Introduction

The family of unconventional superconductors, that do not fit into the framework of the Bardeen-Cooper-Schrieffer (BCS) theory, has grown considerably over the last couple of decades and now includes cuprates1, heavy-fermions2, organic superconductors3 and most recently also iron pnictides4. They all share a similar phase diagram5,6 − superconductivity emerges through doping or applied pressure when the competing magnetic state is suppressed. Remarkably, even after more than 20 years of intensive research the superconducting pairing mechanism is still not fully understood for these compounds7. However, unlike in the phonon-driven BCS superconductors, strong electron correlations and magnetic interactions are believed to be important for the Cooper pairing mechanism.

A comparable phase diagram has been recently established for the cubic alkali fullerides A3C60 (A = alkali metal)8,9,10 with the unit cell volume per fulleride ion, V, as a controlling parameter. For small V, short distances between neighbouring C603− anions result in a strong overlap of the highest occupied triply-degenerate t1u molecular orbitals and stabilize a Fermi-liquid metallic state from which the superconductivity emerges11,12. Tc at first increases with V but then for the optimal Vmax it reaches a maximum Tcmax = 35 and 38 K for the face-centred (f.c.c.) and A15 cubic polymorphs, respectively8,9,10. For even larger V, which are accessible only with the Cs3C60 composition under pressure, Tc starts to decrease with increasing V. At the critical volume Vm, the on-site electron-electron Coulomb repulsion energy (U) prevails over the electronic kinetic energy (measured by the bandwidth W) and the metal/superconductor-to-Mott Jahn Teller-insulator transition (MIT) takes place. The importance of the Jahn-Teller effect arises from the intrinsic orbital degeneracy of the fullerides13.

The size and symmetry of the superconducting gap, Δ, characterize the superconducting state. In the case of cuprates, the d-wave symmetry of the superconducting gap14 and its large amplitude (2Δ0/kBTc is much larger than the weak-coupling BCS value of 3.5215,16; here Δ0 is the value of Δ extrapolated to 0 K) are considered as fingerprints of an unconventional superconducting state. On the other hand, isotropic (s-wave) gaps with substantially different sizes for different Fermi pockets are found in iron-pnictides17, where Hund's rules play an important role in controlling the electronic structure18. Despite these differences, the scaling of Δ0 with Tc in the underdoped regime in both families16,17 suggests that Δ0 is controlled by the pairing strength. In the case of the extensively studied A3C60 molecular superconductors with V < Vmax, such as K3C60 and Rb3C60, the isotropic Δ found in numerous NMR or μSR experiments11,19,20,21,22 has been for many years regarded as a result of a standard type-II BCS superconductivity in the weak coupling limit. However, the strong on-site electron-electron repulsion (U ~ 1 eV)23, which is comparable with or even larger than the bandwidth (W ~ 0.5 eV)11,12, suggests the importance of electron correlations and casts doubts on the applicability of the BCS formalism24 thus bringing forward models of superconductivity which explicitly include correlations25,26,27,28. For V > Vmax, the decrease of Tc with V found for both polymorphs of Cs3C60 under pressure9,10 is a strong indication of the growing importance of electron correlations. Although experiments on expanded Cs3C60 have precisely determined the locations of the superconducting, normal and Mott-insulating states on the phase diagram9,10,29,30,31, the key information about the size and the symmetry of the superconducting gap in the vicinity of the parent Mott-Jahn-Teller insulating state is still missing.

In this work, we have used the local-probes 13C and 133Cs NMR at high hydrostatic pressures to study the f.c.c. Cs3C60 polymorph as it is driven back from the ambient-pressure Mott-insulating state to a metallic/high-Tc superconducting state by compression. We find that as the pressure is released and V increases, the decrease in Tc in the vicinity of the metal-insulator (MI) boundary is accompanied by the significant enhancement of the superconducting ratio, 2Δ0/kBTc while the s-wave symmetry of the superconducting order parameter is retained for all V. The BCS theory fails to account for these results, which thus provide very stringent tests for the fundamental mechanisms that are responsible for superconductivity in f.c.c. A3C60.

