Large single crystals of Pb/La-doped Bi2201 were taken from boules grown independently at two sites via the floating-zone technique. The doping level of the crystals used in this study (p = 0.215 ± 0.005) was estimated from the measured T _c using the Presland relation 1 − T c / T c max = 82.6 ( p − 0.16 ) 2 [29], with T c max = 36 K. Recently, it was shown that when using this relation, normalized ρ _ab (T) curves for Bi2201 and LSCO single crystals of the same p (or T _c ) value collapse on top of one another [30]. The crystals were cleaved along the ab-plane and cut into shape along the c-axis using a wire saw which leaves no burring of the edges. Typical sample dimensions were approximately 1,000 × 200 × 8–40 μm³ (the thicknesses having been determined by a scanning electron microscope).

For the resistivity measurements, electrical contacts were made using 25 μm Au wire and fixed using Dupont 6838 paint before being annealed in flowing oxygen for 10 min at 450°C. Typical contact resistances were between 1 and 10 Ω. All samples were cooled using a ⁴He flow cryostat and their in-plane resistivity measured using a standard four-point ac lock-in detection technique. While the dimensions of the samples could be determined to a high degree of accuracy, the absolute magnitudes of ρ _ab (T) were subject to an uncertainty of ±25% due to the uncertainty in estimating the distance between the voltage electrodes. Complementary magnetization measurements were carried out in a commercial SQUID magnetometer.

Figure 1 shows ρ _ab (T) curves for 6 Pb/La-doped Bi2201 crystals grown in Amsterdam (top panel) and Sendai (bottom panel). For both sets of crystals T _c ≃ 27 K and p ≃ 0.20. According to a recent combined transport and angle-resolved photoemission spectroscopy (ARPES) study [30], this doping level lies very close to the doping level p* at which the pseudogap regime terminates in Bi2201. Correspondingly, all ρ _ab (T) curves display a quasi-T-linear dependence from room temperature down to T _c albeit shifted with respect to each other due to the difference in their respective ρ ₀ values.

In certain crystals (labelled ♯1B and ♯1C in Figure 1A), ρ _ab (T) displays a small upward deviation from T-linearity below around 75 K. One possible origin for this upward deviation is contamination of the signal from c-axis mixing, whereby a proportion of the current flows between the CuO₂ planes. The anisotropy in the resistivity ρ _c /ρ _ab is extremely large in Bi2201 (≈10⁵–10⁶) [31] and ρ _c (T) is known to exhibit only weakly metallic behaviour in the OD regime. Hence, any c-axis mixing would lead to a marked increase in the absolute value of the as-measured resistivity relative to the intrinsic ab-plane response as well as a different T-dependence. In order to minimise this possibility, each sample was mounted in a ‘floating’ configuration, i.e., elevated above the substrate with the silver paint fully extending across the sample thickness, in order to isolate the in-plane current response and avoid any contamination from current along the c-axis. Using only samples in which the resistivities on opposite sides of the crystal were identical, we obtained a series of ρ _ab (T) curves that exhibit the same T-linear slope between 75 K and 300 K (to within our geometrical uncertainty), suggesting that c-axis mixing is indeed negligible in these crystals.

The most striking feature of Figure 1 is the invariance of T _c (= 27 ± 1 K), irrespective of the magnitude of ρ ₀, that itself varies by over 250 μΩcm. (For each crystal, ρ ₀ is obtained by extrapolating a fit to ρ _ab (T) between 50 K and 150 K). Moreover, any small variations in the value of T _c do not appear to be correlated with ρ ₀. As we will discuss in the following section, this level of impurity scattering should be enough to destroy superconductivity many times over. Such robustness is contrary to expectations within dirty d-wave theory in which changes in T _c and ρ ₀ are strongly correlated (via Γ _n ).

Dirty d-wave theory, as applied to cuprate superconductivity, is an extension of an original treatment of paramagnetic impurities in a conventional (s-wave) BCS superconductor [27]. Due to the sign change of the SC order parameter Δ occurring at the nodes, even non-magnetic impurities can induce pair breaking in a d-wave superconductor. This in turn leads to a residual density of zero-energy states and a suppression of both T _c and the superfluid density n _s . More quantitatively, Δ closes at a T _c value that is reduced in the presence of disorder from its optimal value T _c0 according to the Abrikosov-Gor’kov (AG) equation: l n T c 0 T c = ψ 0 1 2 + ℏ Γ n 2 π k B T c − ψ 0 1 2 , (1) where ψ ₀ is the usual digamma function, Γ _n is the normal-state impurity scattering rate, and ℏ and k _B are the reduced Planck constant and Boltzmann constant, respectively.

