Abstract
The physics of underdoped copper oxide superconductors, including the pseudogap, spin and charge ordering and their relation to superconductivity1,2,3, is intensely debated. The overdoped copper oxides are perceived as simpler, with strongly correlated fermion physics evolving smoothly into the conventional Bardeen–Cooper–Schrieffer behaviour. Pioneering studies on a few overdoped samples4,5,6,7,8,9,10,11 indicated that the superfluid density was much lower than expected, but this was attributed to pair-breaking, disorder and phase separation. Here we report the way in which the magnetic penetration depth and the phase stiffness depend on temperature and doping by investigating the entire overdoped side of the La2−xSrxCuO4 phase diagram. We measured the absolute values of the magnetic penetration depth and the phase stiffness to an accuracy of one per cent in thousands of samples; the large statistics reveal clear trends and intrinsic properties. The films are homogeneous; variations in the critical superconducting temperature within a film are very small (less than one kelvin). At every level of doping the phase stiffness decreases linearly with temperature. The dependence of the zero-temperature phase stiffness on the critical superconducting temperature is generally linear, but with an offset; however, close to the origin this dependence becomes parabolic. This scaling law is incompatible with the standard Bardeen–Cooper–Schrieffer description.
This is a preview of subscription content, access via your institution
Access options
Subscribe to this journal
Receive 51 print issues and online access
$199.00 per year
only $3.90 per issue
Rent or buy this article
Prices vary by article type
from$1.95
to$39.95
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
References
Lee, P. A., Nagaosa, N. & Wen, X. G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006)
Zaanen, J. et al. Towards a complete theory of high T c . Nat. Phys. 2, 138–143 (2006)
Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature 518, 179–186 (2015)
Uemura, Y. J. et al. Universal correlations between T c and n s /m* (carrier density over effective mass) in high-T c cuprate superconductors. Phys. Rev. Lett. 62, 2317–2320 (1989)
Uemura, Y. J. et al. Magnetic-field penetration depth in TI2Ba2CuO6+δ in the overdoped regime. Nature 364, 605–607 (1993)
Niedermayer, C. et al. Muon spin rotation study of the correlation between T c and n s /m* in overdoped Tl2Ba2CuO6+δ . Phys. Rev. Lett. 71, 1764–1767 (1993)
Bernhard, C. et al. Magnetic penetration depth and condensate density of cuprate high-T c superconductors determined by muon-spin-rotation experiments. Phys. Rev. B 52, 10488–10498 (1995)
Panagopoulos, C. et al. Superfluid response in monolayer high-T c cuprates. Phys. Rev. B 67, 220502 (2003)
Locquet, J. P. et al. Variation of the in-plane penetration depth λ ab as a function of doping in La2−x Sr x CuO4+δ thin films on SrTiO3: implications for the overdoped state. Phys. Rev. B 54, 7481–7488 (1996)
Lemberger, T. R. et al. Superconductor-to-metal quantum phase transition in overdoped La2−x Sr x CuO4 . Phys. Rev. B 83, 140507 (2011)
Rourke, P. et al. Phase fluctuating superconductivity in overdoped La2−x Sr x CuO4 . Nat. Phys. 7, 455–458 (2011)
Bozovic, I. Atomic-layer engineering of superconducting oxides: yesterday, today, tomorrow. IEEE Trans. Appl. Supercond. 11, 2686–2695 (2001)
Logvenov, G., Gozar, A. & Bozovic, I. High-temperature superconductivity in a single copper-oxygen plane. Science 326, 699–702 (2009)
