In recent years, perovskite materials have garnered significant attention due to their potential to rival the efficiency of silicon-based solar cells^[1–7]. Determining trap density is imperative in improving the quality of thin films, which is critical to enhancing the efficiency of solar cells^[8–10]. Numerous techniques have been employed to quantify this parameter, such as transient capacitance, photoconductivity decay, and thermal admittance spectroscopy^[11–13]. However, extracting parameters from photovoltaic measurements can be a complex process due to various factors that may affect the accuracy of the results, including the deactivation of excitons at the cathode, non-ohmic electrode contacts, filter effects, and pinhole defects in the photoactive layers^[14]. In order to perform a quantitative analysis of traps, it is essential to introduce an optical characterization strategy rather than an electrical detection method. This nondestructive optical approach not only preserves the film’s integrity but also enables monitoring of its quality prior to device fabrication. Notably, the optical method does not necessitate a specific sample area size, rendering it suitable for the characterization of large-area thin films. It can efficiently assess film quality by individually examining small regions, and perform a lattice-based inspection to evaluate the uniformity of the entire film, thereby maximizing its commercial value.

Time-resolved photoluminescence (TRPL) is an optical tool that researchers extensively use to study the carrier dynamics of various materials, including semiconductors, metallic oxides, two-dimensional nanomaterials, nanowires, and quantum wells^[15–19]. Moreover, this technique has been widely used to assess the quality of perovskite materials. TRPL offers the flexibility to be performed under various excitation conditions, including different wavelengths, intensities, and temperatures. As a result, it has gained widespread usage in the field as a reliable tool for evaluating perovskite quality and identifying avenues for performance enhancement^[20–23].

Generally, the information that can be directly extracted from TRPL images is fluorescence lifetime rather than trap density. The lifetime can be obtained by analyzing the fluorescence decay curve and performing a bi-exponential fit, I=A1e−tτ1+A2e−tτ2,(1)resulting in two fluorescence lifetimes, τ1 and τ2. Researchers typically assessed trap density indirectly by comparing τ2, which is longer than τ1, of different films because the longer τ2 is considered to indicate a smaller proportion of nonradiative recombination, and thus a smaller trap density^[24].

However, there are two issues here: a principal issue and a precision issue. In terms of the principal issue, τ2 alone does not provide a straightforward indication of the magnitude of trap density. The two fluorescence lifetimes do not correspond one-to-one with the decay rates of carriers associated with different recombination paths; they are intrinsically coupled^[25]. In addition, electrons captured by trap states could undergo transitions to the conduction band, increasing the likelihood of recombination with holes through band-to-band recombination, a process known as detrapping, which can extend the value of τ2 and potentially complicate the determination of trap density. Concerning the precision issue, the presence of parameter adjustment space in bi-exponential fitting lacks a standardized method for selecting the most reliable set of parameters. As a result, the same curve can yield different interpretations, diminishing the method’s credibility. Therefore, it is necessary to reexamine the processing procedure of TRPL data to enable accurate quantification of defect parameters.

In this study, we addressed the principal concern by establishing corresponding models based on carrier recombination and transition mechanisms in perovskite materials. Furthermore, we introduced a genetic algorithm (GA) combined with statistical methods to address the precision issue^[26]. The parameters obtained through this new fitting strategy are more accurate than conventional fitting results, as they represent the most probable values extracted from the distribution of collected data. Consequently, this innovative approach holds significant importance in the field of trap density engineering, as it enables precise determination of trap density, and optimization of device performance, and provides a more accurate and efficient method for assessing the quality of perovskite films.

In this study, methylammonium lead triiodide (MAPbI3) thin films were used as test samples, and anilinium hypophosphite (AHP) was used as an additive for film modification. Previous research has confirmed the effective improvement of film quality through this modification method^[27]. Two types of film samples were prepared: a control group without AHP and a target group with AHP. Two randomly selected samples were taken from each group, labeled as Control 1 and 2 for the control group, and Target 1 and 2 for the target group. Their trap density of states (tDOS) was measured using thermal admittance spectroscopy (TAS); the results are shown in Fig. 1(a). Since tDOS can be integrated to obtain trap density, we can infer from these results that the overall trap density of the devices decreased after the addition of AHP. Considering that the only difference in the device structure is the absorbing layer film, it can be concluded that the main reason for the decrease in trap density is the reduction in the trap density of the thin film^[28].

