Main

Using atomic-layer-by-layer molecular beam epitaxy (ALL-MBE; refs 12, 13), we synthesized single-crystal films of La2−xSrxCuO4 (LSCO), the simplest copper oxide that we can dope all the way to a non-superconducting metal state. As single-layer LSCO sustains high-temperature superconductivity (HTS) with a Tc value as high as is found in bulk samples13, the physics is quasi-two-dimensional (2D) and we focus on the in-plane properties.

We used the mutual inductance technique9,10,14,15,16 (Fig. 1a–c and Extended Data Figs 1 and 2), improved (see Methods) to resolve the absolute value of λ with a ±1% accuracy. For this, it is critical to accurately determine the superconducting layer thickness, which we achieve by engineering the samples at the atomic-layer level as illustrated in Fig. 1d. The sharpness of the peak in ImVp(T), or equivalently in ImM(T), where Vp is the voltage on the pick-up coil and M is the mutual inductance, puts an upper bound of about 0.1 K on the spread in Tc (defined by the onset of the Meissner effect, that is, the expulsion of the magnetic field) in this 10 × 10 mm2 film.

Figure 1: Synthesis and characterization techniques.
figure 1

a, The real (in-phase) component of Vp, the voltage across the pickup coil (proportional to the mutual inductance), showing diamagnetic screening (the Meissner effect) when the film becomes superconducting. Inset, schematic of the experiment. b, The imaginary part of Vp shows that in this 10 × 10 mm2 film, Tc is homogeneous to better than 0.1 K. Inset, the same, near Tc. c, The penetration depth, λ, and the real part of the complex a.c. (ν = 40 kHz) conductivity, Reσ, derived from the complex impedance. d, Schematic of a sample engineered for this study at the atomic-layer level. LSAO denotes the LaSrAlO4 substrate.

PowerPoint slide

Full size image

We have studied over 2,000 LSCO films in great detail (see Extended Data Figs 3, 4, 5, 6, 7). The film thickness was varied from d = 0.66 nm (half of the unit cell height) to over 100 nm, and the composition was varied across the entire phase diagram. This was crucial for obtaining definitive conclusions—copper oxides are complex compounds, HTS has largely been a materials science endeavour, and good statistics are essential.

In Fig. 2 we show our key experimental data extracted directly from the measured inductance with dense coverage of the entire overdoped LSCO region. Figure 2a shows the doping dependence of λ(T) for the hundred most homogeneous films, which are likely to represent the intrinsic LSCO properties. Figure 2b, c shows the 2D superfluid phase stiffness ρs ≡ A/λ2, which is directly proportional to the 2D superfluid density . Here A = ħ2d/4μ0kBe2 = 3.55 × 10−12 m2 K−1, kB is the Boltzmann constant, m* is the electron effective mass, ħ is the reduced Planck constant, μ0 = 4π × 10−7 N A−2 is the vacuum permeability and e is the electron charge. The ρs(T) dependence is essentially linear, even in the heavily overdoped LSCO films (Fig. 2c). A crossover to a T2-dependence occurs below a very low sample-dependent temperature T**. Figure 2d shows the dependence of Tc on ρs0. (The subscript 0 refers to the T → 0 limit; this extrapolation is justified because our measurements extend down to T = 300 mK.) The Tc(ρs0) scaling in Fig. 2d is largely linear, Tc = T0 + αρs0, with T0 = (7.0 ± 0.1) K and the proportionality coefficient α = 0.37 ± 0.02, except very close to the origin where the curve fits closely to , where γ = (4.2 ± 0.5) K1/2. The data are as accurate as depicted in the figure; the error bars are smaller than the marker size.

Figure 2: The evolution of the superfluid with temperature and doping.
figure 2

a, The penetration depth λ(T) measured in the 100 most homogeneous LSCO films synthesized by ALL-MBE. The red and green dashed lines, as well as the coloured shading, are visual aids. b, The corresponding phase stiffness ρs(T). c, ρs(T) for the most overdoped samples, measured down to T = 0.3 K. d, The dependence of Tc on ρs0 ≡ ρs(T → 0). The experimental data are represented by the blue diamonds; the green dashed line is the fit to Tc = T0 + αρs0 with α = 0.37 ± 0.02 for ρs0 > 15 K and the red dashed line is the fit to Tc = γ  with γ = (4.2 ± 0.5) K1/2 for ρs0 < 12 K. Source Data used to generate the lines in b–d are available online.

