Identifying dominant recombination mechanisms in perovskite solar cells by measuring the transient ideality factor

The ideality factor determined by measuring the open circuit voltage (VOC) as function of light intensity is often used as a means to identify the dominant recombination mechanism in solar cells. However, applying this Suns-VOC technique to perovskite cells is problematic since the VOC evolves with time in a way which depends on the previously applied bias (Vpre), the light intensity, and the device architecture/processing. Here we show that the dominant recombination mechanism in two structurally similar CH3NH3PbI3 devices containing either mesoporous Al2O3 or TiO2 layers can be identified from the signature of the transient ideality factor following application of a forward bias, Vpre, to the device in the dark. The transient ideality factor, is measured by monitoring the temporal evolution of VOC at different light intensities. The initial values of the transient ideality were consistent with corresponding photoluminescence vs intensity as well as electroluminescence vs current density measurements. Time-dependent simulations of the measurement on modelled devices, which include the effects of mobile ionic charge, show that Shockley Read Hall (SRH) recombination through deep traps at the charge collection interfaces is dominant in both devices. Using transient photovoltage measurements superimposed on the evolving VOC of bifacial devices we further show that the charge collection interface extends throughout the mesoporous TiO2 layer, consistent with a transient ideality signature corresponding to SRH recombination in the bulk of the film. This information could not be inferred from an ideality factor determined from only steady-state VOC values. The method we have developed will be useful for identifying performance bottlenecks in new variants of perovskite devices by comparison with the transient ideality signatures we have predicted for a range of possible recombination schemes.


Introduction
Identifying the dominant recombination mechanisms in hybrid perovskite solar cells is necessary to rationally optimise device architectures for performance and stability. The 'light' ideality factor, nid (also known as the 'quality factor'), determined from the open circuit voltage (VOC) dependence on light intensity, has classically been employed as a means to identify the dominant recombination mechanism in silicon and organic absorber solar cells. [1][2][3] However, applying this technique to perovskite cells is problematic since these devices often show hysteresis in their current-voltage (J-V) characteristics, with an associated slow transient evolution of the VOC. 4 This evolution depends on the light and voltage bias device preconditioning. Accordingly, for many perovskite cells, ideality factors derived from these measurements depend on the time at which the VOC is evaluated after illuminating the device as well as the state the cell was in prior to illumination.
Pockett et al. circumvented the issue of VOC evolution by using open circuit voltage values after the cell has reached steady-state as function of light intensity to determine ideality factors; a number of devices in their study showed anomalously high values of nid > 5. 5 Tress et al. conducted light intensity scanning sweeps in both increasing and decreasing directions and took the VOC value after 3 seconds as a compromise between stabilisation of the VOC and irreversible degradation of devices. 6 Optical measurements of the ideality factor on perovskite monolayers and heterojunctions have also been made under steady-state conditions. 7 To date, however, a detailed understanding of the timedependent ideality factor in perovskite solar cells has remained elusive.
There is now considerable evidence to indicate that J-V hysteresis in hybrid perovskite devices results from a combination of the effects of redistribution of mobile ionic defects and high rates of recombination at the perovskite-transport layer interfaces. 4,[8][9][10][11][12] Consequently, it is tempting to speculate that the dominant recombination mechanism in a device evolves with the VOC evolution, or that the recombination mechanism at steady-state could be a function of light intensity. The results presented herein show that such changes are not required to explain much of the behaviour observed when examining the ideality factor of perovskite solar cells.
In this study we present a method for determining the transient ideality factor, nid(t), of a device as it evolves from an initial state, defined by a voltage preconditioning protocol in the dark, towards a steady-state value after illumination. The measurement, made by monitoring the evolution of VOC for different light intensities, (which we will refer to as a 'Transient Suns-VOC' measurement) is presented for two different device architectures which contain either a mesoporous TiO2 or Al2O3 layer. While a mesoporous TiO2 layer is expected to contribute to electron collection and possibly recombination, the electrically insulating nature of mesoporous Al2O3 should not allow charge transfer across its interface. Comparing the two architectures thus provides a means to control the details of possible recombination mechanisms. To cross-check the Transient Suns-VOC measurement we also present corresponding instantaneous ideality factor measurements determined from electroluminescence vs current density (J-EL) and photoluminescence vs light intensity (Suns-PL). These observations are supplemented with fast, wavelength dependent, transient photovoltage measurements superimposed on the evolving VOC enabling us to probe the behaviour of charge carriers near the interfaces. In order to interpret the experimental findings we perform drift-diffusion device simulations describing the time-dependent evolution of the free electron, hole and ionic defect concentration profiles using an existing open source simulation tool, Driftfusion. 4,13 The results show that devices with different recombination schemes exhibit unique nid(t) 'signatures', determined by the change in electron and hole population overlap resulting from the screening of internal electric fields by mobile ionic charge. Following forward bias the initial and final values of the nid(t) can be related to the established quasi-zero dimensional model, enabling the dominant recombination mechanism to be inferred in devices. It is not possible to infer this information from the steady-state ideality factor alone.

