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Fig. 12.5. Displacement of the carbon atoms from Cg2 to [email protected], as a function of the La-carbon distance [12.60].

12.7. Electronic Structure of Fullerenes in the Solid State

In this section we first review unusual aspects of the electronic structure for fullerenes in the solid state and approaches that have been taken to treat the electronic structure of these unusual materials. In the previous sections it has been emphasized that Qo is a molecular solid, thereby implying that the electronic structure is primarily determined by that of the free molecule, although the intermolecular interaction is of course significant, as evidenced by the magenta color of C60 in solution and the yellow color of the solid film [12.62]. To date, most of the experimental and theoretical work on the electronic structure has been directed to C60 and doped C60, and this historical emphasis is reflected in the review presented here. A summary of progress in understanding the electronic structure of higher fullerenes is also included.

12.7.1. Overview of the Electronic Structure in the Solid State

Numerous experiments regarding the lattice vibrations (§11.4), optical properties (§13.3), photoemission and inverse photoemission (§17.1), and Auger spectroscopy (§17.3) indicate the molecular nature of fullerite solids, based on the narrow bandwidths, the weak dispersion of these energy bands, and the strong intramolecular Coulomb interaction of holes and electrons. From an experimental standpoint, the highly molecular nature of fullerite solids makes it possible to carry out classes of experiments that normally are not possible in other crystalline materials, such as the study of harmonics and combination modes in the vibrational spectra (§11.5.3 and §11.5.4).

To treat the electronic structure of fullerite solids, two approaches have been taken. One approach involves one-electron band calculations, exploiting the periodicity of the crystal lattice of fullerenes in the solid state. In this approach the molecular levels of the fullerenes are treated much as the electronic levels of atoms in more conventional solids, and the energy bands are calculated on the basis of these molecular states. Thus the highest occupied (hu) state of the Qo cluster forms the valence band and the lowest unoccupied (ilu) state forms the conduction band. This approach has the advantage that sophisticated energy band calculational techniques are presently available and many computer codes have been developed, so that many fundamental issues which relate to structure-property relations can be addressed [12.63]. For these reasons band structure calculations have added in a major way toward increasing our understanding of experiments pertaining to the electronic structure and the density of electronic states. Ab initio calculations, using for example the local density approximation (LDA), allow calculation also of crystal structures and lattice constants, although most calculations that have thus far been reported assume experimentally determined crystal structures and lattice constants. Differences between the many ab initio calculations [12.25,64-71] are likely due to small differences in the selection of lattice constants and structural geometry [12.72], and also the calculational techniques that are used [12.5]. Modifications to the calculated band structure due to Coulomb interactions, narrow band features, intramolecular effects, merohedral disorder (see §7.1.4), and other effects described by interaction energies comparable to the Qq-Qo cohesive energy of 1.6 eV are then considered.

A second approach considers the intramolecular interactions to be dominant and band effects to be less important. This more chemical approach has been most helpful in gaining physical understanding of experimental results, such as the relation between optical absorption and luminescence spectra, high-resolution Auger spectra, and photoemission-inverse photoemission spectra [12.73]. The chemical approach has also been helpful in elucidating the magnitudes of many-body effects, such as the Coulomb on-site repulsion U between two holes [12.73] (see §12.7.7). Both approaches have thus provided complementary insights into the electronic structure of fullerite solids, as reviewed below. At the present time, neither approach has by itself been able to explain all the experimental observations.

Both approaches to the calculation of the electronic structure emphasize the narrow band nature of this highly correlated electron system with an electron density of states close to that of the free molecule. Both approaches emphasize the value of photoemission and inverse photoemission experiments for providing determinations of the HOMO-LUMO gap, without complications from excitonic and vibronic effects associated with inter-

band excitations. Arguments, based on the low density of dangling bonds, suggest that surface reconstruction is not as important for fullerite surfaces as for typical semiconductors. Although photoemission and inverse photoemission experiments are generally highly surface-sensitive, they are expected to yield information on bulk properties for the case of fullerenes, because of the low density of dangling bonds and surface states.

