E EEtk

Assuming a parabolic dispersion relation for the three bands, and t,(£) oc EPi, then the diffusion thermopower can be written as where Et refers to the ith band at the Fermi level and r, = crja [14.177]. For impurity scattering r, ~ 1/3, pi — -1/2 and 1 /EF = (1/3) ^ l/£,. Using a 3D free electron model for the carriers, Eq. (14.21) becomes

which is linear in 7. Furthermore, the phonon drag term 5p(7) is proportional to 74 since

where the specific heat term C(q) oc T3 (since the phonon density of states at low 7 has a 73 dependence). The scattering term R{q) in Eq. (14.23) can be expressed in terms of the scattering times t{q) and tc(q) by

so that at low-temperature the scattering rate l/t(q) is dominated by crystallite boundary scattering and is independent of 7, since the transition probabilities are given by

i while

where aq is the scattering rate of the gth phonon due to the electron-phonon interaction, so that R(q) ~ aq/b oc T. At higher temperatures Sp(T) ~ \/Tn where n > 1, since at higher temperatures l/t(q) is controlled by phonon-phonon interaction processes. Thus the Sp contribution gives rise to a hump in the S(T) curve. Below this hump, the temperature dependence of the diffusion thermopower is found from Eq. (14.22), which in turn provides an estimate for EF, yielding EF values of 0.32 ± 0.05 eV and 0.20 ± 0.02 eV, respectively, for K3C60 and Rb3C60 [14.177],

The S(T) results of Fig. 14.23 further show room temperature values of -ll/i,V/K and -18/j.V/K for K3C60 and Rb3C60, respectively, and the linear part of S(T) extrapolates to anomalies at ~19 K and ~28 K, respectively, for K3C60 and Rb3C60, in good agreement with the superconducting Tc values for these compounds, as indicated on Fig. 14.23(a) and (b). From the EF values given above, the ratio of the electronic density of states for these compounds is obtained

which is in good agreement with the observed ratio of [Tc(Rb3C60)/ ^(KjQ,,)] = 1.55 and provides additional confirmation of the empirical relation shown in Fig. 15.3 between Tc and N(EF) [14.37]. If it is assumed that EF is approximately at midband, a bandwidth of ~0.65 eV and 0.4 eV is estimated for K3C60 and Rb3C60, respectively, from the thermopower determination of EF [14.177],

Gelfand and Lu [14.21,97] have considered the effect of orientational disorder and have found the relation

to apply to the doped fullerenes where W is the bandwidth. The same values of 5.4 and 8.8 states/eV-C60-spin were reported for K3C60 and Rb3C60 when merohedral disorder was considered.

Below ~100 K a hump in S(T) is observed for the M3C60 compounds. This behavior has been attributed to a phonon drag contribution Sp to S(T) [14.177]. The magnitude of Sp from Fig. 14.23(a) is -1-1.5 fiV/K for KjCft,,, indicative of a relatively weak electron-phonon interaction. Since the intramolecular vibrations occur at temperatures (~400 K) far above the hump in Sp(T), the phonon drag effect has been identified with electron coupling to intermolecular vibrational modes. This explanation for the hump in S(T) at low T does not explain the change in sign observed in the Hall coefficient at ~220 K [14.45]. If the samples contain conducting particles separated by low activation barriers, then the carrier transport likely involves hopping between conducting particles. Physically, the conducting particles may be Qq molecules and the barrier may be the intermolecular distances over which carriers must hop. If charge transport involves hopping, a fluctuation-induced-tunneling mechanism, rather than a free electron model, should probably be used.

The temperature dependence of the thermopower has also been measured for KxC70 at the maximum conductivity phase (which nominally occurs at the composition K4C70), and the results are shown in Fig. 14.23(c) [14.27]. In this case, 5(7) shows an approximately linear 7 dependence at high 7 and another linear 7 dependence with a larger slope at low 7. In comparison with 5(300 K) for K3C60 given above, the room temperature value of 5(300 K) for K4C70 is ~ 20-35/xV/K, and the slope at low 7 gives a value for EF ~ 0.2 eV and a bandwidth of ~0.4 eV, using similar assumptions as for K3C60. It is of interest to note that the temperature at which the slope of 5(7) shows a discontinuity is in good agreement with the temperature where the Hall coefficient changes sign, in contrast to the observations for the M3C60 compounds. The data for 5(7) in K4C70 show no clear evidence for a phonon drag anomaly, from which we conclude that the electron-phonon coupling in K4C70 is weaker than for the M3C60 compounds, in agreement with Raman scattering studies on K4C70 and with the absence of a superconducting transition down to 1.35 K [14.178].

For K4C70 the thermopower (5) is negative and metallic, indicative of charge carriers that are electrons. An anomaly in 5(7) for K4C70 is found at approximately 100 K [14.178], For the same sample as was used for the thermopower measurements, the temperature dependence of the magneto-conductance was found to fit weak localization (WL) and electron-electron (e-e) interaction theories [14.179-181],

14.13. Internal Friction

Internal friction experiments are sensitive to vibrational damping, and measurements of the internal friction can be carried out by preparing a fullerene film on a quartz piezoelectric oscillator [14.182], The temperature dependence of the internal friction shows temperature-independent behavior at low-temperature and a peak in the magnitude of the internal friction at higher temperatures. This peak has been related to the magnitude of one of the elastic constants. The magnitude of the low-temperature internal friction decreases with increasing crystallite size. With films having a crystallite size of ~1500 A, as observed by scanning tunneling microscopy (STM), the magnitude of the internal friction becomes too small to support a glassy phase at low-temperature, consistent with other studies (see §7.1.3).

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