Background
The origins of ideality factors We first examine the origins of the ideality factor without explicitly considering the influence of mobile ions. The J-V characteristics of a solar cell can be described by the non-ideal photodiode equation in which the exponent contains an ideality factor to account for the deviation of the current from the value predicted by Boltzmann statistics: where J is the current density, JSC is the short-circuit current density, J0 is the dark saturation current density, V is the cell voltage, kB is Boltzmann's constant and T is the temperature. Conventionally, the ideality factor is measured either from the slope of the dark ln(J) vs V characteristics (where JSC = 0) or, in order to avoid the effects of series resistance, by measuring the open circuit voltage as a function of light intensity, or incident photon flux φ (known as a 'Suns-VOC' measurement). Using equation 1 and assuming φ ∝ JSC, the ideality factor can be obtained from the VOC vs ln(φ) curve: At open circuit the rate of charge carrier generation is equal to the rate of recombination, U, and thus proportional to φ, allowing the ideality factor to be expressed in terms of the recombination rate and VOC: Ideality factors are typically related to recombination mechanisms and their associated reaction orders using a zero-dimensional theoretical framework. First, a relationship between the quasi Fermi-level splitting, ΔEF (which is equivalent to qVOC in zero dimensions) and free electron (or hole) carrier density n (or p) is defined such that: where ni is the equilibrium charge carrier density and β is a parameter defining the relationship between the charge carrier density and the perturbation of the quasi Fermi-levels from equilibrium EF0. If n ≈ p, then np  n 2 and β  2. In this case ΔEF is split equally between the individual electron (EFn) and hole (EFp) quasi Fermi-levels (see Figure 1a). Where a majority carrier type exists, β → 1. This can be shown by expressing the carrier concentrations in terms of their change from equilibrium values (n0 and p0) such that n = Δn + n0 and p = Δp + p0. If p0 >> Δp = Δn >> n0 then the charge density ratio in the logarithm of equation 4 can be written: In this case it is apparent that ΔEF will be almost exclusively influenced by the changes in the minority carrier, n. Under the assumption that a single recombination reaction order, γ, dominates the behaviour of the device such that U = kn γ (where k is the order dependent recombination rate coefficient) and by substituting equation 4 into equation 3, the relationship between nid, β, and γ can be derived (see supporting information, Note 1) such that: The ideality factor is therefore given by a ratio of the relationship between carrier density and quasi Fermi-level splitting, β, and the dominant recombination reaction order, γ. A device that shows a change in ideality as a function of time following preconditioning, implies that there has either been a shift in the dominant recombination reaction order (for example from SRH to band-to-band dominated) or a change in the charge density dependence of the quasi Fermi-level splitting as defined by the parameter β.
In working solar cells, non-radiative recombination via defect states in the bandgap is consdered the most important recombination pathway since it can be mitigated via passivation of defects, particularly at grain boundaries and interfaces. 15,16 In p-i-n devices with a built-in electric field, the overlap in the populations of electrons and holes at the interfaces is likely to be small, hence idealities close to 1 are often correlated with surface recombination while idealities closer to 2 are identified with bulk recombination via traps (see Table 1). In organic solar cells, experimental values of between 1 and 2 have frequently been observed suggesting a range of behaviours from surface to bulk recombination. 17,18 Idealities of > 2 have previously been attributed to a tail of states in the bandgap 19 or non-linear shunt resistance, 20 although Kirchartz et al. have shown these processes are unnecessary to reproduce unusually high idealities in thin devices where the distribution of the carrier population overalp is highly dependent on the magnitude of the carrier densities. 21 Given this framework for understanding the origins of ideality factors we now examine both measurements and simulations of time-dependent ideality factors in perovskite solar cells with multiple architectures and simulated recombination schemes. In particular, we focus on the influence of mobile ionic charge on localised electron and hole population overlaps, and how this alters the measured ideality factor with time.
This allows a comparison of recombination in a device containing a planar electron transporting material/perovskite interface with one in which the interface is likely to be extended.

Suns-VOC
The history-dependent nature of the open circuit voltage in perovskite devices required an adaptation of the conventional Suns-VOC method for determining the ideality factor. Figure 1b shows the experimental timeline for the measurement. The TRAnsient and Charge Extraction Robot (TRACER) system, described previously, 24 was used to precondition the measured cell for a set time (typically 100 s) at a specified prebias voltage, Vpre, prior to the measurement of VOC at each light condition. The cell was then simultaneously switched (with microsecond precision) to open circuit and a continuous light intensity (with photon flux φ) generated by a bank of white LEDs was switched on. The evolution of VOC(t) was then recorded using a National Instruments USB-6251 data acquisition card where t is the time after illumination. This procedure was repeated sequentially for each light intensity used in the measurement from low to high intensity. The transient ideality factor, nid(t), of the solar cell for a given precondition was evaluated from the fit to the slope of the VOC(t) against ln(φ) (equation 2) between φ = 0.2 -2 sun equivalents (calibrated with the photocurrent generated by a solar simulator) determined at different delay times t.

