Understanding Transient Photoluminescence in Halide Perovskite Layer Stacks and Solar Cells

While transient photoluminescence measurements are a very popular tool to monitor the charge-carrier dynamics in the field of halide perovskite photovoltaics, interpretation of data obtained on multilayer samples is highly challenging due to the superposition of various effects that modulate the charge-carrier concentration in the perovskite layer and thereby the measured PL. These effects include bulk and interfacial recombination, charge transfer to electron or hole transport layers and capacitive charging or discharging. Here, numerical simulations with Sentauraus TCAD, analytical solutions and experimental data with a dynamic range of ~7 orders of magnitude on a variety of different sample geometries from perovskite films on glass to full devices are combined to present an improved understanding of this method. A presentation of the decay time of the TPL decay that follows from taking the derivative of the photoluminescence at every time is proposed. Plotting this decay time as a function of the time-dependent quasi-Fermi level splitting enables distinguishing between the different contributions of radiative and non-radiative recombination as well as charge extraction and capacitive effects to the decay.


Introduction
Technological development of halide-perovskite solar cells towards even higher efficiencies requires ways of understanding and quantitatively analyzing the main loss processes. [1][2][3] Non-radiative recombination is one of the main loss processes in basically any solar-cell technology [4,5] including perovskites that leads to reduced open-circuit voltages at a given illumination condition. [6][7][8][9] Transient photoluminescence (TPL) is a frequently used tool to monitor the charge-carrier dynamics and investigate these recombination losses. [10][11][12] While TPL measured on bare perovskite films on glass is a well-understood and frequently-used method to derive charge-carrier lifetimes and recombination coefficients, [13,14] it is typically the recombination at interfaces between the absorber and charge transfer layers that is the dominant sources of recombination in a complete device. [15][16][17][18][19][20][21] However, adding contact layers to the perovskite film not only adds additional recombination paths but also leads to effects like charge-carrier separation or interface-charging effects [21][22][23] that may change the recombination kinetics fundamentally. Thus, the information on recombination may be totally obscured by interfacial effects such that, e.g., a faster photoluminescence (PL) decay cannot anymore be interpreted simply as an increased recombination coefficient. Hence, misinterpretation of experimental data becomes likely, especially if the information contained in the decay curve is reduced to a single valuethe characteristic decay time of a mono-exponential decay. The present paper introduces a method to analyse the differential PL decay that uses the derivative of the photoluminescence at every time during the transient. [22] We propose to plot this decay time as a function of the corresponding quasi-Fermi level splitting allowing us to better understand the complex interplay between charge extraction and interface or contact charging with radiative and non-radiative recombination. We use a combination of numerical simulations with Sentauraus TCAD, analytical models and experimental data to illustrate how the different effects influence the PL transients and the resulting decay times.
In the following, we investigate transient photoluminescence measurements on the different sample geometries shown in the overview in Figure 1 that start with films on glass and continue via layer stacks to full devices and discuss their respective peculiarities. We show how the addition of further layers and interfaces modifies the transients and adds physical effects that must be considered. With this step-bystep description we aim to create an understanding of which processes dominate and are important for these different sample types and how they affect the transient PL decay and the extracted decay time.
This step-by-step approach is conceptually similar to the already well-understood analysis of steady state PL (SSPL) as a function of sample type that allows screening contact layers for minimum recombination losses [15,24] and to also analyze and quantify changes in ideality factor [20,24] and resistive effects. [25] In SSPL any change in absolute intensity between different samples suggests that the difference between the samples (e.g. an additional interface and layer) must be the cause of the loss in PL which is linked to a loss in quasi-Femi level splitting. In contrast, a clear correlation between the decay of a transient PL signal and the amount of recombination is not necessarily present, because in a transient experiment currents can temporarily flow even at open circuit.

Figure 1:
Overview over the different halide perovskite sample types that are investigated with transient photoluminescence. Depending on the sample type (i.e., absorber layer on glass or absorber layer with one or two contact layers attached), the complexity of the interpretation of data increases from left to right. The second row illustrates typical mechanisms that affect the PL decay in different sample geometries. While the film on glass is only affected by bulk and surface recombination, samples with interfaces to charge extraction layers are affected by charge accumulation and recombination at the interfaces between absorber and charge-extraction layer. On fully contacted devices, charging and discharging of the electrodes finally affects the global band diagram and therefore the transients measured on full devices.

