1.2 Molecular Aspects of Solid C₆₀ Compounds

Solid C₆₀ and its alkali intercalated compounds are molecular solids, i.e. their solid state properties bear a strong relationship to the properties of their isolated molecular constituents. Therefore, this section begins with a discussion of some general features of the alkali fullerides which originate in their molecularity: i) their electronic band structures and ii) the structure of their phonon spectra.

The electronic band structure of pure C₆₀, within a few eV of the Fermi Energy (E_F), consists of bands composed of the $\pi$molecular orbitals discussed in the previous section. The bands are narrow because the intermolecular overlap is small. Due to the high electron affinity of C₆₀, doping with alkali metals results in essentially complete charge transfer (ionization of the alkali). In the alkali fullerides, bands in which the electron is associated with the alkali ion lie far above E_F. Thus the band structure near E_F is again determined by the C₆₀ molecular orbitals. The simplest model of the band structure of the alkali-fullerides is the ``rigid band'' picture. In this model the intermolecular overlap is assumed to change little upon doping, and the bands are determined by those of the undoped solid C<sub>60</sub>. By changing the doping level, one simply alters the filling of the ``conduction'' band derived from the LUMO of C₆₀, so the filling can be tuned from empty to full as the alkali stoichiometry varies from 0 to 6 per C₆₀. C₆₀ and alkali fulleride band structure calculations are reviewed in several places, e.g. [4,6,12]. There is also abundant experimental confirmation of the abovementioned general features of the band structures which are reviewed in, e.g. [4,6,13].

While this simplistic picture explains the semiconducting nature of pure solid C₆₀, the insulating nature of A₆C₆₀, and the metallic nature of A₃C₆₀, it does not account for the observed non-metallic behaviour of, e.g. K₄C₆₀[14], since 4 electrons per C₆₀ should yield a partially filled conduction band and thus metallic behaviour. This brings us to another general aspect of the electronic structure of metallic molecular solids, that is, their tendency to undergo metal-insulator transitions which compete with the superconducting transition. An excellent discussion of this topic may be found in the article by Rosseinsky[15] which places the conducting alkali fullerides in the context of other molecular metals. The repulsive coulomb interaction between electrons competes with their kinetic energy (e.g. [16]). When the kinetic term dominates, the band (or Fermi liquid theory) description of the conduction electrons is valid, but if the Coulomb repulsion is sufficiently strong, the electrons may become localized, and form a Mott-Hubbard insulator. Recent experimental work suggests that this is the case in A₄C₆₀ [17]. Generally, the limited kinetic energy of narrow band metals allows any interaction that tends to localize the carriers (e.g. electron-electron, electron-phonon, electron-defect interactions) to have a better chance of doing so.

Another class of metal-insulator transitions is also important in quasi-one-dimensional molecular metals. In this case, nesting of the Fermi surface causes a feature (divergence in the ideal 1d case) in the electron susceptibility at a particular wavevector called the nesting wavevector, ${\bf q_N}$ (discussed below). The effect can occur in any dimension, but is most pronounced in 1d. Transitions of this class are known as Peierls, Spin-Peierls, Charge Density Wave, and Spin Density Wave (SDW) (see, e.g. [18]). Such transitions are probably not important in the A₄C₆₀ situation, but may be relevant to the metal-insulator transitions in Rb₁C₆₀ and Cs₁C₆₀ (see section 1.2.2). For this reason, a brief description of the SDW state is included here.

In contrast to the localization of conduction electrons in a Mott-Hubbard insulator, in the SDW (and similar states), the conduction electrons remain delocalized, but their spectrum becomes gapped at E_F. The phenomenon was first suggested by Overhauser[19] in the context of the free electron gas. The Fermi surface is found to be unstable to perturbations at the wavevector ${\bf q_N}$, and the Coulomb interaction (between electrons) has a component at this wavevector, since the (unscreened) q-space form of this interaction is

Screening, however, eliminates the SDW instability in most metals (e.g., see footnote 27 on p684 of ref.[20]). The nesting wavevector is just a wavevector that maps one part of the Fermi Surface into another. For a spherical (3d free electron) Fermi Surface such a wavevector exists for any state; however, that wavevector only maps a single state into a state on the opposite side of the Fermi Surface. In this situation, the Fermi Surface is not nested. The nesting criterion is just that there exists a wavevector that maps one part of the Fermi Surface into another for a significant fraction of the Surface. Nesting is said to be ``perfect'' in 1d, where ${\bf q_N} = 2{\bf k_F}$ maps the entirety of one branch of the Fermi surface into the other branch. In higher dimensions, nesting is never perfect, and the degree of nesting depends on the geometry of the Fermi Surface. The origin of the SDW instability is the degeneracy of states separated by ${\bf q_N}$, i.e. states at the Fermi Surface are all degenerate in energy, thus any perturbation with a Fourier component at ${\bf q_N}$which mixes these states will lead to an instability (as in degenerate perturbation theory).

SDWs have been found for example in such 3d systems as chromium and its alloys(e.g. [21,22]) and 1d molecular metals such as the Bechgaard salt (TMTSF)₂PF₆, where TMTSF is tetramethyltetraselenafulvalene (e.g. [23]). In a SDW, the density of conduction electrons contains periodic spatial modulations at the nesting wavevector which are not in phase for the two spin states, the consequence is a continuously modulated net magnetic moment, which can be quite small, e.g. $10^{-2}\mu_B$.A SDW is thus related to the antiferromagnetic state of local moments, in the sense that its ordering wavevector is away from the Brillouin zone centre. The SDW wavevector will not, in general, be commensurate with the underlying lattice.

The acoustic properties of solid C₆₀ and its alkali salts also exhibit conspicuous molecular features. For pure C₆₀, the basis of the crystal lattice consists of (at least) the 60 carbon atoms, and, thus, there are at least $3 \times 60$ modes. For simple solids with a small basis, the phonon modes are conveniently categorized into Longitudinal/Transverse/Optical/Acoustic modes. For a molecular solid, the appropriate classification scheme is to connect the intramolecular phonon branches to their molecular vibration counterparts. The remaining intermolecular modes are of two types: vibrational modes which are of the standard 1 atom per basis varieties (with the C₆₀ molecule taking the role of a large atom), and the rotational modes (librons). One can make this molecular picture more concrete by considering the well discussed example of the diatomic linear chain (e.g. chapter 4 of [24]). If we take the molecular limit, where the bonding is much stronger within a molecule than between molecules, 2 of the optical branches correspond to the intramolecular modes, and the characteristic frequency is near that of the vibrational modes of the free molecule. The 3 longitudinal modes and remaining optical mode correspond to the intermolecular and librational modes, respectively. The libronic mode is optical (e.g. [25]) and its frequency can be found from an effective orientational potential for the molecule in the solid.