Recombination of free charge carriers in materials with a low mobility is often described with the Langevin recombination rate [Langevin 1903 (Ann. Chim. Phys. 28, 433)] (Update 3.12.2008: wrong reference previously, sorry.) Generally, if electron and holes – being potential recombination partners – wish to recombine, the effective recombination rate is proportional to
- the “direct” recombination rate
- finding each other
In high mobility semiconductors, the former is dominant. However, in disordered solids, and particularly disordered organic semiconductors, the low mobility limits the effective recombination rate. The process of finding each other can be described as diffusion limited, which is proportional to the charge carrier mobility when considering the Einstein relation.The idea is as follows (after [Pope & Swenberg 1999]): we assume a negative charge to be fixed, and a positive mobile charge (moving with mobility of electron plus hole), both being attracted by the Coulomb force. The hole can avoid recombination at zero field only if the thermal energy is sufficient to overcome the Coulomb potential. As demarcation line, the Coulomb radius is defined by equating
If we consider now a drift current density for the mobile hole,
bearing in mind that electric field is the spatial derivative of the potential (energy), we get
Now considering that recombination of the two charges takes only place if they find each other, i.e., the hole comes to within the Coulomb radius, we can calculate the hole recombination current. That is the current density flowing into the sphere of radius around the electron,
with gamma being the Langevin recombination strength
Now, indeed, the recombination is proportional to the charge carrier mobility of electron and hole, i.e., to the two carriers finding each other.
The Langevin recombination strength time electron density times hole density is then the Langevin recombination rate, as used in a typical continuity equation
with generation term G0, a monomolecular recombination term, and the bimolecular Langevin recombination rate.
Bimolecular recombination in disordered plastic solar cells is usually described with the Langevin theory. There seem to be some adjustments to be made, which we’ll discuss another time;-)
(Update 3.2.2009: support by WordPress, excellent! My splendid MarsEditblog editor transfers it, but does not parse it… but that would have been asked too much indeed! No, I am glad it works as is:-)
(Update 18.1.2011: Wapf asks, Carsten answers;-) The thermal energy with is only an estimate as compared to the for three degrees of freedom. Also, if you are wondering why , but : I did not transform into ;-) Concerning : consider that a current . Therefore “disappears” going from current to continuity equation. Concerning the “appearing” : the recombination current as calculated by Langevin considers particles which flow into the -Sphere around the center charge. By definition of Langevin, all charges (here, holes) flowing into the sphere recombine with the center center charge (here, one electron). Going from microscopic consideration to macroscopic equation, this one charge of course has to be translated to a charge density, as this can happen with every electron. (And all of this is of course true for the opposite case, electrons flowing into the “hole sphere”, but this is generalised already in the final rate equation.)