Persistence of vortexlike phase fluctuations in underdoped to heavily overdoped cuprates

Domain: physics
Published: 2025-12-16
Editorial outcome: minor_revision

Abstract
The mechanism controlling the superconducting transition temperature Tc as a function of doping is a central question in cuprate high-temperature superconductors. While in underdoped cuprates Tc is set by global phase coherence rather than the scale of pairing, the role of superconducting phase fluctuations in the overdoped region remains controversial. Here, transport measurements in perpendicular magnetic fields (H) on Bi2+xSr2−x−yLayCuO6+δ (Bi-2201) cuprates reveal in the underdoped region immeasurably small Hall response for T > Tc(H) as a signature of a superconducting regime with vortexlike phase fluctuations. The extent of this regime in T and H is suppressed near optimal doping but strongly enhanced in heavily overdoped Bi-2201. These results demonstrate that vortexlike phase fluctuations play a key role in the field-tuned superconducting transition in the heavily overdoped region, in contrast to conventional mean-field Bardeen-Cooper-Schrieffer description. Their unexpected nonmonotonic doping dependence provides a new perspective on the superconducting transition in cuprates.

Introduction
In copper oxides, superconducting state emerges by doping holes or electrons into an antiferromagnetic (AF) Mott insulator. The doping is accomplished either by varying the oxygen concentration or by substitution with nonisovalent elements, suppressing the AF order of the parent compound, and giving rise to a dome-shaped Tc line with increasing concentration of charge carriers (p) in the CuO2 planes. In the underdoped region, where Tc(p) increases, the superfluid density is low due to the proximity to the Mott insulator1–3, implying a significant role of superconducting (SC) phase fluctuations. The fluctuations are further enhanced by the effective two-dimensional (2D) nature of underdoped cuprates. As a consequence, the thermodynamic SC transition occurs at Tc where long-range phase coherence is established, whereas the onset of SC pairing takes place at some higher, crossover temperature4. This is in contrast to conventional superconductors, where the SC transition is well-described by the mean-field Bardeen–Cooper–Schrieffer (BCS) theory, according to which the SC phase coherence and the formation of Cooper pairs occurs simultaneously. Within that framework, fluctuations are negligible and, indeed, thermal SC fluctuations are observable within a small temperature range δT such that δT/Tc ≪ 1 (ref. 5). On the other hand, the BCS theory was long-believed to provide an adequate description of the cuprate overdoped region, where the number of charge carriers is high and the anisotropy of the material is reduced. However, this view has been challenged recently by several experiments performed at H = 0 on heavily hole-doped cuprates6–9. Although arguments have been put forward that experimental observations can still be explained within the context of the BCS theory by properly taking into account the inevitable presence of disorder10,11 ("dirty d-wave" theory), others have argued that disorder in a d-wave superconductor such as cuprates will lead to an emergent granular SC state12, implying the importance of phase fluctuations beyond the mean-field treatment. Real-space imaging in (Pb, Bi)2Sr2CuO6+δ (Bi-2201) at H = 0 has shown9 that a state with SC grains embedded in a metallic matrix does indeed develop at high doping, but the results were inconsistent with the BCS theory for spatially heterogeneous systems and also seemed to rule out the effects of thermal phase fluctuations. Therefore, the role of both thermal and quantum phase fluctuations in the overdoped region remains an open question13.

On the underdoped side, the existence of a broad phase fluctuation regime (at T > Tc) was widely accepted, but its precise extent in temperature remained a subject of debate for many years4. Likewise, in magnetic fields H applied perpendicular to CuO2 planes, the extent of a SC regime with vortices, as well as its interplay with various charge and spin orders, were long controversial, especially at high H as T → 0. Those issues were essentially resolved in a series of linear and nonlinear transport experiments on La-214 cuprates14–16, which determined the in-plane T–H vortex phase diagrams over an unprecedented range of T and H. It was shown that the vortex lattice with zero in-plane resistivity (ρxx = 0) is separated from the anomalous high-field normal state by a wide regime of a viscous vortex liquid (VL). In this intermediate field range, the viscous VL freezes into a vortex glass at Tc(H) = 0. Thus, at low T → 0, increasing H destroys superconductivity by quantum phase fluctuations14,16. At low fields, below the quantum melting field of the vortex lattice where Tc(H) → 0, VL is observable at T > Tc(H) within a temperature range δT(H) such that δT(H)/Tc(H) > 1. Here, including at H = 0 (refs. 17,18), thermal phase fluctuations lead to the destruction of superconductivity. These results also demonstrated that the vortex phase diagrams of La-214 underdoped cuprates are qualitatively the same regardless of the type or strength of charge orders14–16, strongly suggesting that qualitatively the same vortex phase diagrams should apply to all underdoped cuprates. Hence, similar experiments on another cuprate family, preferably one that does not exhibit spin order coexisting with superconductivity in contrast to underdoped La-214, are needed to verify that conclusion. The most important question, though, is whether the VL regime persists to the heavily overdoped regions of the cuprate phase diagram where the (normal state) pseudogap closes. Bi2+xSr2−x−yLayCuO6+δ (Bi-2201) shows in the ρxx(T) data for various fields (Fig. 1a) that it has several properties that make it an ideal candidate for this study. Like La-214, Bi-2201 has a single CuO2 layer per unit cell, and its relatively low Tc allows us to fully suppress superconductivity with H and probe deep into the field-revealed normal state. Bi-2201 has a relatively large residual resistivity20–23, suggesting the importance of disorder, and in contrast to La-214, only short-range charge order has been reported24–26, with no evidence of spin order27. We combine linear in-plane resistivity ρxx and Hall resistivity ρyx with nonlinear in-plane transport or voltage–current (V–I) characteristics to probe both charge and vortex matter over a wide range of 10−2 ≲ T/Tc ≲ 10 down to mK temperatures and fields up to H/Tc ≳ 10 T/K for several values of doping (Fig. 1a). Our main results are summarized in Fig. 1b-e. In the underdoped region, we find that, qualitatively, the T–H vortex phase diagram (Fig. 1b) is the same as that in the La-214 family14–16 and, thus, independent of the presence of spin order. This result establishes the universality of the vortex phase diagram in underdoped cuprates. Using complementary transport techniques allows us also to reveal a vanishing Hall response (ρyx = 0) as a signature of the viscous VL regime in which ρxx ≠ 0; ρyx = 0 is thus attributed to the increasing viscosity of the vortex motion with decreasing T. By tracking the evolution of such a regime with doping (Fig. 1c, d), we obtain our key results. First, the VL does persist in the heavily overdoped region, such that it is observable up to δT(H)/Tc(H) > 1 (Fig. 1d, e). Therefore, complementing previous H = 0 studies6–9 that used other techniques, our magnetotransport measurements indicate that the SC transition in heavily overdoped cuprates is of a non-mean-field type. Second, in contrast to a commonly held belief, the doping dependence of δT(H)/Tc(H) is non-monotonic (Fig. 1e): vortexlike excitations are considerably suppressed near optimal doping compared to both underdoped and heavily overdoped regions.

