Abstract
The cuprate hightemperature superconductors exhibit many unexplained electronic phases, but the superconductivity at high doping is often believed to be governed by conventional meanfield Bardeen–Cooper–Schrieffer theory^{1}. However, it was shown that the superfluid density vanishes when the transition temperature goes to zero^{2,3}, in contradiction to expectations from Bardeen–Cooper–Schrieffer theory. Our scanning tunnelling spectroscopy measurements in the overdoped regime of the (Pb,Bi)_{2}Sr_{2}CuO_{6+δ} hightemperature superconductor show that this is due to the emergence of nanoscale superconducting puddles in a metallic matrix^{4,5}. Our measurements further reveal that this puddling is driven by gap filling instead of gap closing. The important implication is that it is not a diminishing pairing interaction that causes the breakdown of superconductivity. Unexpectedly, the measured gaptofilling correlation also reveals that pair breaking by disorder does not play a dominant role and that the mechanism of superconductivity in overdoped cuprate superconductors is qualitatively different from conventional meanfield theory.
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The essence of hightemperature superconductivity in the cuprates revolves around doping a Mott insulator. Superconductivity emerges when hole doping is greater than 5% per lattice site; the transition temperature T_{c} initially increases through the underdoped region of the phase diagram, before it decreases again in the overdoped region^{1}. Superconductivity breaks down completely at roughly 27% doping. For the strongly overdoped region, it is often assumed that screening sufficiently reduces electron–electron correlations, enough for a Fermi liquid to appear^{6,7,8}. The superconducting state is then of the Bardeen–Cooper–Schrieffer (BCS) type, and the suppression of superconductivity is a consequence of a diminishing pairing interaction. Evidence for such conventional behaviour in the overdoped regime comes from photoemission experiments, which suggest the existence of a full Fermi surface with superconductivity, as indicated by an energy gap that opens up in a BCS fashion below T_{c}^{9,10}. As a caveat, very recent magnetotransport experiments indicate that even at high doping, the normal state has strange metal features^{11}.
The first surprise in this regard was the discovery that the superfluid density decreases linearly to zero with doping beyond optimal doping^{2,3,12}, contrary to the BCS expectation that it should be of the order of the total carrier density and hence proportional to the doping level^{1,2}. Additionally, optical conductivity measurements revealed a large density of metallic carriers below T_{c}^{12}, suggesting a filling of the superconducting gap due to pair breaking. One possible explanation for these observations involves potential disorder, reducing the electron mean free path, at length scales comparable to the small coherence length that is typical for the cuprates^{4,5}. According to Bogoliubov–de Gennes (BdG) theory (that is, BCS in spatially heterogeneous systems), disorder at these length scales leads to emergent granular superconductivity^{4,13,14,15,16}, that is, puddles of welldeveloped superconductivity with a size set by the coherence length, separated by regions where the gap is suppressed. The resulting weaklink superconductor will show a low superfluid density.
We investigate these issues using scanning tunnelling spectroscopy, which yields the spatial distribution of the electron density of states with atomicscale precision. Our measurements show that in (Pb,Bi)_{2}Sr_{2}CuO_{6+δ} (Bi2201) which has one CuO_{2} layer per unit cell and is known to have a high residual resistivity^{11}, such a ‘puddled’ superconductor does indeed develop at high doping (Figs. 1 and 2). The typical spatial extent of the puddles is a few nanometres, of the order of the small coherence length in this system (Fig. 3). Our measurements additionally reveal that the superconducting gap persists beyond the dome, and that instead the heterogeneity is driven by gap filling (Fig. 4). This strongly suggests that the breakdown of superconductivity is not a result of a vanishing pairing interaction. A comparison with BdG simulations suggests that this filling is likely due to the decay of the Cooper pairs in surrounding metallic areas, which in turn explains the observation of a large density of metallic carriers. Unexpectedly, we also find a striking violation of a basic BdG rule. Within BdG theory, pair breaking goes hand in hand with gap closing, because depletion of the number of Cooper pairs in a superconductor leads to a diminishment of the gap magnitude ∆ as well: \({\it{\varDelta }} = V\mathop {\sum}\nolimits_k {\left\langle {c_{k \uparrow }^ + c_{  k \downarrow }^ + } \right\rangle }\), where V is the attractive interaction and c^{+} variables are electron field operators. Instead, our data show that the puddles characterized by the largest gap magnitudes also exhibit the largest gap filling (Fig. 4c), and that the average gap magnitudes barely depend on doping (Fig. 4a). We therefore conclude that the physics governing the superconducting transition is of a different, nonmeanfield kind.
