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

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 participation of 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.


Introduction
Metal halide perovskites (hereinafter perovskites) in all their forms (nanocrystals, thin films of microcrystals, single crystals) demonstrate a large variety of the photoluminescence (PL) lifetimes. In particular, in case of perovskite thin films and nanocrystals the PL decay times span over more than 3 orders of magnitude, from about one nanosecond to more than several microseconds [1][2][3][4]. Besides, far from always the PL lifetimes correlate with quantum yields (QY), especially in the case of nanocrystals: formally same nanocrystals produced in different laboratories demonstrate orders of magnitude different PL lifetimes with comparable PLQY [5][6][7].
Moreover, PL lifetime of the same nanocrystals can depend on the degree of their aggregation [6].
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 reliable interpretation, mainly due to contradictory conclusions made by different authors. In this work, we describe practical approaches how one can interpret PL decay kinetics for most widely investigated metal halide perovskite structures, such as nanocrystals with well passivated surfaces and thin polycrystalline films. In the first case, the contradiction between the observed long PL lifetimes and the expected short exciton radiative lifetime (high radiative recombination rate determined by its high absorption cross section) in perovskites is overcome by means of a threelevel energy diagram including shallow non-quenching traps, as a reason of the PL kinetics retardation. In the second case, the non-radiative recombination at the film surfaces is shown to be the process mostly contributing to the PL decay kinetics, which can be satisfactorily described in terms of one-dimensional diffusion equation enabling determination of the diffusion coefficient D and the surface recombination velocity S. As far as the single perovskite crystals is concerned, they will not be considered here because their PL kinetics is strongly influenced by the PL reabsorption in bulk which in turn depends on excitation/detection experimental conditions [8,9].

PL decay kinetics in perovskite nanocrystals
In terms of phenomenological photophysics, the PL decay kinetics of perovskite nanocrystals is always considered in literature in the framework of a two-level diagram including the ground (unexcited) and excited states (the states 0 and 1 in Fig. 1a) and radiative and nonradiative deactivation channels (solid and dashed lines, rate constants kr and knr, respectively).
Within this diagram, the simple phenomenological formulas (1) and (2) link the PL lifetime PL and the quantum yield  with the radiative and non-radiative rate constants: Following this approach, the values of kr and knr can be easily calculated on the basis of the experimentally measurable PL and  values. As an easy numerical example, if we suppose PL = 25 ns and  = 1.0 (it results in kr = (25 ns) -1 = 4·10 7 s -1 and knr = 0) and then introduce some nonradiative process which decreases PL lifetime and PLQY down to 12.5 ns and 0.5, respectively, it results in kr = knr = (25 ns) -1 = 4·10 7 s -1 .