As mentioned in the Introduction, a recent series of studies based on the self-consistent T-matrix approximation (SCTMA) have indicated that a number of experimental observations in LSCO and Tl2201 can be successfully accounted for by carefully treating the scattering potentials arising from different types of out-of-plane disorder within the dirty d-wave formalism [15–19]. Within their picture, the momentum relaxation rate is assumed to be the same as the single-particle scattering rate, as deduced, for example, from ARPES. Accordingly, Γ _n can be estimated from the Drude expression for the (residual) dc resistivity: ρ 0 = m * n e 2 Γ n . (2) Here m* is the effective mass, n is the carrier density and e is the electronic charge. Within the SCTMA, Born and unitarity scattering contribute additively to Γ _n and thus, from the perspective of Eq. 1, the suppression of T _c is largely independent of the impurity phase shift. Note too that the above expression does not take into account the effects of small-angle scattering, which can cause the momentum relaxation rate to be substantially smaller than the single-particle scattering rate. Hence, the estimate of Γ _n from Eq. 2 is in fact a lower bound for input into Eq. 1.

Figure 2 reproduced from Ref. [17]–shows how different T _c (p) domes for LSCO (red line) and Tl2201 (blue line) can be derived from a singular form of T _c0(p) using estimates for Γ _n that are consistent with experimental observations. The quoted values of Γ _n = 6π (18π) K correspond to ρ ₀ = 6 (20) μΩcm for Tl2201 (LSCO), respectively. (As mentioned above, in a subsequent study [18], slightly modified T _c0(p) domes for Tl2201 and LSCO were incorporated into the model, though the differences were only minor.) The essential feature of Figure 2 is that the reduction in T _c of ∼ 20 K (60 K) from its inferred T _c0 value in optimally doped Tl2201 (LSCO) is attributed to a normal state scattering rate of approximately the same magnitude. At the same time, the extent of the SC dome in LSCO (on the overdoped side) is reduced by Δp ≈ 0.06 for the same level of impurity scattering.

In order to link these estimates for Γ _n to ρ ₀, we must also derive estimates for m* and n = (1 + p)/V _cell into Eq. 1, where V _cell is the volume of the unit cell. First, let us consider LSCO. For p = x = 0.20, we obtain m* ∼ 10 m _e (the bare electron mass) from the electronic specific heat [32] and n = 1.3 × 10²⁸ m⁻³ (using 1 + p instead of p). Hence, Γ _n = 55 K corresponds to ρ ₀ ∼ 20 μΩcm, as quoted above. Similarly for Tl2201 (p = 0.27), m* ∼ 5 m _e [33, 34], n = 7.4 × 10²⁷ m⁻³ and Γ _n = 18 K, giving ρ ₀ ∼ 6 μΩcm.

For Bi2201, m* ∼ 7–10 m _e for p ∼ 0.23 (T _c = 18 K) [35] and n = 7.0 × 10²⁷ m⁻³. Hence, the spread in ρ ₀ shown in Figure 1 (40–290 μΩcm) corresponds to 70 K ≲ Γ _n ≲ 500 K. In other words, while the impurity scattering rate deduced from ρ ₀ varies on the scale of 500 K, the superconducting transition temperature is found to be constant to within 1 K. Such extreme inequality is clearly at odds with expectations from dirty d-wave theory but is likely, at least in part, to reflect the presence of some form of defect that contributes to an enhanced ρ ₀ while creating, by itself, little or no pair-breaking. Before addressing the viability of the BCS pair-breaking picture, therefore, let us first consider alternative explanations for this surprising finding. (Recall that we have already dismissed c-axis mixing in the current flow as a possible cause of this variation in ρ ₀.)

