# Efficient Energy Transfer Between Ligand-Free FAPbBr3 Nanocrystals in a Mesoporous SiO2 Matrix

Victor Guilloux, Thierry Barisien, Laurent Legrand, Carlos Romero-Perez, Juan Galisteo-Lopez, Mauricio Calvo, Hernán Miguez, Maria Chamarro

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Victor Guilloux, Thierry Barisien, Laurent Legrand, Carlos Romero-Perez, Juan Galisteo-Lopez, et al. Efficient Energy Transfer Between Ligand-Free FAPbBr3 Nanocrystals in a Mesoporous SiO2 Matrix. Advanced Optical Materials, 2025, 13 (36), pp.e02066. 10.1002/adom.202502066. hal-05521131

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Efficient Energy Transfer between Ligand-Free FAPbBr₃ Nanocrystals
in a Mesoporous SiO₂ Matrix

V. Guilloux¹, T. Barisien¹, L. Legrand¹, C. Romero-Perez², J. F. Galisteo-Lopez², M. E. Calvo²,
H. Miguez² and M. Chamarro¹*

¹ Sorbonne Université, CNRS, Institut des Nanosciences de Paris, F-75005, Paris, France.

² Instituto de Ciencia de Materiales de Sevilla (ICMS)

Consejo Superior de Investigaciones Científicas – Universidad de Sevilla (CSIC-US)

Américo Vespucio, 49, Sevilla 41092, Spain.

**Keywords:** Perovskite Nanocrystals, Exciton Transport, Förster Energy Transfer, Mesoporous Silica Matrix, Photoluminescence Spectroscopy.

Lead halide perovskite nanocrystals (NCs) are promising materials for next-generation optoelectronic devices due to their exceptional optical properties. However, poor long-term stability remains a major challenge. In this study, we embed FAPbBr₃ NCs into a mesoporous silica matrix to enhance stability and explore exciton transport mechanisms. These NCs display a narrow photoluminescence (PL) linewidth of 25 meV at 7 K. The absence of surface ligands leads to reduced interparticle spacing, favoring non-radiative Förster resonance energy transfer (FRET) as the dominant exciton transport mechanism. Using time-resolved and spectrally-resolved PL spectroscopy at cryogenic temperatures, we observe significant spectral redistribution over time, indicating energy transfer from higher-energy to lower-energy NCs. To quantitatively interpret these dynamics, we employ a theoretical model based on a two-dimensional array of coupled NCs, incorporating Förster’s theory to simulate exciton diffusion. Our model successfully reproduces the experimentally observed PL decay behavior, confirming FRET-mediated exciton transport with an upper-limit efficiency close to 100% and a transfer rate of 105 ns⁻¹. These findings offer key insights into energy transfer processes in ligand-free perovskite NC systems and underscore the potential of mesoporous silica matrices for improving stability and enabling control over excitonic interactions in perovskite-based optoelectronic applications.

*corresponding author: maria.chamarro@insp.jussieu.fr
# 1. Introduction

In recent years, organic and inorganic lead halide perovskites have been the focus of numerous studies, demonstrating their exceptional optoelectronic and photovoltaic properties [1]. These materials are promising candidates for multiple applications, including - but not limited to - solar cells [2], light-emitting diodes [3,4], lasers [5], photodetectors [6] as well as for more advanced applications in the spintronics [7-9] and quantum information domains [10]. To these outstanding properties must be added the very easy and low-cost chemical manufacturing methods used to obtain these materials.

However, films made from deposited lead halide perovskite NCs are highly sensitive to environmental factors and can degrade rapidly under exposure to moisture, oxygen, light or heat [11]. This instability poses challenges for the long-term reliability of optoelectronic devices based on these materials. To address these challenges, new fabrication strategies have been developed or adapted. The strategy that consists of manufacturing complex core-shell type structures as in the homologues II/VI semiconductor-based NCs is still facing difficulties [12,13]. Another promising technique is the encapsulation, where NCs are embedded within a protective matrix composed of materials such as polymers (polymethyl metacrylate, PMMA, or polystyrene), silica or alumina, which shields the NCs from aggressive environmental conditions and enhances their stability. Among these encapsulation approaches, mesoporous silica matrices have emerged as a promising solution. Indeed, some research groups have demonstrated that NCs can be synthesized directly within their pores without the need for capping ligands, thereby protecting the NCs from the environment [14, 15]. Further protection can also be realized by covering the impregnated matrices with PMMA to seal off the pores [16], which can be advantageous especially on the shallow layers of the material.

Optical absorption of photons generates excitons—electron-hole pairs bound by Coulomb interaction and confined within NCs. A central question, with both fundamental and technological implications, is how these primary excitations propagate through the sample or device, a process governed by the connectivity and coupling between NCs. This question becomes particularly critical in strongly confined systems, such as nanoplatelets and small NCs, where excitons possess high binding energies and remain stable even at room temperature. Specifically, when the exciton binding energy is smaller than or comparable to the thermal energy at room temperature, two distinct regimes must be distinguished: (i) room temperature
under high excitation density, and (ii) low temperature under very low excitation density. In the former case, de-excitation dynamics are governed primarily by free electrons and holes, whereas in the latter, excitons dominate the dynamical behavior. More complex scenarios emerge in intermediate regimes, where free carriers and excitons may coexist and compete, resulting in rich and mixed transport and recombination dynamics.

Understanding these distinct regimes is essential for interpreting charge and energy transport in different environments. In mesoporous matrices containing NCs, sufficiently high loadings levels reduce the interparticle distance compared to colloidal NC films of similar average size, primarily due to the absence of surface ligands. Recent studies have demonstrated that, at room temperature, strong connectivity between ligand-free FAPbBr₃ NCs [17] significantly enhances the diffusion of free electrons and holes. However, this improved transport also increases the likelihood of charge carriers encountering defects, thereby promoting non-radiative recombination.

