Singlet exciton fission is an effect in organic semiconductors where a singlet exciton spontaneously splits into two triplet excitons of half the energy of the parent singlet and have complementary angular momenta. This effect is present in nature and is a charge generation mechanism in systems such as bacteriochlorophyl [1].
Singlet exciton fission is the dissociation of a singlet state exciton to a pair of triplet states whose spins are coupled to an overall singlet state. The net angular momentum remains zero and the singlet can be expressed,
where the subscripts on T indicate the z projection of the magnetic moment. This process is thought to happen on a single molecule and is more specifically referred to as intramolecular homofission. [1] It should be noted that the parent singlet state in the above superposition equation is labeled S* rather than S1 because the state from which fission happens and that which is optically excited are not necessarily one in the same. There is always some state with singlet character which is the superposition of triplets with net zero angular momentum and energy 2T1. This is more a matter of notation than one of special physics, since we can always talk about composite systems of particles following the normal rules of momentum and energy conservation. The real question is, how close is this state to the S1 state (eg. how easy is it to populate this state so that fission may happen)?
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The determining factor in how closely the S1 state couples to the S* state is how close their energies are. Since the S* state is the superposition of its constituent triplet excitons, the S* energy is 2T1. Therefore, the ideal case for S1-S* coupling is E(S1) = 2E(T1). [2] As a practical matter, in order to ensure that a majority of the excitons created have energy S* or greater, one has to pump the sample with energy greater than S*. This is due to the fact that thermalization is an ultrafast process and can compete with fission. Pumping to a higher energy means that singlet excitons have sufficient energy to be in the S* state after some vibrational relaxation has occurred. This was evidenced experimentally by Wohlgenannt et al, who measured triplet photogeneration quantum efficiency versus pump energy and show that at higher energies, the S* state is more efficiently populated and the quantum yield increases to approximately 200%. [3, 4] Figure 1 shows an example of their data that illustrates this principle.
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| Figure 1: Plot of exciton generation quantum efficiency vs pump energy for an experiment similar to the one Wohlgenannt et al performed. [4] |
The singlet-triplet energy splitting is determined by the exchange energy correction to the electron-electron coulomb interaction. Specifically, S1-T1 = 2J where J is the exchange energy. [5] We can see why this is the case if we look at the structure of the exchange energy. The exchange energy is proportional to the overlap in the wavefunctions of the two electrons in the exciton, namely the HOMO (π) and LUMO (π*) wavefunctions. The exact structure of the exchange energy is
and is described well by the Hartree-Fock approximation. [6] Here, r and r' refer to the position of electrons 1 and 2 respectively and the wavefunctions, ψ1 and ψ2 are the position wavefunctions that describe electrons 1 and 2. s1 and s2, the arguments of the delta function, refer to the spins of the two particles. This is present simply to ensure that the particles are truly identical. As a system of fermions, we know that the total wavefunction of an exciton must have fermionic nature (e.g. be antisymmetric under particle exchange). Because the singlet has an antisymmetric spin wavefunction, its position wavefunction is symmetric under exchange. We can therefore see that the singlet exchange energy correction carries a positive sign. The triplet spin wavefunctions, however, are symmetric under exchange and therefore, the position wavefunctions are antisymmetric under particle exchange, and thus the mixed terms, ψ1(r') and ψ2(r) contribute a negative sign to the exchange energy for the triplets. Thus, the singlet-triplet energy splitting (S1-T1) is twice the exchange energy, or 2J.
Aside from the energy requirement, there are also geometrical symmetry requirements that determine if it is more favorable for the S* state to decompose into its constituent triplet states or to remain in its mixed state. There are no cut and dry rules for describing these symmetry requirements, however, it has been suggested that a certain degree of symmetry breaking distortion is required to decompose the S* state into two triplet states. Tavan and Schulten talk about local distortions to a conjugated chain encouraging this triplet serparation effect in polyenes. Gradinaru talks about larger scale conformational distortion to carotenoid molecules (provided by the highly anisotropic biological environment) providing this symmetry breaking effect. [1,7]
© 2007 G. F. Burkhard. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] C. C. Gradinaru et al., "An unusual pathway of excitation energy deactivation in carotenoids: Singlet-to-triplet conversion on an ultrafast timescale in a photosynthetic antenna," Proc. Nat. Acad. Sci. 98, 2364 (2001).
[2] I. Paci et al., "Singlet Fission for Dye-Sensitized Solar Cells: Can a Suitable Sensitizer Be Found?" J. Am. Chem. Soc. 128, 16546 (2006).
[3] M. Wohlgenannt et al., "Singlet Fission in Luminescent and Nonluminescent Pi-Conjugated Polymers," Syn. Metals 101, 267 (1999).
[4] M. Wohlgenannt, W. Graupner, G. Leising and Z. V. Vardeny, "Photogeneration Action Spectroscopy of Neutral and Charged Excitations in Films of a Ladder-Type Poly(Para-Phenylene)," Phys. Rev. Lett. 82, 3344 (1999).
[5] N. J. Turro, Modern Molecular Photochemistry (University Science Books, 1991).
[6] N. W. Ashcroft and N. D. Mermin, Solid State Physics (Brooks Cole, 1976).
[7] P. Tavan and K. Schulten, "Electronic excitations in finite and infinite polyenes," Phys. Rev. B 36, 4337 (1987).