The transport resistance in organic solar cells

In one of my last posts on the diode ideality factor (6 years ago…), I promised to talk about the transport resistance in organic solar cells. I came across it already during my time at IMEC in Leuven, Belgium, around 2004: my colleagues and I worked on an analytic model of the open circuit voltage in organic bilayer solar cells. The corresponding paper was published a few years later, [Cheyns et al 2008], but I have to admit that I did not grasp its importance as a relevant loss mechanism for organic solar cells in general, focussing on geminate and nongeminate recombination – until this paper by [Würfel/Neher et al 2015] came out. I think now I have;)

The transport resistance is an internal resistance in the active layer (or transport layer(s)) of the solar cell, acting like an internal series resistance: It changes the slope of the current density–voltage characteristics – for instance around the open circuit voltage – and thus reduces the fill factor.

If you can live with an effective conductivity $\sigma_\text{eff} = e n_\text{eff} \mu_\text{eff}$ for now, I’d say that the transport resistance is given as

$R_{tr}= \frac{1}{e} \int_0^L \frac{1}{n_\text{eff}(x) \mu_\text{eff}(x)} \text{d}x \approx \frac{L}{\sigma_\text{eff}}$

with the effective carrier concentration $n_\text{eff}$, the effective mobility $\mu_\text{eff}$, and the active layer thickness $L$. For a finite current $j$, the externally applied voltage $V_\text{ext}$ of the $j(V_\text{ext})$ curve (red in the sketch) is reduced by $\Delta V = V_\text{ext} - V_\text{int} =j R_{tr}$ as compared to the transport resistance free case which I call $j(V_\text{int})$ (violet). The reason is that the electron and hole energy levels equivalent to conduction and valence band are more tilted due to the influence of the low conductivity. Rod MacKenzie from Durham used his drift–diffusion simulation programme gpvdm – which among many other features allows to use the multiple-trapping-and-release model to account for energetic disorder – to calculate an example. In the figure, the energy levels corresponding to conduction and valence band are shown, and the yellow shading between the band edges correspond to the trap populations which I will ignore. So focus on the edges! The upper left inset features the band bending of the transport resistance free case at a current–voltage point in the fourth quadrant of an illuminated solar cell, $V_\text{int}$ being at 0.8 V. The red lines show that in the bulk, the bands are perfectly flat despite not being at any “flat band” voltage. In contrast, the lower left inset shows the band bending for the illuminated current–voltage point at the same current, but at $V_\text{ext}$ of 0.69 V. The voltage difference $\Delta V$ of 0.11 V is due to the transport resistance. The two perfectly horizontal red lines, just shifted down from above, now show a discrepancy due to the band bending in the bulk. This tilt sums up to the voltage loss $\Delta V$, and corresponds to a gradient of the quasi-Fermi levels, as discussed clearly in [Würfel/Neher et al 2015] around Eqn (8). Let me note that the denomination of $V_\text{int}$ as opposed to $V_\text{ext}$ is a bit unfair, as the latter still drops across the whole (active) layer thickness $L$ – but the gradient is different. So, can the two different band bendings (upper left and upper right) occur at the same time? No of course not. The lower right shows you how a simulated organic solar cell with realistic parameters looks inside: it is limited by a transport resistance due to low effective conductivity in the active layer, which shows up as tilt in the bands, and implies a gradient in the quasi-Fermi levels. The upper right shows you how the bands looked if the solar cell were not limited by a transport resistance (and how they do look under the special case of open circuit conditions).

Still, the voltage drop due to the transport resistance – coming from the low conductivity and leading to the tilted transport levels – is real and does reduce the fill factor.

In my recent talk at the MRS Spring meeting (invited! in person! on Hawaii! slides here:), I showed the approximate difference of the internal and external voltage of a fresh, inverted PM6:Y6 bulk heterojunction solar cell ($L$ = 200 nm, but qualitatively similar for 100 nm) at different light intensities, from 3 micro-suns to 1 sun (300 nW/cm2 to 100 mW/cm2). The ideal diagonal case where $V_\text{int}=V_\text{ext}$ is only seen for the lowest light intensities. In contrast, at 1 sun illumination, when 0.4 V are applied, the internal voltage is still at around 0.7 V. The fill factor at 1 sun is, by the way, somewhat above 60%, whereas the transport resistance free pseudo fill factor (determined using the Suns-Voc method, see below) is above 80%. As If you are interested in the details of this study, please have a look at our preprint. [Update 2022-07-01]: now published in Nature Communications.

