Energy (eV)

Energy (eV)

Fig. 12.18. The density of states of the /,„ LUMO band for the Fm3, Pa3, and isotropically merohedrally disordered (labeled mero) structures, computed from quasiparticle GW band structure calculations [12.9], much change in the width of the density of states feature [12.9]. While the quasiparticle GW approach considers many-body corrections to the LDA approach, little work has thus far been published linking this approach to calculations starting from a many-body point of view, discussed below.

The second approach used to consider many-body effects starts from a many-body point of view. A number of many-body calculations have focused on calculations of quantities such as the Coulomb repulsion U of two holes on the same C6n molecule. Many different calculational techniques have been used by different groups, yielding values of U in the ~0.8-2.1 eV range [12.33,73,138,140-147],

A widely quoted determination of U comes from analysis of the valence photoemission spectra with the Auger electron spectra, which was first carried out for C60 [12.73] and then extended to K3C60 and K^C^,, yielding similar Hubbard correlation energies U = 1.4 ± 0.2 eV for C60 and K3C60 and U = 1.5±0.2 eV for K^Qo [12.150]. In the initial work on C60 [12.73], a shift of 0.6 ± 0.3 eV was measured between the Auger spectra and the self-convoluted valence photoemission spectrum for C60. If the core hole lifetime broadening is much smaller than the half-width of the LUMO band, the screening of the core hole will occur before the Auger process takes place [12.150]. The authors thus conclude that the intermolecular screening does not occur during the XPS process, but it occurs before the Auger decay [12.150],

Many direct calculations of U have been reported. Both theoretical studies and the more empirical investigations which combine experimental measurements with many-body theory conclude that U is greater than the electronic bandwidth W, which is in the 0.2-0.5 eV range, indicating that fullerenes are strongly correlated systems and that level degeneracies and electron filling may significantly affect the electronic properties of fullerite and fulleride materials.

Realistic LDA and semiempirical calculations have been used to obtain theoretical estimations for the effective on-site Coulomb repulsion U. Antropov et al. [12.143] performed linear-muffin-tin-orbital (LMTO) calculations within the atomic-sphere approximation (ASA) and the local density approximation (LDA) to obtain an unscreened value of U0 = 2.7 eV for the isolated C60 molecule. To obtain U for the fee solid, the screening due to polarization of the surrounding molecules was then included by assigning a polarizability to each molecule. A corrected value for U = 0.8-1.3 eV was then obtained for fee C60, in good agreement with the semiempirical approach discussed above. Explicit investigation of screening effects was made [12.144] by calculating the dielectric function for fee C60 within the LDA technique and a value of U = 2.1 eV was obtained, somewhat larger than the values quoted above. Semiempirical modified-neglect-of-differential-overlap (MNDO) quantum chemistry calculations have yielded U ~ 3 eV for the isolated C60 molecule [12.145], which was then corrected for the polarization in the solid state, using an empirical model for the polarization, to obtain U = 1.3 eV for K3C60, in very good agreement with the semiempirical approach [12.73] discussed above.

Referring to Fig. 12.17 for the photoemission (PES) and inverse photoemission (IPES) spectra for C60 [12.73], we see an energy difference of 3.5 eV between the IPES and PES peaks and an energy difference of 2.3 eV between the onsets, which are determined as the point of maximum change in slope in the onset region. This difference in energy (1.2 eV) is close to the calculated value of U. If we then use the value of 1.84 eV for the optical absorption edge (see §17.1.1), a value of 0.46 eV is obtained for the exciton binding energy.

Merohedral disorder or local misorientation of the C60 icosahedra has been shown [12.93] to give rise to large shifts in the energy bands (~1 eV), also indicating a need to consider many-body effects. The disorder in molecular orientations leads to random hopping matrices, which tend to smooth out the spikes in the density of states curves (as shown in Fig. 12.18), and these orientational effects have a significant bearing on measurement of various electronic and transport properties. Because of localization and correlation effects, the Fermi surface of doped C60 may not be well defined.

The short electron lifetime (h/r ~ EF) also introduces uncertainty into the quasiparticle concept used to describe the superconducting state.

One many-body approach to the fullerene electronic structure describes the optical absorption or emission process in terms of a two-band Hubbard Hamiltonian

X = £ + E + U I>/.„"/-a + ml^m^), (12.12)

where nka = ckack(T and mka = d.\adka are, respectively, the number densities for the /i„-derived hole band and the /lu-derived electron band, where ek and rjk are the corresponding energies for these levels and U is the on-site Coulomb interaction between two electrons on the same fullerene molecule. This Hubbard model in Eq. (12.12) has been applied to account for the intensities of the various features in the KW Auger spectrum (see §17.3) and to the interpretation of the optical spectra in solid C60 (see §13.3) in terms of excitonic states in strongly correlated bands where U » W, and W is the level width [12.73]. The Coulomb repulsion U is related to the ionization potential E,, the electron affinity EA, and the HOMO-LUMO splitting in the free molecule A by

in which A = — e^, where the bars denote average values. To obtain estimates for U in the solid state, it is necessary to introduce polarization effects, since E, = 7.6 eV and EA = 2.65 eV are usually measured in the gas-phase. For the free molecule, E, = 7.6 eV [12.151-153] and E4 = 2.65 eV [12.154], and using an estimate of A = 1.9 eV [12.7] from Fig. 12.2 for the free molecule, we obtain Umo[ = 3.1 eV.

Polarization effects lower the ionization energy E, by the polarization energy EP = ze2a/2RA, where z is the number of nearest-neighbors (in the fee lattice 2 = 12), a ~ 85 Â3 is the polarizability [12.155-159], and R = 10.02 Â is the distance between C60 molecules, yielding EP = 0.69 eV. Correspondingly, the polarization of the medium increases the electron affinity EA by the same polarization energy EP. Thus U in the solid is decreased by 2EP = 1.4 eV relative to Umo] = 3.1 eV for the free molecule, U = Um01 - 2EP, yielding a value for U in the solid of 1.7 eV, in good agreement with the measured value of 1.6 eV [12.73].

Furthermore, the band gap Em in the solid is related to the HOMO-LUMO splitting in the molecule A through the relation [12.73]

where it is assumed that the bandwidths for the HOMO- and LUMO-derived bands are approximately equal, Whu ~ W,u. Average values from the literature for the pertinent energies for C60 are: A = 1.8 eV [12.140], U = 0.8 to 1.6 eV [12.73,141,160-166], EA = 2.65 eV [12.154], E, = 7.58 eV [12.167] (where E, and EA are given for free CM molecules), and W = 0.2 - 0.5 eV. Thus using U = 1.2 eV, A = 1.8 eV, and W = 0.4 eV, we obtain an estimated value for Egap ~ 2.6 eV, roughly consistent with the value of 2.3 eV obtained from the photoemission and inverse photoemission data shown in Fig. 12.17 [12.140], For discussion of the application of Auger spectroscopy for the determination of U see §17.3.

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