Results
Underdoped region
Underdoped Bi-2201 samples are single crystals of Bi2Sr2−xLaxCuO6+δ with x = 0.84 and p = 0.10 ± 0.02 (see “Methods”). Similar to earlier measurements28 of the in-plane resistivity down to T = 0.7 K, our study down to much lower ~0.05 K finds that ρxx(H) at low enough T increases (Fig. 2a), indicating a suppression of superconductivity by H. This is followed by a weak field dependence of ρxx at higher H, representing the normal-state behavior. The ρxx(H) data were used to determine Tc(H), the melting temperature of the vortex lattice in which ρxx = 0, as well as H0c(T), the onset of Gaussian fluctuations of the SC amplitude and phase. As usual14,16,29,30, H0c(T) is defined as the field above which the magnetoresistance increases as H2, as expected in the high-T normal state (Supplementary Fig. 1).

At the highest T, the Hall resistivity ρyx ∝ H, as expected in conventional metals (Fig. 2b; see also Supplementary Fig. 2a–d for measurements up to 41 T, Supplementary Figs. 3a–d for the corresponding Hall coefficient RH = ρyx(H)/H, as well as Supplementary Fig. 4 for ρyx(H) and RH(H) at various T.) At lower T, as H increases and suppresses superconductivity, the Hall signal increases and then recovers its conventional ρyx ∝ H behavior at high H (Supplementary Figs. 2a–d), i.e., RH becomes independent of the field (Supplementary Figs. 3a–d; also Supplementary Fig. 4). The results are in good agreement with previous Hall measurements in underdoped Bi-2201, which focused on exploring the Hall coefficient in the normal state31. Here, in contrast, we focus on the low-field regime, where ρxx starts to deviate from zero. Surprisingly, we observe a large, up to several teslas, difference in the onset fields of nonzero ρyx, which defines Tc for that field, and nonzero ρxx, which is shown in the ρxx(T) data for various fields (solid line in Fig. 2c). Also, which has several properties that make it an ideal candidate for this study. At low T, the onset of ρxx(T) for various H, which appears at intermediate fields. In underdoped La-214 cuprates, Tc(H) was identified14,16 as the onset of the viscous VL observed at T < Tc(H). From the T–H phase diagram in Fig. 1b, it is obvious that there is a fairly wide regime with ρxx ≠ 0 and RH = 0 observed at T > Tc for a given H, which is distinct from the pinned vortex lattice characterized by ρxx(T < Tc) = 0 and RH = 0. At low T, the boundary of the RH = 0 regime indeed approaches Tc(H).

To confirm the existence of the viscous VL at intermediate fields, between Tc(H) and Tpeakc(H), we use two methods. First, we consider the T-dependence of the linear (I → 0) resistivity ρxx(T). We establish that, for T < Tpeakc(H), it is described best with the power-law fits ρxx(H, T) = k(H)Tα(H), where α(H) decreases to zero with H (Supplementary Fig. 5a, b). The power-law dependence suggests that, at intermediate fields, a true SC state (ρxx = 0) exists only at T = 0 when the vortices are frozen. Similar behavior was found14,16 in underdoped La-214 cuprates, consistent with the expectations for a viscous VL above its glass freezing temperature32–34 Tg = 0. Second, we measure nonlinear transport, i.e., differential resistance dV/dI as a function of the dc current excitation Idc (“Methods”). In the same field range at low T, we find that dV/dI is non-Ohmic for Idc ≠ 0, although the linear resistance (dV/dI for Idc → 0) is not zero (Supplementary Fig. 5c, d). This type of behavior is precisely what is expected from the motion of vortices in the presence of disorder, i.e., it is a signature of a viscous VL35. At higher H, Ohmic behavior is recovered (Supplementary Fig. 5d), allowing us to determine fields H*(T) that separate the viscous VL regime at lower H from the Ohmic regime characteristic of the normal state at high H (Fig. 1b). Just like in underdoped La-214 cuprates14,16, as T → 0 there is a quantitative agreement between H*(T), the boundary of the viscous VL obtained from nonlinear transport, with the values of Tpeakc(H) obtained from the linear resistivity measurements. Moreover, here these two characteristic energy scales, which indicate the vanishing of the SC regime with vortices, also follow the “h/4e2” line (Fig. 1b), where the sheet resistance R□/layer ≈ RQ, with RQ = h/(2e)2 the quantum resistance for Cooper pairs. This suggests the onset of localization of Cooper pairs in CuO2 planes36 at T = 0, i.e., a quantum phase transition from a frozen viscous VL (vortex glass) to the well-known, albeit little-understood anomalous normal state (Fig. 1b) with a weak, insulating ρxx/ln(1/T) behavior28,37 for H > H0c (Fig. 2c). Therefore, by using complementary transport techniques that are sensitive to global phase coherence, we have established that RH = 0 with ρxx ≠ 0 clearly represents a signature of the SC regime with vortexlike phase fluctuations.

Our results demonstrate that the vortex phase diagram in underdoped Bi-2201 (p = 0.10 ± 0.02) is qualitatively the same as that in the La-214 family and, thus, independent of the details of either spin or charge orders. This universality observed in the underdoped region is further supported by ρxx vs Idc measurements38 in Bi-2201 with p ≈ 0.125 for μ0H/Tc ≈ 1.2 and YBa2Cu4O8 with p = 0.14 for μ0H/Tc ≈ 0.6, which provided evidence for the presence of a vortex liquid. Furthermore, we show that Hall effect measurements and, in particular, the regime with RH = 0 and ρxx ≠ 0, can be used to identify the presence of a viscous vortex liquid in Bi-2201. Therefore, we focus on this method, employed along other transport techniques, to determine vortex phase diagrams in Bi-2201 at higher doping, all the way to the heavily overdoped region.

Weakly overdoped region
Weakly overdoped Bi-2201 samples are single crystals of Bi2.1Sr1.9CuO6+δ with hole concentration p = 0.18 ± 0.02 (see “Methods”). As in the underdoped region, we measure ρxx(H) and ρyx(H) at different T [Fig. 3a, b, respectively; see also Supplementary Figs. 2e–h for additional ρyx(H) data, i.e., Supplementary Figs. 3e–h for the corresponding RH(H)]. Here we also observe a clear difference in the onset fields of nonzero ρyx and nonzero ρxx (Fig. 3b inset). The onsets of RH = 0 are shown in the ρxx(T) data for various fields (solid line in Fig. 3c), as well as in the T–H phase diagram (Fig. 1c).

