Vibrational Modes

In this chapter we review the lattice mode structure for the isolated fullerene molecules and for the corresponding molecular solid. Explicit results are given for C60, C70, and higher fullerenes. The effects of doping, photopolymerization, and pressure on the vibrational spectra are reviewed.

11.1. Overview of Mode Classifications

Because solid C60 is very nearly an ideal molecular solid, its vibrational modes can be subdivided naturally into two classes: intermolecular vibrations (or lattice modes) and intramolecular vibrations (or simply "molecular" modes). The intermolecular modes can be further divided into three subclasses: acoustic, optical, and librational modes. All solids have three acoustic modes, whose frequencies vanish at q = 0. In addition, optic "lattice" modes are found in all atomic solids with two or more atoms per primitive unit cell; in the case of solid C60, the intermolecular optical lattice modes are associated with two or more C60 molecules per unit cell. For molecular solids, a third subclass of modes is identified with librational modes, which stem from the moment of inertia of the C60 molecule and involve a hindered rotation (rocking) of the C60 molecules about their equilibrium lattice positions. The frequencies of the librational modes are quite low because the molecules are only weakly coupled and also because the moment of inertia of the fullerene molecules is large. These distinctions are traditionally made for C^, (as shown in Fig. 11.1) [11.1] but can also be made for higher fullerenes and for doped fullerenes. A number of authors have discussed the vibrational spectra of molecules with icosahedral symmetry both experimentally [11.2-4] and theoretically [11.5-9],

Fig. 11.1. Schematic view of the various classes of vibrations in MjC«, compounds (see text). At low frequencies, the compounds exhibit librational modes of individual Cw molecules (a), intermolecular optic modes (b), and IR-active optic modes (c). At higher frequencies (above ~ 200 cm"1), the intramolecular modes are dominant and follow the schematic vibrational density of states shown in the figure. These "intramolecular" modes have predominantly radial character (d) at lower frequencies (to < 700 cm"1) [for example the Hg( 1) mode], and at higher frequencies (a> > 700 cm-1) the mode displacements have predominantly a tangential character (e) [for example, the Hg(7) mode] [11.1].

Fig. 11.1. Schematic view of the various classes of vibrations in MjC«, compounds (see text). At low frequencies, the compounds exhibit librational modes of individual Cw molecules (a), intermolecular optic modes (b), and IR-active optic modes (c). At higher frequencies (above ~ 200 cm"1), the intramolecular modes are dominant and follow the schematic vibrational density of states shown in the figure. These "intramolecular" modes have predominantly radial character (d) at lower frequencies (to < 700 cm"1) [for example the Hg( 1) mode], and at higher frequencies (a> > 700 cm-1) the mode displacements have predominantly a tangential character (e) [for example, the Hg(7) mode] [11.1].

The introduction of alkaline earth or alkali metal atoms (M) into the fullerene lattice leaves the molecules intact but may rearrange and/or reorient the fullerene molecules with respect to each other (see §8.5 and §8.6). Despite these rearrangements and the concomitant charge transfer of electrons from these metal dopant atoms to create Qq molecular anions and metal cations, the vibrational mode picture described above for solid C60 remains largely intact. However, a new subclass of intermolecular modes, involving the relative motion of the cation (M+) with respect to the molecular anion sublattices, must now be included.

