Are Shockley-Read-Hall and ABC models valid for lead halide perovskites?

Metal halide perovskites are an important class of emerging semiconductors. Their charge carrier dynamics is poorly understood due to limited knowledge of defect physics and charge carrier recombination mechanisms. Nevertheless, classical ABC and Shockley-Read-Hall (SRH) models are ubiquitously applied to perovskites without considering their validity. Herein, an advanced technique mapping photoluminescence quantum yield (PLQY) as a function of both the excitation pulse energy and repetition frequency is developed and employed to examine the validity of these models. While ABC and SRH fail to explain the charge dynamics in a broad range of conditions, the addition of Auger recombination and trapping to the SRH model enables a quantitative fitting of PLQY maps and low-power PL decay kinetics, and extracting trap concentrations and efficacies. However, PL kinetics at high power are too fast and cannot be explained. The proposed PLQY mapping technique is ideal for a comprehensive testing of theories and applicable to any semiconductor.


Supplementary Note 1. Photoluminescence Microscopy Setup
Supplementary Figure 1.1. Schematic of the PL microscopy setup designed for measurements of PLQY(f,P) maps. The bottom panel shows the list of pre-defined parameters of the setup used by the automated control system to measure the 85 data points (numbered 1 to 85) of the PLQY(f,P) map. Points 1, 40 and 85 are highlighted in the exemplary PLQY(f,P) plot. PL decays are also measured for selected f and P combinations, by redirecting part of the emission with a beamsplitter to a hybrid photodetector (HPD) connected to a time correlated single photon counting (TCSPC) module. PL spectra were measured by adding a transmission diffraction grating in front of the CCD camera.
The PLQY(f,P) map and PL decays were measured in a custom-built photoluminescence microscopy setup ( Supplementary Figure 1.1). A pulsed 485 nm laser (Pico Quant, 150 ps pulse width) was used to excite the sample through an objective lens (Olympus 40X, NA = 0.6) of a wide-field fluorescence microscope (Olympus IX-71). The emission of the sample was then collected by the same objective and captured by one of two detectors. The first detector is a EM CCD (Princeton Inst. ProEM 512B) for measuring PL(f,P)) (See Supplementary Note 3 on how PL(f,P) is converted to PLQY(f,P)). The second detector is a hybrid photomultiplier detector (HPD, Picoquant PMA Hybrid-42) which is connected to a time correlated single photon counting (TCSPC) module (Picoquant, Picoharp 300) for the measurement of PL decays kinetics. The instrumental response function minimum width is 200 ps.
In Supplementary Table 2.1 we present the five power fluences P1-P5 in alternative units of charge carrier density -0 (cm -3 ), calculated using the following equation: where 0 is the charge carrier density (cm -3 ), P is the pulse fluence (photons/cm 2 ), d is the thickness of the perovskite layer (260 nm) and Absorptance is the fraction of the excitation light absorbed by the sample, which was estimated to 65% (the remainder light was mostly scattered and reflected). To measure a full PL(f,P) map, the CCD acquires a series of PL images, one for each combination of the laser repetition rate f and the pulse fluence P. This is achieved by utilizing a custom LabVIEW software that controls all parts of the setup. The software reads from an input table (Supplementary Table 2.2) each row containing a combination of parameters for each individual measurement: (i) repetition rate of the laser (f), (ii) optical filter in the laser beam to regulate the pulse power fluence (Exc.OD), (iii) optical attenuation of the emission to protect the detector from saturation (Em.OD) and (iv) exposure time of the CCD (Exp.t). The pulse power fluence and the emission attenuation is controlled by two motorized filter wheels placed in the excitation and emission path, respectively, each containing a set of neutral optical density (OD) filters (Supplementary Figure 1.1).