It is apparent that the regime with RH = 0 and ρxx ≠ 0, indicative of the viscous VL, is substantially suppressed compared to the underdoped region (cf. Figs. 2c and 1b), in agreement with general expectations2. In particular, here RH = 0 with ρxx ≠ 0 is observed only at lower H ≲ 20 T, i.e., for T > Tc(H) (Fig. 1c). In this range of fields, ρxx(T) (Fig. 3c; also Supplementary Fig. 6) is more reminiscent of the behavior of a BCS-type superconductor, indicating relatively weak thermal SC fluctuations. At intermediate fields 20 ≲ H[T] ≲ H0c(T=0) = 24 where Gaussian SC fluctuations are still observed (Fig. 1c) a peak in ρxx(T) is not found at low T (Fig. 3c). Instead, at these fields at the lowest T < 0.1 K, ρxx seems to drop with decreasing T in a manner similar to that at low fields for T > Tc(H) (Supplementary Fig. 6), in contrast to the behavior in the underdoped region (Supplementary Fig. 5a, b). In addition, nonlinear dV/dI was not observed either, suggesting the absence of a viscous VL at intermediate fields. Current-dependent ρxx was also not observed38 in Bi2Sr2−xLaxCuO6+δ with p ≈ 0.19 for μ0H/Tc ≈ 1.3. Finally, at the highest fields [H > H0c(T=0) = 24 T] where SC fluctuations are no longer observed (Fig. 1c), the T-dependence of ρxx is negligible, indicative of the metallic normal state, in agreement with the Hall effect measurements (Supplementary Fig. 7) and previous work28.

Heavily overdoped region
Heavily overdoped Bi-2201 samples are thin films of Bi2Sr2CuO6+δ (see “Methods”). Here ρxx(H) and ρyx(H) measurements at different T on a film with p = 0.25 ± 0.01 [Fig. 4a, b, respectively; see also Supplementary Figs. 3i–l for the corresponding RH(H)] also reveal a large difference in the onset fields of nonzero ρyx and nonzero ρxx (Fig. 4b inset), i.e., a regime with RH = 0 and ρxx ≠ 0 observed over a wider range of T > Tc(H) and H < 13 T (Figs. 4c and 1d). Surprisingly, the suppression of this regime by H (solid line in Fig. 4c) seems to occur less rapidly than in the weakly overdoped region (cf. solid line in Fig. 3c).

At intermediate fields (13 T ≲ H ≲ 19 T), a peak in ρxx(T) emerges (Fig. 4c). Although not as pronounced as in the underdoped region (Fig. 2c), here we also find a power-law dependence characteristic of a viscous VL: ρxx(H, T) = k(H)Tα(H) for T < Tpeakc(H), with α(H) decreasing to zero with H (Supplementary Fig. 5d), allowing us to determine fields H*(T) that separate the viscous VL regime at lower H from the Ohmic regime characteristic of the normal state at high H (Fig. 1b). Just like in underdoped La-214 cuprates14,16, as T → 0 there is a quantitative agreement between H*(T), the boundary of the viscous VL obtained from nonlinear transport, with the values of Tpeakc(H) obtained from the linear resistivity measurements. Moreover, here these two characteristic energy scales, which indicate the vanishing of the SC regime with vortices, also follow the “h/4e2” line (Fig. 1b), where the sheet resistance R□/layer ≈ RQ, with RQ = h/(2e)2 the quantum resistance for Cooper pairs. This suggests the onset of localization of Cooper pairs in CuO2 planes36 at T = 0, i.e., a quantum phase transition from a frozen viscous VL (vortex glass) to the well-known, albeit little-understood anomalous normal state (Fig. 1b) with a weak, insulating ρxx/ln(1/T) behavior28,37 for H > H0c (Fig. 2c). Therefore, by using complementary transport techniques that are sensitive to global phase coherence, we have established that RH = 0 with ρxx ≠ 0 clearly represents a signature of the SC regime with vortexlike phase fluctuations.

Our results demonstrate that the vortex phase diagram in underdoped Bi-2201 (p = 0.10 ± 0.02) is qualitatively the same as that in the La-214 family and, thus, independent of the details of either spin or charge orders. This universality observed in the underdoped region is further supported by ρxx vs Idc measurements38 in Bi-2201 with p ≈ 0.125 for μ0H/Tc ≈ 1.2 and YBa2Cu4O8 with p = 0.14 for μ0H/Tc ≈ 0.6, which provided evidence for the presence of a vortex liquid. Furthermore, we show that Hall effect measurements and, in particular, the regime with RH = 0 and ρxx ≠ 0, can be used to identify the presence of a viscous vortex liquid in Bi-2201. Therefore, we focus on this method, employed along other transport techniques, to determine vortex phase diagrams in Bi-2201 at higher doping, all the way to the heavily overdoped region.

Weakly overdoped region
Weakly overdoped Bi-2201 samples are single crystals of Bi2.1Sr1.9CuO6+δ with hole concentration p = 0.18 ± 0.02 (see “Methods”). As in the underdoped region, we measure ρxx(H) and ρyx(H) at different T [Fig. 3a, b, respectively; see also Supplementary Figs. 2e–h for additional ρyx(H) data, i.e., Supplementary Figs. 3e–h for the corresponding RH(H)]. Here we also observe a clear difference in the onset fields of nonzero ρyx and nonzero ρxx (Fig. 3b inset). The onsets of RH = 0 are shown in the ρxx(T) data for various fields (solid line in Fig. 3c), as well as in the T–H phase diagram (Fig. 1c).

It is apparent that the regime with RH = 0 and ρxx ≠ 0, indicative of the viscous VL, is substantially suppressed compared to the underdoped region (cf. Figs. 2c and 1b), in agreement with general expectations2. In particular, here RH = 0 with ρxx ≠ 0 is observed only at lower H ≲ 20 T, i.e., for T > Tc(H) (Fig. 1c). In this range of fields, ρxx(T) (Fig. 3c; also Supplementary Fig. 6) is more reminiscent of the behavior of a BCS-type superconductor, indicating relatively weak thermal SC fluctuations. At intermediate fields 20 ≲ H[T] ≲ H0c(T=0) = 24 where Gaussian SC fluctuations are still observed (Fig. 1c) a peak in ρxx(T) is not found at low T (Fig. 3c). Instead, at these fields at the lowest T < 0.1 K, ρxx seems to drop with decreasing T in a manner similar to that at low fields for T > Tc(H) (Supplementary Fig. 6), in contrast to the behavior in the underdoped region (Supplementary Fig. 5a, b). In addition, nonlinear dV/dI was not observed either, suggesting the absence of a viscous VL at intermediate fields. Current-dependent ρxx was also not observed38 in Bi2Sr2−xLaxCuO6+δ with p ≈ 0.19 for μ0H/Tc ≈ 1.3. Finally, at the highest fields [H > H0c(T=0) = 24 T] where SC fluctuations are no longer observed (Fig. 1c), the T-dependence of ρxx is negligible, indicative of the metallic normal state, in agreement with the Hall effect measurements (Supplementary Fig. 7) and previous work28.

