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.