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Fig. 14.7. The dependence of resistivity on deposition time for a six-layer A1/CM superlattice sample, consisting of an alternating sequence of 52 A of C,m followed by 58 A of Al. The vertical dashed (dotted) lines delineate the Al (Cm) shutter openings. The expanded data of the inset show the increase in resistance at the beginning of the N = 4 plateau when C^ is deposited after the CM shutter is opened [14.57],

C60 is formed with up to six electrons transferred from the Al to the C60 and this doped monolayer also contributes to the conduction process. Referring to Fig. 14.7, it is seen that the resistance decreases only when the Al is deposited but remains constant while the CM is deposited. The resistance values at each plateau show a dependence {\/RN) = N/R0 where N is the number of superlattice periods and the change in resistance SRN at each C60/A1 interface is proportional to l/N2. These observations suggest that each Al layer contributes equally to the conductance, and the effect of the addition of each C60 layer is similar. In the limit of large N, the resistivity of the superlattice approaches that of aluminum.

Another group [14.58] used Sn, Ba, Ga, In, and Ag as metals for forming metal/C60 multilayer films on sapphire substrates and these authors used four-terminal resistance measurements similar to those described above [14.57]. Typical superlattice thicknesses were C60 (5 nm) and metal (1.5 nm). Whereas Ag and In showed little effect on sample resistance, the normalized surface resistance of the multilayers with Sn, Ba, and Ga showed a decrease in resistance with increasing numbers of superlattice layers, qualitatively similar to the behavior shown in Fig. 14.7, but with the magnitudes of the decrease in resistance strongly dependent on the metal species [14.58], Further studies on metal bilayers [14.59] showed that the resistance drop occurred mainly within the deposition of the first C60 monolayer and the decrease in resistance was greater when the metal species had a higher intrinsic resistance. The metal species for very thin layers normally deposits in disconnected islands, and the presence of the C60 provides a conduction path between the islands through charge transfer to the fullerenes. Measurements on the Cu-C60 bilayer system showed a metallic temperature dependence of resistance and that the strongest metal-C60 interaction was with the interface doped C60 monolayer [14.59].

14.1.6. Lewis Acid-Doped C60

Although it is generally reported that acceptor dopants do not transfer charge to fullerenes, there is one report of good conductivity of C60 reacted with the Lewis acids SbCl5, AsF5, and SbF5 [14.60]. Measurements of the temperature dependence of the resistance (see Fig. 14.8) over a small temperature range (200-260 K) were fit to an Arrhenius plot yielding values of the thermal band gap of 1.1 eV for SbCl5, 0.31 eV for SbF5, and 0.16 eV for AsF5. Because of the tendency for these Lewis acids to disproportionate [14.61,62], the species in the compound may well be different from the initial dopant. By analogy with the behavior of these Lewis acids as acceptor intercalate species for graphite intercalation compounds [14.61], the implication is that conduction in these Lewis acid-doped füllendes is

Fig. 14.8. The resistance of CM reacted with SbCl5 as a function of temperature showing good conduction near room temperature for this Lewis acid-doped Cm material [14.60].

by holes [14.60], although direct verification of hole conduction has not yet been reported.

14.2. Electron-Phonon Interaction

Since the electron-phonon interaction is presently believed to play an important role in electron pairing for superconducting doped fullerenes [14.63], the electron-phonon interaction for doped fullerenes has been intensively discussed. This discussion has focused on the unique properties of doped fullerenes regarding the electron-phonon interaction, estimates for the magnitude of this interaction, and the relative importance of the various phonons in their coupling to the electrons. In this section we also raise the possibility that transport properties are more strongly influenced by intermolecular charge scattering than by intramolecular vibrations, whereas the superconducting pairing is believed to be predominantly associated with intramolecular processes.

14.2.1. Special Properties of Fullerenes

The electron-phonon coupling in superconducting compounds such as K3C60 has some unusual properties. The first unusual property arises from the low average electron density, the unusually low Thomas-Fermi screening length of these electrons, and the narrow widths of the energy bands. Although the nearest-neighbor C-C intramolecular and intermolecular distances are small (with average values of 1.4 A and 3.0 A, respectively), the lattice constants are large (14.24 A for K3C60 and 14.38 A for Rb3C60),

Temperature (K)

leading to a low average density of conduction electrons n — 4.2 x 1021/cm3 for K3C60 and n = 4.0 x 1021/cm3 for Rb3Cft0 (see Table 14.1).

On the basis of a simple free electron model, the Fermi energy of an electron gas of this density would be E°F = (h2/2m)(3ir2n)2/3 = 0.9 eV, with a density of states at the Fermi energy of N°(EF) = 0.5 states/eV-C60, a Fermi velocity vF — 6 x 107 cm/s, and a Fermi wave vector of kF = 3.4 x 107 cm-1. The electron density of the crystalline K3C60 is, however, not homogeneously distributed, but is largely confined to the icosahe-dral surfaces of the fullerene molecules. This distinction is of great importance in discussing both the normal state and superconducting properties, and this inhomogeneous charge distribution has not always been fully appreciated in the literature. Thus local density approximation (LDA) band structure calculations yield rather different values from those quoted above for a uniform free electron gas of the same density. For example, LDA calculations yield values of EF ~ 0.25 eV, N(EF) = 13 states/eV-Qo and (vj-)1/2 = 1.8 x 107 cm/s [14.9], consistent with the narrow widths of the energy bands implied by the molecular nature of this highly correlated solid. Despite the low average electron density, the high density of states at the Fermi level suggests that screening effects are important. An estimate of the static Thomas-Fermi screening length lsc is given by lsc = [4ne2N(EF)]~^2 (14.3)

yielding lsc ~ 0.5 Á for K3C60 as compared to that (l°sc ~ 2.5 Á) for a homogeneous electron gas of the same electron density, which has an average electron-electron separation of ~ 7.5 Á.