Hebard, A. F. & Fiory, A. T. Evidence for the Kosterlitz–Thouless transition in thin superconducting aluminum films. Phys. Rev. Lett. 44, 291–294 (1980)
Claassen, J. H., Reeves, M. E. & Soulen, R. J. Jr. A contactless method for measurement of the critical current density and critical temperature of superconducting rings. Rev. Sci. Instrum. 62, 996–1004 (1991)
Clem, J. R. & Coffey, M. W. Vortex dynamics in a type-II superconducting film and complex linear-response functions. Phys. Rev. B 46, 14662–14674 (1992)
Vignolle, B. et al. Quantum oscillations in an overdoped high-T c superconductor. Nature 455, 952–955 (2008)
Bangura, A. F. et al. Fermi surface and electronic homogeneity of the overdoped cuprate superconductor Tl2Ba2CuO6+δ as revealed by quantum oscillations. Phys. Rev. B 82, 140501(R) (2010)
Deepwell, D. et al. Microwave conductivity and superfluid density in strongly overdoped Tl2Ba2CuO6+δ . Phys. Rev. B 88, 214509 (2013)
Homes, C. C. et al. A universal scaling relation in high-temperature superconductors. Nature 430, 539–541 (2004)
Hosseini, A. et al. Microwave spectroscopy of thermally excited quasiparticles in YBa2Cu3O6.99 . Phys. Rev. B 60, 1349–1359 (1999)
Broun, D. M. et al. Superfluid density in a highly underdoped YBa2Cu3O6+y superconductor. Phys. Rev. Lett. 99, 237003 (2007)
Leggett, A. Quantum Liquids Ch. 3.3–3.4 (Oxford Univ. Press, 2006)
Leggett, A. On the superfluid fraction of an arbitrary many-body system at T = 0. J. Stat. Phys. 93, 927–941 (1998)
Zaanen, J. Superconductivity: why the temperature is high. Nature 430, 512–513 (2004)
Hirschfeld, P. J. & Goldenfeld, N. Effect of strong scattering on the low-temperature penetration depth of a d-wave superconductor. Phys. Rev. B 48, 4219–4222 (1993)
Alldredge, J. W. et al. Evolution of the electronic excitation spectrum with strongly diminishing hole density in superconducting Bi2Sr2CaCu2O8+δ . Nat. Phys. 4, 319–326 (2008)
Cooper, R. A. et al. Anomalous criticality in the electrical resistivity of La2−x Sr x CuO4 . Science 323, 603–607 (2009)
Bozovic, I., Logvenov, G., Belca, I., Narimbetov, B. & Sveklo, I. Epitaxial strain and superconductivity in La2−x Sr x CuO4 thin films. Phys. Rev. Lett. 89, 107001 (2002)
Emery, V. & Kivelson, S. A. Importance of phase fluctuations in superconductors with small superfluid density. Nature 374, 434–437 (1994)
Gozar, A. et al. Interface superconductivity between a metal and a Mott insulator. Nature 455, 782–785 (2008)
Bollinger, A. T. et al. Superconductor–insulator transition in La2−xSrxCuO4 at the pair quantum resistance. Nature 472, 458–460 (2011)
Wu, J. et al. Anomalous independence of interface superconductivity on carrier density. Nat. Mater. 12, 877–881 (2013)
Morenzoni, E. et al. The Meissner effect in a strongly underdoped cuprate above its critical temperature. Nat. Commun. 2, 272 (2010)
Bonn, D. A. & Hardy, W. N. in Physical Properties of High Temperature Superconductors Vol. V (ed. Ginsberg, D. M. ) 7–98 (World Scientific, 1996)
Prozorov, R. & Giannetta, R. W. Magnetic penetration depth in unconventional superconductors. Supercond. Sci. Technol. 19, R41–R67 (2006)
Prozorov, R. & Kogan, V. G. London penetration depth in iron-based superconductors. Rep. Prog. Phys. 74, 124505 (2011)
Lee, J. Y. & Lemberger, T. R. Penetration depth λ(T) of YBa2Cu3O7-δ films determined from the kinetic inductance. Appl. Phys. Lett. 62, 2419–2421 (1993)
