1.4 ± 0.5

67% ,2C, 33% 13C


"The listed Tc value corresponds to 100% abundance of dominant isotope.

"The listed Tc value corresponds to 100% abundance of dominant isotope.

values quoted in Table 15.3. These data are significant in showing the wide deviations from a = 0.5 that are observed in high-cuprate materials. These measurements, taken for a wide range of lsO and lfiO isotopic abundances, show that a is not sensitive to isotopic oxygen substitutions [15.89], Similar results have been obtained using the Laj 85Sr015Cu1_JNiJ04 system as x was varied from x = 0 (a — 0.12) to x = 0.3 (a = 0.45), with a monotonic increase in a observed with increasing x [15.90]. Other pertinent results on the dependence of a on stoichiometry of high-Tc cuprate materials have been reported for the YBa2Cu307_s system giving a values in the range 0.55-0.10 as Pr was substituted for Y [15.91] and for the Nd-Ce-CuO system a < 0.05 where the 180:160 ratio was varied [15.92].

Measurements of the isotope effect on doped C60 samples with only a 33% enrichment of 13C gave a much larger value of a = 1.4±0.5 [15.86,93]. The large range in the values of the exponent a reported by the various groups seems to be associated with the difficulty in determining Tc with sufficient accuracy when the normal-superconducting transition is not sharp. The importance of using samples with high isotopic enrichment can be seen from the following argument. Since Tc = CM~a, we can differentiate this expression and write where M is the isotopic mass and / is the fractional isotopic substitution 0 < / < 1, assuming that AM is related to the difference between the average elemental mass and that of a particular isotope. Equation (15.18)

thus shows that A Tc is measured more accurately when the isotope effect is measured on light mass species and when the fractional abundance / of the isotope under investigation is large.

To provide another perspective on the experimental determination of the isotope effect, two distinct substitutions of ~50% 13C were prepared in the Rb3C60 compound [15.94], In one sample Rb3(13C1_x12Ct)60 called the mass-averaged sample, each C60 molecule had ~50% of 12C and 13C, while in the other sample Rb3[(13C60)1_i(12C60)i)] called the mass-differentiated sample, approximately half of the C60 molecules were prepared from the 12C isotopes and the other half from 13C isotopes. The samples thus prepared were characterized by infrared spectroscopy to confirm that the C60 molecules were mass averaged in the first sample and mass differentiated in the second sample. Measurements of the isotope effect in the mass-averaged sample confirmed the a = 0.3 previously reported for K3C60 [15.85], while the mass-differentiated sample yielded a value of a = 0.7 [15.94], No explanation has yet been given for the different values of a obtained for these two kinds of samples.

On the basis of the superconducting energy gap equation [Eq. (15.9)], all of the reported values of the exponent a suggest that phonons are involved in the pairing mechanism for superconductivity and that the electron-phonon coupling constant is relatively large [15.69,86]. Future work is needed to clarify the experimental picture of the isotope effect in the M3C60 compounds, and if the large value of a — 1.4 were to be confirmed experimentally, detailed theoretical work is needed to explain the significance of the large a value.

15.6. Pressure-Dependent Effects

Of interest also is the dependence of the superconducting parameters on pressure. Closely related to the high compressibility of C60 [15.96] and 60 (M = K, Rb) [15.17] is the large (approximately linear) decrease in Tc with pressure observed in K3C60 [15.17] and Rb3C60 [15.17] for measurements up to a pressure of 1.9 GPa (see Table 15.1). Figure 15.15 is a plot of the temperature-dependent susceptibility for several values of pressure for a pressed powder sample of K3C60 [15.23]. This figure clearly shows a negative pressure dependence of Tc, which is more clearly seen in the pressure-dependent plot of Tc in Fig. 15.16. The pressure coefficients of Tc for K3C60 and Rb3C60 are similar to those measured for the high-Tc superconductors in the La2__c(Ba,Sr)xCu04 system where (dTJdp) ~ +0.64 K/kbar [15.9799], much greater than those for the A15 superconductors (+0.024 K/kbar), but also less than those for BEDT-TTF salt organic superconductors for which {dTJdp) ~ -3 K/kbar [15.29], t £

Fig. 15.15. Plot of the temperature dependence of the susceptibility of a pressed KjC^, sample at the various pressures indicated [15.23].

