The influence of a trap state on the photoluminescence decay times under single pulse excitation

We numerically calculated the time-resolved photoluminescence spectra using the bimolecular trapping-detrapping model. The variations of carrier lifetimes are investigated by changing the carrier recombination and trapping rate constants, as well as the concentration of available trapping states.


Introduction
Perovskite materials are recently attracted a lot of attentions in the research area of solar cell applications. Their outstanding intrinsic properties, such as high absorption coefficient , tunable band gap (Noh et al. 2013;Snaith 2013), large carrier diffusionlength (Edri et al. 2014;Zhao et al. 2014;Dong et al. 2015;Chen et al. 2017) and carrier mobility (Stranks et al. 2013;Stoumpos et al. 2013;Ponseca et al. 2014), show the potential improvement in the photovoltaic performance of a solar cell. However, to the practical purpose in designing the optoelectronic device and improving its photovoltaic efficiency, it needs to understand the fundamental interactions between photons and the carriers in the material. Time-resolved photoluminescence (TRPL) spectroscopy is one of the powerful techniques to investigate the mechanism of electron transition processes in photoluminescence (PL) materials (Lin et al. 2017;Cai et al. 2020). In this work, we will numerically analyze and discuss the evolution of the excitonic and non-radiative trap state electron populations in the perovskite material with respect to time using the bimolecular trappingdetrapping model (Chirvony et al. 2016(Chirvony et al. , 2017Péan et al. 2020). From the results of time evolution populations, it is possible to imitate the TRPL spectra of the sample.
To the best of our knowledge, most of the researches related to the analyses of TRPL spectra usually based on known lifetimes, which are obtained from directly fitting experimentally measured TRPL spectra or from other experimental measurements, and further to 2 Numerical calculation model In the following calculations, we consider the phenomenological model depicted in Fig. 1. We want to imitate the properties of PL in a three-level system. There are two distinct states for electrons: the excited state with energy level slightly above but comparable to the conduction band minimum (CBM) and the non-radiative trap state with energy level below the band edge (Stranks et al. 2014;Yamada et al. 2016;Mandal et al. 2020). The photo-generated electrons in the excited state can recombine with the holes in the valance band through three different transition processes: the radiative electron-hole recombination process with transition rate constant k R (cm 3 ns −1 ), the non-radiative electron-hole recombination process with transition rate constant k N (cm 3 ns −1 ), and the non-radiative trapping-detrapping process through the trap state with electron trapping and detrapping rate constants k T (cm 3 ns −1 ) and k D (cm 3 ns −1 ), respectively. The PL light comes only from the radiative recombination of electrons in the excited state with the holes in the valance band.
The physical mechanism of the radiative electron-hole recombination is associated with the excitonic or band-to-band recombinations (Stranks et al. 2014;Yamada et al. 2016). And the mechanism of non-radiative electron-hole recombination is resulted from shallow trap state-mediated Auger recombination (Stranks et al. 2014;Péan et al. 2020). And, we consider the subgap trap state, which is due to the defect or thermally activated atomic vacancies, as to provide another non-radiative charge-trapping pathways in the carrier transition process (Shkrob et al. 2014;Stranks et al. 2014;Yin et al. 2014;Talite et al. 2016;Péan et al. 2020). There is likely to be a distribution of nonradiative trap states at different subgap energies, but they are shown here at a single energy for simplicity (Stranks et al. 2014;Péan et al. 2020). For the luminescence due to excitonic or band-to-band radiative recombination, the TRPL spectrum can properly be fitted by a distributed exponential decay function with two or three decay times.
The positive doping is considered to be an impurity-induced doping which might affect the hole energy levels in the valance band or act as an acceptor (Meggiolaro et al. 2018;Sabino et al. 2022). The energy level of the acceptor is assumed to be located above the valence band maximum. Therefore, the positive doping considered here should affect mainly on the relative portion of radiative and non-radiative recombination of the electron-hole pairs. Otherwise, the doping could also affect the energy level of trap states, Fermi energy and even the energy band gap, which will in consequence affect the PL wavelength as well as the TRPL lifetime (Pavesi 1996;Huang et al. 2004;Timmerman et al. 2012;Mir et al. 2020). On the other hand, the positively doping center can also lead to the fluctuations of the local nanoenvironment, which plays a role in affecting the transition rate constants and in consequence the inverse power law behavior of the TRPL decay (Kuno et al. 2001(Kuno et al. , 2003Peterson et al. 2009). However, since the effects of recombination energies and the environmental dependence of the transition rate constants are still not involved in our homemade calculation program. The influence of different doping concentrations on the TRPL decay is not investigated in the present work.
