March 18, 2007

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 S_{1} 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 2T_{1}. 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
S_{1} state (eg. how easy is it to populate this state so that
fission may happen)?

The determining factor in how closely the S_{1} 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
2T_{1}. Therefore, the ideal case for S_{1}-S* coupling
is E(S_{1}) = 2E(T_{1}). [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.

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, S_{1}-T_{1} = 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, ψ_{1} and
ψ_{2} are the position wavefunctions that describe electrons
1 and 2. s_{1} and s_{2}, 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, ψ_{1}(**r'**) and
ψ_{2}(**r**) contribute a negative sign to the exchange
energy for the triplets. Thus, the singlet-triplet energy splitting
(S_{1}-T_{1}) 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).