Month: May 2025

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;)