Perovskite film on glass
The simplest sample geometry is that of a perovskite film on glass. Therefore, we want to use this device geometry as a starting point to recapitulate the basic theory of TPL measurements and data interpretation that is already quite well understood and briefly introduce the terminology and the physical parameters that we will use during the discussion. Furthermore, we introduce the representation of the decay time τTPL, which results from the derivative of the photoluminescence with respect to time that we plot as a function of the corresponding time-dependent quasi-Fermi level splitting ∆EF.

Perovskite film on glass with a passivated surface
At first, we consider charge-carrier recombination only in the bulk and leave out the effect of surface recombination, which is discussed separately in section 2.2. A perovskite film on glass whose surface is passivated, e.g. by an organic passivation layer, [26,27] is suitable as a corresponding test structure, because the interface between perovskite layers and glass substrates is typically quite inert and not particularly recombination active. [13,28] When the semiconducting perovskite film is photoexcited by a laser pulse, free excess-electrons n  and equal density of holes p = n are generated. The initial generation profile of these excess-charge carriers depends on the optical properties of the perovskite, its thickness, and the excitation wavelength. The initially inhomogeneous distribution of the charge-carrier concentration is flattened by diffusion. Since the perovskite film is typically only a few hundred nanometers thick and the charge-carrier mobility is typically > 1 cm²/Vs, [29] this carrier equilibration occurs in the first hundreds of pico-to several nanoseconds after the laser pulse. For simplicity, we therefore assume that the system has no spatial gradients of electron or hole concentrations or the electrostatic potential. This assumption leads us to the convenient situation that diffusion and drift currents can be neglected and that the photogenerated charge carriers n  vanish over time only through different recombination processes. The rate equation  (1) accounts for the rates of these different competing processes and describes the change of excess-charge carrier concentration as a function of time. Here, ni is the intrinsic charge-carrier density, n is the total electron and p the total hole concentrations, being defined as n = n0+∆n and p = p0+∆n, respectively, with the corresponding equilibrium concentrations n0 and p0. For the moment, we keep the condition p = n and therefore do not distinguish between the rate equations for electrons and holes. For a perovskite film on glass three different bulk-recombination mechanisms need to be considered, which are radiative band-to-band recombination [30,31] and non-radiative recombination via Auger [32] or firstorder Shockley-Read Hall (SRH). [33,34] The radiative recombination coefficient is given by krad, τn and τp are the non-radiative SRH lifetime for electrons and holes, respectively. For the Auger recombination rates, Cn represents the Auger coefficient for electrons and Cp is the Auger coefficient for holes. Note that the effect of photon recycling [35] is not explicitly considered in this paper, so the radiative coefficient krad is the corresponding external quantity. Photoluminescence is based on measuring the emitted photons that have been created by radiative recombination with a rate proportional to np. Accordingly, the PL intensity TPL  depends on the product np of electron and hole concentrations. The time-resolved photoluminescence itself contains information about these entire recombination processes because the decrease in n and p results from the sum of all recombination processes.
While the radiative recombination coefficient and the Auger coefficients are important material properties, the SRH lifetimes can vary substantially from sample to sample within a material system.
which can be solved analytically. The respective result and solutions for further problems are listed in Table S1 in the Supporting Information. Here, we only want to have a look at the illustrated solutions being shown in Figure 2a. In high level injection, different recombination mechanisms will lead to differently shaped transient decays. Among the three typically studied bulk-recombination mechanisms (SRH, radiative and Auger) only SRH would lead to a mono-exponential decay. Hence, the situation where only SRH recombination occurs can be described by a single lifetime value that is given for n = p by the sum bulk lifetimes, namely 100 ns (yellow), 500 ns (red) and 2 µs (blue). Both, Auger, and radiative recombination lead to faster initial decay, with a higher radiative recombination coefficient krad and higher Auger coefficients Cn and Cp enhancing the effect. As an example, the krad is varied in Figure 2b for the SRH lifetime of 2 µs. At longer times, the slope approaches the one from SRH-only scenario.  Fermi-level splitting E F (eV) . [13] Consequently, if any other recombination mechanism plays a role, the decay is no longer monoexponential and cannot be described by a single lifetime. Thus, we promote the use of a carrier-density-   [37][38][39] and open-circuit voltage decay (OCVD) [40][41][42] measurements. These measurements also allow deriving differential time decay that are however frequently denoted as "lifetimes" in the literature. [39,41,43,44] Radiative and Auger recombination define the shape of the decay time at high Fermi-level splitting, i.e. directly after the laser pulse. The Fermi-level splitting E F (eV)