In this present study, T _c values are quoted based on ρ _ab (T) measurements. Resistivity is effectively a one-dimensional probe of superconductivity, in the sense that a transition to zero resistivity requires only a single, filamentary SC path to be realized. Hence, if a sliver of nominally pristine Bi2201 (i.e., with minimal disorder) permeates each crystal, the apparent robustness of T _c may be illusory. In the normal state, by contrast, the current distribution will be sensitive to all regions of the sample and indeed, if the SC filament is sufficiently thin, it will be dominated by those non-SC regions with higher ρ ₀. Such a scenario may help explain why T _c is so insensitive to marked variations in ρ ₀.

Simulations presented in Supplementary Appendix SA indicate that for such a scenario to be applicable, the SC region must occupy ∼ 1% of the total volume of the sample; otherwise the T-dependence of ρ _ab (T) will visibly deviate downwards from its intrinsic (T-linear) behaviour, which is not observed. In order to estimate the SC volume fraction of our crystals, we measured the dc magnetisation of two of them (with ρ ₀ values of 80 μΩcm (ρ(T) data not shown) and 300 μΩcm (Sample #1A in Figure 1), respectively) using a SQUID magnetometer with the magnetic field applied parallel to the ab-plane (where the demagnetisation factor is minimised). The results are shown in Supplementary Appendix SB and reveal an estimated volume fraction in both crystals of ∼ 100%. Thus, it seems unlikely that the presence of a filamentary SC path (which would have to be very similar in form in all crystals studied) can account for the observed robustness of the SC transition.

A more plausible origin of this insensitivity of T _c on ρ ₀ is the presence of specific (extended) forms of microstructural defects (e.g., dislocations, columnar defects, grain boundaries, etc., … ) that adversely affect ρ ₀ while contributing minimally to pair breaking. An example of such extended defects having a profound effect on ρ _ab (T) but a minimal effect on T _c can be found in the infinite-layer nickelates [36, 37]. Structural and electronic nanoscale inhomogeneity is a well-known feature of Bi-based cuprates [14, 38], but this inhomogeneity tends to be more point-like than extended and as such, should also have a similar effect on both T _c and ρ ₀. Larger defects, such as microcracks, could also cause an increased resistivity though in this case, one might expect to see Arrhenius-type behaviour (ρ(T) ∝ exp(−Δ/k _B T)) due to tunneling across the crack, as one finds in polycrystalline samples with multiple grain boundaries. While we noted earlier that there were small upturns observed in the ρ _ab (T) curves of some of our Bi2201 crystals, there did not appear to be any correlation between the value of ρ ₀ and the presence of an upturn.

A dedicated transmission electron microscopy (TEM) study is currently underway to look for evidence for the type of defect that might cause this dichotomy between T _c and ρ ₀, the results of which will be published elsewhere. Although we cannot rule out the presence of such extended defects affecting ρ ₀ without impacting T _c , in the following section we present a body of evidence on the LSCO family that provides arguably a greater challenge to the viability of the dirty d-wave scenario to cuprates within the strange metal regime.

It is well known that superconductivity in cuprates is strongly suppressed upon Zn substitution on the planar Cu site. In LSCO, for example, 4% Zn substitution can destroy superconductivity entirely while at the same time raising ρ ₀ by ∼ 50 μΩcm [39]. The origin of this suppression is not entirely clear. Despite being a non-magnetic impurity, Zn dopants appear to influence strongly the magnetic environment within the CuO₂ plane as well as act as unitarity-limit scatterers [39].

Mahmood et al. [40] recently reported a study on OD LSCO thin films irradiated using 1 MeV oxygen ions. Ion irradiation is believed to create narrow columnar defect tracks throughout the film. Figure 3A shows a series of ρ _ab (T) curves obtained on an optimally doped LSCO film exposed to different fluences using a flux gradient to produce a spread in defect density. Irradiation leads to a maximal increase in ρ ₀ of 36–40 μΩcm without the T-dependence or the slope of the ρ _ab (T) curves changing. Such adherence to Matthiessen’s rule, coupled with accompanying measurements of the low-frequency Drude response, suggests that the irradiation is simply creating additional elastic scattering centres. According to the calculations in Section 2.2, the magnitude of Δρ ₀ corresponds to ΔΓ _n > 100 K, and given that Eqs 1, 2 predict a ∼ 1 K drop in T _c for every 1 K increase in Γ _n , clearly such a level of disorder should be sufficient to remove all vestiges of superconductivity in the film. Yet T _c itself is found to drop by less than 5 K.