In Fig. 11.1 we show schematically the natural separation of the vibrational modes of a particular crystalline compound, M3C60 (M = alkali metal) solid [11.1], which is superconducting for M = potassium (Tc = 18 K) or M = rubidium (Tc = 29 K). The general nature of the schematic vibrational mode structure shown in Fig. 11.1 should be independent of the extent of the alkali metal doping (i.e., 1 < x < 6 in M^Cft0). Note that for clarity the acoustic lattice modes have been omitted from the figure and that the density of vibrational states has been plotted against a logarithmic frequency scale covering more than two decades in frequency. The characteristic atomic or molecular motion associated with each of the four mode types is also indicated above the respective band of frequencies. The librational (a) modes lie lowest in frequency (10-30 cm-1), followed by the intermolecular optic (b) modes involving relative motion between neutral C60 molecules or C3m anions (30-60 cm-1), then the infrared-active optic modes (c) involving relative motion between the anions and the M+ cations (50-120 cm-1), and finally the C60 intramolecular or "molecular" (d, e) modes (270-1700 cm'1), which involve relative C atom displacements on a single molecule or anion. (The conversion between units commonly used for the vibrational mode frequencies is: 1 meV is 8.0668 cm"1 = 11.606 K = 0.24184 THz, and 1 THz = 33.356 cm"1.) In the limit that the molecule is treated as a "point," the molecular moment of inertia I approaches zero, and the librational modes are lost from the spectrum. The intermolecular optic modes (b) are generally treated in the molecular limit [11.10,11], as is discussed in §11.4.

In general, the low-frequency (a> < 700 cm-1) and high-frequency (w > 700 cm-1) intramolecular modes tend to exhibit approximately radial (d) and tangential (e) displacements, respectively, as indicated for the specific Hg symmetry modes shown in the figure. Some of the molecular modes have vibrational quanta as large as 0.1 to 0.2 eV, which is significant when compared to the electronic bandwidths (~0.5 eV) for energy bands located near the Fermi level and has implications for electrical transport in metallic fullerene solids (see §14.1).

In treating the vibrational spectra for other fullerene molecules or crystalline solids, it is necessary to consider the shape, mass, and moment of inertia for each fullerene species and isomer (C70, C76, C82, C84, etc.). For each of these heavier fullerenes, the four types of vibrational modes, shown schematically in Fig. 11.1, are also expected, as are the schematic phonon density of states curves for the fullerene solid.

11.2. Experimental Techniques

The major experimental techniques for studying the vibrational spectra include Raman and infrared spectroscopy, inelastic neutron scattering, and electron energy loss spectroscopy. Of these techniques, the most precise values for vibrational frequencies are obtained from Raman and infrared spectroscopies. In most crystalline solids, Raman and infrared spectroscopy measurements focus primarily on first-order spectra which are confined to zone-center (q = 0) phonons. Because of the molecular nature of fullerene solids, higher-order Raman and infrared spectra also give sharp spectral features, as shown in §11.5.3 and §11.5.4. Analysis of the second-order Raman and infrared spectra for Cm provides a good determination of the 32 silent mode frequencies and their symmetries. If large single crystals are available, then the most general technique for studying solid-state phonon dispersion relations is inelastic neutron scattering. Unfortunately, large single crystals are not yet available even for Qg, so that almost all the reported inelastic neutron scattering measurements have been done on poly-crystalline samples. Much of the emphasis of the inelastic neutron scattering studies thus far has been on the lowest-energy intermolecular dispersion relations, which have been studied on both single-crystal and polycrystalline samples [11.12,13]. Electron energy loss spectroscopy (EELS), although generally having less resolution than the other techniques, is especially useful for the study of silent modes because of differences in selection rules. An unexpected rich source of information about molecular C60 modes comes from singlet oxygen photoluminescence side bands, which have been especially useful for studying the intramolecular vibrational spectrum of C60 [11.14], since the usual IR or Raman selection rules do not apply to these spectra.

11.3. ^ Intramolecular Modes

In this section we consider theoretical issues affecting the vibrational intramolecular spectra. The room temperature experimental Raman spectra [11.3,4,15,16] suggest that solid-state effects due to the (or Pa3) space group are very weak or give rise to very small unresolved splittings of the 10 main Raman-allowed modes in the isolated C60 molecule. For this reason it is appropriate to start the discussion of the vibrational spectra for the intramolecular modes with those for the free molecule. However, additional lines are observed in Raman and infrared spectroscopy, and the theoretical explanations for such effects are discussed in this section.