Pippard, A. B. Magnetic penetration depth through a superconducting film. Supercond. Sci. Technol. 7, 696–699 (1994)
Turneaure, S. J., Ulm, E. R. & Lemberger, T. R. Numerical modeling of a two-coil apparatus for measuring the magnetic penetration depth in superconducting films and arrays. J. Appl. Phys. 79, 4221–4227 (1996)
Fuchs, A., Prusseit, W., Berberich, P. & Kinder, H. High-precision penetration-depth measurement of YBa2Cu3O7−δ as a function of oxygen content. Phys. Rev. B 53, R14745–R14748 (1996)
Gilchrist, J. & Brandt, E. H. Screening effect of Ohmic and superconducting planar thin films. Phys. Rev. B 54, 3530–3544 (1996)
Lee, J. Y., Kim, Y. H., Hahn, T.-S. & Choi, S. S. Determining the absolute value of penetration depth of large area films. Appl. Phys. Lett. 69, 1637–1639 (1996)
Claassen, J. H., Wilson, M. L., Byers, J. M. & Adrian, S. Optimizing the two-coil mutual inductance measurement of the superconducting penetration depth in thin films. J. Appl. Phys. 82, 3028–3034 (1997)
Turneaure, S. J., Pesetski, A. A. & Lemberger, T. R. Numerical modeling and experimental considerations for a two-coil apparatus to measure the complex conductivity of superconducting films. J. Appl. Phys. 83, 4334–4343 (1998)
Paget, K. M. et al. Magnetic penetration depth in superconducting La2−x Sr x CuO4 films. Phys. Rev. B 59, 641–646 (1999)
Wang, R. F., Zhao, S. P., Chen, G. H. & Yang, Q. S. Absolute measurement of penetration depth in a superconducting film by the two-coil technique. Appl. Phys. Lett. 75, 3865–3867 (1999)
Coffey, M. W. Analyzing mutual inductance measurements to determine the London penetration depth. J. Appl. Phys. 87, 4344–4351 (2000)
Coffey, M. W. Mutual inductance of superconducting thin films. J. Appl. Phys. 89, 5570–5577 (2001)
Rüfenacht, A., Locquet, J. P., Fompeyrine, J., Caimi, D. & Martinoli, P. Electrostatic modulation of the superfluid density in an ultrathin La2−x Sr x CuO4 film. Phys. Rev. Lett. 96, 227002 (2006)
Lemberger, T. R., Hetel, I., Tsukada, A. & Naito, M. Anomalously sharp superconducting transitions in overdoped La2−x Sr x CuO4 films. Phys. Rev. B 82, 214513 (2010)
Gauzzi, A. et al. Very high resolution measurement of the penetration depth of superconductors by a novel single-coil inductance technique. Rev. Sci. Instrum. 71, 2147–2153 (2000)
Gasparov, V. A. & Bozovic, I. Magnetic field and temperature dependence of complex conductance of ultrathin La1.55Sr0.45CuO4/La2CuO4 films. Phys. Rev. B 86, 094523 (2012)
Došlić, M., Pelc, D. & Požek, M. Contactless measurement of nonlinear conductivity in the radio-frequency range. Rev. Sci. Instrum. 85, 073905 (2014)
Dubuis, G., He, X. & Božović, I. Ultra-thermal-stabilization of a closed cycle cryocooler. Rev. Sci. Instrum. 85, 103902 (2014)
Bilbro, L. S. et al. Temporal correlations of superconductivity above the transition temperature in La2-x Sr x CuO4 probed by terahertz spectroscopy. Nat. Phys. 7, 298–302 (2011)
Donnelly, R. J. in Physics Vade Mecum (ed. Anderson, H. L. ) 121, Table E (AIP New York 1981)
Homes, C. C., Dordevic, S. V., Bonn, D. A., Liang, R. & Hardy, W. N. Sum rules and energy scales in the high-temperature superconductor YBa2Cu3O6+x . Phys. Rev. B 69, 024514 (2004)
Homes, C. C., Dordevic, S. V., Valla, T. & Strongin, M. Scaling of the superfluid density in high-temperature superconductors. Phys. Rev. B 72, 134517 (2005)
Tallon, J. L., Cooper, J. R., Naqib, S. H. & Loram, J. W. Scaling relation for the superfluid density of cuprate superconductors: origins and limits. Phys. Rev. B 73, 180504(R) (2006)
