Abstract
Multi-exponential fitting has long been the standard approach for luminescence decay analysis, not because it is physically meaningful but because it offers empirical convenience. Rigorous nonlinear rate-equation models have existed for decades, yet their application was hindered by computational costs. Recent advances in artificial intelligence and high-performance computing now make such models tractable, enabling physically grounded analyses of complex relaxation dynamics. Here we examine donor–acceptor interactions in a prototypical Eu2+-activated multi-site phosphor (La2.544Ca1.456Si12O4.456N16.544:Eu2+) that exhibits wavelength quenching. Metaheuristic-driven Runge–Kutta simulations enabled the extraction of quantitative radiative and non-radiative rate constants, while physics-informed neural networks provided a complementary framework that independently reproduced the experimental decay dynamics. Both approaches converged to consistent rate constants, establishing donor–acceptor transfer as the dominant relaxation pathway over radiative and same-species interactions, offering quantitative, physics-based insight into complex dynamical behaviors beyond phosphors.
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Introduction
There are many phosphors whose host structures provide activators with two or more distinct crystallographic sites, which lead to emission peak broadening and multi-peak emission behavior1,2,3,4,5,6,7,8,9,10,11,12,13,14. Numerous commercially available LED phosphors exhibit multi-peak emission due to distinct activator sites15,16,17,18,19,20,21,22,23,24. The activator distribution over these sites substantially influences luminescent properties, making site engineering a key issue18,25,26,27,28,29,30,31.
Although so-called multi-peak emission behavior induced by multi-site host structures can be frequently observed among well-known LED phosphors, few in-depth analyses of non-radiative site-to-site energy transfer have been extensively investigated. In this regard, it is necessary to investigate the site-to-site energy transfer in terms of measured decay behavior. However, systematic decay curve analyses for energy donors and acceptors can hardly be found despite the simple presentation of decay curves in numerous literature reports.
The emission from both donor and acceptor sites is mostly overlapped, so complete separation between donor and acceptor could be infeasible in measurements. However, evident traces of site-to-site (donor-to-acceptor) energy transfer have been detected and described as so-called wavelength quenching32. Wavelength quenching indicates that the decay rate becomes faster when the probe wavelength for decay measurement shifts to the high-energy (shorter wavelength) side, i.e., the donor side always decays faster than the acceptor decay. We have reported wavelength quenching behavior for several renowned phosphors used in LED applications7,8, and ultimately identified it as a clear indicator of the presence of distinct activator sites in the host structure7,8.
In the context of decay curve analysis for LED phosphors, multi-exponential fitting has long been the default choice, often bypassing explicit modeling of non-radiative processes that lead to quadratic terms in the rate equation. However, in systems where brisk non-radiative processes—such as energy transfer between excited and ground states and cross-relaxation between excited states—are actively involved, multi-exponential fitting provides no physical insight and merely serves as a mathematical visualization.
While multi-exponential decay functions are widely used for decay curve analyses in the phosphor research domain, the multi-exponential fitting—though empirically convenient—cannot capture the true physical nature of decay processes and should be regarded as a purely phenomenological approximation. The multi-exponential model implies the sum of solutions to several independent first-order linear differential equations. If one assumes that a few luminescent centers operate independently without interaction, then it might be justifiable. For example, if activators at two or multiple distinct sites produced luminescence with no interaction, the multi-exponential modeling would make sense, but this is not the case. In addition, if it could be a solution to a single linear differential equation of higher order, the multi-exponential decay model would be legitimate. However, such a higher order rate equation rarely reflects the true physical processes at play. Actual relaxation dynamics are more appropriately described by first-order nonlinear rate equations, which incorporate quadratic interaction terms to account for inter-activator energy transfer.
Although a number of phenomenological mathematical functions have been proposed as alternatives to simple multi-exponential functions for constructing luminescence decay laws33,34,35, we instead adopt a rate-equation framework that provides a more comprehensive and physically grounded interpretation of non-linear interactions. Therefore, for a physically meaningful analysis of decay curves—especially in systems with multiple activator sites—the first-order nonlinear rate equation modeling remains indispensable despite its computational complexity. The rate equation model includes multiple quadratic terms representing donor–acceptor interactions. This physically grounded model offers the only viable pathway for accurately interpreting decay curves in phosphors.
In spite of such legitimacy for the rate equation analysis, there is a considerable hurdle in handling rate equations in real-world decay analysis. It is the computational complexity and the corresponding costs that make the thorough rate equation analysis intractable. Specifically, the lack of an analytical closed-form solution for these nonlinear coupled differential equations necessitates iterative numerical integration (e.g., Runge–Kutta) for every unknown parameter estimation step. This creates a high-dimensional inverse problem where determining multiple unknown rate constants simultaneously becomes computationally expensive and prone to local minima. The primary challenge lies in the fact that there are at least several unidentified rate constants that reside in the rate equations.
Recent rate equation modeling studies on organic luminescent materials have dealt with an independent protocol for rate constant identification36,37,38. Although some of them relied on ab initio calculations36,37, it is infeasible to follow these approaches since neither ab initio calculations nor molecular dynamics calculations enable accurate estimation of rate constants for our inorganic luminescent material systems, i.e., LED phosphors, at the current state of development. In particular, determining the 4f-5d transition constant and especially the interaction constant for the quadratic term would be virtually impossible. Evaluating the unknown rate constants through experimental decay curve data is the only option for inorganic phosphors doped with Eu2+ or Ce3+.
A well-established numerical method such as the Runge-Kutta method39 is used for the complex rate equations involving quadratic interaction terms, which are absent in simpler linear rate equations for organic luminescent materials36,37. Optimization algorithms are inevitably required to evaluate the unknown rate constants in conjunction with the Runge-Kutta method. Metaheuristic algorithms would be a plausible option to achieve the rate constant evaluation. We employ genetic algorithm (GA)40 and particle swarm optimization (PSO)41 for this purpose. In other words, the Runge-Kutta method hybridized with GA and PSO provides relevant rate constant values within a reasonable range and reproducing experimental decay curves for both donor and acceptor.
In addition to the well-established decay analysis protocol based on a numerical process such as the Runge-Kutta method hybridized with GA and PSO, we also introduce an artificial intelligence (AI)-driven strategy. The rate equation is nothing but a first-order nonlinear differential equation. A physics-informed neural network (PINN) approach42 would be applicable for solving it. In this regard, the decay behavior can be analyzed based upon appropriate PINN models such as a simple multi-layer perceptron (MLP) and long short-term memory (LSTM) models.
Three loss functions are introduced: the physics loss, the experimental loss (difference between the measured and PINN-simulated decay curves), and initial/boundary condition loss terms. The PINN-driven rate equation with fully learned parameters is in good agreement with the numerical solution, which in turn matches the experimental data. The non-radiative site-to-site energy transfer is systematically analyzed using the PINN-driven methods, via a reliable end-to-end parameter learning that leads to an alternative interpretation protocol for decay behavior.
To demonstrate both of the decay analyses for real-world decay measurements, i.e., metaheuristic-driven numerical and PINN approaches, we employ a representative model system, La4-xCaxSi12O3+xN18-x:Eu2+ (x = 1.456) phosphor that exhibits two distinct Eu2+ activator sites in the host structure with a monoclinic lattice in the C2 space group. For the La2.544Ca1.456Si12O4.456N16.544:Eu2+ phosphor43 that exhibits the wavelength quenching behavior, the donor and acceptor decay curves are separately measured and eventually reproduced by solving the rate equations through both the decay analysis processes.
