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"The parameters a>Pj, ai0/, and Ty (eV) denote the Lorentz oscillator strength, frequency, and linewidth [see Eq. (13.14)].

From these data a static refractive index n(0) = 1.94 [13.179] or a static dielectric constant e,(0) = n2(0) = 3.76 was determined, somewhat lower than the corresponding values for solid C60.

It can be seen in Fig. 13.39 that the optical density log10(l/5") of C70:Suprasil (d = 280 A) predicted from VASE measurements (+ + +) compares favorably to that obtained directly via normal incidence transmission (J") data (solid curve). The dashed curve in Fig. 13.38 shows the optical density for molecular C-,Q in decalin (a solution spectrum), which was obtained from transmission measurements [13.180]. Overall, a striking similarity is found between the absorption spectrum in the Oj0 solid film and C70 in a decalin solution [13.180], consistent with the view that solid Cyo is a molecular solid. Presumably, the intermolecular interaction accounts for the broadening in the film spectrum of the narrower features observed in solution. The inset to Fig. 13.39 shows the absorbance spectrum s#(co) = 1 - [3ft(a>) + ST(«)] (solid curve) for solid C70:Suprasil (d = 280 A) near the threshold for optical absorption E0 in C7q. The rise of above the baseline indicates that E0 = 1.25 ± 0.05 eV for solid C70. The calculated HOMO-LUMO gap between molecular orbitals for an isolated C70 molecule is higher than the observed absorption edge threshold: 1.76 eV [13.181], 1.68 eV [13.182], and 1.65 eV [13.183]. Intermolecular interactions in the solid state, and the on-ball, electron-hole (exciton) interaction would both be expected to lower the optical absorption threshold

Fig. 13.39. Optical density of C70 dissolved in decalin (dashed curve) [13.180], the optical density of a C70 film (280 A thick) deduced from ellipsometry ( + + + ) and that measured directly (solid line) for the same film by normal incidence transmission spectroscopy [13.179], The inset shows the absorbance [.s/(<u)] on a magnified energy scale near the absorption edge for a thin solid C70 film on quartz. The onset for absorption E0 is seen to be near 1.25 ± 0.05 eV for solid C70 [13.179],

Fig. 13.39. Optical density of C70 dissolved in decalin (dashed curve) [13.180], the optical density of a C70 film (280 A thick) deduced from ellipsometry ( + + + ) and that measured directly (solid line) for the same film by normal incidence transmission spectroscopy [13.179], The inset shows the absorbance [.s/(<u)] on a magnified energy scale near the absorption edge for a thin solid C70 film on quartz. The onset for absorption E0 is seen to be near 1.25 ± 0.05 eV for solid C70 [13.179], below the calculated HOMO-LUMO gap. Localization of the electron and hole on a single C70 molecule leads to a relatively strong exciton binding energy, just as for C60 films (see §13.3.2). Molecular orbital calculations for C70 [13.12] reveal a large number of closely spaced orbitals both above and below the calculated HOMO-LUMO gap [13.12], The large number of orbitals for higher-mass fullerenes with lower symmetry makes it difficult to assign particular groups of transitions to the structure that is observed in the solution spectra of C70.

13.7. Dynamic and Nonlinear Optical Properties of Fullerenes

In this section we discuss dynamic optical properties of C60 and related materials in the solid state (§13.7.1). The corresponding discussion for fullerenes in solution is presented in §13.2.3. The nonlinear optical properties of fullerenes in both solution and crystalline phases are summarized in §13.7.2.

13.7.1. Dynamical Properties

A variety of behavior has been reported for the dynamics of the optical response of C60 in the solid state. The two most important techniques that have been used for these studies are pump-probe transmission/reflection and photoinduced optical absorption. These two techniques have also been important for studies of the dynamic properties of CU) in solution (see §13.2.3). In pump-probe experiments, a system is excited by a short light pulse (the pump pulse) and is then interrogated at a later time (the delay time) by the probe pulse. The probe pulse is normally much weaker in intensity and can be at the same or at a different frequency from the pump pulse. In photoinduced absorption spectroscopy, the absorption spectrum of the photoexcited system is probed. Photoinduced optical absorption is of particular interest in fullerenes because the absorption from the ground state is forbidden, while certain excited states are both long-lived and participate in subsequent dipole-allowed transitions. Reasons for the differences in dynamic behavior reported by the various groups are presumably related to the oxygen (and other impurity) content of the fullerene samples and the magnitude of the light intensity of the excitation pulses as discussed below.

