"LDA method of Ref. [14.75,77]. 6Bond charge method of Ref. [14.75,79], cForce constant method of Ref. [14.80].

^Deformation potential coupling contributions from various phonons in Ref. [14.81].

e Linear muffin tin orbital (LMTO) method of Ref. [14.82],

"LDA method of Ref. [14.75,77]. 6Bond charge method of Ref. [14.75,79], cForce constant method of Ref. [14.80].

^Deformation potential coupling contributions from various phonons in Ref. [14.81].

e Linear muffin tin orbital (LMTO) method of Ref. [14.82], by several other models is also included in the table. The results for several calculations cover a range of V from 32.2 to 82.2 meV and ¿Dph from 786 to 1320 cm"1 [14.77], where the logarithmically averaged phonon frequency, <wph is defined by

Here \p is the contribution of the /?th vibration to the electron-phonon coupling coefficient A (see Table 14.3) and is given by

We will return to a discussion of the parameters V, wph, and \ep in Chapter 15 (§15.7) in connection with the superconducting properties of doped fullerenes.

14.2.3. Symmetry Considerations

From a symmetry standpoint, the coupling of two electrons in the LUMO tlu level is through vibrations with symmetries arising from the direct product of the corresponding irreducible representations Tlu <g> Tlu = Ag + Tlg + Hg, using the irreducible representations of the icosahedral Ih group, where T and F are used interchangeably to label the threefold irreducible representations. Since TXg is asymmetric under interchange of two electrons, the electrons do not in lowest order couple through vibrations with Tlg symmetry, and for this reason only the Hg and Ag modes for C^ are listed in Table 14.3. Coupling to Ag modes does not change the symmetry of the molecule and therefore cannot give rise to Jahn-Teller distortions, but the eight Hg modes can deform the fullerene dynamically. It is for this reason that the discussion of the phonon contribution to the electron-phonon interaction has focused on the Hg modes. Furthermore, two electrons in the tlg level (as occurs for the alkaline earth compounds) couple predominantly through the same vibrations as do two tlu electrons, since the direct product for the two tlg electrons yields the same irreducible representations as for two tlu electrons: Tlg ® TXg = Ag + Tig + Hg. At the present time there is no strong consensus about whether it is the high- or low-frequency phonons that are dominant in determining kep, or whether both low- and high-frequency modes might perhaps contribute in a significant way to kep. Almost all the attention given to the electron-phonon interaction in doped fullerene solids has been directed toward gaining a better understanding of the superconductivity pairing mechanism. Relatively less attention has been given to the electron-phonon interaction as it affects normal state transport in these systems.

14.2.4. Jahn-Teller Effects

Regarding the significance of the Jahn-Teller (JT) effect for the electron-phonon interaction, both static and dynamic JT effects must be considered. In the static JT effect, a structural distortion lowers the symmetry of the system and lifts the degeneracy of the ground state. For a partially filled band, such a distortion leads to a lowering of the energy of the system as the lower energy states of the multiplet are occupied and the higher-lying states remain empty. An example of the static JT effect is the bond alternation or Kekule structure of the hexagonal rings in neutral C60 and its ions, which has been discussed [14.83-85] in terms of the Su-Schrieffer-Heeger (SSH) model, originally developed for polyacetylene (CH)X [14.86]. It should, however, be noted that the distortions of the Kekule structure due to bond alternation in neutral C60 do not lower the icosahedral symmetry of the molecule. In the case of the C60 ions, a change of bond alternation does, however, change the degeneracy of the electronic levels. The degeneracy is lowered when additional electrons are introduced to partially fill the tlu levels.

The dynamic JT effect [14.87] can occur when there is more than one possible distortion that could lead to a lowering of the symmetry (and consequently also a lowering of the energy) of the system. If the potential minima of the adiabatic potential are degenerate for some symmetry-lowered states of a molecule, the electrons will jump from one potential minimum to another, utilizing their vibrational energy, and if this hopping occurs on the same time scale as atomic or molecular vibrations, then no static distortion will be observed by most experimental probes for the dynamic JT effect. Those vibrational modes which induce the dynamic JT effect contribute strongly to the electron-phonon coupling.