Layer with surface recombination
TPL measurements on unpassivated perovskite films on glass are often used to characterize not only the bulk but also the surface. If the bulk properties of the perovskite are already known, this sample type can be used to extract the corresponding surface properties from the TPL measurement. In general, the rate of surface recombination may be limited either by the transport of charge carriers to the surface or by the surface recombination velocities Sn and Sp of electrons and holes at the interface itself. Due to the relatively low thickness of halide perovskite thin films (typically < 1µm) and the typically high mobilities (µ > 1cm²/Vs), the transport of electrons and holes to the surface can be considered fast compared to the rate of recombination at the surface, [13] which can be quite low relative to many other semiconductors. Therefore, we do not have to distinguish between the average carrier concentration in the bulk of the perovskite-thin film and the carrier concentrations at either of the surface. This finding implies that the rates of bulk and surface recombination can just be added up and it is not necessary to calculate the concentration of electrons and holes at the film surface. The surface recombination rate per unit area is given by whereby the approximation sign is valid for n = p and np >> ni 2 .
We note that in Equation 6, the bulk SRH term and the surface recombination term are both linear in electron concentration which suggests that we could use the concept of surface lifetime τsurf and an effective SRH lifetime which would allow us to rewrite Equation 7 to obtain Table S1 in the Supporting Information. . We conclude that the numerical results from Sentaurus TCAD and the analytical description fit well and only deviate for particularly large S. Therefore, the analytical model is suitable to analyse experimental TPL data of a perovskite film on glass with an unpassivated surface. The surface recombination velocity S can then be extracted from the plateau value of the decay time if the SRH bulk lifetime is already known from a measurement of a passivated sample.

Perovskite layer with charge extraction layer
Surface recombination is an important loss mechanism but usually the surface to ambient air is not of special interest. We are rather interested in the analysis of interfaces between the perovskite film and charge-extracting layers to evaluate and compare the quality of contact materials and assign recombination losses in the actual solar cell. There are a range of additional effects -not related to recombination -that will affect the shape of the TPL decay and that we will illustrate in the following using the example of a perovskite/PCBM bilayer. After the laser pulse has hit the sample, the equilibrium Fermi level EF splits up into quasi-Fermilevels EFp for holes and EFn for electrons (see Figure 5b) and charge is transferred to the PCBM, which is visible by the increase of the Fermi-level split inside the PCBM. Due to the reasonably low mobility of the PCBM (µPCBM = 5×10 -2 cm²/Vs), [45] this increase is slower than in the perovskite and position dependent (compare Figure 5b with 5c). The injection of electrons into the PCBM leads to a reduction of the np-product in the perovskite and thereby to a reduction in the PL at early times that is not caused by recombination. In Figure Table S3 and a detailed description of the simulation implementation can be found in the Supporting Information. . A comparison allows us to distinguish between pure recombination processes and the impact of additional effects e.g. interfacial charging, reinjection, or diffusion limitation. Additional simulation parameter can be found in Table S3 in the Supporting Information.  Figure 5d). The effect of charge accumulation gets more severe for higher band offsets, because the misalignment of the conduction bands leads to larger electron densities in the PCBM and reduces the rate of re-injection into the perovskite.
In addition, the thickness of the PCBM layer is varied between 10 -150 nm for a medium-large energy offset of 150 meV. The respective decay times TPL,HLI  , being presented in Figure 6c in different shades of purple with lighter colors representing thinner layers, point out that the first plateau gets more pronounced for thicker PCBM layers. Higher PCBM thickness also allows more impact of band bending, because of electrostatic arguments. For a given charge-carrier density, any change in electrostatic potential of kBT/q will occur over one Debye length, where kB is the Boltzmann constant, T the temperature and q the elementary charge. Hence, thicker layers will allow a larger change in electrostatic potential over the PCBM and therefore stronger band bending. The saturation value of TPL,HLI  (second plateau) at long times and small EF is also strongly affected by PCBM thickness. At longer times, recombination will be limited by the time the electrons take to diffuse through part of the PCBM layer before they can recombine at the interface. This effect leads to longer TPL,HLI  at small EF. Figure 6b shows the case, where interfacial recombination is negligibly small and the decay at longer times must be due to re-injection of electrons into the perovskite absorber layer (scenario (ii)). Now, TPL,HLI  at longer times or small EF can substantially exceed the bulk SRH lifetime of 1 µs. This is because electrons are injected into the PCBM and cannot escape from there other than by re-injection over a barrier of a certain height. This re-injection process is the rate limiting step at longer times and therefore leads to prolonged decay times at small EF. Again, PCBM thickness has the same effect as before for the saturation value of the decay time. Larger thicknesses lead to longer TPL,HLI  due to the diffusion time through the PCBM (Figure 6d).