Results and discussion
Structure and photoluminescence properties
Figure 1a showcases the crystal structure of yellow-light-emitting La2.544Ca1.456Si12O4.456N16.544:Eu2+ phosphor, which exhibits nearly the same luminescent efficacy as the commercially available YAG phosphors43. The host structure of this efficient phosphor exhibits two distinct La/Ca sites (4c) for the Eu2+ accommodation. The bottom of Fig. 1a showcases zoomed-in donor and acceptor sites, where the La/Ca1 site is assigned as the donor and the La/Ca2 site as the acceptor. Both the La/Ca1 and La/Ca2 sites exhibit the same coordination number7 but the average bond length for the La/Ca1 site (2.70 Å) is longer than that for the La/Ca2 site (2.62 Å).
a Crystal structure with monoclinic C2 (no. 5) symmetry; lattice parameters: a = 18.54268(4) Å, b = 4.840398(11) Å, c = 10.700719(18) Å, β = 108.25660(17) Å. b Continuous-wave photoluminescence (PL) spectra were measured under 450 nm excitation using InGaN LEDs, whereas photoluminescence excitation (PLE) spectra were recorded by monitoring the emission at the peak wavelength while scanning the excitation wavelength. c Time-resolved PL (TRPL) spectra in the delay-time range from 10 to 1000 ns, excited at 355 nm with a Nd:YAG laser. d Decay curves measured at 485 and 545 nm. e TRPL spectrum at a 10 ns delay time, along with Gaussian-deconvoluted donor and acceptor peaks.
The anion ligands that constitute the polyhedron around the La/Ca1 and La/Ca2 sites are categorized into two different types: one is pure nitrogen and the other is N/O shared. The number of pure nitrogen ligands matters when assigning them as donor or acceptor. The La/Ca1 site is surrounded by 3 pure nitrogen ligands and the La/Ca2 site by 4. Integrating the polyhedron information such as the average bond length, the coordination number, and the ligand type, we can determine the donor and acceptor sites based on the nephelauxetic (covalency) and crystal field strength effects44,45. The donor site should have a slightly higher energy than the acceptor, so that the longer bond length (larger polyhedron) and the fewer pure nitrogen ligands would give higher energy at the same coordination number7. According to the criteria described above, we assigned the La/Ca1 and La/Ca2 sites to be donor and acceptor, respectively.
Figure 1b showcases PL and PLE spectra for La2.544Ca1.456Si12O4.456N16.544:Eu2+ phosphor. The favorable PLE profile in the blue to near-UV range indicates that this phosphor would be suitable for LED-driven applications. Figure 1c showcases time resolved PL (TRPL) spectra over a wide range of delay times from 10 to 1000 ns. The spectral range for the TRPL spectra might look blue-shifted from the continuous wave (CW) PL spectrum presented in Fig. 1b. It is evident, however, that by integrating all the TRPL spectra it would coincide with the CWPL spectrum. The site-to-site energy transfer makes the shorter wavelength component originating from the donor (Eu2+ activators at the La/Ca1 site) decay rapidly and the longer wavelength component from the acceptor (Eu2+ activators at the La/Ca2 site) gradually become enhanced.
This is typical wavelength quenching behavior represents a brisk energy transfer from donors to acceptors. The decay rate was accelerated at the shorter probe wavelength (485 nm) but retarded at 545 nm, as clearly shown in Fig. 1d. We selected these extreme wavelengths for separate measurements of the donor and acceptor decay curves. Figure 1e showcases an early stage TRPL spectrum measured at 10 ns, which exhibits an archetypic two-peak emission along with a multi-Gaussian deconvolution result. The deconvoluted peak at the higher energy side represents the donor emission from the La/Ca1 site and the lower energy side peak represents the acceptor emission from the La/Ca2 site. The acceptor signal at 545 nm includes no donor component but the donor signal at 485 nm involves a certain fraction of acceptor contribution, which is around a 22% ratio as indicated by the straight line.
Rate equation framework
To strictly describe the donor–acceptor interactions, we adopt a nonlinear rate-equation framework involving four activator types: regular and defective donors/acceptors, as detailed in our previous report46. The governing equations are.
\({D}^{{{\rm{e}}}}\) is the population of regular donors at the excited state (5d level of Eu2+), \({\bar{D}}^{{{\rm{e}}}}\) is the population of defective donors at the excited state, \({A}^{{{\rm{e}}}}\) is the population of regular acceptors at the excited state, \({\bar{A}}^{{{\rm{e}}}}\) is the population of defective acceptors at the excited state, \({D}_{{{\rm{g}}}}\) is the population of regular donors at the ground state (4f level of Eu2+), \({\bar{D}}_{{{\rm{g}}}}\) is the population of defective donors at the ground state, \({A}_{{{\rm{g}}}}\) is the population of regular acceptors at the ground state, and \({\bar{A}}_{{{\rm{g}}}}\) is the population of defective acceptors at the ground state. N is the total population (× 10−30 cm⁻³) and \({D}^{{{\rm{e}}}}\) + \({\bar{D}}^{{{\rm{e}}}}\,\)+ \({A}^{{{\rm{e}}}}\) + \({\bar{A}}^{{{\rm{e}}}}\) + \({D}_{{{\rm{g}}}}\) + \({\bar{D}}_{{{\rm{g}}}}\) + \({A}_{{{\rm{g}}}}\) + \({\bar{A}}_{{{\rm{g}}}}\) = N.
kr is the radiative rate of the 5d–4f transition for Eu2+, which can be obtained from an ideal dilute system based on the assumption that no non-radiative process is available. Instead of the experimental evaluation of kr, however, we set kr as a learnable parameter (decision variable in the metaheuristic process). kDA, kAA, and kDD are non-radiative energy transfer rates for donor to acceptor, acceptor to acceptor, and donor to donor, respectively. These energy-transfer rates are common to both regular and defective activator, and kn is the quenching rate that takes effect for defective activators only. G is the excitation rate. The pulse duration of the laser light source was 3 ns and the energy was 8 mJ, which allowed G to take effect only at t = 0 and then vanish immediately.
The excitation rate (G), which affects the overall level of the population for each state, was not refined but was predetermined to range from 10−2 to 10 s−1. Because we used normalized donor and acceptor decay curves, it was reasonable to exclude G from the GA and PSO processes. Additionally, since G intrinsically involves the excitation fluence, varying G indirectly reflects the experimental fluence variation, which was not performed in this study. The excitation fluence used for the actual measurements was sufficiently high (~113 mJ cm⁻²) to activate all nonlinear interaction terms across the tested G range. Under such conditions, the fluence primarily influences the overall population level rather than the rate constants themselves.
The present rate equation model allows for the energy transfer from defective donors to regular acceptors and donors, which were prohibited in the previous rate equation model46. Of course, the backward energy transfer from acceptors to donors is thoroughly prohibited regardless of whether they are regular or defective. This model demonstrates that the donor–donor and acceptor–acceptor transfer terms inherently account for microscopic energy migration. Our current framework assumes spatially uniform excitation and therefore neglects explicit ∇2 diffusion terms. This approximation remains valid for homogeneously excited bulk phosphors but may require extension for systems exhibiting strong spatial excitation gradients.