How did we determine the approximate internal voltage, or $j(V_\text{int})$, when we only have direct access to the external (i.e., applied) voltage? Similar to [Schiefer et al 2014], we used the Suns-Voc method as a measure for the series and transport resistance free current–voltage characteristics. The idea is in the post on the diode ideality factor, but before sending you away again I repeat it here: the current density, as given by the diode equation under assumption of the superposition principle,

$j = j_0 \left( \exp\left(\frac{q(V_\text{ext}-jR_s)}{nkT} \right) -1 \right) - \frac{V_\text{ext}-jR_s}{R_p} - j_\text{gen}$ (*)

is influenced by the series resistance $R_s$ and parallel resistance $R_p$. At open circuit, $V_\text{ext} = V_\text{int} = V_{oc}$, the current $j(V_{oc})$ is zero,

$0 = j_0 \left( \exp\left(\frac{q(V_{oc}-0 R_s)}{nkT} \right) -1 \right) - \frac{V_{oc}- 0 R_s}{R_p} - j_\text{gen}$,

so that the terms including the series resistance (but not the parallel resistance!) become zero as well. Reordering the equation,

$j_\text{gen} = j_0 \left( \exp\left(\frac{qV_{oc}}{nkT} \right) -1 \right) - \frac{V_{oc}}{R_p}$,

we have an equation very similar to the diode equation (*). The pairs $j_\text{gen}(V_{oc})$ measured under a wide range of light intensities can, when shifted down by $j_{sc}(\text{1 sun})$, correspond to the so called Suns-Voc curve. It represents the $j(V_\text{int})$ curve, as the open circuit voltage is transport resistance free so that (only) there, $\Delta V=0$. The Suns-Voc curve is equivalent to the illuminated $j(V_\text{ext})$ curve, but lacks the influence of the series and transport resistance. The generation current $j_\text{gen} \approx eGL$ with the generation rate $G$ is sometimes (ok, often..) approximated by the short circuit current $j_{sc}$.
An example of how a Suns-Voc curve looks is shown in my earlier post on the diode ideality factor in the third figure (red = illuminated $j(V_\text{ext}$), green symbols = Suns-Voc curve).

Of course there is a better way by measuring both the voltage dependent current and luminescence of a solar cell, as done by [Rau et al 2020] on Cu(In,Ga)Se2 solar cells: from the luminescence, which depends on the quasi-Fermi level splitting, the internal voltage can be determined. For organic solar cells, as singlet and charge transfer exciton photoluminescence need to be separated – the internal voltage will be proportional to the latter – this is harder and has not been done yet, as far as I know.

How can the impact of the transport resistance on the fill factor, and thus performance, of organic solar cells be minimised? To reduce the transport resistance, the active layer conductivity needs to be increased. This can be done by, either, increasing the charge carrier mobility – for a molecular hopping system this means reducing static (low $\sigma$) and/or dynamic disorder (low $\lambda$). Or, increasing the carrier concentration – e.g. by doping, which is kind of hard in a bulk heterojunction, but starting with the material phase with the lower conductivity could be a viable approach.

Some hat tips here beyond the very nice cooperation of our study: Maria and Rod for feedback to an earlier version of the post, and Julien for being the first commenter on it:)

Some links collected over the last months.

I will be at the ISCPAC 2016 meeting next week. In case you are also there, meet up:-)

[2016-06-07 Some Updates in the afternoon;-)]

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”

Interaction of light with solids in experiment and simulation

Hi there, sorry for not getting back to you but starting a new group and having new responsibilities (e.g. involvement in new degree programmes for Material Science) can take (part of) the blame. Just as brief progress indicator, here a link to an interview of the Chemnitz University of Technology press office with me. (Photo: Uwe Meinhold)

The official short name of my group is OPKM, for Optics and Photonics of Condensed Matter. For the (very) long official name I refer you to the web page of the Institute of Physics at the TUC;-) The size of my group is growing slowly but steadily, and the lab building shows progress as well: setups for time correlated single photon counting to measure photoluminescence transients – e.g. to determine charge carrier recombination in perovskite solar cells – and for confocal measurements of luminescence are already available from my predecessor’s group: we just adapt them to our needs. Other setups, time resolved and steady state, are being built and come along nicely. Solar cell preparation is still improvised, using the glovebox system of a colleague and the evaporation chamber of another, until we get our own integrated glovebox/evaporator system. One of my main interests is still Organic Photovoltaics, and with my (PhD) background in inorganic photovoltaics I also look at the hybrid perovskite solar cell hype (as a hype is not necessarily a bad thing;-). What also remains is my joy to combine experiments and simulations (macroscopic device simulations, kinetic Monte Carlo simulations) to understand these systems.

If you are interested in joining us: I have two PhD positions available at present. Please check out the job offer (german; computer-translated here) and contact me.

Cheers!

Restarting in Chemnitz

Just a brief note, I moved from Würzburg to the Institute of Physics at Chemnitz University of Technology this March, starting a new group. At present I have one PhD position open on Organic Photovoltaics – funded by the University, therefore including some teaching duty in German. Have a look here (in German) or drop me a line if you are interested. Cheers,

Carsten

Some links I found interesting since the last time… partly older stuff, though.

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”

New life – again

On 11th of February (my birthday, incidentally), our 2nd child was born: Nicolas Jacob. We are so happy, as is our 2 year old daughter Chiara.

Even with only one kid, it was already rather quiet on this blog in the last months (well, 2 years…). Still, I do have some hope to be able to dedicate one or the other quiet minute to write some urgently needed updates to previous posts. All the best, Carsten

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.