At low temperatures—such as those of liquid helium— excitation transport between NCs is expected to be mediated by exciton incoherent, long-range dipolar interactions, specifically through non-radiative Förster resonance energy transfer (FRET). In this scenario, the exciton in a donor NC couples to an absorbing transition in a nearby acceptor NC, typically within a distance of 1 to 10 nm.

Regarding optoelectronics, FRET can be desirable or detrimental depending on the application. For example, in photovoltaics, large diffusion lengths can facilitate the transport of excitons towards the charge separation interfaces of the device, which can improve the performances of a solar cell [18-20]. Currently, the highest values of FRET-mediated diffusion length and diffusivity ($l = 200$ nm, $D = 0.1$ cm²/s) have been observed in highly ordered monolayers of "isoenergetic" NCs made of CsPbBr₃ [21]. In the context of light-emitting devices optimization, excessive exciton diffusion can be detrimental, as it inhibits radiative recombination within the NC where the exciton was generated, thereby increasing the likelihood of non-radiative trapping in adjacent layers [22]. To another view, FRET can be advantageous in light-emitted diodes (LEDs), where it has been successfully exploited to achieve a broader emission spectrum and enhance color tunability [23, 24]. FRET-based photodetectors and sensors are also associated to enhance accuracy and selectivity of analytical detection across a broader range [25, 26]. Either way, the understanding of exciton diffusion by FRET processes in new optoelectronic systems such as lead halide perovskite NCs is of great importance [27].

Several groups have investigated energy transfers between II-VI semiconductor NCs [28-32] but much less studies have been published so far on lead halide perovskite NCs, even though their
defect-tolerant properties are expected to limit the impact of trap states on the exciton diffusion properties. Among these studies De Weerd and coauthors [33] were the first to report FRET between CsPbBr₃ NCs. In 2D CsPbBr₃ NC films, Penzo *et al.* measured an exciton diffusion length of about 200 nm [21], while Giovanni *et al.* [34] demonstrated that energy transfer *via* FRET is the dominant exciton transport process at the microscopic length scale in MAPbBr₃ NC films and measured an exciton diffusion length larger than 1 µm typically, a value unexpectedly higher than what was measured in the bulk. Their result shows that the energy transfer between NCs counters the usual limitation of the exciton transport properties attributed to the surrounding ligands and the quantum confinement. The CsPbBr₃ core-CsPbI₃ shell nanostructure was designed to minimize the distance between perovskite donor (CsPbBr₃) and acceptor (CsPbI₃), leading to intense red photoluminescence (PL) with a quantum yield of 100 % [35]. A. Singldinger and coworkers achieve a 70% efficiency for FRET mechanism and a transfer rate of 0.99 ns⁻¹ by precisely controlling the thickness of two-dimensional (2D) CsPbBr₃ nanoplatelets [36]. A two orders of magnitude higher transfer rate was reported in (PEA)₂(MA)ₙ⁻₁PbₙI₃n+1 2D layered perovskites, likely due to the quasi-perfect spectral overlap and optimal orientational alignment of adjacent layers [37].

In this work, we investigate exciton transport in ligand-free FAPbBr₃ NCs embedded in a mesoporous silica matrix. Using time- and spectrally-resolved PL spectroscopy at cryogenic temperatures, we demonstrate that the energy migration process can perfectly be described by non-radiative FRET exclusively. A model based on a 2D array of coupled NCs successfully reproduces the observed PL dynamics, confirming efficient exciton diffusion on a sub-nanosecond timescale. These findings provide valuable insights into exciton transport in perovskite NC ensembles and highlight the potential of mesoporous silica matrices for stabilizing and optimizing perovskite-based optoelectronic devices.

## 2. Results and discussion

### 2.1 Sample fabrication and characterization.

The system investigated in this work consists of a 200 nm-thick nanoparticle based mesoporous silica matrix, deposited by dip coating from a suspension of 30 nm diameter SiO₂ colloids diluted in methanol. [16] In all cases, low-fluorescence glass was used as substrate. A solution containing ABX₃ precursors (two ionic halide salts: FABr and PbBr) in dimethylsulfoxide (DMSO) is briefly spin coated on the prepared mesoporous SiO₂ film. After this infiltration
step, the system is thermally treated at 100°C leading to the formation of ligand-free NCs of FAPbBr₃ embedded into SiO₂ porous matrix. The content of the precursors in the original solution is denoted by X, representing the weight fraction, in %(m/v): X = 0.5, 1.0, 3.0 or 30. **Figure 1a** provides an illustration of the system prepared following the aforementioned method. **Figure 1b** presents the absorption and PL spectra of a 30 % sample measured at 7 K. The PL peak is centered at 2.22 eV, with a linewidth of 25 meV. The absorption maximum is blue-shifted by approximately 40 meV relative to the PL peak. This shift between the absorption and PL spectra is a common feature observed in perovskite nanocrystals (NCs) and other families of NCs [38, 39]. The magnitude of this shift is influenced by the NC size and tends to increase as the size decreases. For instance, shifts of around 35-40 meV have been previously reported for cubic-shaped CsPbBr₃ NCs with a size of 9 nm [38]. For the sake of completeness, the evolution of the absorption response as a function of temperature is provided in the Supporting Information section 1.

