## 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 $j_{sc}$. Continue reading “The diode ideality factor in organic solar cells: basics”

## Nongeminate recombination in organic solar cells – slower than expected

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 $n$ and holes $p$ are photogenerated pair wise, the respective excess charge carrier concentrations are symmetric, $n=p$. Then a recombination rate $R$ is

$R = k n p = k n^2$,

where the recombination prefactor $k$ could be a Langevin prefactor – more on that later. In a transient experiment with a photogenerating, short laser pulse at $t=0$, 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)

$\frac{dn}{dt} = G-R$

becomes

$\frac{dn}{dt} = -R$

for $t>0$ (as the generation was only at $t=0$).

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 $t>0$ yield

$\frac{dn}{dt} = k n^2$ Continue reading “Nongeminate recombination in organic solar cells – slower than expected”

## Belectric aquires German Konarka Daughter

Related to this post on Konarka’s bankruptcy: According to a range of news sites, including pv-tech.org, the german company Belectric has acquired Konarka Technologies. Find the press release here (pdf).

The system integrator Belectric is situated in Lower Franconia, less than 50km from Würzburg and less than 10km from where I live. Let’s keep our fingers crossed!

## Nongeminate Recombination: Langevin (again) and beyond (later;-)

Nongeminate recombination is the major loss mechanism for state-of-the-art organic solar cells. In an early blog post, I showed how the Langevin recombination was derived. Although there is more to nongeminate recombination than just this mechanism, it is still instructive and also relevant to trap-assisted recombination mechanisms, due to its mobility-containing prefactor.

[Nenashev 2010] pointed out that in the derivation of the Langevin recombination,

since the electric field scales as r−2 and the surface area of the sphere scales as 2, the value of r chosen is unimportant, leading to a simple solution with constant electron den- sity, thus justifying the neglect of diffusion.

## Konarka bankrupt

Via Juan Bisquert’s post: solar cell company Konarka filed for bankrupty yesterday according to Businessweek. It is always hard to be the first… Konarka received its first venture capital in mid 2001.

Howard Berke, CEO of Konarka:

This is a tragedy for Konarka’s shareholders and employees and for the development of alternative energy in the U.S.

Let’s hope that our friends from Konarka find other suitable positions, and that other companies such as Heliatek take up the lead!

and nothing else.

Photovoltaics:

Global warming:

• Climate sceptics on the go, via /.: Don’t Worry About Global Warming, Say 16 Scientists in the WSJ; I am no climate scientist, but what I read usually points in the other direction… at least judging from most scientists with peer reviewed publications in contrast to non-peer reviewed “scientists”. Nevertheless, the scientists cited above seem to be real ones, although (mostly?) not with scientific background related to the global climate
• we have a similar discussion here in Germany, with RWE manager Fritz Vahrenholt writing a book trying to confute evidence of global warming, relating any temperature change to the solar activity: summary by Die Zeit (german, google translate) and an article (again Die Zeit) by Toralf Staud, refuting the seven main theses of Vahrenholt ( german, google translate).

Other stuff:

## Pseudosymmetry of the photocurrent physically relevant?

Two days ago, a paper considering the role of the “quasiflat band” case in bulk heterojunction solar cells by device simulations was published online [Petersen 2012]. It is critical of the pseudosymmetric photocurrent found and interpreted by [Ooi 2008] and later also ourselves [Limpinsel 2010]. In order to focus on the physical relevance of the (non)symmetry of the photocurrent, the paper by Petersen et al neglects a field dependent photogeneration. As some good points are raised, read the new paper if you are interested in the photocurrent.

[Update 2.4.2012] Another paper showing that band bending is not needed to explain the particular shape of the photocurrent: [Wehenkel 2012].

I will come back to field dependent photogeneration later, it is still intruiging: also here, the photocurrent should (and will be) complemented by pulsed measurements such as time delayed collection field, see e.g. [Kniepert 2011].

## Charge transport in disordered organic matter: hopping transport

As I won a proposal today, I feel up to contributing once again some physics to this blog… I know, it has been a long long wait. So today it is time to consider some fundamentals of charge transport, as this is not only important for the extraction of charge carriers from the device (see earlier posts on mobility and efficiency, surface recombination velocity and photocurrent) but also the nongeminate recombination (see e.g. photocurrent part 2 and 3).

In disordered systems without long range order – such as an organic semiconductor which is processed into a thin film by sin coating – in which charge carriers are localised on different molecular sites, charge transport occurs by a hopping process. Due to the disorder, you can imagine that adjacent molecules are differently aligned and have varying distances across the device. Then, the charge carriers can only move by a combination of tunneling to cover the distance, and thermal activation to jump up in energy. In the 1950s, Rudolph A. Marcus proposed a hopping rate (jumps per second), which is suitable to describe the local charge transport. By the way, he received the 1992 Nobel prize in chemistry for his contributions to this theory of electron transfer reactions in chemical systems. Continue reading “Charge transport in disordered organic matter: hopping transport”

## 2012

Hi there, I am late again, but nevertheless: a happy and successful year 2012!

I have collected a few links which might or might not interest you. Also, I plan to start with some scientific (background) posts again. Let’s see how this works out:-)

Press release of Heliatek: Heliatek achieves new world record for organic solar cells with certified 9.8 % cell efficiency. Evaporated small molecule tandem with area above 1cm2. Very good! Also, Mitsubishi Chemical has reached 10.1% efficiency on solution processed small molecules.

Nature looks back at the science year 2011: 365 days: Images of the year.

Interesting, although not related to physics: Syllabus for David Foster Wallace’s class “English 102-Literary Analysis: Prose Fiction Fall ’94”. Clear rules, yeah! Forgot who linked to it, sorry. Continue reading “2012”

## SPIE Pickings

Already 8 weeks past, recently some Videos (well, stills of the slides plus audio) of the Solar and LED Session of the SPIE Optics and Photonics 2011, San Diego went online.

Here are two or three which might interest you (well, they got my attention;-) but there is more to be found on the above mentioned web site – although I had to modify the settings of my ad blocker to be able to watch. No, there are no ads; still…

Before you scroll down, let me mention some other “findings” of potential interest:

But now to these SPIE presentations [Update: WordPress does not accept the embedded vidos, so here just the links to the videos].

James Durrant, Imperial: Charge photogeneration and recombination in organic solar cells