Possible superconductivity from incoherent carriers in overdoped cuprates

Dear Editor,

We thank the Referees for their expert report(s). Following our response to the first round of reviewing, Referee #1 recommended publication of the revised version which we have now submitted following the addition of some minor revisions made in response to the final round of refereeing. Our response to Referee #1’s second set of comments as well as the points raised by Referee #2 are addressed here, followed by a list of changes to the manuscript. The changes themselves are highlighted in blue in the revised manuscript.

Reply to the second set of Comments from Referee #1

Referee comment:
The reply contains numerous relevant and satisfactory answers. I think the paper has been much improved and I would like to recommend the publication of the revised version. Nevertheless, I would like to add two comments:

Response:
We thank the Referee for recommending the publication of the revised version of the manuscript, which we have attached with this second rebuttal. We address the final two comments below.

Referee comment #1.1:
There are two well-understood ways to get a T-linear resistivity. One is electron-phonon scattering above the Debye temperature in a Bloch-Gruneisen picture (like copper). The second is electron-electron scattering when one of the two colliding electrons is classical (like Sr2Ru3O7). In both, the perfector of the T-linear resistivity is of the order of k_B/hbar. The first is obviously not relevant here as argued by the author. I am less sure about the second.

Response:
We thank the Referee for raising this point. In our recent article [Culo et al., Phys. Rev. Res. vol. 3, 023069 (21)], we discussed at length the possible relevance of the ‘cold-cold-to-cold-hot’ (cc-ch) scattering model of Mousatov et al. [PNAS vol. 117, 2852 (20)] – originally proposed to explain T-linear resistivity in Sr3Ru2O7 - to the strange metal phase of the iron chalcogenide family FeSe1-xSx. The key point in the Mousatov picture is the presence of a van Hove singularity (vHs), or region of high density of states, lying just below the Fermi level E_F . Above a certain temperature (determined by the distance \epsilon_h of the vHs from E_F and its width W_h), the carriers on the small, heavy FS become nondegenerate, i.e., classical and ‘hot’. As a result, electrons on the ‘cold’ FS are more likely to be scattered into these hot spots. In this circumstance, T -linear resistivity is realized due to the nondegenerate nature of the hot electrons. Once k_BT < \epsilon_h + W_h, electrons at the hot spots also become degenerate and the usual T^2 behavior is restored.

There are two key differences between the Mousatov picture for Sr3Ru2O7 and the cuprates. Firstly, while in LSCO, the Fermi level is tuned through the vHs at a doping level around p = 0.19, in the Tl2201 system, the vHs is not expected to be crossed until a much higher doping p ~ 0.45 [see Putzke et al., Nat. Phys. AOP (21)]. Hence, as one dopes across the strange metal region (0.20 < p < 0.30), E_F in the LSCO system is tuned away from the vHs and while in Tl2201, E_F is tuned towards it (though never crossing). The evolution of the T-linear coefficient with doping in both systems, however, is very similar. Moreover, the T-linear resistivity persists down to the lowest temperatures at all doping levels studied, meaning that there is no crossover in the form of the resistivity to a T^2 dependence in either LSCO or Tl2201, as one might expect when one is tuning away from or towards the vHs. Finally, the separation of E_F from the vHs in superconducting Tl2201 is such that the density of states in Tl2201 can be modelled as essentially isotropic within the (kx, ky) plane, as indicated in Figure 1A of the main manuscript. For these reasons, we do not believe that the Mousatov picture is applicable to overdoped cuprates. Electron-phonon scattering is not relevant here either, given the persistence of the T-linear resistivity to temperatures up to three orders of magnitude smaller than the Debye temperature. Hence, the most appropriate interpretation for the T-linear resistivity and other strange metal properties is indeed the one based on maximum dissipation, presumably due to proximity to a quantum critical point or phase, and the incoherent transport associated with it. Nevertheless, we agree with the Referee that it would be helpful to discuss different possible origins of T-linear resistivity and to examine their applicability in overdoped cuprates. We have therefore added one paragraph at the top of page 3 with a discussion about different origins of T-linear resistivity and a short explanation as to why we think that incoherent transport at the maximum dissipation limit is the most plausible.

Changes made to manuscript:
New paragraph added at the top of page 3 describing the different possible origins of a Planckian T-linear resistivity, as well as several new references: Bruin et al, Science vol. 339, 804 (13), Mousatov et al, PNAS vol. 117, 2852 (20), Zaanen, Nature vol. 430, 512 (04), Hartnoll, Nat. Phys. vol. 11, 54 (14), Zaanen, SciPost Phys. vol. 6, 061 (19) and Ledbetter, Physica C vol. 235-240, 1325 (94). Several minor changes throughout the introduction were also made for clarity.

