As promised, here a glimpse of why I believe that recombination in organic bulk heterojunction solar cells under short circuit conditions (and also at Voc) is not necessarily monomolecular.
Sometimes, the short circuit current density vs light intensity is measured, and from the linear scaling a dominant monomolecular recombination is concluded. In (partial) answer, we have performed some relevant device simulations (thanx to wapf). In short, we varied the generation of free charges over four orders of magnitude, assuming different polaron recombination mechanisms.
In the first figure, we assumed monomolecular recombination with different polaron lifetimes, so that the continuity equation for electrons reads 
In all cases, even with negligible monomolecular recombination (longer lifetimes), is the scaling linear.
In the second figure we assumed bimolecular (Langevin type) recombination, using the continuity equation 
The static part of the latter (for details read the paper) is about 0.1. In the graph we used a range of
By the way, experimentally the recombination mechanism in these devices is usually found to be bimolecular. See for instance [Shuttle 2008], [Foertig 2009] or as an overview the review [Deibel 2010] – the latter is now accepted by Reports on Progress in Physics with minor revisions :-)
My conclusion: under typical conditions and for the bulk heterojunction solar cells I know, is it impossible to distinguish from the short circuit current vs light intensity whether or not recombination is monomolecular or bimolecular, respectively. I am interested in your comments!
[Update 2.7.2010] I moved my previous comment from here to a new blog post, with a brief overview on whether or not the whole voltage-dependent photocurrent, as opposed to the short circuit current, gives information on the dominant type of loss in bulk heterojunction solar cells.
For details about the simulation, have a look at these papers [Deibel 2008, Wagenpfahl 2010].
Notes on Disordered Matter 
Thanks, Carsten. I just wanted to throw out there that both with the bilayers and BHJ’s that I’ve made, I’ve always observed a slope of nearly one (0.93 – 0.98) of log Jsc vs. log light intensity.
I would be very interested to see your simulated Voc vs. log intensity plots. That is where I’ve seen interesting behavior in the past in direct disagreement with the work of Blom and company from either 2005 or 2003. I think you may have mentioned their work in your MRS talk. In any case, I’m guessing you may be writing that work up so any time after you guys submit would be great. =)
Cheyns, et. al., whom I know for a fact you mentioned in your talk, predicted a slope change in Voc vs. log intensity depending on whether bimolecular or Shockley-Hall-Read recombination was dominant. By the way, I am of the opinion that it should be debated whether the SHR model can be applied to disordered semiconductors with potentially no mobility edge and hence no extended states at all (unless we’re talking crystalline P3HT, etc.). Anyway, I would also be very interested to see the results of yours guys’ simulations of how the Voc vs. log intensity response changes as you vary the relative contributions of the two recombination processes.
That’s my two cents. Thanks again for sharing your thoughts.
Hi Alex, thanks for your comment! In brief: the behaviour of Voc vs light intensity is similar, in as far as only a very dominant recombination is seen. A difference: whereas the slope 1 behaviour in Jsc vs light is one for any weak recombination as well as for strong monomolecular recombination, in Voc vs light intensity a slope 1 is seen for any weak recombination as well as for strong bimolecular recombination. We go more into detail these days, you are quite right;-)
Concerning Cheyns, I also believe that SRH is very questionable in our disordered organic systems (despite the fact that I am coathour of that – otherwise nice – paper ;-)
Monomolecular recombination = geminate recombination, right? Can’t say I’ve ever heard much about Shockley-Read-Hall recombination–would you care to offer a chemist-friendly explanation?
Thanks for adding to my reading list, by the way. The more physics-heavy side of organic photovoltaics is still quite unfamiliar ground for me, but it’s very interesting.
Hi! Nice username;-)
First, let me point out that I use the term monomolecular recombination as in first order charge carrier decay, where the recombination rate is proportional to either electron or hole. Similarly, I consider bimolecular recombination to be proportional to the product of electron and hole concentration.
Monomolecular recombination (MR) “=” geminate recombination (GR) is almost right, but not quite… Indeed, GR is monomolecular, but MR is not necessarily geminate. Recombination of a free, mobile electron with a trapped hole can be an example for nongeminate recombination which is monomolecular (actually, the latter is similar to SRH recombination… have to consider this at some point more carefully!)