Perovskite layer with charge extraction layer and electrode
In the previous section, we have discussed the relevant case of a type II heterojunction between the perovskite and a charge-transport layerin this case the ETL. Similar simulations could of course be done for hole transport layers that form a type II interface with the perovskite. Two aspects are however missing in section 3 that may appear in practice. One is the case where type I heterojunction is formed, and the second aspect is that an electrode (anode or cathode) may be present in addition to the ETL or HTL. In this section we are combining these two issues and simulate the layer stack glass/ITO/PTAA/perovskite. UPS measurements suggest that PTAA may have a negligible valence band offset to MAPI at least for the samples prepared with the recipe presented in ref. [46] . As in the previous section, we first have a look at the corresponding band diagram in equilibrium and at different delay times after laser pulse excitation being represented in Figure 7a-d. When comparing the equilibrium-band diagrams of Figure 5a and 7a, it is directly apparent that the equilibrium Fermi-level EF is no longer roughly in the middle of the perovskite band gap but is rather pinned by the ITO contact.
When different layers are brought into contact charge-carrier injection from layers with higher electron (or hole concentration) into layers with lower concentrations occurs. Because ITO is much more conductive than PTAA and perovskite, the work function of the ITO sets the position of the Fermi level in the layer stack and holes are transferred into the PTAA and the perovskite. Therefore, the Fermi-level in the perovskite is much closer to the valence band edge EV than to the conduction band edge EC, which implies that the equilibrium hole concentration p0 is substantially higher than ni. Thus, the situation is comparable to the one in a doped semiconductor even though the perovskite itself is assumed to be perfectly intrinsic.
This behavior caused by the ITO contact leads to a transition between high-and low-level injection during the TPL measurement. Directly after (Figure 7b) Table S3 in the Supporting Information.
This lack of information complicates the interpretation of the decay time in terms of bulk or surface lifetimes. In order to better see the transition from HLI to LLI, we use asymmetric SRH lifetimes in Figure 7 which are set to n 500 ns  = and p 2 µs  = . In high-level injection (n ≈ p) the lifetime from SRH recombination is given by n + p=2.5 µs. At lower Fermi-level splitting (LLI), the situation in this example can be described in a simplified manner by the rate equation  At the PTAA/perovskite interface, electrons are the minority carriers in low-level injection and hence the reduction of electron concentration leads to a slight increase in the decay time in low-level injection relative to high-level injection (see Supporting Figure S10).   Table S3 in the Supporting Information.

