It is certainly due to their apparently ordinary phe- to the calculation of the electron-phonon interaction in C60 nomenology that superconductivity in C60 materials has materials. We summarize in Fig. 1 several published results been often assumed to be consistently described by the on A3C60 compounds obtained by using different calculation conventional Migdal–Eliashberg (ME) theory of phonon- schemes [8]. In the figure, V = Vi is the total interaction i mediated superconductivity [1,2]. In favor of this point of arising from the coupling of the eight Hg C60 phonons to view we can enlist several features such as Fermi liquidthe t1u electrons and ph = Vii/V where i is the like normal state properties, order parameter of s-wave i symmetry, sizeable carbon isotope effect, etc. [3]. How- frequency associated to the i-th phonon mode. Fig. 1 is quite ever, despite of these reassuring properties, fullerene-based illuminating since, although the discrepancies in the value of superconductors display also less ordinary features making V among the different calculations are rather important, all the ME picture problematic. In fact, like the high-Tc these calculations agree in estimating the ratio ph/EF to be copper-oxides, C60 compounds have extremely low charge larger than about 0.4. This result is in contradiction with carrier density [4], have a significant electron correlation, the adiabatic hypothesis on which the entire ME framework and are close to a metal-insulator transition showing a strong rests. The ME equations of superconductivity are, in fact, dependence of Tc upon doping and disorder [3,5]. Within the defined only in the adiabatic limit ph/EF 0 in which ordinary ME framework, all these features tend to degrade all the additional nonadiabatic vertex corrections can be superconductivity. neglected in virtue of Migdal’s theorem [1].

The recent discovery of superconductivity at Tc = 52 K The analysis of the energy scales and the data of Fig. in hole doped C60 [6] rises even more doubts on the validity represent a first clue of the inadequacy of the ME theory of the ME picture. In fact, Tc = 52 K is the highest of superconductivity in fullerides. Another important incritical temperature among non-cuprates superconductors dication stems from the analysis of the experimental data (it exceeds also Tc = 39 K of the recently discovered MgB2 of Rb3C60 (Tc = 30 K) for which only recently very superconductivity [7]) and it is difficult to understand why accurate measurement of the carbon isotope coefficient C60 should represent the best optimized ME material and at C = -d ln(Tc)/d ln(M), where M is the isotopic carbon the same time display properties which are degrading for a mass, has been available [9]. In fact, the measured value ME superconductor. C = 0.21 ± 0.012 is sufficiently accurate to permit In addition to the above conceptual difficulties, there are to test the consistency of the ME theory by estimating actually several hints against the ME scenario disseminated which values of the electron-phonon coupling, the phonon in both experimental and theoretical published works on frequency ph and the Coulomb pseudopotential µ are fullerides. Let us consider for example what is known on required to obtain the experimental values of Tc and C for the relevant energy scales involved in the electron-phonon Rb3C60. To this end, we have considered different models interaction. The C60 molecule has a rather wide range for the electron-phonon spectral function 2F() and have of phonon modes of energy extending from min = 400 numerically solved the ME equations by inserting the values up to max = 2300 K. The relevant bandwidth for both of, ph, and µ which reproduce the experimental data electron and hole doped materials is W = 0.5eV= 5800 K Tc = 30 K and C = 0.21. In Fig. 2 we show the results so that for the A3C60 half-filled compounds the Fermi (filled squares) obtained by employing an Einstein phonon energy is EF = 0.25 eV = 2900 K [3]. The value of spectrum 2F() =(ph/2)( - ph). The main point EF for the optimum hole doping (Tc = 52 K) is even of Fig. 2 is that the calculated ph (lower panel) depends lower. Note that these are very small values compared to strongly on the electron-phonon constant. For large values those of conventional superconductors for which EF is of of, Tc = 30 K and C = 0.21 are reproduced only for several eVs. A great effort has been devoted in the past quite small phonon frequencies while decreasing quickly 438 C. Grimaldi, E. Cappelluti, L. Pietronero, S. Strssler shapes of 2F() which eventually include contributions from the lowest intermolecular phonon modes always lead to P > 0.4. We conclude therefore that the conventional phonon-mediated superconductivity is not a self-consistent picture of Rb3C60 since the values of and ph needed to fit Tc = 30 K and C = 0.21 strongly violate Migdal’s theorem [11].

