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.
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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.
The authors declare no competing financial interests.
Extended data figures and tables
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.
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.
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).
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  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.
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.
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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
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