PowerPoint slide

Source data

Full size image

Our findings reinforce pioneering observations4,5,6,7 of the diminishing ρs0 in a few overdoped Tl2Ba2CuO6-δ (Tl-2201) samples, and subsequently also in overdoped LSCO8,9,10,11. We confirm this result for thousands of LSCO films and establish that ρs0 decreases monotonously from the optimal doping until it vanishes and Tc drops to zero.

The disappearance of ρs0 with overdoping was originally attributed to spontaneous electronic phase separation4. However, quantum oscillations observed in overdoped Tl-2201 with Tc = 10 K indicate electronic homogeneity on the 100 nm scale17,18. We show that LSCO films grown by ALL-MBE are also intrinsically homogeneous, at every doping level.

A nearly linear ρs(T) dependence was observed19 by a microwave technique in an overdoped Tl-2201 crystal with Tc ≈ 25 K and very low T** (2 K). We show that the same is true in LSCO films grown by ALL-MBE at every level of doping, with the slope independent of the carrier density p except very close to the edge of the dome-shaped Tc(p) curve.

However, our central result, the Tc(ρs0) scaling law shown in Fig. 2d, differs qualitatively from these early inferences. The well-known Uemura’s law, Tc ∝ ns0/m*, inferred from muon spin rotation (μSR) measurements4, refers to the underdoped side. On the overdoped side, the scarce μSR data were interpolated by a concave ‘boomerang’ shape5,6,7. In contrast, the Tc(ρs0) curve shown in Fig. 2d is markedly convex. The same discrepancy is found with Homes’ law, ρs0 ∝ σdcTc (where σdc is the normal-state conductivity measured close to Tc), based on optics data20. According to Homes’ law and our measured Tc and ρs0 data, ρdc ≡ 1/σdc should increase with doping and diverge as when ρs0 → 0, because under these assumptions (Extended Data Fig. 8). However, our measured values of ρdc decrease monotonously (essentially linearly) with doping. These discrepancies may originate from differences in the quality and homogeneity of the samples—copper oxides are complex materials and call for advanced synthesis techniques—and in much larger error bars for the ρs0 values extracted from μSR and optics data.

A study of high-quality, heavily underdoped YBa2Cu3O7−x (YBCO) crystals by a microwave cavity-perturbation technique21,22 reported ρs(T) curves that are linear down to T** ≈ 4 K. The scaling was seen over a tiny range, 0.054 < p < 0.058 (where p is the doping level), near the transition from superconductor to insulator and was attributed to quantum critical fluctuations. At higher doping levels, the Tc(ρs0) dependence became linear with a clear offset T0 ≈ 4–5 K. This is reminiscent of our results in Fig. 2, but in a different compound, indicating that this behaviour may be universal for the hole-doped copper oxides. Moreover, the fact that the results for underdoped YBCO are similar to those for overdoped LSCO is suggestive of underdoped/overdoped symmetry.

Our results, inferred from the raw data without any assumptions, challenge current thinking. In a clean Bardeen–Cooper–Schrieffer (BCS) superconductor, ρs0 should be equal to the total particle density. (In fact, Leggett’s theorem23,24 asserts that this is true as long as superfluidity conforms to the two-fluid scenario, and more generally for any single-species system that is translation- and time-reversal-invariant.) Pair-breaking due to impurities and disorder can reduce ρs0, but we doubt that our data can be quantitatively explained using the standard ‘dirty’ d-wave BCS formalism, which includes the effects of pair breaking on impurities and other defects, because in that case the Glover–Ferrell–Tinkham sum rule implies that Homes’ law should apply20,25, contrary to what we observe. Qualitatively, for ρs0 to vanish because of disorder the sample must get extremely dirty and a superconductor-to-insulator transition would be observed, while in fact Fig. 3a shows LSCO becoming more metallic.