Supplementary
MAPbI3 films often exhibit photobleaching and photobrightening effects upon exposure to light. We paid great attention to minimizing and monitoring these effects by (i) keeping the samples under a nitrogen environment throughout all measurements, (ii) adding a motorized shutter, also automatically controlled by the software, blocking the excitation beam during the time necessary for the filter wheels to rotate between each acquisition, (iii) introducing a reference point (P1, 80MHz) which is measured multiple times during the acquisition of a full PL(f,P) map (Supplementary Note 4) and (iv) measuring additional PL(f,P) maps on the same spot on the sample.

Supplementary Figure 2.1.
The sequence in which one complete PL(f,P) map is measured. The entire scan is performed in a sequence of 5 measurements as schematically shown: a) P=P1, f scanned, with the reference points 1 and 2 measured before and after the scan respectively; b) P=P2, f scanned followed by the reference point 3; c) P=P3, f scanned followed by the reference point 4; d) P=P4, f scanned followed by the reference point 5; e) P=P5, f scanned and the full round is finished by measuring the final reference point 6. The change in the reference points in the figure is exaggerated to make it clearly distinguishable.

Measurements of the absolute PLQY using an integrating sphere
By definition, external PLQY is the ratio of the number of photons emitted to the total number of photons absorbed: The number of photons absorbed and emitted by the samples under excitation with the same laser source as used for PLQY(f,P) mapping were obtained using an integrating sphere (Horiba ,Quanta-φ) coupled through a fiber to a compact CCD spectrometer (Thorlabs CCS200/M). The measurements of the laser intensity (spectrum) with and without the sample as well as the sample PL gives: where ( ), ( ) are the measured spectra of the laser without and with the sample in the sphere, respectively; ( ) is the spectrum of the sample emission, ( ) is the total light collection coefficient for the wavelength λ and ( ) presents the spectral sensitivity of the integrating sphere, fiber and spectrometer combined. It is important that all spectra in Eqs.

Spectral sensitivity calibration of the PLQY measurement with integrating sphere
To calibrate the spectral sensitivity of the setup, we used the spectrometer (via the sphere and the fiber) to measure the light of a Tungsten Halogen calibration lamp (OceanOptics LS-1-CAL.). Then, ( ) was obtained by dividing the known lamp spectrum with the one measured. Since the standard lamp spectrum is given in the units of power/nm, we multiply all measured spectra by to compensate for the relation between energy and number of photons (Eqs. (3.4)-(3.6)).
Thus, the final calibrated spectra ′ ( ), ′ ( ) and ′( ) in units of photons/nm are obtained as: Conversion of the PL intensity measured by the microscope to the absolute PLQY measured by the integrating sphere Supplementary Figure 3.2. Propagation of PL light emitted by a thin film with a thickness smaller than the wavelength. Collection of PL by a fluorescence microscope and by an integrating sphere. The escape probability of the light emitted along the film is very low because of re-absorption. Since PLQY <<1, a very small fraction of photons originally emitted along the film will be able to leave the film from its edges.
The integrating sphere collects all light which was able to escape. The objective lens collects only the light within the collection angle. There is a ratio between the light collected by the objective and by the detector sitting in the integrating sphere. Below, it will be described how we measured this coefficient.
Using the setup described in the Supplementary Note 1, we first measured the evolution of the PL intensity as a function of the laser repetition rate and pulse fluence (PL(f,P) map). To convert PL intensity to absolute PLQY, we have developed the procedure described below.
We have the following equations: where is the total light detection efficiency of the micro-PL setup for a given sample. It is determined by the CCD quantum efficiency, light collection geometry of the microscope objective, the emission angular diagram of the sample and the transmission coefficient of all optical components; is the number of measured counts by the CCD over an exposure time of , is the fraction of the laser light absorbed by the sample, is the excitation wavelength and is the excitation power in Watts.
In equations 3.12, constant combines all these factors together. By knowing the value of , we can convert the measured PL intensity to an absolute PLQY.

