Notes on Disordered Matter

I’ve talked a lot about polaron pairs and excitons lately, and will continue to do so, that this time I’ll give short explanations of what I am actually talking about. Call it definitions…

A polaron is a charge, i.e., an electron or a hole, plus a distortion of the charge’s surroundings. In a crystalline inorganic material, setting a charge onto a site does not change the surroundings, as the crystal lattice is rigid. Not so in many disordered organic materials. Putting a charge onto a certain molecular site can deform the whole molecule. Moving the charge from this to another molecule means that first the energy for the deformation – the polaron binding energy or reorganisation energy – has to be mustered. The implication is that charge transport becomes more difficult, the charge carrier mobility becomes lower, … This process is also described as self-trapping. As a side note, it is often difficult to distinguish between the influence of polaronic self-trapping and of gaussian disorder, as both have a similar impact on the charge transport properties. This similarity is also reflected in the corresponding hopping rates used to calculate charge transport: Marcus theory is a function of the reorganisation energy, where as the Miller Abrahams rate [Miller 1960] is related to the energetic disorder of the density of states. The polaronic deformation can be quantified in terms of a (lattice) polarisation, or a phonon cloud, or just as the above-mentioned polaron binding energy. Mostly, however, when hearing polaron, think charge;-) See also what wikipedia has to say about polarons.

A polaron pair is a Coulomb bound pair of a negative and a positive polarons, situated on different molecules. Usually, polaron pairs are the intermediate step from an exciton to a pair of free polarons &ndash far enough apart not to feel the attraction of one another &ndash and therefore important in order to understand photogeneration in organic semiconductors.

An exciton is an excited quasiparticle in a solid, which is formed by a Coulomb-bound electron-hole pair. It is more prominent in organic semiconductors as compared to their inorganic counterparts: as the dielectric constant is lower in organics, the screening length is larger. In this case, the name Frenkel exciton is applied, whereas the weakly bound type is called Wannier-Mott. Thus, in organic materials, the two charges feel a strong mutual attraction, and usually reside on one molecule. There seem to be special cases, however, in which the two particles reside on adjacent molecules – of the same kind, in contrast to polaron pairs. The spin-state of the two charges is quite important. Without going into too much detail: when the two spin-vectors add up to zero, we have a singlet exciton. Singlet excitons are the only ones which are generated upon illumination, which is due to the specific selection rules. The other exciton type, triplet excitons, have a nonzero spin vector, which is possible in three different combinations – thus the name triplet. Singlet and triplet excitons can also be formed due to interaction following charge injection; theoretically, this follows a one-to-three ratio, i.e., only a quarter is of singlet type. Some features of singlet excitons and their relevance for organic photovoltaics was discussed here. The exciton binding energy of singlets is around 0.3eV in organics (compared to ~0.01eV in classical semiconductors). Excitons have a certain lifetime, typically of the order of ns in organic semiconductors, after which they recombine radiatively; this is called photoluminescence. Triplet excitons generally have lower energies and longer lifetimes. For photovoltaics, they are not yet import (though might be following some novel concepts), instead they can act as loss mechanisms (by intersystem crossing or electran back transfer) under certain conditions as their energy is too low to generate free charge carriers. Radiative recombination after the triplet long lifetime of maybe some milliseconds – the transition is actually spin forbidden – phosphorescence occurs. As a side note, phosphorescence can be applied to high usefulness in so called triplet emitters, being an important concept for organic light emitting diodes. Maybe we’ll detail this another time. Wikipedia on excitons here.

An exciplex is just an exciton which is located at the interface of its “host” molecular material – indeed it still resides on one molecule – as indicated in the image. Due to the influence of the surface, the exciplex experiences a different environment as compared to a bulk exciton. This leads to photoluminescence which is slighlty red shifted. Also, the lifetime can be prolonged in comparison to the bulk exciton, as it is stabilised by the surface states.

Did you miss bipolarons? I didn’t;-)

Thanks to JG for the exciplex!

[Update 27.4.2010 to answer the question of Jenna] In organic bulk heterojunction solar cells, the path from singlet excitons in P3HT to free charges usually goes via charge transfer complexes of the donor-acceptor system. (See for instance here.) I often refer to these as polaron pairs. However, naming conventions are not that simple. Here a brief excerpt from an unpublished review I recently wrote (accepted for publication by Adv Mater).

The commonly used names for CT states and complexes are diverse, either used alternatively or to define special cases. Examples are polaron pairs, [Dyakonov 1998] intermolecular radical pairs (with the radical cation on the polymer and the radical anion on the fullerene) [ Scharber2003], interfacial charge pairs [Westenhoff 2008], geminate pairs [Arkhipov2003], charge transfer excitons [Veldman2008] and exciplexes [Morteani2004].

Huang et al. [Huang 2008] found by theoretical considerations for polymer-polymer heterojunctions that a range of Coulombically bound CT states with both, emissive and non-emissive character, exist. The different states are a result of the specific features of the intermolecular overlap between donor and acceptor moieties. In order to strive for a more precise nomenclature, they point out that polaron pairs can be considered as one special instance of the more general exciplex. From this point of view, the distincitve property of the polaron pair excitation is that it is due to a complete charge transfer from donor to acceptor, as opposed to a partial CT. Thus, an exciplex can generally be regarded as a hybrid state with partly CT character and a certain fraction of a local excitation on one (or both) molecules of the donor–acceptor system. Already earlier, Gould et al. [Gould 1994] pointed out that the character of the emitting species of an exciplex depends on the relative contributions of pure ion-pair and locally excited states. In their definition, an exciplex with beyond 90% CT character represents a pure contact radical-ion pair. They suggested that it can be identified experimentally by verifying that the emission maximum lies about 5000/cm (100meV) below the singlet exciton photoluminescence.

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