Acknowledgements
A. Gozar, J. Zhang and J. Yoon contributed to developing the characterization techniques during the early stages of this work. R. Sundling developed the software for the inversion of the inductance data. We also benefited from the electrolyte-gating experiments and X-ray diffraction studies by X. Leng, and from numerical simulations by N. Božović. The research was done at BNL and was supported by the US Department of Energy, Basic Energy Sciences, Materials Sciences and Engineering Division. X.H. is supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF4410. I.B. acknowledges discussions with J. Zaanen, G. Deutscher, A. Leggett, P. Littlewood, C.-B. Eom, J. Mannhart, P. Coleman, R. Prozorov, D. van der Marel, A. McKenzie, V. Kogan, P. Armitage, J.-M. Triscone, P. Canfield, A. Chubukov, B. Halperin, P. Kim, T. Lemberger, M. V. Sadovskii, D. Pavuna, Z. Radović and M. Vanević.
Author information
Authors and Affiliations
Contributions
I.B. conceived the project, synthesized the films using ALL-MBE, measured the inductance, analysed the data and wrote the text. X.H. synthesized the films, performed AFM imaging and measured the inductance. A.T.B. fabricated the devices by lithography and performed the inductance measurements in the 3He system. J.W. performed the transport measurements.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Extended data figures and tables
Extended Data Figure 1 The dependence of Tc on ρs0 in several LSCO films.
The films have the same nominal doping near the optimal (p = 0.16) in the active (superconducting) layer, but with different thickness D = nd, where d = 0.662 nm and n = 1, 2, 4, 10, 40 and 80. Although individually both Tc and ρs0 show some random variations, in part due to imperfect control of the doping level and the density of the oxygen vacancies, their ratio apparently stays almost constant, to about ±1%. This reinforces the conclusion that Tc is indeed essentially controlled by ρs0, a purely kinematic quantity.
Extended Data Figure 2 Mutual inductance (raw data) measured on a (275 ± 12)-nm-thick Nb film deposited on standard 10 × 10 × 1 mm3 LaSrAlO4 substrates.
The film was measured on 22 March 2015 (red lines) and 20 January 2016 (black lines). The thermal stabilization is better than ±1 mK and the overall reproducibility is better than ±0.3% on a one-year scale. The inferred value of λ0 = (41 ± 5) nm agrees with values in the literature57; the error here largely comes from the uncertainty in the film thickness. This error is much smaller (down to less than ±1%) in the case of our LSCO films, where we employ atomic-layer deposition, which provides digital control of the film thickness.
Extended Data Figure 3 RHEED recorded during growth of LSCO films by ALL-MBE.
Top, an optimally doped (p = 0.16, Tc = 40 K) LSCO film after the end of growth process. Bottom, a strongly overdoped LSCO film (p = 0.24, Tc = 7.5 K). The stronger main streaks correspond to Bragg-rod reflections at very shallow angles from a terraced surface. The four weaker sidebands in between every pair of main streaks indicate a ubiquitous 5a0 × 5a0 surface reconstruction (where a0 = 0.38 nm is the in-plane lattice constant). The diagonal streaks are so-called Kikuchi lines that are formed by inelastically scattered electrons; they are observable only from atomically perfect surfaces.
Extended Data Figure 4 RHEED oscillations recorded during the growth of an LSCO film by ALL-MBE.
