The Diffusion of Triplet Excitons in Perylenediimide Derivative Crystals

Abstract

Perylenediimide derivatives are materials that exhibit singlet fission (SF), capable of absorbing a single photon to generate multiple triplet excitons. This exciton multiplication process holds the potential to surpass the Shockley-Queisser limit. To effectively harness the energy of triplet excitons, they must possess sufficient diffusion capability. However, the diffusion of triplet excitons in perylenediimide derivatives has rarely been studied. In this work, we synthesized perylenediimide derivative crystals (C5) and fabricated composites (C5-Pe-QDs) by incorporating surface-ligand-functionalized quantum dots (Pe-QDs) at varying concentrations. The Pe-QDs act as traps within the C5 crystals, capturing triplet excitons when they diffuse into their capture range. The experimental and computational results indicate that the diffusion coefficient of triplet excitons in C5 crystals is approximately 3.58 × 10⁻⁵ cm² s⁻¹, with a diffusion length of about 50.9 nm. Using Monte Carlo simulations, we estimated the triplet exciton capture probability by Pe-QDs under ideal distribution conditions to be around 79.5%. The above findings indicate that, in the C5-Pe-QDs composites, triplet excitons can efficiently diffuse to the quantum dots, providing a novel and viable pathway for the effective utilization of triplet exciton energy in silicon-based photovoltaic systems.

1. Introduction

Solar energy represents a vital source of power, and photovoltaic conversion is one of the most significant means of harnessing it. A pressing challenge for contemporary scientists is to establish and employ novel physical mechanisms capable of overcoming the inherent energy losses associated with conventional processes. As research progresses, it has been found that numerous physical phenomena hold the potential to surpass the traditional limits of photovoltaic conversion efficiency—among them, the process of singlet fission (SF). SF is a process of multiple exciton generation, typically occurring in organic materials. In this process, molecules in the material absorb a high-energy photon to form a singlet exciton, which, upon transferring its energy to adjacent molecules, generates two triplet excitons [1]. SF materials offer two main advantages: first, they enable efficient use of high-energy photon energy, thereby reducing lattice thermalization losses in photovoltaic devices [2,3]. Second, they allow for exciton multiplication, which holds the potential to surpass the Shockley-Queisser limit in terms of energy conversion efficiency [4]. The application prospects of SF materials are vast, extending beyond the realm of organic photovoltaics [5], with significant potential to enhance performance in various fields, including quantum dot solar cells [6], photocatalysis [7], and photoelectrochemistry [8]. However, for most SF materials, the energy of triplet excitons is difficult to harness practically due to factors such as their relatively low energy and dark-state characteristics [9,10].

To address the challenge of effectively utilizing triplet exciton energy, quantum dots (QDs) can serve as mediators for triplet energy transfer between organic SF materials and silicon-based photovoltaic devices [11,12]. We previously selected a perylenediimide derivative (C5) as the SF material and donor material, and surface-ligand-functionalized quantum dots (Pe-QDs) as the acceptor material, thereby fabricating the composites (C5-Pe-QDs) [13]. Dark triplet excitons can transfer energy to QDs, forming bright states that can be absorbed by silicon-based photovoltaic devices, thereby enabling the transfer of triplet exciton energy from the SF material to the photovoltaic device [14]. Previous studies have shown that in the C5-Pe-QDs composites, triplet exciton energy from C5 can efficiently transfer to QDs with an emission peak surpassing 1.1 eV [13]. However, we have not yet investigated the diffusion of triplet excitons within C5 crystals or the C5-Pe-QDs composites. The C5-Pe-QDs composites can be regarded as a system where multiple Pe-QDs are encapsulated within the C5 crystals. After the C5 molecules undergo the SF process to generate triplet excitons, these triplet excitons typically diffuse via the Dexter energy transfer mechanism [15]. In this context, Pe-QDs act as traps within the C5 crystals, and when the triplet excitons are captured by the Pe-QDs, there exists an opportunity for the energy of these triplet excitons to be transferred to the QDs (Figure 1) [16]. Therefore, investigating the diffusion of triplet excitons is essential, especially for determining key parameters such as diffusion coefficients and diffusion lengths. This understanding will help clarify triplet exciton transport process in C5-Pe-QDs composites and provide valuable insights for material design to enhance the triplet exciton capture efficiency.