Heavily overdoped region
Heavily overdoped Bi-2201 samples are thin films of Bi2Sr2CuO6+δ (see “Methods”). Here ρxx(H) and ρyx(H) measurements at different T on a film with p = 0.25 ± 0.01 [Fig. 4a, b, respectively; see also Supplementary Figs. 3i–l for the corresponding RH(H)] also reveal a large difference in the onset fields of nonzero ρyx and nonzero ρxx (Fig. 4b inset), i.e., a regime with RH = 0 and ρxx ≠ 0 observed over a wider range of T > Tc(H) and H < 13 T (Figs. 4c and 1d). Surprisingly, the suppression of this regime by H (solid line in Fig. 4c) seems to occur less rapidly than in the weakly overdoped region (cf. solid line in Fig. 3c).

At intermediate fields (13 T ≲ H ≲ 19 T), a peak in ρxx(T) emerges (Fig. 4c). Although not as pronounced as in the underdoped region (Fig. 2c), here we also find a power-law dependence characteristic of a viscous VL: ρxx(H, T) = k(H)Tα(H) for T < Tpeakc(H), with α(H) decreasing to zero with H (Supplementary Fig. 8). While any nonlinear dV/dI was too weak to be observed in this sample, the peak in ρxx(T) becomes more pronounced with further increase in doping (see Supplementary Fig. 9 for a film with p = 0.27 ± 0.01). This confirms the presence of the intermediate viscous VL regime in heavily overdoped samples, outside of the pseudogap regime, and its nonmonotonic dependence on doping.

At the highest fields H > H0c(T=0) ≈ 27 T where SC fluctuations are fully suppressed (Fig. 1d), the normal-state ρxx(T) is weak, indicative of a metallic ground state, with dρxx/dT < 0 suggesting the effects of disorder. These findings are also in agreement with our Hall effect measurements (Supplementary Fig. 10). Moreover, we note that the high-field, normal-state value of RH obtained on the p = (0.25 ± 0.01) Bi2Sr2CuO6+δ film for T → 0 (Supplementary Fig. 10b) is comparable to that found21 at higher T = (15–54) K on a Bi2−y+xPbySr2−xCuO6+δ single crystal with a similar p ≈ 0.245, thus demonstrating consistent behavior between films and single crystals.

Evolution of vortexlike phase fluctuations with doping
To examine the dependence of the RH = 0, ρxx ≠ 0 regime on doping, we analyze its extent in T above Tc(H), i.e., its relative width δT(H)/Tc(H) = [T(RH=0) − Tc(H)]/Tc(H). Figure 1e shows that in the underdoped region at low H, the relative width of the viscous VL, as determined from RH = 0 with ρxx ≠ 0, extends even up to δT(H)/Tc(H) ~ 7, consistent with the findings in underdoped La-214 cuprates using different probes of vortex matter (refs. 14,16,39 and refs. therein). Following its initial suppression by H, δT(H)/Tc(H) increases to comparably large values [δT(H)/Tc(H) ~ 8] at high fields, reflecting the presence of the viscous VL at T < Tc(H) (see Fig. 2c and Fig. 1b). We recall that the viscous VL with Tc(H) = 0 (vortex glass) persists to even higher H ≈ H* (Fig. 1b, not shown in Fig. 1e), a regime where quantum phase fluctuations dominate as T → 0 similar to underdoped La-214 cuprates14,16. In the weakly overdoped region, the RH = 0, ρxx ≠ 0 regime is considerably smaller (Fig. 1e), with δT(H)/Tc(H) < 1 at higher fields, indicating that here vortexlike phase fluctuations do not play an important role in the field-tuned destruction of superconductivity. In the heavily overdoped region, on the other hand, δT(H)/Tc(H) at the lowest H is almost as large as in the underdoped region, but it is then quickly suppressed by the field (Fig. 1e). However, it remains much larger than in the weakly overdoped region for all H. In particular, δT(H)/Tc(H) ~ 2, indicating that the SC transition in heavily overdoped Bi-2201 cannot be described within the BCS, mean-field treatment. We note that, although here δT(H)/Tc(H) for p = 0.25 ± 0.01 appear to decrease towards zero as Tc(H) → 0 (see highest H in Fig. 1e) in agreement with general expectations for a phase transition driven by thermal fluctuations40 at a fixed H, it is not possible to confirm the deviation from δT(H)/Tc(H) ~ 2 within error.

Methods
Samples
Several Bi-2201 samples, both single crystals and thin films, covering underdoped, weakly overdoped, and heavily overdoped hole concentration regions were studied. In order to span such a wide range of doping levels, we have used three representative types of Bi-2201 subfamilies. The heavily overdoped thin film samples were prepared from the “pure” Bi2Sr2CuO6+δ family, in which the carrier concentration is controlled solely by the oxygen content. However, unlike other cuprate systems such as La2−xSrxCuO4, La2−xBaxCuO4, or YBa2Cu3O6+x, oxygen in this structure is relatively stable. As a result, it is difficult to significantly change the carrier concentration by post-annealing alone. Therefore, to vary the carrier density in the weakly overdoped region, we used the method of tuning the Bi:Sr nonstoichiometric ratio. This approach, however, has a limited doping range and cannot reach the underdoped region. To address this issue, we employed single crystals of La-doped Bi-2201 (Bi2Sr2−xLaxCuO6+δ), in which Sr is partially substituted with La to control the carrier concentration. This system is known to have a lower residual resistivity and to exhibit a significantly higher optimal Tc (Tc,max) compared to the other two Bi-2201 subfamilies. Indeed, prior work has demonstrated that disorder due to (Sr, La) substitution is weaker than in the case of (Sr, Bi) substitution20,59,60 because the ionic radius of La3+ is more similar to that of Sr2+ than the ionic radius of Bi3+ (ref. 59).

Single crystals of underdoped Bi-2201, Bi2Sr1.16La0.84CuO6+δ (BSLCO) with p ≈ 0.10, in which La substitution for Sr is used to control hole doping concentration, were grown using the floating-zone technique20. The maximum Tc for the Bi2Sr2−xLaxCuO6+δ family20 is Tc,max ≈ 38 K for p ~ 0.16. Weakly overdoped Bi2.1Sr1.9CuO6+δ (p ≈ 0.18) single crystals were also grown by the floating-zone technique, with Tc,max ≈ 10 K for Bi2+xSr2−xCuO6+δ (refs. 59–61). Inductively coupled plasma spectroscopy was carried out to determine the chemical compositions of the crystals. The hole doping concentration for the underdoped crystal was determined20 by comparing the Hall coefficient, normalized to the unit cell volume, to those of La2−xSrxCuO4 at different doping levels62,63. In weakly overdoped Bi2+xSr2−xCuO6+δ crystals, p was determined by comparing the Hall coefficients to those of Bi2Sr2−xLaxCuO6+δ (Supplementary Fig. 11a, b). In both cases, the uncertainty Δp = ±0.02 (refs. 62,63).