Supplementary Figure 3.3.
Normalized spectra of a Pyridine 2 dye (red), Pyridine 2 dye low energy tail measured using a 695 nm long-pass filter to detect only the low energy tail of the emission spectrum (yellow). PL spectrum of MAPbI3 (blue) is shown for comparison.
The simplest approach of estimating is by measuring the PL intensity and the absolute PLQY of a calibration sample. Since is dependent on the sample properties, the calibration sample should have similar optical properties to the MAPbI3 films under investigation. We selected two types of calibration samples: first, the MAPbI3 film itself and second, a Pyridine 2 laser grade dye (Lambda FYSIK) dispersed in a thin (ca 100 -200 nm) PMMA film. Pyridine 2 has strong absorption at our excitation wavelength and a broad PL extending to the near infrared region. The absorption measured in the integrating sphere was 64.6% for MAPbI3 and 63.7% for the Pyridine 2 sample at an excitation wavelength of 485 nm. PLQY of the Pyridine sample was 12%. Because Pyridine 2 has a broad PL band, we placed a longpass filter in front of the CCD detector to measure only the low energy tail emission which is spectrally close to the emission band of MAPbI3 (Supplementary Figure 3.3). This filtered emission was used to calculate the coefficient. Supplementary Table 3.1 shows the values obtained using the dye calibration sample, compared to those obtained using the MAPbI3 film. We observe larger fluctuations in for the MAPbI3 samples, due to following factors:

Supplementary
i) MAPbI3 films exhibit worse spatial PL homogeneity compared to the spin cast dye. Even though the excitation spot size for the PLQY measurement using the integrating sphere is 14 mm 2 and gives an averaged PLQY over almost the whole sample, the PL microscope spot is probing only a much smaller 900 µm 2 area within the film and is thus more sensitive to any local PL inhomogeneities. ii) MAPbI3 film is less stable under light illumination and exposure to air/humidity. iii) PLQY of MAPbI3 is strongly dependent on the excitation power density. Any mismatch of excitation power densities in the two measurements (PL and PLQY) manifests as error in estimation of . The PLQY of the dye on the other hand shows no dependence on excitation power density. Considering that we noticed that the maximum changes occur at low pulse fluence, we selected the reference point at P1 in order to maximize the apparent effect of enhancement/bleaching. As can be seen in Supplementary Figure 4.1, the maximum change of the PLQY always below a factor of two for the entire experiment. This should be noted when assessing the relative accuracy of the entire measurement. Note also that the different samples showed different response to light which was also dependent on the power density. PL enhancement was dominant for all samples except of G/P/MAPI/P film which showed very slight bleaching. In general, such effects were only apparent in the low power density regime, and were mostly negligible in the middle and high power excitation regimes. In Supplementary Figure 4.3, we also present the measured PLQY(f,P) maps for two different spots/regions of the G/MAPI sample, which shows that the sample exhibits analogous results for different regions.

