Mobility and Efficiency of Polymer Solar Cells

Joshua TreeDisordered organic materials inhibit charge carrier mobilities which are orders of magnitude lower than for inorganic crystals. First thing missing in disordered matter is the regularly ordered lattice of atoms, where the charge carriers can delocalise, leading to band transport. Second thing is the generally lower interaction between adjacent molecules, which is due to weaker bonding and larger distances. The transfer integral, the value of which goes exponentially down with distance, to get from one to the other molecule is too low for delocalisation. Thus, in terms of charge carrier mobility, think 10-2cm2/Vs for disordered organics (if you are lucky) vs. at least 102cm2/Vs for ordered inorganics.

How much does a weak charge transport limit the performance of organic solar cells? How bad is it?

Luckily, not as bad as one might think! It turns out that a certain charge carrier mobility is important to get good power conversion efficiencies, but looking at further improvements, there are other more pressing issues. But one after the other.

Several processes in organic solar cells (the function of which was detailed before in three parts) do strongly depend on the mobility:

  • the polaron pair dissociation (Braun-Onsager theory [Braun 1984], describing the escape from the mutual Coulomb attraction)
  • the charge transport
  • polaron recombination (possibly Langevin recombination, but with a reduced rate, as found experimentally [Deibel 2008b, arxiv:0810.0542])
  • and finally, the charge extraction (which is directly related to charge transport, and possibly influenced by surface recombination)

Using our macroscopic device simulator, we looked at the influence of charge carrier mobility on the solar cell parameters (short circuit current, open circuit voltage, fill factor, and of course the efficiency) [Deibel 2008a, arxiv:0806.2249], following the idea of [Mandoc 2007], but considering a more realistic (=reduced) polaron recombination as well as injection barriers at the electrodes. Voc+jsc+FF-vs-mu.png Due to polaron pair dissociation, the short circuit current jsc increases with mobility (here equal for electrons and holes) until saturation is reached. The open circuit voltage Voc, however, decreases steadily. Actually, the slope steepness is maximum due to our implicit assumption of ideal charge extraction; for a realistic charge extraction (= finite surface recombination), the Voc slope with mobility is weaker… or even constant for zero surface recombination. The fill factor is maximum at intermediate charge carrier mobilities, not far from the experimentally found values!

Looking at the power conversion efficiency, there is indeed a maximum value at rather low mobilities, eta-vs-mu.png just a bit higher as compared to the values found in state-of-the-art polymer solar cells (shown by a vertical dashed line). The parameter zeta shown in the graph is indicative of wether normal (1) or reduced (0.01) Langevin recombination has been considered.

So, what does all that mean?

  • the charge carrier mobility has to be reasonable for good solar cells
  • however, there is not much room for improvement; even if surface recombination is rather small (which is to be expected in materials without dangling bonds;-), the maximum efficiency is reached already at low mobilities
  • this is due to very low polaron recombination rates, i.e., even though slow, the charges are extracted at some time (if they do not recombine, which they almost never do), leading to photocurrent
  • a brief note: the decreasing efficiency at high mobilities is overestimated, as mentioned before; for realistic extraction, it will only be weakly decreasing or even remaining constant… but not increasing after approx. 10-6m2/Vs (10-2cm2/Vs)!

So, finally, how to get higher efficiencies? What can be optimised?

  • very important, but achieved for some material combinations: a donor-acceptor phase separation which is fine-grained enough for good exciton dissociation, and coarse enough for good charge transport
  • polaron pair dissociation: better at low fields than previously thought, but still limiting… more basic understanding is needed
  • the narrow absorption bands are a major issue, limiting the photocurrent and thus the short circuit current
  • the exciton binding energy and the relative acceptor energy offset: the energy needed for exciton dissociation limits the open circuit voltage

Some of these points I had mentioned already earlier. So, for light absorption, tandem solar cells might be a solution (with new problems arising, e.g., current matching with angle dependence of the incident light), or design/synthesis of novel materials. Same goes for exciton dissociation. But I believe there are many more ideas still out there which need to be implemented and tested;-)

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12 thoughts on “Mobility and Efficiency of Polymer Solar Cells”

  1. some scientific questions if that’s ok. Why does Voc drop with increasing mobility, is it to do with dark current losses?