The calculated exciton absorption energy, E, minus the band gap energy at low temperature of FAPbBr₃, *E*<sub>g</sub> = 2.233 eV [40], normalized by the exciton binding energy, *E*<sub>x</sub>(FAPbBr₃) = 22 meV is shown in **Figure 1c** as a function of the ratio of the NC size, *L*, and the exciton Borhr radius, *a*<sub>x</sub> (*a*<sub>x</sub> = 3.88 nm for FAPbBr₃) [40]. Three cases are considered: strong and weak confinement, as well as, the intermediate confinement [41] within the hypothesis of cubic shape NCs. We observe that as the ratio *L*/*a*<sub>x</sub> approaches ten, the value of (*E* - *E*<sub>g</sub>)/*E*<sub>x</sub> from intermediate confinement calculations converges to that of weak confinement. In the opposite limit, as *L*/*a*<sub>x</sub> tends to 1, (*E*-*E*<sub>g</sub>)/*E*<sub>x</sub> approaches the value obtained in the strong confinement. The horizontal dashed line in **Figure 1c** represents the normalized absorption energy at its peak in the studied sample. The average NC size estimated in the case of intermediate confinement (*L* ≈ *a*<sub>x</sub>) is around 11 nm.

Tamarat and coworkers [42] investigated colloidal FAPbBr₃ NCs deposited on films with an average size of *L* ≈ 9.2 nm and a size dispersion characterized by a full width at half maximum of ≈ 1.5 nm. They found that the PL energy of individual FAPbBr₃ NCs ranges between 2.21 eV and 2.23 eV. Accounting for all aspects, we conclude that FAPbBr₃ NCs embedded in the silica scaffold have an average size of approximately 10 nm (very close to the theoretically estimated value), indeed placing their excitonic properties within the intermediate confinement regime. Considering an energy dispersion corresponding to the FWHM of the PL peak (ie ≈ 25 meV) it is finally possible to roughly estimate a size dispersion (FWHM again) in this regime using the curve of **Figure 1c**. We find it equal to ± 2 nm indicating that a large amount of NCs have their sizes between 8 and 12 nm typically, with a non-negligible 'size spreading' on the
edges.

As already mentioned, due to the absent of ligands the small inter-NC distance is favorable to

<figure><img src="image_2.png" /><figcaption>**Figure 1.** (a) Illustration of the mesoporous matrix filled with FAPbBr₃ NCs. The gray spheres of diameter 30 nm represent the silica beads while the NCs are depicted in green as smaller green cubes of average length L = 10 nm. (b) Low-temperature photoluminescence (PL) (black line) and absorption (red line) spectra of FAPbBr₃ nanocrystals (NCs) embedded in a mesoporous silica matrix. The sample was prepared using a solution with a 30 wt% precursor concentration, leading to a high NC density within the pores of the host matrix. The absorption spectrum is corrected for low-energy Rayleight-type scattering; the remaining low-energy tail reflects static disorder (Urbach tail), while some residual scattering contribution may still be present. (c) Calculated exciton shift, $(E-E_g)$, normalized to the exciton binding energy, $E_x$, as a function of the normalized NC size, $L/a_x$, for the different confinement regimes: weak, strong and intermediate (red, blue and purple solid line, respectively).</figcaption></figure>

the coupling of closer NCs via non-radiative FRET. In the following, to explain the experimental observation we exclude the Dexter energy transfer process [43] since this mechanism involves electron exchange between the donor and acceptor, requiring an overlap of their wavefunctions. This overlap corresponds to a maximum range that is too short for the NC system under investigation. Additionally, we exclude photon recycling [44, 45], a mechanism that can contribute to excitation diffusion in densely packed films. D. Giovanni and coworkers [34] demonstrated that the contribution of photon recycling to excitation diffusion is weaker than that of FRET in lead halide perovskite films. In particular, we see thereafter that long range inter-NCs coupling of that type is not required to obtain satisfying simulations the PL dynamics.
## 2.2 Photoluminescence of isolated single FAPbBr₃ NCs

In NC ensembles, size dispersion leads to broad PL and absorption spectra, with each NC size contributing at a specific energy. Smaller NCs emit and absorb on the blue side of the spectrum, while larger NCs do so on the red side. Due to this inhomogeneous broadening, the overall spectrum does not accurately determine spectral overlaps between individual donors and acceptors. To address this, we performed highly spatially resolved PL experiments on samples with similar NC size distribution but at lower concentrations (0.5 or 1 %), allowing us to isolate and analyze the PL of individual NCs. This kind of experiments that we routinely conduct is described in the Experimental section.

Preliminary results indicated that reduced precursor concentrations were advantageous for isolating single-particle behavior; thus, we focused further experiments on the 0.5% sample. However, this sample exhibited highly heterogeneous NC distribution. Certain regions displayed PL emission characteristic of densely packed ensembles, similar to the 3% sample (see Supporting Information, Section 2), whereas most of the sample showed no detectable emission. Although the underlying cause of this phenomenon is not fully understood, it was consistently observed across multiple preparations. This suggests the existence of a threshold precursor concentration below which NC spatial distribution becomes significantly

In contrast, the 1% sample, while still relatively dense in most areas, exhibited regions where emission from individual NCs could be resolved. Representative PL spectra from such regions are shown in **Figure 2a,b**. The presence of well-resolved peaks in these spectra reflects the excitonic fine structure. As previously reported, both doublet and triplet features can appear in the PL of single perovskite NCs [42, 46-48]. The observed energy splittings—1.18 meV for the doublet and 0.69 meV and 1.25 meV for the triplet—are primarily governed by the electron-hole exchange interaction and the crystal field strength associated with the tetragonal or orthorhombic lattice structure [41, 46, 49]. These values, along with the linewidths, are in good agreement with previous studies on isolated FAPbBr₃ NCs of comparable size [42]. Further details on the exciton fine structure are provided in Supporting Information, Section 3.