Figure 3: Overdoped LSCO films synthesized by ALL-MBE are clean superconductors.
figure 3

a, Resistivity in LSCO films with different doping levels (see Methods), top to bottom: p = 0.172, 0.205, 0.211, 0.217, 0.220, 0.224, 0.228, 0.230, 0.233, 0.237, 0.242, 0.245, 0.248, 0.251, 0.254, 0.258, 0.295. b, The dependence of the shape of ρs(T) on disorder. The red line represents an LSCO film that is optimally doped (p = 0.16) but with 0.5% of the Cu substituted by Zn. The blue line indicates a clean LSCO film that is slightly overdoped (p = 0.19) to have the matching Tc (approximately 38 K). In the clean film the ρs(T) dependence is essentially linear, while in the dirty film pair-breaking makes this relation parabolic below about 25 K.

PowerPoint slide

Full size image

The ρs(T) dependence is expected to be linear only in very clean d-wave BCS superconductors; disorder and pair-breaking turn this relation quadratic21,22,26 below T**. In Fig. 2c, T** ≲ 2 K, comparable to the high-quality YBCO crystals21,22 and corresponding within this formalism to a mean-free-path l0 ≳ 4μm and thus to the ‘ultra-clean’ limit. In Fig. 3b, we illustrate the effect of (deliberately added) disorder by comparing two samples with the same Tc (38 K), one slightly overdoped (p = 0.19) and clean, and the other optimally doped (p = 0.16) but with 0.5% of the Cu replaced by Zn, a known pair-breaker. Although this small amount of disorder does not affect Tc greatly (ΔTc ≈ −3 K), it has a dramatic effect on the shape of the ρs(T) curve, which becomes parabolic below 20–25 K. We are therefore able to detect even a small amount of disorder and pair-breaking.

The results we highlight here—the linear ρs(T) dependence and the overall Tc(ρs0) scaling—are robust and not sample-dependent, and are hence likely to be intrinsic. (On the contrary, T** and ρdc are affected by disorder and other extrinsic factors; both can be modified by annealing the film in a vacuum, oxygen, or ozone, thus modifying the concentration of oxygen vacancies, without a substantial effect on Tc.) The extremely dirty BCS picture is also inconsistent with observations of quantum oscillations and other experiments17,18,27.

Homes’ law can be derived by a different line of reasoning25, assuming the ‘Planckian’ dissipation, with the scattering time τ = ħ/kBT as strong as is allowed by the uncertainty relation—which connects it naturally to another mysterious feature of HTS copper oxides, the linear temperature dependence of resistivity. The latter is probably related to the linear ρs0(T) and Tc(ρs0) dependences we observe; as Tc and ρs0 decrease with overdoping, the T-linear term in resistivity decreases as well28, and this may be the reason for the deviation from the Homes relation.

Another important inference is that Tc seems to be principally controlled by the superfluid density. It seems unlikely that the doping level is the primary factor that controls both Tc and ρs0, because Tc strongly increases in LSCO under hydrostatic pressure or compressive epitaxial strain, which increases ρs0 while keeping p constant29. The underdoped/overdoped symmetry, even if only approximate, is another strong indication that ρs0, rather than p, controls Tc. If Tc is in fact essentially determined by the kinematics, this points to local pairing rather than to BCS physics. This notion is further supported by extensive study of the magnetoresistance in thousands of LSCO samples (I.B., J.W., A.T.B. and X.H., unpublished results), which together with the results presented here leads to the conclusion that the pair size is always smaller than their separation. However, this premise alone does not resolve the paradoxes, because Leggett’s theorem23,24 remains valid for arbitrary interaction strengths. The fact that Tc and ρs0 are comparable points to massive phase fluctuations11,30, which calls any mean-field description into question. Our experimental findings therefore challenge the existing theories; the accurate scaling reported here may be a benchmark test.

