Interpretation of the photoluminescence decay kinetics in metal halide perovskite nanocrystals and thin polycrystalline films

Journal Pre-proof Interpretation of the photoluminescence decay kinetics in metal halide perovskite nanocrystals and thin polycrystalline films Vladimir S. Chirvony, Kairolla S. Sekerbayev, Hamid Pashaei Adl, Isaac Suárez, Yerzhan T. Taurbayev, Andrés F. Gualdrón-Reyes, Iván Mora-Seró, Juan P. Martínez-Pastor PII:

S0022-2313(19)31402-4

DOI:

https://doi.org/10.1016/j.jlumin.2020.117092

Reference:

LUMIN 117092

To appear in:

Journal of Luminescence

Received Date: 16 July 2019 Revised Date:

10 December 2019

Accepted Date: 2 February 2020

Please cite this article as: V.S. Chirvony, K.S. Sekerbayev, H.P. Adl, I. Suárez, Y.T. Taurbayev, André.F. Gualdrón-Reyes, Ivá. Mora-Seró, J.P. Martínez-Pastor, Interpretation of the photoluminescence decay kinetics in metal halide perovskite nanocrystals and thin polycrystalline films, Journal of Luminescence (2020), doi: https://doi.org/10.1016/j.jlumin.2020.117092. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.

Interpretation of the Photoluminescence Decay Kinetics in Metal Halide Perovskite Nanocrystals and Thin Polycrystalline Films

Vladimir S. Chirvony,1* Kairolla S. Sekerbayev,2 Hamid Pashaei Adl,1 Isaac Suárez,1,3 Yerzhan T. Taurbayev,2 Andrés F. Gualdrón-Reyes,4 Iván Mora-Seró,4 Juan P. Martínez-Pastor1 1

UMDO (Unidad de Materiales y Dispositivos Optoelectrónicos), Instituto de Ciencia de los Materiales, Universidad de Valencia, Valencia 46071, Spain;2Institute of Experimental and Theoretical Physics, Al-Farabi Kazakh National University, Almaty, 050040 Kazakhstan; 3ETSI Telecomunicación, Universidad Rey Juan Carlos, C/Camino del Molino s/n, 28943 Fuenlabrada, Spain; 4Institute of Advanced Materials (INAM), Universitat Jaume I, Castelló 12006, Spain

1

Abstract In this paper we present critical analysis of different points of view on interpretation of the photoluminescence (PL) decay kinetics in lead halide perovskites prepared in the form of well passivated nanocrystals (PNCs) or thin polycrystalline layers. In addition to the literature data, our own measurements are also considered. For PNCs, a strong dependence of the PL lifetimes on the type of passivating ligand was observed with a consistently high PL quantum yield. It is shown that such ligand effects, as well as a decrease in the PL lifetime with decreasing temperature, are well qualitatively explained by the phenomenological model of thermally activated delayed luminescence, in which the extension of the PL decay time with temperature occurs due to the population of lower-lying shallow non-quenching traps. In the case of thin perovskite layers, we conclude that the PL kinetics under sufficiently low excitation intensity is determined by the excitation quenching on the layer surfaces. We demonstrate that a large variety of possible PL decay kinetics for thin polycrystalline perovskite films can be modelled by means of one-dimensional diffusion equation with use of the diffusion coefficient D and surface recombination velocity S as parameters and conclude that long-lived PL kinetics are formed in case of low D and/or S values.

2

Introduction Metal halide perovskites (hereinafter perovskites) in all their forms (nanocrystals, thin films of microcrystals, single crystals) demonstrate a large variety of photoluminescence (PL) lifetimes. In particular, in case of perovskite nanocrystals (PNCs) of the APbX3 cubic structure (where A is an organic cation, such as methylamine (MA) or formamidinium (FA), or an alkali cation; X is the halide) the experimentally observed PL decay times span over more than 5 orders of magnitude for different X at different temperatures, from tens of picoseconds [1] to more than one microsecond [2]. Even for formally identical PNCs, such as MAPbBr3, which differ only by surface passivating organic ligands, PL lifetimes are known to differ more than two orders of magnitude. At the same time, such kinds of identical PNCs of MAPbBr3 possess comparable and close to unity PL quantum yield (PLQY), which means that this parameter does not correlate with PL lifetimes [2-4]. A wide range of PL lifetimes (from few nanoseconds [5, 6] to several microseconds [7]) can be also found in literature for formally identical perovskite polycrystalline thin films. The abundance of experimental results in the field of perovskite photophysics not only does not help to understand how the PL kinetics is formed in this class of compounds, but rather complicates a realistic interpretation due to contradictory conclusions made by different authors. In this work, we describe several practical approaches, which can be used for interpretation and modeling of the PL decay kinetics for such perovskite structures as (i) nanocrystals with well passivated surface, and (ii) thin polycrystalline films. As far as perovskite single crystals is concerned, they will not be considered here, because their PL transients are strongly influenced by the PL reabsorption, which depends in turn on the excitation/detection experimental conditions [8, 9] that makes difficult to obtain reproducible measurements and the modeling of PL transients.