Results

The high f.c.c. Cs3C60 (86%) phase fraction of the present sample has allowed us to use 13C and 133Cs NMR spectroscopy as a local probe of the normal and superconducting state properties of the f.c.c. Cs3C60 polymorph with applied pressure (Fig. 1, Fig. S1). At ambient pressure, the large paramagnetic susceptibility of the exchange-coupled Mott-insulating state at low temperature dramatically broadens the NMR spectra through the hyperfine interaction10 and, for instance, prevents the clear separation of octahedral and tetrahedral 133Cs signals. The 13C NMR shift calculated from the first moment of the spectra, is shifted by about 190 ppm relative to the TMS standard, a characteristic value of C603− ions (Fig. 1a). At the same time, the 13C spin-lattice relaxation rate (Fig. S2), 1/13T1, is temperature independent consistent with the insulating nature of the electronic ground state down to 4 K10.

Figure 1
figure 1

Phase coexistence.

13C (a) and 133Cs (b) NMR spectra of f.c.c. Cs3C60 measured at 11 K in the superconducting state as a function of pressure. The 13C (133Cs) NMR spectra measured at ambient pressure in the paramagnetic Mott-insulating state are also shown for comparison. At low pressure (0.5 ≤ P ≤ 1.3 kbar), two contributions to the spectrum from the superconducting (red lines) and Mott-insulating (blue lines) phases are present. The grey vertical line in (a) marks the expected (chemical) shift in the superconducting state.

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Slight pressurization of the sample leads to the appearance in the spectra of an additional much narrower component in addition to the broad NMR signal due to the paramagnetic insulator. In the case of the 133Cs NMR spectra measured at 0.5 and 1.3 kbar (Fig. 1b), this is seen as the emergence of a two-peak component with a peak intensity ratio of 2:1 − reflecting the relative occupancy by the Cs+ ions of the tetrahedral and octahedral sites32, respectively, in the crystal structure − superimposed on the broad signal. Similarly, the 13C NMR spectra show low-temperature narrowing (Fig. 1a) below approximately 25 K at 0.9 kbar. This narrowing of the NMR spectra with applied pressure provides the signature of the transition to the superconducting state in which the spin susceptibility vanishes, affording the sharper signals which come from the superconductor. The appearance of the superconducting component is also picked up very sensitively in the spin-lattice relaxation rate data (Fig. S2). For instance, slight pressurization (P = 0.9 kbar) of the f.c.c. Cs3C60 sample leads to an incomplete suppression of 1/13T1T below Tc = 25 K that is consistent with the opening of the superconducting gap, Δ. However, since 1/13T1T does not approach zero even when the temperature is reduced well below Tc, the transition here to the superconducting state is incomplete. The partial suppression of the spin-lattice relaxation rates and the presence of two overlapping NMR components are thus consistent with the presence of superconducting and Mott-insulating phases in different parts of the sample at these pressures. Such phase coexistence close to the MIT boundary has also been detected for the A15 Cs3C60 polymorph9 and implies that the transition between the Mott-insulating and superconducting states is of first order.

As the pressure is gradually increased, the narrow NMR signal due to the superconducting component completely dominates the low-temperature NMR spectra in agreement with the presence of bulk superconductivity for P ≥ 1.7 kbar (Fig. 1). The critical temperature has increased to 26.5(5) K at 1.7 kbar. At this pressure, the temperature dependence of the 13C NMR line shift, calculated from the first moment of the spectra, is rapidly suppressed for T ≤ Tc (Fig. 2). We note that in the superconducting state there are three main contributions to the 13C NMR shift: the temperature-independent chemical shift, the Knight shift, which is proportional to the spin susceptibility and the diamagnetic contribution due to the Meissner effect. The latter is estimated to be very small − around 3.2 ppm33 − and can thus be neglected. The 13C NMR shift approaches ~150 ppm as T → 0, which is the value expected for the C603− isotropic chemical shift33. We thus conclude that the Knight shift vanishes at T = 0 thus providing firm evidence for the vanishing spin susceptibility of the spin-singlet Cooper pairs. The complete suppression of the 13C and 133Cs spin-lattice relaxation rates well below Tc also fully supports such a conclusion. However, there is no characteristic enhancement of the spin-lattice relaxation rates just below Tc that would mark the presence of the Hebel-Slichter coherence peak [Fig. 2b & d].

