Possible superconductivity from incoherent carriers in overdoped cuprates

Dear Editor,

We thank the Referee for their report, for recognizing the originality of our idea and for highlighting some areas of ambiguity and incompleteness that we address in this rebuttal and in the revised manuscript. The Referee also argued for a stronger connection to other strange metals or superconductors. We had chosen to restrict the current paper to just those two cuprate families (Tl2201 and LSCO) (a) which are single-layer, single-band cuprates, (b) whose doping range covers the entire strange metal regime and (c) for which sufficient data on the superfluid and Hall densities exist. The single-band nature of the cuprates makes the presented analysis much more straightforward than for multiband superconductors such as the iron pnictides or chalcogenides Moreover, given the unique character of the cuprates, it is unclear whether the scenario presented here would be applicable to other strange metals. Hence, we do not feel that a stronger connection to other strange metals or superconductors would necessarily strengthen the manuscript and have thus maintained the focus of the manuscript on the two highlighted cuprate families.

In the following, the original comments from the Referee are copied in red, our response is in black, while the action we have taken is summarized in blue. All changes to the actual manuscript and appendices have also been highlighted in blue. Finally, a summary of the corrections is provided at the end of the rebuttal.

Referee comment:
i) What is “an incoherent carrier”? The authors do not define what they mean by this expression but seem to suggest that it refers to entities displaying Planckian dissipation. What about electrons in copper, which copper display a T-linear resistivity with a Planckian prefactor (see ref. 68)?

Response:
We are grateful to the Referee for pointing out that we had not defined the term ‘incoherent carrier’ with sufficient clarity in our original manuscript. Hence, in the introduction of the revised manuscript and below, for the Referee’s convenience, we provide a more extensive discussion of this term.

The precise nature of these incoherent carriers is not known at present and there are many alternative definitions of incoherence in the literature. Hartnoll [Nat. Phys. vol. 11, 54 (15)], for example, defines (non-quasiparticle) incoherent transport as that controlled by the collective diffusion of energy and charge rather than by quasiparticle or momentum relaxation. Other researchers associate the term directly to a T-linear resistivity with a slope consistent with the Planckian dissipation limit. This notion, however, typically refers to high temperature T-linear resistivity and as the Referee correctly pointed out, the scattering rate in copper also exhibits a similar ‘Planckian’ form at high temperatures, in this case due to electron-phonon scattering. As the temperature is lowered and the phonon modes begin to freeze out in Cu, however, the inelastic scattering rate plummets. Indeed, the T-dependent scattering rate in Cu at 10 K is three orders of magnitude smaller than it is in cuprates (previously Ref. [68], now Ref. [71]) and thus bears no relation to the large T-linear coefficient seen in cuprates at low T. Moreover, there is no quadrature magnetoresistance (MR) linked to the Planckian dissipation in Cu, nor any other evidence for strange metal transport. The key point for our discussion is that the T- (and H)-linear MR in overdoped curpates persist both down to the lowest temperatures measured and over a wide doping range.

A coherent quasiparticle is one whose decay rate Γ is smaller than its excitation energy ε which, in a Fermi-liquid (FL), is guaranteed at sufficiently low temperatures (energies) by the relation Γ ~ ε^2. In strange metals, on the other hand, quasiparticle decoherence is implicit in the fact that Γ (or ρ) varies linearly with T and ε at the lowest energy scales. In this regard, it is worth mentioning the angle-resolved photoemission spectroscopy study of overdoped LSCO by Chang et al., [Nat. Commun. vol. 4, 2559 (13)]. Their study showed that true FL quasiparticles with Γ ~ ε^2 exist only around the nodal regions of the Fermi surface, while near the anti-nodal region, Γ ~ ε, signifying low-lying excitations with a vanishing quasiparticle residue. The resulting picture of anisotropic quasiparticle breakdown, including the extent of the FL-like and non-FL-like sectors on the Fermi surface, is strikingly similar to the one invoked in the present manuscript.