Based on this premise, one sample was randomly selected from each group, and TRPL measurements were conducted under high and low excitation intensities. The results are shown in Fig. 1(b). After fitting the TRPL signals in Fig. 1(b) using a bi-exponential fitting method, it was found that under low excitation conditions, the target sample had a shorter τ2 (452.87 ns) compared to the control sample (639.75 ns), while under high excitation conditions, the target sample had a longer τ2 (383.18 ns) compared to the control sample (365.43 ns). This indicates that the numerical value of τ2 cannot be directly used as a basis for judging the trap density under a single excitation intensity.

Considering this phenomenon, we further explore and optimize the use of the TRPL method.

In the model established by Maiberg et al. in 2015, the transition rates derived from Shockley-Read-Hall (SRH) theory are applied to a trap state located within a p-type semiconductor^[29,30]. In order to represent it visually, the carriers’ transition demonstrated by Eq. (2) is illustrated in Fig. 2(a), $$\begin{gathered} ddtn=G−Rn+ee−ec, \\ ddtp=G−Rp+he−hc, \\ ddtnt=ec−ee+he−hc, \end{gathered}$$(2)where G represents the generation rate, R denotes the radiative recombination rate, ee signifies the electron emission rate, ec stands for the electron capture rate, he indicates the hole emission rate, and hc refers to the hole capture rate.

In this study, we simplify the model by using p-doped MAPbI3, which implies that the trap levels are primarily concentrated near the conduction band. For simplicity, we consider all potential wells as a single-level trap state, which we refer to as the trap distribution maximum. Therefore, we will focus on ee, ec, he, and hc related to the potential well at the donor level. By adopting this simplified approach, we aim to accurately capture the trapping dynamics and nonradiative processes that influence the film’s overall performance. In TRPL experiments, the generation term should be zero, and carriers created by excitation can be incorporated into boundary conditions. The recombination terms Rn,p are considered radiative terms based on our previous assumption. Subsequently, each term is expanded, $$\begin{gathered} ddtn=−B(np−n0p0)+σnvnn*nt−σnvnn(Nt−nt), \\ ddtp=−B(np−n0p0)+σpvpp*(Nt−nt)−σpvppnt, \\ ddtnt=σnvnn(Nt−nt)−σnvnn*nt+σpvpp*(Nt−nt)−σpvppnt. \end{gathered}$$(3)

We simplify these equations as follows: $$\begin{gathered} ddtn=−B(np−n0p0)+σnvnn*Ntf−σnvnnNt(1−f), \\ ddtp=−B(np−n0p0)+σpvpp*Nt(1−f)−σpvppNtf, \\ ddtf=σnvnn(1−f)−σnvnn*f+σpvpp*(1−f)−σpvppf, \end{gathered}$$(4)where B represents the radiative recombination rate, Nt denotes the trap density, σn,p signifies the capture cross section, and vn,p indicates the thermal velocity of electrons and holes. Moreover, f represents the probability that a trap will contain an electron while nt equals Nt multiplied by f. Furthermore, n*=Nce−Ec−EtkT is the electron density and p*=Nve−Et−EvkT is the hole density if the Fermi-level lies at the trap level Et, the parameter Nc is referred to as the effective density of states function in the conduction band, while Nv is known as the effective density of states in the valence band. Additionally, Ec represents the conduction band level, Ev describes the valence band level, and Et illustrates the trap level. In the following, we define ΔE as Ec−Et.

Next, we establish the boundary conditions for p-doped MAPbI3, $$\begin{gathered} f(0)=11+12exp(Eg2−ΔE−kTlnn(0)nikT), \\ n(0)=n0+Δn0, \\ p(0)=p0+Δn0, \end{gathered}$$(5)where Eg represents the bandgap, ΔE denotes the trap depth, k is the Boltzmann constant, T is the temperature in Kelvin, ni stands for intrinsic carrier concentration, and n0, p0 represent initial concentrations of electrons and holes, respectively. Δn0 and Δp0 indicate generated concentrations of electrons and holes by excitation.

Additionally, Table 1 presents the fixed parameters utilized in the simulation. By incorporating these fixed parameters, we significantly enhance the accuracy and reliability of the fitting results.

The present model can be utilized for conducting simulation analysis. The parameters can be classified into two major groups, each of which can be further subdivided into two subgroups based on trap density. This classification facilitates a comprehensive analysis of TRPL performance across diverse parameter settings. The parameters for the main groups can be found in Table S1 (Supplementary Material), where the arrow within the trap distribution maximum column (ΔE) distinguishes between the two parameter sets, providing clear differentiation.