A totally different kind of electronic structure calculation is the phe-nomenological tight-binding approach that employs crystal symmetry and uses experimental measurements to determine the coefficients (band parameters) of the model. Such calculations have now been carried out for Qo [12.74,75] and the expansion coefficients of these models have been evaluated by comparison to experiment [12.74-77]. Because of the relative simplicity of the tight-binding calculation, more complicated physical cases can be handled, such as the quadridirectional orientational ordering of C60 below Tm (see Fig. 7.5). Optimization of intermolecular bond lengths and geometrical representations can be handled relatively easily by such an approach [12.78-80]. Since tight binding calculations are so closely related to group theoretical approaches, they can be used in conjunction with ab initio calculations in desired ways, by parameterizing the ab initio calculations and extending them to include additional interactions, using parameters that are evaluated by comparison with experiment.

The discussion given here for the various approaches for calculating the electronic structure of crystalline C60 also applies to calculating the electronic structure for both lower-mass and higher-mass fullerenes (see §12.7.6), as well as to doped fullerenes (see §12.7.3), as discussed below.

12.7.2. Band Calculations for Solid C^

The band approach to the electronic structure of Cm gained early favor because it provided important insights into important experimental progress in the field [12.25,81]. Early band calculations found a band gap for C60 of ~1.5 eV, close to the onset of the experimental optical absorption and close to the low-energy electron energy loss peak at 1.55 eV [12.82-84]. Although later experiments showed the optical absorption edge to be more complicated and not simply connected to the HOMO-LUMO gap (see §13.1.2), the apparent early successes of the LDA approach stimulated many detailed band calculations on fullerene-based solids, which have since provided many valuable insights, even if answers to all the questions raised by the many experiments reported thus far have not yet been found.

A typical and widely referenced band calculation for C60 is the total energy electronic structure calculation using a norm-conserving pseudopotential, an LDA in density functional theory, and a Gaussian-orbital basis set

[12.25], The results of this calculation for C60 are shown in Fig. 12.2(b), including the dispersion relations associated with the HOMO and LUMO levels of the C60 crystalline solid, where we note the calculated HOMO-LUMO band gap is 1.5 eV, and the bandwidth is ~0.4 eV for each of these bands. The calculation of the dispersion relations is sensitive to the crystal structure of solid Qq and the lattice constant. This small calculated bandwidth is in agreement with optical and transport experiments, although the calculated band gap by the LDA method is now known to be underestimated [12.5,9,25]. One success of the LDA approach to the C60 molecule has been the good agreement of the bond lengths along the pentagon edges as and those shared by two hexagons a6 from the LDA calculation (a5 = 1.45 A and a6 = 1.40 A) [12.25] and from nuclear magnetic resonance (NMR) measurements (a5 = 1.46 A and a6 = 1.40 A) [12.85,86]. Since these bond lengths are unchanged as the molecule enters the solid phase, this result shows that the band calculations retain the molecular integrity of the fullerenes, as experiment requires.

The agreement of the calculated Qq-Qq cohesive binding energy (1.6 eV) with thermodynamic experiments (1.7 eV/Qo) [12.87] for the heat of sublimation/vaporization is another correct feature of band theory. Furthermore, LDA band theory gives a value of the Qo-Qo intermolecular distance that is only 4% smaller [12.25] and a pseudopotential approach yields a distance 1 to 2% smaller [12.64] than measured by x-rays [12.88]. Using the energy band dispersion relations calculated from a pseudopotential approach, an equation of state has been calculated [12.64] from which a bulk modulus of 16.5 to 18.5 GPa was obtained, in good agreement with early experimental results, but larger than experimental values reported later (see Table 7.1).

Electronic energy band calculations [12.25,64,69,89] also offer predictions valuable for the interpretation of experiments, such as the energies of one-electron optical transitions and electron density contours (see Fig. 12.6). The charge density contours are important for understanding intermolecular charge transport during conduction (§14.1). Figure 12.2(b) further predicts Qq to be a direct gap semiconductor (gap at the Z-point in the fee Brillouin zone), in agreement with other LDA calculations, yielding values for the band gaps of 1.5 eV [12.7], 1.4 eV [12.90] and 1.2 eV [12.91]. These and other related calculations all locate the band extrema at the AT-point in the Brillouin zone.

Band structure calculations for simple cubic (the low-temperature structure below T01) and fee Qq (the room temperature phase) have been carried out using the tight-binding calculation, showing the sensitivity of the band dispersion to the crystal structure [12.25,75,92], These calculations show a narrowing of the hu and ilu-derived bands below T01, and the density of

Fig. 12.6. Charge density map for a uni-directionally oriented array of CM molecules normal to the (110) direction, showing weak coupling between neighboring Cw molecules [12.25],

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