Figure 2
figure 2

Critical temperatures.

(a), (b) Temperature dependence of the 13C NMR shifts (green squares) and the spin-lattice relaxation rates, 1/13T1T (orange circles) measured at 7.8 and 1.7 kbar, respectively. (c), (d) Temperature dependence of the 133Cs NMR shifts (light green squares) and the spin-lattice relaxation rates, 1/133T1T (light orange circles) measured at 5.3 and 1.7 kbar, respectively. The 13C (133Cs) NMR shifts were obtained from the first moments, M1, of the 13C (133Cs) NMR spectra. The dashed vertical lines mark the onset temperatures at which M1 becomes suppressed in the superconducting state. At high pressure [(a), (c)], 1/T1T is suppressed at a slightly lower temperature than M1, which is the signature of a damped Hebel-Slichter coherence peak. At low pressure [(b), (d)], the two onset temperatures coincide, thus implying the absence of a coherence peak.

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The temperature dependence of the spin-lattice relaxation rates below Tc can provide information on the size and symmetry of the superconducting gap, Δ. Plotting (1/13T1) against 1/T as a semilog plot for the 1.7 kbar data yields a straight line [Fig. 3a & Fig. S3]. This implies that the temperature dependence of the spin-lattice relaxation rate follows an activated behaviour with a single isotropic BCS-like (s-wave) superconducting gap, Δ0 as described by the equation: 1/13T1 ∝ exp[−Δ0/kBT], where kB is the Boltzmann constant. This result rules out other singlet-pairing symmetries, such as that of a d-wave for which a power-law dependence 1/13T1 ∝ T3 is expected34. However, at the lowest temperatures, the T1 values are unambiguously longer by at least a factor of four than those expected in the BCS weak-coupling limit implying strong enhancement of the superconducting gap. Taking into account data for (Tc/T) > 1.25, the magnitude of Δ0 at 1.7 kbar is found to be 6.0(2) meV. At the same time, the normalised gap value, 2Δ0/kBTc = 5.3(2) is significantly enhanced relative to that expected for a weakly coupled BCS superconductor (2Δ0/kBTc = 3.52). The single s-wave symmetry superconducting gap in f.c.c. Cs3C60 contrasts with the behaviour of other high-temperature superconducting families. The cuprates, which have a parent Mott insulating state like the fullerides and are accepted as strongly correlated, universally show d-wave superconductivity14. MgB2, which is not correlated and has multiple bands at the Fermi level that can be compared with the three electronically active t1u orbitals in Cs3C60, is s-wave but displays multiple gaps35,36. Similarly, the iron pnictides, which are weakly to moderately correlated systems with multiple bands, also show multiple s-wave gap behaviour18.

Figure 3
figure 3

Superconducting gap.

(a) 13C spin-lattice relaxation rate, 1/13T1 normalized to its value at Tcvs inverse temperature, Tc/T for three characteristic pressures 1.7 (orange circles), 2.9 (green circles) and 14.2 kbar (blue circles). Solid lines through the points are fits to the equation, 1/13T1 = A exp[−Δ0/kBT], where the fitting parameters are the amplitude A and the value of the superconducting gap at T = 0 K, Δ0. Only data for (Tc/T) > 1.25 are included in the fits. Thin solid and dashed lines mark the expected slopes for 2Δ0/kBTc ratios of 3.52 and 5, respectively. The dot-dashed line is the power law dependence, 1/13T1 ∝ T3 anticipated for d-wave superconductivity. (b) The low-temperature phase diagram of f.c.c. Cs3C60 as derived from the shifts of the NMR spectra in the superconducting state. Squares and circles mark the onset of superconductivity as deduced from the 13C and 133Cs NMR data, respectively. The experiments were conducted at a 9.39 T magnetic field. The volume (pressure, top scale) dependence of Tc represented by the solid green line is that obtained from the low-field magnetization studies on the same sample10. The thick grey vertical line indicates the critical volume, Vm for the metal/superconductor-to-Mott insulator transition. The antiferromagnetic transition temperature, TN = 2.2 K, of the Mott-insulating phase is taken from Ref. [10] (AFI denotes antiferromagnetic insulating phase). (c) The volume per C603−, V, dependence of the superconducting gap divided by the superconducting critical temperature, 2Δ0/kBTc, obtained from the 13C (squares) and 133Cs (circles) spin-lattice relaxation rate data in the superconducting state. The solid thick line is a guide to the eye, while the dashed blue line marks the BCS value, 2Δ0/kBTc = 3.52. The thick grey vertical line marks the metal/superconductor-to-Mott insulator critical volume, Vm.

