Abstract
Controlled access to the border of the Mott insulating state by variation of control parameters offers exotic electronic states such as anomalous and possibly hightransitiontemperature (T_{c}) superconductivity. The alkalidoped fullerides show a transition from a Mott insulator to a superconductor for the first time in threedimensional 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 H_{c2} 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 strongcoupling 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 highT_{c} and highH_{c2}.
Introduction
The interplay between superconductivity and electron correlations is one of the central issues in condensed matter physics. Superconducting (SC) materials based on Mott insulators, such as twodimensional (2D) cuprates^{1} and organic chargetransfer salts^{2}, are model platforms that have been extensively studied thus far. A domelike dependence of the SC transition temperature T_{c} as a function of tuning parameters, such as carrier doping and pressure, has been discussed as a fingerprint of unconventional superconductivity^{3}. Recent physical and chemical pressure studies of Cs_{3}C_{60} have revealed that the family of cubic fullerides A_{3}C_{60} (A: alkali metal), where superconductivity emerges from the Mott insulating state driven by dynamical intramolecular Jahn–Teller (JT) distortions and strong Coulomb repulsion, is a new example of superconductors that show a domelike SC phase diagram as a function of unitcell volume V (refs 4, 5, 6, 7, 8, 9). This suggests the importance of strong electron correlation to SC mechanisms^{10} and the need for further treatment beyond conventional framework of theory^{11}. Recent study has revealed a crossover in the normal state from the conventional Fermi liquid to a nontrivial metallic state where JT distortions persist (JT metal)^{9,12}. There, localized electrons coexist with itinerant electrons microscopically and heterogeneously.
The dependence of the upper critical field H_{c2} on T_{c} is relevant to the understanding of the domelike SC phase because H_{c2} is determined by the coherence length (the size of the Cooper pair) as well as the strength of the pairing potential. Therefore, H_{c2} is also important to understand the underlying mechanism of the superconductivity. However, for the fullerides, H_{c2} as a function of V has not as yet been determined due to the very large H_{c2} and the need for high pressure to access superconductivity in Cs_{3}C_{60}.
Here we report measurements of H_{c2} using a pulsed magnetic field in Rb_{x}Cs_{3−x}C_{60}, where superconductivity appears near the Mott transition even at ambient pressure^{9}. In proximity to the Mott transition, H_{c2} is enhanced up to ∼90 T, which is the highest among cubic superconductors. We uncovered that H_{c2} and the pairing strength increase concomitantly with increasing lattice volume near the Mott transition, suggesting that molecular characteristics as well as electron correlations play important roles for realizing superconductivity with highT_{c} and highH_{c2} in molecular materials.
Results
Temperature dependence of upper critical field
H_{c2} of the fulleride superconductors (Fig. 1a) Na_{2}CsC_{60}, K_{3}C_{60}, and Rb_{x}Cs_{3−x}C_{60} (0<x3), has been measured by a radiofrequency technique in pulsed magnetic fields^{13} up to 62 T (see Methods). In Rb_{x}Cs_{3−x}C_{60} with x1, the dynamical JT distortions (Fig. 1b) persist down to low temperature and coexist with the metallic state, and superconductivity emerges from this JT metal state (V_{max}<V<V_{cr}, in Fig. 1c). Figure 2 shows temperature (T) variations of frequency shift Δf as a function of the magnetic field H for Rb_{x}Cs_{3−x}C_{60} (x=2, 0.75, and 0.35) (see also Supplementary Fig. 1). The T dependence of H_{c2}, H_{c2}(T), was determined as a point at which Δf intercepts the normalstate background (arrows in Fig. 2). H_{c2}(T) curves for A_{3}C_{60} are plotted in Fig. 3a,b for VV_{max} and V_{max}<V<V_{cr} in the proximity of the Mott transition, respectively. H_{c2}(T) increases linearly with decreasing T near T_{c} and has a tendency to saturate at low temperatures. No obvious upturn of H_{c2}(T) is found in any of the samples measured, implying that H_{c2}(T) can be understood within a simple singleband picture despite the multiband nature of the triply degenerate t_{1u} orbitals of anions, in contrast to MgB_{2} and iron pnictides where multiband and multigap behaviour with upturn or quasilinear T dependence down to T∼0 is commonly observed.
