Recombination in low mobility semiconductors: Langevin theory

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, Langevin recombinaton basics.png 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

$E_\text{Coulomb} = E_\text{thermal}$

$\frac{q^2}{4\pi\epsilon r_c} = kT$

$r_c = \frac{q^2}{4\pi\epsilon kT}$

If we consider now a drift current density for the mobile hole,

$j_\text{hole} = qp\mu F$

bearing in mind that electric field is the spatial derivative of the potential (energy), we get

$j_\text{hole} = \frac{q^2}{4\pi\epsilon r_c^2}p\mu$

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 $j_\text{hole}$ flowing into the sphere of radius $r_c$ around the electron,

$I_\text{hole,rec} = j_\text{hole}\cdot 4\pi r_c^2 = \frac{q^2}{\epsilon}\mu p = \gamma qp$

with gamma being the Langevin recombination strength

$\gamma = \frac{q}{\epsilon}\mu$

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

$\frac{dn}{dt} = G_0 - \frac{n}{\tau} - \gamma np$

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: $\LaTeX$ 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:-)

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This entry was posted on Friday, April 4th, 2008 at 16:11 and is filed under physics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

One Response to “Recombination in low mobility semiconductors: Langevin theory”

Pedro Says:
2. March 2009 at 16:49 | Log in to Reply
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