Time-resolved PL measurements were also conducted on the 1% sample to probe the emission dynamics of 'non-interacting' FAPbBr₃ NCs at various emission energies within the spectral range characteristic of higher concentration samples. **Figure 2c** shows the time-integrated PL
spectrum at 7 K, while **Figure 2d** presents the corresponding spectro-temporal PL map and representative decay curves at selected emission energies. The PL decay profiles are well described by mono-exponential fits, yielding characteristic lifetimes ($\tau$) with slight variation across the spectral window ($\tau \approx 215 \pm 45$ ps), as shown in **Figure 2e**.

<figure><img src="image_3.png" /></figure>

**Figure 2.** PL spectra of individual emitters acquired on the X = 1 % sample at T = 7 K (excitation at ~400 nm): (a) response under the form of a doublet, (b) triplet-like response. The colored solid lines are Lorentzian fits. (c)-(e) Typical emission dynamics pattern of non-interacting NCs (low concentration sample) measured across the spectral range corresponding to the inhomogeneous PL of higher concentration samples. (c) Time integrated PL spectrum. (d) PL spectro-temporal map (streak-camera measurement). Several illustrative plots of the decays have been superimposed (black lines) for a selection of emission energies (indicated by the arrows). (e) Decay constant, $\tau$, extracted from the mono-exponential fit of the PL signal as a function of the emission energy. $\tau$ is found slightly fluctuating in the explored energy range ($\tau \approx 215 \pm 45$ ps).

Reported values for the radiative lifetime of FAPbBr₃ NCs with sizes ranging from 8 to 12 nm indicate a size-dependent trend, with a minimum lifetime of 630 ps for NCs averaging 10 nm in diameter [49]. Accordingly, the non-radiative decay rate for 'non-interacting' NCs decreases with increasing NC size and is estimated to be $\Gamma_{NR} = 3 \pm 0.5$ ns⁻¹ for NCs with L = 10 ± 2 nm. This finding indicates the presence of non-radiative recombination channels even in the absence of inter-NC interactions.
It is worth noting that the invariance of the emission lifetime with energy in isolated nanocrystals constitutes a significant advantage for the interpretation of spectro-temporal dynamics in denser phases, where lifetime variations will, in contrast, be evident and will necessitate consideration of interactions between emissive centers.

## 2.3 Time- and spectrally-Resolved Photoluminescence results

To investigate exciton coupling dynamics mediated by Förster Resonance Energy Transfer (FRET) within a quasi-homogeneous ensemble of FAPbBr₃ nanocrystals (NCs) embedded in a silica scaffold, we conducted time- and spectrally-resolved PL measurements. **Figure 3** shows a representative energy-time PL intensity map for a sample fabricated using a precursor solution with a high concentration (X = 30%) of FAPbBr₃. The upper panel displays the time-integrated PL spectrum reconstructed from the energy-time dataset. Measurements were performed at a cryogenic temperature of 7 K under non-resonant excitation at 397 nm, achieved via the second harmonic generation of a Ti:Sapphire laser, with an excitation power maintained at approximately 1 µW to minimize photo-induced effects. This energy-time PL representation provides a rapid visual assessment of the underlying emission dynamics within the system. A pronounced temporal asymmetry is evident in the PL response: a fast-decaying high-energy tail precedes a more intense, longer-lived emission centered at approximately 2.22 eV. This behavior suggests a combined influence of the intrinsic size distribution of the NCs and inter-nanocrystal energy transfer processes on the PL dynamics.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Energy-time domain photoluminescence (PL) intensity map recorded from a sample</figcaption></figure>
consisting of FAPbBr₃ nanocrystals (NCs) embedded in a mesoporous SiO₂ matrix, synthesized using a high precursor concentration (X = 30%) (main panel). The PL spectrum shown at the top (blue solid line) corresponds to the time-integrated emission over the first 50 ps around the excitation at t = 0. A Gaussian fit (dark red dashed line) is applied to the low-energy portion of this spectrum to highlight its asymmetric shape. The PL decay curves displayed on the right (semi-logarithmic scale) are obtained by integrating the PL signal over 2 meV-wide energy intervals centered at the energies indicated in the legend.

From the PL map, the time evolution of the emission intensity at specific energies was extracted and is shown on the right side of **Figure 3**. The high-energy spectral region was sampled using six equally spaced energy windows, each 2 meV in width, to enhance the signal-to-noise ratio in the resulting decay curves. Two key trends emerge from this analysis. First, the PL decay becomes progressively slower at lower emission energies. For example, at 2.26 eV the emission exhibits a half-life of approximately 55 ps with a multi-exponential decay profile, while at 2.23 eV the half-life extends to roughly 165 ps. Second, the time to reach peak emission intensity is shorter for higher-energy NCs, with a temporal shift of approximately 30 ps observed between the 2.26 eV and 2.22 eV emissions. It is important to note that the PL decay traces display complex kinetics, precluding accurate modeling with simple mono- or bi-exponential functions due to the likely superposition of multiple exponential contributions. Consequently, the temporal dynamics are characterized here in terms of effective half-lives.

The observed behavior—consistent with reports in other dense NC films—is indicative of FRET-mediated energy transfer from higher-energy (smaller) NCs to lower-energy (larger) neighbors [29, 31, 33]. Upon photoexcitation, high-energy NCs may relax radiatively or non-radiatively, or they may transfer their excitation energy to adjacent NCs with lower-energy exciton transitions. This transfer results in shorter effective lifetimes for the donor NCs due to the increased likelihood of non-radiative transfer pathways. To validate this interpretation, we performed numerical simulations based on a FRET model, which were then fitted to the experimental spectro-temporal PL data. For this purpose, we selected the dataset shown in **Figure 3** as a reference, as it represents a typical response observed in spatially homogeneous regions of the sample where minimal variation in emission characteristics was detected.

