Charge Trapping and Defect Dynamics as Origin of Memory Effects in Metal Halide Perovskite Memlumors

Large language models for artificial intelligence applications require energy-efficient computing. Neuromorphic photonics has the potential to reach significantly lower energy consumption in comparison with classical electronics. A recently proposed memlumor device uses photoluminescence output that carries information about its excitation history via the excited state dynamics of the material. Solution-processed metal halide perovskites can be used as efficient memlumors. We show that trapping of photogenerated charge carriers modulated by photoinduced dynamics of the trapping states themselves explains the memory response of perovskite memlumors on time scales from nanoseconds to minutes. The memlumor concept shifts the paradigm of the detrimental role of charge traps and their dynamics in metal halide perovskite semiconductors by enabling new applications based on these trap states. The appropriate control of defect dynamics in perovskites allows these materials to enter the field of energy-efficient photonic neuromorphic computing, which we illustrate by proposing several possible realizations of such systems.


Preparation of perovskite precursor solution:
CsBr (62 mg) and PbBr2 (110 mg) were mixed and dissolved in DMSO (1 mL) by shaking without heating to obtain a clear 0.3M solution.The solution was filtered through a 0.45 μm PTFE syringe adapter right before the procedure of perovskite film deposition.

CsPbBr3 polycrystalline film preparation:
The glass substrates (15 x 15 mm 2 ) were cleaned by sonication in NaHCO3 solution, deionized water, acetone, and 2-propanol for 10 min consecutively, and then exposed to UV-generated ozone for 15 minutes to obtain a hydrophilic surface.Afterwards, substrates were transferred in the dry glovebox filled with N2 gas.The deposition of perovskite films was conducted on the substrates by single-step spin-coating method at 3000 rpm for 5 minutes.Thereafter, the samples were gradually annealed on a hot plate from 50 o C up to 130 o C for 15 min to remove dimethyl sulfoxide residues and complete the crystallization of perovskite.

CsPbBr3 microwires preparation:
For the CsPbBr3 microwires (MWs) synthesis, the temperature difference-triggered growth method was used. 1 For this, a furnace (PZ 28-3TD High-Temperature Titanium Hotplate and Program Regler PR5-3T) was employed to control the temperature during the MWs growth.The CsPbBr3 perovskite material was sublimated from a source substrate to a target substrate.As a source substrate, the CsPbBr3 polycrystalline film (prepared by the method described above) was used.The target sapphire substrate (10 x 10 mm 2 ) was cleaned by sonication in deionized water, acetone, and 2-propanol for 10 min consecutively.Two substrates were separated with an air gap of 0.5 cm.The temperature of both substrates was controlled by the furnace temperature.The synthesis was initiated at a furnace temperature of 350 • C. Afterwards, the temperature was increased up to 520 • C for 10 min and kept unchanged for 10 min.As a result, the sublimated CsPbBr3 microwires were grown in three crystallographic directions of the sapphite.

CsPbBr3 NPs on GaP nanowaveguides preparation:
Cs2CO3 (0.407 g) was loaded into a 100 mL flask along with octadecene (20 mL) and oleic acid (OA, 1.25 mL), dried for 1 h at 120 °C, and then heated up to 150 °C to give a clear CsOA 0.125M solution.All manipulations were conducted in a N2-filled glove box.Perovskite nanoparticles (NPs) were synthesized using a Schlenk line by modified hot-injection method.PbBr2 (35 mg) was added to a 50 ml two-neck flask and dried in DE solvent at 120 o C under vacuum for 40 min.Then, OA (150 μL) and OLAm (150 μL) were added dropwise to it.Thereafter, the flask was purged with N2 and the mixture was heated up to 150 o C to obtain a clear solution.The solution was cooled down to 88 o C before the injection of preheated at 120 o C CsOA solution (0.5 mL) followed by incubation for 2 h at 88 o C. Afterwards, the solution was heated up to 150 o C, incubated for 30 min, and, finally, quenched by using an ice bath.The product was centrifuged at 3000 rpm for 5 min, separated from supernatant solution, and redispersed in 10 mL of n-hexane.Large particles were settled down in 5 min and upper fraction of the solution (3 mL) was pipetted for further manipulations.This fraction contains a wide dispersion of NPs with the mean size of about 360 nm according to dynamic laser scattering measurements.
GaP NWs were grown on a Si substrate by molecular beam epitaxy using a protocol reported by Trofimov et al. 2 NWs were transferred into suspension by ultrasonication of 0.5x0.5 cm substrate in 1 mL of 2-propanol, drop-casted onto a glass substrate, and rinsed with hot acetone two times.Then, the solution containing CsPbBr3 NPs was drop-casted over GaP NWs to give nanowaveguides decorated with subwavelength light emitters.