    Why does the fill factor peak?

    How does this change with thickness? You know if devices could be thicker then a lot more light could be absorbed.

    Personally I have seen evidence that charge recombination is not mobility controlled but instead D-A interface surface area limited. Does your model account for that possibility?

    Thanks for the blog, I never thought such a blog could exist and now its my favourite read. ;-)

  2. @ineverwantedtobeascientistiwantedtobealumberjack: Thanks for the “favourite read” and the comment:-)

    Voc drops with higher mobility, as the charge extraction becomes more and more efficient. This means that in steady state, e.g., holes at the cathode have very low concentrations, but high concentrations at the anode (which is hole injecting). This steep gradient concentration translates into a strongly bent band due to the Poisson equation. In a steep band, however, the quasi-Fermi levels cannot be splitted very far (see Fig. 4(c) in [Deibel 2008a]). Therefore, the higher the mobility, the lower the Voc. However, as mentioned in the post already, this was calculated for ideal charge extraction. The Voc will decrease less (but still will decrease) when the extraction is more realistic.

    Very brief (and incomplete;-) explanation for why the fill factor peaks: for the extreme cases, low mobility and high mobility, respectively, you have either zero voltage or zero current. Both mean zero fill factor. The maximum is inbetween.

    Thickness: higher thickness means higher absorption (mor excitons) and lower internal field (less polaron pair dissociation). This means that there is an optimum thickness… which is a trade-off once again;-) This does change some values, but not the general behaviour of efficiency vs mobility.

    Concerning polaron recombination, indeed mobility alone does not determine it in a two-phase system. However, there is no single model to explain the experimental recombination rates right now. We use a prefactor (zeta) which “manually”, but in accordance with experiment, reduces the Langevin recombination rate. Part of zeta’s origin might be due to the D-A interface. Another part certainly is energetic (and spatial) disorder: Langevin just considered a continuum, no hopping system. This or that way, our macroscopic model considers a blend as a single effective, ambipolar material, with a linear correction for the Langevin recombination.

    If you have more information on recombination, I’d be glad to hear it.

  3. Thanks for the reply. Don’t know if I understand the voltage dropping to zero thing but I should brush up on my equations.

    If you are interested I did my thesis on charge recombination in polymer-fullerene solar cells monitored using pump -probe transient absorption spectroscopy. The data is unpublished as of yet but you are welcome to look at it.

    http://www.lulu.com/content/375725
    (download is free)

    Skip straight to chapter 3 on recombination in P3HT:PCBM. Not sure the other chapters will interest you much however. :-)

  4. Can you explain why this phenomenon of reduced quasi-fermi level splitting with increased mobility is not seen with inorganic PVs? (see for example Sen, Srivastava, Joshi, and Goyal, Performance of Polycrystalline Solar Cells, Phys. Stat. Sol. A 75, 657, 1983.)

    Thanks

  5. Hi rlunt, thanks for your comment! As you see, I wrote a blog post as comment to your question, which also serves as an update/extension to the post here: https://blog.disorderedmatter.eu/2009/08/26/influence-of-finite-surface-on-efficiency-vs-mobility-of-polymer-solar-cells/ . Somewhat circuitous way of “answering” your question, sorry ;-) Concerning the paper you cited, I think a different effect is looked at there, efficiency vs grain size. Large grain size means less losses due to the grain boundary, with the efficiency saturating at the “normal” bulk mobility.