J-EL measurement
An alternative method to determine nid was also checked using electroluminescence vs current density measurements. Electroluminescence (EL) was measured using a Shamrock 303 spectrograph combined with an iDUS InGaAs array detector cooled to -90 °C. Before measurement of EL at each injection current density, the device was preconditioned at a voltage, Vpre, as described above for the Suns-VOC measurements. After this poling step, the device was set to a constant injection current level ranging between J = 1.25 and 1250 mA cm -2 . A pair of optical lenses were used to focus the resulting EL signal allowing sampling at time intervals of between 0.02 -0.1 s. The EL signals were normalised relative to the value at t = 1 s. The relative emission flux for each injection current was subsequently determined by integration over the whole spectrum. The EL emission flux φEL can be described as: which is proportional to the radiative component of the total recombination. Here, J0,rad is the dark saturation current for the radiative recombination only. Since the radiative component of recombination comes from band-to-band transitions, equation 7 has an effective ideality factor of unity (c.f. Table 1). We can combine equation 7 with the diode equation 1 (where, with no illumination, JSC = 0) to obtain an expression for ideality factor nid of the device: The ideality factor can therefore be derived from the slope of the logarithm plot of the EL intensity against injection current density.

Suns-PL measurement
As a further cross-check for the determination of nid we used photoluminescence (PL) intensity as a function of excitation intensity measurements. PL from perovskite devices was measured with the same spectrograph and detector system as the EL setup described above. A 473 nm diode laser was used as the excitation source, and a set of neutral density filters were applied to adjust the laser intensity. Prior to each measurement, the devices were preconditioned at Vpre in the dark (as described for the Suns-VOC and EL-J measurements). The sample was then illuminated and the integrated PL signal recorded with a sampling interval of 0.1 s. A small fraction of the total recombination is in the form of radiative emission, where PL emission flux from the solar cell is related to the quasi Fermi-level splitting (qVOC) in the device such that: During the PL measurement at open circuit, the total recombination flux, U, in the device will be proportional to the illumination flux from the laser, φ, so that using equation 1 for open circuit conditions (J = 0 and JSC = qφ) we again arrive at equation 2. By combining equation 9 with equation 2 we can derive the following expression: The cell ideality can therefore be evaluated from the slope of the logarithm of PL intensity (flux) plotted against the logarithm of the laser intensity (flux) at different times (t) following illumination.

Transient photovoltage measurements
To probe the direction of charge transport within different regions of the device we used the TRansient Of The TRansient Photovoltage (TROTTR) technique previously introduced in reference 4 , using a 488 nm laser excitation source superimposed on the white LED bias light. The measurement is similar to the Transient Suns-VOC measurement, performed at 1 sun equivalent intensity, with the superposition of short (0.5 -10 μs) 488 nm laser pulses at periodic intervals (1 second) throughout the background illumination stage to induce small perturbation photovoltage transients ΔV. The experimental timeline is reproduced from reference 4 in supporting information, Figure S7 where the experimental setup is also given ( Figure S8). The relatively short penetration depth of the laser results in charge carriers generated close to the electrode which is directly illuminated by the pulse.
A negative ΔV indicates the presence of a reverse electric field in the region of the active layer probed by the excitation pulse.

Simulation
The DrIFtFUSION simulation tool was used to solve for electron, hole and mobile ion densities as a function of position and time in a p-i-n device structure. Here, a mirrored-cell approach with open circuit (J = 0) boundary conditions, was used to enable the open circuit voltage to be solved for directly as a function of time. 4 Prior to solving with illumination, the equilbrium solution for the cell in the dark with an applied voltage of Vpre was established to give the initial condition for the subsequent VOC(t) solutions. For simplicity a uniform generation profile was used during the VOC simuations and solutions were caculated for 9 different constant light intensities ranging from 0.1 to 25.6 sun equivalents where (1 sun equivalent was set to be a generation rate 2.5 × 10 21 cm -3 s -1 throughout the 400 nm thickness of the perovskite layer). A single positive mobile ionic defect species was considered with a uniform static distribution of negative counter-ions in the active layer. Low rates of second order band-to-band recombination (Ubtb ∝ np) were implemented in all layers, whereas SRH recombination (USRH) was used in different locations of the device dependent on the recombination scheme. Five different recombination schemes were examined summarised in Table  2. The parameteres used and a more complete description of the model are given in the supporting information Table S1 and reference 4 . The MATLAB code is freely available to be downloaded at https://github.com/barnesgroupICL/Driftfusion. Table 2. Recombination mechanisms simulated in the study. Shallow traps were set to be in the bandgap and 0.2 eV away from the conduction and valence bands in the n and p-type layers respectively. Mid-gap traps were defined to be in the centre of the 1.6 eV bandgap.

Dominant recombination mechanism
Trap energy The recombination coefficients were adjusted dependent on the trap energy to yield VOC values approximately similar to those observed experimentally. Where trap levels are described as 'shallow' the energies were set to be in the bandgap, 0.2 eV away from the conduction and valence bands in the n and p-type layers respectively. While symmetrical trap distributions such as this are highly improbable in reality, this approach enabled the study of location-dependent recombination in isolation of asymmetric electron and hole recombination rates.
The simulated solutions for VOC(t) determined for the different photon fluxes were used evaluate nid(t) from the slope of VOC vs the logarithm of φ. Since this slope varied somewhat over the range of φ, particularly with low Vpre values, the average value is quoted in this study (see supporting information, Figure S10).

