# Characterization and Simulation

## Transient photoluminescence

The insights obtained from transient photoluminescence (PL) measurements have contributed to an improved understanding of recombination and transport in a wide range of semiconductors. Transient photoluminescence is attractive because it allows contactless measurements of films on glass, layer stacks or complete devices while studying processes on different time- and length-scales. In particular, it permits analysis of the various recombination processes that happen in photovoltaic absorber materials and that may reduce the open-circuit voltage and subsequently the efficiency of solar cells made from these materials. However, analyzing the transients is challenging because of the multitude of (non-linear) effects that contribute to the shape of the PL transient. Recent work was therefore focused on combining measurements of metal-halide perovskite films with one contact layer (e.g. electron contact) with simulations to study the amount of interfacial recombination.

Figure 1 shows the different mechanisms that take place during a transient PL measurement. Initially electrons are transferred to the contact material (panel a) and the luminescence is decreased (red area labelled transfer in panels c and d). Then charge accumulates at the interface (panel b) and repels further transfer of electrons to the contact material leading to a reduction in the decrease of the PL transient (panel c), i.e. a longer differential lifetime (panel d). The value of the lifetime for longer times then depends on the light intensity and the interfacial recombination rate both of which control the amount of charge accumulation at the interface. Combining these effects in a numerical model allows us to quantitatively evaluate the interfacial recombination velocity at interfaces between the perovskite absorber layer and a contact layer.

A unique selling point of our work in Jülich is to make measurements with an extremely high dynamic range (see Figure 2 a) that enables us to determine a differential decay time that is a function of time, carrier density or quasi-Fermi level splitting (= chemical potential of the electron/hole pairs). Thus, we can determine not one lifetime (e.g. by using an exponential fit) but we can instead translate the complete and complex dynamics of the photoluminescence decay into a decay time (see Figure 2 b) that can be interpreted using analytical^{1} or numerical models.^{2}

## Transient and frequency dependent optoelectronic methods

The group uses a range of optoelectronic methods dedicated to better understand recombination, transport and the density of states in organic and perovskite-based semiconductors based on measurements on devices. These measurements are either transient, using the voltage or current transient of a solar cell after excitation with a laser pulse, or they are frequency dependent, i.e. based on a measurement of the admittance or impedance of a semiconductor as a function of frequency, temperature and DC voltage. These families of measurements have in common that they are affected by a multitude of effects, and it is rather challenging to find situations where one can safely assume that the result of the measurement will be entirely dominated by e.g. the charge-carrier mobility or alternatively the charge-carrier lifetime. However, the methods provide a wealth of information if they are combined with other methods and with numerical simulations to check whether a particular way of analyzing the data is likely to be applicable. Specific topics we have recently worked on are the above-mentioned comparison between luminescence and voltage-based transient techniques,^{1} and the interpretation of capacitance-based techniques for devices that contain more than one layer that significantly contributes to the capacitance and resistance of the total device.^{3-5}

An important recent finding related to the application of frequency dependent measurements is shown in Figure 3 . So far, defect densities in halide perovskites have been mostly obtained experimentally using methods that are sensitive to the charge density of the defects. However, charge is not directly detected but by its influence on the capacitance or the current density of a device at a certain voltage. Thus, the charge has to have an impact on the steady state or time-dependent electrostatic potential to be visible. However, given that photovoltaic devices are basically always sandwich-type devices, where the electrode distance scales with the thickness of the absorber layer, volume charge on defects will only be visible if it significantly influences the electrostatic potential that is otherwise dominated by the electrode charge that behaves similarly to a plate capacitor. Therefore, each of these measurements have thickness-dependent detectivity thresholds that have so far been largely ignored in the literature. Nearly all data on thin films lie close to the detectivity threshold as shown in Figure 3 a and b. This implies that a substantial part of the scientific literature on defects in halide perovskites is a discussion of measurement artifacts.^{3-5}

## Device simulations

Device simulations using drift-diffusion models in steady state, time domain and frequency domain are important tools for better understanding and analyzing experimental data on devices, layers or layer stacks. In addition, device simulations can be used to better plan experiments by making us aware of the potential challenges and they allow estimating the potential of certain technologies, improvements or the importance of certain physical effects such as photon recycling in perovskites.^{28} Therefore, simulations are frequently used in the group and we apply software tools such as SCAPS, ASA, or SETFOS.

