# The diode ideality factor in organic solar cells: basics

Where does one start after so long an absence — meaning only the blog abstinence; I have been working and publishing since last time;-) One of the things which have been on my mind is the ideality factor, a figure of merit for the charge carrier recombination mechanism in a semiconductor diode. In short, a diode ideality factor of 1 is interpreted as direct recombination of electrons and holes across the bandgap. An ideality factor of 2 is interpreted as recombination through defects states, i.e. recombination centres. More on that in a later post, let’s start with the basics.

A couple of years ago, I wrote about some general properties of current-voltage characteristics of organic solar cells, but did not describe the ideality factor.1 I think the ideality factor was mentioned only once, and then without details.

The Shockley diode equation describes the current–voltage characteristics of a diode,

$j=j_0 \left(\exp\left(\frac{eV}{n_{id}kT}\right)-1\right) - j_{gen}$.

Here, $j$ current, $V$ the voltage, $e$ elementary charge, $kT$ thermal voltage, $j_0$ the dark saturation current, and $j_{gen}$ the photogenerated current. If the ideality factor $n_{id}$ was equal to one, one could call this the ideal Shockley equation. It derivation can be found in semiconductor text books, but it can also be derived based on thermodynamic arguments (see Peter Würfel’s excellent book on the physics of solar cells).

The current $j$ flowing out of the diode is defined to be negative. Essentially, the charge carriers which can flow out are the generated ones (e.g. $j_{gen}$), but reduced by the recombination current. That means,

$j=\underbrace{j_0 \left(\exp\left(\frac{eV}{n_{id}kT}\right)-1\right)}_{j_{rec}} - j_{gen}$.

However, the term $j_{rec}$ contains also a negative contribution, $j_0$ times the $-1$ from the bracket. This is the thermal generation current $j_{gen,th} \equiv j_0$, i.e. charge carriers excited across the bandgap just by thermal energy — and therefore very little. Still, the term is very important, as it is the prefactor of the whole $j(V)$ curve. Without light, i.e. with photocurrent $j_{gen}=0$, we can clarify

$j=\underbrace{j_0 \exp\left(\frac{eV}{n_{id}kT}\right)}_{j_{rec,dark}} - \underbrace{j_0}_{j_{gen,th}}$.

so that at negative voltages, $j=-j_0$. (Please note that under realistic conditions, $j_0$ is not only pretty small and difficult to measure in principle, it is also hidden behind shunt currents in the device. ) At zero volt, $j=j_0-j_0=0$. Thus, generation = recombination — or more specifically, thermal generation current = recombination current — which essentially implies that 0V correspond to the open circuit voltage in the dark.

How can one determine the ideality factor and the dark saturation current (at least in principle, see below for a better way on real devices)? It is common to neglect the thermal generation current (the term -1, multiplied by $j_0$), which is a good approximation for voltages some $kT/e$ larger than 0. Then, calculate the logarithm of the dark current ($j_{gen}=0$),

$\ln(j) = \ln(j_0) +\frac{e}{n_{id}kT}V$,

so that the ideality factor can be determined from the inverse slope of the ln(current) at forward bias, and the dark saturation current from the current-axis offset. Let me already tell you that I do not recommend this approach, for reasons written below, and as explained in more detail in a recent paper of Kris Tvingstedt and myself [Tvingstedt/Deibel 2016].

Under illumination and at open circuit conditions, $j(V_{oc})=0$, we can rewrite the Shockley equation as

$j_{gen}=j_0 \left(\exp\left(\frac{eV_{oc}}{n_{id}kT}\right)-1\right)$,

which has the same shape as the Shockley equation in the dark. This means that if you measure ($j_{gen}, V_{oc}$) pairs for a (wide) range of different illumination intensities (thus varying $j_{gen}$), the points should overlap with the dark $j(V)$ curve! We’ll come back to this important point further below. Note that for solar cells with good fill factor, $j_{gen}$ can be approximated by the short circuit current $j_{sc}$.