A second unusual aspect of doped fullerenes that is important to the electron-phonon interaction relates to the unusual properties of the vibrational spectra, which include intramolecular vibrations spanning a large frequency range and intermolecular vibrations and librations at very low energies (see §11.4). Furthermore, the highest vibrational frequencies (~0.2 eV) are comparable to the occupied bandwidth (~0.25 eV), so that phonons not only scatter carriers on the Fermi surface but also can scatter carriers far away from EF. Thus, carrier scattering in M3C60 compounds may, in fact, involve most of the electron states of the ilu-derived bands and states throughout the Brillouin zone. The similarity of the energy scales for the electrons and intramolecular phonons may violate Migdal's theorem, as has been discussed by various authors [14.64-68]. The Migdal criterion, that the phonon energy satisfy hw h\F ■ q, is not met for some high-energy phonons with q near the Brillouin zone boundary [14.69]. This criterion is, however, satisfied for most phonon branches and for most q values. Detailed calculations show that the average electron velocity is small only near the band edge and zone boundaries and that orientational (merohe-

dral) disorder (see §7.1.4 and §8.5) does not strongly affect the mean carrier velocity [14.70].

Thus, although the Migdal approximation is not strictly satisfied, many of the electrons and phonons in doped fullerenes do satisfy the criterion, and it is thought that the expressions usually used for the electron-phonon interaction are likely to provide a useful approximation also for the superconducting fullerene-derived compounds [14.9]. It is, however, believed that because the Migdal criterion is not strictly met, caution is needed in interpreting certain experimental data, such as the pressure dependence of Te and the isotope effect [14.71].

Finally, we briefly mention questions concerning the validity of Bardeen-Cooper-Schrieffer (BCS) theory for C60 in terms of Migdal's theory. Migdal's theory states that vertex correction of the electron-phonon interaction may be small (~1/100) in ordinary BCS superconductivity. The factor that specifies the magnitude of the vertex correction is hwD/EF or (m/M)1'2 where m and M are masses of electrons and ions, respectively [14.72], The vertex correction is due to the fact that the sound velocity is much smaller than the Fermi velocity. If we neglect this vertex correction, then the integral equation for the self-energy (or free energy) can be solved approximately using Green's functions without any vertex correction, thus giving the result of the original BCS theory. In the case of C60, the ratio of (m/M)1/2 may still be small, even though h(oD/EF is on the order of unity. Based on this fact, Pietronero and Strassler [14.73] have calculated a vertex correction of the electron-phonon interaction, and they show that a nonadiabatic effect produces a strong enhancement of Tc and implies the breakdown of Migdal's theorem. However, it is still unknown whether vertex corrections from other vertices could contribute to the electron-phonon interaction. This possibility should be examined in the future. Furthermore, in the nonadiabatic case, the renormalization of the electron-phonon interaction and the validity of the Fermi-liquid picture in the nonadiabatic regime based on a realistic electronic structure are important subjects for further investigation.

14.2.2. Magnitudes of the Coupling Constant

Taking into account some of the unusual properties of the doped fullerenes, a number of attempts have been made to estimate the magnitude of the electron-phonon coupling coefficient Kep and to identify the phonons that are most important in the electron pairing that gives rise to superconductivity. The magnitude of \ep enters directly into the BCS equation for Tc and depends strongly on the magnitude of the electron-phonon interaction discussed in this section and the electronic density of states at the Fermi level discussed in §14.5.

In the various estimates of the average electron-phonon coupling coefficient Kep, it is customary to sum over the contributions A^ from various phonon branches and wave vectors qv where nb denotes the number of phonon branches and N is the number of wave vectors in the Brillouin zone [14.74],

In the BCS theory of superconductivity, the dimensionless coupling coefficient, ke , is given by the simplest vertex of the electron-phonon coupling, where e„k denotes the energy of a Bloch electron in band n and wave vector k, and /„k „,k,(p, q) is the matrix element of the electron-phonon interaction between electronic states nk and n'k' mediated by the pth phonon with wave vector q = k — k' [14.75,76]. The delta functions in Eq. (14.5) restrict the sum so that both the wave vectors for the initial and final electronic states k and k' are on the Fermi surface e = 0. Also in Eq. (14.5), N(EF) is the density of states at the Fermi level per spin orientation, Kp(q) = Mw2p(q) is the force constant of the p\h phonon, «^(q) is the frequency of a phonon on the pth branch with wave vector q, and M is the reduced mass. Schlüter et al. [14.76] proceeded to evaluate e„k using the tight-binding representation for the Bloch states and to show that the derivative of an atomic hopping matrix element of an atom at a specific site is proportional to the matrix element /„ii,„<k'(p>q) in Eq. (14.5). These authors further show that all contributions to the electron-phonon coupling that contribute to pairing are small except for the phonons that modulate the strong intramolecular it-it overlap of the tlu states. The calculations also show that the intramolecular vibrations are dominant, especially those with high frequencies and Hg symmetry, modes that contribute to JahnTeller distortions and to symmetry lowering for the M3C60 compounds, as discussed further below.

In the limit that only the intramolecular transfer integral contributes to the electron-phonon coupling coefficient Kep, Schlüter et al. [14.76] have shown that