Referee comment #1.2:
The attached specific heat data is extremely useful. But the figure gives the misleading impression that the constant normal-state electronic specific heat as a function of doping has been directly measured. which is most probably not the case. It would be extremely helpful to tell the reader is that what has been measured is the relative amplitude of the JUMP in specific heat normalized to the normal state assuming that the normal state specific heat remains unchanged after the subtraction of a phononic background. Adding this detail would make the figure less impressive, less surprising, and more understandable. The critical temperature dives towards zero with overdoping and so does the entropy difference between the normal and superconducting states.

Response:
We thank the Referee for raising this important point, but according to the thesis of Matthew Wade, and to the associated published article [Wade et al., J. Supercon. vol. 7, 261 (94)], the constancy of the normal-state electronic specific heat across the entire doping range is in fact a robust finding. To quote from the thesis of Matthew Wade (here, δ refers to the oxygen off-stoichiometry, γ_n is the normal state electronic specific heat coefficient, fig. 6.7 is identical to the main panel of figure 4 in the J. Supercon. article, while fig. 6.9 is the figure uploaded in the previous round of refereeing.):

“Not only does γ_n(T) appear to be independent of temperature, but from fig. 6.7 we can see that it also appears to show no δ-dependence within the scatter. It ought to be pointed out that although Δγ (i.e. the left hand axis on fig. 6.7) is reliably known, the value of γ (i.e. right hand axis on fig. 6.7) is less certain. The fact that γ_n shows no dependence on δ however follows from Δγ rather than γ and so is reliably known.”

“The scatter seen in γ_n(T) as a function of δ may well be the result of an inadequate phonon correction arising from a small error in δ. As a result of the rather involved process used to determine δ it is difficult to estimate the incurred error but it is quite likely to be less than the small ad hoc (|δ| < 0.005) shifts which would be required in order to force γ_s - γ_n = 0 immediately above T_c in fig. 6.7. Reanalysing the raw data letting δ vary slightly to ensure entropy conservation below T_c and γ_s = γ_n above it gives the plot seen in fig. 6.9.”

In light of the Referee’s comment, however, we have added an extra sentence stressing that the doping independence of γ_n is robust to small changes in the phonon specific heat.

Changes made to manuscript:
Extra sentence added to Appendix A.7 clarifying that the doping independence of γ_n is robust to small changes in the phonon specific heat.

Reviewer #2
We thank the Referee for their thoughtful report on our manuscript and for acknowledging the potential interest and impact of our findings. The Referee raised a couple of points that we address below:

Referee comment #2.1:
The argument n_H + n _s = 1+p in Tl2201 is strongly dependent on the validity of the analysis of Hall resistivity in Ref. 14 where the Hall coefficient has been found to change by about a factor of two between p=0.27 and p=0.23, much larger than the relative change in 1+p in the same doping range. The magnitude of the Hall coefficient n_H in Ref 14 has been inferred form the high field measurement which show a strong field and temperature dependence of R_H, with the variance comparable with with the factor of 2 required to distinguish reliably the value of n_H(p) and 1+p at p=0.23. Although the inferred value of n_H(p=0.27) in Ref 14 is consistent with the quantum oscillations measurements (Ref 29) in Tl2201, no quantum oscillation measurements exist at p=0.23 and the the uncertainty of the value of n_H in Ref. 14 cannot be reliably established.

Response:
We thank the Referee for raising their concern that the uncertainty of the value of n_H in Ref. [14] cannot be reliably established. It is important to realize, however, that the field dependence of R_H in Tl2201 (in the field-induced normal state) diminishes with decreasing temperature (Fig. 1 of Ref. [14]) and for most samples, essentially vanishes at the lowest temperatures. Moreover, while the Referee emphasizes the factor of 2 change in n_H between p = 0.27 and 0.23, it should be acknowledged that this variation in n_H is part of a larger variation of a factor of 4 between p = 0.27 and p = 0.20. These changes are much larger than the changes in n_H due to field or temperature (in the relevant ranges of interest – Fig. 2 of Ref. [14]). These large variations in n_H(0) with doping in Tl2201 are found to be consistent with those determined in Bi2201, reported both in Ref. [14] and in Lizaire et al. [arXiv:2008.13692] where the field dependence is even weaker (due to a shorter mean-free-path). Significantly, in the Lizaire study, the change in R_H(0) with doping is much larger than the changes in either temperature or magnetic field strength. Hence, we are confident that the changes in n_H(0) across the strange metal regime reported in Ref. [14] are indeed greater than the changes in 1 + p. In response to the Referee’s comment, we have added a short sentence emphasizing this point.