Methods

Atomic-layer-by-layer molecular beam epitaxy (ALL-MBE) synthesis

In most HTS experiments so far, the main sources of uncertainty were the samples themselves. In complex materials such as copper oxides, some level of inhomogeneity is present in most samples due to extrinsic factors. Most HTS ‘single crystals’ in fact contain stacking faults and intergrowths of other copper oxide phases and polytypes. Moreover, oxygen is volatile in copper oxides, and hence bulk crystals are prone to gradients in the density of the oxygen vacancies or interstitials. Irregular geometries of crystals and/or contacts cause some uncertainty in transport property measurements. In principle, one can alleviate the above problems by working with very thin single-crystal films; however, most HTS films are granular and contain both secondary phase precipitates and pinholes. This calls for some advanced materials science—as well as for large sample sets with sufficient statistics to clearly discern intrinsic behaviour.

For film synthesis we use a custom ALL-MBE system12. It is equipped with 16 metal sources (either K cells or rod-fed electron beam sources), a pure ozone source and a 16-channel real-time rate monitoring system based on atomic absorption spectroscopy. It also contains a dual-deflection reflection high-energy electron diffraction (RHEED) system that can monitor 20 samples in parallel, and a time-of-flight ion scattering and recoil spectroscopy (TOF-ISARS) system for chemical analysis of the film surface. These advanced surface science tools provide real-time information about the morphology of the film surface, the chemical composition and the crystal structure, and are also quintessential to grow atomically smooth and perfect films31,32,33.

Using this system, we have performed over 2,500 LSCO film growth experiments. Each film was characterized in real time by RHEED and ex situ by atomic force microscopy (AFM) and magnetic susceptibility measurements. RHEED oscillations provide a digital count of the atomic layers and real-time control of the film quality. Selected films were also characterized in situ by TOF-ISARS and ex situ by X-ray diffraction (XRD), transport measurements and Rutherford backscattering. Further characterization was undertaken using atomic-resolution scanning transmission electron microscopy and electron energy-loss spectroscopy, resonant elastic and inelastic synchrotron X-ray scattering, synchrotron-based X-ray phase-retrieval techniques such as coherent Bragg rod analysis, muon spin rotation, ultrafast electron diffraction, ultrafast optical and THz pump-probe techniques and so on.

Using ALL-MBE, we synthesize single-crystal thin films of LSCO that are atomically smooth, without any secondary phase precipitates or pinholes. The films can be made ultrathin, down to a single unit cell thick31,32,33; this is advantageous for transmission measurements, because a relatively large transmittance helps to minimize the effect of any small-area defects such as pinholes, which can arise from imperfections in substrate polishing, for example. Much thicker films are also grown for reflectance measurements.

To alleviate the problem of oxygen non-uniformity, we have performed over a thousand experiments that involved annealing in ozone, oxygen, or a vacuum, spanning 13 orders of magnitude in pressure from 10−8 torr to 200 atm. Before and after each annealing step the films were characterized by AFM, transport and XRD measurements. On the basis of these extensive studies, we have developed recipes that involve multiple annealing steps at different temperatures and pressures, yielding the most homogeneous films with the sharpest superconducting transitions.

Atomic-layer engineering

Uncertainty in the film thickness is another key problem that limits the accuracy of some measurements—of transmittance or the critical current density, for example. ALL-MBE solves this problem by providing digital control over the film thickness31,32,33; we count atomic monolayers, while we determine the lattice constant with crystallographic accuracy by X-ray diffraction. However, this still leaves some uncertainty about the actual thickness of the superfluid, because we have found that typically a couple of layers next to the substrate are modified structurally and chemically, and are not superconducting. The same is generally true for a couple of layers near the free film surface once the film has been exposed to contamination from the atmosphere. To eliminate this problem we resort to atomic-layer engineering.

