As an in-between, we’ll talk about a topic which will hopefully become more and more recognised by the organic photvoltaics community: the shortcomings of the established Shockley model, 
made for crystalline inorganic diodes, when applied on fitting organic solar cells.
The most important figures of merit describing the performance of a solar cell are the open circuit voltage, the short circuit current, the fill factor and the (power conversion) efficiency. The fill factor is given by the quotient of maximum power (yellow rectangle in the figure) and the product of open circuit voltage and short circuit current (white rectangle); it therefore decribes the “squareness” of the solar cell’s current-voltage characteristics. The efficiency is the ratio of maximum power to incident radiant power – typically radiated by the sun. E.g., a well-known detailed balance calculation for inorganic single gap solar cells gives a theoretical maximum of about 30% power conversion efficiency [Shockley 1961]. The upper limit for organic solar cells is somewhat lower, but that’s another story.
As you may know, the Shockley diode equation
(which is older than 1961 but also used in the paper) looks as in the equation below when corrected for real inorganic devices with series and shunt resistance:
In the Shockley equation for “real” diodes, an optional photocurrent is included by a parallel shift of the current-voltage curve down the current axis: this is the (constant) photocurrent jph. Now, many people have fitted the current-voltage characteristics of organic solar cells under illumination with this equation, but as one can clearly see from the figure above, the shown j(V) curve for a typical organic solar cells has a strongly field dependent photocurrent. There is for example a crossing point of dark and illuminated curve at approx. 700mV which cannot be explained by the Shockley equation. The reason is, as explained in “How Do Organic Solar Cells Function? – Part One“, that the Coulomb bound polaron pairs (approximately: electron-hole pairs) have to be split by the externally applied electric field. At 700mV (in this instance), however, the internal electric field, which is the contact potential difference minus the external electric field, is zero. That means flat band conditions, and therefore there is not enough driving force for the polaron pairs to be separated: there has to be a crossing point. (Actually, even in inorganic compound semiconductors such as CuInSe2 there are similar crossing points, but their origin is different.)
As you can also see from the upper 
figure, it sometimes happens that the maximum photocurrent is not reached at 0 Volts, i.e., under short circuit conditions, but only at more negative bias, corresponding to a higher internal field. This happens in organic solar cells where the polaron pair dissociation is more difficult, e.g. if the active layer is thicker, and therefore at the same (external) voltage the (internal) field at zero bias is lower.
The details of polaron pair dissociation are not completely understood. Right now, the so called Onsager theory [Onsager 1938] and its somewhat more modern incarnation [Braun 1984] are used to describe its field dependence. According to me, however, the last word is not yet spoken… which might not mean much;-)
Coming back to the Shockley equation. A positive bias leads to the injection of charge carriers into the solar cell: the current increases exponentially, we see a rectifying (=diode-like) behaviour in the ideal case. In real solar cells, however, there are losses, considered in the second equation above by two resistors. The so called series resistance Rs – in series with the diode – describes (amongst others) contact resistances such as injection barriers and sheet resistances. In contrast, the parallel resistance covers the influence of local shunts (=short circuits) between the two electrodes, i.e., additional current paths circumventing the diode. 
Works nicely in silicon solar cells, but in organic solar cells some problems appear: the “parallel resistance” now seems to depend on the voltage and illumination intensity, the “series resistance” also also changes with voltage.
Unfortunately, there is no analytic equation yet to properly describe the peculiarities of organic solar cells. what we’ll settle for now is to describe the known differences between Shockley and real organic cells. As organic semiconductors are usually not as conductive as their inorganic counterparts, at higher voltages (and sometimes also at higher negative internal fields under illumination, even in the 4th quadrant!) space charges can build up, leading to space charge limited currents. Here, the current is proportional to the square of the voltage (an not linearly proportional to the voltage as for resistors). (Actually, in organics, the well-known Childs law or Mott-Gurney law with j being proportional to V2 is also not strictly correct… maybe more on this another time;-) This can lead to the determination of apparently voltage dependent resistors. As mentioned above, space charges under illumination, which can be induced for instance by trapped charges, can superimpose with the “parallel resistance”, which than becomes voltage (and light) dependent as well. And of course, shunts and contact resistances do also exist in organic solar cells.