Changes made to manuscript:
Sentence added to the discussion of the Hall number on page 3 emphasizing that the variation in n_H(0) with doping in Tl2201 and Bi2201 is much greater than the variation in field at the lowest temperatures.

Referee comment #2.2:
The main point argued in this manuscript does present several interesting possibilities for the understanding of the physics of cuprates, and will be met with interest by its readers. However, the argument in the Manuscripts relies on several weak interpretational steps (BCS-like interpretation of n_s, Fermi-liquid-like interpretation of Hall resistivity) to make a strong leap in their interpretation of the nature of the superconducting state. In particular, it is not clear what is the basis for interpretation of penetration depth measurements in terms of electronic density in the absence of any quantitative description of the superconductivity emerging from incoherent excitations. It is also not clear how big is the interpretational error bar on n_H in Ref. 14.
That said, the range of experimental studies collected together in this manuscript will be of broad interest and will stimulate further discussion of the physics of cuprates.

Response:
Again, we thank the Referee for raising this point. The main purpose of our present manuscript was not to provide a microscopic description of the superconductivity emerging from incoherent excitations, but rather to highlight the anti-correlation between n_H(0) and n_s(0) as well as the empirical relation n_s(0) + n_coh = 1 + p. We accept that certain assumptions have been made in order to arrive at these (cor)relations, but throughout the manuscript, we have strived to be fully transparent and explicit in the assumptions that we have made. It is indeed our hope and expectation that these relations will be of broad interest and will stimulate a response from the theoretical community working on the origins of superconductivity in the cuprates and in other unconventional superconductors.

Our estimates of n_s(0) were made using the London equation and thus do not require any recourse to BCS theory. Moreover, the excellent agreement between the estimates of n_s(0) and the change in the electronic specific heat coefficient below T_c suggests strongly that our approach (interpreting penetration depth measurements in terms of electronic density of states) is valid. The conversion of the Hall resistivity into a carrier density may seem at first sight difficult to justify in a system close to the Mott insulating state. However, Ando and co-workers measured the Hall response in lightly-doped LSCO and found that the low-T Hall number n_H(0) ~ x (p) for 0.01 < x (p) < 0.08 [Ando et al., PRL vol. 92, 197001 (04)]. Beyond x = 0.08, this correspondence breaks down, presumably due to the emergence of charge (stripe) order in the intermediate doping range 0.09 < x (p) < 0.16. At high doping (p > 0.27), the relation n_H(0) ~ 1 + p has been confirmed in Tl2201. Hence, at both ends of the phase diagram, the relation between n_H(0) and the number of mobile holes appears to hold and it thus seems reasonable to assume that in the crossover regime 0.16 < p < 0.27, the value of n_H(0) also provides a good estimate of the effective carrier density. The only assumptions made in the present study in determining the density of coherent carriers is that the Fermi surface can be decomposed into distinct coherent and incoherent sections and that the incoherent part of the Fermi surface has no intrinsic Hall response, as speculated in Ref. [14]. This latter assumption is based in part on the observation that the in-plane MR in overdoped cuprates is insensitive to both the level of impurity scattering and the orientation of the magnetic field with respect to the applied current. Both observations imply that the quadrature MR scaling found in overdoped cuprates does not stem from cyclotron motion (i.e. from the Lorentz force).

In response to the Referee’s comment, a new paragraph has been added in Appendix A.1 on page 14 incorporating the above discussion. We have also added a phrase to the sentence at the end of Section 2 emphasizing that the empirical relations introduced here do not rely on the exact microscopic origin of the non-Fermi-liquid, strange metal component.

Changes made to manuscript:
Paragraph added in Appendix A.1 on page 14 discussing the relation between n_H(0) and p in underdoped LSCO and in overdoped Tl2201 and its relevance to estimates of the carrier density extracted from R_H(0) measurements within the strange metal (crossover) regime.
Phrase added to sentence at the end of Section 2 that now reads ‘This simple empirical relation is our central finding, one that does not rely on knowing the exact microscopic origin of the non-FL, strange metal component’.