Supplementary Figure 4.3.
Measurements of the PLQY maps for G/MAPI sample on two different regions of the same film.
Solution preparation: Perovskite precursor was prepared according to the previous work, 1,2 in detail 40 %wt perovskite solution was prepared with 1:3 molar ratio of lead acetate trihydrate and methylammonium iodide dissolving in dimethylformamide. Additionally, 8‰ (HPA/DMF volume ratio) amount of HPA were added into the perovskite precursor. PMMA solution was prepared by dissolving 10 mg PMMA in 1 ml chlorobenzene. Thin films fabrication: Glass substrates (microscope cover slips, 0.17 mm thickness) were ultrasonically cleaned with 2 % hellmanex detergent, deionized water, acetone, and isopropanol, followed by 10 min oxygen plasma treatment. The cleaned substrates were then transferred into a drybox (RH < 1 %) for further manipulations. For samples with PMMA layer beneath the perovskite film, PMMA was spin-coated on the clean substrates with 3000 rpm for 30 s and annealed at 100 °C for 10 min. The perovskite precursor was spin-coated at 2000 rpm for 60 s on glass or glass/PMMA substrates, following by a 25 s dry air blowing, a 5 min room temperature drying and a 10 min 100°C annealing. For the samples with PMMA on top, no further annealing was applied after depositing PMMA (3000 rpm for 30 s) on the top of the perovskite layer.
XRD measurements were conducted on samples prepared on glass and measured at room temperature in ambient on a Rigaku SmartLab diffractometer equipped with a 9 kW rotating copper anode. A 2D HyPix3000 detector in a coupled θ − 2θ scan (beam collimator 0.2 mmϕ) at a detector distance of 110 mm was used to gather 2D-XRD data. The 1D-profile was obtained after integrating a 46° wide central wedge of the 2D-diffraction pattern, correction of the background and removal of contributions of Kα2 using the Rigaku 2DP software. Solar cell preparation. The MAPbI3 films prepared by the methodology described above were used to prepare photovoltaic devices, details are given below.
Pre-patterned indium tin oxide (ITO) coated glass substrates (PsiOTech Ltd., 15 Ohm/sqr) were ultrasonically cleaned with 2 % hellmanex detergent, deionized water, acetone, and isopropanol, followed by 8 min oxygen plasma treatment. In a drybox (RH<3%), PTAA (1.5 mg/ml dissolved in toluene) was spin-coated on the substrates at 4000 rpm for 30 s and annealed at 100 C for 10 minutes. The perovskite active layer was deposited as described in the film preparation section. Next, the samples were transferred into a nitrogen filled glove box, where PCBM (20 mg/ml dissolved in chlorobenzene) was dynamically spin-coated at 2000 rpm 30 s on the perovskite layer followed by a 10 min annealing at 100 C. Finally, BCP (0.5 mg/ml dissolved in isopropanol) was spin-coated at 4000 rpm for 30 s, following by 80 nm thermally evaporated silver.
Solar cell characterisation. The current density-voltage (J-V) curve was measured using a computer controlled Keithley 2450 Source Measure Unit under simulated AM 1.5 sunlight with 100 mW/cm 2 irradiation (Abet Sun 3000 Class AAA solar simulator). The light intensity was calibrated with a Si reference cell (NIST traceable, VLSI) and corrected by measuring the spectral mismatch between the solar spectrum, the spectral response of the perovskite solar cell and the reference cell. The cells were scanned from forward bias to short-circuit and reverse at a rate of 0. Germany) was used to acquire the images at an acceleration voltage of 1 kV using a secondary electron detector.

Supplementary Note 7. PL decays in the single pulse and quasi-CW regimes
For each sample, PL decays were measured for different pulse fluences (from P1 to P5) and at different laser frequencies ranging from 30 kHz to 10 MHz. The purpose of these measurements was to track the evolution of the PL decays during the transition from a single pulse to a quasi-CW regime. In particular, for P4 pulse fluence, at 30 kHz all samples are in the single pulse regime and at 1MHz, all of them are at the quasi-CW regime. As is shown in Supplementary Figure 7.1, there appears to be no significant difference in the PL decays between these two frequencies, rendering the examination of PL decays on their own incapable of distinguishing between the two different excitations regimes. It is interesting that for MAPbI3 without PMMA coverage increasing of the repetition rate leads to a faster PL decay, while for the samples with PMMA coverage the behavior is the opposite. See  Supplementary Note 9. Theoretical calculations and fitting 9.1 Notes about diffusion and photon recycling PLQY calculated in this section is the so-called external PLQY because we operate with the experimentally determined PL and excitation intensities. Photon recycling is included indirectly in the theoretical models, because the rate constants of radiative recombination and Auger trapping can be seen as re-normalized constants corresponding to the light/energy propagation conditions for the particular sample, see Supplementary Note 10 for details.
We do not explicitly include charge diffusion in the model. This is rationalized by the fact that charge carrier diffusion in MAPbI3 is fast, with diffusion coefficients on the order of 1 cm 2 /s reported in literature. 5,6 Thus, assuming 1D diffusion toward the surface, initially inhomogeneous (exponentially distributed due to the excitation light attenuation in the film) charge distribution homogenizes over the 260 nm thickness of the film with the characteristic time t=<x 2 >/2D, t=(260×10 -7 cm) 2 /(2×1 cm 2 /s) = 3.4 10 -10 s = 0.34 ns. In our measurements and calculations, we are interested in the dynamics at timescales longer than 10 ns, that is why we can assume equilibrated homogeneous distribution of charge carriers over the entire thickness of the film.