Figure 1.
Traditional two-level (a) and delayed luminescence three-level (b) energy diagrams which can be used for phenomenological description of photophysical processes in perovskite nanocrystals; 0, 1 and T are the ground (unexcited), excited (excitonic) and shallow trap states, respectively; kr and knr in (a) as well as k1r and k1nr in (b) are the radiative and non-radiative rate constants of the exciton recombination; k1T and kT1 are the rate constants of trapping and de-trapping. The black circular arrows in (b) symbolize the cyclic process of population exchange between the bright excitonic and dark trap states.
However, there are experimental facts in perovskite photophysics which principally contradict the two-level diagram presented in Fig. 1a. First of all, based on the extinction coefficients (absorption cross sections) of perovskite nanocrystals of the excitonic optical transition, the radiative lifetime of the perovskite nanocrystals should be of the order of 1 ns or less [6,7,10,11]. Because the non-radiative processes can only shorten the PL decay kinetics as compared to the radiative lifetime, it means that the experimentally measured excitonic PL lifetimes PL for perovskite nanocrystals in the framework of the diagram in Fig. 1a should be less or much less than 1 ns. However, the experimentally measured PL for 3D perovskite nanocrystals are from a few nanoseconds to several microseconds [6,7]. Thus, the long lifetimes are not compatible either with the direct-gap origin of the perovskite systems or with the two-level model.
The above indicated contradiction can be resolved within the framework of our recently proposed scheme for the origin of PL kinetics in perovskite nanoparticles, as illustrated in Fig. 1b [6,7], which is an analogue of the well-known thermally activated delayed fluorescence in molecular photophysics [12][13][14]. This is the so-called delayed luminescence model including shallow traps (T in Fig. 1b (4) and (5), which describe all possible depopulation processes in the scheme in Fig. 1b, with use of E as parameter. We performed such calculations, and the fitting resulted in E ~ 60 meV. Therefore, having the radiative recombination rate as high as k1r = 5·10 8 s -1 , the process of multiple trapping and de-trapping by the trap with E ~ 60 meV below the emissive state 1 leads to the PL decay kinetics with the PL lifetime as long as 25 ns. ns and from 1.0 to 0.5, respectively, then the quenching rate constant kq is calculated within the two-level scheme as kq = knr = (25 ns) -1 = 0.04·10 9 s -1 . However, in the framework of the delayed luminescence model and with the above mentioned parameters, the two-fold decrease of the PL decay time and quantum yield is due to an appearance of the non-radiative recombination channel with k1nr = kq = 0.5·10 9 s -1 , which is one order of magnitude higher than in the case of the twolevel model, where the radiative exciton lifetime is clearly inconsistent with its high oscillator strength of the (direct gap) perovskite.
Is there any direct evidence that the delayed luminescence model adequately describes the photophysical behavior of perovskite nanocrystals? We believe that the PL lifetime shortening at low temperatures as compared to the room temperature values, that is predicted by the delayed luminescence model, could be such a proof. Indeed, as we described, at room temperature the PL kinetics of perovskite NCs is formed as a result of the effect of multiple trapping (the rate constant k1T) and de-trapping (the rate constant kT1, which notably increases by increasing T after Eq. 3) by shallow non-quenching traps [6,7]. Thus, the experimentally observed PL kinetics is in fact determined by the rate constant kT1 of de-trapping which strongly decreases (increases) with a decrease (increase) of temperature. At very low temperatures the kT1 becomes very small, which means that all shallow traps become completely filled by carriers due to illumination. At these conditions carrier trapping becomes impossible. This is equivalent to the absence of the trap states, so that the system can be considered as two-level one (Fig. 1b) for which the simple expression 1 = 1⁄ is still valid. The predicted effect is really observed for perovskite NCs. As an example, Fig. 2 demonstrates a substantial reduction of the PL lifetime for a thin film of CsPbBr1.5I1.5 nanocrystals we found when temperature decreases from 300 down to 15 [18,19].

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. 3) [20]. 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. 3). In principle, one can imagine two possible morphologies of the layers: (a) individual grains occupy the entire thickness of the layer (Fig. 3a), and (b) several grains stacked on each other to form the layer thickness (Fig. 3b). As recent investigations have shown, the boundaries between the grains inside the layer do not cause quenching of the PL [21][22][23][24] 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). 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 [25,26]. Therefore, according to literature data, the most efficient mechanism responsible for carrier recombination in perovskite layers is non-radiative recombination on the layer surfaces. In such a case, from the point of view of extracting 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 [20].
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. 3). Therefore, the morphology presented in Fig. 3a can be considered as homogeneous from the point of view of diffusion across the layer thickness. It is believed in the literature that in most of the state-of-the-art perovskite films are continuous and monocrystalline in the vertical direction [27,28] (Fig. 3a). However, that the structures shown in ( , ) ∝ ( , ) · − 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 the equation (6), standard boundary conditions describing recombination on the front (equation 8) and rear (equation 9) surfaces should be also used: where SF and SB are the surface recombination velocity, respectively, on the front and back side, d is the layer thickness. The numerical solution of the diffusion equation (6) with conditions (7)(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: 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 [31][32][33]. Only in few works, this approach was used to determine both parameters, diffusivity D and surface recombination velocity S, with the free surface of the perovskite layer [20].
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 microsecond, since passivating only the front surface of the layer usually extends the luminescence lifetime to several microseconds [25,26]. In our calculations (see below), we assumed B = 1 s.
In Fig. 4 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 neglect bimolecular recombination of carriers and assume B = 1 s). As one can see, for large values of D (of the order of 1 cm 2 /s and more), the kinetics no longer depend on D and are determined only by the values of S ( Fig. 4a and 4b). 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, the PL decay can be fitted with a single exponential, and when PL<<B the relation (11) is valid, which allows one to determine the value of S knowing the experimentally measured PL lifetime PL and the layer thickness d [34]:  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 CsPbBr1.5I1.5 colloidal solution was centrifuged at 4700 rpm for 15 min. The nanocrystals pellets were obtained from the supernatant and concentrated to 50 mg mL -1 with hexane.
Sample preparation. CsPbBr1.5I1.5 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 time-resolved PL 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.
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.
PL kinetics modeling. The numerical solution of the differential equation was made in the framework of the Crank-Nicolson difference scheme.