**2.4 Theoretical model and comparison with experimental results**

FRET rate value between donors and acceptors depends on three key factors [50, 51]. First, there must be spectral overlap between the normalized fluorescence spectrum of the donor and the absorption spectrum of the acceptor, as this overlap is essential for efficient energy transfer.
Second, the donor-acceptor distance ($R_{DA}$) plays a critical role in determining the Coulomb coupling strength. According to Förster's theory, this interaction follows an $R^{-6}$ dependence, meaning that efficient energy transfer occurs only at the shortest distances. Finally, the transition dipoles of both the donor and the acceptor must be correctly oriented relative to each other to facilitate energy transfer. These conditions were summarized by Förster in the following equation:

$$ \Gamma_t = \frac{1}{\tau_D} \left( \frac{R_0}{R_{DA}} \right)^6 \quad (1) $$

where $\Gamma_t$ is the transfer rate of the process and depends on the exciton lifetime $\tau_D^{-1} = \Gamma_R + \Gamma_{NR}$ with $\Gamma_R$ and $\Gamma_{NR}$ the radiative and non-radiative rates associated to the exciton recombination, respectively. $R_{DA}$ is the donor-acceptor distance and $R_0$ is the distance at which the transfer has a 50 % probability of occurrence; it is given by:

$$ R_0^6 = \frac{9 \ln(10) \Phi_D \kappa^2}{128 \pi^5 N_A n^4} J(\lambda) \quad \text{with} \quad J(\lambda) = \int_{-\infty}^{+\infty} F_D(\lambda) A(\lambda) \lambda^4 d\lambda. \quad (2) $$

Here, $\Phi_D$ is the PL quantum yield of the donor, $N_A$ is Avogadro's number, $n$ is the refractive index of the surrounding medium, $\kappa^2$ is the dipole-dipole orientation factor, which ranges from 0 to 4. A common assumption is to consider that transition dipole moments in an ensemble of chromophores are isotropically oriented yielding $\langle \kappa^2 \rangle = 2/3$, though in systems with higher-order structures, such as dense films of NCs or superlattices, a higher value could be obtained. Finally, $J(\lambda)$ represents the overlap integral between the normalized fluorescence spectrum of the donor $F_D(\lambda)$ and the absorption of the acceptor $A(\lambda)$, weighted by a function of the optical frequency ($\lambda^4$ term).

In the following, we consider the characteristics of the studied system (NCs embedded in the silica skeleton) to model the dynamics of the NCs emission. We adapt the previously mentioned equations (see below) in order to predict the dynamics and reach a satisfying adjustment of the decay curves presented in **Figure 3**.

The host matrix consists in a packing of silica spheres with an average diameter of approximately 30 nm, arranged without any specific order or symmetry. This means that the pores between the silica spheres likely follow a size distribution, the measurement of which is beyond the scope of this work. However, as a starting point, we estimate the volume of pores in the case of compact arrangements of silica spheres and the number of NCs contained in pores, assuming a 100% filling factor (see Supporting Information). In the model, we assume that the
matrix pores are completely filled with NCs and, to simplify the problem, we consider that the NCs in the pores are distributed in 2D arrays of adjustable sizes. A square lattice thus describes a pore with a number of NC sites defined by the pore size. The corresponding simulation results will be compared to the simulations obtained in the 2D model (see Supporting Information).

As the excitation laser is tuned to higher energy compared to the absorption threshold and focused over a ≈ 1 µm diameter spot, several pores are excited simultaneously. To reproduce this, the simulations are performed with multiple pores excited independently. In these simulations, intra-pore energy transfers are allowed, but the inter-pores transfers are not. Following the work of Miyazaki and Kinoshita [28, 52], the exciton dynamics is governed by a set of first order differential equations:

$$ \frac{d\rho_n}{dt} = \sum_{m \neq n} \Gamma_{t,nm} \rho_m - (\Gamma_{t,nm} + \Gamma_{R,n} + \Gamma_{NR,n}) \rho_n \quad (3) $$

In this expression, $\rho_n$ is the occupation probability of the site n (n=1 to N, the total number of sites in the considered squared lattice). $\Gamma_{R,n}$ and $\Gamma_{NR,n}$ denote the radiative and non-radiative decay rates for the exciton on site n, respectively. The sum of these two quantities is extracted from time-resolved PL of isolated NCs, as discussed in section 2.2. Following the results in Figure 2c-e, we also make the approximation that $\Gamma_{R,n} + \Gamma_{NR,n}$ is constant and independent of the NC size, ie $\Gamma_{R,n} + \Gamma_{NR,n} = 4.8 ns^{-1} = \tau_D^{-1}$.

The energy transfer is accounted for through the terms $\Gamma_{t,nm}$ and $\Gamma_{t,nn}$. The first term represents the transfer rate from site m to site n, with m and n being nearest neighbors, whereas the second term contains all the transfer components from site n to its nearest neighbors. The assumption of an efficient energy transfer only to first neighbors (8 for a cubic lattice, including the diagonal elements) is reasonable given the mean size of the studied NCs (10nm). The transfer rates, $\Gamma_{t,nm}$, between coupled NCs are given by the Förster expression, following Fleming [53]:

$$ \Gamma_{t,nm} = \frac{|c_{n,m}|^2}{2\pi\hbar^2} \int_{-\infty}^{+\infty} F_{m(\omega)} A_n(\omega) d\omega, \quad (4) $$

where the parameter $c_{n,m}$ represents a Coulomb dipole-dipole coupling constant between NCs n and m and $F_{m(\omega)}$, $A_n(\omega)$ are the spectral line shapes of the donor (fluorescence) and acceptor (absorption), respectively. In this model, both spectra are considered to have a Gaussian profile:
$$F_m = \frac{\hbar}{\sqrt{2\pi}\sigma_h} \exp\left[-\frac{(\hbar\omega - \epsilon_m + \Delta_{SS})^2}{2\sigma_h^2}\right] \quad (5)$$

$$A_n = \frac{\hbar}{\sqrt{2\pi}\sigma_h} \exp\left[-\frac{(\hbar\omega - \epsilon_n)^2}{2\sigma_h^2}\right] \quad (6)$$