One of the early discussions of the dynamic Jahn-Teller effect for doped fullerenes was given by Johnson et al. [14.78], who proposed the dynamic JT effect as a pairing mechanism for superconductivity. The Jahn-Teller effect (whether static or dynamic) plays an important role in determining the electronic states for the charged C60 molecules and has been widely discussed as an important mechanism for electron-phonon coupling in the doped fullerenes [14.75,77,81,88], For the case of the addition of three electrons to each fullerene by alkali metal doping, model calculations [14.89] show that the degenerate flu-derived LUMO ground state is split into three non-degenerate levels. This splitting can be understood in terms of a static Jahn-Teller effect for degenerate tlu levels. No geometrical optimization of the structure of the alkali-doped C60 solid has yet been carried out. As for a single C60 molecule, a "Modified Intermediate Neglect of Differential Overlap, version 3.0, Molecular Orbital " (MINDO/3 MO) [14.90] calculation with the unrestricted Hartree-Fock (UHF) approach shows that the singly charged anion C^1 is deformed from Ih to Did symmetry by the Jahn-Teller effect [14.91], It is noted here that the splitting of the tlu levels comes not only from the lattice deformation, but also from the symmetry-lowering effect of the static Coulomb interaction caused by the additional electrons on the fullerene anions due to charge transfer and by the alkali metal ion dopants. In fact, "Modified Neglect of Diatomic Overlap, Configuration Interaction" (MNDO-CI) calculations for C£0~ (0 < n < 5), have been carried out, and the results show that a symmetry-lowering effect on the electronic structure occurs for the various ground states that were degenerate in Ih symmetry [14.92], Anisotropy in the EPR lineshapes provides evidence for Jahn-Teller distortions for negatively charged fullerene anions (see §16.2.2).

14.3. Hall Coefficient

For simple one-carrier type conductors, the Hall coefficient provides a direct measurement of the carrier density through the relation RH = l/ne (in MKS units), where e is the charge on the electron. Since pristine C60 and C70 have a very low carrier density, no Hall effect has been measured on these materials. Thus Hall effect measurements have been limited to doped fullerenes near their minimum resistivity stoichiometrics (see §14.1).

Temperature-dependent Hall effect measurements have been carried out in the temperature range 30 to 260 K on a K3C60 thin film [14.45]. Assuming three electrons per C60, the expected Hall coefficient RH based on a one carrier model (—1.5 x 10~9m3/C) would be about a factor of 4 greater in magnitude than the experimentally observed value RH = —0.35 x 10-9 m3/C at low T [14.45]. Experimentally RH changes sign from negative below 220 K to positive above 220 K (see Fig. 14.9) [14.45]. The small value of the observed Hall coefficient suggests multiple carrier pockets including both electrons and holes, consistent with band structure calculations (see §12.7.3). A reduction in the magnitude of the Hall coefficient could also arise from merohedral disorder effects (see §8.5) [14.23]. Since three electrons occupy states in the three flu (tlu) bands, it is reasonable to expect both electron and hole pockets in the Brillouin zone from elementary considerations. Multiple carrier types are also consistent with the Fermi surface calculations by Erwin and Pederson [14.93] and by Oshiyama et al. [14.13], who also found both electron and hole orbits on the Fermi surface.

Measurements of RH(T) on K4C70 show similar behavior to that found for K3C60 with small magnitudes of RH (-2 x 10_9m3/C at low T) and a crossover from negative to positive RH at ~80 K [14.27]. Incidentally, the Hall coefficient RH of K-intercalated graphite intercalation compounds (GICs) (see §2.13) also changes sign in the same temperature range [14.94].

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