There are numerous experimental methods for studying exciton diffusion, including direct observation delayed luminescence spread [17], measurements in LED configuration [18], photocurrent and microwave conductivity [19,20], remote phosphorescent sensing [21], phosphorescent quenching [22], and modeling of absorption transients [23,24,25,26,27]. All the aforementioned experimental methods have been widely recognized and have been used in numerous studies to obtain triplet exciton diffusion information for various materials. However, most of the available techniques are complex, and each technique has specific requirements for the test materials. For example, direct observation delayed luminescence spread is only suitable for systems where the triplet exciton diffusion length is at the micron scale. Measurements in LED configuration require special multilayer designs. Differentiating the contributions of triplet and singlet excitons to photocurrent and microwave conductivity introduces additional complexity. Remote phosphorescent sensing involves a large number of fitting parameters and a complex theoretical model. Phosphorescent quenching is only applicable to phosphorescent materials. Given that our C5-Pe-QDs composites are already quite complex, we believe that the transient absorption technique is the most suitable for studying our composites. In recent years, for SF materials such as tetracene and perylenediimide derivatives, researchers have determined parameters like exciton diffusion coefficients and diffusion lengths using transient absorption microscopy [28,29]. However, transient absorption microscopy requires focusing the pump light into a very small area, and the excitation flux used is typically much higher than the solar flux [29], which causes significant singlet-singlet annihilation and triplet-triplet annihilation in SF materials. While triplet excitons can extend their diffusion length through singlet-mediated transport, triplet yield is often reduced due to pronounced exciton annihilation [29]. Moreover, transient absorption microscopy typically measures parameters related to the diffusion of triplet excitons mediated by singlet states [29,30]. To date, no studies have reported the diffusion behavior of spatially separated independent triplet excitons in perylenediimide derivative crystals.

In this study, to investigate the diffusion of triplet excitons in C5 crystals and C5-Pe-QDs composites, we employed optical techniques to probe intrinsic material properties and used low photoexcitation fluences to more closely mimic solar illumination conditions [31,32]. Pe-QDs with varying concentrations were introduced into the C5 crystals as traps to terminate the diffusion of triplet excitons, and the exciton lifetime was measured using femtosecond transient absorption (fs-TA) spectroscopy. As the Pe-QD concentration varied, the triplet exciton lifetime also changed. Based on the model of Zhitomirsky et al. [33,34], we calculated the triplet exciton diffusion coefficient and diffusion length. Subsequently, through Monte Carlo simulations [35], we studied the diffusion process of triplet excitons in the C5-Pe-QDs composites, revealing the relationship between triplet exciton capture probability and trap percentage, and calculated the capture probability of triplet excitons in the ideal distribution of Pe-QDs in the C5-Pe-QDs composites. The results indicate that in C5 crystals, the triplet exciton diffusion coefficient is approximately 3.58 × 10⁻⁵ cm² s⁻¹, with a diffusion length of approximately 50.9 nm. Under the ideal distribution of Pe-QDs, the triplet exciton capture probability in the C5-Pe-QDs composites is approximately 79.5%. Our findings suggest that in the C5-Pe-QDs composites, triplet excitons can efficiently diffuse to the Pe-QDs, making the effective utilization of triplet exciton energy possible.

2. Materials and Methods

To obtain information regarding the diffusion of triplet excitons, we selected Pe-QDs as the acceptor material and prepared C5-Pe-QDs composites. Previous studies have shown that the presence of Pe ligands facilitates the uniform dispersion of Pe-QDs within C5 crystals [13]. Additionally, compared to the original oleic acid ligands, using short-linker molecules such as Pe is advantageous in maintaining a shorter donor-acceptor distance [36]. Therefore, Pe ligands were selected to replace the original oleic acid ligands on the surface of the QDs. Specifically, a QD solution (0.5 mg/mL) was prepared in toluene, while a Pe solution (1.25 mg/mL) was prepared in tetrahydrofuran (THF). Prior to ligand exchange, the Pe solution was sonicated to disrupt any aggregates that might have formed. The QD solution was then mixed with the Pe solution and stirred for 1 h and 10 min, followed by purification through sequential precipitation, centrifugation, and resuspension to remove unbound ligands, yielding the Pe-QDs. Subsequently, the Pe-QDs were dispersed in THF to prepare solutions of varying concentrations, which were then individually mixed with THF solutions of C5. Water was then added, and the mixtures underwent bath sonication for 30 min to obtain the C5-Pe-QDs composites. The preparation process is detailed in the Supplementary Information (Section S1).