The evolution of VOC with light intensity and preconditioning bias
Examples of the measured evolution of the VOC following illumination at different constant light intentisities are shown in Figure 2a and Figure 2b pre-biassing at 1.2 V and 0 V respectively in the dark for a mesoporous Al2O3 cell. It is apparent that, following Vpre = 1.2 V, the VOC gradually decreases over the course of 60 s from an initially higher value towards a steady-state value for each light intensity. In contrast, following Vpre = 0 V the VOC rises towards a steady-state value. Figure 2c shows that the slope of VOC plotted against the logarithm of photon flux density increases with time after illuination for the Vpre = 1.2 V case.
We note that when the measurement protocol was applied to devices with poor degradation stability, we observed irreversible losses during the measurement, these typically resulted in variation in initial-value of the ideality (see supporting information Note 4). When the measurements were applied to more stable devices we observed repeatability in the transient ideality factor signature, and were able to correlate the results with our simulations.
We have previously discussed the mechanism underlying the evolution of VOC at 1 sun equivalent intensity following different preconditioning biases in considerable detail. 4 The gradual change in VOC often observed in perovskite cells can be explained by a slow redistribution of ionic charge from an initial state towards a new dynamic equilibrium at open circuit in the light. In the initial state the charge is accumulated at the interfaces, screening the internal potential resulting from the combination of the built-in and preconditioning voltages. Upon illumination, the change in the electric field distribution within the device, resulting from the development of a photovoltage, can result in either enhanced segregation of charges towards the charge collecting contacts (following positive Vpre close to the open circuit potential) or segregation of photogenerated charges towards the opposite contacts (following zero or negative Vpre). If these ion distribution effects are combined with significant interfacial recombination, the VOC either slowly decreases or increases from its initial value as the mobile ionic charge moves to a new equilibrium distribution. At the new equilibrium the internal electric field within the perovskite layer is again screened by an accumulation of ionic charge at the interfaces and a new VOC value is reached. It is likely that this core process of ionic redistribution is present in all the devices examined in this work. The rate of redistribution will depend on the light intensity and magnitude of the preconditioning bias. The details of how this redistribution influences the evolution of the transient ideality factor will, in turn, depend on the details of the recombination processes in the device.  Table 2, band-to-band in bulk and interfacial SRH recombination via midgap states). (f) Simulated VOC vs incident photon flux, φ, at different delay times following illumination for the dark Vpre = 1.1 V initial conditions case shown in (d) also shows an increase in ideality factor with time.
The evolution of the transient ideality factor Figure 3a shows the evolution of nid(t) for the two different types of cell following dark preconditioning at Vpre = 1.2 V (mp-Al2O3 device), Vpre = 1.0 V (mp-TiO2 device). It is apparent that the evolution of nid(t) is significantly different for each of the examples shown. Following Vpre = 1.2 V the ideality starts at nid(t=0) ≈ 1 and increases to approximately 2.1 for the mesoporous Al2O3 cell (green curve, circle markers) during the 30 seconds of the measurement. When the ideality for the same cell was evaluated following Vpre = 0 V, the values of nid were much higher that expected given the classical interpretation, starting at 3.7 and decreasing to approximately 1.8 (see supporting information, Figure S9a). The mesoporous TiO2 cell (light blue curve, triangle markers) also showed and increase in nid following Vpre = 1.0 V, from 1.5 to around 2.7. The curve appears similar in shape to that of the mesoporous Al2O3 device but shifted with respect to the y-axis. Following Vpre = 0 V, the nid again shows much higher values, starting at 3.2 and finishing at 2.6 (see supporting information, Figure S9a). A summary of the initial and final nid values for different measurement techniques and preconditioning voltages is given in Table 3.

Comparison with instantaneous idealities determined from electroluminescence and photoluminescence measurements
To cross-check that the values of nid measured this way are reasonable we made further measurements of the initial value of the ideality factor nid(t≈0) following dark prebiassing using both electroluminescence vs current density measurements (J-EL) and photoluminescence vs illumination intensity measurements (Suns-PL). Figure 4 shows examples of these measurements which are summarised for various cell types in Table 3. There was good agreement between the ideality values determined using the different methods following Vpre > 0. For Vpre = 0, we noted discrepancies between the Suns-PL measurements and Suns-VOC values in some cases leading us concentrate our analysis on transient ideality factors derived for Vpre > 0 (this is discussed further along with possible explanations in the supporting information Note 2). Overall, the agreement between the different techniques suggests the concept of a transient ideality factor measured by the Suns-VOC technique is related to recombination processes and not to charge storage or capacitive effects within the cell.