To arrive at these findings, we study a series of Bi2201 samples with eight different doping levels, from underdoped to beyond the superconductor–metal transition, with an emphasis on the strongly overdoped regime. We chose Bi2201 because it has only one CuO_{2} plane per unit cell, and has a rather large residual resistivity^{17}, suggesting that disorder is exceptionally important. On each sample, we measure the atomicscaleresolved differential conductance g(E,r) as a function of bias energy E and location r, which is proportional to the Bogoliubov quasiparticle density of states.
We first consider the spatially averaged g(E) spectra obtained at 4.2 K (Fig. 1a). Consistent with earlier reports^{18,19}, crossing into the overdoped regime, the spectra acquire an increasingly large nonzero Bogoliubov quasiparticle density of states at the Fermi level. This is remarkable as this quantity should go to zero for a standard dwave BCS superconductor, but it is consistent with results from optical conductivity measurements^{12}. It remains to be seen whether angleresolved photoemission spectroscopy (ARPES), if performed in strongly overdoped regions with such small gaps, would observe a similar phenomenology in both Bi2201 and (Pb,Bi)_{2}Sr_{2}CaCu_{2}O_{8}_{+}_{δ} (Bi2212). We investigate this phenomenology using individual spectra, as in a heterogeneous situation like this, the average spectra do not represent the phenomenology adequately (Fig. 1b–d).
Next, we use a phenomenological model to fit all spectra over the whole doping range to extract the superconducting gap and gap filling of each individual spectrum. We calculate the spectral weight on each point k = (k_{x}, k_{y}) on the Fermi surface using a Dynes formula with superconducting gap ∆_{k} = ∆(cos(k_{x}) − cos(k_{y}))/2, where ∆ is the maximal gap, and then average over the Fermi surface^{20}. We use the Dynes formula^{21} as a mere phenomenological description constructed to reveal characteristic scales for the observed gap size and the gap filling, and discuss interpretational concerns after the presentation of the data. Our model yields the following function for the modelled differential conductance:
where 〈〉_{FS} indicates the average over the Fermi surface, P(E) is a thirddegree polynomial function to account for background variation and Dynes(E, ∆_{k}, Γ) is the Dynes function with the pairbreaking parameter Γ. For this study, we concentrate on the superconducting gap and thus restrict ourselves to a ± 15 meV range. (In the underdoped and optimally doped range, which are not the focus of this paper, a pseudogap exists at a larger energy scale, as indicated by the red arrows in Fig. 1a). Lastly, we convolute g(E) with a Gaussian function to account for spectrum broadening due to a finite temperature and the lockin modulation.
We define the filling parameter F as the ratio g(E = 0,T → 0)/P(E = 0), which can be expressed in terms of our fitting parameters as:
Figure 2a,b illustrates how the model differentiates between gap closure, controlled by ∆, and gap filling, controlled by Γ or F. Figure 2c shows some typical spectra and fits from various locations. It is clear that, when compared to the scenarios presented in Fig. 2a,b, the measured spectra look more similar to the filling scenario as opposed to the closing scenario. We then fit roughly 10^{5} spectra from eight different doping levels with this model and display the extracted gap size and gap filling in Fig. 3. We note that for the strongly overdoped samples, high a signaltonoise ratio is key for successful fits; the traces shown in Figs. 1b–d and 2c are raw spectra without any averaging. A further challenge is that at higher doping, a substantial fraction of spectra exhibit completely gapless regions. We identify such spectra after fitting and exclude them from subsequent analysis. In the Supplementary Information, we provide details (Supplementary Section IIB and Supplementary Fig. 4) and demonstrate that our key results are independent of these choices (Supplementary Section IIA–C). We also provide a modified version of the model with an alternative definition of the gap filling and show that our results are independent of the precise definition of gap filling (Supplementary Section IID).