Steady-state absorption spectra were measured using a UV-Vis spectrophotometer (SHIMADZU-UV-3600, Shimadzu Corporation, Kyoto, Japan). The films were characterized via X-ray diffraction (XRD, Rigaku SmartLab SE, Rigaku Corporation, Tokyo, Japan) and crossed polarized optical microscopy (POM, OLYMPUS GX71, Olympus Corporation, Tokyo, Japan). Fs-TA measurements were performed utilizing a femtosecond broadband transient absorption spectrometer (TA-100, TIME-TECH SPECTRA, Dalian Chuangrui Spectra Co., Ltd., Dalian, China). Target analysis was employed to process the fs-TA spectral data. The diffusion coefficient and diffusion length of triplet excitons were calculated by referring to the model established by Zhitomirsky et al. [33,34]. The diffusion behavior of triplet excitons in the composites was simulated using the Monte Carlo simulation method. Detailed experimental procedures, computational methodologies, and simulation details can be found in the Supplementary Information (Sections S2–S5).

3. Results and Discussion

In the C5-Pe-QDs composites, Pe-QDs act as traps within the C5 crystals, capturing triplet excitons during their diffusion. To investigate the impact of trap percentage (${\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$) on triplet exciton dynamics, we selected three different concentrations of Pe-QDs relative to C5, resulting in ${\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ values of 0.010%, 0.026%, and 0.052%, respectively. The POM images of C5 crystal film, as well as C5-Pe-QDs (0.010%), C5-Pe-QDs (0.026%), and C5-Pe-QDs (0.052%) composite films, are shown in Figure 2. For the C5 crystal film (Figure 2a), the respective overall brightness of the multiple rod-like structures is consistent, which is similar to previous reports on perylenediimide derivatives [37]. In contrast, for the C5-Pe-QDs composite films (Figure 2b–d), the respective overall brightness of the rod-like structures is inconsistent, reflecting the incorporation of Pe-QDs. As the ${\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ increases, the crystal structure becomes more complex, indicating an increase in nucleation density [16], that is, more Pe-QDs participate in the crystallization process of C5 crystals. XRD patterns exhibit similar results (Figure S2). As ${\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ increases, more crystallographic orientations emerge of C5, and the characteristic diffraction peaks of QDs become more pronounced. These results confirm that Pe-QDs of varying concentrations were successfully incorporated into the C5 crystals, forming the C5-Pe-QDs composites. It has been reported that perylenediimide derivatives can form cocrystals with other molecules through π-π interactions [38], leading us to hypothesize that a structure similar to a donor-acceptor cocrystal may have formed within the C5-Pe-QDs composites synthesized in this study [39,40].

To investigate the diffusion of triplet excitons in C5-Pe-QDs composites, we performed fs-TA measurements with a time window ranging from 0 to 7000 ps. To minimize thermal artifact signals associated with high excitation densities [41], all fs-TA measurements were performed at a low excitation density of 19.1 $\mathrm{\mu }$J/cm². The test results are shown in Figure 3. The fs-TA spectra of C5 crystal film and C5-Pe-QDs composite films are quite complex, encompassing multiple dynamic processes of excited states. Referring to the analysis of absorption and kinetic characteristics of spatially separated, independent triplet (T₁) states in previous studies on perylenediimide derivatives [13,29,42], we attribute the absorption peak centered at 569 nm in our study to the excited-state absorption of T₁ states. The inset in the upper right corner of each spectrum shows the temporal evolution of the T₁ state absorption peak. It is evident that as ${\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ increases, the T₁ state absorption peak exhibits more pronounced attenuation, indicating that the capture probability of T₁ excitons increases with the density of traps.