Comparison with simulated transient ideality factors
Different recombination mechanisms might be expected to dominate different device architectures. If the transient ideality factors (plotted in Figure 3a and Figure 3b) are related to these mechanisms then we hypothesise that the shape of nid(t) gives a signature which can be used to identify the primary recombination pathway for any given device. To test this possibility we now examine simulations of the nid(t) measurement for modelled cells with a range of recombination schemes as summarised in Table 2. In particular, the simulations provide us with a means to assess the likely consequences of mobile ionic defects on our measurements. Furthermore, by examining the evolution of the free electron and hole profiles as the mobile ionic charge redistributes, we can ascertain how the relative concentration of electrons and holes varies with time in regions of the cell where recombination is localised. This allows us to diagnose the specific conditions in Table 1 that give rise to an observed value of nid at a given time in the measurement.  Table 2). The results reproduce the majority of features seen in Figure 2d and Figure 2e, and can be used to find nid following different illumination times (Figure 2f). This process was repeated for the different recombination schemes shown in Table 2 allowing nid(t) to be evaluated from the results for the two initial conditions. The corresponding simulated nid(t) curves are shown in Figure 3b. We note that linearity of the VOC vs ln(φ) curves used to determine nid(t) is considerably greater following the forward bias precondition relative to the zero bias condition (see examples in the supporting information, Figure S10). Combined with the descrepancy between the ideality following Vpre = 0 V for different measurement techniques discussed in supporting information, Note 3 this suggests that nid(t) determined following forward biasing provides a more reliable measure of recombination mechanism.
There are a number of strikingly similar features between particular pairs measured and simulated nid(t) profiles in Figure 3. On the basis of these similarities and the more detailed analysis of the simulation output, discussed below, we assign the evolution of the measured nid(t) profiles examined in our study to likely dominant recombination schemes. We do not assert here that the dominant recombination mechanisms we assign to the devices will always represent the dominant recombination route for the respective architectures since it is very likely that variation in the processing during device fabrication can lead to a number of possible recombination routes and thus device-to-device variation in nid(t) for cells prepared with the same architecture (discussed further in the supporting information, Note 4). Rather, the approach described here presents a means to assess the most likely recombination routes for a particular device.

Recombination mechanism in the mesoporous Al2O3 cell
The transient ideality signatures for the mp-Al2O3 cell ( Figure 3, green curve, circle markers) correspond most closely to a model where interfacial SRH recombination via mid-bandgap states (scheme 3) is the dominant recombination mechanism. This recombination scheme is also consistent with the large degree of J-V hysteresis seen in this device (see supporting information, Figure S13a) since we, and others, have previously shown that a combination of mobile ionic charge and interfacial recombination are necessary to observe hysteresis. 4,7,8,12,25 While computational studies on defective CH3NH3PbI3 crystals have not shown deep trapping states with easily accessible activation energies, 26,27 the interface between CH3NH3PbI3 and the compact TiO2 layer is likely to contain a high density of deep inter-bandgap electronic states as commonly observed at the TiO2 hole transporting material interface in dye sensitised solar cells. 28,29 Furthermore, recent studies have shown that recombination rates at the hole transport layer interface are significantly influenced by doping 30 and that the CH3NH3PbI3 interface with heavily lithium-doped Spiro-OMeTAD is a site of significant recombination. 7 We now consider the likely mechanism underlying the observed changes in the transient ideality factor over the course of the measurement for this device. In the simulation with recombination scheme 3, following Vpre = 1.3 V, nid(t) initially starts at a value close 1.0 and increases to a value of close to 2 by the end of the measurement (Figure 3b, green curve, circle markers). The evolution of the simulated integrated recombination fluxes in the cell indicate that the SRH recombination at the contacts dominates at all times during the transient (see Figure 4d) such that the changes in the ideality factor cannot be explained by a change in the dominant recombination mechanism. We note however that there is significant net reduction in the fraction of recombination arising from band-toband recombination during the simulation, which is consistent with the reduction in PL signal with illumination time observed over a timescale of seconds in the mp-Al2O3 device (Figure 4c and Figure  4d).  (Table 2) was used in the simulation results shown, the green and blue shaded regions indicate the respective p and n-type regions of the device.
Since the primary recombination mechanism does not change during the measurement (and thus γ is constant) the observed variation in nid is likely to be related to variation in β (c.f. equation 7), which defines the relationship between the charge carrier density and the perturbation of the quasi Fermi-levels from equilibrium. Examination of the initial (Figure 5a) and final (Figure 5c) simulated electron and hole profiles for the Vpre = 1.3 V case reveals that there is indeed a change in the relative electron concentrations in the interface regions as the ions redistribute. Initially n >> p in at the n-type interface (and p >> n at the p-type interface) such that for SRH interfacial recombination via mid-bandgap traps β ≈ 1 (see Table 1). In the final state n ≈ p at both interfaces such that β ≈ 2, explaining the change from 1 to 2 in nid(t) during the course of the measurement.
The similarity in electronic charge density profiles following preconditioning at Vpre = 1.3 V ( Figure  5a) and that of a simulated device without mobile ionic charge (Figure 5d) is highly significant; the preconditioning stage reduces the density of ionic charge accumulated at the interface with respect to Vpre = 0 V ( Figure 5b) and results in an electric field in the device at t = 0 s of the measurement which drives charge carriers towards their respective transport layers. Consequently, forward biasing with an applied potential sufficient to separate the electron and hole densities at the interfaces determines an instantaneous ideality factor which can be interpreted using the standard theory (as outlined in the introductory theory section). The steady-state nid value can only be interpreted correctly by considering the fact that n ≈ p (β ≈ 2) throughout the active layer (see Table 1). The evolution of the transient ideality factor following Vpre = 0 V is discussed further in the supporting information Figure S8 caption.