Volume dependence
In spinsinglet superconductors, H_{c2} is determined by two distinct effects, i.e., the orbital and the Pauli paramagnetic effect. The orbital limit and Pauli limit are given by and , respectively (Φ_{0}, ξ_{GL}, Δ_{0}, and μ_{B} are the flux quantum, Ginzburg–Landau (GL) coherence length, superconducting gap and Bohr magneton, respectively)^{14,15}. In a weakcoupling BCS superconductor, the Pauli limit is [T]=1.84T_{c}[K]. A simple estimation from gives ξ_{GL}=1.8–4.6 nm (Supplementary Table 1), which is comparable to the lattice constant. It should be noted that the fulleride superconductors are in the dirty limit, ℓ≲ξ_{0} (ℓ and ξ_{0} are the mean free path and Pippard coherence length, respectively), as demonstrated by transport and optical measurements^{16,17}. The orientational disorder of the anions can account for the short ℓ, which is comparable to the intermolecular separation. The relation in the dirty limit, where ξ_{0}=ħv_{F}/πΔ_{0} and (v_{F}, m, k_{F}, and N are the Fermi velocity, effective mass, Fermi momentum, and number of electrons per C_{60}, respectively) for the parabolic band approximation yield . In the extreme cases ( or ), H_{c2}(0) is determined solely by H_{c2}^{orb} or H_{P}. However, when these two quantities are comparable, H_{c2}(T) can be described by the extended WHH formula^{14}, which considers both the orbital and Pauli paramagnetic effects as well as spin–orbit scattering,
where t=T/T_{c}, =0.281H_{c2}(T)/, , , is the digamma function, and λ_{so} is the spin–orbit scattering constant. With fixed , finite α reduces H_{c2}(0), but it recovers toward the original value with increasing λ_{so}, since spin–orbit scattering suppresses the Pauli paramagnetic effect.
was estimated from the initial slope of H_{c2}(T) since the Pauli paramagnetic effect is not relevant near T_{c} (Supplementary Note 1; Supplementary Fig. 2; and Supplementary Table 1). Then, H_{c2}(T) curves were fitted with H_{P} and λ_{so} as fitting parameters. As shown by the solid lines in Fig. 3a,b, H_{c2}(T) curves are well described by equation (1). Figure 3c shows H_{c2}(T) normalized by as a function of T/T_{c}. The normalized H_{c2}(T) curves collapse into a single curve except for Na_{2}CsC_{60}, implying that the parameters α and λ_{so} are unchanged in a wide V region of the phase diagram, resulting in (α, λ_{so})=(1.5, 4.4). Figure 3d displays the evolution of H_{c2}(0) as a function of V, together with . H_{c2}(0) reaches as high as 88 T in Rb_{x}Cs_{3−x}C_{60} with x1 (VV_{max}) very close to the Mott transition. Moreover, H_{c2}(0) is clearly larger than at V>V_{max}, and the difference between H_{c2}(0) and becomes pronounced with increasing V, although T_{c} is almost unchanged near the Mott transition.
Discussion
H_{c2}(0) values reaching ∼90 T are remarkably high for 3D materials. Typical examples of 3D superconductors are cubic Nb_{3}Sn (H_{c2}(0)=30 T, T_{c}=18 K), which is well known as a material for a SC magnet^{18}, and Ba_{1−x}K_{x}BiO_{3} (H_{c2}(0)=32 T, T_{c}=28 K)^{19}. MgB_{2} exhibits strong anisotropy (H_{c2}(0)=49 T and 34 T parallel to the ab plane and c axis, respectively, T_{c}=39 K)^{18} due to its anisotropic electronic structure. H_{c2}(0) of the fullerides is even higher than that of recently discovered H_{3}S superconductors with likely a cubic structure (H_{c2}(0)≈70 T, T_{c}=203 K)^{20} despite its much higher T_{c}. In 2D systems under inplane applied fields, the orbital effect is quenched and higher H_{c2} can be expected. Very large H_{c2} compared with low T_{c} has been demonstrated in iongated MoS_{2} (H_{c2}(0)=52 T, T_{c}=9.7 K)^{21,22} and monolayer NbSe_{2} (H_{c2}(0)=32 T, T_{c}=3.0 K)^{23}. In the bulk materials, the inplane H_{c2} of the cuprates is exceptionally high at above 100 T. However, H_{c2} is no longer a thermodynamic transition line, but a crossover line due to thermal fluctuations. Contrastingly, H_{c2} in pnictides with T_{c}≃30 K is as large as that of fullerides^{24}. Therefore, our results highlight the uniquely high H_{c2} measured in the fulleride superconductors that are cubic, and thus, 3D.
