T_{o} track each other. For all dopings the THz fluctuation conductivity shows an onset between 5 K and 16 K above T_{c}. This is in strong contrast to measurements such as diamagnetism, where the signal persists in similar samples to almost 100 K above T_{c}. Although fluctuation diamagnetism is a thermodynamic quantity sensitive to spatial correlations and THz conductivity a dynamical quantity sensitive to temporal correlations, the differences are still surprising, as within most models based on diffusive dynamics we expect a close correspondence between them^{18}. An approximately 20 K difference in the onset temperatures exists between diamagnetism and previous THz measurements on BSCCO (refs 11, 21), but the present 80 K discrepancy is much larger (particularly when normalized to the respective T_{c} values) and cannot be easily explained away.
We may draw a few possible conclusions. One obvious possibility is that there is little correspondence of diamagnetism (and Nernst effect) to conductivity because the former class of measurements is sensitive to something other than only superconductivity well above T_{c}. In this regard, it has been shown recently that the onset of densitywave order can give a strong Nernst response^{9}, and that bond current states can in principle exhibit enhanced diamagnetism^{22}. However, if the diamagnetism signal above T_{o} is solely due to superconducting correlations then it is a wellposed theoretical challenge to explain the lack of straightforward correspondence to conductivity.
We have found that at some temperature T_{Q}>T_{c} the fluctuation rate Ω is either overwhelmed by the linearinT normalstate scattering rate or becomes linear in T itself. Calculations that model the normal state as a vortex liquid^{23,24,25} with a characteristic dissipation rate proportional to T favour the latter scenario. Unfortunately, our measurements cannot distinguish these possibilities. It is interesting to note that the regime in which we observe a large fluctuation conductivity is essentially the same as the regime of ‘fragile London rigidity’ in the diamagnetism of Li et al. ^{10,21}. It may be that there are two distinct types of superconducting fluctuation, one of which gives a divergent or neardivergent contribution to the susceptibility in a relatively narrow range of temperatures above T_{c} and the other of which gives a more extended, but less spectacular, contribution to the susceptibility. If the first type makes a more significant contribution to the conductivity than the second, this may resolve the apparent conflict between the two probes. Similarly, the differences may arise in how the quantities in question depend on correlations in space (probed by diamagnetism) versus correlations in time (probed by conductivity). As noted above, within classical diffusive dynamics, we generally expect a correspondence between these quantities. Our data show that if the regime of enhanced diamagnetism is due to superconductivity then a very unconventional relationship must exist between length and time correlations in the fluctuation superconductivity. Among other possibilities, this could arise from phase separation^{26}, unusually fast vortices^{27,28} or the presence of explicitly quantum diffusion. Whatever the reason, our data show that superconducting fluctuations do not make an appreciable contribution to the chargetransport anomalies in the pseudogap regime at temperatures well above T_{c}.
Methods
The complex conductivity was determined by timedomain THz spectroscopy. A femtosecond laser pulse is split along two paths and excites a pair of photoconductive ‘Auston’switch antennae grown on radiationdamaged silicon on sapphire wafers. A broadband THzrange pulse is emitted by one antenna, transmitted through the LSCO film and measured at the other antenna. By varying the length difference of the two paths, we map out the entire electric field of the transmitted pulse as a function of time. Comparing the Fourier transform of the transmission through LSCO with that of a reference resolves the full complex transmission. We then invert the transmission to obtain the complex conductivity through the standard formula for thin films on a substrate: , where Φ_{s} is the phase accumulated from the small difference in thickness between the sample and reference substrates, n is the substrate index of refraction, Z_{0}=377 Ω is the impedance of free space and d is the thinfilm thickness. By measuring both the magnitude and phase of the transmission, this inversion to conductivity is done directly and does not require Kramers–Kronig transformation.