MAPbI3 and MAPbI2.96 films:
MAI and Pb(Ac)2•3(H2O) (at 3:1 or 2.96:1molar ratio) were dissolved in anhydrous N,Ndimethylformamide (DMF) with a concentration of 40 wt% with the addition of hypophosphorous acid solution (6 µL / 1 mL DMF).The perovskite solution was spin coated at 2000 rpm for 60 s in a drybox (RH < 0.5 %).After spin coating, the samples were dried for 20 seconds by a stream of dry air.Afterwards, the samples were kept at room temperature for 5 min and subsequently annealed at 100 °C for 5 min.

Triple cation perovskite films:
The perovskite precursor solution was prepared by dissolving PbI2 and PbBr2 in a solvent mixture (DMF/DMSO = 4/1) and CsI in DMSO at 180 °C for 10 minutes.After cooling down, CsI, PbI2 and PbBr2 solutions were mixed in a volume ratio of 0.05:0.85:0.15, to obtain an inorganic stock solution.MAI and FAI powders in separate vials were then dissolved to form stock solutions.Finally, the MAI-stock-solution was mixed with the FAI-stock-solution in a 1:5 volume ratio to acquire the final 1.2 M Cs0.05(FA5/6MA1/6)0.95Pb(I0.9Br0.1)3perovskite solution in DMF and DMSO.The perovskite layer was deposited via a two-step spin-coating procedure with 1000 rpm for 12 s and 5000 rpm for 28 s.Before spinning, 40 μL of perovskite precursor solution were applied to the sample statically and 150 μL of Trifluorotoluene (TFT) was dripped on the spinning substrate, 5 s before the end of the second spin-coating step in a rapid fashion, in agreement with Taylor et al. 3 Subsequently, the samples were annealed at 100°C for 30 min.

