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 two-dimensional (2D) cuprates1 and organic charge-transfer salts2, are model platforms that have been extensively studied thus far. A dome-like dependence of the SC transition temperature Tc as a function of tuning parameters, such as carrier doping and pressure, has been discussed as a fingerprint of unconventional superconductivity3. Recent physical and chemical pressure studies of Cs3C60 have revealed that the family of cubic fullerides A3C60 (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 dome-like SC phase diagram as a function of unit-cell volume V (refs 4, 5, 6, 7, 8, 9). This suggests the importance of strong electron correlation to SC mechanisms10 and the need for further treatment beyond conventional framework of theory11. 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 Hc2 on Tc is relevant to the understanding of the dome-like SC phase because Hc2 is determined by the coherence length (the size of the Cooper pair) as well as the strength of the pairing potential. Therefore, Hc2 is also important to understand the underlying mechanism of the superconductivity. However, for the fullerides, Hc2 as a function of V has not as yet been determined due to the very large Hc2 and the need for high pressure to access superconductivity in Cs3C60.

Here we report measurements of Hc2 using a pulsed magnetic field in RbxCs3−xC60, where superconductivity appears near the Mott transition even at ambient pressure9. In proximity to the Mott transition, Hc2 is enhanced up to 90 T, which is the highest among cubic superconductors. We uncovered that Hc2 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 high-Tc and high-Hc2 in molecular materials.


Temperature dependence of upper critical field

Hc2 of the fulleride superconductors (Fig. 1a) Na2CsC60, K3C60, and RbxCs3−xC60 (0<x3), has been measured by a radiofrequency technique in pulsed magnetic fields13 up to 62 T (see Methods). In RbxCs3−xC60 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 (Vmax<V<Vcr, in Fig. 1c). Figure 2 shows temperature (T) variations of frequency shift Δf as a function of the magnetic field H for RbxCs3−xC60 (x=2, 0.75, and 0.35) (see also Supplementary Fig. 1). The T dependence of Hc2, Hc2(T), was determined as a point at which Δf intercepts the normal-state background (arrows in Fig. 2). Hc2(T) curves for A3C60 are plotted in Fig. 3a,b for V Vmax and Vmax<V<Vcr in the proximity of the Mott transition, respectively. Hc2(T) increases linearly with decreasing T near Tc and has a tendency to saturate at low temperatures. No obvious upturn of Hc2(T) is found in any of the samples measured, implying that Hc2(T) can be understood within a simple single-band picture despite the multiband nature of the triply degenerate t1u orbitals of anions, in contrast to MgB2 and iron pnictides where multiband and multigap behaviour with upturn or quasilinear T dependence down to T0 is commonly observed.

Figure 1: Crystal structure and electronic phase diagram of fcc fullerides.
figure 1

(a) Crystal structure of fcc A3C60. Orange and black spheres represent A and C atoms, respectively. The anions adopt two orientations related by 90° rotation about the [100] axis. Only one is shown at each site. (b) Schematic structures of anions and molecular t1u orbitals. At low V, anions are isotropic, and t1u orbitals are triply degenerate. At large V, dynamical JT distortions give rise to threefold splitting of the t1u orbitals. (c) Electronic phase diagram of cubic fullerides. Squares and circles are the superconducting (SC) transition temperature Tc for f.c.c. anion packings with Pa symmetry and Fmm symmetry, respectively. In fcc-Cs3C60 at ambient pressure, an electron-correlation-driven insulating state (Mott-Jahn-Teller insulator, MJTI) appears, which is accompanied by an intramolecular dynamical Jahn–Teller (JT) effect distorting the anions and stabilizing the low-spin (S=1/2) states that give rise to an antiferromagnetic insulating (AFI) state at low temperatures. In the metallic regime, gradient shading from green to orange schematically illustrates a crossover from the conventional metal to unusual metallic state where JT distortions persist, which we define as the JT metal (JTM) state. The grey line represents the MJTI-to-JTM crossover line, where the crossover temperatures (crosses) were obtained from X-ray powder diffraction, nuclear magnetic resonance spectroscopy, and infrared spectroscopy9. The ratio of upper critical field at T=0 and Tc, Hc2(0)/Tc (yellow triangles), shows an enhancement in the JTM regime. Error bars represent the s.d. in the values of Hc2(0) estimated from the least-squares fits of equation (1) to Hc2(T) data.

Figure 2: Determination of upper critical fields.
figure 2

Frequency shift (Δf) as a function of magnetic field for RbxCs3−xC60 with (a) x=2, (b) 0.75, and (c) 0.35 at selected temperatures. Open circles are Δf taken at T>Tc as a normal-state background signal. The arrows indicate Hc2(T) determined from the point deviating from the background signal. Inset in a shows a schematic of the experimental set-up. The sample in a capillary was inserted in one coil of the pair wound clockwise and anti-clockwise to compensate induced voltages that are generated during the field pulse.