The available interaction schemes are graphically well-described in Fig. 2. The phonon dissipation was precluded from the rate equation model, as direct multi-phonon relaxation processes have minimal impact on Eu2+ luminescence. While the phonon energy values for La2.544Ca1.456Si12O4.456N16.544 are currently unreported, we can assess roughly the multi-phonon relaxation rate for our system using Si3N4 as a reasonable approximation. Given that the energy gap for the 5d → 4f transition in La2.544Ca1.456Si12O4.456N16.544 is approximately 18,797 cm⁻¹, and the phonon energy for Si3N4 ranges from 300 to 1200 cm⁻¹ in various vibrational modes47,48, the multi-phonon relaxation rate would be negligible due to its exponential decrease with increasing energy gap49.
Only the La/Ca1 and La/Ca2 sites are depicted in the La2.544Ca1.456Si12O4.456N16.544 supercell. Excited states are shown as bright colors, ground states as dim colors, and the four types of activators are indicated in different colors. The excited-state number density follows the calculated data for G = 10 s−1, clearly showing an increase with elapsed delay time.
Decision variables and objective function
There is no way to evaluate the exact values of rate constants in quadratic rate equations (kDA, kAA, and kDD) for La2.544Ca1.456Si12O4.456N16.544:Eu2+ phosphor luminescence. Unlike organic systems where rate constants can often be estimated via ab initio calculations36,37, obtaining quadratic interaction terms for inorganic phosphors via first-principles remains computationally prohibitive50. Therefore, extracting these constants through experimental decay curve fitting is currently the only viable option.
Evaluating those rate constants that constitute the phosphor rate equation would be viable only through semi-empirical approaches; that is, the interaction constants have been evaluated from the experimental data fitting process using traditional models such as Inokuti & Hirayama51, Van Uitert & Johnson52, and Yokota & Tanimoto53. Benchmark comparisons (Supplementary Note 1, Supplementary Fig. 4, and Supplementary Table 2) against the Inokuti–Hirayama51 and Yokota and Tanimoto53 models demonstrate the superior fitting accuracy of our rate equation approach. Even in our rate equation approach, the only option would be regression from the measured decay curves through numerical processes in conjunction with optimization strategies40,41,54. In this context, we adopted the Runge-Kutta method along with representative metaheuristics such as GA and PSO.
When the objective function is defined, the population of donors and acceptors at the excited state was not simply set as the measured decay curves detected at 485 and 545 nm, respectively. In theory, the donor and acceptor decay curves stand for \({D}^{{{\rm{e}}}}\) + \({\bar{D}}^{{{\rm{e}}}}\) and \({A}^{{{\rm{e}}}}\) + \({\bar{A}}^{{{\rm{e}}}}\), respectively. However, the measured decay curves at 485 and 545 nm are referred to as donor-side and acceptor-side decay curves (M485 and M545), respectively, as they do not directly represent the donor and acceptor but could be a weighted sum of donor and acceptor decay components. In this context, one should not be confused between the theoretical donor decay (\({{{\rm{D}}}}^{{{\rm{e}}}}\) + \({\bar{{{\rm{D}}}}}^{{{\rm{e}}}}\)) and the measured donor-side decay (M485 = m (\({D}^{{{\rm{e}}}}\) + \({\bar{D}}^{{{\rm{e}}}}\)) + (1 - m) (\({A}^{{{\rm{e}}}}\) + \({\bar{A}}^{{{\rm{e}}}}\))), where m stands for the donor contribution to the measured decay curve, which is an experimentally measurable parameter that can be determined by the selection of the probe wavelength.
The parameter m was determined to be ~0.7, as inferred from the vertical line at the probe wavelength (485 nm) in Fig. 1e, where the line intersects the donor and acceptor Gaussian components and their relative intensities yield a ratio of approximately 0.78. Although this experimental determination of m was available from the emission spectrum deconvolution, we set m as an adjustable decision variable in the metaheuristic approach so that it could be adjusted (learned) to an optimum value during the GA/PSO and PINN processes, and thereby it can act as a barometer for judging the validity of the metaheuristic-based optimization performance. It was finally proven that the m values learned from the GA/PSO and PINN processes approximately converged to 0.7.
The acceptor side decay is defined as M545 = \({A}^{{{\rm{e}}}}\) + \({\bar{A}}^{{{\rm{e}}}}\), where the donor contribution is nullified in the case of the acceptor-side decay curve measurement by selecting the probe wavelength at the point where the donor contribution fades out, as evidenced from the vertical straight line located at the probe wavelength (545 nm) in Fig. 1e.
The initial condition for the Runge-Kutta process is not stringently fixed but involves two learnable parameters such as p and q. It is reasonable to assume that the initial population mostly resides in the ground state before the laser pulse excitation; then the relative fraction between donor and acceptor populations and the fraction between regular and defective sites would significantly matter. Based on the crystal structure, the site fraction of La/Ca1 (donor) and La/Ca2 (acceptor) is determined to be p = 0.5. However, this equal number of available sites for donor and acceptor does not necessarily imply an equal distribution of Eu²⁺ activators. Therefore, the actual initial donor fraction was set as a decision variable and defined as p, which is also to be adjusted to an optimum value during the metaheuristic algorithm execution and also learnable in the PINN process. In addition, the fraction of defective activators, defined as q, is also involved in the initial condition. Consequently, the initial condition involving two adjustable (learnable) parameters (p and q) for the rate equation is summarized as follows:
We introduce eight learnable parameters (eight decision variables) in total. Five rate constants (kr, kDA, kAA, kDD, and kn) are from the rate equation, one (m) from the measurement condition, and two more (p and q) from the initial condition.
The objective function for the metaheuristic approach is defined as the difference between the measured and Runge-Kutta-method-driven decay curves, as shown below;
The instantaneous populations at every excited state are used to construct the computed decay curves that are compared with the measured ones (M485 and M545). We employed the classical 4th-order Runge-Kutta (RK4) method involving 4 gradients with a weighted average. The detailed description of the Runge-Kutta process is omitted here for brevity but is available in Supplementary Note 2.
Decay curve fitting results
Figure 3a showcases donor- and acceptor-side decay curves measured at 485 and 545 nm (M485 and M545), along with the first and second derivatives, and the rightmost plot showcases the coverture calculated from the first and second derivatives. The initial stage shape of the first and second derivatives shows a dramatic change. In particular, a sign reversal was observed in the first and second derivatives for acceptor-side decay due to the presence of rising part in the acceptor-side decay curve. The first and second derivatives as well as the curvature can be used as a decay curve shape indicator, which can be used for judging the validity of all later-appearing computed decay curves. That is, unless the computed curves follow this shape trend then we cannot trust the calculation scheme. Fortunately, most of the calculated result followed this trend in case of GA/PSO-driven Runge-Kutta process, while some of PINN-reproduced acceptor-side decay curves for high G values (G = 1 and G = 10) slightly deviated from this shape trend.
a Decay curves detected at 485 and 545 nm along with their first/second derivatives and curvature plots. b Reproduced decay curves (M485 and M545) from the GA-driven Runge–Kutta process. c Reproduced decay curves from the PSO-driven Runge–Kutta process. All curves are normalized to the total number of Eu2+ activators.