CsPbI3 QDs perovskite films:
The synthesis of the perovskite QDs was done via the hot injection method, firstly published by Protesescu et al., with several adjustments. 4Before the synthesis, oleic acid (OA, technical grade 90%, Sigma Aldrich) and oleylamine (OLA, technical grade 70 %, Sigma Aldrich) were degassed at 100 °C for 1 h to guarantee high purity of the reactants.Cs-oleate was produced by combining Cs-carbonate with OA.For the preparation of Cs-oleate solution, 0.407 g of Cs2CO3 (TCI, >98 %), 20 mL of octadecene (ODE, technical grade 90 %, Acros Organics), and 1.25 mL of OA were loaded in a 2-neck round-bottom flask and degassed for 1 h at 100°C in vacuum.Thereafter, the flask was filled with nitrogen and heated to 150 °C until all reactants reacted and a clear solution of Cs-oleate was obtained.The Cs-oleate was then stored in nitrogen at 70 °C until usage.For the synthesis of the CsPbI3, 1 g of PbI2 (99.99 %, TCI) and 60 ml of ODE were filled in a 2-neck round-bottom flask and degassed for 1 h at 120 °C in a vacuum.Subsequently, the flask was filled with nitrogen and 6 mL of OLA and 6 ml of OA were mixed in a vial and then injected.The flask was again pumped to vacuum for 30 min, until a yellow transparent solution was obtained.Then, the flask was filled with nitrogen and heated to 170 °C.At the target temperature, 4 mL of Cs-oleate was quickly injected into the flask.The solution turned dark red and after 5 s the reaction was quenched with an ice-water bath.For the purification of the as-prepared CsPbI3 nanocrystals, 12.5 mL of the crude solution was mixed with 37.5 mL of Methyl acetate (MeOAc, 99 %, Acros Organics) and centrifuged for 10 min at 6000 rpm.The supernatant was discarded and the wet CsPbI3 pellets were redispersed in 3 mL of hexane (97%, Acros Organics).The solution was again mixed with 5mL of MeOAc and centrifuged for 10 min at 6000 rpm.The supernatant was removed, and the precipitates of all tubes were combined in one and dispersed in 25 mL of hexane.This solution was centrifuged for 5 min at 4000 rpm and this time the supernatant was collected and stored overnight at 4 °C.After that, the solution was centrifuged again for 5 min at 4000 rpm.Finally, the supernatant was collected and dried by using a rotary evaporator.The obtained CsPbI3 nanocrystals were dispersed in octane (99%, Acros Organics) at a concentration of 75 mg/mL for further use.
Then, for fabrication of CsPbI3 quantum dot samples we followed the procedure reported previously 5 with slight modifications.Glass substrates were cleaned thoroughly by sonicating them in acetone and isopropanol for 15 min respectively.To activate the surface and remove organic remains the samples were subjected to oxygen plasma for 10 min.After that, they were transferred to a nitrogen-filled glovebox for the deposition of the perovskite quantum dots.The quantum dot solution that was obtained from the synthesis was then spin-coated dynamically on the glass substrates at a speed of 1000 rpm for the first 5 s and 2000 rpm for the following 10 s.To exchange the long organic ligands, the film was in immersed in a ligand solution of sodium acetate (NaOAc, 99.995 %, Sigma Aldrich) in MeOAc for 5 s and was spin-dried, followed by soaking and spin-drying the film in MeOAc for 5 s twice.This procedure was repeated four times to obtain a layer of approximately 250 nm.After that the sample was soaked in a solution of phenethylammonium iodide (PEAI, Great Cell Solar) in ethyl acetate for 10 s and spin-dryed.The PEAI treated sample was then washed with ethyl acetate.

PMMA layer deposition:
The PMMA coating of all different types of perovskite films was performed by preparing a solution of polymethyl methacrylate (PMMA, Greatcell Solar) in chlorobenzene (10 mg/ml) that was then spin-coated at 3000 rpm for 30 s on top of each of the layers.

Light Excitation Conditions
Excitation conditions and values of the induced charge carrier densities were the same as described in our recent publication. 6,8e initial value of the photogenerated charge carrier density n0 can be estimated for each of the pulse fluence values   as  0 =   •  , where d is the film thickness and Abs in the film absorption. = (1 − )(1 − ) ( 1 −  − ), where scattering S<<1, R is reflectance and  is the absorption coefficient of the material.In the simplest case R = (−1) 2 (+1) 2 , where n is refractive index of a material at a corresponding wavelength.
For the CsPbBr3 polycrystalline thin film we consider  ≈ 2.3 ,  ≈ 0.45 • 10 5  −1 at a 485nm wavelength. 9,10Thus, for film thickness of 90 nm, Abs=0.28.Therefore,  0 for each of the pulse fluence   is equal to: The calculated values for n0 are listed in Supplementary Table 1 below.