To analyze the T₁ state dynamics from the fs-TA spectra, we applied target analysis (Figure 4) [43]. Based on scientific hypotheses, a compartmental model was established, and the fs-TA data were decomposed into the convolution of exponential decay concentrations for each component with the instrument response function [43]. This approach allows for effective analysis of the fs-TA measurements and the determination of the microscopic rate constants for each component. Further details can be found in Supplementary Information (Section S3). The results of the target analysis are consistent with our previous findings, where S₁, I, T_C1, and T₁ correspond to the singlet state, intermediate state, special triplet state, and spatially separated independent triplet state, respectively [13]. For the T₁ state, which is the primary focus of this study, we observe that as the ${\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ increases, the decay rate of the T₁ state dynamics (represented by the blue curve) significantly accelerates (Figure 4b,d,f,h). By fitting the target analysis results of the fs-TA spectra, we accurately reproduced the T₁ state dynamics for each film (Figure S3) and obtained the kinetic parameters for each film (Table S1) along with their corresponding standard errors (Table S2). The fitting results indicate that the decay rate of the T₁ exciton population (k_TD) increases with ${\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$, with the C5-Pe-QDs (0.052%) composite film exhibiting the highest k_TD, reaching 2.25 $\times$ 10⁻⁴ ps⁻¹. These results confirm that the transfer of T₁ excitons from the C5 crystals to the Pe-QDs occurs in C5-Pe-QDs, where Pe-QDs act as traps, capturing T₁ excitons more rapidly as the trap density increases.

Based on the model established by Zhitomirsky et al. [33,34], the diffusion coefficient of T₁ excitons in C5 crystals was calculated. The T₁ excitons undergo hopping transport between C5 molecules, and their mobility ($\mu$) can be expressed as [44]:

$$\mu ={\displaystyle \frac{{qd}^2}{6{\tau }_{hop}kT}}$$

(1)

Here, q is the charge of the carrier, d is the donor-acceptor separation distance [45], and ${\tau }_{\mathrm{h}\mathrm{o}\mathrm{p}}$ denotes the exciton transfer time between molecules, also referred to as the hopping time. k is the Boltzmann constant, and T is the temperature.

The Shockley-Read-Hall recombination model was employed to simulate the capture of T₁ excitons by Pe-QDs in the C5-Pe-QDs composites. Treating Pe-QDs as traps within the composites, the rate at which T₁ excitons are captured by Pe-QDs ($k_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$) is the inverse of the trapping lifetime (${\tau }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$) [46], expressed as:

$$k_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}={{\tau }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}}^{-1}=V_{\mathrm{t}\mathrm{h}}\sigma N_{\mathrm{t}}$$

(2)

Here, $V_{\mathrm{t}\mathrm{h}}$ represents the thermal velocity in the hopping regime, also expressed as d/${\tau }_{\mathrm{h}\mathrm{o}\mathrm{p}}$. $\sigma$ is the capture cross section of Pe-QDs. $N_{\mathrm{t}}$ denotes the trap density. A portion of the T₁ excitons diffusing within the composites will eventually encounter the Pe-QDs and be captured.

According to the Einstein equation relation concerning the diffusion coefficient (D) and mobility ($\mu$), it is known that [47,48]:

$$D={\displaystyle \frac{kT\mu }{q}}$$

(3)

By combining Equations (1)–(3), we obtain the following expression:

$$D={\displaystyle \frac{d{V}_{\mathrm{t}\mathrm{h}}}{6}}={\displaystyle \frac{d}{6\sigma {\tau }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}{N}_{\mathrm{t}}}}={\displaystyle \frac{d}{6\sigma \left(\frac{{\tau }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}}{{N_{\mathrm{t}}}^{-1}}\right)}}={\displaystyle \frac{d}{6\sigma \left(\frac{{\tau }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}}{{{\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}}^{-1}}\right)\left(\frac{1}{{N_{\mathrm{d}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{r}}}^{-1}}\right)}}$$

(4)

Here, ${\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ refers to the trap percentage in the composites. $N_{\mathrm{d}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{r}}$ denotes the density of C5 molecules in the C5 crystals, calculated by dividing the number of molecules per unit cell by the total volume of the unit cell. $N_{\mathrm{t}}={\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}\times N_{\mathrm{d}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{r}}$.

The root mean square displacement of an exciton from its initial position due to the diffusion process is called diffusion length. The diffusion length of T₁ excitons, L_D, can be calculated using the following equation [49]:

$$L_{\mathrm{D}}=\sqrt{2ZD\tau }$$

(5)