Our scaling analysis includes some uncertainty in setting the overall scale of Ω, as equation (1) only specifies Ω up to a numerical factor. We set the overall scale of Ω so that the loss peak in Fig. 1c is exhibited at a temperature where the probing frequency ω is equal to the fluctuation frequency Ω at that temperature. This is an imprecise procedure and applying it strictly gives a small distribution (40%) in the values of α for different dopings. We choose the normalizations for the data plotted in Fig. 3 such that α=0.4, which is the mean value for all the data. We take the relatively small spread in α values as evidence of the veracity of our procedure, but emphasize that the α values could be revised with a different criterion for the scale factors. As a crosscheck we also fit the conductivity using a Kramers–Kronig consistent fitting routine^{29} and obtain good agreement above T_{c} between the halfwidth of the Lorentzian peak used to model the fluctuation contribution and the rates given in Fig. 3b.
The LSCO films were deposited on 1mmthick singlecrystal LaSrAlO_{4} substrates, epitaxially polished perpendicular to the (001) direction, by atomic layerbylayer molecular beam epitaxy^{30}. The samples were characterized by reflection highenergy electron diffraction, atomic force microscopy, Xray diffraction, and resistivity and magnetization measurements, all of which indicate excellent film quality. For accurate determination of the conductivity, it is critical to know the film thickness accurately. This was measured digitally by counting atomic layers and reflection highenergy electron diffraction oscillations, as well as from socalled Kiessig fringes in smallangle Xray reflectance and from finite thickness oscillations observed in Xray diffraction patterns. The x=0.14, 0.16 and 0.25 films are 80 monolayers thick; the x=0.095 and 0.19 films are 114 monolayers; the x=0.09, 0.10 and 0.12 films are 150 monolayers (one monolayer is ≈6.6 Å).
References
Timusk, T. & Statt, B. The pseudogap in hightemperature superconductors: An experimental survey. Rep. Prog. Phys. 62, 61–122 (1999).
Damascelli, A., Hussain, Z. & Shen, Z. X. Angleresolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473–541 (2003).
Norman, M., Pines, D. & Kallin, C. The pseudogap: Friend or foe of high Tc? Adv. Phys. 54, 715–733 (2005).
Uemura, Y. J. et al. Basic similarities among cuprate, bismuthate, organic, Chevrelphase, and heavyfermion superconductors shown by penetrationdepth measurements. Phys. Rev. Lett. 66, 2665–2668 (1991).
Emery, V. & Kivelson, S. Importance of phase fluctuations in superconductors with small superfluid density. Nature 374, 434–437 (1995).
Wang, Y., Li, L. & Ong, N. P. Nernst effect in highTc superconductors. Phys. Rev. B 73, 024510 (2006).
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).
Lee, P. A., Nagaosa, N. & Wen, X. G. Doping a Mott insulator: Physics of hightemperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).
CyrChoinière, O. et al. Enhancement of the Nernst effect by stripe order in a highTc superconductor. Nature 458, 743–745 (2009).
Li, L. et al. Diamagnetism and Cooper pairing above Tc in cuprates. Phys. Rev. B 81, 054510 (2010).
Corson, J., Mallozzi, R., Orenstein, J., Eckstein, J. N. & Bozovic, I. Vanishing of phase coherence in underdoped Bi2Sr2CaCu2O8+δ . Nature 398, 221–223 (1999).
Crane, R. et al. Survival of superconducting correlations across the twodimensional superconductorinsulator transition: A finitefrequency study. Phys. Rev. B 75, 184530 (2007).
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).
Kitano, H., Ohashi, T., Maeda, A. & Tsukada, I. Critical microwaveconductivity fluctuations across the phase diagram of superconducting La2−xSrxCuO4 thin films. Phys. Rev. B 73, 092504 (2006).
Grbić, M. S. et al. Microwave measurements of the inplane and caxis conductivity in HgBa2CuO4+δ: Discriminating between superconducting fluctuations and pseudogap effects. Phys. Rev. B 80, 094511 (2009).