Figure 3: Upper critical field in fullerene superconductors.
figure 3

Temperature dependence of the upper critical field for (a) V Vmax and for (b) Vmax<V<Vcr. The solid lines represent fits using equation (1). (c) Normalized Hc2, , as a function of normalized temperature T/Tc. The solid and dashed lines represent calculated Hc2(T) using the extended WHH formula (equation (1)) with α=1.5 and λso=4.4 and the conventional WHH formula in the dirty limit, respectively. (d) Hc2(0), obtained from fits using equation (1), are plotted as a function of volume per anions. Error bars represent s.d. of the fit to Hc2(T) curves. Conventional BCS values of the Pauli limiting field are also shown. (e) Evolution of 2Δ0/kBTc and Hc2(0)/Tc with approaching to the Mott transition. Error bars on 2Δ0/kBTc and Hc2(0)/Tc are calculated from the s.d. in the values of Hc2(0) estimated from the least-squares fits of equation (1) to Hc2(T) data. 2Δ0/kBTc obtained from NMR measurements is taken from ref. 9. Inset shows V dependence of derived using , Δ0, and V. Error bars are calculated from the s.d. in the values of Hc2(0) estimated from the least-squares fits of equation (1) to Hc2(T) data..

Volume dependence

In spin-singlet superconductors, Hc2 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 (Φ0, ξGL, Δ0, and μB are the flux quantum, Ginzburg–Landau (GL) coherence length, superconducting gap and Bohr magneton, respectively)14,15. In a weak-coupling BCS superconductor, the Pauli limit is [T]=1.84Tc[K]. A simple estimation from gives ξGL=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, ℓξ0 (ℓ and ξ0 are the mean free path and Pippard coherence length, respectively), as demonstrated by transport and optical measurements16,17. 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 ξ0=ħvF/πΔ0 and (vF, m, kF, and N are the Fermi velocity, effective mass, Fermi momentum, and number of electrons per C60, respectively) for the parabolic band approximation yield . In the extreme cases ( or ), Hc2(0) is determined solely by Hc2orb or HP. However, when these two quantities are comparable, Hc2(T) can be described by the extended WHH formula14, which considers both the orbital and Pauli paramagnetic effects as well as spin–orbit scattering,

where t=T/Tc, =0.281Hc2(T)/, , , is the digamma function, and λso is the spin–orbit scattering constant. With fixed , finite α reduces Hc2(0), but it recovers toward the original value with increasing λso, since spin–orbit scattering suppresses the Pauli paramagnetic effect.

was estimated from the initial slope of Hc2(T) since the Pauli paramagnetic effect is not relevant near Tc (Supplementary Note 1; Supplementary Fig. 2; and Supplementary Table 1). Then, Hc2(T) curves were fitted with HP and λso as fitting parameters. As shown by the solid lines in Fig. 3a,b, Hc2(T) curves are well described by equation (1). Figure 3c shows Hc2(T) normalized by as a function of T/Tc. The normalized Hc2(T) curves collapse into a single curve except for Na2CsC60, implying that the parameters α and λso are unchanged in a wide V region of the phase diagram, resulting in (α, λso)=(1.5, 4.4). Figure 3d displays the evolution of Hc2(0) as a function of V, together with . Hc2(0) reaches as high as 88 T in RbxCs3−xC60 with x1 (V Vmax) very close to the Mott transition. Moreover, Hc2(0) is clearly larger than at V>Vmax, and the difference between Hc2(0) and becomes pronounced with increasing V, although Tc is almost unchanged near the Mott transition.


Hc2(0) values reaching 90 T are remarkably high for 3D materials. Typical examples of 3D superconductors are cubic Nb3Sn (Hc2(0)=30 T, Tc=18 K), which is well known as a material for a SC magnet18, and Ba1−xKxBiO3 (Hc2(0)=32 T, Tc=28 K)19. MgB2 exhibits strong anisotropy (Hc2(0)=49 T and 34 T parallel to the ab plane and c axis, respectively, Tc=39 K)18 due to its anisotropic electronic structure. Hc2(0) of the fullerides is even higher than that of recently discovered H3S superconductors with likely a cubic structure (Hc2(0)≈70 T, Tc=203 K)20 despite its much higher Tc. In 2D systems under in-plane applied fields, the orbital effect is quenched and higher Hc2 can be expected. Very large Hc2 compared with low Tc has been demonstrated in ion-gated MoS2 (Hc2(0)=52 T, Tc=9.7 K)21,22 and monolayer NbSe2 (Hc2(0)=32 T, Tc=3.0 K)23. In the bulk materials, the in-plane Hc2 of the cuprates is exceptionally high at above 100 T. However, Hc2 is no longer a thermodynamic transition line, but a crossover line due to thermal fluctuations. Contrastingly, Hc2 in pnictides with Tc30 K is as large as that of fullerides24. Therefore, our results highlight the uniquely high Hc2 measured in the fulleride superconductors that are cubic, and thus, 3D.