Here, $\sigma_h$ represents the homogeneous broadening of a single NC. We take $\sigma_h = 4$ meV to encompass all the transitions of the exciton fine structure (see single NC response in paragraph 2.2 and Figure 2a,b) and incorporate a slight additional broadening to account for larger interactions (in average) expected in environments characterized by a higher density of NCs. $\epsilon_m$ and $\epsilon_n$ are the exciton transition energies for the NCs located on site m and n, respectively. For simplification purposes, the NC Stokes shift, $\Delta_{SS}$ (gap between the absorption and emission peaking energies), is assumed to be constant i.e. independent of the NC size. This value will be an adjustable parameter in the fitting process.

Note that equation (4) applies under well-defined conditions [53, 54]: the transfer is driven by dipole-dipole interactions between donor and acceptor centers, the intrinsic properties of each NC are preserved (i.e., they remain unchanged by interactions), and the process occurs incoherently, without quantum coherence effects.

In principle, $C_{n,m}$ explicitly depends on $R_{DA}$, as shown in Eq. (1). In our approach, however, we approximate this by assigning a single constant $C_{n,m}$ to all first neighbors at site $(n,m)$, irrespective of the specific $(n,m)$ pair. This treatment effectively corresponds to an average coupling distance, independent of the actual sizes of neighboring NCs. Given the narrow size dispersion in the sample, this simplification is both justified and robust. The explicit $R$-dependence of $C_{n,m}$ would become relevant for second-neighbor interactions, where the larger average separation makes efficient transfer through FRET processes unlikely. Consequently, within the framework of the model, the total transfer rate $\Gamma_{t, nm}$ is governed solely by the degree of spectral overlap between donor emission and acceptor absorption linewidths.

Changing the integration variable and using a proper normalization, the integral of Equation 4 writes:

$$\Gamma_{t,nm} = \Gamma_{t,0} \int_{-\infty}^{+\infty} F'_{m}(\omega) A'_{n}(\omega) d\omega \quad (7)$$

where $\Gamma_{t,0}$ is now the transfer rate between two NCs with perfect overlap between their effective donor and acceptor linewidths (ie $\epsilon_m = \epsilon_n + \Delta_{SS}$). $\Gamma_{t,0}$ will also be one of the adjustable
parameters in the simulation.

We assign each NC in the simulated 2D arrays specific absorption transition energy. To do this, we use the line shape of the ensemble PL spectrum from **Figure 3** considered only at early time (i.e in the first ~50 ps) as a starting point to model the absorption spectrum of the optical transitions for the entire NCs ensemble. It can indeed be considered that transfers still did not occur at that time or played only a minor role so that the PL line ‘mimics’ the whole absorption energy transition distribution. A skew normal distribution, a generalized form of the normal (Gaussian) distribution that incorporates skewness (i.e., asymmetry), is used to replicate the special early time line shape of the PL spectrum and its extension towards higher energies. In agreement with the work of Brennan *et al.* [37], an adjustable Stokes shift towards the higher energies is then applied to the calculated asymmetric PL spectrum to generate an absorption line. The resulting curve is used to generate a histogram of the transition energies for each NC in a large square lattice composed of N objects finally randomly divided in smaller clusters (1 cluster = 1 pore). In our experiments, the excitation is non-resonant (~3.1 eV), leading primarily to the generation of hot carriers. In metal halide perovskites (MHPs), the subsequent hot-carrier cooling plays a key role in determining the charge dynamics, particularly in photovoltaic applications [55, 56]. In this work, we assume that the cooling stage is ultrafast and dynamically decoupled from exciton energy transfer through FRET-type hopping, described by the rate constant Γt,0 (≈10 ps). The experiments were performed under moderate excess energy (0.84 eV) and low excitation fluence (<<1 nJ per pulse), corresponding to carrier densities below 2×10¹⁶ cm⁻³. Under such gentle excitation, mechanisms that can slow carrier cooling, such as hot-phonon bottleneck or Auger heating, are expected to be negligible [55, 57-59]. Consequently, the carrier cooling times remain in the sub-picosecond to picosecond range, in agreement with previous reports on FAPbBr₃ and MAPbBr₃ nanocrystals [57, 60]. These observations support the assumption of dynamic decoupling between carrier relaxation and exciton transfer. Therefore, the emission dynamics observed in our experiments are attributed mainly to band-edge excitons formed after ultrafast cooling and governed by FRET-type energy transfer mechanisms.

