Upper critical field reaches 90 tesla near the Mott transition in fulleride superconductors

Controlled access to the border of the Mott insulating state by variation of control parameters offers exotic electronic states such as anomalous and possibly high-transition-temperature (Tc) superconductivity. The alkali-doped fullerides show a transition from a Mott insulator to a superconductor for the first time in three-dimensional materials, but the impact of dimensionality and electron correlation on superconducting properties has remained unclear. Here we show that, near the Mott insulating phase, the upper critical field Hc2 of the fulleride superconductors reaches values as high as ∼90 T—the highest among cubic crystals. This is accompanied by a crossover from weak- to strong-coupling superconductivity and appears upon entering the metallic state with the dynamical Jahn–Teller effect as the Mott transition is approached. These results suggest that the cooperative interplay between molecular electronic structure and strong electron correlations plays a key role in realizing robust superconductivity with high-Tc and high-Hc2.

Let us start with Fig. 1c. This summarizes much of the previous work of the authors and others and it is important as a point of orientation. But the meaning of the various things are not well and explicitly explained. What is the grey line bounding the Mott insulating regime? Is it a line of first order transitions or is it a crossover line? This should be clearly stated, and the criterion for drawing it should be stated. In particular, the points marked by + are never defined -or if they are I missed it. Indeed, since the "Mott Insulator," in the sense used here, is not a distinct phase of matter, it needs to be defined by some criterion. It would be great if the precise criterion used could be stated. The JT metal is likewise a crossover phenomenon -one that I know the authors are very fond of. I am not sure I am convinced that it is a useful notion, but useful or not it should be sharply defined. Suppose you show me the requisite piece of experimental data -I should be told how to look at that piece of data and tell whether I am looking at a characteristic measurement of the metal regime of the JT metal regime. The sharper the definition, the happier I would be. I think I know what the definition of AFI is in this phase diagram, but even that is not stated and should be.
Still on Fig. 1c, let me make one more suggestion. Phase diagrams are very useful ways of summarizing knowledge. I think it would be good to add the new data to this figure. What I would suggest is having a curve showing H_{c2}(0) on the same figure, with a scale shown on the right edge. Then this one figure would contain the principle take-away message of the paper plus a summary of what is known from before.
On the analysis -I am not exactly sure why it is obvious that BCS theory in the "dirty limit" is what we should be thinking about. Indeed, I would suggest that, in order to incorporate some of the explicit discussion I suggested, cut out much of the standard BCS related discussion and include a good reference instead. Both H_c2^orb and H_P^BCS can be defined without justifying the details H_c2^orb \equiv 0.69 T_c H^\prime_c and H_P^BCS \equiv 1.84 T_c. (By the way, I would suggest a better symbol for this -how about just H_P for "Pauli limit" and also in this defining equation it should be said that this is for T_c measured in K and H_P in Tesla.) There are a few other features of the SC state that would be great to mention if the data exists. Firstly, it would be nice to have a more direct (spectroscopic) measure of \Delta_0 either from tunneling or from optics. Does such a thing exist? Secondly, it would be wonderful to have information about the superfluid stiffness -especially at T \to 0. Does this exist? Is there any sense that the high inferred values of 2\Delta/T_c could be associated with a growing importance of phase fluctuations, or is this simply a peculiar strong-coupling limit of BCS theory.
Finally, let me put in a complaint -this is my own perspective and out of step with most people in the field, so the authors should feel free to neglect this. I think that many of the battles in the filed are based on false dichotomies. For instance, in the present problem, there can be no doubt that both strong correlations and electron-phonon coupling play a significant role in this. Let me illustrate this by reference to the published literature -a reference I include NOT to urge the authors to reference it, but that this is a place that makes the general point clear. In PRL 69, 212-212 (1992), the issue of the selfconsistency of a pure Eliashberg-based treatment of the problem was considered, and it was shown that because the bands in C_60 are so narrow, that there is not sufficient retardation to permit an effective attraction to emerge from a problem with net repulsive interactions -in other words, a model that ignores strong correlation effects is internally inconsistent. Similarly, the strong renormalization of phonon JT phonon modes proves that they are coupled to the interesting electronic degrees of freedom. There are surely multi-band effects. There is no doubt that there are strong coupling effects, but the fact that the chemical potential does not move from the band center (as best I understand) proves that system is nowhere near the real-space pairing (on-molecule pairs) limit. The issue should not always be framed as a sharp either-or. The issue is what is the simplest context to obtain a qualitative understanding of the basic physics. For instance, even if the electron-phonon coupling and JT distortion play a quantitative role, can