Resistance

Even a very good real solar cell does not exactly follow the Shockley equation as stated at the beginning. Contact resistances and small shunt currents flowing from electrode to electrode in parallel to the diode (i.e. without rectification) have to be considered. These effects can be approximated by considering a series resistance $R_s$ and a parallel (shunt) resistance $R_p$,

$j=j_0 \left(\exp\left(\frac{e(V-jR_s)}{n_{id}kT}\right)-1\right) + \frac{V-jR_s}{R_p}- j_{gen}$.

That means, the internal voltage at the solar cell is reduced by a voltage drop across the series resistance, and the diode current is essentially superpositioned on a shunt current. Here, indeed, the dark current in reverse voltage direction is not $j_0$, but dominated by the shunt current. The exponential regime of the current–voltage characteristics, from which we determined both the ideality factor and the dark saturation current above, is now partly hidden: at low voltages the shunt resistance dominates the current, and at high voltages the series resistance drags the exponential current into a linear one. The ideality factor could only be determined from the dark characteristics using the “remaining” part of the exponential current–voltage regime. Again, this is not the recommended way of determining the ideality factor.

If we again look at what happens for $j(V_{oc})=0$, we get

$j_{gen}=j_0 \left(\exp\left(\frac{eV_{oc}}{n_{id}kT}\right)-1\right) + \frac{V}{R_p}$ :

the term $j R_s$ becomes zero as the open circuit voltage is “measured” without current flow, so the series resistance does not apply. (Note, although pretty evident I think: all figures in this post show calculated data, not measurements!) However, the shunt resistance still does! Nevertheless, this implies that while the ideality factor determination from the dark current–voltage characteristics under real conditions is limited by series and shunt resistance, the method using ($j_{gen}, V_{oc}$) pairs is at least not limited by the series resistance! This can also be seen when comparing the dark current-voltage characteristics for an internal voltage $V_{int}$ with the same current plotted at the external voltage $V_{applied}$, which is reduced compared to the internal one by the (series) resistance. As shown in the figure, the fill factor for a measured device (which happens always with the applied voltage, of course;-) is clearly lower as compared to the one plotted against the internal voltage. However, the ($j_{gen}, V_{oc}$) pairs (in the figure approximated by ($j_{sc}, V_{oc}$) are not limited by the (series) resistance and therefore show the higher fill factor. Similarly, ideality factor should be determined with the ($j_{gen}, V_{oc}$) pairs (yielding $n_{oc}$ in the figure, which corresponds to the “reference” for the internal voltage $n_{int}$) and not from the dark characteristics $j(V_{applied})$ (yielding $n_{ext}$. Often less extreme overestimation, but just the same: do not do it;-).

So, what’s next. I plan to write two more posts on the ideality factor, one on its relation to the recombination rate, and one the transport resistance (see recent papers by [Würfel/Neher et al 2015] and [Neher/Koster et al 2016].

P.S. Not only finding time to write a blog post is more difficult these days: I have been taking only photographs of my kids – which I do not post on the internet – since 2011, but almost no nature or architecture photographs. Thus, not much to lighten the text and equations, but also less distractions ;-)

P.P.S. [Update 2016-05-15] added “-” everywhere, terribly sorry!

1. Revisiting these old posts makes me acutely aware of what I did not know then and do know now a bit more about. E.g. the explanation that crossing point is due to the field dependent separation of polaron pairs is not correct.
1. Thanks, good point. I’d say a good rule of thumb is: if the slope of the current–voltage characteristics at short circuit is (close to) zero (i.e., $dj(V)/dV|_{V=0}\approx 0$), then $j_{sc}\approx j_{gen}$ is a good assumption. This is even true if losses of singlet excitons reduce the charge carrier generation rate (for a given singlet exciton generation rate), as these losses are pretty independent of voltage.