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Mechanisms of the PL transient formation in APbX3 PNCs Before starting a consideration of the physical mechanisms that form the kinetics of PL decay in PNCs, we should mention the role of optical effects, such as the photon recycling well known for perovskite layers, which involves multiple re-absorption and re-emission of the PL photons within the sample. Layers of PNCs on glass (or similar substrate materials with low refractive index) are especially prone to exhibit such a behavior due to the waveguiding effect, when, due to internal reflection, the main part of the emitted PL is directed along the high refractive index layer of a perovskite that greatly increases reabsorption. This effect leads to both a distortion of the PL spectrum and to some lengthening of the observed PL kinetics, which is less documented [10]. In case of PNC films and suspensions with high PNC concentrations the reabsorption effect is also well recognized by a red shift of their PL spectra [11]. Therefore, sufficiently thin PNC films and low concentration suspensions should be used to avoid artificial lengthening of the PL transients due to photon recycling effect. In the ideal case, it would be desirable to perform time resolved PL (TRPL) measurements at the single nanocrystal level [12]. The most extensive studies of TRPL have so far been performed for MAPbBr3 PNCs (see, for example, [2-4, 13-16]). The PL transients measured in these works can be mainly divided into two groups: (A) kinetics with short average lifetimes of the order of 10-20 ns [3, 4, 13], and (B) kinetics with long average lifetimes of the order of about 500 -1000 ns [2, 14, 15]. Note that in all these cases the PLQY was quite high, about 0.7-1.0. The chemical differences between MAPbBr3 PNCs were that an alkylamine (oleylamine or n-octylamine) together with oleic acid were used as passivating ligands in the group (A), whereas ammonium bromide salts were used instead of alkylamines in case of the group (B), the salts include octylammonium 4

bromide (together with oleic acid) [2], 2-adamantylammonium bromide (without oleic acid) [14], octylammonium bromide and octadecylammonium bromide (without oleic acid) [15]. Note also that long PL lifetimes of the order of 100 ns were reported in [16], where octadecylammonium bromide with oleic acid were used as passivating ligands, but the low PLQY (~0.20) does not allow a quantitative comparison of this case with the data obtained in [2-4, 13-15]. Thus, a first reading conclusion would be that more than one order of magnitude differing PL lifetimes in MAPbBr3 PNCs are due to different interactions of the active groups of passivating ligands, alkylamines on the one hand and ammonium bromide salts on the other hand, with the nanocrystal surface. We should also stress that PL lifetimes in these two PNC groups do not correlate with PLQYs, because they are all close to 1.0 for both types of ligands. A few TRPL studies were carried out on FAPbBr3 PNCs, but available data definitely suggest that also in this case PL lifetimes would depend on the type of passivating ligands. With a high PLQY for all used ligands, the PL lifetimes of FAPbBr3 PNCs are about 50-55 ns for oleic acid/octylamine and oleic acid/hexylamine passivating ligands [17-19] and 80 ns for oleic acid/butylamine pair [18]. In the case of CsPbX3 NCs (X = Cl, Br, I), practically in all studies oleic acid/octylamine passivating ligands were used. Thus, the above analysis shows that PL lifetimes of PNCs can vary by more than an order of magnitude depending on the type of surface passivating ligands. In principle, the idea of a significant influence of nanocrystal surface on the exciton recombination dynamics is not new and was previously illustrated for semiconductor core/shell QDs, such as CdSe/CdS/ZnS [20, 21]. In the case of MAPbBr3 PNCs, the surface effect occurs to be even stronger and seems to actually determine the dynamics of exciton recombination. How can it be possible if all other physical characteristics of PNCs possessing different passivating ligands are practically equal? 5