  6. Hi.. I started reading the blog a few days back and I have learned from it.

    I have a doubt regarding this post. You mention that the Voc decreases with increasing mobility because the concentraion gradient of charge carriers increases. I understand why a more steep concentration gradient would lead to a lower Voc. What I do not get is why there is a concentration gradient at Voc (no internal electric field)?

    This is what I think about Voc and mobility (after reading some literature):

    At low mobilities the probability for polaron pair dissociation is low, so decaying polaron pairs will reduce the quasi-Fermi level splitting. However, a low bimolecular recombination at low mobiliies compensates for this loss, and a high Voc is obtained at low mobilities. As you increase the mobilitiy the polaron pair dissociation process is more efficient, but the bimolecular recombination also increases. The increased bimolecular recombination at high mobility reduces the quasi-Fermi level splitting. Even if the polaron pair dissociation is very efficient at high mobilities, the high bimolecuar recombination outweights it and results in lower quasi-Fermi level splitting and hence a low Voc. Is this correct?

    1. Hi Jose, thanks for your comment. There are different ways of looking at this, either by considering carrier concentration and its gradient, or in terms of recombination. Both are of course valid and indeed equivalent.

      You are right, polaron pair dissociation is improved by a higher charge carrier mobility (if there is anything to improve), but at some point you have perfect photogeneration and even higher mobility does not make a difference any more. Thus, this is indeed unrelated to the reduction of Voc at high mobilities.

      More crucial is the nongeminate recombination, which (as pointed out above) influences the carrier concentration and also its gradient. It is important to understand that the open circuit voltage corresponds to a current free situation, _not_ a field free case. Thus, you certainly can have a finite internal electric field at open circuit. The injection barriers play an important role here. For low injection barriers and thus good injection, the electron injecting contact will determine the carrier concentration at this side (similar for the hole contact): this high concentration is not influenced by recombination! Recombination will of course lower the overall carrier concentration, and due to the boundary condition of injection at the same time the carrier concentration gradients will become steeper. Simultaneously, Voc will be reduced.

      It is important to point out that our 2008 paper considered only Braun-Onsager for photogeneration and Langevin recombination for already free charge carriers. The latter was taken into account even for charge carrier mobilities beyond about 1 cm2/Vs, although for high mobilities the finding of charge carriers is not the step limiting the recombination rate any more: Langevin is only valid for low mobility materials. Therefore, check out our 2009 followup paper, also considering surface recombination in addition to bulk recombination: [Wagenpfahl 2009]. Best,

      Carsten

  7. Very impressive and useful blog site I came across :).

    Dr. Deibel, I could not understand the details about how imbalance mobilities of D/A polymers in organic solar cell adversely affect the performance, although I read many literatures on this. Let’s assume two cases of mobilities: (i) hole:=10^-6 cm2/V.s and electron:=10^-6 cm2/V.s, (ii) hole:=10^-6 cm^2/V.s and electron:=10^-2 cm2/V.s
    How do you compare the performance of these two? Will the second case (highly imbalanced) show lower power conversion efficiency due to space charge build-up? I believe, since the performance is mainly governed by the lower mobility, the higher electron mobility (in this case) does not improve the PCE or at least it should not not make it worse. Please throw some light on it.

    1. Hi Anup! It depends a bit on the details, even for your two cases, but: in general, a mobility imbalance is unfavourable.

      Even if not considering recombination, for your case (ii) the electrons leave the solar cell 4 orders of magnitude faster than the holes after a photogeneration event. This will lead to the build-up of space charge due to the (on average) higher concentration of holes in the device, which can lead to an s-shaped current-voltage curve and a correspondingly low fill factor. Thus, the effect will be similar to what we showed in [Wagenpfahl 2010]. I believe W. Tress, than at IAPP Dresden, modelled I(V)s for imbalanced mobilities, but as he did not cite us I forgot the paper reference;-)

      Recombination may also be influenced, depending on the actual recombination rate, in addition to the charge transport / extraction issue above. Best,

      Carsten

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