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Having established the fundamental behaviour of f.c.c. Cs3C60 close to the metal/superconductor-to-insulator transition, we next address how the superconducting gap relates to the superconducting Tc upon tuning the bandwidth by pressure. As the applied pressure increases and the interfullerene separation gradually decreases, Tc (Fig. 3b) first increases (29(1) K at 2.9 kbar), reaches a maximum at 33(1) K at 7.8 kbar and then begins to decrease to 30.5(8) K at the highest pressure of the present experiments (14.2 kbar) tracking the dome-shaped, Tc(P) response established before by low-field magnetisation measurements10. However, contrary to this behaviour of Tc, the gap, Δ0 does not show a maximum value but rather decreases monotonically (Fig. S3) with increasing bandwidth (decreasing V). When the normalised value of the gap, 2Δ0/kBTc = 4.7(2) (at 2.9 kbar) is considered (Fig. 3c), we find that this smoothly decreases towards the BCS weak coupling value of 3.52 at 7.8 kbar and above, implying a continuous reduction in the coupling strength as the system moves away from the metal-to-insulator boundary.

Discussion

The enhancement of 2Δ0/kBTc (Fig. 3c) with a concomitant decrease in Tc (Fig. 3b) for the large V region of the phase diagram provides an unprecedented opportunity to test the applicability of the BCS theory in alkali fullerides. In principle, the maximal value, 2Δ0/kBTc ≈ 5 close to the MIT boundary, could be obtained for strong electron-phonon coupling, but this would require optical or intermolecular phonons (ωph ~100 cm−1)37 to take part in the superconducting pairing mechanism. The weak-coupling BCS value of 3.52 found at high P (small V) can only be obtained by the involvement of the intramolecular phonons (ωph ~1000–1500 cm−1) in the pairing interaction. The V dependence of 2Δ0/kBTc would thus then require that phonon modes in distinctly different spectral regions are active in different parts of the electronic phase diagram. This is unlikely as the intramolecular phonon modes are always present and cannot be active only in one part of the phase diagram. Therefore, these arguments rule out the conventional BCS-like explanation of superconductivity in f.c.c. Cs3C60, despite the retention of s-wave symmetry over the entire phase diagram.

The failure of the BCS theory therefore necessitates the presence of an additional parameter responsible for the strong coupling superconductivity as we approach the MIT boundary from the low volume per C603−, V, side of the phase diagram. In this part of the phase diagram, the screening of the Coulomb interactions is not so effective anymore and the effective U is expected to rapidly increase thus making electron correlations progressively more important. For instance, they are directly reflected in the non-Korringa temperature dependence of 1/T1T in the normal state for large V31. Therefore, we propose that the additional parameter in the superconducting mechanism is most likely the increased importance of correlations relative to the electronic bandwidth.

The s-wave nature of the superconducting gap and the anomalously large values of 2Δ0/kBTc set the fullerides out as an unusual class of strongly correlated superconductors in which the s-wave nature is retained right across the entire V range but the non-BCS nature is strongly indicated by the V (or bandwidth) dependence of the superconducting gap. This conclusion is in qualitative agreement with the recent application of density functional theory for superconductors within the local density approximation (LDA) to the alkali fullerides, A3C60 (A = K, Rb, Cs)24. This work strongly suggests the necessity to go beyond the framework of the Migdal-Eliashberg theory and calls for direct comparisons with the other families of high-temperature superconductors to determine whether they share the same purely electronic pairing mechanism. In the strongly correlated cuprates, the superconducting gap has a d-wave symmetry in momentum space with the gap values at the “antinodes” that place 2Δ0/kBTc well in the strong coupling regime15. The gap generally first increases with doping in the underdoped regime, varying in a similar manner to Tc, but then it soon reaches a maximum and remains at the same size for larger doping levels16. On the other hand, the pnictides are less correlated systems, but like the fullerides they are multiband superconductors. Their superconducting gaps also scale linearly with Tc in the underdoped regime17 and show many similarities with the behaviour observed in underdoped cuprates. The opposing volume dependences of 2Δ0/kBTc and Tc observed for the f.c.c. Cs3C60 compound are thus qualitatively different from these two families. Finally, we note that a preprint recently appeared on the arXiv server38 reporting NMR results on the A15 Cs3C60 polymorph that show a comparable V dependence of the superconducting gap thus implying a lattice-independent superconductivity mechanism in both these systems.