To explain such photophysical behavior of PNCs, and first of all the lack of correlation between the PLQY and the PL decay time when various ligands are passivating the surface, it should be recognized that this contradiction cannot be resolved in the frame of the standard two-level energy diagram (Fig. 1a). For this simple scheme, the PL lifetime, τPL, and the PLQY, Φ, are related to the radiative and non-radiative rate constants by the well-known phenomenological formulas:
 = Φ=

    

  



(1)

(2)

Within this scheme τPL and Φ are always proportional each other. For the discussed MAPbBr3 PNCs this is not the case, and it can only mean that the experimentally measured PL decay time

τPL is not determined by the radiation lifetime (kr)-1 (formula (1)). Such behavior of luminescence is known in photophysics, this is the so-called delayed luminescence (fluorescence), which is described by a three-level scheme (see below). In the case of delayed luminescence, only formula (2) remains valid, whereas formula (1) for PL lifetime is incorrect: τPL is much longer than the value obtained by formula (1) that is a result of the carriers capture by a long-lived (and often dark) state and their subsequent return to the radiative state. The most known example of this kind of delayed emission is thermally activated delayed fluorescence in molecular systems, where the triplet electronic state plays the role of a long-lived trap [22-24]. Recently, delayed luminescence has been identified for semiconductor quantum dots [25, 26]. Figure 1b shows the general scheme for the formation of delayed luminescence in various systems by trapping carriers by traps T (trapping rate constant ktr) and then their release from the trap back to the emitting state with the detrapping rate constant kdt. Two main mechanisms of detrapping are 6

considered in the literature: (1) tunneling from a trap into an emitting state, and (2) thermally activated detrapping. For semiconductor quantum dots, it was found that detrapping (the rate constant kdt in Fig. 1b) is accomplished by tunneling through a barrier separating the trap and the emitting state, since the PL decay kinetics does not exhibit temperature dependence [26]. To explain the long PL decay times in perovskite nanocrystals, a three-level scheme of PL formation with the participation of long-lived shallow non-quenching traps has been recently proposed; within this framework, detrapping is a thermally activated process (Fig. 1c) [27, 28].

Figure 1. (a) Traditional two-level, (b) generalized three-level, and (c-d) thermally-activated three-level energy diagrams describing the process of the prompt (a) and delayed (b-d) PL formation in semiconductor nanocrystals; the ground (unexcited), excited (excitonic) and trap states are noted as 0, 1 and T, respectively; kr and knr are the radiative and non-radiative rate constants of the exciton recombination; k1T (ktr) and kT1 (kdt) are the rate constants for trapping and detrapping. The black circular arrows in (c) symbolize the cyclic process of population exchange

7

between the bright excitonic and dark trap states. In (d) different positions of shallow trap states TA and TB are shown which are formed as a result of the midgap quenching traps passivation by the ligands of the types A and (B), respectively (see the text).

This phenomenological scheme of delayed luminescence includes shallow traps (indicated as T in Fig. 1c), which capture and store carriers during some time followed by their returning to the emissive (excitonic) state 1. In the framework of the delayed luminescence model the experimentally measured long PL lifetimes τPL of PNCs (tens – hundreds of ns) [27] are well consistent with high radiative rate constants kr (about 109 s-1 and higher) typical for direct gap semiconductors. Indeed, as one can see in Fig. 1c, if the condition k1T>>kr, knr is fulfilled, a majority of photoexcited carriers are preferably trapped with the time constant k1T in the trap state T, and only small part of them recombine radiatively or non-radiatively with the rate constants kr and knr, respectively. The rate constants for carrier trapping and detrapping, k1T and kT1, are related to each other by the expression ([27] and references therein):  =  exp (−Δ ⁄  )

(3)

where ∆E, kB and T are the energy difference between the emissive exciton and trap states, the Boltzmann constant and the absolute temperature, respectively. The proposed delayed luminescence model reconciles the short radiative lifetime τr=(kr)-1 with very long PL lifetime τPL, which is determined by long detrapping times, τT1 = (kT1)-1. It is important to emphasize that although the delayed luminescence model in its simplest form (that is with one trap state [27]) results in monoexponential delayed PL decay, in case of using broad distribution of the trap states by energy instead of a single trap, the model can result in the power-law PL decay because a power-law function can be approximated with a finite sum of weighted exponentials [29].