Consistent with the typical energy transfer theory, the donor-side decay is faster than the acceptor-side decay. Moreover, it should be noted that the decay curve shape at the early stage for the acceptor-side decay curve results in a negative curvature (the negative second derivative), which implies that there should be a certain degree of rising component contribution. This is direct evidence for the active energy transfer from donors to acceptors immediately following the intense pulse laser excitation.
It is not recommendable to analyze the acceptor-side decay behavior (M545) by simply adopting multi-exponential decay functions, because a rising component is present. Well-established analytical decay curve equations51,52,53 also do not work for describing the rising part. Although the deconvolution into multi-exponential decay functions would seem available for the donor-side decay curve (M485), it is also unrecommendable to employ the multi-exponential deconvolution even for the donor-side decay curve that has positive curvature over the entire time range, as it is impossible to systematically account for the non-linear interaction terms in the rate equation, as already discussed in the earlier subsections.
The GA/PSO-driven Runge-Kutta process gave rise to more reasonable decay curve fitting results for both the donor and acceptor sides. Figure 3b and c showcase the curve fitting results for the GA- and PSO-driven processes, respectively. Relatively good agreement is achieved between measured and computed curves. As mentioned above, the excitation rate (G) was fixed to several predetermined values from 10−2 to 10 s−1. The calculated decay curves are in good agreement with the measured ones for both the donor and acceptor sides. It is a rare case to achieve a perfect match between calculated and measured decay curves simultaneously for both the donor and acceptor decay curves using a single constitutive equation. The GA/PSO-driven Runge-Kutta process made it possible to reproduce decay curves based on the rate equation model that involves non-linear interaction terms.
In addition to the reproduction of the measured decay curves (M485 and M545), the rate equation modeling also enabled us to obtain the time evolution of the population of all existing energy levels, which are normally unavailable for experimental measurement. Figure 4a and b showcase the time evolution of regular and defective Eu2+ populations at excited states at La/Ca1 (donor site) and La/Ca2 (acceptor) sites. The overall shape of the calculated curves never deviated from the curvature-based shape trend, as discussed in the previous subsection. In addition, the overall curve shape is consistent irrespective of the choice of G values but the overall population level differs considerably at different G values. The initial population as well as the instantaneous population level increase with the G value, which is explained by the definition of G, which involves excitation fluence. The higher the excitation pumping power (the excitation fluence), the higher the G value, which eventually pumps more population to the excited state and thereby the overall decay curve level is raised.
a Estimated from GA-driven and b PSO-driven Runge–Kutta processes. The curves represent regular donors (\({D}^{{{\rm{e}}}}\)), defective donors (\({\bar{D}}^{{{\rm{e}}}}\)), regular acceptors (\({A}^{{{\rm{e}}}}\)), defective acceptors (\({\bar{A}}^{{{\rm{e}}}}\)).
Metaheuristic evaluation of parameters
Two well-known metaheuristics such as GA and PSO were utilized to evaluate the eight unknown parameters including five rate constants of interest. Brief wrap-ups for GA and PSO are described in Supplementary Note 3, and the codes for the GA/PSO-driven Runge-Kutta process are available at our GitHub site55. Figures 3 and 4 showcases computed time evolution curves for \({{{\rm{D}}}}^{{{\rm{e}}}}\), \({\bar{{{\rm{D}}}}}^{{{\rm{e}}}}\), \({{{\rm{A}}}}^{{{\rm{e}}}}\), and \({\bar{{{\rm{A}}}}}^{{{\rm{e}}}}\) and also shows reproduced donor- and acceptor-side decay curves (M485 = m (\({D}^{{{\rm{e}}}}\) + \({\bar{D}}^{{{\rm{e}}}}\)) + (1 - m) (\({A}^{{{\rm{e}}}}\) + \({\bar{A}}^{{{\rm{e}}}}\)), M545 = \({A}^{{{\rm{e}}}}\) + \({\bar{A}}^{{{\rm{e}}}}\)) using the best-fit rate constants obtained from the GA/PSO iteration.
The fitting quality in Fig. 3 is generally acceptable for all G values, which implies that the predetermined G setting was appropriate. Figure 5a and Supplementary Table 1 showcase the best-fit rate constants for all G values. Within a limited variance, all the rate constants did not vary significantly with G, which means that every constant was determined within the same order of magnitude for all G values. The rate constants kr, kn, kDA, kAA, and kDD are of particular interest for comprehensive understanding of energy transfer behavior. For reasonable comparison between the rate constants representing the linear and non-linear terms, the non-linear terms should be normalized by the total number of activators (N). That is, the three interaction constants (energy transfer rates) representing the non-linear terms such as kDA, kAA, and kDD in the unit of cm3s−1 are not directly comparable with the other rate constants in the unit of s−1, which represent the linear terms in the rate equation. To make it reasonable to compare them, a unified unit is prerequisite and thereby kDA, kAA, and kDD should be multiplied by the total number of activators (N).
a Results from the GA/PSO-driven Runge–Kutta process. b Results from the PINN process. c The standard deviation of MAE for OAT perturbation test.
With unified units, direct comparison between rate constants becomes meaningful. The radiative rate kr is on the order of 106 s−1, consistent with reported intrinsic Eu2+ radiative rates (0.56–2.5 × 106 s−1)56 and substantially smaller than the non-radiative rate constants. When comparing kr and kn, kr is much smaller than kn. It is natural that the non-radiative quenching rate is faster than the radiative rate.
More intriguing rate constants are kDA, kAA, and kDD, which represent the non-linear interaction terms in the rate equation. These interaction (inter-activator energy transfer) constants presented in Fig. 5a and Supplementary Table 1 are actually those multiplied by N. These N·kDA, N·kAA, and N·kDD values are in the range from 105 to 107, wherein the most interesting finding is that the donor–acceptor transfer rate (N·kDA) is much greater than the donor–donor and acceptor–acceptor transfer rates (N·kAA and N·kDD). It can be postulated that the donor–acceptor transfer overwhelmed the others, so that the N·kAA and N·kDD terms can be ignored. When comparing N·kDA with kr and kn, we can reach a consistent inequality relation: kn ≈ N·kDA > kr and, among the interaction terms, N·kDA ≫ N·kAA > N·kDD, where “≫” denotes “much greater than” and “≈” indicates “similar to.” This indicates that the non-radiative processes (kn and N·kDA) occur at much faster rates than the radiative process (kr) and the same-species interactions (N·kAA and N·kDD).
For sensitivity analysis, we performed a simple one-at-a-time (OAT) perturbation test. Each of the five optimized rate parameters was independently varied by ±10% while the remaining parameters were fixed, and the corresponding change in the mean-absolute-error (MAE) of the decay-curve fitting was recorded. The results are summarized in Fig. 5c. The overall trend shows that kr and kDA produce the largest change in MAE (ΔMAE < 5%), whereas the remaining parameters have negligible influence (<2%). These results confirm that the fitted solution is reasonably stable and that the model does not rely excessively on any single rate constant.