We start our discussion with the spatial maps of the gap size ∆(r) as a function of doping (Fig. 3b–d). Strikingly, while more spectra are fully filled at higher doping, the average gap size remains roughly constant on the strongly overdoped side (Fig. 3a). Initially, the gap size increases when moving from underdoped to optimally doped. Beyond optimal doping, the gap size barely decreases anymore when going through the overdoped and strongly overdoped side, and instead remains roughly constant—even beyond the superconductortometal transition. In particular, throughout the strongly overdoped region, we observe an almost constant average gap amplitude even though T_{c} is rapidly decreasing. Our study thus excludes a homogeneously diminishing pairing interaction as the cause of the superconductortometal transition.
Given a constant gap, what drives the changes in spectra on the overdoped side? Our analysis indicates that it is the gap filling. We extract the gap filling, F, using equation (2), for each measured sample, and present the distribution of the gap fillings in Fig. 3f–h and their histograms in Fig. 3e. Remarkably, the mean gap filling changes considerably over the doping range. In the optimally doped region, the spectra have a filling close to zero; that is, they are fully gapped. Crossing into the overdoped regime, a subset of spectra starts to develop a finite gap filling. This subset grows with further doping, with all spectra having a finite filling in the strongly overdoped regime. The values of F shift markedly in this doping regime from nearly fully gapped (F = 0) near optimal doping to almost fully filled (F = 1) towards the strongly overdoped regime and extending into the metallic regime. The trends in gap closing and gap filling are summarized in Fig. 4a,b: as the doping is increased into the overdoped regime, the gap size remains roughly constant; by contrast, the gap filling increases rapidly. Thus, a first key result of this paper is that it is not a decaying gap width Δ, but an increasing gap filling F that is responsible for the diminishing superconductivity and that eventually drives the superconductortometal transition. We also note that the persistence of the superconducting gap we observe is remarkably similar to the persistent spin gap observed in a similar doping range^{22}.
Notably, the gap filling is highly heterogeneous, as can be seen from the width of the distributions in Fig. 3e and in the spatial maps in Fig. 3f–h. We observe areas both with and without a gap, each existing at a length scale consistent with the coherence length (~1.5 nm). Some spectra exhibit a peak that can be associated with a van Hove singularity, as reported previously^{18,19}, but we note that it is a highly anomalous one: both the energy and the amplitude of the peak vary in space on length scales that are not consistent with the spatially averaged antinodal signature observed in photoemission^{23}. Further, we find this peak only in the strongly overdoped regime, whereas ARPES measurements suggest that the van Hove singularity should be observable in lowerdoped samples as well, at energies still easily measurable by scanning tunnelling microscopy. The question of the van Hove singularity in scanning tunnelling microscopy data remains open. Notwithstanding the van Hove singularity, our observations indicate that the breakdown of superconductivity in the overdoped regime of the singlelayer bismuth cuprate is likely caused by an emergent strongly inhomogeneous superconductivity, leading to an effective weaklink physics that explains the diminishing superfluid density. Hence, at first glance, our data suggest that theoretical models involving the disorderdriven breakdown of superconductivity in the BdG framework^{4,5,13,14,16} are a good description of the physics of strongly overdoped Bi2201, with the additional information that it is the gap filling that drives the formation of the superconducting puddles.
Next, we focus on the origin of the gap filling. According to BdG theory, the excitations that fill the gap are quasiparticles of the Fermi liquid normal state that are released by breaking up Cooper pairs. Wellknown causes for pair breaking are potential disorder^{4,5} (for a dwave superconductor) and thermal phase fluctuations^{24,25,26}. However, if potential disorder were the only culprit, the areas where the pair breaking is smallest (where superconductivity survives best) should have the largest gaps, which is not what we observe. We demonstrate this in Fig. 4c, where we show the local relationship between the gap size Δ and the pairbreaking Γ, and find a clear positive correlation between the two. Further, we can exclude thermal phase fluctuations based on our temperaturedependent measurements, up to 20 K for the overdoped sample with T_{c} = 9 K (OD9K). Thermal phase fluctuations should lead to a strongly temperaturedependent filling, in contrast to our observations (Supplementary Section III).
