V

1.1»

hu i,„

~2.0"

l.T

1.8«'

',„ hu

K -> 'is

2.8°

3.0e

2 Jb,d,e

hs + 8S hu

3.6°

3.6e

3.6M'e

hu -» hs

I"

2.7b,d,e

hu hg

4.7"

4.7e

4.5M'e

hs+8g-+ hu

5.5°

5.7e

5.5w,e

"Ref. [13.97,99], "I" denotes transitions induced by alkali doping. fcRef. [13.101]. cRef. [13.73], dRef. [13.102]. cRef. [13.103],

"Ref. [13.97,99], "I" denotes transitions induced by alkali doping. fcRef. [13.101]. cRef. [13.73], dRef. [13.102]. cRef. [13.103], hopping integral tw ~ 0.1/, where t is a C-C hopping integral on a single molecule (t — 1.8 eV). Dipole-allowed transitions for a four-molecule cluster were examined to determine to what extent the charge in the pho-toexcited state is localized on the same molecule (Frenkel exciton) or de-localized over neighboring molecules [charge transfer (CT) exciton] in the photon absorption process [13.95], An example of the first (or band) approach can be found in the work of Xu and co-workers [13.105], who determined the contribution to the dielectric function by calculating the matrix elements for vertical (^-conserving) transitions throughout the Brillouin zone, in addition to calculating the joint density of states.

In the band approach, the band structure for solid C60 was computed in the local density approximation using an orthogonalized, linear combination of atomic orbitals (OLCAO), and the results are in good agreement with other LDA calculations [13.12,80,106-109], In Fig. 13.14 [13.104] we show the resulting electronic density of states (DOS), where the labels Vj and C, in the figure refer to the various valence and conduction bands. Note that the electronic energy bandwidths are small ~0.5-0.6 eV and that the bands are well separated by gaps. A direct gap of 1.34 eV was calculated between the bottom of the conduction band and the top of the valence band at the X-point in the Brillouin zone. However, direct transitions are forbidden at X between the Vx and Q bands of Fig. 13.14. These calculations [13.104] show that the flu-derived band splits into three bands in the vicinity of the Appoint and that transitions are allowed between the top of the valence band and the second highest conduction band near X, giving rise to a calculated optical threshold at 1.46 eV. It is common for LDA-based

ENERGY(eV)

Fig. 13.14. Calculated density of states (DOS) for fee C60 using an orthogonalized linear combination of atomic orbitals within the LDA approximation [13.104]. The valence (V,) and conduction (C,) bonds are labeled relative to the Fermi level which is located between V, and C,.

ENERGY(eV)

Fig. 13.14. Calculated density of states (DOS) for fee C60 using an orthogonalized linear combination of atomic orbitals within the LDA approximation [13.104]. The valence (V,) and conduction (C,) bonds are labeled relative to the Fermi level which is located between V, and C,.

band structure calculations to yield absorption thresholds at lower energies than experimental observations [13.87]. In Fig. 13.15 [13.104] we show the results of these calculations for the imaginary part e2(w) (a) and the real part £[(«) (b) of the dielectric function using both the matrix elements and DOS derived from their band structure calculations. Labeled A through E in Fig. 13.15(a) are five distinct optical absorption bands in the visible-UV range which were assigned by Xu and co-workers [13.106] to particular groups of interband transitions following the notation of Fig. 13.14: (A) Vx Cy\ (B) Vx C2 and V2 Cx; (C) Vx C3 and V2 C2; (D) V2 C3 and Vx C4; (E) Vx -* C5 and V2 C4. In the inset to Fig. 13.15(a), a comparison is made between their calculated optical conductivity o-x(u>) = (to/4ir)e2(w) (solid line, arbitrary units) and the experimental absorption spectrum (dashed line, arbitrary units) for a thin solid C60 film vacuum-deposited on quartz. The relation between the assignments of Xu et al. [13.106] in Fig. 13.14 and Srdanov et al. [13.99] in Table 13.4 is: Vx = hu, V2 — hg + g?, C, = tXu, C2 = tXg, C3 = t2u, C4 = hg, and the C5 level is not identified with a one-electron band (see Fig. 12.1). However, the assignments of the optical absorption bonds are still tentative.

It should be noted that the observed oscillator strength for the Vx C, transition is much lower than that identified with feature A in Fig. 13.15, where Vx and Cx are derived from the HOMO (hu) and LUMO (tXu) molecular orbitals, respectively. The results of Fig. 13.15 [13.104] show that the intermolecular (or solid-state) interaction also admixes gerade character into the conduction and valence band states, thereby activating transitions between HOMO- and LUMO-derived states in the fee solid. We recall that in the free molecule, a vibronic interaction was needed to activate transitions between the hu and tXu orbitals because of optical selection rules. It

Fig. 13.15. Calculated (a) imaginary part of the dielectric function e2(o>), (b) real part of the dielectric function e,(oj), and (c) energy-loss function Im[-l/e(<u)j, all plotted as a function of photon energy. The inset to (a) shows the calculated optical conductivity <7, (A) curve (solid line) compared with experimental visible-UV absorption spectra from Kratschmer and Huffman [13.110] (dashed line) in arbitrary units [13.104], and plotted as a function of wavelength.