one understand the basic phenomena ignoring this? Or conversely, even though the usual pseudo-potential analysis of the Coulomb interactions is clearly flawed, if one simply considered a model in which the Coulomb repulsions were set to be small, would this lead to qualitative predictions that can be falsified. (I think they can be, but this is how I would frame the discussion.) Reviewer #2 (Remarks to the Author): The authors present an experimental and theoretical study of the temperature dependence of the upper critical field Hc2 in superconducting fullerides. Hc2(T) was measured in different fullerides by a radio-frequency technique in a pulsed magnetic field up to 62T and the T dependence was extrapolated to T=0 following the WHH theory and assuming the dirty limit. The values found (up to 90T) are among the highest found for 3D superconductors. In particular, the study of RbxCs3-xC60 systems which realize superconductivity near the Mott transition, showed that electron correlations appear to strengthen the superconducting state enhancing the coupling, Tc and Hc2. The article is interesting, the achieved results are sound and clearly presented. As such it is suitable for publication in Nature Comm. However an issue must be clarified by the authors: the compounds displaying the most interesting behaviour are the fullerides RbxCs3-xC60 which are located in the highly correlated JT metal region and which display the high value of Hc2(0) and consequently a quite small coherence length, which is, as remarked in the manuscript, of the order of the lattice parameter. However their fractional stoichiometry creates local compositional inhomogeneities which, if on a length-scale larger or comparable with the coherence length, as in this case, can deeply affect the superconducting phase. The influence of this structural disorder has been ignored by the authors. Therefore they should suitably justify their procedure and show that this disorder is not expected to affect their results.
After this issue has been duly clarified in the manuscript I can recommend it for publication in Nature Comm.
I like to think that the changes made in response to my suggestions helped improve it, but the main thing is that I do not want (and did not want to) delay publication of beautiful and important results. Congratulations on a lovely paper.
Reviewer #2 (Remarks to the Author): I consider the authors' explanation satisfactory so I recommend the revised version of the manuscript for publication on Nature Communications. We thank Reviewer #1 for the in depth review and constructive comments that are highly encouraging for our study. Our reply to the questions is the following: However, I think the presentation could be improved by removing some of the theoretical discussion -which I think is largely extraneous -and focusing in more closely on making clear and explicit the central new points. I will try to explain what I have in mind, but first let me make it clear that I think the results are so interesting that the authors have "earned" the right to present them any way they want, so these suggestions should be viewed as just SUGGESTIONS, not requirements for acceptance.
We agree that the theoretical discussions are largely extraneous. This is a very helpful suggestion for improving our manuscript.
We removed the theoretical discussions in the last paragraph of the original manuscript. Then, we explicitly stated the central new points of this work, i.e., our results imply that the presence of both molecular characteristics that is absent in the atom-based superconductors, involving the dynamical Jahn-Teller effect and resulting renormalization of electronic structure, and electron correlation effects play a significant role for both high-Tc and high-Hc2 in the fullerides. We will explain this point later as a response to the last comments by the referee. We agree that the explanation of Fig. 1c is largely missing. In accordance with the Reviewer #1's comments, we revised Fig. 1c and its caption.
The grey line in Fig. 1c represents the crossover line, not a line of phase transition, from Mott-Jahn-Teller insulator (MJTI) to Jahn-teller (JT) metal (We changed "Mott insulator" to MJTI. We will explain the definition of MJTI later). We explicitly mentioned this in the caption of Fig. 1c in the revised manuscript.
The points marked by + are the crossover temperature obtained form the X-ray powder diffraction, nuclear magnetic resonance spectroscopy, and infrared spectroscopy (Ref. 9).
We defined MJTI as an electron-correlation-driven insulating state accompanied by the intramolecular dynamic JT effect distorting the C60 -3 anions and stabilizing the low-spin (S = 1/2) states that give rise to an antiferromagnetic insulating (AFI) state at low temperatures. JT metal is defined as a metallic state where the dynamical JT distortions persist.
Before change: (c) Electronic phase diagram of fcc fullerides. In the metallic regime, gradient shading from green to orange schematically illustrates the conventional metal to Jahn-Teller metal crossover. We completely agree with the ideas put forward for improving Fig. 1c. We revised Fig. 1c accordingly.
On the analysis -I am not exactly sure why it is obvious that BCS theory in the "dirty limit" is what we should be thinking about.
The mean free path estimated form the transport and optical measurements is as small as, or even smaller than, the intermolecular distance. Therefore, the mean free path is much smaller than the estimated GL coherence length, indicating that the fullerides are in the dirty limit. We explicitly mention this in the revised manuscript.
In the second paragraph of "Results" section, we added sentences as below.