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The phenomenological model of thermally activated delayed luminescence well explains many experimental facts. In particular, the model explains how different passivating ligands can lead to similar high PLQYs, but completely different τPL values. Indeed, it may happen if knr is significantly lower than kr in case of both A and B ligands (it determines high PLQY, Φ =    

similarly to the scheme in Fig. 1a), but ∆E are very different for the corresponding two

types of ligands [27]. It should not be thought that the shallow traps forming the kinetics of the PL have some special nature that is different from the nature of the deep quenching traps lying in the middle of the band gap (midgap states). We speculate that passivation does not completely eliminate midgap traps, but makes them shallower [30], and their depth would depend on the type of passivating ligand (see Fig. 1c). As a result, the proposed model would explain how the PL lifetime could depend on the type of passivating ligands. In addition, the model of delayed thermally activated luminescence well explains the unusual temperature dependence of the PL decay kinetics in PNCs. Indeed, there is debate in literature on the temperature dependence of the PL transients for PNCs, the dependence is found to be opposite to the one known for conventional semiconductor quantum dots (QDs) such as CdSe, where a significant lengthening of the PL lifetimes occurs with a decrease of T. Such behavior in case of QDs is caused by the well-established fact that there are two types of the exciton lowest states in semiconductor QDs: (i) higher lying and effectively emitting “bright” states with ∼10 ns lifetimes [31] and (ii) lower lying and weakly emitting “dark” states with lifetimes of the order of 1 µs [32, 33]. Since the splitting between the lowest (dark) and the higher (bright) states is only a few meV [34], the average radiative lifetime is strongly temperature dependent: it is of ∼20-30 ns at room temperature [35, 36] increasing toward the dark-state radiative lifetime of about 1 µs at 4 K. In case of PNCs the temperature dependence is 9

opposite: PL lifetimes significantly decrease when the temperature decreases to the cryogenic one. An example of such behavior is shown in Fig. 2 for the case of CsPbI3 PNCs, where the average PL lifetime decreases from 24 ns at 300 K to 2.5 ns at 15 K. The trend is in line with the temperature dependences found in literature for CsPbX3 PNCs [1, 12, 37, 38] (note that in all cases oleic acid and oleylamine were used as passivating ligands). In particular, for CsPbI3 NCs in a thin film the PL lifetime decreased from 55 ns at 300 K to 2.0 ns at 2.7 K [1]. Note that absolute values of PL lifetimes are not identical for PNCs of similar size and PLQY as well as at the same temperature: for individual CsPbI3 PNCs the PL lifetime at room temperature was measured as long as 80-90 ns [12] that is 4 times longer than in our case (Fig. 2) and about 1.5 times longer than in [1]. In case of CsPbBr3 (CsPbCl3) PNCs the PL lifetime decreases from 7 ns to 120 ps (from 2.6 ns to 30 ps) when temperature decreases from 300 to 2.7K [1]. How such “inverted” temperature dependence of the PL lifetimes can be explained?

PL Intensity, norm.

1

15K experiment 15K fitting 300K experiment 300K fitting

τPL=24 ns

0.1

τPL=2.5 ns

0

20 40 60 80 100 Time after excitation, ns

120

10

Figure 2. PL decay transients measured for thin films of CsPbI3 NCs at 300 and 15 K. Symbols are experimental points, solid lines are bi-exponential fitting curves. Average lifetimes τav are 24 and 2.5 ns, respectively.

Possible answers to this question were recently considered in ref. 1, where it was noted that the experimentally detected temperature dependences of the PL transients for PNCs cannot be explained either by the presence of a low-lying dark excitonic state, or by a combination of direct and indirect bandgaps, or by changes in oscillator strength for NC forms of these materials. The authors concluded that a qualitative explanation of the detected effects is possible in the framework of the exciton fission model (i.e., the temperature-induced exciton dissociation). The authors [1] argue that the distance between recombining charges are higher in case of free electron-hole pairs that should result in longer recombination lifetimes than in case of excitons. As a result, an increase of electron-hole pairs population with temperature should lead to a lengthening of the observed PL decay kinetics. We performed a quantitative analysis of the exciton fission model and concluded that this approach also cannot explain the observed strong (about ten-fold) increase of the PL lifetime of perovskite nanocrystals with increasing temperature from the cryogenic ones to 300 K.