Although it has been argued recently that the widely used method of comparing the Hall coefficients of Bi-2201 to La2−xSrxCuO4 may not provide a reliable estimate of p, the question about a proper method to determine p in Bi-2201 is still not fully resolved22. In ref. 21, p was determined using the phenomenological formula proposed by Presland et al.64: Tc = Tc,max [1 − 82.6(p − 0.16)2]. While the same formula with Tc,max ≈ 10 K predicts p ≈ 0.24 for our Bi2.1Sr1.9CuO6+δ crystal, a comparison of the Hall number per Cu atom nH ≡ Vcell/eRH (here Vcell = 3.8 Å × 3.8 Å × 12.3 Å) obtained from our precise measurements at low T and high fields (nH ≈ 0.65 from Supplementary Fig. 7b) to nH in ref. 21, yield p ~ 0.20. This apparent discrepancy is attributed to the uncertainty in Tc,max, which is known to depend strongly on the Bi:Sr ratio59,60. We note, however, that our main conclusions do not depend on the precise values of p, but only on the relative positions of our samples with respect to the pseudogap. Therefore, we have determined the pseudogap temperature TPG in Bi2.1Sr1.9CuO6+δ from both the out-of-plane resistivity ρc(T), as usual (see, e.g., ref. 65), and ρxx(T) (see, e.g., ref. 21). Supplementary Fig. 11c, d shows that consistent values of TPG ~ 50–75 K are obtained, placing our weakly overdoped crystal within the pseudogap region. According to ref. 22, in which p in single crystals of Bi2+z−yPbySr2−x−zLaxCuO6+δ (Bi-2201) was found (to within Δp = 0.01) by comparing ρxx(T) curves to those of La2−xSrxCuO4, TPG ~ 50–75 K corresponds to p ≈ 0.20, in a good agreement with p = 0.18 ± 0.02 determined from the Hall coefficient (Supplementary Fig. 11a, b; also Supplementary Fig. 7b and ref. 21). Importantly, the pseudogap closes22 at p ~ 0.21. For p ≈ 0.10, the pseudogap temperature TPG ~ 250 K (refs. 22,66).

The p ≈ 0.10 and p ≈ 0.18 samples were shaped as rectangular bars suitable for direct measurements of the longitudinal and transverse (Hall) resistivity, ρxx and ρyx, respectively. Detailed measurements of ρxx and ρyx were performed on a p ≈ 0.10 sample with dimensions 1.48 × 0.87 × 0.05 mm3 (a × b × c, i.e., length × width × thickness) and on a p ≈ 0.18 sample with dimensions 1.27 × 0.55 × 0.03 mm3 (a × b × c). Both single crystal samples were measured in various systems, magnets and cryostats, over the period of ~3 years, and they were found to be stable and consistent in their behavior. The data are shown for the voltage contacts separated by 0.62 mm for Bi2Sr1.16La0.84CuO6+δ and 0.53 mm for Bi2.1Sr1.9CuO6+δ. The current contacts were painted to cover two opposing side faces of the platelet-shaped crystals to ensure a uniform current flow and the voltage contacts were painted on the two remaining side faces of the crystals. The contact pads were hand-drawn with gold paint, followed by a heat treatment at 400 °C for 30 min (i.e., for a week in case of a p ≈ 0.18 crystal) in the air, which makes the gold particles adhere well to the sample surface. After this heat treatment to cure the gold contact pads, the samples were annealed in flowing air at 650 °C for 48 h to control the oxygen concentration, and they were quenched to room temperature at the end of the annealing. Finally, gold wires were attached to the contact pads using EPO-TEK® silver epoxy, which was cured at a relatively low temperature, 130 °C. The resulting contact resistances were less than 1 Ω at room temperature20. The out-of-plane resistivity ρc shown in Supplementary Fig. 11d was measured on a Bi2.1Sr1.9CuO6+δ (p = 0.18 ± 0.02) sample, where the voltage contact distance was 0.053 mm (i.e., sample thickness along c) and the cross-sectional area was 0.91 × 1.55 mm2. The sample was prepared by hand-painting ring-shaped current contacts and small circular voltage contacts in the center of the current-contact ring on the opposing ab faces of the crystal20. The contacts were subsequently treated as described above.

Heavily overdoped thin films were prepared from the “pure” Bi2Sr2CuO6+δ family by RF magnetron sputtering on c-axis oriented SrTiO3 single crystal substrates67,68 and also on MgO substrates for composition analysis using Rutherford backscattering spectrometry69. (The values of Tc are the same for both types of substrates.) Oxygen annealing treatments were used to control hole doping concentration. The films were patterned mechanically into Hall bars, parallel to the a crystallographic axis, with six gold contacts for ρxx and ρyx measurements. Detailed measurements were performed on two heavily overdoped thin films: sample F2 with the actual cationic composition Bi2.14Sr1.86Cu1.13O6+δ, dimensions 2 × 0.365 mm2 (a × b) and thickness of 310 nm, and sample F0 with the composition Bi2.15Sr2Cu1.16O6+δ, dimensions 2 × 0.332 mm2 (a × b) and thickness of 78 nm. The data are shown for the voltage contacts separated by 0.62 mm for the F2 sample and 0.61 mm for the F0 sample. The hole concentrations determined using the Presland et al. formula64 with Tc,max = (16 ± 2) K (refs. 67–70) were p ≈ 0.26 and p ≈ 0.27 for samples F2 and F0, respectively. However, the Hall number nH ≈ 0.8 obtained for sample F2 (Supplementary Fig. 10b) corresponds to the hole concentration of ~0.24 in ref. 21, which also used the Presland et al. formula64 to determine p. Therefore, we conclude that p = 0.25 ± 0.01 for sample F2, with at least the same uncertainty Δp = ±0.01 for sample F0 (p ≈ 0.27).

For each sample, Tc was defined as the temperature at which the linear resistivity becomes zero, i.e., falls below the experimental noise floor (~0.7 mΩ), giving Tc = (1.8 ± 0.2) K for p ≈ 0.10, Tc = (4.8 ± 0.3) K for p ≈ 0.18, Tc = (3.3 ± 0.2) K for p ≈ 0.25, and Tc < 0.4 K for p ≈ 0.27 (T = 0.4 K was the lowest measurement temperature for this sample).