Measurements of time-resolved PL under pulsed burst excitation using TCSPC
In our experiments we used a standard time-correlated single photon counting (TCSPC) setup in a non-standard regime.The setup is based on a picosecond diode laser, multichannel picosecond diode laser driver SEPIA 828 (PicoQuant) and time counting device PicoHarp 300 (PicoQuant).
In the standard TCSPC scheme, the sample is excited by a pulsed laser with a fixed repetition rate.For example, if the repetition rate is 100 kHz, sample is excited each T=10 µs.For each period T, a trigger signal from the laser driver arrives to the first channel (start pulse) of the time TCSPC device.After detecting this signal, the device measures the time until the stop pulse arrives to its second channel from the photodetector (one detected photon generates an electrical pulse).The histogram of the delays between the start and the stop pulses builds over time and shows the shape of the PL decay. 11 Supplementary Fig. 2. TCSPC technique with excitations by a burst of laser pulses.Contrary to this standard regime, the memlumors are excited by a burst of several closely separated laser pulses and this burst is repeated with a repetition period T. Such pulse timepattern is created by a Sepia 828 oscillator triggering the laser diode.For example, the experiment shown in Fig. 2a was carried out using the following time sequence of the pulses per one period T (see also Supplementary Fig. 2): The basic frequency was set to 20 MHz resulting in a 50 ns separation between the pulses.The entire pulse sequence is the following: -The 1 st pulse creates the start signal for the TCSPC electronics; -The next 40 pulses are sent to the laser driver and produce the burst of 40 laser pulses which excites the sample; -The next 10 pulses are skipped creating a delay of 550 ns; -The next single pulse is sent to the laser driver and produces one laser pulse which also excites the sample; -The next 8000 pulses are skipped to create a large delay.
The entire sequence is repeated over 100 s to accumulate the PL response curve.
We would like to stress several important practical issues concerning the possible photon count rates in such an excitation scheme.Older generations of TCSPC electronics operated in such a way that after detection of a stop signal the system was waiting until the next start pulse to start the counting again.Thus, not more than just one stop pulse could be detected.As a consequence, in order to obtain a histogram which would reflect the real PL decay curve without disturbance by the pile-up effect, one needed to limit the number of stop pulses per start to a value that was much less than one.The practical "rule of thumb" was to set the probability to detect a stop photon less than 5% per an excitation pulse.Thus, the photon counting rate was < 5 % of the repetition rate of the laser pulses.
If we would follow this rule with the burst excitation scheme discussed above, the maximum count rate of the signal should be limited to 5%×2488 Hz= 124 counts per second only.Note that the dark counts of the detector (PMA Hybrid detector, PicoQuant) is approximately 200 s - 1 .Thus, with this limited detection count rate it is not possible to obtain a curve with a high signal to noise ratio for low excitation intensities, such as the one presented in Supplementary Fig. 2.However, fortunately, this rule is not applicable to the experimental conditions applied to the memlumors.As we will explain in the following, a much higher signal count rates can be used.This is possible because the current generations of TCSPC systems are able to count the time difference relatively to the start pulse for many detection events per repetition period as long as the photons do not arrive during the detection system dead time (each photon detection event locks the system for the deadtime).So, the pile-up effect in its classical sense is only present when the repetition period of the laser pulses or the fluorescence decay itself is shorter than the deadtime.For example, for our system (PicoHarp 300) the deadtime is approximately 100 ns.Based on this, only for the laser repetition rates larger than 1 MHz (time gap to the next pulse is < 100 ns) does the 5% rule apply.
In the experiments with memlumors, the repetition rate is as low as 2.5 kHz (400 µs repetition period).At the same time, the PL signal is spread over the time window of approximately 3 µs.These times are substantially larger than the detection deadtime (100 ns).Therefore, several photons can be detected per repetition period without any pile-up effect.This leads to a count rate of several kHz, which is much larger than the noise counts.The maximum safe count rate depends on the shape of the PL response curve.In practice, test experiments should be performed by increasing the count rate to identify the point when a distortion of the PL response due to the pile up effect becomes visible.Then a rate of at least 10 times smaller should be used.