In addition to the above, very general definition, we also consider here a more empirical definition for incoherent carriers, namely electronic states that exhibit signatures of non-orbital, but nonetheless ‘metallic’ transport; specifically a H-linear MR at high field strengths that is insensitive to both field orientation and impurity scattering rate and a vanishing Hall conductivity. This insensitivity to field orientation and disorder is in marked contrast to expectations from conventional Drude or Boltzmann transport theory in which the magnitude of the MR is dictated both by the projection of the Lorentz force and by the product of the cyclotron frequency and scattering rate ωcτ, where τ includes both the inelastic and elastic (i.e. impurity) scattering rates. The observed H/T scaling of the MR also implies a direct link between this non-orbital MR and the T-linear resistivity, and by association, a direct link between the incoherent carriers and Planckian dissipation. In this way, the various definitions of incoherent transport (signatures of non-orbital response in MR and linearity of Γ/ρ on T and ε at low T, ε) become inter-connected. While Planckian T-linear resistivity is not, by itself, a sufficient condition for incoherence, we hope that it is now clear how it can be associated with the incoherent carriers.

Revision to manuscript:
A separate paragraph outlining of our definition of the term ‘incoherent carrier’ is now included in the introduction of the revised manuscript. A couple of sentences have also been added discussing the similarity between ARPES on LSCO by Chang et al..

Referee comment:
ii) Berg et al. in PNAS 117, 2852 (2020) invoke incoherent carriers in a well-defined fashion. These are “hot” carriers, which are almost classical due to their lower degeneracy temperature or shorter lifetime. Interestingly, the strange metal Sr3Ru2O7, like overdoped cuprates displays T-linear resistivity with a Planckian prefactor (ref. 68) despite its well-defined Fermi surface.

Response:
Inspection of the article by Mousatov, Berg and Hartnoll reveals no mention of the term ‘incoherent’ nor ‘incoherence’. Hence, one cannot use the description of ‘hot’ carriers in this paper as a precise definition of incoherent carriers. The key element of the Mousatov model is a large density of states (DOS) on one of the bands, as found near a van Hove singularity (vHs). When the Fermi level is tuned through this region of high DOS, the resistivity can remain T-linear down to the lowest temperatures. In this picture, the high DOS in the hot (h) regions generate significant scattering of the ‘cold’ (c) quasiparticles located elsewhere on the Fermi surface through what the authors termed cc-ch scattering.

In the cuprates, while the Fermi level is known to cross the vHs in LSCO near p*, thereby generating similar conditions for cc-ch scattering, the vHs in Tl2201 is located at a doping level far beyond the end of the superconducting dome (see Ref. [14] of present manuscript). Moreover, while in overdoped LSCO, the Fermi level is tuned away from the vHs with increasing hole doping, the opposite is true in Tl2201, yet the resistivity of both families evolves in a very similar way with doping. This implies that the Mousatov mechanism for generating T-linear resistivity is not applicable to overdoped cuprates.

Revision to manuscript:
No change to the manuscript has been made in response to this comment.

Referee comment:
iii) Strontium titanate has also a superconducting dome. Like cuprates, its superfluid density as a function of doping decreases as soon as the peak Tc is attained (Collignon et al., PRB, 96, 224506 (2017)). This appears to be a generic feature of any superconducting dome. If for whatever reason, adding an electron, instead of enhancing Tc (because of enhanced density of states) pulls it down, then you expect the superfluid density to do the same, because adding an electron reduces foremost the ratio of the superconducting gap to the carrier lifetime. Why should one exempt cuprates from this general rule?

Response:
We thank the Referee for bringing to our attention the interesting paper by Collignon et al. This very careful study of the variation of the superfluid density as a function of doping in SrTi1-xNbxO3 shows clearly how ns(0) decreases relative to nH(0) – the Hall carrier density – due to a crossover from the clean to the dirty limit. Such a crossover has frequently been invoked to explain the drop in ns(0) with doping in overdoped (OD) cuprates, as described in detail in appendix C. However, the evolution of ns(0) and nH(0) in SrTi1-xNbxO3 and in the cuprates has one crucial difference. While in both cases nH(0) increases with increasing doping beyond the optimal doping, in contrast to SrTi1-xNbxO3, the value of nH(0) in cuprates (Ref. [14]), is significantly smaller than the total carrier density expected from the full Luttinger count. It is this feature of the Hall response in OD cuprates that suggests the presence of two distinct charge sectors; something that has not, to our knowledge, been reported in SrTiO3. The different scenarios for SrTiO3 and the cuprates likely stem from the fact that the parent state is a band insulator in the former and a Mott insulator in the latter. One of the key motivations of the present manuscript is to present an alternative scenario to the dirty-limit BCS case that, for the reasons highlighted above, may be inapplicable to the OD cuprates. The seminal work by Božović et al. (Ref. [7]) also raises a serious challenge to the appropriateness of the dirty-limit picture for OD cuprates, as does the quantitative comparison of the effects of disorder (via the residual resistivity) on Tc, again highlighted in appendix C.