"It should be noted that the fulleride superconductors are in the dirty limit, ℓ ≲ (ℓ and are the mean free path and Pippard coherence length, respectively), as demonstrated by transport and optical measurements [16,17]. The orientational disorder of the C 60 -3 anions can account for the short ℓ , which is comparable to the intermolecular separation." Indeed, I would suggest that, in order to incorporate some of the explicit discussion I suggested, cut out much of the standard BCS related discussion and include a good reference instead. Both H_c2^orb and H_P^BCS can be defined without justifying the details H_c2^orb ¥equiv

T_c H^¥prime_c and H_P^BCS ¥equiv 1.84 T_c. (By the way, I would suggest a better symbol for this -how about just H_P for "Pauli limit" and also in this defining equation it should be said that this is for T_c measured in K and H_P in Tesla.)
We agree the referee's suggestions. We revised the standard BCS related discussion and cited references.  As the referee pointed out, we should take into account both the electron-phonon coupling and electron correlations. Indeed, the observed strong-coupling effect cannot be explained solely by the electron-phonon coupling or electron-electron interactions. This is explained as follows.
First, in the scenario of electron-phonon coupling, intermolecular phonons (with energy ~ 100 cm -1 ) are required as a pairing interaction to account for very large 2Δ0/kBTc up to 6 at large V. On the other hand, the weak-coupling value (2Δ0/kBTc ~ 3.5) can be explained only by intramolecular phonons, involving JT phonon modes (with energy ~ 1000-1500 cm -1 ). Therefore, large enhancement of 2Δ0/kBTc requires a crossover of the distinct phonon modes that are active for the pairing. This is highly unlikely since the intramolecular phonon modes are always present over the phase diagram.
Second, the analysis of the previous specific heat measurements (ref.9  We thank Reviewer #2 for the in depth review and constructive comments that are highly encouraging for our study. Our reply to the questions is the following: However an issue must be clarified by the authors: the compounds displaying the most interesting behaviour are the fullerides RbxCs3-xC60 which are located in the highly correlated JT metal region and which display the high value of Hc2(0) and consequently a quite small coherence length, which is, as remarked in the manuscript, of the order of the lattice parameter. However their fractional stoichiometry creates local compositional inhomogeneities which, if on a length-scale larger or comparable with the coherence length, as in this case, can deeply affect the superconducting phase. The influence of this structural disorder has been ignored by the authors. Therefore they should suitably justify their procedure and show that this disorder is not expected to affect their results.
After this issue has been duly clarified in the manuscript I can recommend it for publication in Nature Comm.
This is a helpful and new perspective which we have included in the paper as detailed below.
Physical and chemical pressure studies on fcc-Cs3C60 show essentially the same behaviors in the superconducting properties, including Tc and 2Δ0/kBTc (Refs. 7,9,25). Moreover, pressure studies on A15-Cs3C60 and fcc-Cs3C60 also exhibits the comparable behaviors to those in chemically-pressurized fcc-Cs3C60, i.e., RbxCs3-xC60, in both the normal state properties, such as the presence of JTM state near the Mott transition, and superconducting properties (Ref. 9,12,25,26). In contrast to RbxCs3-xC60, A15-Cs3C60 is free from both the fractional stoichiometry and orientational disorder of C60 -3 anions. Therefore, we conclude that fractional stoichiometry does not affect the present results.
On the other hand, the shortest length scale of the structural disorder is most likely determined by the orientational disorder (merohedral disorder) of C60 -3 anions (Potocnik et al., Chem. Sci. 5, 3008 (2014)). There are two equal C60 -3 orientations, and they are randomly and equally distributed at low temperatures. The orientational disorder is present even in the stoichiometric compounds such as K3C60. The length scale of the orientational disorder is roughly estimated to be of the order of the intermolecular distance (about 9 Å) or intermolecular C-C spacing (about 3 Å). This is consistent with the mean free path ℓ estimated form the transport and optical measurements, where ℓ is found to be smaller than the estimated GL coherence length. We explicitly mention this in the revised manuscript.
The length scale of structural disorder in the alkali-metal sites is estimated as follows. In the fcc structure of the fullerides, there are tetrahedral and octahedral sites for the alkali atoms at (1/4,1/4,1/4) and (1/2,0,0), respectively.
In RbxCs3-xC60 with x < 1, the octahedral sites are occupied only by Rb atoms and the tetrahedral sites by both Rb and Cs atoms. Therefore, the length scale of the fractional stoichiometry is comparable to the distance of the tetrahedral site, which is about 10 Å. This is also comparable to the intermolecular distance and is smaller than the GL coherence length.
In the second paragraph of "Results" section, we added sentences as below.
"It should be noted that the fulleride superconductors are in the dirty limit, ℓ≲ (ℓ and are the mean free path and Pippard coherence length, respectively), as demonstrated by transport and optical measurements [16,17]. The orientational disorder of the C 60 -3 anions can account for the short ℓ , which is comparable to the intermolecular separation."