Fig. 15.15. Plot of the temperature dependence of the susceptibility of a pressed KjC^, sample at the various pressures indicated [15.23].

10 is

Temperature (K)

10 is

Temperature (K)

Fig. 15.16. Pressure dependence of Tcl (transition onset) and of Tc2 [the morphology-dependent kink in the x(T) curves] (see Fig. 15.15) for a pressed powder sample of KjC«, [15.23],

Fig. 15.16. Pressure dependence of Tcl (transition onset) and of Tc2 [the morphology-dependent kink in the x(T) curves] (see Fig. 15.15) for a pressed powder sample of KjC«, [15.23],

10 15

Pressure (kbar)

10 15

Pressure (kbar)

When the pressure dependence of Tc for K3C60 and Rb3C60 is expressed as (\/Tc)(dTc/dp)p=0 in the limit of p = 0, a common value of -0.35 GPa"1 is obtained for both K3C50 and Rb3C60 [15.60,100]. In this interpretation, the smaller size of the K+ ion relative to Rb+ (0.186 Â) is attributed to an effective relative "chemical pressure" of 1.06 GPa, which is found by considering the compressibilities of the two compounds (see Table 15.1). By displacing the pressure scale of the Tc{p) data for K3C60 by 1.06 GPa, the pressure dependence of Tc for K3C60 and Rb3C60 could be made to coincide, and the results for both compounds could be fit to the same functional form rc(p) = rc(0)exp(-yp) (15.19)

with the same value of y = 0.44 ±0.03 GPa"1 [15.17], A plot of Tc vs. pressure obtained from curves (such as in Fig. 15.15) is presented in Fig. 15.16, showing a nearly linear decrease of Tc with pressure. The result given by Eq. (15.19) can be interpreted in terms of a Tc that depends only on the lattice constant for the M3C60 compound and not on the identity of the alkali metal dopant (see §15.1). The large effect of pressure on Tc is due to the sensitivity of the intermolecular coupling to the overlap between nearest-neighbor carbon atoms on adjacent molecules, which are separated by 3.18 A in the absence of pressure. When this intermolecular carboncarbon distance becomes comparable to the nearest-neighbor intramolecular C-C distance (~1.5 A), a transition to another structural phase takes place [15.17].

15.7. Mechanism for Superconductivity

The observation of a 13C isotope effect on Tc indicates that C-related vibrational modes are involved in the pairing mechanism. Several other experimental observations suggest that the role of the alkali metal dopant is simply to transfer electrons to C60-derived states (/,„ band) and to expand the lattice. This is consistent with the absence of an isotope effect for Rb in Rb3C60 and with the strong correlation of Tc with lattice constant rather than, for example, with the mass of the alkali metal ion. If metal ion displacements were important in the pairing mechanism, then Rb3C60 would be expected to have a lower Tc than K3C60, contrary to observations.

Which C60 vibrational modes are most important to Tc is not completely resolved. The dependence of Tc not on the specific alkali metal species but rather on the lattice constant a0 of the crystal implies that the coupling mechanism for superconductivity is likely through lattice vibrations and is more closely related to intramolecular than to intermolecular processes. Raman scattering studies (see §11.6.1) of the //^-derived, intramolecular modes in M3C60 show large increases in the Raman linewidths of many of these modes relative to their counterparts in the insulating parent material C60 or in the doped and insulating phase M6C60. This linewidth broadening strongly suggests that an important contribution to the electron-phonon interaction comes from the intramolecular modes.

Theoretical support for the broadening of certain intramolecular vibrational modes as a result of alkali metal doping has been provided by considering the effect of the Jahn-Teller mechanism on the line broadening through both an adiabatic and a nonadiabatic electron-intramolecular vibration coupling process [15.101]. Broadening was found in modes with Hg symmetry, as well as in some modes with Hu and Tlu symmetries, and the broadening results from a reduction in symmetry due to a Jahn-Teller distortion. An especially large effect on the highest Hg mode due to doping has been predicted [15.101], The broadening effect appears to be in agreement with phonon spectra on alkali metal-doped fullerenes as observed in neutron scattering studies (see §11.5.8).