Complete device stack
In case of the TPL of complete perovskite solar cells, the charging and discharging of the device electrodes adds an additional effect modifying the shape of the TPL decay. The mathematical problem can still be approximately described by an ordinary differential equation in time without any spatial dependences. This differential equation (neglecting Auger recombination for simplicity) could be written as where Carea is the area-related capacitance in F/cm² and V the voltage between the electrodes.  [40,41,43,44] To illustrate the effects occurring in complete devices, Figure 9a -f presents the band diagrams simulated, using the full differential equations which include charge transport, at different delay times during a TPL measurement. Figure 9b depicts the band diagram directly after the pulse has hit the sample showing a substantial internal quasi-Fermi level splitting in the perovskite layer while the external voltage is still zero (see the Fermi levels in the ITO relative to the one in the Ag in Figure 9b). Figure 9c pictures the situation 5 ns after the pulse. At this stage, the quasi-Fermi levels are substantially split in the perovskite and also an external voltage has now been built up. However, there is still a substantial difference between the quasi-Fermi level splitting inside the perovskite absorber and the external quasi- ITO PTAA perovskite PCBM Ag   As an illustration, we picked a cell that has good interface properties and a long bulk life. The entire simulation parameters can be found in Table S3 in the Supporting Information. If we simulate the photoluminescence transients for the device geometry presented in Figure 9a-f, we observe a strong dependence of the decay curve on the pulse fluence of the laser. Figure 9g shows the decays normalized to their maximum. For the low pulse fluence of 10 nJ/cm 2 , the PL decays quickly directly after the laser pulse has hit sample. In this case, nearly all electrons and holes generated by the laser pulse are necessary to change the amount of charge on the device electrodes for a sufficient external voltage to build up. If, however, the pulse energy is higher, no abrupt fast decay at early times is visible because only a small fraction of photogenerated electrons and holes are needed to build up the external voltage. However, the initial decay is still not monoexponential because Auger-and radiative recombination in the absorber will now matter at early times. The combination of these effects leads to a difficult to interpret situation, where fast decays at early times are visible for very low and very high pulse energies, however, for completely different reasons. Figure  ∆EF consists of three regions, which will be discussed in the following.   whether there is a limitation due to capacitive effects or due to fast radiative recombination. This means that the determination of the recombination processes becomes particularly difficult for very good devices with long SRH lifetimes.   Table S3 in the Supporting Information.

23
With Figure 11 we confirm that the trend of the analytical relation can also be obtained in numerical simulations of a solar cell. In Figure 11a a variation of the SRH bulk lifetime (100 ns-40 µs) is illustrated. In this example, the interfaces between the perovskite absorber and the charge-extracting layers are nearly ideal (low S, no or small band offsets) and similar to the simulation parameters used in Figure 9 (see Table S3 in the Supporting Information). Here again, the SRH bulk lifetime correlates with the value of the decay time TPL,HLI  near the inflection point of the curve and also the radiative limitation for high τn + τp becomes apparent. A direct comparison of the analytical and numerical solution can be found in Figure S14 in the Supporting Information. In the presence of surface recombination (dot-dashed dark grey line) the decay time TPL,HLI  at low EF is reduced. In case of a perovskite/ETL bilayer (blue curves), the shape of the decay time is fundamentally changed due to effects such as interface charging (dash-dotted blue line) and electron reinjection (solid blue line) from the ETL into the perovskite. In addition to interfacial charging, we also observe electrode charging and discharging if we analyze complete devices (red line). In this case, we observe an exponential increase of TPL,HLI  towards lower values of EF.
Finally, Figure 12 shows a summary of some exemplary effects observed so far in the simulations and how they affect the charge-density-dependent decay time. Since the recombination resistance of the solar cell increases exponentially towards lower voltages, the decay time associated with electrode discharging also increases exponentially.