PLQY(f,P) mapping
For the PLQY(f,P) mapping experiments 7,12 the sample is excited by a pulsed laser at repetition rate f [s -1 ] and pulse fluence Pi [photons/cm 2 ] which are controlled by the laser driver and a neutral density filter wheel, respectively.The sample PL intensity is measured for each combination of f and P. We used a CCD camera to measure the PL integrated over 30 µm laser excitation spot.Because the entire system is calibrated, the PL intensity can be converted to an external PLQY. 7,12To acquire the PLQY(f,P) map, PL is measured for a laser excitation spanning over 4 orders of magnitude (from ca. 10 8 to 10 12 photons/cm 2 /pulse) in 4 steps with power fluences Pi -P1, P2, P3, P4 and P5 (each step changes the fluence approximately 10 times, see Supplementary Table 1) and almost 7 orders of magnitude in pulse repetition rate, i.e., from 10 Hz to 80 MHz (Supplementary Table 2, Supplementary Table 3).For each pulse fluence P, the repetition rate, f, is scanned across the entire range.Usually, a complete map consists of 50-150 data points.
All the data points of the PLQY(f, P) map are acquired automatically because the setup is fully controlled by home-developed LabVIEW program.It executes the experiment according to a pre-loaded table of parameters for the data acquisition (f, P, shutter timing, acquisition time of the camera, filters, etc.).To minimize the sample exposure, the shutter is synchronized with camera acquisition to allow the laser beam to irradiate the sample during the PL acquisition only.The program automatically saves the PL images, which are later processed using another program to yield the PLQY(f,P) map.This ensures that the data acquisition conditions are fully reproducible to be able to repeat exactly the same experiment with another sample.This is of curial importance for light-sensitive materials like metal halide perovskites and memlumors in general.The complete measurement takes from 1 to 3 hours, where the longest time is required to acquire data for low P and low f values since exposure times as long as several minutes per data point are often essential.combination of f and P) is larger than 1 ms.So, PLQY(f,P) mapping makes it possible to obtain the components of the state vector  ⃗ ⃗ = (  ,   ,   ) assuming that these components are not affected by the experimental procedure, which mathematically means that: However, this assumption contradicts the knowledge of CsPbBr3 as a photosensitive material, where the parameters of the defect states are light sensitive.In other words, it is known that the experiment itself may influence the result of the measurement (the so-called observer effect, see details in reference 12 ).Considering that the PLQY(f,P) measurements involve both very low and very high excitation power densities, extra care should be taken so that the results are not affected by a constant evolution of the material under light irradiation (excitation history).This means it is important to consider that the same data point measured with different irradiation time (exposure time) might result in a different value of PLQY.
Supplementary Fig. 3a shows the PLQY(f,P) map obtained for the CsPbBr3 film studied here using the standard experimental protocol used previously for these type of measurements. 7The parameters used to measure each experimental point are shown in Supplementary Table 2.A simple visual inspection of the PLQY(f,P) plot reveals the presence of a significant sample instability. 12This is evidenced by the appearance of a non-constant PLQY where the single pulse excitation regime is expected and the absence of a common quasi-CW regime for all the pulse fluences (see Supplementary Note 5 and references 7,12 as well as the labels in Supplementary Fig. 3a).
To understand this, it is helpful to roughly split the entire PLQY(f,P) map into two regions: Region 1, in which the averaged excitation power density is mild or low and Region 2, in which the averaged power density is high.Supplementary Table 2 lists the experimental parameters for the measurements.The vector  ⃗ ⃗ = (  ,   ,   ) is likely to change in Region 2, however the impact of this change is more clearly observed in Region 1, most likely due to the fact that non-radiative recombination is dominant in this region.Hence, to track the change in the state of the memlumor during the measurements, a reference point is measured at a fluence P2 at 80 MHz (inside Region 1) after each point of the PLQY (f,P) map.
The PL intensity at this reference point should be constant if the sample is stable, however, this is clearly not the case as can be clearly seen in Supplementary Fig. 3b.The first and the last value of the reference point varies by two orders of magnitude.The evolution of the reference point (Supplementary Fig. 3b) shows that entering Region 2 influences the sample significantly.Hence the values of the  ⃗ ⃗ vector for Region 1 and Region 2 must be very different, and as a result, the PLQY(f,P) map is affected by the sample scanning history and cannot be fitted correctly because equation S3.1 does not hold ( (,) ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗  ≠ 0).Indeed, the shape of the PLQY (f,P) map contains features that are not compatible with any charge recombination model with constant parameters (see further explanation in Supplementary Fig. 3a).