Here, Z denotes the dimensionality, where Z = 1, 2, or 3 for one-, two-, or three-dimensional diffusion, respectively. $\tau$ denotes the lifetime of T₁ exciton in the pure C5 crystals, defined as ${\tau }^{-1}={{\tau }_{\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}}^{-1}+{{\tau }_{\mathrm{C}5 \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}}^{-1}$, where ${{\tau }_{\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}}^{-1}$ and ${{\tau }_{\mathrm{C}5 \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}}^{-1}$ correspond to the phosphorescent radiative decay rate of the T₁ state and the rate of T₁ exciton capture by intrinsic traps in the pure C5 crystals, respectively [50]. In some publications, the factor of 2 in the expression for L_D is omitted. When the factor of 2 is omitted, the value of L_D is approximately equal to the average displacement of the exciton from its initial position. However, when the factor of 2 is included, the L_D value represents the root-mean-square displacement of the exciton from its initial position. To make L_D more suitable for the three-dimensional Monte Carlo simulations presented later in our study, we chose to retain this coefficient. The T₁ exciton lifetime in pure C5 crystals equals the reciprocal of the population decay rate, which we previously determined from fitting the results of the target analysis of the fs-TA spectra of C5 crystal film, yielding ${\tau }^{-1}=$ k_TD $=$ 8.29 × 10⁻⁶ ps⁻¹ (Table S1). Considering that T₁ state phosphorescence radiative primarily occurs on the microsecond timescale, far beyond the timescale of k_TD [30], the ${{\tau }_{\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}}^{-1}$ term can be neglected. Consequently, ${{{\tau }_{\mathrm{C}5 \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}}^{-1}\approx \tau }^{-1}=$ 8.29 × 10⁻⁶ ps⁻¹.

In the C5-Pe-QDs composites, the reciprocal of the T₁ exciton lifetime can be approximated as ${{\tau }_{\mathrm{C}5-\mathrm{P}\mathrm{e}-\mathrm{Q}\mathrm{D}}}^{-1}\approx {{\tau }_{\mathrm{C}5 \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}}^{-1}+{{\tau }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}}^{-1}$. This indicates that the T₁ exciton lifetime is governed by both the capture rate of T₁ excitons by intrinsic traps of the C5 crystals and the capture rate of T₁ excitons by Pe-QDs [13]. For the C5-Pe-QDs (0.052%) composite film, ${{\tau }_{\mathrm{C}5-\mathrm{P}\mathrm{e}-\mathrm{Q}\mathrm{D}}}^{-1}=$ k_TD $=$ 2.25 × 10⁻⁴ ps⁻¹, which we obtained through fitting the results of the target analysis of its fs-TA spectra (Table S1). With ${{\tau }_{\mathrm{C}5 \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}}^{-1}=$8.29 $\times$ 10⁻⁶ ps⁻¹, ${\tau }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ can then be derived. Substituting ${\tau }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ into Equation (4) yields the D of T₁ excitons, and subsequently inserting D into Equation (5) gives the L_D. Ultimately, we calculated the D of T₁ excitons to be approximately 3.58 × 10⁻⁵ cm² s⁻¹ and the L_D to be about 50.9 nm. The detailed calculation process is provided in the Supplementary Information (Section S4). In reference to studies on triplet exciton diffusion [49], our calculated D and L_D fall within a reasonable range.

Compared with related studies on similar perylenediimide derivatives [29], the D and L_D values we calculated are smaller. This is because our calculated D and L_D correspond to triplet excitons that are fully spatially separated after the SF process, with their population decay occurring at least after 1500 ps (Figure 5a,b). In contrast, the triplet excitons in the related studies can undergo singlet-mediated transport via bimolecular triplet exciton fusion, a process that occurs on a much shorter timescale [29]. Singlet-mediated transport can effectively enhance the D and L_D of triplet excitons; however, the triplet exciton fusion process competes with the SF process. As singlet-mediated transport improves, the yield of spatially independent triplet excitons decreases, indicating a trade-off between triplet yield and singlet-mediated transport [29]. Our experiments were performed with a very low excitation density, which more closely simulates solar illumination conditions and minimizes triplet exciton fusion, thereby ensuring a high triplet yield. The triplet yield of the C5 crystals and C5-Pe-QDs composites we prepared remains between 166% and 170% [13].