Comparison to experimental data
Finally, we want to apply and transfer our findings from the simulations to experimental data.  Figure S18). The sample series, we show, always include a sample (glass/MAPI/TOPO) that should be well passivated. The molecule n-trioctylphosphine oxide (TOPO) has been shown to strongly reduce surface recombination velocities. [26,27] In addition, the sample series include one sample with the hole transport layer PTAA (glass/PTAA/MAPI/TOPO) that serves to characterize recombination losses at the PTAA/MAPI interface. Finally, the complete cells are included which feature in addition the MAPI/ETL interface as an additional source of recombination. Figure 13a and b compare the normalized transient PL intensities ϕTPL and (c-d) the respective decay times TPL,HLI  vs. ∆EF for the two sample series described above. Each data set results from stitching several measurements recorded with a gated CCD camera starting at different delay times after the laser pulse and using different gains and integration times. This approach enables a very high dynamic range of up to 7 orders of magnitude that is necessary to observe the wide range of physical phenomena discussed in the simulation sections of this paper. Typical combined measurement times were several hours. Given the differences in Voc of the two cells, we expect that the two sample series should differ in the recombination losses that occur either in the bulk or at interfaces and that these differences should be reflected in differences in the decay times TPL,HLI  . This qualitative expectation is confirmed already by studying the PL transients for the two sample series (Figure 13a and b) which show substantially faster TPL decays for the samples based on coevaporated perovskite layers.
In order to obtain additional insights, we first determine the decay times shown in Figure 13c and d.
While taking the derivative defined by Equation 3 is a simple task for smooth, simulated data, it is quite challenging to extract meaningful derivatives from noisy experimental data. Therefore, it is advisable to not or not only take the derivative of the background corrected raw data but to first fit the data with a function and then differentiate the fit. We note that given the multitude of non-exponential features affecting the transients, we opted to identify functions for which the fit algorithm converges easily and leads to a good agreement with the experimental transients even though the functional form of the fit functions bears no physical meaning. We observed that good candidates for fit functions are high order polynomials that can be fitted to the logarithm of the PL. Alternatives are rational functions, i.e. ratios of higher order polynomials. In Figure 13c and d, we therefore see symbols and lines, where the symbols represent the derivative of the background-corrected and stitched raw data while the lines are the derivative of the fits to the data. Both agree within the accuracy of the method, which isat least partly a consequence of the raw data being fairly high quality (low noise level) due to long integration and high measurement times. The EF axis for the experimental data was determined from the knowledge of the laser fluences used in the measurement (provides the initial EF at time zero after the pulse with Equation 4) and the knowledge that the PL intensity scales with exp(EF/kBT). [47] This proportionality implies that any order of magnitude decrease in PL leads to a 58 mV decrease in EF. From comparing the glass/MAPI/TOPO (grey) samples of the coevaporated and the solution-processed perovskite sample series in Figure 13c and d, it is directly apparent that the quality of the bulk differs substantially. In addition, the coevaporated cell suffers from increased interface recombination relative to the solutionprocessed cell, which we conclude from the substantial reduction in TPL,HLI  for the samples with interfaces (green and red) relative to the passivated layer on glass (grey). Furthermore, the comparison in Figure 13 demonstrates that the representation of the decay time TPL,HLI  via Fermi-level-splitting EF is advantageous compared to the usual representation of the decay itself. This new type of graph highlights differences and similarities between the samples more clearly and allows estimating recombination parameters.   Figure S21, we show how the simulations with lower SRH lifetimes look like. From Figure S21, we conclude for lifetimes τn + τp < 40 µs, the agreement between simulation and experiment deteriorates substantially. SRH bulk lifetimes as high as 80 µs would allow an open-circuit voltage of 1.31 V under AM1.5g illumination, i.e. a value very close to the radiative limit of 1.32 V. [13] The solution-processed samples with PTAA (green, yellow) do not suffer from additional losses. The decay times are similar to the passivated sample with the glass/perovskite interface. The band offset between perovskite and PTAA and the surface recombination velocity must therefore be negligibly small. In the Supporting Figure S21 we demonstrate that a surface recombination velocity must be around SPTAA=1 cm/s. Also, the band offset and surface recombination at the PCBM/perovskite interface must be quite small to explain the data. Nevertheless, the PCBM/perovskite interface is the only interface that causes visible deviations of the PL transients from the behaviour expected in the radiative limit. We find that a surface recombination velocity SPCBM=17 cm/s and an offset of 70 meV lead to the best agreement with the experimental data.
Note that the decay time curve for the solution-processed cell (red line, Figure 12c) reproduces the typical S-shape predicted by Equation 11. It should also be noted that the simulated JV-curve of a simulated cell with the stated parameters agrees well with the measured JV-characteristic (Supporting Figure S19). Another implication from the data shown in Figure 14a is that for perovskite films and layer stacks with small recombination losses, the measured differential time constants may assume nearly any value (from tens of ns to tens of ms), depending on the range of carrier concentrations and Fermi-level splitting that are set by the laser fluence.  Figure 14: Experimental data of transient photoluminescence measurements of the solution-processed sample series and fits from Sentaurus TCAD, which allow us to state the material parameters that describe the sample behavior best. All simulation parameters are listed in Table S4 and S5 in the Supporting Information.
Without the additional information on the laser fluence and without high dynamic range data as shown here, the information obtained from TPL data on many high-quality layers or layer stacks would be either difficult to compare or entirely meaningless.
In the following we want to discuss the fitting of the coevaporated sample series, representing an example of samples with a higher degree of non-radiative recombination. A SRH bulk lifetime of about τn + τp = 750 ns best describes the experimental TPL data of coevaporated bulk passivated with TOPO in the glass/perovskite/TOPO stack (grey). This SRH bulk lifetime would allow much higher opencircuit voltages than 1.05 V. Combined with an effective radiative recombination coefficient krad = 2×10 -10 cm 3 s -1 , a Voc of about 1.23 V would still be possible for the coevaporated bulk if no additional recombination losses would occur in the stack. The simulations suggest that for the coevaporated cell, these additional recombination losses are caused by misaligned energy levels and increased recombination at both interfaces (see Figure 6). Interface recombination at the PTAA/MAPI interface (coevaporated sample) leads to slightly shorter decay times TPL,HLI  at small Fermi-level splitting (green) as opposed to the samples without charge-extracting layers attached (grey). Note that these two decay time curves nicely overlap at high ∆EF, where radiative and Auger recombination dominate the τTPL,HLI.
The trend of PTAA/perovskite interface can be best explained by a band offset of ∆χ = 100 meV and a surface recombination velocity of Sn = Sp = 100 cm/s. Since the PTAA layer is very thin we would expect that TPL,HLI  saturates for small ∆EF around ~400 ns (discussion section 3). However, the decay time at small ∆EF is much higher and increases beyond the bulk lifetime. We observe this behaviour in our simulations if we assume shallow, neutral defects. Thus, in the simulation that best fits the data the defects are shallow and positioned only 150 meV away from the VB band edge (Supporting Figure S22).
These parameters would still allow a Fermi-level splitting of 1.18 V in simulated PTAA/perovskite stack. Also, the properties of the perovskite/C60 interface in the solar cell (red) are quite complex. In this case, the decay time is roughly two orders of magnitudes lower suggesting the additional interfacial losses in Voc must be higher. Only for small ∆EF the decay time increases again indicating that the loss is caused by a huge conduction band offset e.g. ∆χ = 200 meV. To explain the shape of τTPL,HLI vs. ∆EF that deviates from the typical S-shape we introduced in the discussion of the TPL on solar cells, we had to assume at least two different defect state. A deep level defect is responsible for the loss in Voc and another shallow defect must be introduced to match the shape of the differential decay curve. While we note here that various defect properties such as its charge, concentration and energetic and spatial position affect the TPL, it is beyond the scope of this work to present a systematic investigation on the influence of these parameters on the decay time TPL,HLI  .

