The PPI method is based on a detailed balance approach, considering that in steady state, the current density measured at the outer solar cell contacts, J(V), can be expressed by the current density of the photo-generated charge carriers, Jgen, and the (internal) recombination in the perovskite layer and at respective interfaces, Jrec(V),
Here, we assume that in the considered operation range (0 V ≤ V≤ Voc), the internal generation current Jgen is not affected by the applied bias. Moreover, without loss of generality, we assume that the photon flux to illuminate the sample is constant over time and spatially homogeneous.
The internal recombination processes comprise non-radiative and radiative components, Jn.r. and Jrad, respectively: Jrec = Jn.r. + Jrad. They can be related to each other by
Equation (2) is generally valid if k is a function of the applied bias, k = k(V). Here, we show that, if the probed sample has a diode ideality factor close to one and displays negligible resistive losses (high shunt resistance, low series resistance), then it is justified to assume a linear relationship between Jn.r.(V) and Jrad(V) and hence k = const., which we will assume in the following. Empirical indications for this circumstance in perovskite solar cells have already been presented by Stolterfoht and coworkers.23 A detailed theoretical discussion of this assumption as well as a general expression are presented in the Supporting Information (SI section B).
Now, Equation (1) can be expressed as
J rad(V) can be related to the signal of a photodetector SPL(V) by
where c describes the probability that photons created by radiative recombination enter the detector area and are translated into a detector signal.
Using Equation (3), and (4), we can now relate the electrical photocurrent to the difference between the voltage-dependent PL intensity PL(V) and the PL at open circuit. By normalizing the term, the expression becomes independent of setup-specific factors which makes the technique independent from elaborate calibration measures. We find that
This relation shows that photoluminescence microscopy can be used to derive spatially as well as time-resolved images of the local J(V) performance of a photovoltaic device. Two applications are especially interesting: First, by recording only two PL images, one at open circuit and one at short circuit, the image of the local short-circuit photocurrent density Jsc can be derived. Secondly, by recording PL images at various voltages, the local J(V) of specific spots on the cell can be investigated. These approaches are investigated in the following. Thereby, the J(V)/Jgen results determined by PL microscopy will be denoted as J(V)/Jgen |PL.