The D and L_D of the T₁ excitons we calculated were derived from experimental data obtained from C5 crystals and C5-Pe-QDs (0.052%) composites. In fact, similar calculations for ${\tau }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ can also be performed using experimental data from C5-Pe-QDs (0.010%) and C5-Pe-QDs (0.026%) composites, but their associated errors are relatively large. This is because the triplet lifetimes in C5-Pe-QDs (0.010%) and C5-Pe-QDs (0.026%) are longer and exhibit greater errors compared to C5-Pe-QDs (0.052%). As a result, the errors in the calculated ${\tau }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ values for these two groups of data are also larger (Figure 5c). The detailed calculation process is provided in the Supplementary Information (Section S4). By plotting ${\tau }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ on the x-axis and ${{\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}}^{-1}$ on the y-axis, a linear correlation is observed between the two variables (Figure 5c), which indicates that no significant aggregation of Pe-QDs occurs as the concentration increases. If aggregation were to happen, it would reduce the effective ${\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$, and the measured ${\tau }_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ would be longer than expected based on the linear trend [32]. Therefore, this linear correlation actually suggests that Pe-QDs are uniformly dispersed throughout the film rather than aggregating. This linear relationship further confirms the validity of our calculation method.

Given the known values of D and L_D, we employed the Monte Carlo simulation method to model the diffusion of T₁ excitons in the C5-Pe-QDs composites within three-dimensional space, in order to calculate the T₁ exciton capture probability. Triplet exciton diffusion follows the Dexter energy transfer mechanism [49], involving hopping transport between donor and acceptor molecules [51]. Accordingly, we utilized a random walk model to simulate the diffusion of triplet excitons [52,53,54]. In the model, the T₁ excitons have two possible outcomes: either they survive until the set stop time and recombine, or they reach a location within the capture range of the Pe-QDs and undergo T₁ state energy transfer. In these simulations, Pe-QDs are randomly distributed within the three-dimensional space, and the excitons perform 1000 random walks. Detailed information regarding the Monte Carlo simulation is provided in the Supplementary Information (Section S5). Through these simulations, we obtained a plot of the T₁ exciton capture probability as a function of ${\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ (Figure 6a), which presents the simulation results for fifteen different ${\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ values of C5-Pe-QDs composites. The capture probability results for C5-Pe-QDs (0.010%), C5-Pe-QDs (0.026%), and C5-Pe-QDs (0.052%) composites are specifically highlighted. Representative random walk trajectory simulations for the T₁ excitons are shown in Figures S4–S6.

As ${\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ increases, the capture probability of T₁ excitons rises from 7.0% to approximately 92.0%, with the specific probabilities and standard errors provided in Table S3. However, it is important to note that when PbS QDs are closely packed, fluorescence quenching may occur due to inter-dot energy transfer [55,56]. Therefore, although the capture probability of T₁ excitons increases with ${\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$, this does not necessarily imply that the photoluminescence quantum yield of the C5-Pe-QDs composites will also increase. During the composite preparation, it is crucial to ensure that the QDs are uniformly distributed and sufficiently spaced apart to avoid the occurrence of fluorescence quenching. Accordingly, we simulated the capture probability of T₁ excitons in the C5-Pe-QDs composites under the ideal distribution of Pe-QDs. Figure 6b shows a simulation of the three-dimensional random walk trajectory of one T₁ exciton that is ultimately captured under this ideal distribution, with additional random walk trajectory simulations provided in Figure S7. Detailed information is available in the Supplementary Information (Section S5). The results indicate that under this ideal distribution, the triplet exciton capture probability is 79.5% (±1.9%). Combining this with the high triplet yield (166–170%) [13], it can be calculated that under the ideal distribution of Pe-QDs, the transfer efficiency of T₁ excitons in the C5-Pe-QDs composites can reach up to approximately 129–138%.

4. Conclusions

To investigate triplet exciton diffusion in perylenediimide derivatives, we synthesized C5 crystals and C5-Pe-QDs composites. Using fs-TA measurements, target analysis, and a series of calculations, we determined the D (3.58 × 10⁻⁵ cm² s⁻¹) and L_D (50.9 nm) of T₁ excitons in C5 crystals. Using Monte Carlo simulations, we established the relationship between the T₁ exciton capture probability and ${\%}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ in the C5-Pe-QDs composites. The results demonstrate that, under the ideal distribution of Pe-QDs, T₁ excitons in the C5-Pe-QDs composites can efficiently diffuse to the Pe-QDs, with a T₁ exciton capture probability of 79.5% (±1.9%). This suggests that the energy of triplet excitons has the potential to be efficiently harnessed. However, several challenges remain for the practical application of these composites: first, ensuring the orderly and uniform arrangement of Pe-QDs within the C5 crystals, and second, improving the fluorescence efficiency of the QDs [57]. Addressing these issues will require further optimization of the fabrication process. Once these challenges are overcome, C5-Pe-QDs composites hold great potential for applications in the field of photovoltaics.