In light of the Referee’s comment, an extra discussion section has been added highlighting the difference between SrTiO3 and the cuprates, along with a citation to the paper by Collignon et al.

Revision to manuscript:
Extra sentences added at the bottom of page 5 highlighting the difference between SrTiO3 and the cuprates, along with a reference to the paper by Collignon et al..

Referee comment:
iv) The very first sentence of the abstract qualifies the non-superconducting state of overdoped cuprates as a strange metal. Doesn’t this contradict previous works by Hussey and collaborators, such as PRB 68, 100502 (2003)? I am also surprised by the logical flow. The paper begins with a conjecture and presents arguments in favor of this conjecture, instead of starting with observations and ending with a conclusion.

Response:
We apologise for the confusion in the original wording of our opening sentence ‘The non-superconducting state of overdoped cuprates is conjectured to be a strange metal comprising two distinct charge sectors’. Firstly, we are not referring to the state beyond the end of the superconducting dome (i.e. a non-superconducting cuprate, which does indeed show correlated but conventional Fermi-liquid behaviour). Rather, we are referring to the normal state of overdoped but nonetheless superconducting cuprates, i.e. the state above Tc or above Hc2. Hence, there is no contradiction with the previous work reported in PRB 68, 100502 (2003), which focused on a doping level beyond the superconducting dome. Secondly, in our opening statement, we used the word ‘conjecture’ inappropriately. There is now compelling evidence for the existence of two distinct contributions to the in-plane transport in overdoped cuprates, firstly from our high-field study of the Hall number [Nat. Phys. (2021) doi:10.1038/s41567-021-01197-0] and secondly from our more recent magnetoresistance scaling study [Nature, in press (2021) [2012.01208] (arxiv.org)].

As stated clearly in the Conclusion section, we took these observations as our initial motivation and then set out to address the question: which sector or sectors form the superconducting condensate? Hence, we do not believe there is any issue with the logical flow of our paper. However, in response to the Referee’s welcome comment, we have reworded the opening sentence of the abstract.

Revision to manuscript:
Opening sentence of the abstract changed to: ‘There is now compelling evidence that the normal state of superconducting overdoped cuprates is a strange metal comprising two distinct charge sectors,…’

Referee comment:
v) Fig. 1A compares two normalized quantities and finds a striking correlation. However, it raises numerous questions. What is the amplitude of superfluid density in physical units (cm-3 for example)? This information cannot be found in ref. 7. So can one compare it with the Hall density of the normal state? What is the amplitude of the residual zero-temperature specific heat and how does it evolve with doping? Was it measured on single crystals? At what temperature? Inside the superconducting state? How can rule out that this zero-temperature residual term in not caused by uncontrolled disorder? Given that the source of this information is unpublished (ref. 31) and the importance of this figure in the authors’ scenario, I recommend a more detailed presentation of the data condensed in this figure.

Response:
We thank the Referee for highlighting our oversight in not providing a more detailed description of the specific heat data contained within Figure 1A. In response to this, we have added a new appendix (A7) describing in more detail the specific heat analysis plotted in Figure 1A of the main manuscript. In support of this, a new table (Table 5) has been added showing the absolute values of the residual specific heat reported by Wade in Ref. [31]. The relevant figure is attached (Wade_specific_heat_plot.pdf) for the benefit of the Referee and of the interested reader.