To understand the underlying mechanisms for the evolution of Hc2(0), we estimated unknown parameters that determine HP and (Supplementary Fig. 1), that is, Δ0 and the product of parameters in the normal state . Δ0 can be directly estimated from HP. In Fig. 3e, the V dependences of 2Δ0/kBTc, which is related to the strength of the pairing interaction, and (m0 is the bare electron mass) are shown. At low V, 2Δ0/kBTc is comparable to the BCS weak-coupling limit value of 3.52. In contrast with the dome-shaped Tc, 2Δ0/kBTc continuously increases with increasing V and reaches values as large as 6, indicating a crossover from weak- to strong-coupling superconductivity on approaching the Mott transition. This is in good agreement with the previous nuclear magnetic resonance results for RbxCs3−xC60 at ambient pressure9 and for both fcc- and A15-Cs3C60 under pressure25,26, implying universal behaviour in the fullerides. On the other hand, is almost constant, indicating that both HP/Tc and /Tc are solely proportional to 2Δ0/kBTc. These results lead to the conclusion that the enhancement of Hc2(0) is dominated by the strong-coupling effect developing near the Mott transition.

We here recall Hc2(0) of other families of high-Tc or strongly correlated superconductors, i.e., cuprates, organic κ-(ET)2X, and pnictides24,27,28,29,30,31, having a dome-like SC phase and a proximate antiferromagnetic phase. In Fig. 4, Hc2(0)/Tc is displayed as a function of the relevant tuning parameter for each materials family. We show Hc2(0) for the in-plane field (H c), where the Pauli paramagnetic effect is dominating, in κ-(ET)2X and pnictides but show Hc2(0) for the out-of-plane field (H || c) in cuprates since there are no reliable estimates of Hc2(0) for H c. A remarkable feature of the fullerides is that Hc2(0)/Tc appears to be strongly enhanced at x≤1, where the JT metal phase emerges (Fig. 1c), with retaining nearly optimal Tc and Hc2(0) values near the Mott transition. This is in marked contrast to the pnictides and cuprates. In the pnictides, Hc2(0)/Tc is almost constant across the SC dome. This is ascribed to the variation of Δ0, which linearly scales with Tc (ref. 32), implying constant coupling strength. Moreover, in pnictides, Tc and Hc2(0) are strongly reduced upon decreasing doping, associated with the appearance of the antiferromagnetic phase. Non-monotonic behaviour in cuprates appears with mass enhancement near p=0.08 and 0.18, which originates from phase competition between superconductivity and Fermi-surface reconstruction or charge-density-wave order27. This is distinct from the continuous evolution of in the fullerides (Supplementary Fig. 2), suggesting the absence of such competing states. In κ-(ET)2X, 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 electron-electron interactions33,34. κ-(ET)2X shows qualitatively similar behaviour with the strong-coupling effects near the antiferromagnetic phase35. However, the enhancement of Hc2(0)/Tc is much weaker than that in the fullerides. Therefore, the steep enhancement of Hc2(0)/Tc and 2Δ0/kBTc upon entering the JT metal phase cannot be explained solely by the electron correlation effect, highlighting the uniqueness of fullerides among the high-Tc or strongly correlated superconductors. We also emphasize that it is difficult to reconcile the strong-coupling effect with the electron–phonon coupling alone25. Our results establish the importance of both molecular characteristics, absent in the atom-based superconductors, involving the dynamical JT effect and the resulting renormalization of the electronic structure and electron correlation effects for both the high-Tc and the high-Hc2 in the fullerides, as supported by the recent theoretical calculations33. This provides a new perspective on realizing robust superconductivity with high Tc and Hc2 in molecular materials.

Figure 4: Comparison of upper critical field and Tc as a function of control parameters in high-Tc superconductors.
figure 4

Variations of (a) Hc2(0)/Tc and (b) Tc in high-Tc superconductors, including fullerides A3C60, cuprates YBa2Cu3Oy (YBCO) (ref. 27), iron-pnictides BaFe2−xNixAs2 (ref. 24), and organic charge-transfer salts κ-(BEDT-TTF)2X (X=Cu(NCS)2 and Cu[N(CN)2]Br) (refs 28, 29, 30), plotted as a function of control parameters, i.e., lattice volume per (V), hole concentration (p), Ni content (x), and effective pressure measured from κ-(BEDT-TTF)2Cu[N(CN)2]Cl (ref. 31). Error bars on Hc2(0)/Tc for fullerides represent the s.d. in the values of Hc2(0) estimated from the least-squares fits of equation (1) to Hc2(T) data.


Sample synthesis and characterization

Fullerene superconductors Na2CsC60, K3C60, and RbxCs3−xC60 (0<x3) were synthesized by solid-vapor reaction method as described in ref. 9. The samples used here were identical to those in ref. 9. For RbxCs3−xC60 with x=0.5, 1, and 2, our samples correspond to Rb0.5Cs2.5C60 (Sample I), RbCs2C60 (Sample I), and Rb2CsC60 (Sample II) in ref. 9, respectively. The samples were characterized by synchrotron X-ray powder diffraction and magnetization measurements. The phase fraction of the fcc phase was larger than 70% and typical shielding fraction was 90%.

Measurements of Hc2

Contactless radiofrequency (r.f.) penetration depth measurements were performed using a proximity detector oscillator technique13 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 Hc2 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/fΔλ/λ (ref. 13). Upper critical field Hc2 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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