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Figure 3. (a) The diagram explaining the exciton fission model and including the ground (0), excitonic (1) and band (2) states and transitions between them. (b) the excitonic state population decay curves obtained by solving the rate equations (4-6) in the text with the following parameters: k20=107 s-1, k10=109 s-1, k21=1012 s-1, ∆E=10 meV, T=300K (curve 1) or 10K (curve 2).

  " 

= − 



= 

− 



−   ∆%



− 

 =  exp #  ' &

!  



(4) (5) (6)

The energy diagram corresponding to the exciton fission model is shown in Fig. 3a, where 0, 1 and 2 mean the ground, excitonic and band states, respectively. The rate equations (4-5)

12

determine the population dynamics for excitonic (n1) and free-carrier (n2) states, where exciton formation (k21) and dissociation (k12) rate coefficients are linked by equation (6). The solution of the equation system (4-5) was undertaken under the following reasonable conditions: k20=107 s-1; k10=109 s-1; k21=1012 s-1; ∆E=10 meV; T=300K and 10K; at t=0 n2=1 and n1=0. The resulting calculated excitonic PL transients are shown in Fig. 3b. Indeed, due to the much slower charge recombination in the band state 2, a growth of temperature results in the excitonic PL lengthening from 1 ns at 10K to 1.66 ns at 300 K, but this effect is minor as compared to the about ten-fold increase experimentally observed for all PNCs. It is worth to note that changes in the model parameters, such as, for example, a further increase of the lifetime of the band state 1, or an increase of the exciton binding energy ∆E, do not result in further extension of the PL decay kinetics. Thus, the possibilities of the exciton fission model to extend the PL transients with increasing temperature compared with PL lifetimes at cryogenic temperatures are limited by less than a twofold increase. It should be noted that in all literature models proposed for explaining the effect of temperature on the PNC PL kinetics, the underlying physical mechanisms were related to the semiconductor material rather than to a system including a near-bulk nanocrystal semiconductor, its surface and the passivating ligands interacting with the surface. We believe, however, that surface effects, in particular shallow surface traps, should play a significant role in the formation of the PL transients. In this sense, the model of thermally-activated delayed luminescence (Fig. 1c) [27] qualitatively well describes the temperature dependence of the PL kinetics for PNCs. Indeed, according to this model, the PNC PL lifetime τPL at room temperature is determined by the depth of surface traps and can exceed the radiative lifetime (k1r)-1 by several orders of magnitude. As the temperature decreases, the process of carrier detrapping towards the radiative 13

exciton state slows down more and more (the constant kT1 drops following Equation (3)) so that the surface trap filling occurs that means that the surface trapping process is blocked. As a result, the only possible processes in the excitonic state 1 are the radiative and (usually as a minor process at low temperature) nonradiative exciton recombination paths, which occur on the nanosecond or subnanosecond time scales. We cannot exclude, however, that there is some contribution of one more effect to the PL decay shortening with the temperature decrease, namely a growth of the value of the radiative recombination constant [39], although this effect alone cannot explain the observed about ten-fold shortening of the PL lifetime with temperature decrease from 300 to about 10K [1]. PL decay kinetics in thin perovskite layers At first glance, it may seem that the PL decay kinetics in thin (hundreds of nanometers) layers of perovskites are extremely difficult to interpret due to the large number of recombination processes, which can take part in their formation. First of all, these are free carrier bulk recombination processes, such as the quasi-monomolecular deep-trap assisted non-radiative recombination, bimolecular radiative recombination, and Auger recombination (at high carrier concentrations). In addition, it is customary in the literature to consider separately the processes of the non-radiative carrier recombination on a layer surfaces. Fortunately, in the case of metal halide perovskite layers the situation turns out to be favorable for the PL kinetics interpretation because the bulk recombination rates are essentially lower than the non-radiative recombination rates on the surface. Due to this, in most cases the PL decay kinetics in perovskite thin polycrystalline layers is determined by the carrier diffusion across the layer thickness to both front and back surfaces followed by the non-radiative recombination on them (see Fig. 4) [40].

