In a “recent” post (just 3 posts but 10 months ago;-) I wrote once again on the derivation of the Langevin recombination rate for nongeminate recombination. The question is, is Langevin recombination really what governs the charge carrier loss rate in organic solar cells?
Recombination of electrons with holes is usually a 2nd order decay. As electrons and holes are photogenerated pair wise, the respective excess charge carrier concentrations are symmetric, . Then a recombination rate is
where the recombination prefactor could be a Langevin prefactor – more on that later. In a transient experiment with a photogenerating, short laser pulse at , the continuity equation for charge carriers (here, e.g. electrons) under open circuti conditions (no external current flow, for instance if the experiment is done on a thin film without electrodes)
for (as the generation was only at ).
If all electrons and hole are available for recombination (i.e., can reach all other charge carriers and can be reached by them), then the recombination rate and the continuity equation for yield
which can be solved analytically:
That means, on a log-log plot (or for a transient absorption experiment), the transient due to charge carrier recombination in a second order decay has a slope of -1.
It is no news that slower decays
with higher recombination order are reported in literature by transient absorption experiments [Montanari 2002, Offermans 2003] and recently also using other techniques, see references in this post. This corresponds to recombination rates of the empirical form
Clearly, this is not a second order recombination. However, it is very unlikely that more than two charge carriers are involved (c.f. this post). Thus, can it actually be Langevin recombination?
[Shuttle 2010] proposed that this could still be Langevin recombination, where the order of decay higher than originates from the charge carrier mobility. That implies that all charge carriers can recombine with one another, but the charge carrier mobility in the Langevin prefactor depends on the charge carrier concentration. The latter is credible, as the macroscopic mobility in a system with a lot of charge carrier trapping (and there is! [Schafferhans 2010]) can indeed depent on the carrier concentration [Tanase 2003, Pasveer 2005] – depending on the density of states distribution [Oelerich 2012]. Then,
That was easy;)
But is it the complete explanation? That may not be the case. We showed last year [Rauh 2012], that often
opposing the above statement (1). However, a weakness of both studies is that the mobility is not measured directly – still to be done.
In the meantime, a very interesting paper [Kirchartz 2011] (you need to take some time for it, though;-) – and slightly later also [Kuik 2011] – showed that Langevin recombination and trapping should be combined in a more detailed way. If you consider that some charge carriers, say electrons, in the disordered organic semiconductor are mobile, with density , and some are trapped, , their sum still equals the overall electron concentration $n$. As long as the charge carriers remain trapped, they are immobile and cannot move towards their recombination partners. That means, they cannot actively contribute to recombination, but can be “found” by another (oppositely charged) charge carriers. The recombination rate for recombination of a mobile electron with a trapped hole can be written as
or, if electron and hole concentrations as well as their trap distributions are assumed to be the same,
Importantly, the mobility is not the measured macroscopic mobility, but the maximum possible mobility without trapping.
With two assumptions, this equation can be brought into the Langevin recombination shape again: (1) the concentration of trapped carriers by far exceeds the mobile ones (which makes sensse as charge carrriers tend to relax down in energy), , so that $n_t \approx n$, and (2) the mobility and the actual recombination process are governed by the same density of (trap) states distribution. Then, from the multiple-trapping-and-release model, which I’ll hopefully write more on another time, the macroscopic mobility is connected to the high local mobility by , and at the same time the so-called trapping factor ! Therefore, the last equation can be written as
Thus, in principle the nongeminate recombination of charge carriers in organic solar cells, even with trapping,
could be described by the Langevin recombination rate. Under these conditions, it is very similar to Shockley-Read-Hall recombination (in contrast to what I wrote in a comment a few years back!).
However, the discrepancy (1) mentioned above, , still remains and needs to be resolved. While I believe that Langevin recombination can also in the future be used to describe nongeminate losses in organic solar cells, I think it will have to be further adapted (i.e., also the last equation) – for instance in view of probably partial phase separation, where not every mobile charge carrier can actually meet a trapped one, if the latter is deep within one material phase [Baumann 2011].