One expects that an optimally doped film will possess a more robust SC state than those at a higher doping level with a lower T _c . In reality, Mahmood et al. observed the opposite trend. As shown in Figure 3B, for a pristine film with T _c ∼ 10 K, there was no discernible change in T _c for the same level of fluence that induced a ∼ 5 K reduction in T _c in the optimally-doped film. It seems that the more overdoped the pristine film is and the lower its initial superfluid density, the more robust is the superconductivity to similar levels of irradiation.

Two further examples of the insensitivity of T _c to changes in ρ ₀ in OD LSCO are shown in Panels C and D of Figure 3. Panel C shows ρ _ab (T) curves for two single crystals (x = 0.24) whose ρ ₀ values differ by ∼ 35 μΩcm, corresponding to ΔΓ _n ∼ 100 K [41], yet the difference in T _c in the two crystals is less than 2 K. Panel D shows ρ _ab (T) for a single crystal [2] and thin film [42] with x = 0.23. Note that the form of ρ _ab (T), as well as their derivatives [43], are the same, implying that their doping levels are essentially equivalent. The difference in ρ ₀ of the two samples (after normalising their slopes [43]) is such that ΔΓ _n ∼ 85 K, but yet again, their T _c values are almost indistinguishable.

In the previous section, we discussed the possibility that in Bi2201, the insensitivity of T _c on ρ ₀ reflects the fact that elastic scattering is dominated by extended defects that contribute largely to ρ ₀ but do not, by themselves, break pairs and thereby cause a suppression in T _c . In the study by Mahmood et al., extended (columnar) defects were found to cause an increase in scattering, as deduced from the width of the Drude conductivity peak. We note too that the superfluid density also decreased, implying that such defects do indeed cause pair breaking, yet T _c itself remained remarkably robust.

According to Figure 2 and the corresponding relation between Γ _n and ρ ₀, dirty d-wave theory predicts that for every 1 μΩcm increase in ρ ₀, T _c in optimally doped LSCO should decrease by around 2.5 K [17], irrespective of the phase shift of the dominant scattering process. At the same time, the full extent of the SC dome diminishes by Δp ∼ 0.0075 per 1 μΩcm increase in ρ ₀. This extreme sensitivity of the T _c (p) dome to small changes in ρ ₀ represents arguably the greatest challenge to the theory’s applicability. Indeed, one of the most striking and largely overlooked features of cuprate research is the immutability of the T _c (p) dome.

The six panels in Figure 4 reproduce full T _c (p) domes for bulk LSCO reported over a period of 2 decades (1989–2009) [44–49]. The samples in question are both poly- and single crystalline and were prepared by various techniques, including flux and travelling-solvent floating-zone growth of single crystals and spray-drying or powder-mixing procedures for the ceramic powders. Note that all T _c values, bar those in panel (E), were determined by magnetisation or susceptibility measurements. The dashed green line in each panel represents the Presland formula [29] with T c max = 38 K and assuming p = x (the Sr content). The consistent overlap between the dashed lines and the data indicates that the T _c (p) dome in bulk, as-grown LSCO is invariant, independent of the quality of the starting materials or the way in which the samples have been synthesized. Moreover, as highlighted by the single horizontal line spanning all six panels, T c max is the same for all series of samples to within 1 K. For such consistency to be accounted for within the dirty d-wave scenario, ρ ₀ would have to be identical to within 0.4 μΩcm ( < 2 % ) for every sample at every single doping level.

Such extreme levels of reproducibility are clearly beyond all reasonable expectations (requiring as it does that all LSCO samples have identical values of Γ _n to within 1 K) and highlight a key feature of the superconductivity in LSCO that remains unresolved. It appears that the T _c (p) dome in bulk LSCO is not, as has been argued [15], set by the level of disorder in the material, but by some other driving mechanism, such as the strength of next-nearest hopping [50, 51]. Certainly, it would be remarkable if the drop from T _c0(p) to T _c (p) for LSCO in Figure 2 were due to a scattering rate Γ _n whose magnitude is fixed and commensurate with a ρ ₀ value equivalent to 20 μΩcm and that all other contributions to ρ ₀, e.g., flux or crucible inclusions, were extraneous and had no further pair-breaking effect. A similar fundamental limit to ρ ₀ would also have to exist for Bi2201, despite the fact that Bi2201 may contain multiple elements (e.g., Bi/Pb, La/Sr) in its formula unit.