An example is illustrated in Fig. 1d. The active (superconducting) part of the sample under study is an exactly 5-unit-cell- (5-UC-) thick HTS layer. It is protected on both sides by a metallic (M = La1.60Sr0.40CuO4) buffer and a cover layer of fixed thickness. However, without further sample engineering, there would be some hole depletion from M and accumulation in the nearest HTS layers31,33. To minimize this interfacial effect, we ‘sculpt’ the charge profile near the interfaces by graded Sr doping in the four relevant layers. Moreover, we also dope these transition layers by substituting 3% of the Cu with Zn; this is known13 to suppress Tc by a factor of two and ns even more dramatically, by four–five times. In this way, we quench any potential residual superconductivity and eliminate any interface contributions. One is then left with exactly 5 UC of superconducting material—10 HTS CuO2 planes. We have double-checked the validity of this approach by synthesizing a series of heterostructures in which we kept the composition of the constituent materials unaltered, but varied the thickness of the active HTS layers and verified that the sheet superfluid density scaled linearly with the number of HTS CuO2 planes (see Extended Data Fig. 1).

Note that this is just one example; indeed, we have synthesized a large number of samples, varying the layering scheme, the composition and thickness of individual layers, as well as the synthesis and post-annealing conditions. The results presented in this Letter are valid generally and not restricted to any of these choices.

Penetration depth measurements

The λ is typically measured using μSR4,5,6,7,8,34, microwave resonance (cavity-perturbation)20,21,22,35,36,37, or mutual inductance9,10,15,16,17,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54 techniques. Because of the cost and duration, μSR experiments are usually restricted to just a few compositions and temperatures. The microwave technique offers an unsurpassed relative accuracy but the absolute accuracy is limited by the uncertainty in geometric factors. For these reasons, and because it is best suited for thin-film studies, we have chosen the mutual inductance method, which has been pioneered by several groups10,14,15 (Fig 1a, b). A detailed theoretical treatment16 of a superconducting film of thickness d and infinite radius, characterized by the ac conductivity σ(ω) = σ1(ω) − iσ2(ω), where ω is the measurement (angular) frequency and i the imaginary unit, placed between two coils of radii R1 and R2, respectively, parallel to one another, and separated by a distance D, provided the following expression for the mutual inductance:

where μ0 = 4π × 10−7 F m–1, q is the wave vector, J1(x) is the first-order Bessel function, Q2 = q2 + (1/λ2) − iμ0ωσ1 and σ2 = 1/μ0ωλ2. This is readily generalized to the case of two solenoids with N1 and N2 turns, respectively, by the summation over each pair of coils, one from each side. Once the values of ReM and ImM are measured experimentally, using equation (1) one can determine the values of λ and σ1.

In practice, this procedure is subject to some uncertainties because of imprecision in the coil geometry, run-to-run variations in the film position, the finite size of the film that allows some flux to ‘leak’ around it, other parasitic coupling between the two solenoids and inaccuracy in the film thickness. In addition, the measurements are done down to some finite temperature, most frequently T = 4.2 K, and the limiting value λ0 is obtained by extrapolation. These uncertainties have been analysed in detail in the literature, and various solutions were suggested to reduce them. In what follows, we explain briefly the improvements that we made in the measuring apparatus and technique to tighten the error bars.

Parasitic field coupling around the film and through the electronics

We use inductance coils with a large number of turns (300–1,500) but very small inner radius (250 μm), much smaller than the film size (10 × 10 mm2), so that field leakage around the film is minimal (<0.3%) to begin with. We nevertheless subtract it accurately based on measurements of films of Nb, Pb and Al that are thick enough for transmittance to be negligible, and deposited on identical substrates. Alternatively, we deposit a thick Al overlayer on top of the HTS film and switch the diamagnetic screening in Al (at T = 0.3 K) between total (zero transmittance) and essentially none (total transmittance) by applying a small d.c. magnetic field (typically 100 G) that drives Al normal but does not affect the HTS film. As a consistency check, we verified that the corrected Ns0 scales linearly with the film thickness, while being independent of frequency. (In contrast, the leakage contribution varies with frequency ν, linearly for 10 kHz < ν < 100 kHz, and thus can be clearly identified by repeating measurements on the same film at several frequencies.) To minimize eddy currents, we built the sample holder out of a single block of sapphire crystal. The coils are fixed rigidly and the sample is spring-loaded so that the film surface always gets to exactly the same position. Overall, we have achieved reproducibility and precision of ±0.3%, or better.

