On the basis of the vibrational spectroscopic studies discussed in Chapter 11, it is concluded that fullerenes form highly molecular solids, which imply narrow electronic energy bands. In this chapter it will be shown that these bandwidths are only a factor of 2 to 3 wider than the highest vibrational energies (see §11.3.1). Thus, the electronic structures of the crystalline phases are expected to be closely related to the electronic levels of the isolated fullerene molecules [12.1].
For this reason we start the discussion of the electronic structure of fullerene solids by reviewing the electronic levels for free C60 molecules, the fullerene that has been most extensively investigated. We then discuss modification to the electronic structure of the free molecules through intermolecular interactions which occur in the solid state, as well as modifications to the electronic structure arising from doping. Because of the weak hybridization between the dopant and carbon levels, doping effects are first considered in terms of the electronic states of the C^ molecular ions, and their solid-state effects are then considered, including the case of the metallic alkali metal M3C60 compounds. The molecular states are considered from both a one-electron and a many-electron point of view, and the relation between these approaches is discussed. Also, the electronic structure for the solid is considered from both the standpoint of a highly correlated molecular solid and as a one-electron band solid, and the relation between these approaches is also discussed.
Theoretical work on the electronic structure of fullerenes had relied heavily on comparisons of these calculations with photoemission experiments [12.2,3], because, in principle, photoemission and inverse photoemission experiments are, respectively, sensitive to the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) electronic densities of states. Photoemission studies are reviewed in §17.1. Optical studies provide another powerful tool for studying the electronic structure (see §13.2). Since optical studies involve excitonic effects, the electronic structure of exciton states is reviewed in this chapter.
12.1. Electronic Levels for Free C^ Molecules
Each carbon atom in C60 has two single bonds along adjacent sides of a pentagon and one double bond between two adjoining hexagons (see §3.1). If these bonds were coplanar, they would be very similar to the sp2 trigonal bonding in graphite. The curvature of the C60 surface causes the planar-derived trigonal orbitals to hybridize, thereby admixing some sp3 character into the dominant sp2 planar bonding. The shortening of the double bonds to 1.40 Â and lengthening of the single bonds to 1.46 Â in the hexagonal rings of the C60 molecule also strongly influence the electronic structure. The details of the structural arrangement are important for detailed calculations of the electronic levels.
In this section, we first review the general framework of the electronic structure imposed by symmetry considerations, which is followed by a discussion of the molecular ground state and the corresponding excited states, including the electronic structure of CM as well as higher mass fullerene molecules.
Several models for the electronic structure of fullerene molecules have been developed, ranging from one-electron Hiickel calculations or tight-binding models [12.4] to first principles models [12.5]. While sophisticated models yield somewhat more quantitative agreement with optical, photoemission, and other experiments sensitive to the electronic structure, the simple Hiickel models lead to the same level ordering near the Fermi level and for this reason are often used for physical discussions of the electronic structure.
In the simple models, each carbon atom in C«, is equivalent to every other carbon atom, each being located at an equivalent site at the vertices of the truncated icosahedron which describes the structure of the C60 molecule. By introducing the atomic potential for the isolated carbon atom at each of these sites, the one-electron energy levels in a tight-binding model can be calculated. Corresponding to each carbon atom, which is in column IV of the periodic table, are three distorted (sp2) bonds which are predominantly confined to the shell of the C60 molecule, coupling each carbon atom to its three nearest neighbors and giving rise to three occupied bonding a orbitals and one occupied bonding it orbital normal to the shell, thus accounting for the four valence electrons. The a orbitals lie low in energy, roughly 3 to 6 eV below the Fermi level EF (roughly estimated from the C-C cohesive energy [12.6]), and therefore do not contribute significantly to the electronic transport or optical properties, which are dominated by the tt orbitals, lying near EF.
The ordering of the tt molecular energy levels and their icosahedral symmetry identifications based on a Hiickel calculation [12.4] are shown in Fig. 12.1, where the filled levels are indicated as '+' and the unoccupied states by '-'. Figure 12.1 and the results of a local density calculation of the molecular energy levels for the C60 molecule shown in Fig. 12.2(a) are both widely used in the literature for discussion of the electronic structure of the C60 molecule. For example, the molecular orbital calculation in Fig. 12.2(a) implies that the bonding cr electrons lie more than 7 eV below the HOMO level of C50 and the antibonding a electron states lie more than 6 eV above the Fermi level. According to the energy scale of the a(sp2) bonding and antibonding states in fullerenes, the bandwidth of the molecular levels in Fig. 12.2 is large (~25 eV) and of similar magnitude to the corresponding states in graphite.
A number of calculations of the electronic states for the free Qq molecule have been carried out using a variety of calculational methods [12.5,8-10], and general agreement is found with the level ordering shown in Figs. 12.1 and 12.2(a). Some models have even been successful in obtaining bond lengths of 1.45 A for as and 1.40 A for a6 [12.11], in good agreement with experimental values (see §3.1) [12.12,13], Another relevant energy is the ionization potential for the free molecule (7.6 eV), denoting the energy needed to remove an electron from neutral C60 to create the CJ, ion. This energy is to be compared with the much smaller electron affinity (2.65 eV [12.14]), the energy needed to add one electron, giving rise to the Q"0 ion. Whereas the electron affinity for C^ is similar to that for other acceptor molecules (e.g., TCNQ, for which the electron affinity is 2.82 eV), the large ionization potential for C60 makes it unlikely that C60 would behave as an electron donor.
Since Qq and other closed cage fullerene molecules have an approximately spherical shape, some authors have used a phenomenological approach to the electronic structure based on symmetry considerations [12.15, 16]. According to this spherical shell approach, the fullerene eigenstates can be simply described by spherical harmonics and classified by their angular momentum quantum numbers. The icosahedral (or lower) symmetry of the fullerene molecule is then imposed and treated as a perturbation, causing
HÖCKEL MOLECULAR ORBITALS
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