[Nenashev 2010] pointed out that in the derivation of the Langevin recombination,
since the electric field scales as r−2 and the surface area of the sphere scales as 2, the value of r chosen is unimportant, leading to a simple solution with constant electron den- sity, thus justifying the neglect of diffusion.
In Langevin’s derivation, the radius rc where Coulomb energy=thermal energy was chosen, but indeed the terms with rc cancel out so that any radius can be selected. Thus, it may make more sense to consider the Langevin recombination in the following way (as I did not use eqnation numbers in the previous post, I will recount the whole stuff again;-):
We consider a mobile hole and a fixed electron. The former drifts towards the latter, just driven by the Coulomb attraction, leading to an electric field felt by the hole of
with elementary charge , the dielectric constant of the medium times vacuum permittivity, , and the (arbitrary!) electron-hole distance r. The corresponding drift current density for holes is
Consequently, the recombination current of holes flowing into a single sphere around the fixed electron with the radius being the same as the r from above becomes
[Update 13.11.2012, thanks Gebi]
which defines the Langevin recombination prefactor .
In order to define the Langevin recombination rate , we can consider that a recombination current density is defined as
with active layer thickness . The recombination current per electron, can be generalised to consider a number of electrons , . Then
Replacing and , we get
Comparing this equation to (*), we see that indeed
This is the classical result by Langevin, but derived with arbitrary radius as explained in the quote of [Nenashev 2010]. Thus, the explanation of the Coulomb radius for letting all charge carriers within recombine, and all without getting away, is not quite correct. In the paper by Nenashev et al., even though considering recombination (including diffusion!) in two dimensional systems, a possibly more intuitive approach for calculating an equivalent recombination rate is shown via the time until two charge carriers recombine.
However, Langevin recombination is only a start, as trapping of charge carriers is rather important in disordered organic semiconductors and its applications, e.g. organic solar cells. As a starter, see [Kirchartz 2011], [Street 2012], [Foertig 2012] etc.
P.S. Thanks to Jens L for making me aware of the “useless” and the Nenashev paper.