Recombination mechanism in the mesoporous TiO2 cell
The transient ideality factor profile of the mp-TiO2 (Figure 3a, light blue curve, triangle markers) following Vpre = 1 V shows an intermediate behaviour between surface (Figure 3b, pink curve, circle markers) and bulk SRH recombination (Figure 3b, dark blue curve, triangle markers) via mid-bandgap traps within the bulk of the active layer.
The shape of the simulated SRH mid-gap recombination in the bulk nid(t) signatures can be understood by examining the charge carrier concentration profiles during the simulations. These were qualitatively similar to those shown in Figure 5 both for the Vpre = 1.3 V and Vpre = 0 V cases. For the measurement following forward bias preconditioning, the initial recombination flux has significant contributions from regions of the bulk where n ≠ p such that 1 < nid < 2. In the final state n ≈ p throughout the bulk such that nid  2. The simulated measurement following Vpre = 0 V in this case resulted in VOC vs ln(φ) slopes that were very non-linear so the relevance of the derived nid(t) values to recombination values are questionable. We observe a curious case where the initial recombination is dominated by the central region where n ≈ p (so that γ ≈ 1) but Fermi level splitting is dominated by changes in concentration near the interfaces (β ≈ 1) so that the mean ideality goes from 1 to 2 as the β  2 during the redistribution of ionic charge to a steady state (see supporting information, Figure S11).

Determining the spatial extent of the charge collection interface/recombination region
In order to further investigate the origin of the initial measured ideality value of nid(t = 0) = 1.5, we probed the internal electric field by performing Transient of the Transient Photovoltage measurements (see Methods) on similar mp-TiO2 and mp-Al2O3 architecture devices fabricated using a Transparent Conductive Adhesive (TCA) top electrode 31 with a short wavelength laser excitation source. The absorption coefficient for CH3NH3PbI3 at 488 nm is 1.84 x 10 5 cm -1 , giving a characteristic penetration depth of 54 nm (see optical modelling in supporting information Figure S14). Thus, only charge carriers generated close to the contact through which the device is illuminated are expected contribute to change in photovoltage (ΔV).  Figure 6 shows the results of the TPV experiments combined with schematics of the likely band configuration on the mp-TiO2 and mp-Al2O3 devices. In both cases the cells were preconditioned at Vpre = 0 V for 60 s prior to illumination at 1 sun equivalent intensity. The configuration of ions under these conditions would be expected to initially induce a reverse field in the devices similar to that seen in the simulated device following Vpre = 0 V (Figure 5b). 9,10 At early times (<10 s) following illumination through the TCA side of the device, the TPV signal for the mp-TiO2 showed a distinct negative deflection (Figure 6a). This negative response was significantly less pronounced when the device was instead illuminated through the glass/FTO side (Figure 6b). We interpret this as evidence that the electric field is predominantly confined to the pure perovskite phase in the device containing mp-TiO2 as shown in Figure 6c. Here, carriers generated close to the hole transporter are driven in the wrong direction with respect to charge collection, leading to an increased accumulation of minority charge carriers at the Spiro-OMeTAD/Perovskite interface, increased recombination, and an associated negative TPV deflection and reduced magnitude as compared to later times in the measurement. Carriers generated in the mesoporous region instead diffuse with towards the correct contact and generate a positive signal. After 50 s the majority of the mobile ionic charge has migrated to screen out the field once more and the signal becomes similarly positive independent of illumination side (Figure 6a and Figure 6b, blue curves).
When the same experiment was repeated on the mp-Al2O3 device, a negative deflection was apparent in the initial TPV signals independent of illumination side (Figure 6d and Figure 6e), suggesting that the field is extended throughout the perovskite in the mesoporous region (Figure 6f).
During illumination under open circuit conditions, the TPV signal followed a similar evolution to a more positive signal independent of illumination side in the proceeding 50 s. The persistent negative signal during the pulse after 50 s implies a long-lived polarisation in the device that could result from slower moving ionic species. 32 Following forward bias preconditioning at Vpre = 1 V, the negative component of the signal was diminished in all cases indicating that the strength of the inverted field at open circuit had been reduced (see supporting information, Figure S15), consistent with the simulation following forward bias preconditioning in Figure 5a.
Based on these results we hypothesis that in the mp-TiO2 device, at the Spiro-OMeTAD interface the charge populations are separated (β ≈ 1), while within the mesoporous region and at the c-TiO2 interface, the charge populations are approximately equal (β ≈ 2) following a forward bias precondition. The scaffold is likely to introduce deep trapping states in the region of this extended interface, leading to a reaction order of γ ≈ 1. If both interfaces have similar SRH time constants then an ideality of 1 < nid < 2 is expected: While the device is still dominated by surface recombination, the overlapping electron and hole populations (n ≈ p) throughout the mesoporous region leads to an ideality factor greater than 1. Further work incorporating the scaffold region into models will be required to test the above hypotheses. The reduced hysteresis in the cell's J-V curve by comparison with the mp-Al2O3 device (see supporting information, Figure S13a and Figure S13c) also suggests that recombination is dominated by a mix of surface and bulk effects. 8 To summarise, these observations imply that the extended interface introduced by the mesoporous TiO2 allows for recombination of electrons and holes via deep traps throughout the mesoporous region which can be considered 'bulk like'. In contrast, cells with mesoporous Al2O3 appear to be dominated by deep trap recombination primarily confined to the charge collecting interfaces, and not extending through the insulating mesoporous layer.