Under these conditions, all NCs emit from the outset, and FRET manifests as a low-amplitude contribution superimposed on the intrinsic PL band. This contribution is what we extract from the simulations, starting from a uniformly excited NC population with no initial selectivity.
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Comparison between simulated decay profiles using the 2D grid model and experimental TRPL data, presented on a semi-logarithmic scale. The upper panel displays five decay curves corresponding to selected energies on the low-energy side of the photoluminescence (PL) emission spectrum, with specific energy values indicated in the legend. The lower panel shows decay curves acquired at various energy positions on the high-energy side of the PL spectrum. The system of differential equations governing the simulation (equations set (3)) was solved using the `solve_ivp` function from the `integrate` module of the `SciPy` library in Python, employing the Runge-Kutta method of order 5(4), denoted as "RK45".</figcaption></figure>

After integration of the differential equations given in **Eqs (3)**, we obtain the time evolution of the exciton population of each site of the 2D arrays. The simulated evolution is then sorted into energy classes with a width of 2 meV, allowing us to reconstruct the exciton population decay at specific energies. They are displayed in **Figure 4**. The calculated decays are plotted alongside their experimental counterparts and divided into high (top graph) and low (bottom graph) energy sides of the PL signal. For both sets of curves, the NC 2D arrays were made of 5×5 elements with the number of excited pores set at 600 in order to obtain a satisfying averaging. Two key
parameters, $\Delta_{ss}$ and $\Gamma_{t,0}$, are adjusted to obtain a good agreement between the numerical work and the experiments. It is found that $\Delta_{ss}$ pilots the manner the 'rapid' short-time decays evolve as a function of energy on the high-energy side of the PL (with decays slowing down as the energy decreases). $\Delta_{ss} = 32 \pm 3$ meV yields good results. The average $\Delta_{ss}$ value is therefore in consistent agreement with the ensemble measurements shown in Figure 1. We further emphasize that $\Delta_{ss}$ is substantially larger- by roughly one order of magnitude- than the homogeneous linewidth of a single NC, which rules out any possibility of self-quenching of the emission. Finally, $\Gamma_{t,0}$ is fine-tuned to achieve an optimal fit across the entire energy range. A value of $\Gamma_{t,0} = 50 \pm 10$ ns$^{-1}$ provides a satisfactory agreement with the experimental data. However, a correction is required to account for orientation effects. For simplicity, the orientation factor, commonly denoted $\kappa^2$, was initially set to 1 in the calculations, corresponding to perfectly parallel dipoles. In reality, the NCs are randomly dispersed within the matrix, and the system is better described as a frozen solution, where dipole orientations are randomized but in which rotational Brownian motion, as well as the translational Brownian motion, are completely frozen throughout the transfer prior to transfer (with $\kappa^2 = 0.476$ [61]). Accordingly, $\Gamma_{t,0}$ must be scaled proportionally, resulting in a characteristic value of $\Gamma_{t,0} \approx 105$ ns$^{-1}$ for this system. This value is comparable to the reported one for II-VI based NCs in which resonance conditions - for ET - have been finely tuned [62, 63] or are quasi-naturally achieved [64] and larger than the value obtained in CsPbBr$_3$ nanoplatelets [36] but smaller than the value obtained in $(PEA)_2(MA)_{n-1}$Pb$_n$I$_{3n+1}$ 2D layered perovskites [37]. Independently of the model, we can identify within the emission band those NCs for which the transfer rate matches the decay time of uncoupled NCs. These correspond to NCs emitting at ~2.248 eV, with an excitonic transition (after accounting for the Stokes shift) at ~2.28 eV, i.e. a size of ~9.6 nm. This gives an average center-to-center distance of ~10 nm, which we regard as a reasonable rough estimate of the Förster radius for the system.

Overall, these simulations successfully reproduce the main experimental observations with a relatively simple model. At high energy, the initial fast-decay regime is explained by the decline of the exciton population that transfers to lower energy (or larger size) acceptors in its vicinity, occurring on a timescale shorter than the radiative lifetime. On the low energy side, the decays are initially slowed down by the constant 'feeding' of the exciton population from higher energy donors before recovering a mono-exponential behavior after all the exciton transfers have occurred. Interestingly, the NCs emitting at energies located near the center of the PL spectrum maintain a well-defined mono-exponential behavior throughout, explained by the larger population of these NCs compared to those at higher or lower energies, and their role as
intermediates between the two energy sides. In average, the additional exciton population they receive from donors with higher confinement is transferred directly to the acceptors, resulting in an exciton population that remains almost unaffected by this process over time.

We can finally provide an upper value for the FRET efficiency for quasi-perfectly tuned NCs, $\eta_{max}$, in this particular system (environment) using the following simple equation [35]:

$$ \eta_{max} = 1 - \frac{\Gamma_D}{\Gamma_{t,0}}, \quad (8) $$

where $\Gamma_D = \tau_D^{-1}$ is the experimental decay time of the isolated NC (found to be nearly constant with the size - see text above). The extracted parameters demonstrate that the efficiency is close to its ultimate value (≈ 100%), pointing to the potentiality of the system for further energy transfer engineering. We note that this critical value is not reached even for the those NCs with the highest energy in the distribution (at 2.26 eV for example), which have a greater number of potential acceptors. That might be explained by (i) the statistics itself (few NCs have a perfectly matching acceptor as a first neighbor) and, (ii) the fact that $\Delta_{ss}$ was kept constant in the simulation and is very likely underestimated for ‘higher energy’ NCs and overestimated for the ‘lower energy’ ones [37]. This probably also explains the poorer adjustment for decays associated to NCs having their transitions slightly below the PL maximum, that cannot find enough acceptors in numbers in the simulation compared to the real situation.

The fact that $\frac{\Gamma_{NR}}{\Gamma_{t,0}} \ll 1$ deserves attention; the ratio can be considered as a decisive figure of merit for the whole system (matrix plus NCs) for it illustrates how the non-radiative dissipative channels could be efficiently ‘bypassed’ during the first hundreds of ps by tailoring the FRET properties, placing them as close as possible from the ideal tuning situation.

Finally let us note that many refinements and/or extensions are possible like examining the role of a size dependent Stokes-Shift (to the best of our knowledge, experimental data for the FAPbBr₃ material are not available yet) or including longer range FRET couplings but we estimate they should not drastically change the features that are highlighted throughout this work.