While keeping these fundamental characteristics in mind, theorists have been trying to identify the dominant electron pairing mechanism for superconducting fullerenes. A number of approaches have been attempted and are summarized below. In the first, only the electron-phonon interaction is considered and electron-electron interactions are neglected. This main approach has been followed by most workers, with different possible side branches pursued, as outlined below. The second approach considers a negative electron-electron interaction whereby superconductivity is explained by an attractive Hubbard model. Returning to the various branches of the first approach, one branch considers a BCS version of the electron-phonon interaction without vertex corrections, while another branch includes vertex corrections for the enhancement of Tc.

Within the first approach, much effort has gone into the identification of the phonon modes which are most important in the electron-phonon interaction. Much of the discussion has focused on the use of the Eliashberg equations to yield the frequency-dependent a2eF(o>) spectral functions from which the electron-phonon coupling parameter \ep can be determined by the relation

It is readily seen that a given spectral weight a2F(«) at low frequency contributes more to kep than the corresponding weight at high o> because of the o) factor in the denominator of Eqs. (15.20) and (15.21).

Since Tc depends predominantly on the lattice constant, which in turn depends on the density of states, as discussed in §15.1, Tc is most sensitive to molecular properties. Thus, it is generally believed that the intramolecular phonons are dominant in the electron-phonon interaction [15.50]. This conclusion is based on detailed studies of both the temperature dependence of the transport properties [15.50] (see §14.1) and the electron-phonon

and the average phonon frequency w h is then determined by

coupling constant directly [15.51,52], Although most authors favor dominance by the high-frequency intramolecular vibrations, some authors have emphasized the low-frequency intramolecular vibrations [15.102] for special reasons, such as an effort to explain possible strong coupling in K3C60 and Rb3C60 as implied by the STM tunneling experiments to determine the superconducting energy gap [15.54], Thus, it may be said that no firm conclusion about the dominant modes has yet been reached [15.84,103,104]. If both high- and low-frequency phonons were to contribute strongly to the electron-phonon interaction, then two peaks in the Eliashberg spectral function a2eF(u)) would occur in Eqs. (15.20) and (15.21), such that wPh = w2 > (15.22)

where \ep is obtained from Eq. (15.20) by integration of 2a2F(a>)/to over all phonon frequencies and where A, and A2 are related to the contribution from each peak in a2eF((x)) [15.84]. According to this two-peak approach, phonons with energies ho)ph greater than irkBTc are effective at pair breaking, while the low-energy phonons are not.

In narrowband systems, such as in M3Cgg, we expect a Coulomb repulsion to be present between electrons that are added to the C60 molecule in the degenerate /lu (or tlu) levels or energy bands. In the limit of a strong Coulomb repulsive interaction, the tlu energy band is split into upper and lower Hubbard bands, and thus the solid becomes a magnetic insulator [15.105,106] or acts as a heavy fermion system. If M3C60 is a Mott insulator, then the superconducting phase must have off-stoichiometric compositions M3_sC60, where a value of S = 0.001 could account for the normal state conductivity. It has been argued that the success of LDA band calculations for KjQq and Rb3C60 in predicting a linear dependence of Tc on pressure

[15.107] and a variety of experimental efforts to look for anomalous behavior very close to the x = 3 stoichiometry indicates that K3C60 and Rb3C60 are metals and not Mott insulators.

In the case of solid C60, the competition between the electron-electron interaction and the transfer energy from molecule to molecule must be considered in treating the superconductivity of n electrons in the flu (tlu) energy band. Below we also review efforts to treat the pairing of two electrons on the basis of the electron-electron interaction.

White and his co-workers discussed the attraction between two electrons localized on a single C60 anion by defining the pair-binding energy,

£;air = 2$,. - 3Vi - <Di+1 (i = 1,3,5), (15.23)

where <t>, is the total energy of a molecule when the molecule has i additional electrons. If £pair is positive, it is energetically favorable for two adjacent molecules to have (i + 1) electrons on one molecule and (i - 1) electrons on the other. This transfer of electrons has been proposed as a possible mechanism for superconductivity in fullerenes [15.109,110].

This situation has been examined by a numerical calculation using the extended Hubbard Hamiltonian,

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