Conclusions
We have introduced a method to evaluate the transient ideality factor of perovskite solar cells following a voltage preconditioning period in the dark. The approach allows meaningful information about the dominant recombination mechanism to be derived in the presence of mobile ionic defects. The mobile ions result in an evolving electrostatic profile throughout the thickness of a perovskite solar cell and consequently a time varying VOC which hampers measurement of an ideality factor by conventional means. By preconditioning cells at a forward bias potential sufficient to separate the charge carrier densities at the interfaces, the cell can temporarily be set in a state which would yield a similar ideality factor to a similar device which contained no mobile ions and could thus be analysed using the convention theories applied to ideality factors.
With the aid of time-dependent device simulations, which include the effects of mobile ions, we have shown that the evolution of the ideality factor following different voltage preconditions can be used as a signature to identify the dominant recombination mechanism in a perovskite solar cell. We demonstrated this idea with standard perovskite device architectures incorporating mesoporous TiO2 and mesoporous Al2O3 scaffolds. For these examples we assigned surface recombination via deep traps at the charge collection layer interfaces as the dominant recombination mechanism in both device types. Transient photovoltage measurements on devices with dual transparent electrodes indicated that, distinct from the mp-Al2O3 device, the electric field was predominantly confined to the pure perovskite region in the mp-TiO2 architecture. The homogeneity of electron and hole populations throughout the extended mesoporous TiO2 interface resulted in a measured initial ideality following forward bias of 1.5 despite surface recombination remaining the dominant mechanism in the device.
We emphasise that the dominant recombination mechanism can vary between devices which were nominally prepared in the same way, an observation we attribute to small differences in the processing conditions and microstructure of the cells. We also note that our observations do not preclude the possibility that trap energy and spatial distribution in the device may be dependent on the distribution of ionic defects. Together our findings provide new insight on the meaningful interpretation of ideality measurements in perovskite solar cells and a valuable new tool for assigning the dominant recombination mechanism in devices.

Supporting information
Note 1: Derivation of ideality factor, nid as a function of the charge carrier density relationship to Fermi level splitting, β and the reaction order, γ Using the Shockley diode equation, the ideality factor, nid can be expressed in terms of the quasi-Fermi level splitting ΔEF (identical to qVOC in the zero dimensional model) and recombination rate, U such that: where q is the elementary charge, kB is Boltzmann's constant and T is the temperature.

Note 2: Derivation of reaction orders
Here we examine how the reaction order γ for the general form of the equation describing the recombination rate U in terms of the electron concentration, n, and the reaction order dependent recombination rate coefficient: = depends on the recombination mechanism and relative concentration of electrons and holes as listed in table 1 of the main text.

Band-to-band recombination
For n = p the recombination rate, U, is given by: where kbtb is the band-to-band recombination rate constant hence γ = 2.
For n >> p, the majority carrier density can be considered as effectively constant, leading to: ∝ hence γ = 1.

Shockley Read Hall recombination
The compact form of the SRH recombination expression is: τn and τn are the trap electron and hole recombination lifetimes and nt and pt are the electron and hole densities when the Fermi level is at the trap level, Et, for example for electrons: where ECB is the conduction band energy.

Shallow traps, n = p
For shallow traps close to the conduction band, in most cases EFn < Et, and n << nt, p >> pt, and p << nt. Where n = p the rate of recombination is determined by both charge carrier densities:

Mid-bandgap traps, n = p
For deep traps where, in the presence of a bias EFn > Et and as a result n >> nt and p >> pt. If n = p then:

Shallow traps, n >> p
For shallow traps close to the conduction band, in most cases EFn < Et, and n << nt, p >> pt, and p << nt. If n >> p and n is approximately constant (for example if n is pinned by the contact concentration ncontact) then recombination is limited by the availability of holes: hence γ = 1. However this does not necessarily hold generally. In the cases considered in the main paper this implies that the above approximation will be valid following forward bias preconditioning, but not zero or reverse bias preconditioning.

Mid-bandgap traps, n >> p
For deep traps in the presence of a bias where EFn > Et, then n >> nt and p >> pt. In a situation where n >> p, recombination is limited by the availability of holes:  Hence where SRH = btb = 10 -10 as in Scheme 5, provided n < 10 20 cm -3 , which is guaranteed by the choice of 10 20 cm -3 for the effective density of states, SRH will dominate.