The ABC model
Derivation of the ABC model from the SRH+ model In the limit of → ∞ the density of the trapped electrons equals to 0 as follows from Eq.(9.2). Thus ( ) = ( ), which is the condition for the ABC model. From Eq. (9.1) we get Or in another form where = , = + , and = . PL intensity can be found as

Pulse excitation experiment in the ABC model
In order to find PL intensity in the pulse excitation regime the following equation has to be solved: with the periodic boundary condition (periodic solution): where 0 is the carrier density generated by one pulse, is the time period between pulses.

Quasi-CW regime of the ABC model
As seen from the periodic boundary condition, the variation of the ( ) value during the period between the laser pulses is equal to 0 . Thus, if the condition ( ) ≫ 0 satisfies, we can set ( ) equal to its averaged value, which means that the system is in quasi-CW regime

( ) =̃
Substituting this expression to the right-hand side of Eq. (9.7) we get: Integrating Eq. (9.9) with respect to time from 0 to T we obtain: Introducing the averaged density of the photogenerated carriers per second = 0 , we get the following equation: =̃+̃2 +̃3 (9.10) Solving Eq. (9.10) at a given value we can get ̃ and then can calculate the PL quantum yield:

=̃2
(9.11) PL kinetics in the ABC model in the low excitation limit.
If the following inequalities apply ≪ and ≪ Eq. (9.7) can be rewritten as The PLQY can be found as:

Quasi-CW regime of the SRH+ model
As it is seen from the periodic boundary conditions, the variations of the values of ( ), ( ) and ( ) are not larger than 0 during the period between the laser pulses. However, in order for the system to be in a quasi-CW regime, it is not required for all these values to be much greater than 0 . As it is seen from Figure 3a the PL intensity could decay for more than an order of magnitude during the time period within a quasi-CW regime. It can be explained by a fast decay of the electron density As these equations shows, that PL during the period T between the laser pulses can decay to any level which is determined on the total decay rate , see the cartoon in Fig. 1 d,e,f in the main text.

SRH model
Kinetic equations of the SRH model can be obtained from Eqs. (9.12-9.20) by setting = 0 and = 0: These equations can be considered as low excitation approximation of SRH+ model when the following inequalities apply (negligible rates of Auger-assisted processes): 2 ≪ and ≪

Low excitation intensity in the SRH model when PLQY is low and there is no trap filling
Let's consider the SRH model when additional conditions are applied: 1) PLQY is low (radiative rate<<non-radiative radiative) ≪ (9.41) 2) There is no trap filling, which means that ≪ , (9.42) In general, conditions (9.41) and (9.42) correspond to low excitation conditions, however, before we solve the equations, we cannot write the condition for the generation rate G explicitly.
In this limit the equations (9.38-9.40) can be simplified:

Explicit conditions for generation rate G and their consequences
After solving the equations of the SRH model we can write explicitly the conditions (9.41) and (9.42). The first condition (9.41) is that PLQY is low is equivalent to: ≪ where The second condition (9.42) is absence of trap filling condition, which can be presented in the form: Thus, to observe the square root PLQY dependence on the excitation power density which was just discussed above the generation rate must be smaller than and and : ≪ and ≪ and ≪ In general behavior of the PLQY(W) upon creasing the power density depends on which of the values , or is the smallest, or, in other words, which of the conditions breaks first upon increasing of the generation rate (which of the limiting values , or is the smallest).