The Referee also asks what is the amplitude of the superfluid density in Figure 1A in physical units, but then refers to Ref. [7] of the manuscript. Figure 1A presents data on overdoped Tl2201 while Ref. [7] describes measurements of the superfluid density in overdoped LSCO. For the sake of completeness, we address this question for both systems.

In OD Tl2201, measurements of 1/λab(0)^2 – the inverse square of the in-plane penetration depth - have been reported by three separate groups and their absolute values are listed in Table 4. Eq. (2) of the main manuscript then gives the equation linking the value of 1/λab(0)^2 to the superfluid density ns(0), normalized to a single unit cell. Again, these values are listed in Table 4. In all cases, we have assumed, as explained in the main text, that the effective mass m* = 5.2 me consistent with both quantum oscillation and specific heat studies. Hence, we do not believe there is any ambiguity around how the values of ns(0)/(1+p) plotted in Figure 1A were obtained. In response to the comment about units, however, wherever ns(0), nH(0) and ncoh are introduced in the text as stand-alone quantities, we now stress that these values have been normalised to a single Cu site.

For OD LSCO, the values of 1/λab(0)^2 listed in Table 9 (now Table 10, see below)) and quoted from Ref. [37] are the as-measured values. In Ref. [7], values for the superfluid phase stiffness ρs are quoted in units of Kelvin, but these are determined directly from the as-measured 1/λab(0)^2 values and the scaling prefactor (A) is quoted in Ref. [7], making it possible to ‘re-engineer’ the 1/λab(0)^2 values directly from the ρs values plotted in Figure 2 of Ref. [7]. Conversion of 1/λab(0)^2 into ns(0) is not as straightforward in LSCO as it is in Tl2201, due to the strong curvature of the Fermi surface, proximity to the vHs and resultant strong anisotropy of the effective mass. Our method of doing this is explained at length in the manuscript.

Revision to manuscript:
New appendix (A.7) has been added describing in more detail the specific heat analysis plotted in Figure 1A of the main manuscript. In support of this, a new table (Table 5) has been added showing the absolute values of the residual specific heat reported by Wade in Ref. [31]. All subsequent tables have been re-labeled. For further clarification of our analyses of LSCO, an extra paragraph and two citations have been added to appendix B.1 to discuss the correction factor used to account for discrepancies between ARPES- and tight-binding derived Fermi surfaces. Finally, wherever ns(0), nH(0) and ncoh are introduced in the text as stand-alone quantities, we now stress that these values have been normalised to a single Cu site.

List of corrections:

In response to Referee’s comments (labelled by roman numerals):

(i) Separate paragraph on page 3 outlining of our definition of the term ‘incoherent carrier’ is now included in the introduction of the revised manuscript.

(ii) No change to the manuscript has been made in response to this comment.

(iii) Extra sentences added on pages 5 and 6 highlighting the difference between SrTiO3 and the cuprates, along with a reference to the paper by Collignon et al..
(iv) Abstract: Line 1: changed to: ‘There is now compelling evidence that the normal state of superconducting overdoped cuprates is a strange metal comprising two distinct charge sectors,…’

(v) New appendix (A.7) added describing the specific heat analysis plotted in Figure 1A of the main manuscript. In support of this, a new table (Table 5) also added showing the absolute values of the residual specific heat reported by Wade in Ref. [31]. All subsequent tables have been re-labeled.

(v) Wherever ns(0), nH(0) and ncoh are introduced in the text as stand-alone quantities, we now stress that these values have been normalised to a single Cu site.

Additional modifications to improve clarity

Page 8: A couple of sentences added discussing the similarity between ARPES on LSCO by Chang et al.. Sentence referring to this work also added to the figure caption of Figure 2.

Page 11: Extra sentences and references added in discussion of the observation of quantum oscillations in highly overdoped Tl2201 to emphasise the fact that the observation of QOs is no longer seen as a ‘smoking gun’ for the existence of a fully coherent Fermi surface.

Page 23: Extra sentences and references added to appendix B.1 to discuss the correction factor used to account for discrepancies between ARPES- and tight-binding derived Fermi surfaces. Finally, wherever ns(0), nH(0) and ncoh are introduced in the text as stand-alone quantities, we now stress that these values have been normalised to a single Cu site.

Attachment:

Wade_specific_heat_plot.pdf