Grbić, M. S. et al. Temperature range of superconducting fluctuations above Tc in YBa2Cu3O7−δ single crystals. Preprint at http://arxiv.org/abs/1005.4789v2 (2010).
Maeda, A., Nakamura, D., Shibuya, Y., Imai, Y. & Tsukada, I. THz conductivity of La2−xSrxCuO4 in the pseudogap region and in the superconductivity state. Physica C 470, 1018–1020 (2010).
Halperin, B. I. & Nelson, D. R. Resistive transition in superconducting films. J. Low Temp. Phys. 36, 599–616 (1979).
Fisher, D. S., Fisher, M. P. A. & Huse, D. A. Thermal fluctuations, quenched disorder, phase transitions, and transport in typeII superconductors. Phys. Rev. B 43, 130–159 (1991).
Molegraaf, H. J. A., Presura, C., van der Marel, D., Kes, P. H. & Li, M. Superconductivityinduced transfer of inplane spectral weight in Bi2Sr2CaCu2O8+δ . Science 295, 2239–2241 (2002).
Li, L. et al. Strongly nonlinear magnetization above Tc in Bi2Sr2CaCu2O8+δ . Europhys. Lett. 72, 451–457 (2005).
Sau, J. D. & Tewari, S. Diamagnetism from the 6vertex model and implications for the cuprate superconductors. Preprint at http://arxiv.org/abs/1009.5926v2 (2010).
Vafek, O. & Tešanović, Z. Quantum criticality of dwave quasiparticles and superconducting phase fluctuations. Phys. Rev. Lett. 91, 237001 (2003).
Melikyan, A. & Tešanović, Z. Model of phase fluctuations in a lattice dwave superconductor: Application to the Cooperpair chargedensity wave in underdoped cuprates. Phys. Rev. B 71, 214511 (2005).
Anderson, P. W. Bose fluids above Tc: Incompressible vortex fluids and ‘supersolidity’. Phys. Rev. Lett. 100, 215301 (2008).
Martin, I. & Panagopoulos, C. Nernst effect and diamagnetic response in a stripe model of superconducting cuprates. Europhys. Lett. 91, 67001 (2010).
Ioffe, L. B. & Millis, A. J. Big fast vortices in the dwave resonating valence bond theory of hightemperature superconductivity. Phys. Rev. B 66, 094513 (2002).
Lee, P. A. Orbital currents and cheap vortices in underdoped cuprates. Physica C 388–389, 7–10 (2003).
Kuzmenko, A. B. Kramers–Kronig constrained variational analysis of optical spectra. Rev. Sci. Instrum. 76, 083108 (2005).
Bozovic, I. Atomiclayer engineering of superconducting oxides: Yesterday, today, tomorrow. IEEE Trans. Appl. Supercon. 11, 2686–2695 (2001).
Acknowledgements
The authors would like to thank P. W. Anderson, A. Auerbach, A. Dorsey, N. Drichko, S. Kivelson, L. Li, W. Liu, V. Oganesyan, N. P. Ong, J. Orenstein, F. Ronning, O. Tchernyshyov, Z. Tes̆anović, A. Tsvelik, D. van der Marel and J. Zaanen for discussions and/or correspondence. Support for the measurements at The Johns Hopkins University was provided under the auspices of the Institute for Quantum Matter, Department of Energy DEFG0208ER46544. The work at Brookhaven National Laboratory was supported by the US Department of Energy under project No MA509MACA.
Author information
Authors and Affiliations
Contributions
L.S.B. designed and built the THz spectrometer; L.S.B. and R.V.A. made the THz measurements; L.S.B. analysed the data; G.L. and I.B. synthesized the films (using reflection highenergy electron diffraction for in situ characterization) and measured mutual inductance; O.P. took the Xray diffraction and atomic force microscopy data; L.S.B., R.V.A., I.B. and N.P.A. wrote and revised the manuscript; N.P.A. devised the experiment.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Information (PDF 426 kb)
Rights and permissions
About this article
Cite this article
Bilbro, L., Aguilar, R., Logvenov, G. et al. Temporal correlations of superconductivity above the transition temperature in La_{2−x}Sr_{x}CuO_{4} probed by terahertz spectroscopy. Nature Phys 7, 298–302 (2011). https://doi.org/10.1038/nphys1912
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys1912
This article is cited by

Evidence for dwave superconductivity of infinitelayer nickelates from lowenergy electrodynamics
Nature Materials (2024)

Terahertz control of manybody dynamics in quantum materials
Nature Reviews Materials (2023)

Absence of a BCSBEC crossover in the cuprate superconductors
npj Quantum Materials (2023)

Electronic nature of the pseudogap in electrondoped Sr2IrO4
npj Quantum Materials (2022)

Paramagnons and hightemperature superconductivity in a model family of cuprates
Nature Communications (2022)