The results of Figs. 1 and 2 imply that the description of superconductivity in C60 materials should be formulated beyond the ME theory. In particular, the low value of EF suggests that the adiabatic hypothesis and Migdal’s theorem should be abandoned from the start and that a consistent theory should be formulated by allowing ph/EF to have values sensibly larger than zero. In this perspective, in the past years we have constructed a theory of nonadiabatic superconductivity by including explicitly vertex Figure 1. Adiabatic parameter ph/EF versus the electron-phonon and other diagrammatic terms arising in the nonadiabatic pairing interaction V as extracted from various calculations of the regime [12,13]. A primary role is played by the vertex intramolecular electron-phonon pairing in fullerides [8]. V is related diagrams, which have been shown to strongly depend on the to the electron-phonon coupling constant via V = /N0, where exchanged phonon momentum and frequency, respectively q N0 is the density of states at the Fermi level. The Fermi energy has and [13]. The momentum and frequency structure of the been set equal to EF = 0.25 ± 0.05 eV [13].

vertex diagrams is quite complex and it is hard in principle to determine in which way these nonadiabatic terms affect the superconducting properties, in particular if they favour enhances ph. The C60 phonon spectrum is however limited or disfavour the superconductivity onset. In this regards the by a maximum phonon frequency of 2300 K [3,8], so that microscopic characteristics of real materials are important.

cannot be less than about 1.25. By using different shapes In particular the strong degree of electronic correlation of 2F() and of the frequency cut-off in µ, we can lower in fullerenes has been shown to have an important and the minimum allowed value of to about min 1.0. Note positive effect in nonadiabatic regime by favouring small q that the obtained values of the Coulomb pseudopotential µ scattering [14] where vertex corrections mainly enhance (upper panel, filled squares) are always quite large compared the superconducting pairing [12,13]. In this context, the to the standart value µ 0.1 [10].

electron-phonon coupling does no longer characterize the According to the ME analysis, Rb3C60 is therefore an intermediate to strong coupling superconductor. Let us address now whether this conclusion is consistent or not with the ME framework [1,2,10]. As already pointed out before, the assumption at the basis of the ME framework is Migdal’s theorem which states that, as long as the phonons have a much slower dynamics than that of the electrons, the nonadiabatic interference effects (vertex corrections) can be neglected [1]. We can test whether the data of Fig. 2 are consistent with Migdal’s theorem by evaluating the order of magnitude P of the first nonadiabatic electron-phonon vertex correction. By following Migdal [1], P is given by ph P =. (1) EF Conventional superconductors such as Pb, Al etc., have Fermi energies of order EF 5-10 eV, phonon frequencies usually not exceeding 50 meV, and less than about 1-1.5 [10]. Hence for conventional materials, P 1, the vertex corrections are negligibly small and the ME framework is well defined. To estimate the value of P in Rb3C60, we insert in Eq. (1) the value of and ph Figure 2. Coulomb pseudopotential µ (upper panel) and phonon resulting from our solution of the ME equations. We obtain frequency ph (lower panel) as a function of the electron-phonon that for all couples of values, ph of Fig. 2 (lower panel, coupling. Both the ME (filled squares) and the nonadiabatic filled squares) the Migdal parameter P is always larger than (open triangles) equations have been solved in order to fit the about 0.4. This result is remarkably robust and different experimental data Tc = 30 K and C = 0.21.