SI page 13
Supplementary Table 2.The conditions for acquiring of the experimental points listed in the measurement order (from 1 to 125) in the experiment shown in Supplementary Fig. 3.The two regions of low and high intensity are separated.The rows showing the reference points are filled with the white color to separate them from the rows (filled by light green) showing the actual points of the PLQY(f, P) map.The font color highlights the fluences P1-P5 according to the color scheme used in Supplementary Fig. 3a and Supplementary Table 1.SI page 14 Supplementary Table 3.The experimental points located in the order of their acquisition (from 1 to 42) in the experiment shown in Supplementary Fig. 4a, b.The two regions of low and high intensity are separated.The rows showing the reference points are filled with white color to separate them from the rows (filled with light green) showing the actual points of the PLQY(f, P) map.The font color highlights the fluences P1-P5 according to the standard color scheme used in Supplementary Fig. 4a, b and Supplementary Table 1. is prone to easily change into another state  ⃗ ⃗ i during the measurements even when taking significant precautions.
It was found that by exposing the sample to the 160 W/cm 2 for 20 s, it was possible to induce a state (designated as  ⃗ ⃗ 1) which is much more stable as the initial state of the sample.Several experiments shown in the main text (Fig. 2a, 3d, 3e) were carried out for samples prepared in this way.The PLQY(f,P) plot corresponding to this state is shown in Supplementary Fig. 4b.
In such a state, the reference point was stable and the PLQY(f,P) map did not exhibit any features that could not be well described by the SRH+ model.This data was utilized for modelling using the SRH+ model to extract the model parameters (see Supplementary Note 5.6 and Supplementary Fig. 10).Note, however, that the parameters extracted from this data cannot be completely assigned to the  ⃗ ⃗ 1 state.The PLQY(f,P) mapping experiment lasts about 1 h and some relaxation processes of the sample state during this time cannot be excluded.
where, g(t) is the density of charge carriers generated by the excitation light per second, kr is the radiative recombination rate constant, kt is the electron trapping rate constant, Nt is the electron trap density and, kn is the non-radiative recombination rate constant.The traps are considered to be deep enough to make de-trapping of electrons negligible.
The generation rate g(t) [excitations/(cm 3 s)] is a function of the excitation intensity I(t) [W/cm 2 ].In the simplest case considered here 7 we assume that the sample is rather thin and the charge carriers redistribute very quickly after their generation over the thickness d of the sample.In this case () =  () ( ℎ) ⁄ =  () , where d -sample thickness and ℎexcitation photon energy, and Abs -absorption coefficient of the sample (see Supplementary Note 2.2) and  =  (ℎ) ⁄ (Supplementary Fig. 6a).
Since an intrinsic semiconductor is charge neutral, the following condition takes place: () +   () = () S5.4 In the presence of chemical doping the condition for charge neutrality becomes () +   () +  = () Where cd is the doping density, cd should be negative in the case of n-doping or positive in the case of p-doping.
The PL intensity in this model is given by () =   ()() S5.5 Trapping of electrons by the trap states leads to an excess of holes in the valence band and results in the effect commonly referred to as photodoping.
In the equations above electron trapping was assumed.However, the same equations also apply for hole trapping if one replaces n with p and vice versa.Consequently, neither the model nor this type of experiment make it possible to distinguish between photodoping induced by electron trapping or by hole trapping.

Extended Shockley-Read-Hall model (SRH+) with added Auger processes.
The model described in the section above is valid at low excitation conditions only, such that the third order Auger-assisted processes can be neglected.To explain the carrier recombination at high excitation conditions, the so-called SRH+ model 7 is applied.This model includes Auger recombination and Auger-assisted charge trapping (Supplementary Fig. 7).
The kinetic equations of the SRH+ model can be written as follows: Here, ke, and ka are the Auger-assisted electron trapping rate constant and, Auger-assisted recombination coefficient, respectively.
In this work, the SRH+ model was used to fit the PLQY (f, P) data and the PL decays (Supplementary Fig. 10).These data sets were obtained for a very broad range of excitation conditions making it necessary to consider the third order processes.Modeling of this experimental data made it possible to obtain all the parameters of the model (all rates and trap concentration) (Supplementary Table 4).