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Figure 4. A schematic representation of two possible morphologies of a polycrystalline perovskite layer of thickness d: (a) individual grains occupy the entire thickness of the layer, and (b) several grains stacked on each other form the thickness of the layer. Initial inhomogeneous distribution of photoexcited carriers over the layer thickness formed due to high absorption coefficient of perovskite is shown as a gradient red color. Wide yellow-brown vertical arrows, directed to both the front and back surfaces, show the directions of carrier diffusion across the layer thickness caused by the initial inhomogeneous distribution of carriers and carrier recombination on the surfaces.

Wherein, one should remember that the perovskite layers, which are usually fabricated either by spin coating of precursor solutions (solution-deposited layers) or by vacuum codeposition of sublimated precursors (vacuum-deposited layers) consist of grains, which are tightly adjacent to each other (see Fig. 4). In principle, one can imagine two possible morphologies of the layers: (a) individual grains occupy the entire thickness of the layer (Fig. 4a), and (b) several grains stacked on each other to form the layer thickness (Fig. 4b). As recent investigations have shown, the boundaries between the grains inside the layer do not cause quenching of the PL [41-44] and, therefore, do not participate in the formation of the PL decay kinetics. Only grain boundaries forming external surfaces of layer (both front and back surfaces) are usually responsible for the perovskite PL quenching (carrier non-radiative recombination).

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Numerous data on the significant (ten-fold and more) lengthening of the PL lifetime in a perovskite layer as a result of chemical passivation of their front surface serve as an evidence of a leading role of the layer surfaces in the PL kinetics formation [7, 45]. Therefore, according to literature data, the most efficient mechanism responsible for carrier recombination in perovskite layers is non-radiative recombination at the layer surfaces. In such a case, from the point of view of extracting the maximum of physical information, the most reasonable method of analyzing PL kinetics is fitting them with use of a one-dimensional diffusion equation. It makes possible to obtain simultaneously the values of the carrier ambivalent diffusion coefficient D and the surface recombination velocity S [40]. It should be noted that, generally speaking, the diffusion equation is applicable only for homogeneous materials. It is obvious that polycrystalline perovskite layers are not homogeneous, since they contain not only bulk material, which in the first approximation can be considered the same in different grains, but also grain boundaries. However, in reality, the situation here is quite favorable from the point of view of interpretation of the luminescence kinetics, because after photoexcitation the diffusion moves the charges only in the vertical direction through the layer thickness, since only in this direction there is an initial photoinduced concentration gradient (the gradient is shown by the red color in Fig. 4). Therefore, the morphology presented in Fig. 4a can be considered as homogeneous from the point of view of diffusion across the layer thickness. It is believed in the literature that most of the state-of-the-art perovskite films are continuous and monocrystalline in the vertical direction [46, 47] (Fig. 3a). However, the structures shown in Fig. 3b are also possible, for example in case of films employed in the fabrication of LEDs where small grain sizes are preferred. In such structures, perovskite layers consist of grains stacking on top of each other and this is a non-homogeneous material because, in course of diffusion, carriers

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have to cross several internal boundaries to reach the layer surfaces that can essentially retards the diffusion [48]. Therefore, in case of morphology shown in Fig. 3b the values of D, obtained as a result of the PL kinetics fitting by the diffusion equation (see below), are consequently effective (averaged) values. Another comment concerns the consideration of the bimolecular recombination of carriers in the bulk film. From our own experience, it is reasonable to minimize a contribution of this mechanism to the kinetics because it makes the diffusion equation much more complex and thus decreases the accuracy of determination of D and S values. The bimolecular recombination exclusion can be done by reducing the concentration of photoexcited carriers below a certain value, approximately 1017 cm-3 [49]. The time dependence of the concentration n of photogenerated charge carriers along the layer thickness can be described by the one-dimensional diffusion equation in the following form: (), (

=+

(  ), () 



), ,&

- ., / ∝ 1 ., / · 3 45)

+ -., /

(6)

(7)

where D is the diffusion coefficient, τB is the bulk carrier lifetime and G (x, t) is the generation rate upon a light pulse, α is the absorption coefficient at the excitation wavelength, and I (x, t) is the excitation pulse profile. To solve equation (6), standard boundary conditions describing recombination on the front (equation 8) and rear (equation 9) surfaces should be also used: (), ()

6

)7

=

89 :

0, /

(8)

17

(), ()

6

)7

=

8& :

<, /

(9)

where SF and SB are the front and back surface recombination velocities, respectively,and d is the layer thickness. The numerical solution of the diffusion equation (6-7) with conditions (8-9) enables to obtain the distribution of charge carriers over the layer thickness as a function of time. The relationship between the carrier distribution and the experimentally measured kinetics of the PL decay IPL(t) can be done by the equation: 

1 / = =



., /<.