Measurements
The standard four-probe ac method (~13 Hz) was used for measurements of the longitudinal and transverse resistances, Rxx and Ryx, respectively, with the magnetic field parallel and antiparallel to the c axis. The Hall resistance was determined from the transverse voltage by extracting the component antisymmetric in the magnetic field. The Hall coefficient RH = Ryxd/H = ρyx/H, where d is the sample thickness. The resistance per square per CuO2 layer R□/layer = ρxx/l, where l = 12.3 Å is the spacing between layers. (The ~2% error introduced into the calculation of R□/layer for Bi2Sr1.16La0.84CuO6+δ by taking l = 12.3 Å instead of the more precise68 l ≈ 12.0 Å is negligible compared to the ~30% geometrical uncertainty in ρxx, typical of cuprate samples22, and it does not affect any of our conclusions.) Depending on the temperature and magnetic field, the excitation current (density) of 1 μA to 100 μA (2.3 × 10−3 A/cm2 to 0.230 A/cm2 for p ≈ 0.10 sample and 6 × 10−3 A/cm2 to 0.606 A/cm2 for p ≈ 0.18 sample) was used for the single crystals, and 100 nA to 300 nA (0.088 A/cm2 to 0.265 A/cm2) for thin films. The excitation currents were adjusted to ensure Ohmic behavior, i.e., to avoid Joule heating. dV/dI measurements were performed by applying a dc current bias Idc and a small ac current excitation Iac (~13 Hz) through the sample while measuring the ac voltage across the sample. For each value of Idc, the ac voltage was monitored for 300 s, and the average value was recorded16,39. A 1 kΩ resistor in series with a π filter [5 dB (60 dB) noise reduction at 10 MHz (1 GHz)] was placed in each wire at the room temperature end of the cryostat to reduce the noise and heating by radiation in all measurements.