The relative magnitude order for the evaluated rate constants holds consistently for both the GA- and PSO-driven evaluation processes. The non-radiative processes are faster than the radiative process, which is common sense, and the inter-activator energy transfer rate is as fast as the quenching rate to the non-radiative trap, which implies that the energy migration among the activators is a key factor in the decay process. This leads to the notion that the activator concentration control (i.e., the inter-activator distance control) is very important as well as the defect control when synthesizing phosphor samples.
Neural network rate equation analysis
The rate equations were also solved using PINNs. The results are summarized in Figs. 5b and 6. The PINN process was completed using typical machine learning training procedures by constructing appropriate artificial neural networks (ANNs) that represent the population dynamics of all the activator states (specifically, \({D}^{{{\rm{e}}}}\), \({\bar{D}}^{{{\rm{e}}}}\), \({A}^{{{\rm{e}}}}\), \({\bar{A}}^{{{\rm{e}}}}\), \({D}_{{{\rm{g}}}}\), \({\bar{D}}_{{{\rm{g}}}}\), \({A}_{{{\rm{g}}}}\), and \({\bar{A}}_{{{\rm{g}}}}\)), which are approximated as simple MLP57 and LSTM58 models. Since the solutions obtained via the numerical process (i.e., the GA/PSO-driven Runge–Kutta process) are tentatively regarded as benchmark reference solutions, the PINN should be trained to reproduce the time evolution curves of populations, such that the reproduced curves fit as closely to the reference solutions as possible.
a Time evolution of Eu2+ populations at excited states estimated from the PINN process: regular donors (\({D}^{{{\rm{e}}}}\)), defective donors (\({\bar{D}}^{{{\rm{e}}}}\)), regular acceptors (\({A}^{{{\rm{e}}}}\)), defective acceptors (\({\bar{A}}^{{{\rm{e}}}}\)). b Normalized decay curves (M485 and M545) reproduced by the PINN model, compared with experimental measurements.
It is, however, crucial to note that the loss function is not simply the difference between the PINN-calculated and experimental decay curves, but rather a composite loss function consisting of physics-informed loss, initial condition loss, and experimental data loss. In this case, the physics loss terms are derived from the governing rate equations, which are systems of ordinary differential equations (ODEs). The model was trained by generating a substantial number of collocation points across the entire time range, and the analytically differentiable MLP and LSTM models were highly beneficial during the PINN training process. The training process results in the refinement of parameters in both the ANN and LSTM models, thereby yielding appropriate time evolution functions that also accommodate the experimentally available decay curves through the data loss component.
The recently emerging transformer model based on the attention mechanism59 was not considered, since the decay curves represent smooth, low-dimensional sequential time series with only weak long-range correlations, for which the LSTM architecture is generally more appropriate. Moreover, as shown in Supplementary Fig. 4, there was no significant difference in PINN performance between the MLP and LSTM models. This similarity in performance is attributed to the relatively simple shapes of the decay curves (population time evolution curves of excited states) and other energy-level populations. Thus, the present study focused on the MLP result only and the LSTM result is available in the Supplementary Information.
While training the PINN models, both PINN parameters and physical parameters (e.g., rate constants in the rate equations) can be learned simultaneously within a unified training framework. This end-to-end capability is a representative advantage of PINNs over typical numerical processes, eliminating the need to separate the rate-equation solving step from the parameter estimation step. Another key advantage is that PINNs do not require large, pre-labeled training datasets. More detailed discussions of this advantage are provided in Supplementary Note 4, together with the results (Supplementary Fig. 5) obtained using a drastically reduced experimental dataset, which still yielded consistent rate-constant evaluations without any deterioration in performance.
These two advantages—end-to-end parameter learning and physics-informed training without extensive labeled data—make PINNs uniquely suited for scientific computing applications, especially in scenarios where experimental data are scarce or expensive to obtain. In practice, however, successful PINN training often requires additional strategies (e.g., careful loss weighting or adaptive sampling) beyond these inherent advantages. The PINN training process will be more comprehensively discussed in association with the loss setting for the PINN process as described below.
The PINN training was implemented using three loss functions. The learning curves for each loss function are present in Supplementary Fig. 2. First, the physics loss is the primary component and is defined as the square of the difference between the right- and left-hand terms of the rate equations shown in Eq.(1)–(6). When using the physics loss alone, the final result was incomplete, instead leading to a large discrepancy between the measured and PINN-calculated decay curves. In this context, we included experimental data loss in the PINN training process. The data loss was defined as the difference between PINN-computed and experimentally measured decay curves. The data loss is similar to the objective function used in the GA/PSO-driven Runge–Kutta process, with the key difference that the time evolution of the excited-state populations (decay curves) is approximated through deep learning models such as MLPs.
The initial condition for the rate equation was incorporated into the loss function used in training the PINN model. Specifically, the initial condition defined in Eq. (7) and (8) was directly used to construct the initial-condition loss term. However, an additional artifact was introduced into this loss due to the abrupt population rise in the excited states (\({D}^{{{\rm{e}}}}\), \({\bar{D}}^{{{\rm{e}}}}\), \({A}^{{{\rm{e}}}}\), and \({\bar{A}}^{{{\rm{e}}}}\)) immediately after excitation. This sharp increase occurs within approximately 1 × 10−8 s, which approaches the time-step discretization limit of the Runge–Kutta integration. Such an ultrafast event is difficult for the PINN to capture accurately. To address this, we included the population values at 1 × 10−8 s as auxiliary initial conditions. Furthermore, population data at the terminal time (1.45 × 10−6 s) were also included to reinforce the correct decaying behavior. As an alternative to directly constraining the initial and terminal population values in the loss function, we introduced an auxiliary loss term that preserves the difference between these two values. This formulation serves as a guiding signal that shapes the overall decay curve. Although this strategy led to slower convergence compared to directly enforcing the initial and terminal conditions, the final decay curves converged successfully and yielded results essentially consistent with the direct-constraint approach. Once the training stabilized, these terms had negligible influence on the final results and did not affect the fitted rate constants. This strategy remains fully consistent with the principles of PINN training and is analogous to the common practice in phosphor research of approximating decay curves by multi-exponential functions, where the ultrafast rising phase is effectively neglected.
The rate constants were also determined within physically reasonable ranges from the PINN process, starting from randomly chosen initial guesses. When the initial guesses were far from the GA- and PSO-driven Runge–Kutta results, an adaptive learning-rate strategy was employed: the learning rate was initially set higher and then gradually reduced to ensure stable and efficient convergence. The optimized rate constants obtained from the PINN training were all within expected bounds and closely matched those derived from the GA/PSO-driven Runge–Kutta process. Figure 5b presents the optimized rate constants learned by the PINN, which are in close agreement with the GA/PSO-driven Runge–Kutta results.
The previous relationships obtained from the GA/PSO-driven Runge–Kutta process were consistently reproduced in the PINN results, namely kn ≈ N·kDA > kr and N·kDA ≫ N·kAA > N·kDD. The overall magnitudes of all the rate constants remained within physically reasonable ranges, as was also the case for the GA/PSO-driven Runge–Kutta process. The predetermined value of G was likewise adopted in the PINN framework. The population curves of the excited states (\({D}^{{{\rm{e}}}}\), \({\bar{D}}^{{{\rm{e}}}}\), \({A}^{{{\rm{e}}}}\), and \({\bar{A}}^{{{\rm{e}}}}\)) exhibited overall shapes that closely resembled those calculated with the GA/PSO-driven Runge–Kutta process, although minor discrepancies remained, as shown in Fig. 6.