11.3.1. The Role of Symmetry and Theoretical Models

The vibrational modes of an isolated C60 molecule may be classified according to their symmetry using standard group theoretical methods (see §4.2 and Table 4.6) [11.17-19], by which it is found that the 46 distinct intramolecular mode frequencies correspond to the following symmetries:

rmol = 2Ag+3Flg+4F2g+6Gg+8Hg+Au+4Flu+5Flu+6Gu+7Hu, (11.1)

where the subscripts g (gerade or even) and u (ungerade or odd) refer to the symmetry of the eigenvector under the action of the inversion opera tor, and the symmetry labels (e.g., Fu H) refer to irreducible representations of the icosahedral symmetry group (Ih) (see §4.2). (Note that some authors use the notation T, and T2 in place of the Ft and F2 employed here for the three-dimensional representations of the icosahedral group, and for this reason, we use both notations.) The degeneracy for each mode symmetry (given in parentheses) also follows from group theory: Ag( 1), Au( 1); Flg(3), F2g(3), Flu(3), F2u(3); Gg(4), G„(4); and Hg(5), Hu(5) (see §4.1). Thus Eq. (11.1) enumerates a total of 174 normal mode eigenvectors corresponding to 46 distinct mode frequencies, such that two distinct eigen-frequencies have Ag symmetry, three have Flg symmetry, etc. Starting with 60x3=180 total degrees of freedom for an isolated C60 molecule, and subtracting the six degrees of freedom corresponding to three translations and three rotations, results in 174 vibrational degrees of freedom. Group theory, furthermore, indicates that 10 of the 46 mode frequencies are Raman-active (2Ag + 8Hg) in first order, 4 are infrared (IR)-active (4F,„) in first order, and the remaining 32 are optically silent. Experimental values for all of the 46 mode frequencies, determined by first- and second-order Raman [11.20] and IR [11.21] spectroscopic features, are displayed in Table 11.1 [11.21], where the mode frequencies are identified with their appropriate symmetries and are listed in order of increasing vibrational frequency for each symmetry type.

Despite the high symmetry of the C60 molecule, the eigenvectors are, in general, somewhat difficult to visualize. In Fig. 11.2 we display eigenvectors for the 10 Raman-active and the 4 infrared-active modes [11.27]. The nondegenerate Ag modes, because of the higher symmetry of their eigenvectors, are easier to visualize. As shown in Fig. 11.3, the Ag( 1) "breathing" mode (492 cm-1) involves identical radial displacements for all 60 carbon atoms, whereas the higher-frequency Ag(2) mode, or "pentagonal pinch" mode (1469 cm"1), involves primarily tangential displacements, with a contraction of the pentagonal rings and an expansion of the hexagonal rings. Because these modes have the same symmetry, any linear combination of these modes also has A„ symmetry. Physically, the curvature of the C6g cage gives rise to a small admixture of the radial and tangential modes. As can be seen in Fig. 11.2, the 8 fivefold degenerate Hg modes are much more complex and span the frequency range from 273 [//g(l)] to 1578 cm-1 [//g(8)]. The tendency of the lower-frequency molecular modes to have displacements that are more radial in character than those at higher frequency can be seen in Fig. 11.2. It should be recalled that the eigenvectors shown for the Hg modes are not unique, as is the case for degenerate modes, generally. Each Hg mode is fivefold degenerate with five partners, so that equally valid sets of eigenvectors for a common frequency can be constructed by forming orthonormal linear combinations of these five partners.

Table 11.1

Intramolecular vibrational frequencies of the Cm molecule and their symmetries: experiment and models." The Raman-active modes have Ag and Hg symmetry and the IR-active modes hare F,u symmetries. The remaining 32 modes are silent modes.

Even-parity Odd-parity

01,(32) Frequency (cnr1) Frequency (cm"')

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