To understand the underlying mechanisms for the evolution of H_{c2}(0), we estimated unknown parameters that determine H_{P} and (Supplementary Fig. 1), that is, Δ_{0} and the product of parameters in the normal state . Δ_{0} can be directly estimated from H_{P}. In Fig. 3e, the V dependences of 2Δ_{0}/k_{B}T_{c}, which is related to the strength of the pairing interaction, and (m_{0} is the bare electron mass) are shown. At low V, 2Δ_{0}/k_{B}T_{c} is comparable to the BCS weakcoupling limit value of 3.52. In contrast with the domeshaped T_{c}, 2Δ_{0}/k_{B}T_{c} continuously increases with increasing V and reaches values as large as 6, indicating a crossover from weak to strongcoupling superconductivity on approaching the Mott transition. This is in good agreement with the previous nuclear magnetic resonance results for Rb_{x}Cs_{3−x}C_{60} at ambient pressure^{9} and for both fcc and A15Cs_{3}C_{60} under pressure^{25,26}, implying universal behaviour in the fullerides. On the other hand, is almost constant, indicating that both H_{P}/T_{c} and /T_{c} are solely proportional to 2Δ_{0}/k_{B}T_{c}. These results lead to the conclusion that the enhancement of H_{c2}(0) is dominated by the strongcoupling effect developing near the Mott transition.
We here recall H_{c2}(0) of other families of highT_{c} or strongly correlated superconductors, i.e., cuprates, organic κ(ET)_{2}X, and pnictides^{24,27,28,29,30,31}, having a domelike SC phase and a proximate antiferromagnetic phase. In Fig. 4, H_{c2}(0)/T_{c} is displayed as a function of the relevant tuning parameter for each materials family. We show H_{c2}(0) for the inplane field (H ⊥ c), where the Pauli paramagnetic effect is dominating, in κ(ET)_{2}X and pnictides but show H_{c2}(0) for the outofplane field (H  c) in cuprates since there are no reliable estimates of H_{c2}(0) for H ⊥ c. A remarkable feature of the fullerides is that H_{c2}(0)/T_{c} appears to be strongly enhanced at x≤1, where the JT metal phase emerges (Fig. 1c), with retaining nearly optimal T_{c} and H_{c2}(0) values near the Mott transition. This is in marked contrast to the pnictides and cuprates. In the pnictides, H_{c2}(0)/T_{c} is almost constant across the SC dome. This is ascribed to the variation of Δ_{0}, which linearly scales with T_{c} (ref. 32), implying constant coupling strength. Moreover, in pnictides, T_{c} and H_{c2}(0) are strongly reduced upon decreasing doping, associated with the appearance of the antiferromagnetic phase. Nonmonotonic behaviour in cuprates appears with mass enhancement near p=0.08 and 0.18, which originates from phase competition between superconductivity and Fermisurface reconstruction or chargedensitywave order^{27}. This is distinct from the continuous evolution of in the fullerides (Supplementary Fig. 2), suggesting the absence of such competing states. In κ(ET)_{2}X, there is no competing phase near the Mott transition and the molecular degrees of freedom are not relevant to the superconductivity in contrast to the fullerides. Moreover, the SC pairing is most likely mediated by purely electronic interaction, in contrast to the fullerides, where there is considerable controversy because of comparable energy scales in the electron–phonon and electronelectron interactions^{33,34}. κ(ET)_{2}X shows qualitatively similar behaviour with the strongcoupling effects near the antiferromagnetic phase^{35}. However, the enhancement of H_{c2}(0)/T_{c} is much weaker than that in the fullerides. Therefore, the steep enhancement of H_{c2}(0)/T_{c} and 2Δ_{0}/k_{B}T_{c} upon entering the JT metal phase cannot be explained solely by the electron correlation effect, highlighting the uniqueness of fullerides among the highT_{c} or strongly correlated superconductors. We also emphasize that it is difficult to reconcile the strongcoupling effect with the electron–phonon coupling alone^{25}. Our results establish the importance of both molecular characteristics, absent in the atombased superconductors, involving the dynamical JT effect and the resulting renormalization of the electronic structure and electron correlation effects for both the highT_{c} and the highH_{c2} in the fullerides, as supported by the recent theoretical calculations^{33}. This provides a new perspective on realizing robust superconductivity with high T_{c} and H_{c2} in molecular materials.