The experiments were conducted in several different magnets at the National High Magnetic Field Laboratory: a dilution refrigerator (0.016 K ⩽ T ⩽ 1.1 K) in superconducting magnets with H up to 18 T and 28 T, a 3He system (0.3 K ⩽ T ⩽ 60 K) in a superconducting magnet with H up to 18 T, using 0.1–0.2 T/min sweep rates; a 3He system (0.3 K ⩽ T ⩽ 15 K) in a 35 T resistive magnet, using 1–2 T/min sweep rate; a 3He system (0.3 K ⩽ T ⩽ 30 K) in a 41 T resistive magnet, using 1 T/min sweep rate; and a 3He system (0.3 K ⩽ T ⩽ 12 K) in a 45 T hybrid magnet, using 1–2 T/min sweep rates. The fields were swept at constant temperatures, and the sweep rates were low enough to avoid eddy current heating of the samples. The results obtained in different magnets and cryostats agree well.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The present manuscript presents some careful, systematic transport measurements (resistivity and Hall coefficient) in the single-layered cuprate Bi2201 on either side of the optimal doping concentration. Comparison of the field onset scale for finite longitudinal and Hall resistivities, coupled with non-linear IV characteristics, has led the authors to infer the persistence of phase-fluctuating vortex-like excitations on both sides of the superconducting peak. This result challenges current preconceptions that the overdoped cuprates in particular conform to the standard (dirty d-wave) BCS paradigm. This is a very interesting and important study that, in principle, could warrant publication in Nature Communications. There are a few things that need to be modified or confirmed, however, before the paper can be considered to be ready for publication. The main issue with the paper is given below: 1) The main conclusion of this work - that the propensity for SC phase-fluctuations is minimized at near optimal doping but grows on either side of this - rests entirely on the validity of the doping level of their 'p = 0.18 sample' (which is claimed to represent the behavior near optimal doping). The Tc(0) value of this sample is 4.8 K. It is stated that the hole doping concentration for the 'p = 0.18 crystal' was determined from the Hall coefficient, but no details were given. It is also stated that the p value for the heavily overdoped (thin-film) sample (p = 0.25) is determined using the Presland formula. According to the authors, Tc(0)^max for the crystal was around 11 K. Inserting the 4.8 K value for Tc(0) into the Presland formula returns a doping level of p = 0.24, i.e. almost the same as the film! If the latter is the real doping level, then the central thesis of this work is invalidated, since their main claim relies on the fact that the 'p = 0.18 sample', which shows very little evidence for phase-fluctuating superconductivity, lies close to optimal doping. If p = 0.24 for this crystal, then there appears to be a difference in the response of single crystals and thin films and nothing more. For the characterisation of the samples to support their main claim, the doping level for the 'p = 0.18 sample' needs to be more rigorously shown. A reason for the vastly reduced Tc(0) value also needs to be provided. I note that the low-T Hall coefficient for the 'p = 0.18 sample' is also close to the value shown for the p = 0.25 sample (1.4 mm^3/C vs. 1.7 mm^3/C). Other less serious issues are listed here: 2) Two of the main conclusions of the paper (b) the persistence of pairing without phase coherence to elevated temperatures in the heavily overdoped cuprates, and (c) the importance of SC phase fluctuations depend on doping in a nonmonotonic manner, were previously reported by Rourke et al. in the LSCO family of cuprates (Ref. 28 of the present manuscript). While this paper is referenced in the current work, it is not referenced for these observations. It would seem more appropriate to include an explicit reference and acknowledgement of their prior claim. 3) The paper would benefit from a clear explanation at the start of the paper about why \\rho_xy becoming finite at a later field scale to \\rho_xx is a signature of a vortex liquid. 4) The discrepancy (in temperature) between these two onset fields is labelled \\delta T(H)/T_c(H). Unfortunately, this label appears at the top of page 4 before an explanation of what it actually means. This needs to be rectified. 5) The phase diagrams in Fig. 1d) could benefit from being bigger. I realize that the authors have tried to be uniform with their figure layouts, but there is a lot more detail in Fig. 1d) than in Fig. 2d) or Fig. 3d). Currently, it is hard to make out what is going on. 6) At the start of the discussion section, it is stated that 'the behavior with R_H = 0 and \\rho_xx =/= 0 was observed in the viscous VL regime, and thus attributed to the freezing or pinning of the vortex motion.' This statement is confusing. The onset of a finite \\rho_xx is a signature of vortex melting or depinning, not freezing or pinning. Please clarify.", "reviewer_2": "The manuscript by Terzic et al. presents high-field, low-temperature, transport measurements on the high-Tc superconductor BSCCO (2201) family, which has a single CuO2 layer per unit cell structure. The choice of the material permits the confirmation of existing results in other high Tc materials which however ‘suffer’ from structural or electronic transitions in the region of the explored phase diagram, by excluding such extrinsic origins for the measured behavior. The measurements are made on single crystals for the underdoped and near optimal samples and on thin films for the overdoped samples. This data is then analyzed to establish a (T,H) phase diagram of each sample permitting the identification of regions. At very high fields and/or high temperatures a normal state is found, as expected. At low temperatures and low fields the authors identify a vortex lattice state which corresponds to vortices pinned by disorder. Between these two regions they find a region corresponding to Gaussian superconducting fluctuations on the high H/ high T side. Close to the vortex lattice phase boundary they identify a viscous Vortex Liquid phase which exists as a narrow sliver in this phase diagram, with a phase boundary on the ‘other side’ (red diamonds in Figures 1,2,3 d) which roughly follows the vortex lattice phase boundary (black diamonds in the same figures). They then extract the ‘height’ (a vertical cut which is a temperature) of this sliver, normalized to the corresponding vortex lattice phase boundary temperature (fig 4). Since this corresponds to a temperature range over which the vortex liquid exists, they argue that a non-zero value establishes its existence and a substantial (>2) value corresponds to an anomalous condition with respect to the BCS condition. Overall, the data seems to be of good quality and the measurements, which are new, have been performed in extreme conditions of field and temperature. The analysis is difficult to follow and one tends to get confused between the different phases and their identification. Below we put forward some questions concerning the basics of the analysis and also some more technical points that the authors should address. 1) The authors make the point that their measurements exclude the BCS mean-field situation for the understanding of high Tc superconductors. This is not a new finding, even possibly in the highly overdoped region in high-Tc superconductors. A more central question is the 2D nature of the sample. Though the samples themselves are not single or few layer, does the intrinsic 2D nature of Bi-2201 suffice according to the authors to envisage a 2D scenario? The point is that fluctuations and transitions associated with these would be expected in 2D following the well-known BKT prediction. But this is never invoked by the authors. Is it not relevant to their findings? If it is, have they tested any specific predictions of this model? 2) Figure 4, condenses the findings of the authors, notably Figure 4c. The underdoped region which is characterized by the pseudogap exhibits an extended vortex liquid phase, a result which is not surprising. However, an extended vortex liquid region in the highly overdoped region which is considered closer to a normal Fermi liquid, is an unexpected finding according to the authors. However, if the system is 2D, this might be related to the reduced dimension and the related phase stiffness. In particular, in the abstract the authors repeatedly cite phase fluctuations and ‘vortexlike’ phase fluctuations in the overdoped region. It is not clear how these superconducting phase fluctuations are related to magnetic field induced vortex dynamics. What is meant by the phrase (bottom of page 3): ‘a viscous vortex liquid (VL), i.e. SC phase fluctuations’. The authors should clarify these points. 3) There is no discussion about the different sub-families of the Bi2201 used in the study. The overdoped thin films are from the “pure” Bi2201 family where the doping is only controlled by oxygen, while the lightly overdoped crystal is from the Bi-rich Bi2201 family, and the underdoped sample is from the La-doped Bi2201 family which is known for having a much higher optimal critical temperature than the two others. Readers not familiar with the details of 40 years of cuprate research would need brief comments on these differences, and the rationale for using these specific samples. Furthermore, do these differences affect the conclusions, notably on the establishment of universal phase boundaries? The specific question is, could the phase boundaries be affected by impurity differences in the samples given the different doping strategies in each family? 4) The hole doping determination of each sample is briefly touched upon, only in the methods section. Although it is not the center of the discussion and is the subject of decade old debates in cuprates and Bi2201 in particular, the readers are mostly referred to the literature without further justification. For instance, the “hole doping concentration for both underdoped and overdoped single crystals was determined from the Hall coefficient (refs. 51&52)”, however in those references, which only really apply to the underdoped BSLCO sample, the doping of similar samples is inferred, not directly from the Hall coefficient, but by comparing the Hall coefficient to the ones of LSCO at different dopings. The doping of the lightly overdoped sample is not justified by another reference. The doping of the overdoped thin film is determined from the transition temperature itself using the Presland et al. formula, with an optimal Tc of 11.5K. Neither the use of an “alternative” method for this sample nor the value of the optimal maximum temperature are explained or justified by the authors. The authors should clarify these aspects. Some technical points follow: 1) Both the figure legends and the text are difficult to follow for the non-specialist. An effort to make them easier to read would be welcomed. 2) Figure 1d is quite dense and the authors may consider removing less relevant parts of the phase boundaries and showing them in another figure, perhaps in the extended data. 3) The color bar orientation in Fig 3d is the opposite of the one in Figs 1 and 2 d. 4) In the Methods section sometimes the samples are noted as Bi2+xSr2−xCuO4 (instead of O6+d). 5) The thin films are grown on SrTiO3 substrates and readers are pointed for more explanations to ref. 53, in which samples are grown on MgO without further comments. 