Although a number of SC properties of OD LSCO and Tl2201 have been successfully modeled by considering the differences in the impurity potential, its phase shift and its location relative to the CuO₂ plane [15–19], the relation between T _c and Γ _n is, by and large, independent of these details and as such, should be a robust test of the theory’s applicability. The inability of dirty d-wave theory to account for the remarkable insensitivity of T _c to changes in ρ ₀–highlighted in Figures 1, 4 – thus implies that either we do not understand the true causes of residual resistivity in cuprates (i.e., that the correspondence between Γ _n and ρ ₀ is somehow lost), or that the basis of the theory is not the right framework to describe the transition from strange metal to superconductor.

Motivated by these findings, we present below an alternative (non-BCS) scenario for the robust T _c (p) domes in OD cuprates in which the non-FL nature of the cuprate strange-metal plays a central role. In the process, we offer an alternative explanation as to why the SC dome in cleaner Tl2201 extends to a higher p-value than in Bi2201 and LSCO and suggest ways to test the validity of such a scenario.

In the previous section, we highlighted various types of extended defects that could, in principle, enhance ρ ₀ without necessarily inducing substantial pair breaking within the CuO₂ plane. In order to investigate whether such defects are indeed the root cause of this behaviour, more detailed microstructural studies of each of the relevant cuprate families are strongly advocated. Certainly, it is something that has been largely overlooked by the community. Until such time, however, it is worthwhile to at least consider alternative explanations for the demise of superconductivity on the overdoped side.

Franz et al. [52] have argued that the predicted drop in T _c with disorder is higher than observed experimentally due to the fact that within AG theory, the order parameter is spatially averaged, an assumption that may not be applicable to high-T _c cuprates by virtue of their short coherence lengths. Allowing for the spatial variation of the order parameter within a Bogoliubov-de Gennes formalism leads to a suppression of T _c that is indeed weaker than that predicted by the AG theory, but only by a factor of 2. Moreover, naively, one would expect the coherence length to diverge as p → p _sc , yet according to the study of Mahmood et al., T _c becomes even more robust at higher doping levels [40].

The SCTMA used by Lee-Hone et al. [15] treats disorder using an effective medium theory in which the SC state is also assumed to be homogeneous. Other treatments, however, have considered inhomogeneity or granularity in the SC state [53, 54]. When the Cooper pair coherence length becomes comparable to the correlation length of the disorder potential, the order parameter is found to vary spatially while the superconductor segregates into regions of high and low superfluid density, that in turn enhances the propensity for SC phase fluctuations. Indeed, evidence has emerged for both granular superconductivity [14, 55] and enhanced SC phase fluctuations [56] in OD cuprates.

While the granular model described in Ref. [54] captures a number of key observations, it cannot be the complete picture. What this model–and indeed the majority of disorder models that consider this problem–assumes it that the OD cuprates are essentially Fermi-liquids that transition into a BCS superconductor (homogeneous or otherwise) below T _c . Yet, as stressed elsewhere, there is now mounting evidence that OD cuprates are in fact strange metals, with a dominant non-FL T-linear resistivity extending over the entire doping region [2]. Moreover, this strange metal is claimed to exhibit dual character [9, 10] with coexisting but spatially separated regions with FL and non-FL character, respectively [57].

The natural question that arises is whether the non-FL sector possesses the necessary qualities to preserve the size of the pairing amplitude in the presence of disorder. While there is currently no microscopic picture that addresses this, we consider here a simple ‘patchwork’ model for OD cuprates in which intrinsic superconductivity emerges uniquely from the non-FL sector and is resilient to large changes in Γ _n . Figure 5A shows a schematic of such a patchwork cuprate comprising distinct regions of non-FL (in blue) and FL (in red). In a related article [59], we applied both effective medium theory and random resistor networks to a binary mixture of FL and non-FL patches to capture the evolution of the low-T resistivity from purely T ² at high dopings to T-linear near p* (≈0.19) with an increasing fraction f of non-FL sector. The same model also explains the correlation between the T-linear resistivity coefficient and the slope of the H-linear magnetoresistance [59]. In Figure 5A, the doping level is set such that there is no percolation path available for the non-FL component. Supercurrent could, in principle, flow through the FL regions via the proximity effect and thus maintain the SC state. In inhomogeneous systems like LSCO and Bi2201, however, the FL sector will be susceptible to strong pair-breaking effects due to scattering off such inhomogeneities, thereby inhibiting the flow of supercurrent between the SC patches. Hence, as soon as the percolation limit is exceeded, the zero-resistance state is lost. For LSCO and Bi2201, this percolation limit is assumed to coincide with the end of the SC dome at p _sc = 0.27.

