Mark, many researchers would tend to disagree with your description. As far as I have seen, most researchers do not describe exciton diffusion as the actual movement of the electron and hole. Instead, exciton diffusion is usually thought to occur through an energy transfer process. Instead of the electron and hole moving, the energy absorbed by a particular molecular site is transferred to another nearby molecular site. I recommend looking up papers about Förster Resonance Energy Transfer (FRET) for more information about how the mechanism works. This doesn't mean that electron motion cannot occur, as Dexter Electron Exchange is another mechanism in which the actually move by swapping positions. However, in this mechanism, two electrons are exchanged with each other in one step. These two mechanisms have different distance dependencies and in reality I would guess that both may occur to some extent depending on the material. I still think that FRET dominates in most organic semiconductor films.

The problem I see with your description, as I understand it, is that you describe a two step mechanism, where first the electron moves and then the hole follows it. Since electron transfer is field dependent, I would then expect that in your first step the electron would tend to hop in the direction of an electric field, which would result in field dependent exciton diffusion. I have never seen a study showing field dependent exciton diffusion or describing exciton diffusion in this way. If you have a reference to support your explanation I would be very interested to read it.

Best of luck to all in understanding a very complex subject.

Chem. Rev. 2010, 110, 6736–6767

ACCOUNTS OF CHEMICAL RESEARCH,42,1691-1699.

These two review articles describe overall electron transfer processes in OSC and give a basic idea about exciton diffusion in those materials.

Exciton diffusion is quite similar to normal electron (or hole) diffusion - the difference being that the nearby opposite charge changes the local potential of various orbitals/wavefunctions.

In an exciton, there is a positive charge on an atomic nucleus (the hole) and an electron that's in a delocalized state (conduction band). These positive and negative charges are close together so they mutually affect one another. However, due to the electronic structure of the semiconductor, they won't immediately recombine like they would in a metal.

Similar to how an electron diffuses normally, the conduction band electron can hop around to nearby conduction band orbitals. The difference being, in an exciton, there is a nearby positive charge that will lower the electronic potential of the orbitals that are close to the positively-charge nucleus. Thus, the electron will tend to stay closer to the nucleus, although since it is delocalized it will still tend to hop around a bit.

And of course, the hole can also move. Once the delocalized electron hops from a state relatively overlapping with the atom that holds the hole (let's call it atom A) to a state more overlapping a different atom (let's call it atom B,) it becomes attractive for valence band electrons localized on atom B to jump and fill the hole at atom A. This is because atom B now has a partial negative charge that pushes away the valence electrons, whereas A has a positive charge that's attracting the valence electrons, so the valence band of A is at a lower electronic potential than the valence band of B.

So, that's how exciton diffusion works - the conduction band electron hops around (because it is delocalized) and the hole tends to follow it (because of the differing electronic potentials of the valence bands caused by the proximity (more technically wavefunction overlap) of the delocalized electron.)

Now, why is the exciton diffusion so different in different materials? In materials that can quickly and effectively "screen" the positive and negative charges by re-orienting dipoles, the electron and the hole feel less Coulombic attraction. To first order, you can estimate the exciton diffusion from the dielectric constant, although this does not take the kinetics/dynamics of the reorientation into account (because the dielectric constant is measured at static conditions.) For example, polymers tend to have small dipoles which are hard to re-orient, so polymer are very poor at screening charges. This leads to polymers having very low exciton diffusion properties, due to the strong attraction between electron and hole. Adding a species that can create a relatively strong induced dipole, such as iodine, can improve the charge screening and increase the exciton diffusion length. Inorganic semiconductors have strong dipoles and tend to have good exciton diffusion characteristics.

Mark, many researchers would tend to disagree with your description. As far as I have seen, most researchers do not describe exciton diffusion as the actual movement of the electron and hole. Instead, exciton diffusion is usually thought to occur through an energy transfer process. Instead of the electron and hole moving, the energy absorbed by a particular molecular site is transferred to another nearby molecular site. I recommend looking up papers about Förster Resonance Energy Transfer (FRET) for more information about how the mechanism works. This doesn't mean that electron motion cannot occur, as Dexter Electron Exchange is another mechanism in which the actually move by swapping positions. However, in this mechanism, two electrons are exchanged with each other in one step. These two mechanisms have different distance dependencies and in reality I would guess that both may occur to some extent depending on the material. I still think that FRET dominates in most organic semiconductor films.

The problem I see with your description, as I understand it, is that you describe a two step mechanism, where first the electron moves and then the hole follows it. Since electron transfer is field dependent, I would then expect that in your first step the electron would tend to hop in the direction of an electric field, which would result in field dependent exciton diffusion. I have never seen a study showing field dependent exciton diffusion or describing exciton diffusion in this way. If you have a reference to support your explanation I would be very interested to read it.

Best of luck to all in understanding a very complex subject.

Thank you Michael, I had neglected homo-FRET, which no doubt enhances the diffusion rates, and may indeed dominate in many organic semiconductors. In general though, I don't think it can be assumed to be the primary exciton diffusion process.

Take for example the case of exciton diffusion in an indirect band gap semiconductor such as silicon. Due to the poor fluorescence quantum yield, and also because the absorption coefficient is so small at the band gap energy, I would expect the FRET efficiency to be negligible. However, exciton diffusion rates in silicon are still very high - for example, the paper "Excitons in silicon diodes and solar cells: A three-particle theory" estimates the exciton diffusion coefficient in silicon as 17 cm^2 s^-1, orders of magnitude higher than the value for P3HT of 5x10^-4 cm^2 s^-1 as reported in the paper "Exciton Diffusion Measurements in Poly(3-Hexylthiophene)".

Also, a quick search shows that Dexter electron transfer can happen as two steps. What is the evidence that Dexter transfer is a one-step process?

I don't think the two-step process that I described is dependent on the electric field. For the exciton to move via the process I described, both electron transfer and hole transfer have to occur. While the electron transfer would be aided by the electric field, the hole transfer would be impeded by the same electric field. Thus, the net effect should cancel out, and I don't see why this process would be field dependent.

It should be noted as well, that in hetero-systems which have two components with different optical properties, hetero-FRET should be distinguished from "exciton diffusion" because hetero-FRET is a directional process (from donor to acceptor) whereas diffusion by definition is non-directional. It is possible to overestimate the true exciton diffusion coefficient when hetero-FRET is involved - see the paper "Long-Range Resonant Energy Transfer for Enhanced Exciton Harvesting for Organic Solar Cells". It is only homo-FRET when the donor and acceptor are the same species that is accurately termed exciton diffusion, because this process is not directional. For example, in a pure P3HT film homo-FRET would enhance the exciton diffusion, but if you add an energy-accepting dye to your system you might overestimate the exciton diffusion coefficient of the P3HT by grouping the hetero-FRET together with exciton diffusion.

Hi guys,

Thanks for the getting back on this important question. As you might have noticed, the term "exciton diffusion" is thrown around loosely. In particular the diffusion in semiconducting polymer is completely different to the band-model developed earlier borrowing concepts from inorganic semiconductor physics. As Soumer mentioned the proper theory is developed considering the random hopping model where the exciton is getting delocalized into the surrounding sites with lower energy (but significantly more than kt ) in a random fashion till it encounters a trap or an hetero interface where it can undergo dissociation of simply diffuse till it completely relaxes (called the diffusion length) governed by the exciton lifetime.

thanks

santosh