6) In the Methods section it is said that “The pseudogap temperatures are ∼ 250 K for p ≈ 0.10 (ref. 55) [...]”, however in ref. 55 samples with p ≈ 0.10 are in the AF phase and not superconducting, which may be an inconsistency with how the hole doping is determined in that ref. compared to this paper. This may also give an erroneous pseudo-gap temperature. 7) Extended data figure 2: error bars are missing. 8) Extended data figure 9: color bar on panel b has no usable scale.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "This experimental manuscript presents transport measurements of the longitudinal (\\rho_{xx}) and transverse (\\rho_{xy}) resistivities in underdoped, weakly overdoped, and heavily overdoped Bi-2201 cuprate superconductors. In addition to the “pure” superconducting state (\\rho_{xx}=0, \\rho_{xy}=0), the authors identify three additional states on the non-superconducting side as the magnetic field or temperature increases: (1) a vortexlike phase fluctuation state (\\rho_{xx} \\neq 0, \\rho_{xy}=0), (2) a Gaussian fluctuation state (\\rho_{xx} \\neq 0, \\rho_{xy} \\neq 0), and (3) a normal state, where \\rho_{xx} follows an H^2-dependent magnetoresistance. They emphasize that vortexlike phase fluctuations are pronounced in both the underdoped and heavily overdoped regimes but are significantly suppressed near optimal doping. Based on this, they conclude that the BCS mechanism is more applicable near optimal doping but fails to describe the underdoped and heavily overdoped cases. While the experimental results are intriguing, the presentation and interpretations are disorganized, and the underlying mechanisms are highly unclear. In particular, the claim and interpretation regarding the emergence of vortexlike phase fluctuations are not strongly supported by the experimental data. (1) Regarding the key results, why do the authors classify the state with \\rho_{xx} \\neq 0 and \\rho_{xy}=0 as a vortexlike phase fluctuation state? Can the authors provide experimental evidence demonstrating vortexlike features (short-range excitations) in this regime? (2) A nonzero Hall signal is typically associated with vortex generation due to magnetic fields or temperature, revealing singular superconducting fluctuations on the non-superconducting side [Nat. Phys. 20, 269 (2024)]. Clearly, the authors' interpretation of the \\rho_{xy} = 0 state as indicative of vortexlike fluctuations contradicts this understanding, and should address more discussion and evidence to support this interpretation. (3) The authors claim that \\rho_{xx} \\neq 0 and \\rho_{xy} = 0 indicate a nearly frozen or pinned vortex liquid. However, if vortices are pinned or frozen, leading to a vanishing \\rho_{xy}, it remains unclear why \\rho_{xx} remains nonzero. The origin of nonzero \\rho_{xx} in this scenario should be clarified. (4) There is inconsistency in how the phase fluctuation regime is described throughout the manuscript. The abstract, introduction, and discussion sections suggest that the \\rho_{xx} \\neq 0 and \\rho_{xy} = 0 state corresponds to the breaking of global phase coherence, while the main text associates it with vortexlike phase fluctuations and further classifies it as a frozen/pinned vortex liquid. These interpretations should be reconciled, as the breaking of global phase coherence is related to long-range phase fluctuations rather than short-range (vortexlike) ones. (5) \"Our results strongly suggest that, in Bi-2201, strong vortex pinning that gives rise to R_H=0 with \\rho_{xx} is due to disorder\". How can the authors arrive at this statement directly based on their experimental data? There is no disorder tuning in this experiment. (6) In the overdoped regime, as revealed by Ref.~12, short-range phase and amplitude fluctuations associated with disorder are expected to be strongly coupled as they should be according to the phase stiffness. How do the authors address this to explain the soley existing vortex phase fluctuations? (7) The statement, “Our results strongly suggest that it is thermal phase fluctuations that are predominantly responsible for the loss of phase coherence near the superconductor-to-metal transition in sufficiently disordered systems”, lacks clear justification from the presented data. How do the authors arrive at this conclusion to distinguish the role of quantum and thermal phase fluctuations? (8) The authors extract H_c' from \\rho_{xx}. Can a similar value be extracted from \\rho_{xy}, which is more directly related to vortex fluctuations? (9) Even near optimal doping, the Gaussian fluctuation regime remains significant. It is therefore misleading to claim that the optimal doping regime is reminiscent of a BCS-type superconductor, as BCS theory does not account for fluctuations, partly due to the Anderson-Higgs mechanism. (10) The figures in the manuscript are also poorly organized: The open black circles in Figs.~1c, 2c, and 3c do not aid in data interpretation. The open green circles in Fig.~1d do not provide meaningful information relevant to the manuscript’s conclusions. The results in Figs.~4a, 4b, and 4c are redundant with those in Fig.~4d. Given these concerns, I cannot recommend the manuscript for publication in its current form."}, "author_response": "Reply to the Reviewers\nWe are grateful to the reviewers for reading our manuscript and for their constructive suggestions\non content and clarity. We welcome the opportunity to improve our manuscript. In the revised\nmanuscript, we have addressed all of the reviewers’ comments. The changes are summarized\nbelow (comments in blue, replies to comments in black). Unless noted otherwise, the reference\nnumbers correspond to those in the current version of the manuscript.\nReviewer #1 (Remarks to the Author):\n[...]\nWe thank the reviewer for the overall positive assessment of our accomplishments and for\nappreciating the significance of our work. As described below, we have addressed all of the\nreviewer’s comments and suggestions and, therefore, we hope that the reviewer will find the\nrevised manuscript ready for publication in Nature Communications.\n[...]\n(See full text in provided document)"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I am satisfied with the response to all the referees' comments and the corresponding changes to the manuscript. I am therefore happy to propose publication of the revised article in Nature Communications. I do recommend one final change related to the first comment of my original review: namely to quote the uncertainty in the p values (i.e. 0.18 $\\pm$ 0.02 and 0.25 $\\pm$ 0.01) in the main text, not just in the Methods and SI. Without quoting this uncertainty, the reader may be minded to consider the doping levels as precise, even with the $\\approx$ sign.", "reviewer_2": "The authors have considerably revised the manuscript and its presentation. The points of discussion have been mostly cleared and the manuscript is now close to an acceptable form. We would like to point out the following points that should be addressed in the final version: a) On page 9 of their reply, while discussing a possible BKT interpretation the authors say: “In addition, our work shows that, in our intermediate field regime where viscous vortex liquid (VL) behavior is clearly observed, the in-plane resistivity follows a characteristic simple power-law temperature dependence (Supplementary Fig. 5a, b for the underdoped, and Supplementary Fig. 8 for the heavily overdoped region), in contrast to the exponential behavior characteristic of the (H=0) BKT transition.” - This remark could, in some form, be included in the manuscript as it would put such questions to rest. b) End of \"Weakly overdoped Sample\" section (page 11 bottom) \"Finally, at the highest fields [H > Hc′(T = 0) = 24 T] where SC fluctuations are no longer observed, ρxx(T ) is negligible, indicative of the metallic normal state, in agreement with the Hall effect measurements (Supplementary Fig. 7) and previous work28.\" - \"ρxx(T ) is negligible\" is unclear: is the absolute value of the resistivity negligible ? Or its variation with temperature? c) Figure 1 caption: \"b H'(T) (neon squares):\" \"light green\" would be a more appropriate color name than \"neon\"", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "I thank the authors for their revised manuscript and their detailed response to my previous report, and I appreciate the care they have taken. I now agree that the manuscript can be published, as the experimental data are solid and the authors have significantly improved the presentation. Moreover, these experimental results are indeed noteworthy and should be brought to the attention of the community. While I appreciate the authors’ efforts in revising the manuscript and responding to the comments, I find that certain key concerns about the interpretation of the experimental findings remain insufficiently addressed. The authors tend to dismiss the key concerns raised by myself and Referees 1 and 2 by simply stating that ``this result is a new and unanticipated finding of our study, it is not possible to provide such an explanation''. They attribute the findings to disorder or pinning effects, while citing various disorder-related papers without providing convincing evidence. In fact, I do not see a clear connection, given that disorder is expected to be weaker in the underdoped regime and to increase with doping. Yet the reported new finding (the regime with finite \\rho_xx and vanishing \\rho_xy) appear to be weakest near optimal doping, while remaining strong in both the underdoped and overdoped regions. I think the authors may have overlooked the following picture suggested by BKT-type physics in two-dimensional superconductors: just above $T_c$, vortices and antivortices unbind and proliferate in the system. Their motion naturally gives rise to a finite longitudinal resistivity \\rho_{xx}, while the symmetric contributions of vortices and antivortices cancel in the transverse direction, leading to \\rho_{xy}=0. Only when an external magnetic field generates a sufficient net vortex density can a finite Hall resistivity \\rho_{xy} appear. The BKT-type scenario is highly sensitive to the superconducting phase stiffness and coherence length, which varies substantially with doping, and this may provide a natural explanation for the experimental results. The authors seem to associate a sheet resistance exceeding the so-called “quantum resistance” $R_Q=h/4e^2$ with the localization of Cooper pairs. In my view, this issue is not central to the main message of the paper. I would caution against taking this criterion too literally. A sheet resistance exceeding $R_Q = h/4e^2$ does not by itself establish Cooper-pair localization; it only indicates entry into an insulating regime, which may correspond either to a Bose insulator with localized Cooper pairs or to a Fermi insulator with broken pairs, and distinguishing between these possibilities requires additional evidence and measurement. I would therefore encourage the authors to rephrase this conclusion in a more cautious manner."}, "author_response": "Second Reply to the Reviewers\nWe are grateful to the reviewers for reading our revised manuscript and for their constructive\nsuggestions. In this version of the manuscript, we have addressed all of their remaining comments,\nwhich have helped us to improve our manuscript. The changes are summarized below (comments\nin blue, replies to comments in black).\n[...]\n(See full text in provided document)"}]

Decision letter