Shockley-Read-Hall recombination (SRH) is to my knowledge typically found in inorganic crystalline semiconductors (where, just as an additional note, the term (non)geminate loses its importance). In contrast to (bimolecular) band-band recombination or (bimolecular) Langevin recombination, SRH is recombination via an energy level due to impurity or defect. The rate of SRH is generally
; thus, although it depends on the product of electron and hole concentration, it is practically monomolecular (see details here and elsewhere, or keep up the discussion;-)
Hi, Carsten, thank you and your blog,that’s very very useful!
I started to do dynamics 5 months ago, i’ve got a question. You said “In a few view days, there are another publication about recombination of free polaron (free carriers)—also called nongeminate bicombination, more specificially, trimolecular recombination”, and today you wrote “Recombination of a free, mobile electron with a trapped hole can be an example for nongeminate recombination which is monomolecular. ” I’m so confused! Is nongeminate recombination a monomolecular or trimolecular? Or there are other similar recombination mechanism we can call it nongeminat recombination? thank you very much!
Dear Linda, proper naming of the recombination mechanisms is not quite consistent in literature – and unfortunately also not consistent in my blog. Make sure to read the more recent posts, such as this one for definitions. Best is probably to use the term nongeminate recombination for the annihilation of free electrons and holes with one another. In the cited blog post above I call this bimolecular recombination, with a recombination order of usually two if electrons and holes have similar concentrations. However, for trap-assisted recombination (e.g. comparable to Shockley-Read-Hall) the recombination order can also become one – which I called monomolecular in the old post. Also, recombination orders higher than two (eg. three, which I called trimolecular back then) where reported. Cheers, Carsetn
Thanks! That’s quite useful. Your example of a mobile electron recombining with a trapped hole, though…how/why is that different from the trap-limited bimolecular recombination discussed by Shuttle & friends? (Sorry, no ref–I only know of this work because he’s here for a postdoc and has given a talk or two about it.)
And what exactly do you mean by band-band recombination?
Apologies for the stupid chemist questions. I’d really love to ply a few people from your side of the field with drinks until they explain things in more accessible terms, but I haven’t found anyone suitable yet. :)
Briefly only, it’s sunday morning (well, more or less;-)
Shuttle [Shuttle 2008] and also we [Foertig 2009] have published recombination with order of decay larger than two, and explain this by bimolecular recombination (order two) plus trapped charges. If the latter do not actively participate much in the recombination process with the mobile charges, they are detrapped at some point during the measurement and extracted. Therefore, this is not recombination via traps as intermediate states I disussed before.
With band-to-band, a term used for crystalline inorganics, I mean direct recombination of mobile carriers.
Never mind asking questions! Now, concerning the drinks… ;-)
OK, that makes more sense. Now I owe you beer if/when we end up at the same conference :)
Hi, Carsten. I share the same opinion with you. According to the drift-diffusion modeling [L. Koster et al., Phys. Rev. B 72, 85205 (2005)], at short-circuit charge carriers are more likely to distribute at the electrodes rather than in the bulk, therefore the recombination dependencies at short-circuit are relatively weaker than at open-circuit.
In my opinion the reduced prefactor of bimolecular recombination may originate from the 1D modeling. It will be interesting to compare the reduced prefactor of 1D version of drift-diffusion model to the 3D one.
Hi hsjufeng, thanks for your comment!
Concerning the reduced prefactor: indeed, 1d simulation implies an effective medium without real world morphology. The latter, i.e. donor-acceptor phase separation will certainly influence the recombination prefactor.
Two other factors are carrier concentration gradients [Deibel 2009] (already mentioned in the post), which is present even in 1d, and the temperature dependence of mobility and polaron pair lifetime [Hilczer 2010]. A nice mix of all three will be pretty close to reality. Unfortunately, all these contributions are not that easily accessible.
I vote for donor-acceptor phase separation being most important control on the recombination of freely moving hole and electron or to put it another way recombination is governed by interface surface area and charge concentration.