In the atomic-layer growth mode, the intensity of the specular beam oscillates. When 2D islands form on the surface, the diffuse reflectance increases as the specular reflectance decreases, until about half of the surface is covered. Then the specular reflectance increases again, reaching a new maximum at the full coverage. The fact that the amplitude of the oscillations does not decrease indicates perfect atomic-layer growth. The number of periods provides digital information on the film thickness, expressed in the units of the lattice constant (which we know accurately from XRD).
Extended Data Figure 5 AFM images showing the quality of the film surfaces.
Left, an LSAO substrate. The steps, 0.5 UC (0.65 nm) high, occur because the polished surface is unintentionally (but unavoidably) oriented slightly (typically by less than 0.3°) off the desired crystallographic plane perpendicular to the [001] direction. Right, a 225-Å-thick LSCO film grown on the same substrate. The steps in the substrate are projected onto the film and persist all the way to the film surface, indicating atomic-layer growth. The overall root-mean-square (r.m.s.) surface roughness is about 0.24 nm; the terraces between steps are atomically smooth.
Extended Data Figure 6 A wide-angle 2θ XRD pattern of an LSCO film grown on an LSAO substrate by ALL-MBE.
The top panel shows a pristine LSAO substrate (black) and an LSCO film grown on the same substrate (red). Only even-order reflections are allowed by the space-group symmetry. The substrate peaks are labelled S. There are no traces of any other phases. The bottom panel is an expanded view near the (004) LSCO reflection. The side-bands between the LSCO Bragg reflections are the so-called Laüe (or finite-thickness) fringes that originate from the interference between X-rays reflected from the film surface and the substrate–film interface.
Extended Data Figure 7 Low-angle X-ray reflectivity measured from an LSCO film grown on an LSAO substrate by ALL-MBE.
The oscillations are so-called Kiesig fringes that originate from interference between X-rays reflected from the film surface and the substrate–film interface. They are analogous to a Fabry–Perot interferogram, and indicate that the two ‘mirrors’ are smooth and parallel on the scale of the wavelength of light (here, 1.54 Å). By comparing with simulated interferograms, one can estimate the film thickness and roughness; the estimates agree well with the thickness inferred from the digital count of the unit cells by RHEED and the surface roughness as determined by AFM.
Extended Data Figure 8 Failure of the dirty BCS model to account for experimental data.
The red diamonds represent ρdc calculated from our measured Tc and ρs0 values by applying Homes’ law, ρs0 ∝ σdcT, which follows from the Ferrell–Glover–Tinkham sum rule for dirty BCS superconductors23,28,58,59,60. As at high overdoping, the predicted ρdc value diverges as , which should trigger a superconductor-to-insulator transition. The red dashed line is a fit to f(p) = c1+c2p+c3/(0.26 − p). Blue circles are the measured ρdc values (from the data shown in Fig. 3a) showing that the samples in fact get more metallic. The blue dashed line is a fit to f(p) = c1 − c2p. The gross discrepancy with the experiment implies that the original premise—the dirty BCS scenario—is incorrect.
Source data
Rights and permissions
About this article
Cite this article
Božović, I., He, X., Wu, J. et al. Dependence of the critical temperature in overdoped copper oxides on superfluid density. Nature 536, 309–311 (2016). https://doi.org/10.1038/nature19061
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nature19061
This article is cited by
-
Puddle formation and persistent gaps across the non-mean-field breakdown of superconductivity in overdoped (Pb,Bi)2Sr2CuO6+δ
Nature Materials (2023)
-
Superfluid response of an atomically thin gate-tuned van der Waals superconductor
Nature Communications (2023)
-
The emergence of global phase coherence from local pairing in underdoped cuprates
Nature Physics (2023)
-
Impact of high growth rates on the microstructure and vortex pinning of high-temperature superconducting coated conductors
Nature Reviews Physics (2023)
-
Electronic origin of high superconducting critical temperature in trilayer cuprates
Nature Physics (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.