I like the graphics for the I-V curves you used. Did you make them yourself or do you know where you found them. Thanks.
Thanks for your comment. All images are self-made, of course:-) I’ll email you a pdf. If you use them, I’d be grateful for a citation. Regards, Carsten
Let me congratulate you on this excellent blog. Like Chris said on 19th July it is so helpful… we learn things we are not able to acquire from papers.
I would like to ask two rather unprofound questions: first concerning your Current-Voltage characteristics of OSCs. Why is the behavior of the dark current in a general OSC so similar to a Si:c pn junction? Here the concentration gradient from p to n of the available charges allows for a (exponential) positive current as soon as a positive ext voltage lowers the internal potential that exists at equilibrium. The charges themselves are available since kT is of the same order of the ionization energy of the dopants. In a dark OSC, it seems to me you have no available free charges for transport (even after doping). But you mentioned in your note Polaron,…,Exciplex: ”Singlet and triplet excitons can also be formed due to interaction following charge injection…”. Is it possible that charges injected externally into the OSC (in opposite sense to the photocurrent that would flow under illumination) will somehow generate excitons (in a reversal of the dissociation mechanism exciton to Polaron Pair)? This being the case the more you lower the internal electrostatic field the less dissociation of excitons would take place (contributing to oppose the injected current) and a positive current charge carrier might hop from molecule to molecule… would there be radiative photoluminescence then?
Second (even more boring) question: concerning this figure of 1,3 Kw/m2 of radiation on a 500km orbit satellite (ACRIM program). It seems to me that if you resort to a pyrheliometer you are basically interpreting the power of radiation (irradiance) as a heating power. Since photons transmit chemical energy and brightness (illuminance) in addition to heating a surface, probably there will be far less photons reaching a unit surface (photon flux) than calculated in this way…
Carlos, thanks:)
In the dark, you do have free charges available in an organic semiconductor device, no matter if made of one semiconductor (e.g., LED) or two (a solar cell). Usually, these carriers are not intrinsic, but injected. If you can inject both electrons and holes, you may have a certain fraction of them interacting, generating excitons (singlet excitons and CT singlet excitons for diode and solar cell, respectively). The latter may recombine, also radiatively. This is the basis for organic light emitting diodes, where the resulting emission is significant. For solar cells, you will also have radiative recombination, but that cannot be seen by eye only, you will probably need a good detector;-) This emission is called electroluminescence, as it results from injection, not photoexcitation.
Concerning radiation, I do not think that you will have a (significant) discrepancy between the “thermodynamic” measurement and the photon flux on the solar cell surface. Solar irradiation is based on a distribution of photon energies, thus a spectral distribution, with a certain flux depending on the photon energy. For solar irradiation, black body radiation of 5000K or so works pretty well, therefore your detector does not need to measure every part of the spectral range separately. Best, Carsten
Thank you, for your prompt answer which I thought I would be notified of by wordpress.com or other but only now by chance did I discover it.
Sorry to insist: why is the dark current in the OSC, in reverse bias, which increases the internal field, so low, then? … which is the same thing as asking why does the non-illuminated device act as a rectifier.
Best wishes,
Carlos
About my last question, you would probably mention the Schottky barrier.
If you do have the time to come again on the Child’s law (J~ V1.5), could you please comment on:”[…]exponent larger than 2, which is commonly explained by transport with an exponential distribution of traps[…]”, from a paper on trimolecular recombination that mentions your work, by Schubert, M., Steyrleuthner, R., Bange, S., Sellinger, A., Neher, D., Charge transport and recombination in bulk heterojunction solar cells containing a dicyanoimidazole-based acceptor, Phys. Status Solidi A 206, n.º12, 2743-2749 (2009)?
Best wishes,
Carlos
There is rectification in the dark solar cell, because it is an ambipolar device with asymmetric injection barriers. In forward bias, electrons and holes are injected (from anode resp cathode), in reverse bias the corresponding injection barriers are too high: electrons (holes) cannot easily enter at the cathode (anode). Illumination gives you charge generation within the organic blend, thus generating an additional (negative/extraction) current, but the other boundary conditions remain the same.