We therefore consider an alternative candidate for pair breaking: the decay of Cooper pairs into smaller gap or metallic regions, as previously suggested^{27,28}. This can be seen as akin to an inverse proximity effect^{29}. We are not aware of selfconsistent simulations for this scenario in the literature, but they are possible with stateoftheart numerical methods. We start with a large realspace supercell implementing a realistic tightbinding band structure. We then introduce the superconducting puddles by switching on a local pairing interaction characterized by a linear dimension L that is approaching the (bulk) coherence length. The BdG equations are then solved selfconsistently (Supplementary Section IV for further details), and typical outcomes are shown in Fig. 1e. The simulated spectra are surprisingly similar to the experimental ones, and one might wonder whether this gapfillingdominated physics is connected to certain disordered superconductors^{15,30} and interface superconductors^{31} with a local density of states phenomenology that is not dissimilar from what we observe here.
However, there is one aspect of our data that is markedly inconsistent with the BdG description of granular superconductors. Our data show that the largest gaps also exhibit the strongest gap filling (Fig. 4c), while within BdG, gap filling should always go hand in hand with a decrease of the gap magnitude. Our selfconsistent simulations confirm that this is indeed also valid for the heterogeneous case: upon application of our fitting model to the calculated spectra shown in Fig. 1e, we find that the regions with the largest gaps show the least amount of pair breaking, as shown in Fig. 4d. Selfconsistency in the calculations is necessary here; fixing the gap magnitude artificially would obscure any effect from pair breaking on the gap magnitude. The comparison in Fig. 4c,d shows a striking inconsistency between the experiment and BdG expectation.
In summary, our realspace imaging reveals a strongly heterogeneous superconductivity consisting of superconducting puddles with a size set by the coherence length immersed in a metallic matrix. This explains the diminishing superfluid density^{2,3} and the origin of the large fraction of metallic carriers^{12}: it stems from the filling of the gap. Our data further demonstrate that superconductivity does not, as it is often assumed, become conventional in the strongly overdoped regime. The breakdown of superconductivity is not the consequence of a vanishing pairing interaction and does not follow the BdG description. Furthermore, the gap filling is entirely different from simple quasiparticles populating the gap and counting the number of broken BCS Cooper pairs. Instead, what fills the gap might be related to the strange normal state^{11}, for example, collective excitations of an unknown kind rooted in the ‘strange metal’ physics, which at present cannot be calculated, or to electrons from a different sector, but not by means of simple pair breaking. Last but not least, this unconventional physics may not be limited to the lowtemperature, overdoped regime. Scanning tunnelling microscopy studies at an optimal doping of Bi2212 showed a rather similar puddling effect upon approaching the superconducting transition temperature^{27,28}. This may imply that the physics of the thermal transition—the ‘high T_{c}’ problem itself—is governed by unknown physics. Therefore, it would be interesting to revisit the hightemperature regime in Bi2212 to make this more precise.
Data availability
The full threedimensional datasets used for this paper, and the data used to generate Figs. 1–4 and Supplementary Figs. 1–13, will be available on Zenodo, https://doi.org/10.5281/zenodo.7643856.
Code availability
The code used for fitting and generating figures is available upon reasonable request to the authors.
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Acknowledgements
We thank E. Hudson, D. Cho, R. Fermin, E. Stolte, L. Bawden, K. Schalm, M. Berben, N. Hussey, G. Verdoes, K. van Oosten and F. Groenewoud for their help and insightful discussions. This work was supported by the European Research Council (ERC StG SpinMelt). M.A.S. acknowledges funding support from the National Science Foundation (NSFDMR1849751) and the Intelligence Advanced Research Projects Activity (IARPA) SuperTools programme.
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W.O.T., T.B., J.F.G., K.M.B., D.C. and Y.Y. performed scanning tunnelling microscopy experiments. S.S. and M.S.G. performed ARPES measurements. W.O.T., I.B. and A.V. did data analysis. J.Z. and M.A.S. performed the theoretical simulations. W.O.T., T.B., J.F.G., J.Z. and M.P.A. wrote the manuscript. Y.H., T.K. and E.v.H. made and characterized the samples. M.P.A. supervised the project. All authors commented on and agreed on the manuscript.
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Supplementary sections I–VII (figs. 1–14 and discussion).
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Tromp, W.O., Benschop, T., Ge, JF. et al. Puddle formation and persistent gaps across the nonmeanfield breakdown of superconductivity in overdoped (Pb,Bi)_{2}Sr_{2}CuO_{6+δ}. Nat. Mater. 22, 703–709 (2023). https://doi.org/10.1038/s41563023014971
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DOI: https://doi.org/10.1038/s41563023014971
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