The final problem in the area of characterization of recombination and transport in solar cells and other optoelectronic devices is the method of data analysis and parameter estimation. Data analysis approaches in semiconductor physics are often based on analytical equations such as the Mott-Gurney law or the Mott-Schottky law that often work well in idealized situations with a high relevance for certain technologies. For instance, the Mott-Schottky method requires the depletion approximation to be true and is very useful for thick, doped semiconductor-semiconductor or semiconductor-metal junctions. However, for weakly doped, very thin semiconductors it is often not applicable as seen in Figure 3 .

In the absence of simple analytical models, one can use numerical models that require fewer assumptions but add more unknown parameters. Thus, the key challenge originating from the complexity associated with thin, weakly doped and furthermore ionic semiconductors such as halide perovskites is to extract material parameters from methods that cannot be analyzed analytically. Currently, there is a lack of data analysis approaches that take several measurements into account that can only be described using numerical (and typically, highly non-linear) models and that are able to quantify the confidence in the result. The current strategy of fitting experimental data with a parameter set only provides a best fit that may or may not represent the true material parameters. A best fit result will normally not reveal any level of confidence in the result except for the error associated with the fit. An alternative promising strategy that circumvents the downsides of fitting but maintains the key advantage of the higher accuracy of numerical models that has so far not been widely explored for perovskite solar cells is Bayesian parameter estimation (BPE).^{6}

The general idea of BPE is to first replace time-consuming numerical simulation models with surrogate models based on convolutional neural networks (CNNs). Those are trained on the simulation data and can then replace the slow numerical simulation tools at least for a certain range of parameters that was contained in the training data. The essential idea is that the knowledge contained in a large number of simulations (typically 10^{5} or more) is stored in the weights of a neural network so that it can be used time-efficiently if needed for data analysis. Then, the multidimensional material parameter space is sampled using Markov Chain Monte Carlo (MCMC) approaches.

## References

[1] Krückemeier, L., Liu, Z., Krogmeier, B., Rau, U., & Kirchartz, T. (2021). Consistent Interpretation of Electrical and Optical Transients in Halide Perovskite Layers and Solar Cells. *Advanced Energy Materials, 11*(n/a), 2102290. doi:https://doi.org/10.1002/aenm.202102290

[2] Krückemeier, L., Krogmeier, B., Liu, Z., Rau, U., & Kirchartz, T. (2021). Understanding Transient Photoluminescence in Halide Perovskite Layer Stacks and Solar Cells. *Advanced Energy Materials, 11*, 2003489. doi:https://doi.org/10.1002/aenm.202003489

[3] Ravishankar, S., Liu, Z., Rau, U., & Kirchartz, T. (2022). Multilayer Capacitances: How Selective Contacts Affect Capacitance Measurements of Perovskite Solar Cells. *PRX Energy, 1*(1), 013003. doi:https://doi.org/10.1103/PRXEnergy.1.013003

[4] Ravishankar, S., Unold, T., & Kirchartz, T. (2021). Comment on “Resolving spatial and energetic distributions of trap states in metal halide perovskite solar cells”. *Science, 371*(6532), eabd8014. doi:https://doi.org/10.1126/science.abd8014

[5] Siekmann, J., Ravishankar, S., & Kirchartz, T. (2021). Apparent Defect Densities in Halide Perovskite Thin Films and Single Crystals. *ACS Energy Letters, 6*, 3244-3251. doi:https://doi.org/10.1021/acsenergylett.1c01449

[6] Brandt, R. E., Kurchin, R. C., Steinmann, V., Kitchaev, D., Roat, C., Levcenco, S., . . . Buonassisi, T. (2017). Rapid photovoltaic device characterization through bayesian parameter estimation. *Joule, 1*(4), 843-856. doi:https://doi.org/10.1016/j.joule.2017.10.001

[7] Kirchartz, T., Bisquert, J., Mora-Sero, I., & Garcia-Belmonte, G. (2015). Classification of solar cells according to mechanisms of charge separation and charge collection. *Physical Chemistry Chemical Physics, 17*(6), 4007-4014. doi:https://doi.org/10.1039/C4CP05174B

[8] Hüpkes, J., Rau, U., & Kirchartz, T. (2022). Dielectric Junction: Electrostatic Design for Charge Carrier Collection in Solar Cells. *Solar RRL, 6*(1), 2100720. doi:https://doi.org/10.1002/solr.202100720

## Prof. Dr. Thomas Kirchartz

Stellvertretender Direktor