Space charge limited currents (SCLC, with Child’s law as the trap-free case) is unrelated to recombination. For details, have a look at [Arkhipov 2001], in which the behaviour is well explained for gaussian DOS. The principle is the same for exponential DOS.
Best, C
First many thanks for keeping this blog, it has been very, very useful to me…
Just wanted to point out a paper from Durrant claiming that in some polymer cells current generation does not seem to be field dependant (they use Transient absorption techniques to compare different devices with very different J-V characteristics:|J. Phys. Chem. Lett. 2010, 1, 3306–3310).
I’m very new to this field and there are some things I don’t understand like the space charge concept. About space charge build up in organic solar cells (suposedly due to lower mobility and higher trap concentration, right?), is there any way to characterise such space charge? Some charge extraction experiements have been reported on polymer-based solar cells where the excess charge in the devices often follows an steep increase with light bias (suposedly exponential)(see:Shuttle et al. Appl.Phys LEtt.,93,183501, 2008; Shuttle et al. Appl.Phys Lett.,92,093311 (2008); Hamilton et al. J.Phys. Chem. Lett., 2010, 1, 1432). Is this charge build such “space charge build up”? From you paper, phys. stat. sol. (RRL), 2, 175 (2008), I understand that “efficient charge extraction at high mobility” limits Open circuit voltage, because there won’t be any charge build up and if mobility is too low charge build up will increase giving high VOC but low current. However some studies we are carrying up on small molecule (DPP(TBFu)2)-based BHJ devices (first described by N’Guyen et al.: adv. func. mater. 2009, 19, 3063) seem to show a different trend, as Open circuit voltage is very high (very close to theoretical values, and currents are very high). In contrast to reported devices the charge density show a complete linear dependance on light bias (no charge build up)! Could this mean that band bending in such devices is significant, then?
Hi Aurel, thanks!
Durrant et al are right: for many “good” organic solar cells at room temperature, the charge generation seems to be field independent. The same is seen by Street et al (PRB 2010, do not have the reference at hand). There are not that many studies on this field. We were able to model the photocurrent of P3HT:PCBM solar cells by a charge generation yield of 60% at short circuit and 50% under open circuit conditions, which is also almost field independent [Limpinsel 2010].
Space charge build up happens in low mobility materials in the (dark) injection regime (see my reference in the previous comment in this thread), and sometimes also in the photocurrent [Mihailetchi 2005] [Wagenpfahl 2010]. I do not think that the “charge buildup” seen by the Durrant group is similar, but have not checked their papers again to make sure.
Our phys stat sol RRL was made under the condition that even at high mobilities the recombination occurs with the Langevin recombination rate. This is not a very good assumption, I am sorry to say. Please find a more applicable simulation here [Wagenpfahl 2010a]. In any case, it is difficult to get into this high mobility regime with organic semiconductors. Concerning your studies, more important to Voc are probably the energy of the charge transfer complex (the “effective bandgap” or “maximum Voc”) which is, in view of Voc, reduced by charge carrier recombination. What do you mean with “charge buildup” in view of the linear Voc dependence on light intensity? The term does not quite fit here, it seems to me, more important is the recombination regime (1st order recombination or 2nd order, and even then it depends on the “strength” of the recombination”. Details on request;-) Best, C
I appreciate your courage to dwell in these matters in the circumstances…
I suppose anode (+) should be substituted for cathode (-) (or electrons by holes)…
When you have the time and had some sleep (according to my experience, things will get steadily worse from now on, until one reaches the IT summit…), referring to the denominator of mobility in CELIV, shouldn’t it be tmax2 in the Schubert et al. paper I mentioned previously, as in your paper Phys. Status Solidi A 206, No. 12, 2731–2736 (2009) ?
Thank you so much for the answer, your thoughts and the Arkhipov paper,
Congratulations, Good Luck and very
Best wishes
Carlos
Thanks for the good wishes! The definition of anode and cathode is not very specific, it seems to me, and depends also on the action (i.e. can be different for diode and solar cell, light generation vs current generation). Typically, the electrode with the lower work function is called anode in the papers about organic solar cells I know. Concerning the Schubert paper, I think they have a typo, but I am pretty sure they use the correct equation (including the correct tmax2) for data evaluation.