Optical Properties

In this chapter we discuss primarily the optical response of fullerenes over a photon energy range from the near infrared (IR) through the ultraviolet (UV) region of the spectrum (1-6 eV). This region of the spectrum is used to explore the nature of the electronic states of fullerenes, whether they are studied in the gas phase, in solution, or assembled in the solid state. Results of experiments to obtain the linear, as well as the nonlinear, optical response are considered and compared, when possible, with theoretical models and calculations. Most of the work to date has focused on C60. A much smaller body of information is available on the optical properties of C70 and higher fullerenes. Also included in this chapter is a review of current knowledge of the optical properties of doped and phototransformed fullerenes.

13.1. Optical Response of Isolated C60 Molecules

This section provides a framework for discussion of the absorption and luminescence spectra for isolated C60 molecules, either in the gas phase or in solution. The symmetries of the ground state and excitonic states associated with photoabsorption are considered. It is shown that vibronic states which couple the excitonic states to vibrational states are needed for electric dipole transitions near the optical absorption edge, and the symmetries for vibronic states involved in absorption, luminescence, intersystem crossings, and phosphorescence are discussed. Optical absorption at higher photon energies to dipole-allowed states is also reviewed.

13.1.1. Introduction to Molecular Photophysics

In this subsection we discuss the photophysical properties of isolated C60 molecules. This molecular isolation can be realized in the gas phase and can also be achieved approximately for molecules in solution. Since C60 is soluble in a wide variety of solvents (see Table 5.1), numerous optical studies on solutions have been carried out to investigate the "photophysics" of this unusual molecule. By a "photophysical" process, it is usually meant that an electronic excitation is achieved through the absorption of a photon, without a resulting chemical change (e.g., fragmentation or bond rearrangement). Photophysical processes can either be unimolecular or bimolecular (i.e., involve one or multiple molecules), and they can involve the absorption or emission of one or several photons. In the case of CM, and to some degree C70, photochemical transformation to a dimer or oligomer has been reported [13.1-5]. However, these reports refer to fullerenes in the solid state, where the molecules are in closer contact. Thus, these photochemical changes are much less likely (but also possible) in solution.

Photoexcited electronic states of molecules, in general, include spin triplet (5 = 1) and spin singlet (5 = 0) states arising from the excited electron and the hole that is left behind. The notion of triplet states (e.g., spin-paired electronic states with a multiplicity of 25 + 1 = 3) is quite familiar to chemists, and possibly less familiar to solid-state physicists and material scientists. The conventional notation denoting these states in photophysical processes is as follows [13.6]:

Because of the coupling of vibrational and electronic levels by the electron-molecular vibration (EMV) interaction, a ladder of "vibronic" states with approximate energy is realized, where £, and hwj are, respectively, particular (renormalized) electronic and vibrational energies, and n is an integer n — 0,1,2, — In Fig. 13.1, we show schematically optical transitions between the ground state vibronic ladder and an excited state vibronic ladder for a generic molecule. The transition 1 -> 0 is commonly called a "hot band" because elevated temperatures are needed to populate the n — 1 vibrational level in the ground electronic state (e.g., a 250 cm-1 molecular vibration corre-

ground singlet state first excited singlet state

(p > 1) higher excited singlet states first excited triplet state (q > 1) higher excited triplet states.

Fig. 13.1. Schematic level diagram for vibronic multiphonon absorption transitions from the singlet S0 electronic ground state to the first excited singlet electronic state S,. The symbols n m denote the optical transitions between vibronic levels associated with the initial and final electronic states E0 and E'0, respectively.

13. Optical Properties 3

So sponds to a temperature of 360 K). Note that the vibrational frequency of a particular vibrational mode in the electronic ground state should differ slightly from the frequency of this mode in the excited electronic states, because of a softening of the harmonic potential due to the excited state electronic configuration. The energy Eofy (see Fig. 13.1) denotes the energy difference between the excited and ground electronic zero vibrational states, while En m, denotes the energy difference between the vibrational states of En and E'm, where the prime notation is used to denote the excited electronic state (as shown in Fig. 13.1). These vibronic ladders can be generated from either triplet or singlet electronic states, and, as we shall see in §13.1.3, these vibronic states may be required in the case of C60 to understand the low-lying weak, optical absorption, near the absorption edge.