Uncertainty in the coil geometry

Our measured mutual inductance M(T) differs a little (typically by a couple of per cent) from the calculated value due to some deviations in the geometry of the actual coil from its idealized mathematical model. To account for this (multiplicative) factor, we normalize the measured M(T) by its value Mhigh just above Tc, or equivalently by Msub of a pristine substrate (the latter is essentially independent of temperature and equal to Mhigh).

Field penetration through secondary phase precipitates, scratches and pinholes

This is particularly dangerous for thicker films where the intrinsic transmission is small. For this reason, we synthesize very thin films by ALL-MBE, down to 0.5 UC thick. Also, we have a very large statistics, so we can easily recognize extrinsic behaviour and factors.

Uncertainty in the film thickness

This is probably the single largest source of error40,45 in the measurements of λ by inductance techniques reported so far. We have minimized this using digital synthesis as described in the atomic-layer engineering section above, and illustrated in Fig. 1d. We also compared films of the same composition and Tc but of different thickness, as illustrated in Extended Data Fig. 1, and verified that the results are consistent. This constitutes the best confirmation that we can achieve at present. Regrettably, the technique described in Fig. 1d would not work on the underdoped side, but that does not affect our main goal here, which was to study the evolution of LSCO physics from the overdoped metal to the optimal doping level.

Extrapolation to T → 0 of measurements done only down to T = 4 K

Although this has been done routinely in the literature, it may introduce error, especially for films with very low Tc—say, below about 10 K. As we are interested in behaviour near the quantum critical points where Tc vanishes, we have built a 3He-based set-up that extends the temperature range down to 300 mK. In the same set-up, it is also possible to apply a d.c. magnetic field up to 9 T. To ensure the accuracy in temperature reading, we measure on both cooling and heating, at a very low rate of 0.1–0.2 K min−1, in a set-up where we achieve55 a thermal stability better than 1 mK.

We measure λ(T) with a reproducibility better than ±0.3%, as illustrated for a Nb film in Extended Data Fig. 2. Altogether, the accuracy in the absolute value of λ is better than ±1%. Moreover, some of our conclusions derive from the temperature dependence of λ, which is measured with a relative accuracy better than ±0.1%. The same is true of ρs, as ρs = A/λ2, and A = ħ2d/4μ0kBe2 = 3.55 × 10−12 m2 K−1.

On selected films, we also compared our results with in-house high-frequency (0.5–50 MHz) inductance measurements in the reflectance geometry52,53, as well as with the results of terahertz pump-probe56 and μSR experiments34 performed by our collaborators, and found good agreement.

So far, we have performed inductance measurements on more than 2,500 LSCO films, a number of which were measured dozens of times and on multiple (a total of ten so far) set-ups. Mining this large database allows one to identify clear statistical trends and uncover intrinsic behaviour. In this Letter, we focus on the films with the sharpest transitions; in the best ones, near Tc we see σ1(ω,T) rising exponentially on the scale of 0.1–0.2 K. This puts an upper bound on any inhomogeneity in Tc, as the transition width comes largely from thermal fluctuations. Clearly these samples are very homogeneous and hence they can be presumed to display intrinsic properties and behaviour.

The mobile carrier density

Note that the values of p quoted here, and in the literature, are only approximately equal to the mobile carrier (hole) density. A prevailing convention in the field is to infer p from the measured Tc, assuming a parabolic Tc(p) relation. This label p should roughly indicate the doping state in a particular sample. To make it easier for the reader to compare our data with various phase diagrams in the literature, in Figs 2 and 3 we provide the nominal p values inferred by assuming that in LSCO Tc = A(p − pc1)(pc2 − p), with pc1 = 0.06, pc2 = 0.26 and A = 4.15 × 103 K, so that p ≡ 0.16 + (0.01 − 2.4 × 10−4Tc)1/2. However, we stress that all of our conclusions are in fact only based on the quantities (Tc, λ, ρs and so on) that we measure directly and accurately.