Standard square rood dependence with saturation
In the simplest case (straight line in the quasi-CW regime) the smallest is . This is also equivalent to ≥ , In this case the square root dependence turns to the saturation with ~1 at ≫ See the curve labeled SRH in Figure 4 c in the main text.

Reaching an ABC-like behavior at higher excitation in the SRH model
Let's consider the case, when is the smallest of , . This is equivalent to: See the series of curves labeled "kn increases" in Figure 4 d in the main text which shows how the square-root dependence changes to the linear dependence (ABC -like behavior).
We can use these considerations to estimate the value which is required to make ABC model valid for the excitation power density larger than min = 4 × 10 −6 W/cm 2 (the smallest value used in our experiments) and for 1 Sun power density sun = 0.1 W/cm 2 . The corresponding values are: min = min • ℎ ≈ 2.5 × 10 17 cm −3 s −1 sun ≈ 6.2 × 10 21 cm −3 s −1 Using the parameter = 5 × 10 6 s −1 found for G/P/MAPI/P sample we obtain for the first case

Chemical doping and its influence on PLQY in quasi-CW regime
Let us assume that there is chemical doping in the system. Doping can be n-doping (extra electrons) or p-doping (extra holes).
Thus, the charge conservation condition in the system of Eq. (9.48-9.50) has to be substituted by: where is a density of additional charges due to chemical doping. Density is positive for pdoped sample and negative for n-doped sample.
−̃̃= 0 (9.64) ̄−̃̃= 0 (9.65) ̄+̃+ =̃ (9.66) The system is equivalent to the following equation: That means that the qualitative behavior of the PLQY in the quasi-CW regime as a function of the renormalized excitation fluence ̂= / 2 depends only on the values , , , while the trap concentration is a scaling parameter. In other words, the shape of the PLQY(W) depends only on the rate constants, while changing of N moves the dependence (or the whole PLQY(f,P) map) along the excitation power axis.
We are going to present a detailed analysis of the possible PLQY(W) regimes in a forthcoming theoretical work.

Photon reabsorption in thick (>>λ) films
Photon reabsorption/recycling is considered as an important process influencing the charge dynamics. 7 In our experimental study we compare samples of very similar geometries ensuring the effects of photon reabsorption/recycling to be similar, such that they cannot serve as the reasons for the differences between PLQY(f,P) maps and PL decay kinetics. Moreover, since the thickness of the studied MAPbI3 films is only 260 nm, which is smaller than PL wavelength (760 nm), strictly speaking it is not possible to talk about photon reflection from film surfaces in terms of geometrical optics. Instead emission of a spatially-limited nano-scale media, the standard problem of nano-optics, 8 should be solved using explicit electrodynamic calculations. As we discuss below in detail, all influences on the charge carrier dynamics related to "photon recycling" in broad terms (both far field (photon reabsorption) and near field (energy transfer) effects), are included in our SRH+ model via "renormalized" radiative rate constant and the Auger trapping rate, respectively.
Photon re-absorption is a process which is widely discussed in relation to the photo-physics of MHP films and single crystals. It is generally assumed that a photon created inside a material (for example, a film), then propagates as a wave and reaches an interface with another dielectric media (e.g. air or glass). At this interface, the wave partially or fully (total internal reflection) reflects toward the bulk of the crystal. In this way the wave becomes partially trapped inside the film and photons get reabsorbed by the semiconductor again because of the long propagation length in the material. Note here, that all these considerations are based on the geometrical optics which is valid only if the size of the crystal (thickness of the film) is substantially larger than the wavelength of light λ. Figure 10.1. Photon re-absorption in a semiconductor when a photon generates a free electron and hole. This process is considered in the SRH+ model due to effective renormalization of the radiative recombination rate constant (it is lower due to re-absorption).