Pulsed photoexcitation in the framework of the SRH+ model
In a typical experiment, the PL is excited by a very short laser pulse at time t=0.Assuming that the pulse width is much shorter than all relaxation times in the system, the generation g(t) can be approximated as a -function centered around t=0 creating the initial concentration of electron-hole pairs n0.However, it is necessary to consider that practically no experiments are carried out with just one excitation pulse, instead the excitation pulse is repeated with a certain repetition period T. So, it is important to consider that the excitations created by the previous pulse may not have yet decayed by the moment of the next pulse.Under such circumstances, the charge carrier kinetic equations can be expressed as follows (see more details in SI to ref the time-averaged power density (W/cm 2 ), which is proportional to the product f×P.This regime is called quasi-CW excitation regime. 7en the SRH+ model is used to calculate the parameters which can be compared with experimental data, e.g. with PL(t) measured by the TCSPC method, it is important to realize that such experimental data are measured in the quasi-steady state condition in the sense that the sample is repeatedly excited with laser pulses.Because the accumulation time of the PL decay (tens of seconds) is much larger than the time required for PL to reach the quasi equilibrium, the TCSPC measures an equilibrated response of the system to the pulsed excitation.
If the excitation is the single-pulse regime, this equilibrated response is the same as the response to one single pulse.In other cases, one needs to apply multiple pulses until the system comes to an equilibrium.Experimentally, this is always the case because the accumulation time for PL decays is usually tens of seconds which is several orders of magnitude longer than the time required to reach the equilibrium (Supplementary Fig. 8).To reproduce the experimental conditions in the calculations, it is necessary to sequentially calculate as many repeated pulses as needed to establish the quasi-steady state values of the electron and hole densities.This is illustrated in Supplementary Fig. 8 where one can see that the time-averaged concentrations of electrons (n) trapped electrons (nt) and free holes (p) are increasing steadily after each excitation pulse until they reach certain equilibrated values.At this quasi-equilibrium the concentrations still go up and down in response to the laser pulses, however, these dependencies are stable and fully periodic.When the model reaches this quasi-equilibrium condition, PLQY and PL decays are calculated and used to fit the PLQY(f,P) map (see Supplementary Note 5.5 and 5.6) and the PL lifetime data.
Taking this into consideration, we must conclude that the time  ′ in equation S5.12 (the time from which the integration commences) should be larger than the time required for the system to reach the quasi-steady state condition.For calculations of the PL response to bursts of excitation pulses, the bursts were repeated several times with the needed repetition period until the quasi-steady state response is reached.

Solution of the model equations of SRH+ radiative model in the special case of low excitation fluence
An analytical solution of the equations of the SRH+ radiative model is possible at the limiting case of very low excitation density.In this case, the radiative recombination rate becomes negligible compared to the charge trapping rate.In addition, it is assumed that nt is significantly smaller than the trap density Nt (   ≪        ≪   ).In this case, equations S5.1 -S5.3 can be approximated as: Then, the periodic solution of S5.13 can be given as: Supplementary Fig. 12 shows that among Nt, kt and kn parameters the largest effect on M is due to kn.Decreasing of kn makes the trapped electrons live longer and thus increasing M (25 times decrease kn increases M by 10 times).Changing of the trapping rate kt does not influence M almost at all.Dependence on Nt is also rather weak.Changing of Nt by 25 times leads to only 2 times change in M. Dependence of M on Nt over a broader range is shown in Supplementary Fig. 13.
Supplementary Fig. 12. Memory strength (M) dependence on the state vector  ⃗ ⃗  parameters.400 pulses are used as a Write burst with 80 MHz internal repetition rate (12.5 ns between pulses).The Read pulse comes after 1 microsecond delay.The signal integration time t1 (see Supplementary Fig. 11) is equal to the arrival time of the Read pulse, while the length of integration t2-t1 equals to 10 microseconds.For all panels the pulse fluence P2 is used (see Supplementary Table 1).Panels a-g show the PL response calculated for different state vector  ⃗ ⃗ .a)  ⃗ ⃗ =  ⃗ ⃗  = ( 1 ,  1 ,  1 ) (Supplementary Table 5), this is the condition of our experiments.The other panels show the situation where one of the parameters deviates from the reference condition, namely: b)   =  1 /5, c)   = 5 1 , d)   =  1 /5, e)   = 5 1 , f)   =  1 /5 and g)   = 5 1 . Photoluminescence

. 3 .
a) Standard 8 measurement of the PLQY(f,P) map for CsPbBr3 polycrystalline film.Two regions are separated: Region 1, where the input power density is low, and Region 2 where the input power density is high (> 0.05 W/cm 2 ).The map exhibits very strong artifacts due to sample instability (see notes in the plot).b) Evolution of the reference point.

Table 1 .
Values of the five different pulse fluences P1-P5 mainly used in the experiments, the corresponding charge carrier densities  0 , pulse energy densities, and average power densities.The color scheme shown in the table is used in Fig.1bof the paper.