Conclusions
Transient photoluminescence experiments are abundantly used in the field of halide perovskite photovoltaics to study charge-carrier recombination in the bulk and at interfaces. While the interpretation of TPL on thin films on glass has been thoroughly discussed and used in the literature, [12,13,48] the most important recombination losses are often occurring at the interfaces between the absorber and the charge-transfer layers. [15] In addition, the presence of charge-transfer layers can also have an impact on how the perovskite films grow and hence affect the bulk and surface quality of the perovskite layer. Thus, there is a clear need to extend our theoretical understanding to TPL measurements done on a variety of sample geometries including zero, one or two charge-transfer layers in contact to the perovskite absorber. In addition, it is important to also understand how contact layers such as ITO or Ag affect the TPL decay and to be able to understand measurements done on complete devices. The present paper provides an extensive account based on a combination of experimental data, numerical simulations with Sentaurus TCAD and analytical solutions to differential equations that allows the reader to understand the mechanisms affecting a TPL decay in a variety of sample geometries. As a key tool to analyze the data, we introduce the concept of a decay time TPL,HLI  displayed as a function of the time- diagram. Along with this paper, the reader finds a video collection of the band diagram during the TPL simulation for the different layer stacks. After introducing the general concepts using numerical simulations, we show experimental data sets on different sample geometries and absorber deposition methods (solution-processed vs. coevaporated). We determine the TPL decays over seven orders of magnitude in dynamic range and show that our previously presented recipe for MAPI layers allows bulk lifetimes of several tens of µs (best fits are obtained for 80 µs the sum of electron and hole lifetimes).
In addition, the TPL transients clearly indicate negligible losses at the MAPI/PTAA interface and only moderate losses with surface recombination velocities of 17 cm/s for the MAPI/PCBM interface.

Data availability
The data that support the findings of this study are available from the corresponding author on request.

Code availability
The code and data sets generated and analyzed during this study are available from the corresponding author on reasonable request.