In Fig. 13.2, we display a generic energy level diagram for an isolated molecule, showing only the n = 0 vibronic states (i.e., only the base of each vibronic ladder of Fig. 13.1). For clarity, the singlet and triplet states are formally separated in Fig. 13.2; singlet and triplet states are drawn to the left and right, respectively. The molecule depicted in Fig. 13.2 has a singlet ground state (S0), as does C60. Of particular interest to us here are the following transitions:

Tt T3

Fig. 13.2. Level diagram for optical absorption and emission transitions between singlet and triplet molecular states. Solid lines denote optical transitions, and dashed lines denote radiationless transitions. The dotted lines denote intersystem crossings.

1. Excited state absorption (5, Sp and Tx Tp; p > 1), which are spin-conserving transitions, and therefore can be quite strong (shown in Fig. 13.2 as solid lines).

3. Phosphorescence (Tx 50).

4. Internal conversion (Sp —> Sp_x and Tp -v Tp_]), which are radiation-less transitions (shown as dashed vertical lines in Fig. 13.2).

5. Intersystem crossing (Sp -*■ Tq for p > 0, or Tq —Sp for q > 0), which requires the spin-orbit interaction to mix levels with differing spin [the selection rule is that there is no symmetry change for the electronic wave function in an intersystem crossing, i.e., T,(5) -> r,(T)].

Typically, internal conversion is a rapid, spin-conserving process or series of processes which returns the photoexcited molecule to either an 5) or Tt state. From these states, luminescence, either fluorescence (Sj) or phosphorescence (Tj), can occur. Because phosphorescence is not a spin-conserving transition (and the spin-orbit interaction is needed to couple the 7\ and S0 levels), the radiative lifetime of the Tx state is long. Duration times for phosphorescent emission from a collection of molecules are typically ~ 1 ms to 10 s, while typical duration times for fluorescent luminescence are considerably shorter, ~ 1 ns to 1 /xs.

In thinking about the nature of the photoexcited states discussed above, it is important to realize that, in principle, all relevant interactions are included in determining the eigenfunctions and eigenenergies, such as electronic many-body effects. We discuss this point further in §13.1.2 in connection with the appropriate excited states in C60.

13.1.2. Dipole-Allowed Transitions

In this section we consider the photoexcited states for molecular C60. Near the threshold for optical absorption, the oscillator strengths for optical absorption are very weak because the optical transitions are forbidden by the electric dipole absorption process. In this regime, the absorption and luminescence processes are described in terms of vibronic states associated with excitonic ladders shown in Figs. 13.1 and 13.2. At higher photon energies, allowed electric dipole transitions can occur and a simple approach is used to describe the dipole-allowed optical transitions. To illustrate the two regimes for the optical transitions (forbidden transitions at low photon energies and allowed transitions at higher energies), it is instructive to consider the photon energy dependence of the electric dipole oscillator strength which denotes the square of the matrix element between initial and final states in an optical transition.

In Fig. 13.3 we compare the electric dipole oscillator strength for transitions from the ground state of a C60 molecule as calculated by Westin and Rosén [13.7,8] and by Negri et al. [13.9], The Westin and Rosén calculation uses one-electron wave functions (i.e., molecular orbitals) while the Negri calculation is a quantum chemical calculation using the full molecular many-electron wave function with configuration mixing. In these calculations, the area under an optical absorption band for an isolated molecule is approximately proportional to hwf where hu> is the photon energy and / is the oscillator strength of the particular transition as shown in Fig. 13.3. The photon energy weight magnifies the absorption in the UV relative to the visible region of the spectrum. Optical bands identified with a particular excitation may be broadened by several interactions: molecule-solvent, electron-molecular vibration, or intermolecular interactions, the latter being more important in the solid state.

Both calculations in Fig. 13.3 consider only first-order optical processes; i.e., the excited state has one hole and one electron. The calculations of Negri et al. explicitly take into account the electron-hole interaction in the excited state, whereas the single-particle calculations of Westin and Rosén

13.1. Optical Response of Isolated Cm Molecules 5.0

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