Supplementary
Let us consider a film with thickness >> λ. If the emitted photon generated by recombination of an eh pair is reabsorbed, it generates a new e-h pair (Supplementary Figure 10.1). Thus, density of the free electrons and holes remains unchanged. Photon reabsorption can be considered in the SRH model by replacing the internal radiative recombination rate constant by its renormalized value which is lower than the value of the case in which the reabsorption was absent: where is the probability of generation of a free electron and a hole by photon reabsorption. So, in the limit of the applicability of the geometrical optics, photon reabsorption decreases the effective radiative recombination rate. Because it needs to compete with NR recombination (unchanged), decreasing of due to re-absorption decreases PLQY of the sample.
Parasitic absorption without formation any charge carriers or optical excitation of an electron directly to the trap states are not considered due to very low absorption coefficients these processes in comparison with the absorption above the bandgap for our sampled prepared on glass.

Thin (<λ) perovskite thin films: energy transfer vs photon reabsorption
Consideration of processes such as the emission and absorption of photons inside an object with a size of the order of the wavelength seems to be very problematic. The formation of an electromagnetic field (photon) occurs in the near-field region, the size of which is several times larger than the wavelength. This means that one cannot say that a photon is emitted from a "point" inside the perovskite film of ~260 nm thickness (the thickness of our samples and a typical thickness for solar VB CB cells) and then the wave travels until the interface. In other words, the bulk of the perovskite film and its interfaces cannot be treated independently, instead, the problem of the emission of the whole layer must be solved explicitly using quantum electrodynamics. The solution of the problem will lead to a certain constant of radiative recombination which then has to be employed in the modelling of charge dynamics. Note that we in principle do not consider an "external" and "internal" PLQY herein. PLQY is inherent to the particular sample geometry and, if the characteristic dimensions of the sample are smaller than the wavelength, cannot be split into "internal" and "external". However, there is a process that occurs at the nanoscale (near field) which results in similar effects to those of photon reabsorption in the far-field -the Förster resonance energy transfer (FRET) (schematically shown in Supplementary Figure 10 As is shown in Supplementary Figure 10.2a, the energy released from recombination of an e-h pair is transferred over a few nanometer distance and creates another e-h pair. Obviously, this process does not affect the carrier density and does not change the radiative recombination rate and hence the PLQY. This process is "invisible" in the SRH model and can influence the charge dynamics only indirectly via increasing the overall charge diffusion. However, the energy can be transferred to a trap state directly as shown in Supplementary Figure 10.2b and c. The energy of a recombining e-h pair is used to excite an electron from the valence band to a trap. One can view this process as an Auger assisted trapping, 9 because it involves three particles: free electron and hole near the band edges and a deep electron form the valence band. Since this process results in a trapped electron, it will eventually lead to NR recombination. Thus, energy transfer and Auger trapping have the identical dependencies on the charge carrier concentrations and can be both included to the SRH theory as a third order trapping term:  Figure 12.2) we estimate value using Eq. (9.17).

2.
Repetition rate dependence of PLQY at high pulse fluence (P5) is fitted by the theoretical dependence obtained numerically by solving Eqs. (9.7-9.8). The fitting parameters are , and .
The fitting procedure for SRH+ model 1.
We estimate product value by fitting the decay curves at low intensities with an exponential function using Eqs.

3.
Power density dependence of PLQY in Quasi-CW regime is fitted by the theoretical dependence obtained numerically by solving Eqs. (9.32-9.36). The fitting parameters are , , , and .

3) SRH.
To plot the results of the SRH model we use the parameters obtained for the SRH+ model. In other words, the fit by the SRH model is obtained using the best fit parameters from the SRH+ with kA and kE set to zero. That is why the predictions of the SRH and SRH+ are identical at the low power regime.