The total concentration of the available trap state is N T (cm −3 ). The concentration of photo-generated electrons produced by illuminating a single pumping pulse is N 0 (cm −3 ). After the single pulse excitation, some of the photo-generated electrons will recombine with the holes in the valance band, and some of them will transit into the trap state. The trapped electrons then recombine with the holes in the valance band through the detrapping process. The time dependent concentrations of electrons in the excited and trap states are n E (cm −3 ) and n T (cm −3 ), respectively. And the associated time dependent concentration of photo-generated holes in the valance band is n H (cm −3 ). Considering the sample is positively doped with doping concentration p 0 (cm −3 ). Therefore, the total concentration of holes in the valance band is (n H + p 0 ) after the pumping pulse. In our calculations, the bimolecular trapping-detrapping model is used to describe the time rate of change of each carrier concentration. The carrier concentrations of n E , n H and n T after single pulse excitation can be described by the following set of differential equations.
The rate constant k B is the combination of radiative and non-radiative recombination rate constants, i.e. k B = (k R + k N ). The electron trapping and detrapping rate constants are k T and k D , respectively. We use the fifth order Runge-Kutta method (Cheney and Kincaid 2008) to solve (1) with initial values n E (0) = n H (0) = N 0 , n T (0) = 0. And the time dependent evolutions of n E (t), n H (t) and n T (t) for different values of k B , k D , k T and N T are calculated in our following studies. To investigate the influence of the power of the pumping pulse, different values of N 0 is also used in our calculations. In order to investigate the properties of TRPL spectrum corresponding to the system, the TRPL intensity is estimated by The default parameters used in our calculations are N 0 = 500 × 10 12 (cm −3 ), N T = 60 × 10 12 (cm −3 ), p 0 = 65 × 10 12 (cm −3 ), k B = 60 × 10 -20 (cm 3 ns −1 ), k R = 0.8k B , k D = 80 × 10 -20 (cm 3 ns −1 ), and k T = 6000 × 10 -20 (cm 3 ns −1 ). These default parameters and the variation range of the parameters used in the following calculations are mainly referred to the reference (Péan 2020) and the references in it (Hutter et al. 2015;Brenes et al. 2017).
With changing single parameter once a time, the TRPL spectra for different parameters are calculated using (2) after solving the time evolutions of the carrier concentrations from (1). And the associated luminescence lifetimes are obtained by fitting each calculated TRPL spectrum using a combination of exponentially decaying functions. Although, in a real material, the rate constants are usually as a function of excitation intensity, pulse duration, temperature of the surroundings etc. (Greben et al. 2017;Wright et al. 2017;Mandal et al. 2020;Ha et al. 2021;Chen et al. 2021) These effects are not involved in our present calculation program.

Results and discussion
The results shown in Fig. 2a-d are the calculated TRPL spectra using the default parameters but changing the value of k B , k D , k T , and N T , respectively. The increasing in k B affects the TRPL spectra more obviously in the medium and long-time duration after the excitation pulse. And, the increasing in k D and k T mainly affect the behavior of TRPL spectra in the long-time duration and in the time duration just after the excitation pulse, respectively. The increasing in N T seems to affect the TRPL spectra more obviously in full time duration of the calculations. To quantitatively analyze the variations of the carrier lifetimes with different parameters, the TRPL spectra are fitted by a tri-exponential decaying function A 1 exp(t/ 1 ) + A 2 exp(t/ 2 ) + A 1 exp(t/ 3 ). In the analysis of a time-resolved PL spectrum, the most usually used fitting functions are bi-exponential and tri-exponential decay functions. In our calculated results, a bi-exponential decay function cannot well fit all the TRPL spectra we have calculated for different parameters. Therefore, we choose a tri-exponential decay function, which can well fit all the calculated spectrum results, to fit our calculated TRPL spectra to obtain the carrier lifetimes.
The variations of the fitted carrier lifetimes with increasing k B , k D , and k T are shown in Fig. 3. With increasing k B , the long lifetime ( 3 and medium lifetime ( 2 decrease gradually, but the short lifetime ( 1 remains unchanged at about 35-40 ns. The increasing k D results in slightly decreasing in 2 and 3 , and has almost no effect on 1 . With increasing k T , all the three lifetimes rapidly decrease until k T reaches the value about 1000 × 10 -20 (cm 3 ns −1 ). For even larger k T , each lifetime changes slowly with increasing k T . From the results shown in Fig. 3, we can see that the property of the short lifetime, 1 , is only correlated with the trapping rate constant of the trap state. The property of the medium lifetime, 2 , is strongly affected by both the recombination rate constant and trapping rate constant.