Room temperature pump-probe experiments on C60 films near 2 eV have been carried out by several groups [13.184-186], Whereas pump-probe experiments normally show an exponential decay of the photoexcited state which is expressed in terms of a single decay time for subpicosecond pulses, the pump-probe results for C60 films are unusual and show a nonexponen-tial decay of the pump excitation. Analysis of the decay profile for the excited states of C60 shows contributions ranging from fast decay times (t ~ 1 ps) to decay times longer by several orders of magnitude, indicative of a number of relaxation processes for the excited state, including carrier trapping, carrier tunneling, carrier hopping, singlet-triplet decay, polaron formation, and triplet-triplet annihilation [13.184] (see §13.2.3). Decay times faster than 10~9 s have been identified with the decay of the singlet exciton state, while slow components (r > 10~6 s) have been identified with decay of triplet states, consistent with results for C60 in solution (see §13.2.3). Studies of decay times as a function of laser fluence at constant wavelength (such as 597 nm) show a strong decrease in decay times with increasing laser fluence [13.46,187], which may be connected with a phototransformation of the sample. Such a phototransformation lowers the symmetry, thereby enhancing the cross section for dipole-allowed radiative transitions. It is of interest to note that the pump pulse causes a decrease in optical transmission for both C60 and alkali metal-doped C60, consistent with the dominance of laser-induced excited state absorption processes in pump-probe experiments for these materials [13.46].

In contrast to observations in solid C60, the decay in solid C70 is significantly faster at the same laser excitation frequency on' and intensity, where <a' for both materials is measured relative to their respective absorption thresholds. This effect was already reported for the dynamical properties of fullerenes in solution and is related both to the lower symmetry of C70 which provides more dipole-allowed decay channels, and to higher impurity and defect densities in C70 samples. Whereas the decay of C70 is only weakly dependent on laser fluence for wavelengths well below the absorption edge, the excited state decay in C70 is strongly dependent on laser intensity above the absorption edge [13.187],

However, low-temperature (5-10 K) time-resolved photoluminescence and pump-probe measurements on C60 in the solid state show a simple exponential decay [13.50] with a relaxation time of 1.2 ns for the excited state, corresponding to the lifetime of the singlet .S1, state. The low-temperature luminescence varies as the cube of the laser excitation intensity [13.50,186] and exhibits an Arrhenius behavior, yielding an activation energy of 813 meV. This activation energy is similar in magnitude to the energy difference in the orientational potential between adjacent C60 molecules aligned so that an electron-rich double bond is opposite a hexagon rather than opposite a pentagon (see §7.1.3). The nonexponential decay mentioned above is observed at higher temperatures and at higher laser excitation intensities, where the emission spectrum is strongly shifted to lower photon energies. C60 films show different time-resolved luminescence behavior than single crystals [13.50].

Time-resolved absorption studies have also been carried out for cases where the pump and probe are at different photon energies. In these studies the fullerenes are promoted to an excited state and optical transitions to higher-lying states are made before the fullerenes relax to their ground states. Relaxation time measurements of the induced photoabsorption also reveal a fast component of ~2 ps and a slow component > 2 ns [13.188]. The relaxation time for the fast component was found to decrease and the intensity of the induced photoabsorption to increase with increasing laser excitation intensity as shown in Fig. 13.40. Here the induced photoabsorption is found to saturate at a laser excitation energy corresponding to the excitation of one electron—hole pair per Cjq molecule [13.89]. Little temperature dependence was found, indicating that the induced photoabsorption is associated with the free molecule. It was suggested that the induced photoabsorption arises from a molecular distortion of the excited state (see §16.2.3) which lowers the symmetry and therefore increases the transition probability for optical absorption [13.89].

We now discuss photoinduced absorption studies originating from optically pumped excited states. Figure 13.41 shows a broad photoinduced absorption band from 0.8 to 2.2 eV with a maximum at ~1.8 eV in a C60 film [13.64]. Since the induced photoabsorption showed no initial photo-bleaching and no spectral diffusion, it was concluded that the induced photoabsorption response is dominated by photoinduced absorption originating from previously pumped excited states. Since the photoluminescence and induced photoabsorption exhibit the same dependence on the density of excitations, it appears that the induced photoabsorption occurs on a time scale of 10-12 s and is associated with Frenkel singlet excitons. The onset of

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