The use of PINNs for solving nonlinear differential equations with unknown parameters is advantageous as it provides a unified end-to-end learning framework. Unlike conventional methods that sequentially decouple numerical solving from parameter estimation—often causing computational inefficiencies—PINNs enable the simultaneous determination of solution trajectories and physical parameters through a single gradient-based optimization. This integrated approach eliminates the overhead of repeated numerical integrations, resulting in a more streamlined pathway with enhanced convergence stability. Moreover, the framework utilizes automatic differentiation to compute exact gradients, ensuring numerical stability without the discretization errors typical of traditional finite-difference methods. Finally, the neural network–based formulation efficiently leverages modern GPU architectures for parallel processing, yielding substantial computational speedups compared to traditional CPU-based schemes
We note that similar rate-equation–based fitting approaches have also been reported recently in different contexts. For instance, Toyoda et al.38 applied a linear ODE framework to exciton migration kinetics, while Shukla et al.60 introduced the CRNN model to infer rate constants from transient spectroscopy. However, both frameworks are limited to first-order kinetics or independent elementary reactions, whereas the present work uniquely addresses nonlinear rate equations with quadratic donor–acceptor interactions using metaheuristic and PINN-based optimization. This distinction underscores the broader applicability of our method to complex multibody systems beyond the reach of conventional linear or CRNN approaches.
In this study, we established a physically grounded framework for analyzing luminescence decay behavior in multi-site Eu2+-activated phosphors. Unlike conventional multi-exponential fitting—which provides only phenomenological approximations without physical validity—the proposed nonlinear rate-equation model incorporates quadratic interaction terms to account for site-to-site non-radiative energy transfer. This approach enabled us to reproduce donor- and acceptor-side decay curves in La2.544Ca1.456Si12O4.456N16.544:Eu2+ with high fidelity, while simultaneously offering quantitative access to otherwise inaccessible rate constants.
Using a hybrid GA/PSO–Runge–Kutta protocol, we successfully evaluated key radiative and non-radiative rate constants, confirming consistent inequality relations such as kn ≈ N·kDA > kr and N·kDA ≫ N·kAA > N·kDD. These results demonstrate that donor–acceptor transfer and non-radiative quenching dominate the relaxation dynamics, highlighting the critical role of both activator concentration (inter-activator distance) and defect control in phosphor synthesis.
In parallel, we introduced a PINN framework as an alternative strategy for solving rate equations and estimating parameters. By embedding physics loss, initial-condition constraints, and experimental data loss into a unified training scheme, PINNs achieved end-to-end learning of both the solution trajectories and unknown rate constants. The optimized parameters obtained from the PINN training closely matched those from the GA/PSO–Runge–Kutta process, confirming the robustness of the approach. Beyond reproducing experimental decay curves, the PINN methodology provides additional computational advantages, including analytical differentiability for stable derivative evaluation, direct gradient-based optimization, and efficient parallelization on GPU architectures. These features enable scalable analyses of rate-equation models, particularly in parameter studies and uncertainty quantification tasks requiring numerous evaluations.
Although demonstrated here for Eu2+-activated phosphors, the combined GA/PSO–Runge–Kutta and PINN framework is broadly applicable to nonlinear relaxation problems across materials science. Our results establish a transferable, AI–physics integrated paradigm that replaces such phenomenological models with physically justified rate-equation analysis, offering quantitative insight into complex dynamical behaviors beyond phosphors.
Methods
Materials synthesis
The commercially available starting materials in powder form—CaO (Kojundo, 99.9%), SrO (Kojundo, 98%), BaO (Kojundo, 99% UP), La2O3 (Kojundo, 99.99%), Y2O3 (Kojundo, 99.9%), α-Si3N4 (Aldrich, unreported; Ube, unreported) and Eu2O3 (Kojundo, 99.9%)—were manually ground for several hours. The total batch size was approximately 3 g, yielding sufficient phosphor powder for conventional characterizations. The mixed raw materials were placed in a BN crucible and fired at 1525 °C for 4 h under a flowing N2 atmosphere (500 mL min−1) in a sealed tube furnace.
Materials characterization
The time-resolved emission spectra were examined using an in-house photoluminescence system equipped with a picosecond Nd:YAG laser (355 nm excitation, Continuum, Santa Clara, CA) and a CCD detector with a temporal resolution of 10 ns. Both the delay and gate times were fixed at 20 ns. Emission and excitation spectra were recorded using an in-house fabricated continuous-wave (CW) PL system with a xenon lamp and a spectrophotometer (Darsapro-5000, PSI Co., Ltd.).
Runge Kutta method
We employed the classical 4th-order Runge-Kutta (RK4) method to calculate all the population curves (\({D}^{{{\rm{e}}}}\), \({\bar{D}}^{{{\rm{e}}}}\), \({A}^{{{\rm{e}}}}\), \({\bar{A}}^{{{\rm{e}}}}\), \({D}_{{{\rm{g}}}}\), \({\bar{D}}_{{{\rm{g}}}}\), \({A}_{{{\rm{g}}}}\), and \({\bar{A}}_{{{\rm{g}}}}\)), as detailed in Supplementary Note 2. The time step was set to 1 × 10−8 s, and the entire simulation spanned 0–1.45 × 10−6 s. The RK4 code, integrated with metaheuristic algorithms such as GA and PSO, is available in our GitHub repository50.
Metaheuristic optimization execution
The GA- and PSO-driven optimization targeted eight kinetic parameters: fraction of defective activators (p), initial donor fraction (q), radiative decay rate (kr), energy transfer rates (kDA, kDD, kAA), non-radiative quenching rate (kn), and mixing parameter (m). The objective function to be minimized was the mean absolute error (MAE) between normalized experimental and simulated fluorescence intensities for both M485 and M545 channels. All calculations were performed in double precision (float64) to ensure numerical stability and consistent convergence.
For GA optimization, a population of 200 individuals was evolved over 1000 generations. The algorithm used a crossover rate of 0.9, a mutation rate of 0.1, and elitist selection preserving the top 10 individuals in each generation. Parents were chosen from the top 50 performers, and mutations were applied by adding Gaussian noise scaled to 10% of the parameter range. Multiple optimization runs with systematically varied parameter bounds were performed to promote robust global convergence.
For PSO optimization, 200 particles were evolved over 1000 iterations. The inertia weight (w) was set to 0.7, with acceleration coefficients (c1, c2) of 1.5 each to balance exploration and exploitation. Parameter bounds were dynamically adjusted during the search, and velocity clamping was applied to prevent excessive parameter variations.
Neural network architectures
The MLP architecture comprises three fully connected hidden layers, each with 64 neurons and hyperbolic tangent (tanh) activation functions. The network accepts a single time input t and produces eight state variables as outputs. A final softmax layer normalizes the outputs so that they sum to unity, representing a probability distribution over the states. This MLP architecture was selected through a pseudo hyperparameter optimization process, i.e., it was selected among three different architectures, as shown in Supplementary Fig. 3. For temporal modeling, we adopt an LSTM architecture with three stacked LSTM layers, each having a hidden dimension of 64. The time input is reshaped into a sequence format (batch, sequence, feature), enabling the network to capture temporal dependencies in the differential equation system. The final hidden state from the last LSTM layer is passed through a linear output layer, followed by softmax normalization. All network weights are initialized using Xavier uniform initialization to promote stable gradient flow during training.