Methods
Sample synthesis and characterization
Fullerene superconductors Na_{2}CsC_{60}, K_{3}C_{60}, and Rb_{x}Cs_{3−x}C_{60} (0<x3) were synthesized by solidvapor reaction method as described in ref. 9. The samples used here were identical to those in ref. 9. For Rb_{x}Cs_{3−x}C_{60} with x=0.5, 1, and 2, our samples correspond to Rb_{0.5}Cs_{2.5}C_{60} (Sample I), RbCs_{2}C_{60} (Sample I), and Rb_{2}CsC_{60} (Sample II) in ref. 9, respectively. The samples were characterized by synchrotron Xray powder diffraction and magnetization measurements. The phase fraction of the fcc phase was larger than 70% and typical shielding fraction was ∼90%.
Measurements of H _{c2}
Contactless radiofrequency (r.f.) penetration depth measurements were performed using a proximity detector oscillator technique^{13} and a pulsed magnetic field up to 62 T in Los Alamos NHMFL. The typical resonant frequency was ∼28 MHz. The r.f. technique is highly sensitive to small changes (approximately 1–5 nm) in the r.f. penetration depth λ, and thus, it is an accurate method for determining H_{c2} of superconductors. Powder samples were compressed into pellets and sealed in thin glass capillaries with a small amount of He gas. Coils that generate and detect microwave signals are directly wound around the capillary (inset of Fig. 2a). The relative change of λ is proportional to the relative change of the resonating frequency f through the inductance of the coil, that is, Δf/f∝Δλ/λ (ref. 13). Upper critical field H_{c2} was determined from the field dependence of the frequency shift Δf (Supplementary Fig. 1) as the point at which the slope of the r.f. signal in the superconducting state intercepts the slope of the normal state background.
Data availability
The data that support the findings of this study are available on request from the corresponding authors (Y.K. or Y.I.).
Additional information
How to cite this article: Kasahara, Y. et al. Upper critical field reaches 90 Tesla near the Mott transition in fulleride superconductors. Nat. Commun. 8, 14467 doi: 10.1038/ncomms14467 (2017).
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Acknowledgements
We thank Y. Nomura, R. Arita and E. Tosatti for fruitful discussions. This work was supported in part by GrantsinAid for Specially Promoted Research (No 25000003), for Young Scientists (B) (No 2474022), and for Scientific Research on Innovative Areas ‘3D ActiveSite Science’ (No 26105004) and ‘JPhysics' (No 15H05882) from JSPS, Japan, and SICORPLEMSUPER FP7NMP2011EUJapan project (No 283214). This work was also supported by the Mitsubishi Foundation and sponsored by the ‘World Premier International (WPI) Research Center Initiative for Atoms, Molecules and Materials,’ Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan. K.P. and M.J.R. thank EPSRC for support (EP/K027255 and EP/K027212). M.J.R. is a Royal Society Research Professor. RMcD acknowledges support from U.S. Department of Energy Office of Basic Energy Sciences ‘Science at 100 T’ program and that a portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No DMR1157490 and the State of Florida.
Author information
Affiliations
Department of Physics, Kyoto University, Kyoto 6068502, Japan
 Y. Kasahara
QuantumPhase Electronics Center (QPEC) and Department of Applied Physics, University of Tokyo, Tokyo 1138656, Japan
 Y. Takeuchi
 & Y. Iwasa
Department of Chemistry, Durham University, Durham DH1 3LE, UK
 R. H. Zadik
 & R. H. Colman
WPI—Advanced Institute for Materials Research, Tohoku University, Sendai 9808577, Japan
 Y. Takabayashi
 & K. Prassides
NHMFL, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
 R. D. McDonald
Department of Chemistry, University of Liverpool, Liverpool L69 7ZD, UK
 M. J. Rosseinsky
Japan Science and Technology Agency (JST), ERATO Isobe Degenerate πIntegration Project, Tohoku University, Sendai 9808577, Japan
 K. Prassides
RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 3510198, Japan
 Y. Iwasa
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Contributions
Y.K. and Y.I. conceived the experiments. Samples were grown and characterized by R.H.Z., Ya. T., R.H.C., M.J.R. and K.P., and prepared for the measurements of H_{c2} by Y.K. and Yu. T. Pulsefield experiments were performed by Y.K. with help of R.D.M. in National High Magnetic Field Laboratory, Pulsed Field Facility, in Los Alamos National Laboratory, USA. Y.K., K.P. and Y.I. led physical discussions. Y.K. mainly wrote the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Y. Kasahara or Y. Iwasa.
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