Figure 2(b) clearly shows that when different trap depths are set, low-quality films exhibit distinct decay trends compared to high-quality films. To provide a quantitative explanation, we perform curve fitting on the plotted data in the figure, $$\begin{gathered} I=−3.76+1.29e−t72.34+2.44e−t853.41,for low trap density, \\ I=−3.42+1.20e−t59.54+2.16e−t869.31,for high trap density. \end{gathered}$$(6)

It can be observed that the τ2 extracted from the low trap density films is shorter than that of high trap density films. Similarly, we subjected data in Fig. 2(c) to the same processing, obtaining the corresponding fitting equations, $$\begin{gathered} I=−3.72+1.18e−t62.80+2.52e−t745.73,for low trap density, \\ I=−3.07+1.60e−t81.24+1.39e−t611.60,for high trap density. \end{gathered}$$(7)

In contrast to Eq. (6), it is evident that in Eq. (7), the τ2 extracted from low trap density films is longer than that of high trap density films. Through this simulation, it is clear that the traditional method of determining lifetime may not accurately reflect the underlying quality factors, necessitating the development of alternative techniques for assessing sample quality.

In this work, when processing the TRPL signal obtained at a single excitation light intensity, we found that a single TRPL curve can be fitted with many different combinations of parameters. To address this issue, we simultaneously fitted three curves of a sample under different excitation intensities. To improve the fitting efficiency, we introduced a GA to optimize the fitting process and constrain the parameter range.

Figure 3(a) displays the TRPL signals of a sample at different excitation light intensities, which are computed through the fitting module depicted in Fig. 3(c). This module illustrates the process of fitting and parameter optimization using a GA. To achieve success with the TRPL fitting technique and GA, it is crucial to select the appropriate fitness function, and in each iteration, the coefficient of determination (R2) is used to assess the fitness. After inputting the TRPL data, the parameter set in the model is initialized. It is worth noting that in the GA, a large number of parameter sets, referred to as individuals, are generated and their fitness is evaluated. In subsequent iterations, if the fitness exceeds a threshold, the individual is retained. If not, the optimal individual is selected for the next iteration, while the other individuals undergo crossover and mutation operations before entering the next iteration. Through this iterative process, a set of optimal parameter combinations is ultimately obtained. Figure 3(b) showcases the fitted curve generated by this set of parameters.

The process above is iterated multiple times, generating numerous parameter sets that can fit the three curves simultaneously. While individual data points may exhibit randomness, a more extensive data set demonstrates statistical patterns and converges to a specific range. By evaluating the peak of the distribution, we can effectively differentiate the quality disparity between different films.

The fitting process involves optimizing several parameters, including the maximum trap distribution (ΔE), trap density (Nt), electron capture cross section (σn), radiative recombination rate (B), and incident intensity (Δn0, Δp0, where we represent intensity using the carrier density excited by the incident laser). The range of values for these parameters can be found in Table 2, which is a valuable reference for setting up the fitting process.

As shown in Figs. 3(d) and 3(e), after the iterative fitting process, we analyzed the large number of obtained parameter combinations. We calculated the distribution of the parameters within various intervals and extracted the statistical distributions of the trap density and trap depth. Subsequently, we visualized these distributions by creating a histogram.

Three samples were randomly selected from each of the control and target groups for testing. The control samples were labeled as Control 1, Control 2, and Control 3, while the target samples were labeled as Target 1, Target 2, and Target 3. TRPL tests were conducted on these samples under varying excitation intensity conditions, and the obtained signals were fitted using the fitting program above. Figure 4(a) shows the statistical distribution of trap densities obtained from the fitted results. The most probable trap densities for Target 1, Target 2, and Target 3 were all 3×1016cm−3, while for Control 1, Control 2, and Control 3, the most probable trap densities were 7.5×1016cm−3, 7×1016cm−3, and 5×1016cm−3, respectively, consistent with trap densities of the existing literature on undoped MAPbI3 films^[33–35].