(10)

Thus, by fitting an experimental PL decay kinetics IPL(t) with the diffusion equation, one can determine D and S values as fitting parameters. In the literature on perovskites, such a diffusion approach of the PL kinetics interpretation was mainly used to determine the diffusion coefficients for electrons and holes under the conditions of the perovskite surface coating by an electron- or hole-transport layer, respectively [50-52]. Only in a few works this approach was used to determine both parameters, D and S for uncovered surfaces of the perovskite layer [40]. To fit the experimental kinetics of perovskite layers using equation (6), it is necessary to know the carrier lifetime (luminescence lifetime) in the bulk. It is often impossible to measure this value experimentally because of the quenching effect of the layer surfaces. However, it can be assumed that this value is at least 1 µs, since passivating only the front surface of the layer usually extends the luminescence lifetime to several µs [7, 45]. In our calculations (see below), we assumed τB = 1 µs. In Fig. 5 we show the results of modeling of the luminescence decay kinetics we performed by solving the diffusion equation (6) for various combinations of D and S (here we

18

neglect bimolecular recombination of carriers and assume τB = 1 µs). As one can see, for large values of D (of the order of 1 cm2/s and more), the kinetics no longer depend on D and are determined only by the values of S (Fig. 5a and 5b). This means that the diffusion redistribution of carriers over the layer thickness occurs much faster than the carrier recombination on the surface. In this case of very fast diffusion and when τPL << τB, the relation (11) is valid [53], which allows one to determine the value of S knowing the experimentally measured PL lifetime

τPL and the layer thickness d: 

> = ,

?@

(11)

19

Figure 5. Model PL kinetics calculated by the diffusion equation for different D and S values. In all cases absorption coefficient α = 105 cm-1, τB=1000 ns. The model corresponds to the case when bimolecular recombination of carriers is absent (n ~ 1015 - 1016 cm-3).

It should be noted that the values of D obtained as a result of fitting the PL decay kinetics in thin layers of perovskites can provide additional information about the morphology of these layers. Indeed, as the literature data analysis performed in Ref. 54 shows, an average carrier mobility for CH3NH3PbI3 perovskite thin films, measured by THz and microwave methods, which are short-range by their origin and inform only about carrier transport inside (sub)micrometer grains, is µ = 37 cm2/Vs (D = 0.93 cm2/s). This is rather close to the average values of µ = 73 cm2/Vs (D = 1.8 cm2/s) found for single crystals [54]. Recent direct measurements by the Light Induced Transient Grating method also showed that the diffusion coefficient of carriers in the plane of the layer at distances less than the grain size is of the same order of 1-2 cm2/s [55, 56]. Thus, it can be assumed that the transport of carriers inside the individual grains and in the volume of single-crystal perovskites occurs in the same way, with a diffusion coefficient of about 1-2 cm2/s. Therefore, we assume that if the PL kinetics fitting for the perovskite layers result in such high values of D, this indicates that the morphology shown in Fig. 4a is realized in these layers, when perovskite grains occupy the entire thickness of the layer. In this case diffusion, which occurs only in the vertical direction, is realized in the bulk material without participation of grain boundaries, that is as in perovskite single crystal. If fitting of the PL kinetics yields values of D significantly smaller than 1 cm2/s (see the review [54]), then this is likely an indication that the structure of the perovskite layer is closer to that shown in Fig. 4b.

