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:-)

(Update 18.1.2011: Wapf asks, Carsten answers;-) The thermal energy with $kT$ is only an estimate as compared to the $3/2kT$ for three degrees of freedom. Also, if you are wondering why $I_\text{hole,rec}=\gamma qp$ , but $\frac{dn}{dt} \propto \gamma np$ : I did not transform $q$ into $n$ Concerning $q$ : consider that a current $I = dQ/dt = d(q n)/dt = q dn/dt$ . Therefore $q$ “disappears” going from current to continuity equation. Concerning the “appearing” $n$ : the recombination current as calculated by Langevin considers particles which flow into the $r_c$ -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.)

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19 Responses to Recombination in low mobility semiconductors: Langevin theory

Pedro says:

2. March 2009 at 16:49

Clear!

Reply
armismelas says:

9. November 2010 at 13:27

Yet again nicely presented!

I have a few questions. Calculating the Langevin recombination strength we assume that the hole is moving in an electric field equal to that generated by two point charges seperated by Coulumb radius? Why? I realise if they don’t get within Coulumb radius then thermal energy is sufficient to escape recombination and if they do get within Coulumb radius then they recombine. Is that so? Could you clarify this point to me, thank you
And in the last formula why is the recombination proportional to langevin recombination strength times electron density n? Why not n^2? I realise it should depend on both carrier densities but why a simple multiplication?

thank you

Reply
deibel says:

9. November 2010 at 23:48

Hi! Langevin’s theory is very simple but pretty effective. Of course, both electron and hole move, and if they come to within the Coulomb radius, the Coulombic attraction exceeds the thermal energy sufficient for escape. The clever assumption was to make a clear cut: Langevin proposed that every electron-hole pair with a radius smaller than $r_c$ recombines, all others do not. This is then generalised to account for the carrier concentrations, i.e., these assumptions are used to calculate a recombination current, and from that the recombination rate including the Langevin prefactor.

Your question concerning the last equation is a bit unclear to me. The last equation is $dn/dt = \dots - \gamma n p$ , thus the recombination rate is proportional to Langevin recombination strength times electron density times hole density.

Best, C

Reply
armismelas says:

10. November 2010 at 15:07

Thanks, I got it now
Regarding the second question I meant why is it $-\gamma n p$ ? Why not $-\gamma n^{0.8} p$ ?

Reply
- deibel says:
  
  10. November 2010 at 18:58
  
  Not sure where the 0.8 exponent should come from: could you elaborate, please?
  
  Reply
  - armismelas says:
    
    11. November 2010 at 17:12
    
    Calculating the recombination rate: I realise the recombination current is $-\gamma q p$ and in addition the recombination rate should also be proportional to electron density. What I don’t understand is why isn’t there a proportionality factor A $A -\gamma n p$
    I apologize if I am making this really unclear but I just want to understand
armismelas says:

11. November 2010 at 17:13

Apologies, I meant $- A /gamma n p$

Reply
- deibel says:
  
  11. November 2010 at 18:42
  
  I am really sorry, but I just don’t get your point. Why an exponent of only 0.8 (your previous comment), and where should the proportionality constant come from? Of course, you get a prefactor when solving the continuity equation, but that does not change the recombination rate $R$ (which is actually, more precisely, $R_\text{Langevin} = \gamma (np - n_i^2)$ , but that’s another story;-) Experimentally, though, you can of course have deviations from Langevin’s theory; ,in our group, we call this additional prefactor $\zeta$ .
  
  Reply
  - armismelas says:
    
    18. November 2010 at 10:44
    
    I got it now sorry for the confusion. The additional prefactor $\zeta$ you mentioned clarified it for me. Thanks
    
    I am just curious how large those deviations could be? I mean the values of $\zeta$
  - deibel says:
    
    18. November 2010 at 19:27
    
    $\zeta$ is typically in the range between 10-3 and 0.1, as reported by Juska, Durrant or our group. Still fascinating that despite this lowered prefactor, the open circuit voltage is so heaviliy influenced by bimolecular recombination!
armismelas says:

19. November 2010 at 0:58

Wow didn’t expect it to be that small. That’s so interesting, I’ll keep reading!
Thanks for claryfing those points for me

Reply
Carlos Augusto says:

1. September 2011 at 15:51

I jsut didn’t understand what is the Coulomb radius. Can you explain me?

.

Reply
- deibel says:
  
  3. September 2011 at 8:34
  
  I am sorry, there is nothing more to it than written in the post. The particle is free once it does not feel the Coulomb attraction to the other particle any more, which is the case when thermal energy = coulomb energy. That equation is then rearranged to give the Coulomb radius, as described.
  
  Reply
Jiebing says:

1. November 2011 at 16:42

Thanks for your useful post. Can you explain the second term n_i^2 in R_Langevin = \gamma *(np-n_i^2)?

I was thinking some holes may have multiple choices (path) to choose which electron to “marry” to, which causes overestimate of the current. Do this term account for this or for other reasons? Thanks.

Reply
- deibel says:
  
  1. November 2011 at 20:03
  
  $n_i$ is the intrinsic carrier concentration. The contribution proportional to $-n_i^2$ is opposing recombination and is thus a generation term, not a second generation path. Essentially, you can interpret it as the charge carrier densities $n$ and $p$ never being lower than $n_i$ . Once $np=n_i^2$ , there is no more Langevin recombination. This corresponds to the mass action law you find in semiconductor text books. The Langevin recombination equations works for an ensemble of charge carriers, thus is “on average”, so there is no need (and no way) that a charge carrier can “choose” its recombination partner. Regards, C
  
  Reply
  - Jiebing says:
    
    1. November 2011 at 23:03
    
    Thank you very much for your clear explanation.
    
    BTW, do you happen to have an English version of the paper Langevin 1903 (Ann. Chim. Phys. 28, 433)? I want to read the original paper. Thanks.
  - deibel says:
    
    2. November 2011 at 9:16
    
    Unfortunately not, but I recommend you have a look at the book by Pope and Swenberg (1999) for background – or use google translate on Langevin’s original paper;-)
Miriam says:

27. January 2012 at 12:18

Hey,
I am just trying to understand your paper (phys. rev. b 82, 2010) and I was wondering how you get the connection between what you explain here in this (really nice) blog and the equation R = (q/e0er) (µn+µp)(np-ni^2) labeled with equ. 6 in your paper.

Cheers and greetings to my home area Lower Franconia
Miriam

Reply
- deibel says:
  
  27. January 2012 at 13:42
  
  Hi Miriam, thanks for your comment. In the blog post, the derivation of the Langevin equation is shown. In the paper, we expand the original theory somewhat by explaining the prefactor (which we call $\zeta$ ) sometimes necessary to fit experimental polaron dynamics with bimolecular recombination. This prefactor stems in part from spatial gradients of the charge carrier distribution. Thus, if at one end of your device there are many more electrons than holes, the assumption $n(x)\approx p(x) \approx \bar{n}$ (where $\bar{n}$ is the average carrier concentration which can be determined experimentally) does not work any more. In our paper [Deibel 2010] we just show how the additional prefactor $\zeta$ can be calculated if the gradients are known. Does that answer your question? Best wishes, Carsten
  
  Reply

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