Moreover, according to the analysis of trap depth data in Fig. 4(b), it can be observed that the control group exhibits a relatively scattered distribution of trap depths, with a range of up to 0.12 eV. Based on the comprehensive analysis of the obtained parameters, we found two changes in trap density for the target group compared to the control group. First, the overall trap density in the target group decreased, with a reduction ranging from 40% to 60% compared to the control group. Second, the values of trap density became more uniform, with the majority of trap density values distributed within a narrow range close to 3×1016cm−3. Additionally, there were two changes in trap depth for the target group relative to the control group. First, the overall trap depth tended to be closer to the median value compared to the films before AHP modification. Second, the range of trap depths became narrower. In conclusion, the numerical results obtained through the TRPL method indicate that AHP treatment reduces the trap density of the thin film and improves its uniformity. This is consistent with previous literature reports and further confirms the feasibility of this method^[27].

In order to validate the universality of this method, we applied it to process the TRPL signals of several other perovskite materials. The corresponding data can be found in Table S3 (Supplementary Material).

From this point of view, TRPL can facilitate the multidimensional characterization of physical parameters germane to the film, which are pivotal in determining its overall properties. This method enables a comprehensive assessment of the relevant physical parameters to fully comprehend the film’s properties.

This study effectively addressed the principal limitations of the fitting process through the formulation of a robust physical model. Moreover, we enhanced the precision of the fitting process by integrating advanced statistical methodologies. By employing iterative fitting, we successfully identified a prominent peak in the trap density distribution curve, signifying the most probable trap density value. Leveraging these advancements, we accurately quantified the trap density of diverse samples and achieved satisfactory outcomes. Additionally, the observed variation in decay rates among the undoped films can be attributed to the trapping–detrapping phenomenon, which is associated with a broad range of trap depths. In summary, this innovative approach holds significant implications for trap density engineering, as it enables precise determination of trap density, and optimization of device performance, and provides a more accurate and efficient means of assessing the quality of perovskite films.

We procured the following materials: poly(bis(4-phenyl)(2,4,6-trimethylphenyl)amine) (PTAA, Mn 7,000–10,000), bathocuproine (BCP), lead (II) iodide (PbI2, 99.99% trace metals), methylammonium iodide (MAI), and methylammonium chloride (MACl) from Xi’an Polymer Light Technology Corp. N, N-dimethylformamide (DMF, 99.9%), dimethyl sulfoxide (DMSO, 99.9%), 2-methoxyethanol (2-ME, 99.8%), chlorobenzene (CB, 99.8%), anilinium hypophosphite (AHP), and L-α-phosphatidylcholine (L-α-PC) from Sigma-Aldrich. Additionally, we purchased C60 from Advanced Election Technology Co., Ltd. and ITO-coated glass substrates (15Ω·sq−1) from South China Science & Technology Co., Ltd. All materials were utilized without further purification, and copper for thermal evaporation was sourced from Zhongnuo Advanced Material (Beijing) Technology Co., Ltd.

For the PTAA solution, PTAA powder was dissolved in CB at a concentration of 6 mg/mL. The precursor solutions for MAPbI3, with a concentration of 2.0 mol/L, were prepared by dissolving MAI and PbI2 in 2-ME and stirring it overnight. Prior to blade coating, these precursor solutions were diluted to 1.1 mol/L. To create the pristine films, additives such as L-α-PC (0.4 mg/mL), MACl (0.2 mg/mL), and DMSO (2.8%, volume fraction) were introduced into the precursor solution. In the case of the optimized films, an AHP solution (20 mg/mL) was prepared by dissolving AHP in 2-ME and added to the precursor solution as an additional additive (4%, volume fraction).

TRPL measurements of perovskite samples were conducted using a 405-nm picosecond-pulsed diode laser in conjunction with a laser-scanned fluorescence lifetime/intensity imaging system (FLRM300, Time-Tech Spectra, LLC). TRPL lifetime measurements were performed using a time-correlated single-photon counter setup with a laser repetition rate of 0.5 MHz. The samples were tested at room temperature, and a laser with a power of 1.69 µW was used. Two optical filters were employed to obtain three different orders of incident intensity, resulting in intensity values of 1.69 µW, 0.169 µW, and 0.0169 µW, respectively.

During the tDOS measurement, a fixed DC bias of 0 V was accompanied by an AC bias amplitude of 20 mV. The frequency dependence of the capacitance was then recorded within the range of 1 kHz to 5 MHz. The tDOS [NT(Eω)] is calculated from C−f measurements using an equation NT(Eω)=−1qkTωdCdωVbiW. The depletion width (W) and built-in potential (Vbi) are derived from the Mott–Schottky analysis of C−V measurements. The calculation of the demarcation energy Eω involves the use of an equation Eω=kTln(ω0ω), wherein ω0 represents the angular frequency of an attempt-to-escape event, which is equal to 2πv0T2.