Note 3: Discrepancies between nid(t≈0) for Vpre = 0 derived from Suns-VOC and Suns-PL measurements
For the mp-TiO2 cell we note that although the changes in nid values from Suns-PL and Suns-VOC were consistent as Vpre was varied the absolute values of nid for Vpre = 0 were lower with the Suns-PL measurements (nid(t≈0) = 1.8) than the corresponding Suns-VOC measurements (nid(t≈0) = 3.2). This difference in absolute values may be related to the relatively localised generation of charge carriers by the laser spot during the Suns-PL measurement which might avoid localised recombination shunts (dependent on the ion distribution). These shunts would not be avoided during the Suns-VOC measurements since charge carriers are generated over the whole cell area.

Note 4: Device-to-device variability in the transient ideality factor
A significant issue for perovskite solar cells is variation in the properties of devices nominally prepared by the same methods. This issue has been noted by many authors including Pockett et al. who observed significant differences in the steady-state ideality factor observed for devices prepared in an identical way. 5 We also observed a similar phenomenon, with differences in the initial ideality factor values as well as at steady-state. For example, for a batch of 6 planar compact TiO2 cells manufactured in a different laboratory to those described above, initial idealities after forward biasing at Vpre = 1.1 V were 0.97 < nid(t≈0) < 4.22 ( ̅ id ( ≈ 0) = 1.78, σnid = 1.16), while for mp-TiO2 architectures following the same preconditioning, the values were 0.04 < nid(t≈0) < 1.12 ( ̅ id ( ≈ 0) = 0.75, σnid = 0.62, for 6 devices). Mesoporous TiO2 devices with triple cation active layers showed greater consistency with initial idealities after forward biasing of 0.40 < nid(t≈0) < 1.04 ( ̅ id ( ≈ 0) = 0.77, σnid = 0.32, for 3 devices). The values close to 1 here might suggest that recombination at the compact TiO2 layer interface is more significant than at the mp-TiO2/CH3NH3PbI3 interface in these devices. However, the unexpectedly high nid and standard deviation suggest that the recombination kinetics at any given interface type are highly dependent on localised details of the interface formation during device fabrication. Degradation during measurements could account for particularly low ideality factors since the VOC transients were measured in order of intensity from lowest to highest. If degradation was continuous throughout the measurement VOCs would be lower than expected at higher light intensity, reducing the measured nid. Further work on high performing, stable devices will be required to more confidently ascertain whether or not the signature transient idealities presented in this study allow more general statements on identifying the dominant recombination mechanism in different perovskite device architectures and material combinations.  Figure S8. Transient optoelectronic measurement system circuit diagram. Circuit diagram for the transient data acquisition system. The yellow shaded area is the primary circuit, which allows the cell to be electrically biased and switched between open and closed circuit. The light red shaded area is the laser circuit and the light blue shaded region is the white bias LED circuit.

Note 5: Transient system technical specifications
Data acquisition was performed by a Tektronix DPO5104B digital oscilloscope and a National Instruments USB-6361 DAQ. The laser pulse was provided by a digitally modulated Omicron PhoxX+638 nm diode laser with a 100 Hz repetition rate. The laser spot size was expanded to cover the active pixel and the continuous wave intensity over the cell pixel area was approximately 550 mWcm -2 during the pulse. The preconditioning bias was applied using the data acquisition card and the system was controlled by a custom Labview code. with Vpre = 0 V, with the following recombination mechanisms: bulk bandto-band recombination (black curve, diamond markers), interfacial SRH recombination via shallow traps (grey curve, square markers), interfacial SRH recombination via deep traps (pink curve, circle markers), bulk SRH recombination via shallow traps (yellow curve, inverted triangle markers), and bulk SRH recombination via deep traps (dark blue curve, triangle markers). The distribution of ions immediately following illumination leads to a reverse internal electric field which results in an accumulation of free holes at the n-type interface such that p > n and an accumulation of electrons at the p-type interface, n > p (Figure 5b). Although deeper into the n and p-type contact there is a point where n = p which will also contribute to recombination. This is very different from what would be expected if no mobile ionic charge were present. A consequence is that an instantaneous ideality factor with an intermediate value of nid(t≈0) ≈ 1.75 is observed. During the subsequent redistribution of ions the cell tends towards an ideality factor of 2 since n ≈ p right at the interfaces in the final state (β = 2 and γ = 1). However, due to the different rates of ion redistribution which are slower with increasing φ (see VOC transients in both Figure 2b and Figure 2d) the net result is a dip in the transient ideality factor during this process resulting in the observed signature nid(t) seen in (b).

Optical modelling of devices with transparent conductive adhesive (TCA) electrode
The normalised optical power density was modelled for a mesoporous TiO2 device stack with a transparent conductive adhesive (TCA) top electrode using a simple Beer-Lambert function such that the light intensity φ at position x in each layer falls exponentially in each layer: where φ0 is the intensity at the start of the layer and α is the absorption coefficient of the material. For the mesoporous TiO2 the optical constants were calculated using the values for pure perovskite and TiO2 phases in proportion 1:1 by volume fraction. The Polyethylene terephthalate (PET) layer of the TCA was assumed to be completely transparent. The optical constants were obtained from references. 3338