20

In conclusion, simple procedures are described in this work for interpretation of the PL decay kinetics of metal halide perovskite structures. In case of well passivated nanocrystals, an application of the delayed luminescence model can give insight about non-quenching (surface) traps responsible for the PL kinetics formation. In case of thin polycrystalline films limited by interfacial recombination, the PL decay fitting by one-dimensional diffusion equation enables one to determine the diffusivity D and the surface recombination velocity S values and, thus, evaluate qualitatively the layer morphology. Methods Synthesis of Colloidal solution. CsPbI3 halide perovskite nanocrystals were prepared by following the hot-injection method [57, 58] with some modifications. Firstly, Cs-oleate solution was achieved by mixing under constant stirring 0.410 g of Cs2CO3, (99.9 %, Sigma-Aldrich), 1.25 mL of oleic acid (OA, 90 %, Sigma-Aldrich) and 20 mL of 1-octadecene (1-ODE, 90 %, Sigma-Aldrich) into a 50 mL- three neck-flask under vacuum for 1 h. Then, the solution was heated at 150 °C under N2 atmosphere until dissolve Cs2CO3 completely. In order to avoid the precipitation of Cs-oleate, the mixture was stored under N2, keeping the temperature at 100 °C. For carrying out the CsPbI3 colloidal solution, 0.87 g PbI2 (99.99%, TCI) and 50 mL of 1-ODE were mixed into a 100 mL-three neck flask and heated at 120 °C for 1 h under vacuum conditions. Then, both OA and oleylamine (5 mL each one) were separately loaded into the halide solution under N2-purge and quickly heated to reach 170 C, injecting 4 mL of the preheated Cs-oleate solution immediately. Lastly, the reaction was quenched by adding the flask into an ice bath for 5 s. In order to perform the perovskite isolation process, the as-prepared CsPbI3 colloidal solution was centrifuged at 4700 rpm for 15 min. The size of CsPbI3 NCs was

21

about 12 nm. The nanocrystals pellets were obtained from the supernatant and concentrated to 50 mg mL-1 with hexane. Sample preparation. CsPbI3 perovskite nanocrystals were deposited on a commercial borosilicate substrate by means of spin-coating method with post-baking at 100 ºC for 1 minute. Prior to the deposition substrates were carefully cleaned with acetone, ethanol and isopropanol during 10 minutes in an ultrasound bath. Low temperature PL and TRPL measurements. Samples were held in a cold finger of a closed-cycle He cryostat, which can be cooled down to 10K. PL was excited by using a Ti:sapphire mode-locked laser (Coherent Mira 900D, 200 fs pulses with a repetition rate 76 MHz) at a wavelength of 405 nm obtained by doubling initial 810 nm emission with a BBO crystal. The excitation density was ~0.5 µJ/cm2. To measure PL spectra, the PL signal was dispersed by a double 0.3 m focal length grating spectrograph and detected with a back illuminated Si CCD. To measure PL transients, the emitted light is collected on a Si avalanche photodiode connected to a time correlated single photon counting electronics. Room temperature TRPL measurements. Room temperature TRPL experiments were carried out using the fluorescence lifetime spectrometer C11367 Quantaurus-Tau with LED pulse light sources. The measured PL decay kinetics were fitted with a two- or triexponential function of time (t): 1 / = ΣBC exp (−/⁄
C )

(12)

where τi represents the decay time of the ith component and ai represents the amplitude of the ith component. The average PL lifetimes (τav) were estimated with the τi and ai values from the fitted curves data according to eq 13: 22


DE =

∑ DG ,G ∑ DG ,G

(13)

PL kinetics modeling. The numerical solution of the one-dimensional differential equation was made in the framework of the Crank-Nicolson difference scheme. Acknowledgements Financial support by Spanish MINECO through project nº TEC2017-86102-C2-1-R, the European Research Council (ERC) via Consolidator Grant (724424 - No-LIMIT) and Generalitat Valenciana via Prometeo Grant Q-Devices (Prometeo/2018/098) are gratefully acknowledged. SSK and YTT acknowledge financial support from the Ministry of Education and Science of the Republic of Kazakhstan through the Project AP05130083.

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Highlights:

The PL kinetics of perovskite nanocrystals is often longer than the radiative lifetime. Such delayed PL is well described by a model with carrier trapping in shallow traps. The model well explains PL lifetime decrease by filling the traps at low temperature. In thin perovskite films the PL kinetics is usually determined by surface quenching.

Author statement Vladimir Chirvony: Conceptualization, Methodology, Writing - Original Draft. Kairolla Sekerbayev: Software, Investigation. Hamid Pashaei Adl: Software, Investigation. Isaac Suárez: Methodology, Investigation. Yerzhan Taurbayev: Software, Resources. Andrés Gualdrón-Reyes: Investigation. Iván Mora-Seró: Conceptualization, Writing - Review & Editing. Juan Martínez-Pastor: Conceptualization, Methodology, Writing - Review & Editing.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: