Tag: device physics

How to see the temperature dependence of the open-circuit voltage from the ideal diode equation?

The open-circuit voltage is the voltage in the current–voltage characteristics of a solar cell that is defined where the current is zero. That means that the (internal) charge carrier generation and recombination rates are equal, so that no net current can flow out of the device.

We can simply rearrange the ideal diode equation and solve for the open-circuit voltage. The ideal diode equation was discussed with respect to the ideality factor in this post. The current density is given as

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

with V the voltage, e elementary charge, kT thermal voltage, n_\mathrm{id} the recombination ideality factor, j_0 the dark saturation current, and j_\mathrm{gen} the photogenerated current. For simplicity, the latter is chosen to be voltage independent, and therefore is equal to the short-circuit current j_\mathrm{sc}.

As the open-circuit voltage is determined at zero net current, j(V_\mathrm{oc}) = 0, we get

j(V_\mathrm{oc}) = 0 = j_0 \left(\exp\left(\frac{eV_\mathrm{oc}}{n_{id}kT}\right)-1\right) - j_\text{gen},

which we can rearrange to yield the open-circuit voltage

V_\mathrm{oc} = \frac{n_{id}kT}{e} \ln \left( \frac{j_\text{gen} + j_0}{j_0} \right).

Here, j_\text{gen} is the photocurrent due to solar illumination, and the dark saturation current density j_0 is due to excitation of thermal “black body” photons from the ambient at, say, room temperature. In the simplest case – in the dark where j_\text{gen} = 0 – we see that V_\mathrm{oc} = 0, too. Generally, the thermal generation leading to j_0 is much weaker than the solar generation j_\text{gen}, therefore

V_\mathrm{oc} \approx \frac{n_{id}kT}{e} \ln \left( \frac{j_\text{gen}}{j_0} \right).

is usually a very good approximation.

This simple equation to describe the open-circuit voltage is very general and can describe (outside of the shunt region, which is not considered here) very different solar cell technologies correctly. The reason is that many parameters that differ for different semiconductors are accounted for. So what determines the open-circuit voltage?

At a given temperature, the open circuit voltage is determined by
– j_\text{gen},
– j_0, and
– n_\text{id} (which was discussed previously)..

The generation current density

j_\text{gen} = e \int EQE(E)\, b_s(E)\, dE,

which means it is given by how much of the solar photon flux of the sun, b_s, is converted into electrons – which is described by the external quantum efficiency EQE(E). The EQE includes reflection losses, the absorptance (in the simplest case 0 below the bandgap and 1 for energies at or higher than the band gap E_g), and charge collection losses (for instance due to the transport resistance). Essentially, the higher the bandgap E_g, the higher j_\text{gen} – which leads to a higher open-circuit voltage.

The dark saturation current density in an ideal solar cell without non-radiative losses is essentially given by

j_0 = e \int EQE(E)\, b_a(E)\, dE.

The only difference to the equation for j_\text{gen} is that the photons do not come from the sun anymore, but from the ambient (note the subscript a;). This b_a(E) is just Planck’s law to describe thermal radiation from black bodies, so everything “earthly” around the solar cell that emits at the current temperature T (or T_a, the temperature of the ambient). While the sun that generates j_\text{gen} can also be well approximated by a block body and Planck’s law – assuming a black body temperature of the sun of T_s \approx 5800 \text{ K}, usually the temperature of the ambient that leads to j_0 corresponds to the solar cell temperature… for instance, T = T_a \approx 300 \text{ K}.

With this in mind, we can maybe see already what determines the temperature dependence of the open-circuit voltage! In

V_\mathrm{oc}(T) \approx \frac{n_{id}(T)kT}{e} \ln \left( \frac{j_\text{gen}}{j_0(T)} \right),

there is an explicit temperature dependence in the prefactor, kT, which “promises” increasing open-circuit voltages at higher temperatures, but from measurements we know that the opposite happens: the open-circuit voltage increases towards lower temperatures!

Except for temperature-dependent changes in the charge collection (or bandgap), j_\text{gen} is (solar cell) temperature independent. The ideality factor can depend on temperature (and, in organic solar cells, even light intensity dependence [Saladina 2023]), but does not dominate the temperature dependence of V_\mathrm{oc}. The real “culprit” is indeed the very important dark saturation current density j_0, which describes charge carrier generation (and recombination!) across the bandgap by thermal photons (plus, in real systems, non-radiative processes)!

When we approximate the equation for the ideal dark saturation current density (ideal, because we again neglect non-radiative losses) from above – using the Boltzmann approximation instead of the Bose-Einstein statistics that are featured in Planck’s law – we can describe the temperature dependence of j_0:

j_0(T) \approx j_{00} \exp \left( - \frac{E_g}{n(T)kT} \right) \qquad \mathrm{(*)}.

The temperature here is, again, the solar cell’s and the ambient temperature is equal to it. We assume for simplicity (again a pretty good approximation) that the bandgap E_g is not (or only very weakly) temperature dependent. The prefactor j_{00} is essentially the dark saturation current density at infinite temperature, when the effective density of states is (hypothetically) completely filled, which is reduced by lower temperatures: the thermal photons from the ambient generate electron–hole pairs across the bandgap, increasing j_0 and – as it is in the denominator of the equation for V_\mathrm{oc} – decreasing the open-circuit voltage at higher temperatures. We can make this result more transparent when entering the temperature dependent dark saturation current density into the equation for the open-circuit voltage: (For simplicitly, I left the explicit temperature dependence of the ideality factor out, writing n_{id} instead of n_{id}(T).)

V_\mathrm{oc}(T) \approx \frac{n_{id}kT}{e} \ln \left( \frac{j_\text{gen}}{j_{00} \exp \left( - \frac{E_g}{n_{id}kT} \right)} \right)
= \frac{n_{id}kT}{e} \ln \left( \frac{j_\text{gen}}{j_{00}} \exp \left(\frac{E_g}{n_{id}kT} \right) \right)
= \frac{n_{id}kT}{e} \ln \left( \exp \left(\frac{E_g}{n_{id}kT} \right) \right) + \frac{n_{id}kT}{e} \ln \left( \frac{j_\text{gen}}{j_{00}} \right)
= \frac{n_{id}kT}{e} \frac{E_g}{n_{id}kT} + \frac{n_{id}kT}{e} \ln \left( \frac{j_\text{gen}}{j_{00}} \right)
= \frac{E_g}{e} + \frac{n_{id}kT}{e} \ln \left( \frac{j_\text{gen}}{j_{00}} \right)

We see now that the maximum open circuit voltage at zero temperature (in a system described by classical Boltzmann approximation, Voc(T) for a-Si H solar cell.i.e. without the energetic disorder that is an important topic in organic solar cells) is given by the (effective) bandgap of the device – usually the bandgap of the active layer material. The open-circuit voltage is reduced at higher temperatures(!) despite the plus sign after the term E_g/e: the dark saturation current density at infinite temperature j_{00} is larger than j_\mathrm{gen}, and the argument of a natural logarithm with argument smaller than one is negative. This means that the second term on the right-hand side is negative, and V_\mathrm{oc} decreases at higher temperatures.

I’ll leave it here for now, with two comments: first, there is more to be said about why one always should use the ideality factor in the equation \mathrm{(*)} that gives the temperature dependence of the dark saturation current density! And second, the impact of energetic disorder on V_\mathrm{oc} is very important and leads to changes – the devil in the details;)

Pitfalls when measuring recombination lifetimes in organic solar cells

Six years ago, I came across an interesting publication by David Kiermasch and Kristofer Tvingstedt, [Kiermasch et al 2018], Frosch verlässt Seerose. titled Revisiting lifetimes from transient electrical characterization of thin film solar cells; a capacitive concern evaluated for silicon, organic and perovskite devices. It shows that particular in thin film solar cells, the time constant determined by voltage based techniques – open circuit voltage decay (OCVD), transient photovoltage (TPV), intensity modulated photovoltage spectroscopy (IMVS) – is in many cases not the recombination lifetime, but corresponds to an RC-time from the device itself. While the authors did not find this effect, they showed impressively how most modern solar cells are limited in this respect, and it has to be verified carefully whether or not the experimentally determined time constants do correspond to recombination lifetimes!

I took this publication very seriously. Below I show you a summary slide I made for my group seminar in the year of publication, 2018.

Carstens tuesday note 2018.

The shown equation was actually animated, sorry for making your life harder (but mine easier;). Briefly, the idea of why an RC limitation shows up is the following. The charge in the device Q is changing with time during the measurement – no matter if the method is a large signal (OCVD) or small signal method (TPC, IMVS):

\frac{dQ(t)}{dt} = \underbrace{\frac{dQ(t)}{dt}}_{-\frac{Q(t)}{\tau_\text{rec}}} - \underbrace{\frac{\partial Q}{\partial V}}_{C(V)} \frac{d V(t)}{d t},

where the first term on the right hand side represents the recombination rate – of which we want to measure the recombination lifetime \tau_\text{rec} – and the second term a contribution coming from the response of charge due to the measured signal of the experimental technique: the time variation of the voltage as response to a time dependent light signal (pulsed or modulated). If the latter capacitive term becomes dominant, the recombination lifetime cannot be determined anymore, as is hidden behind the RC time.

As you can see in this slide, based on our data, for P3HT:PCBM (bottom left of the slide) the capacitive (RC) times remain lower than the measured time constants. We can state with some confidence that the RC times do not limit the measured recombination lifetimes. Also, the slopes of \tau vs V_{oc} are different and, in this case, a measure of the recombination order.

In contrast, for the PCDTBT:PC70BM solar cell, for which I took the measured time constants at room temperature from literature, you see that the RC time limits the recombination lifetime, as the measured time constants just correspond to the RC times. This implies that the recombination lifetime is too short to be measured for the given RC limitation.

A nice aspect of the paper by Kiermasch et al. is that it gives a comparatively simple way to estimate the RC times:

\tau_{RC} \approx \frac{C(V_{oc})}{j_\text{gen}(V_{oc})} \frac{nkT}{e}

Here, nkT/e is the recombination ideality factor times the thermal voltage, C is the voltage dependent capacitance, which can be estimated (as lower bound) by the geometric capacitance of the active layer, C_\text{geo} = \epsilon_r\epsilon_0/L, with the dielectric constants (relative and vacuum) and L the active layer thickness (you could also include organic transport layers, but probably not PEDOT:PSS as it has a relative dielectric constant \gg 3). The generation current density can be approximated, in most cases, by the short circuit current density j_{sc} (unless the transport resistance loss is too large:).

Please note, that in this particular RC time where both R and C come from the solar cell itself, the active area cancels out (as both j and C contain it and are in denominator and numerator, respectively). So, in order to reduce the RC time for a given solar cell, the only way seems to either 1. increase the current, by increasing the light intensity, measuring up to higher Voc… if possible, or 2. reduce the (geometric) capacitance by increasing the device thickness… a lot.

All of this came to my mind again when looking at IMVS data that we took on PM6:Y12 solar cell devices (made by Chen, measured by Nino with support from Christopher).
PM6-Y12 IMVS RC time vs IMVS time constant.
On the left, you see the estimation of the RC time \tau_{RC} after Kiermasch 2018, compared to the measured time constants by IMVS, \tau_c. Except for, maybe, low temperatures, the RC times dominate the measured signal at high open circuit voltages, whereas the shunt limits the lifetime determination at lower open V_{oc}. A direct comparison is shown on the right hand side, where everything in the shaded triangle is either shunt or RC-limited.

While quite sad, I think this is important: if you want to determine recombination lifetimes in thin film solar cell devices, check for limitations by RC times and shunt.

Two notes

A few weeks ago, Heliatek managed to take the lead for organic solar cell efficiencies, achieving 8.3% confirmed power conversion efficiency on 1.1cm2 active area with vacuum deposited small molecules. Madeira Rainbow in AutumnThe device was a tandem. Thomas Körner, VP of Sales, marketing and Business Development at Heliatek, added

The first products should be coming onto the market at the start of 2012.

Good!

Second, you may remember my post on photocurrent in organic solar cells back in July. It was inspired by a comment I wrote on a paper by Street et al, who proposed monomolecular recombination to dominate the loss of free charges in organic bulk heterojunction solar cells. My comment and Bob Street’s reply to it are now online at Phys Rev B. I’ll not comment this interesting exchange any further (unless requested by you;-), so read and think for yourself!

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