Data availability
The authors declare that all data supporting the findings of this study are available within the paper and its Supplementary Information files. The source data underlying all figures (including those in the Supplementary Information) are provided as a Source Data file. Additionally, all raw data and generated datasets are publicly available in the GitHub repository at [https://github.com/socoolblue/Decay_Analysis]. Source data are provided with this paper.
Code availability
The custom code developed for the GA/PSO-driven Runge–Kutta process and PINN models is publicly available at the GitHub repository [https://github.com/socoolblue/Decay_Analysis] and is archived with a permanent identifier at https://doi.org/10.5281/zenodo.1815967955. The algorithms were implemented using Python (version 3.11 or higher) in conjunction with standard libraries including PyTorch and NumPy. Specific variables and parameters used to generate and process the data are detailed in the ‘Metaheuristic optimization execution’ and ‘Neural network architectures’ subsections of the Methods.
References
Daicho, H. et al. A novel phosphor for glareless white light-emitting diodes. Nat. Commun. 3, 1132 (2012).
Pritzl, R. M., Buda, A. T., Witthaut, K., Schmidt, P. J. & Schnick, W. Adamantane-type nitridophosphate phosphors AExLi10−2xP4N10:Eu2+ via medium-pressure ion-exchange reactions. Angew. Chem. Int. Ed. 64, e202420565 (2024).
Ji, W. Y. et al. A zero-thermal-quenching phosphor. Nat. Mater. 16, 526–531 (2017).
Bachmann, V., Justel, T., Meijerink, A., Ronda, C. & Schmidt, P. J. Luminescence properties of SrSi2O2N2 doped with divalent rare earth ions. J. Lumin. 121, 441–449 (2006).
Horikawa, T., Fujitani, M., Piao, X., Hanzawa, H. & Machida, K. Synthesis and characterization of Sr2Si5N8:Eu2+ phosphor using strontium carboxylate. J. Ceram. Soc. Jpn. 115, 623–627 (2007).
Zhang, M., Wang, J., Zhang, Q., Ding, W. & Su, Q. Optical properties of Ba₂SiO₄:Eu²⁺ phosphor for green light-emitting diode (LED). Mater. Res. Bull. 42, 33–39 (2007).
Sohn, K. S., Lee, S., Xie, R. J. & Hirosaki, N. Time-resolved photoluminescence analysis of two-peak emission behavior in Sr₂Si₅N₈:Eu²⁺. Appl. Phys. Lett. 95, 121903 (2009).
Sohn, K. S., Lee, B., Xie, R. J. & Hirosaki, N. Rate-equation model for energy transfer between activators at different crystallographic sites in Sr2Si5N8:Eu2+. Opt. Lett. 34, 3427–3429 (2009).
Kim, J. S., Kibble, K., Kwon, Y. N. & Sohn, K. S. Rate-equation model for the loading-rate-dependent mechanoluminescence of SrAl2O4:Eu2+,Dy3+. Opt. Lett. 34, 1915–1917 (2009).
Park, W. B., Singh, S. P., Yoon, C. & Sohn, K. S. Eu²⁺ luminescence from five different crystallographic sites in a novel red phosphor, Ca15Si20O10N30:Eu2+. J. Mater. Chem. 22, 14068–14075 (2012).
Park, W. B., Singh, S. P., Yoon, C. & Sohn, K. S. Combinatorial chemistry of oxynitride phosphors and discovery of a novel phosphor for use in light-emitting diodes, Ca1.5Ba0.5Si5N6O3:Eu2+. J. Mater. Chem. C. 1, 1832–1839 (2013).
Lee, J. H. et al. A rate equation model for the energy transfer mechanism of a novel multi-color-emissive phosphor, Ca1.624Sr0.376Si5O3N6:Eu2+. Inorg. Chem. Front. 6, 3493–3500 (2019).
Kim, M., Singh, S. P., Shim, S., Park, W. B. & Sohn, K. S. Discovery of a quaternary sulfide, Ba₂−xLiAlS4:Eu2+, and its potential as a fast-decaying LED phosphor. Chem. Mater. 32, 6697–6705 (2020).
Park, J. et al. A novel sulfide phosphor, BaNaAlS3:Eu2+, discovered via particle swarm optimization. J. Alloy. Compd. 922, 166187 (2022).
Wang, L. et al. Broadband orange-emitting Sr3Si8O4N10:Eu2+ phosphor discovered by a modified single-particle-diagnosis approach. J. Adv. Ceram. 12, 734–746 (2023).
He, C. et al. Powder synthesis and luminescence of a novel yellow-emitting Ba5Si11Al7N25:Eu2+ phosphor discovered by a single-particle-diagnosis approach for warm w-LEDs. Chem. Eng. J. 455, 140932 (2023).
Zhao, M., Yang, Z., Ning, L. & Xia, Z. Tailoring of white luminescence in a NaLi3SiO4:Eu2+ phosphor containing broad-band defect-induced charge-transfer emission. Adv. Mater. 33, 2101428 (2021).
Chen, Y. et al. Site occupation engineering toward giant red-shifted photoluminescence in (Ba,Sr)2LaGaO5:Eu2+ phosphors. Chem. Mater. 35, 8714–8721 (2023).
Takeda, T., Hirosaki, N., Funahashi, S. & Xie, R.-J. Narrow-band green-emitting phosphor Ba2LiSi7AlN12:Eu2+ with high thermal stability discovered by a single-particle diagnosis approach. Chem. Mater. 27, 5892–5898 (2015).
Ten Kate, O. M., Xie, R. J., Wang, C. Y., Funahashi, S. & Hirosaki, N. Eu2+-doped Sr2B2−2xSi2+3xAl2−xN8+x: a boron-containing orange-emitting nitridosilicate with composition-dependent photoluminescence properties. Inorg. Chem. 55, 11331–11336 (2016).
Wang, X. J. et al. Structure and luminescence of a novel orange-yellow-emitting Ca1.62Eu0.38Si5O3N6 phosphor for warm white LEDs discovered by a single-particle-diagnosis approach. J. Mater. Chem. C. 4, 9968–9975 (2016).
Wang, X. J. et al. Blue-emitting Sr3Si8–xAlxO7+xN8–x:Eu2+ discovered by a single-particle-diagnosis approach: crystal structure, luminescence, scale-up synthesis, and abnormal thermal quenching behavior. Chem. Mater. 27, 7689–7697 (2015).
Zhang, Y. et al. Realizing red/orange emission of Eu2+/Ce3+ in La26−xSrxSi41Ox+1N80−x (x = 12.72–12.90) phosphors for high-color-rendition white LEDs. J. Mater. Chem. C. 8, 13458–13466 (2020).
Wang, Z. et al. Mining unexplored chemistries for phosphors for high-color-quality white-light-emitting diodes. Joule 2, 914–926 (2018).
Blasse, G. & Grabmaier, B. C. Luminescent Materials (Springer-Verlag, 1994).
Wen, D. et al. Anomalous orange light-emitting (Sr,Ba)₂SiO₄:Eu²⁺ phosphors for warm white LEDs. ACS Appl. Mater. Interfaces 8, 11615–11620 (2016).