23456789 10 ENERGY(eV)

Fig. 13.15. Calculated (a) imaginary part of the dielectric function e2(o>), (b) real part of the dielectric function e,(oj), and (c) energy-loss function Im[-l/e(<u)j, all plotted as a function of photon energy. The inset to (a) shows the calculated optical conductivity <7, (A) curve (solid line) compared with experimental visible-UV absorption spectra from Kratschmer and Huffman [13.110] (dashed line) in arbitrary units [13.104], and plotted as a function of wavelength.

seems reasonable, therefore, to expect that both vibronic and intermolecular interactions contribute to the dipole oscillator strength between narrow hu- and /lu-derived bands in solid C60.

Turning to the higher-energy features B, C, D, E in Fig. 13.15 for the calculated e2(<o) curve, the optical absorption bands are strongest at low energy and diminish in strength with increasing photon energy. Comparison between theory and experiment is shown in the inset to Fig. 13.15(a) [13.104], where it is seen that the calculations overestimate the relative oscillator strength of feature B and underestimate the strength of the higher-energy features C and D. Interestingly, this situation was also encountered in theoretical calculations of the oscillator strengths in the free molecule (see §13.1.2). As shown previously in Fig. 13.5, the calculated free-molecule unscreened response also overestimates significantly the optical response in the visible and underestimates the UV response. However, as pointed out by both Westin and Rosén [13.7] and Bertsch et al. [13.22], the inclusion of electronic correlations (or screening) in the calculations for the oscillator strengths for the isolated molecule leads to a dramatic shift of oscillator strength from the visible to the UV, and thus reasonably good agreement with experiment is achieved for the free molecules (see Fig. 13.5). It is reasonable to expect that this theoretical observation for the free molecule applies to C60 in the solid phase as well, suggesting that a proper treatment of electronic correlations in the solid will also shift oscillator strength to the UV region and thereby improve considerably the agreement between theory and experiment for C60 films.

Louie and co-workers have also calculated the energy band structure in relation to the implied optical properties of solid C60 using an ab initio, quasiparticle approach [13.87,111,112]. Their results for the excited band states also include many-electron corrections to the excitation energy, which they find in solid C60 leads to an increase in the direct gap from 1.04 eV to 2.15 eV. One important advance of this calculation is to show how to correct for the underestimate in the A'-point band gap in C60 which is generally found in LDA calculations [13.111]. The band gap value of 2.15 eV is in better agreement with the gap of 2.3 to 2.6 eV obtained from combined photoemission and inverse photoemission data [13.113-115]. Furthermore, Louie and co-workers find that although the empty tXu and t]g bands exhibit 0.8-1.0 eV dispersion, the angular dependence of the inverse photoemission associated with these bands is surprisingly weak, in agreement with experiment [13.114,116].

It is widely believed that the photoemission gap should be larger than an experimental optical gap based on optical absorption edge measurements [13.111], This is a direct consequence of the attractive interaction between excited electron-hole pairs, which creates excitons and leads to excitation energies less than the band gap between the valence and conduction bands because of the binding energy of the exciton (~0.5 eV in CM). While excitons are excited optically, they are not probed in photoemission and inverse photoemission experiments, which instead probe the density of states for the addition of a quasi-hole or quasi-electron to the solid, or equivalently probe the density of states of the valence band and the conduction band. By inclusion of the Coulomb interaction between the quasi-electrons and the quasi-holes in the conduction and valence bands, Louie has calculated a series of narrow exciton bands in the gap between the valence (Hu) and conduction bands (Tlu) [13.112] and has assigned the appropriate symmetries to these excitons using the relation Hu ® Tlu — Tlg + T2g + Gg+Hg, as discussed in §12.5. The excited state wave functions are found to be largely localized on a single molecule; i.e., they are Frenkel excitons. For singlet excitons they find the lowest exciton (Tlg symmetry) at 1.6 eV, close to the threshold observed for optical absorption in solid C60 (see Fig. 13.12) and leading to an exciton binding energy of (2.15 - 1.60) = 0.55 eV. The onset of a series of triplet excitons is reported starting 0.26 eV below the T2g exciton, in good agreement with the experimental value of 0.28 eV for the splitting of the lowest excited singlet and triplet states from EELS measurements [13.103]. However, since the ground state of solid CM in the many-electron notation has Ag symmetry, electric dipole transitions from the ground state Ag to the excitonic states TXg,T2g,Gg, or Hg are forbidden by parity (see §13.1.3), so that a Herzberg-Teller vibronic interaction mechanism must be invoked to admix ungerade symmetry into the excitonic state, as described above for the isolated C60 molecule. However, it should be recalled that the energy of the final excitonic (vibronic) state includes both the electronic (£00) and the vibrational mode energy (hw), which can range from 30 meV to 190 meV. These vibrational energies represent a significant fraction of the singlet exciton binding energy (0.55 eV) and the singlet-triplet exciton splitting (0.28 eV). The exciton binding energy is determined experimentally from the difference between the HOMO-LUMO gap found by photoemission/inverse photoemission experiments and the electronic energy E00 determined optically.

Harigaya and Abe [13.94] have also analyzed the excitonic structure for a linear four-molecule cluster to simulate solid-state effects. Their goal was to separate the contributions to the optical absorption for solid CM from excitons localized on single molecules from somewhat less localized excitations, where charge is transferred to nearest-neighbor molecules. To model their cluster they required an intermolecular hopping integral tw ~ 0.18 eV. In Fig. 13.16(a) their calculated density of states for occupied and unoccupied states is presented, while Fig. 13.16(b) shows the excitation density vs. excitation energy. The shaded vertical bars in Fig. 13.16(b) correspond

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