(2) And the behavior of the long lifetime, 3 , dependents on each of the rate constants. Based   Fig. 3 The variations of fitted lifetimes from the calculated TRPL spectra using the default parameters but with a different k B , b different k D , and c different k T on these observations the lifetime 1 should be correlated with the recombination process that occurs when the electron concentration of the trap state, n T , is low. In this situation, the concentration n E decreases rapidly mainly due to the transition of electrons into the trap state and the effect of detrapping of electrons in the trap state is negligible. The intensity of luminescence is therefore decreasing rapidly, and the larger the trapping rate constant k T the shorter the lifetime 1 .
On the other hand, because of increasing k B and k T both result in reducing the lifetime 2 , it should be correlated with the recombination process that occurs when n T becomes larger but is still unsaturated. In this situation, the rate of decreasing in n E due to the trapping effect becomes smaller and the detrapping effect of the electrons in the trap state is still obscure. Therefore, the larger k B or k T will leads to a smaller 2 , and the increase in k D is only slightly reducing 2 . Finally, the lifetime 3 should be correlated with the recombination process when n T approaches its saturation value and even begins to decrease from the saturation. In this situation, the increase in either k B , k D or k T will increase the rate change of n E and n H , therefore, reducing the lifetime 3 . This can also explain why the 3 is more obviously influenced by the detrapping rate constant in this condition.
We also calculated the TRPL spectra for different concentrations of available trap state, N T . Figure 4 shows the variations of the three fitted lifetimes when changing the value of N T with other parameters are remained as the default values. The inset in Fig. 4 shows the results for 1 1 with a magnified vertical scale. When the available trap state is larger, more electrons in the excited state can be trapped into the trap state. This will result in a larger decreasing rate of n E in the time period of just after the pumping pulse and till n T reaching its saturation. Therefore, the three lifetimes of the TRPL spectra tend to decrease with increasing N T . It is interesting to note that 1 1 and 1 2 seems to change their behavior when N T is larger than 200 × 10 12 (cm −3 ) which is about 0.4N 0 .
To see what happened to cause this change, we demonstrate the evolutions of n E , n H and n T with time for the N T values of 100 × 10 12 , 220 × 10 12 and 400 × 10 12 (cm −3 ), respectively, in Fig. 5. The results in Fig. 5a represent the situation for relatively small N T . We can see that the concentration of trapped electrons, n T , reaches its saturation value quickly and remains at saturation for a long time. The variations of 1 and 2 with different N T are relatively small in this situation because these two lifetimes are correlated with the recombination process that occurs when n T is still unsaturated. The results in Fig. 5b represent the situation for relatively large N T , that is, in the region when the lifetimes suddenly decrease with increasing N T . The concentration n T still can reach its saturation value in this situation, but the concentration of n E has decreased to a value which is comparable to or smaller than the saturation value of n T . The recombination processes associated with 1 and 2 might be seriously interrupting by the process associated with 3 . The results in Fig. 5c represent the situation for even larger N T . We can see that n T cannot reach its saturation value and n E quickly decreases to very small values in this situation.
From the results in Figs. 3c and 4, we can see that the variation property of the lifetimes for small N T (when k T is fixed) behaves similar to that for large k T (when N T is fixed). The changes in the lifetimes are slowly in these situations. This behavior could be attributed to the concentration of trapped electron approaching to its saturation condition completely and there are still abundant photo-generated electrons after the saturation. The recombination processes associated with each lifetime in the system can be proceeded completely. On the other hand, the variation property of the lifetimes for large N T (when k T is fixed) behaves similar to that for small k T (when N T is fixed). The variations of the lifetimes with the corresponding parameter are more obvious in these situations, which could be explained by the reason described in the discussions of the results shown in Fig. 5b, c. 1 3 163 Page 8 of 11

Conclusion
The properties of TRPL spectra corresponding to the carrier recombination processes with one shallow trap state mediated in the energy gap are numerically discussed in this work. The bimolecular trapping-detrapping model are used to describe the transitions of photo-generated carriers in our calculations. Time evaluations of the carrier concentrations involved in the recombination processes are solved by the fifth order Runge-Kutta method, and the TRPL spectra are calculated accordingly. The resulted TRPL spectra can be fitted by a triexponential decaying function to obtain three luminescence lifetimes. The value of the short lifetime is mainly affected by the rate constant k T . and is almost independent of the rate constant k D . The value of the medium lifetime is mainly affected by the rate constants k B and k T . These two luminescence lifetimes could be associated with the carrier recombination processes that occur under the situation when the electrons in the non-radiative trap state are still unsaturated. The value of the long lifetime is affected by either the rate constants k B , k D and k T . It could be associated with the carrier recombination process that occurs when the electrons in the trap state approaches its saturation condition and even after.