### 3. Conclusion

In this study, we investigated exciton transport dynamics in ligand-free FAPbBr₃ NCs embedded within a mesoporous silica matrix, a system that offers both enhanced stability and efficient energy transfer. Time- and spectrally-resolved PL spectroscopy at cryogenic
temperatures revealed a notable spectral redistribution over time, perfectly explainable in a
framework where non-radiative FRET is the dominant exciton migration mechanism. Our
model, based on a 2D lattice of coupled NCs, though setting some limitations to the transfers
by allowing only first neighbors coupling, accurately reproduces the observed PL decay
dynamics, confirming rapid exciton diffusion on a sub-nanosecond timescale. The extracted
resonance-limited transfer rate, Γ<sub>t,0</sub> = 105 ± 15 ns<sup>-1</sup>, characterizes donor-acceptor interactions
in perovskite NC systems tuned for optimal energy overlap.

These results also underscore the crucial role of reduced interparticle distances - enabled by the
absence of capping ligands - in significantly enhancing the energy transfer efficiency and
possibly placing its maximal value close to 100% in this specific system.

This work provides insights into exciton transport in perovskite NC ensembles and reinforce
the interest for mesoporous silica as a stabilizing and highly versatile and configurable
playground for energy transfer studies. Fine-tuning the structural organization (size and
connectivity of the pores) and composition of the host matrix (trying to optimize the energy
cascade) to further optimize excitonic interactions and promote diffusion, gain or even
coherence are among the most challenging axes for future investigations. This should pave the
way towards more stable and improved-performance perovskite-based materials.

**4. Experimental Section**

**Materials**

Lead (II) bromide (PbBr<sub>2</sub>, TCI, 99,99%), formamidinium bromide (FABr, GreatCell Solar Materials, 99,9%), dimethylsulfoxide (DMSO, Merck, anhydrous 99.8%), methanol (MeOH, VWR, 98%) were purchased and used without additional purification steps.

**Preparation of SiO2 nanoparticles porous matrix**

A colloidal suspension of 30 nm SiO₂ nanoparticles (LUDOX-TMA, Sigma-Aldrich) was diluted in an appropriated amount of methanol to reach 3% w/v. SiO₂ colloids were deposited by dip coating of previously cleaned and dried low-fluorescence glass substrates. Substrates were dipped in the suspension during 30 s and then withdrawal at a controlled speed of 120 mm/min. This process was repeated 6 times in total to produce a porous scaffold of thickness around 200 nm. After this, an annealing process at 450°C for 30 min was made to remove any organic component remaining within the matrix and to improve its mechanical stability.
## Synthesis of FAPbBr₃ nanocrystals within nanoporous silica scaffold

Perovskite solution precursors were made dissolving FABr and PbBr₂ powders in a 1:1 molar ratio in DMSO at 0.5, 1, 3 and 30% (m/v). The infiltration of the precursors was performed by spin-coating (5000 rpm for 60 s), followed by heating at 100°C for 1 hour, to obtain FAPbBr₃ nanocrystals within the pores of the matrix.

## Spectroscopy

### Optical absorption and micro-photoluminescence

The optical transmission was measured using a CARY 5000 double-beam spectrophotometer (Agilent company). The spectra were scanned with a resolution of a few Å.

Micro-photoluminescence (μ-PL) experiments were conducted in a reflection configuration on the different samples (see also the Supporting Information, Section 2) and, in order to reach the response of individual FAPbBr₃ nanocrystals, the X = 1 % sample was explored. For the experiments, the second harmonic generation of a Ti:Sapphire laser (82 MHz repetition rate, ≈ 2 picosecond pulses duration) was used as an excitation beam. The excitation power at ≈ 400 nm was maintained and sufficient in the μW range to observe individual objects in less dense regions of the sample. Spectrally highly resolved spectra are obtained working at low temperature. To that end the samples are mounted on the cold finger of a μ-PL Oxford Instruments cryostat with an enhanced mechanic stability enabling acquisitions over time with a good signal to noise ratio which is essential when single objects are addressed. In the process, the excitation laser beam is focused using a long working distance microscope objective (NA = 0.6) optimized to work through the e = 1.5 mm suprasil-made cryostat window, so that ≈ 1 μm spots are generated. The objective is attached to a three-axes piezo stage allowing scanning the sample with a sub-μm resolution and adjusting the optical coupling through fine focalization. The PL was sent and analyzed in a 750 mm focal length ACTON SP2750i Roper Scientific-Princeton Instruments spectrometer coupled to a nitrogen-cooled SPEC10 (RS-PI) CCD camera. Dispersion of the emission with a 1200 lines/mm grating provides the spectral resolution required to investigate single nanocrystals fine structure responses (a ≈ 75 μeV resolution is reached in the investigated range). In the experiments, the scattered laser light was removed using a high-performance dichroic long-pass filter (Semrock, FF01-430/LP-25), positioned along the optical path. The same filter was also employed in time-resolved photoluminescence measurements (see below).
This setup enables the routine characterization of the excitonic fine structure in single
perovskite emitters, typically manifesting as multiplets such as doublets or triplets (see
references [1] and [2] of the Supporting Information and references therein). The unambiguous
identification of such spectral features is based on well-established criteria: the precise emission
energy position, characteristic linewidths and energy splittings between the components,
correlated spectral fluctuations among the individual peaks—indicative of a common emitter—
and polarization-resolved measurements of the multiplet components.

Time-Resolved Photoluminescence.

Time-resolved photoluminescence (TRPL) spectroscopy was achieved mainly on the X = 30 % sample using a streak camera synchronized with the high repetition rate Ti:sapphire laser (C5680 model from Hamamatsu incorporating an M5675 synchroscan unit) and coupled to the Acton spectrometer. All measurements were carried out at low temperature in a confocal geometry (using the previously described setup) at ≈ 400 nm at low excitation power to avoid nonlinear effects.
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