According to Pelc et al., percolation emerges when the fraction of SC patches reaches a critical value of 0.3 (assuming SC and non-SC patches of equivalent size) [60]. Related to this, α ₁–the coefficient of the low-T T-linear resistivity in OD cuprates–is found to grow linearly from p = 0.31 up to its maximum value α 1 max at p* ∼ 0.20 where the pseudogap opens. Thus, at p = 0.27, α 1 / α 1 max ∼ 0.35. In a recent high-field transport study [59], we argued that α 1 / α 1 max is a measure of the fraction of Planckian carriers that are present at a particular doping. If these carriers, and only these carriers, form the superfluid condensate in LSCO and Bi2201, then the reason for the onset of superconductivity at p = p _sc becomes self-evident–it is the point at which the supercurrent can travel percolatively between adjacent patches.

Figure 5B compares the normalised T _c dome for LSCO and Bi2201 (dashed line) with that of Tl2201 (solid line and squares). The yellow shaded area in Figure 5B represents the region of the phase diagram where the superconductivity in Tl2201 is enhanced relative to that seen in LSCO and Bi2201. (This extended region of superconductivity is essentially the same as that shown in Figure 2.) Tl2201 is known to be more homogeneous than LSCO and Bi2201 and possess longer mean-free-paths (as manifest in lower ρ ₀ values) [34]. According to the above picture, the level of disorder scattering in Tl2201 is low enough to allow supercurrent to traverse the FL sectors via the proximity effect (see Figure 5C) and for superconductivity to persist beyond the percolation limit. Once the non-FL sector vanishes at p = 0.31, however, then all traces of superconductivity are lost.

This picture represents a marked departure from the extended BCS description for a disordered d-wave superconductor, yet is clearly nothing more than a toy model at present. Before closing, therefore, let us consider some of the consequences of the proposed picture and how it might be tested experimentally. One such consequence may in fact have been tested already. In an earlier electron irradiation study on OD Tl2201 (T _c = 31 K) [61], an increase in ρ ₀ by 70 μΩcm was found to cause a reduction of 20 K in T _c . Using the values quoted in Section 2.2, we find that ΔΓ _n ∼ 210 K > 10 ΔT _c . Despite this level of T _c reduction being far smaller than expected by modified AG theory (ΔT _c = -(π/4)Γ _n [61]), it is still more than is seen in LSCO or in Bi2201. (In Ref. [61], the authors used p, rather than 1 + p, for the carrier density, making the agreement with their expectations from AG theory appear reasonable.). If superconductivity within the FL sector is susceptible to disorder (in accordance with dirty d-wave theory) but is resilient within the non-FL sector, electron radiation might induce pair-breaking predominantly or uniquely within the FL sector. Moreover, if the doping level sits close to the percolation threshold, superconductivity will be appreciably suppressed. Further irradiation, on the other hand, would not cause a further deterioration in T _c due to the resilience of the superfluid residing the non-FL sector. A more dedicated irradiation study, over a range of dopings and to higher fluences, could thus serve as a robust test of the validity of this proposal.

The other corollary of this picture is the presence of SC droplets beyond p _sc = 0.27. According to Ref. [59], signatures of superconductivity are intimately tied to the existence of the T-linear component in ρ(T) that itself indicates the fraction of carriers that are not standard Landau quasiparticles. Hence, for 0.27 ≤ p ≤ 0.31, one expects SC patches to survive, fluctuating or otherwise. A recent STM study [14] observed gap features persisting in nominally non-SC Bi2201, albeit with a low filling fraction, consistent with this picture. In order to test this idea more rigorously, however, one would need to track the evolution of these features in combination with ρ(T) measurements up to p = 0.31 and beyond.