Conclusions

The superconducting gap size in f.c.c. Cs3C60 increases as the insulator is approached, but the singlet s-wave nature of the superconductivity is maintained across the entire phase diagram. The observation of a single isotropic gap from a molecular s/p system with three degenerate orbitals contributing to the states at the Fermi energy contrasts with the multiband multigap structure of extreme BCS MgB2, which shares the electronic orbital parentage of the states involved but with much broader bands due to its extended non-molecular nature. The weaker overlap between the parent orbitals in the fullerides and the observation of magnetism and electronic correlations in the competing states makes the contrasting variation of Δ0 and Tc from the cuprates and the pnictides important. Hund's rules are not relevant for the cuprates, as there is only one accessible spin state with one hole in the parent insulator, but are thought to control the electronic structure of the pnictides20. In the fullerides, there is a competition between high- and low-spin states involving both Hund's rules and Jahn-Teller electron-phonon coupling, which is decisive in favouring the low-spin state13. The observation of a large non-BCS but s-wave gap in a superconductor which emerges from an onsite correlation-driven Mott insulating state and in which the molecular nature of the electronic states gives strong local Jahn-Teller coupling may indicate that the fullerides are indeed non-BCS but are distinct from other examples of unconventional superconductivity in correlated materials.

Methods

Sample

All measurements were performed on a Cs3C60 sample that contains predominately f.c.c. (86%) polymorph with secondary A15 Cs3C60 (3%), body-centred orthorhombic Cs4C60 (7%) and CsC60 (4%) phases, based on Rietveld refinement of synchrotron powder XRD data10.

NMR measurements

13C (nuclear spin I = 1/2) and 133Cs (nuclear spin I = 7/2) NMR spectra and their spin-lattice relaxation times, T1, were measured between 4 and 300 K at a magnetic field of 9.39 T. As references, tetramethylsilane (TMS) and CsNO3 standards were used with corresponding reference frequencies, ν(13C) = 100.5713 MHz and ν(133Cs) = 52.4609 MHz, respectively. In the 13C NMR lineshape measurements, a Hahn-echo pulse sequence π/2−τ−π−τ− echo was used, with pulse length tw(π/2) = 8 μs and interpulse delay τ = 40 μs. In the 133Cs NMR experiments, a two-pulse solid-echo sequence π/2−τ−π/2−τ− echo was used, with a pulse length tw(π/2) = 5 μs and an interpulse delay τ = 50 μs. The pulse length was optimized for 133Cs nuclei in the high-symmetry octahedral and tetrahedral sites with zero quadrupole frequency (expected for the f.c.c. Cs3C60 polymorph), which further suppressed the already weak signals of minority phases. For T1 measurements, the saturation recovery and inversion recovery techniques were both applied.

High-pressure cell

A home-built clamp-type cell design was used for the high-pressure NMR experiments. The cell body was manufactured from non-magnetic Ni-Cr-Al hardened alloy and 1:1 mixture of fluorinert oils FC-770 and FC-70 was used as pressure medium in order to minimize non-hydrostatic conditions. The ruby N2 luminescence line was measured in-situ in order to follow the evolution of pressure inside the cell at each temperature. The unit-cell volume at each temperature and pressure was calculated from the published f.c.c. Cs3C60 structural, thermal contraction and compressibility data10. Since the in-situ pressure was notably changing during the temperature dependent measurements, all reported pressures are given as those measured at 35 K. The absolute values of pressures are accurate within ±0.5 kbar.