The interface is indeed very important, for both geminate and nongeminate recombination. For the latter, smaller surface (due to larger D-A phases) means less recombination: that is part of the reduction prefactor
for Langevin recombination. Larger phases, however, also mean less spatial disorder, thus can mean larger mobility, and therefore higher recombination. Depends really on the details. For geminate recombination, surface dipoles can “extend better” for larger phases, thus influencing separation and losses. Extreme case here is the planar heterojunction. Again, mobility is important for separation, and again it depends also on the spatial (dis)order.
Why cant you apply a sufficient negative bias to the device in order to aid mobile charge extraction and then measure the light intensity vs current slope. This will give you the losses in the device which are inherent due to geminate pair recombination (probably slope will be linear to start with and then become sub-linear). Then you short circuit the device, assume geminate recombination losses are more or less the same and deconvolute to get the rate of mobile charge recombination.
That is done.
where 
is the generation rate of free charges. It depends somewhat on voltage. Saturation means large negative bias (hoping that no shunt resistance makes determination of that regime impossible). Nevertheless, it is not as easy: in principle, also charge extraction can be field dependent. From just I(V) curves, one cannot tell!
Hi, Carsten. I have a basic question to ask. Why the continuity equation doesn’t have a term of 1/q*(dJ/dx)at the short circuit condition?
Hi llb, it does have the term, I just forgot to include it in the typeset equation (no corrected): sorry. It is certainly included in the simulation:)
Hi, Carsten. It’s so nice to talk to you. BTW, you are really pro at photography. :) :)
I’m a graduate student and now doing some research on the free polaron lifetime of bulk heterojunction solar cell. You blog helps a lot for me.
Regarding the continuity equation, I think the monomolecular recombination rate should be (n-n0)/tao, where n0 is the negative polaron’s density at the equilibrium condition and tao is the lifetime of negative polaron. What do you think?
Thank you for letting me know that it has a term of 1/q*(dJ/dx). What about the Eq. 14 in the reference PHYSICAL REVIEW B 80, 075203 2009? Is it a typo?
Thank you.
Hi llb, thanks for your comment. You like pulling my leg (the “typo” in PRB… ) don’t you ;-) However, in the PRB it is not a typo, but an intended omission — which we indeed could have explained better! For photo-CELIV (or TPV etc), the recombination takes place close to flatband conditions, and assuming that the spatial current gradient is about zero is pretty good. (By the way, why we wrote
instead of 
with 
is beyond my now).
Concerning monomolecular recombination,
is indeed only an approximation. I believe that I explained this somewhere, but did not find it just yet. Your suggestion is in principal the way to go, but 
is not that easily defined. Should equilibrium carriers not be able to recombine? Equilibrium actually included recombination. So, no 
… Thinking about it, charge carriers always need some recombination partner, therefore a recombination of only electrons or holes by themselves is not possible AFAIK. Consequently, there is no monomolecular bulk recombination for free charge carriers, only for excitons, CT complexes etc. We use this simplification only to show the consequences on the device characteristics, even if physically not existant;-)
Looking forward to see you around! C
Thank you. I will digest it first and see what I will figure out. Looking forward to the new posts in this blog. :)
Hi Mr Carsten,
Your blog is quite helpful! However, I still have a silly question on geminate recombination and nongeminate recombination. What is the difference between the two indeed? I hear that the former involves electron and hole polarons originally from the same exciton while the latter from two different excitons. Is it correct? I am also wondering how you can experimentally recognize these two recombination mechanisms? I read some papers on that but didn’t have a clue. Thanks very much~!
Hi Eigen,
thanks; Carsten (without any title) will do;-) You are right: geminate recombination describes a recombination in which the recombination partners come from the same precurcor state. Consequently: nongeminate recombination, different precursors. The former is for instance recombination of a singlet exciton or a charge transfer exciton, which is first order recombination usually on a time scale of below 1 to 10ns. Nongeminate recombination implies that first two different precursors states need to dissociate to generate free species which then find one other to recombine. Sounds more complicated, also takes longer: usually more than 1 to 10ns, and usually 2nd (or higher) order recombination. I belive this is also explained in my review, in which you might find useful references as well. Best,
Carsten