Физика твердого тела, 2002, том 44, вып. Nonadiabatic superconductivity in fullerene-based materials Figure 3. Nonadiabatic Pauli susceptibility (a) and its isotope coefficient (b) as a function of the adiabatic parameter ph/EF for = 0.7. Dashed (solid) lines refer to the first (second) order nonadiabatic approximation (see text). From lower to upper solid lines:

Qc = 0.1, 0.3, 0.5, 0.7, and 1.0.

strength of superconducting pairing whereas it is the opening Tc = 0.2Kfor K8C). Superconductivity in GICs is explained of new nonadiabatic channels of pairing which appears to by a weak to moderate coupling ( 0.3) to carbon be the driving element of large critical temperatures. In phonon modes of energies similar to those of C60. Within simple words, this means that a moderate coupling, which the ME framework, some current theories claim that the in the context of the conventional adiabatic ME theory is enhancement of Tc from GICs to fullerides arises from an expected to yield no or low temperature superconductivity, amplification of (from 0.3 to 1 or more) can actually account for large Tc’s in the new framework on due to the finite curvature of the C60 molecule [15]. In nonadiabatic theory of superconductivity. the nonadiabatic framework, instead, the most striking To illustrate this point, let us re-consider Rb3C60 under difference between GICs and fullerides is the value of EF.

the broader framework of nonadiabatic superconductivity. In the former compounds, in fact, EF is of several eVs and The open triangles in Fig. 2 are the, µ values generated Migdal’s theorem holds true. Hence, in this perspective, by the numerical solutions of the nonadiabatic equations the amplification of Tc from GICs to fullerides stems mainly constrained to fit the experimental values Tc = 30 K and from the opening of electron-phonon nonadiabatic channels C = 0.21 [11]. In order to model the strong correlation, we rather than from a 300% enhancement of.

have used an upper cut-off qc = 0.2kF for the momentum The interpretation of superconductivity in C60-based matransfer of the electron-phonon scattering (kF is the Fermi terials in terms of the nonadiabatic scenario can be susmomentum). The first remarkable difference between the tained by the observation of clear additional fingerprints of ME theory (filled squares) and the nonadiabatic solutions such a nonadiabatic regime. In order to gain robustness, (open triangles) is that much lower (and more realistic) such fingerprints should be sought among those physical values of are now needed to reproduce the experimental quantities for which some well-established properties in the data. In the context of the nonadiabatic superconductivity, ME regime are qualitatively modified in the nonadiabatic high critical temperatures arise thus from conventional values one. Let us consider for example the electron-phonon of ( < 1) embedded in a new theory, rather than renormalized charge carrier mass m. In the ME regime from extremely large values of ( > 1) predicted by m = (1 + )m, where m is the bare mass. A strong the conventional theory. It is also certainly worth stressing prediction of ME theory is that, since is independent of that the nonadiabatic solutions of Fig. 2 lead to values of P the ion mass M, no isotope effect is expected for m. Within always less than 0.25 [11]. This is perfectly compatible the nonadiabatic framework, the situation is completely with the perturbative approach used, which disregard all different. Now, the electron self-energy is modified by the the nonadiabatic irreducible vertex diagrams of order P2 or nonadiabatic vertex correction so that m acquires and addilarger. tional ph/EF 1/M1/2 dependence leading to a non-zero At this point it is illuminating the comparison between isotope effect on m. We have found that, in general, the m fullerides and intercalated graphite compounds (GICs) for isotope coefficient, m = -d ln(m)/d ln(M), is negative which quite low values of Tc are recorded (for example, and that, for example, m -0.2 for ph/EF = 0.4 and Физика твердого тела, 2002, том 44, вып. 440 C. Grimaldi, E. Cappelluti, L. Pietronero, S. Strssler = 1 [16]. The experimental observation of non-zero values and Tc. We believe that experiments in this direction are of of m in fullerides would certainly imply the breakdown of great importance.

As a concluding remark, we think it is interesting to add the ME theory and strongly support the nonadiabatic picture.

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