Amachraa, M. et al. MxLa1–xSiO2–yNz (M = Ca/Sr/Ba): elucidating and tuning the structure and Eu2+ local environments to develop full-visible spectrum phosphors. Chem. Mater. 34, 4039–4049 (2022).
Fang, M. H. et al. High-performance NaK2Li[Li3SiO4]4:Eu green phosphor for backlighting light-emitting diodes. Chem. Mater. 33, 1893–1899 (2021).
Qiao, J., Zhou, Y., Molokeev, M. S., Zhang, Q. & Xia, Z. Narrow bandwidth luminescence in Sr2Li(Al,Ga)O4:Eu2+ by selective site occupancy engineering for high definition displays. Laser Photon. Rev. 15, 2100392 (2021).
Zhao, M., Zhang, Q. & Xia, Z. Structural engineering of Eu2+-doped silicates phosphors for LED applications. Acc. Mater. Res. 1, 137–145 (2020).
Wu, D. et al. Full-visible-spectrum lighting enabled by site-selective occupation in the high efficient and thermal stable (Rb, K)2CaPO4F: Eu2+ solid-solution phosphors. Chem. Eng. J. 430, 133062 (2022).
Asaithambi, S. et al. Decay behavior of Eu2+-activated sulfide phosphors: the pivotal role of unique activator sites. J. Lumin. 263, 120109 (2023).
Berberan-Santos, M. N., Bodunov, E. N. & Valeur, B. Mathematical functions for the analysis of luminescence decays with underlying distributions 1. Kohlrausch decay function (stretched exponential). Chem. Phys. 315, 171–182 (2005).
Potopnyk, M. A. et al. Aggregation/Crystallization-induced emission in naphthyridine-based carbazolyl-modified donor-acceptor boron dyes tunable by fluorine atoms. Chem. Eur. J. 30, e202400004 (2024).
van Driel, A. F. et al. Statistical analysis of time-resolved emission from ensembles of semiconductor quantum dots: interpretation of exponential decay models. Phys. Rev. B. 75, 035323 (2007).
Xiong, S. et al. Waterproof and ultraflexible organic photovoltaics with improved interface adhesion. Nat. Commun. 15, 2267 (2024).
Shizu, K. & Kaji, H. Quantitative prediction of rate constants and its application to organic emitters. Nat. Commun. 15, 4723 (2024).
Toyoda, R. et al. Discrete coordination nanochains based on photoluminescent dyes reveal intrachain exciton migration dynamics. Nat. Commun. 16, 1367 (2025).
Butcher, J. C. Numerical Methods for Ordinary Differential Equations (John Wiley & Sons, 2008).
Holland, J. H. Adaptation in Natural and Artificial Systems (The University of Michigan,1975).
Kennedy, J. & Eberhart, R. Particle Swarm Optimization. In Proc. ICNN’95- International Conference of Neural Network 4, 1942 (IEEE, 1995).
Raissi, M., Perdikaris, P. & Karniadakis, G. E. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019).
Park, W. B. et al. A new paradigm for materials discovery: heuristics-assisted combinatorial chemistry involving parameterization of material novelty. Adv. Funct. Mater. 22, 2258–2266 (2012).
Dorenbos, P. 5d-level energies of Ce3+ and the crystalline environment. I. Fluoride compounds. Phys. Rev. B 62, 15640–15649 (2000).
Dorenbos, P. 5d-level energies of Ce3+ and the crystalline environment. II. Chloride, bromide, and iodide compounds. Phys. Rev. B 62, 15650–15659 (2000).
Park, W. B., Song, Y., Pyo, M. & Sohn, K. S. Nonradiative energy transfer between two different activator sites in La4-xCaxSi12O3+xN18-x:Eu2+. Opt. Lett. 38, 1739–1741 (2013).
Fang, C. M., de Wijs, G. A., Hintzen, H. T. & de With, G. Phonon spectrum and thermal properties of cubic Si3N4 from first-principles calculations. J. Appl. Phys. 93, 5175–5180 (2003).
Bermudez, V. M. & Perkins, F. K. Preparation and properties of clean Si3N4 surfaces. Appl. Surf. Sci. 235, 406–419 (2004).
Layne, C. B. Multiphonon Multiphonon relaxation and excitation transfer in rare-earth doped glasses. (The University of California, 1975).
Shafei, R. et al. Electronic and optical properties of Eu2+-activated narrow-band phosphors of phosphor-converted light-emitting diode application: insights from a theoretical spectroscopy perspective. J. Am. Chem. Soc. 144, 8038–8053 (2022).
Inokuti, M. & Hirayama, F. Influence of energy transfer by the exchange mechanism on donor luminescence. J. Chem. Phys. 43, 1978–1989 (1965).
van Uitert, L. G. & Johnson, L. F. Energy transfer between rare-earth ion. J. Chem. Phys. 44, 3514–3522 (1966).
Yokota, M. & Tanimoto, O. Effects of diffusion on energy transfer by resonance. J. Phys. Soc. Jpn. 22, 779–784 (1967).
Jones, D. R., Schonlau, M. & Welch, W. J. Efficient global optimization of expensive black-box functions. J. Glob. Optim. 13, 455–492 (1998).
Lee, B. D. et al. Resolving energy transfer dynamics in Eu2+-activated multi-site phosphors via metaheuristic optimization and physics-informed neural networks (this paper). Decay_Analysis https://doi.org/10.5281/zenodo.18159679 (2026).
Poort, S. H. M., Meijerink, A. & Blasse, G. Lifetime measurements in Eu²⁺-doped host lattices. J. Phys. Chem. Solids 58, 1451–1456 (1997).
Rumelhart, D. E., Hinton, G. E. & Williams, R. J. Learning representations by back-propagating errors. Nature 323, 533–536 (1986).
Hochreiter, S. & Schmidhuber, J. Long short-term memory. Neural Comput. 9, 1735–1780 (1997).
Vaswani, A. et al. Attention is all you need. Preprint at https://doi.org/10.48550/arXiv.1706.03762 (2017).
Shukla, J. et al. Discovering CO adsorption and desorption pathways from chemical reaction neural network modeling of transient kinetics spectroscopy. J. Phys. Chem. Lett. 16, 3562–3570 (2025).
Acknowledgements
This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2024-00446825 to K.-S.S. and W.B.P.), by the Alchemist Project (20012196 to K.-S.S. and W.B.P.) funded by the Ministry of Trade, Industry and Energy (MOTIE), Korea, and partly by the Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (IITP-2025-RS-2024-00437494 to K.-S.S.). The RS-2024-00446825 grant contributed 50% to this work.
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K.-S.S. conceived and designed the research. B.D.L. and Y.H.S. performed the computations, conducted the numerical simulations (Runge–Kutta with GA/PSO), and developed the PINN models. W.B.P., M.Y.C., and J.S.H. assisted with the phosphor synthesis and photoluminescence measurements. W.B.P., B.D.L., and Y.H.S. analyzed the data and K.-S.S. and W.B.P. wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Lee, B.D., Seo, Y.H., Cho, M.Y. et al. Resolving energy transfer dynamics in Eu²⁺-activated multi-site phosphors via metaheuristic optimization and physics-informed neural networks. Nat Commun 17, 1837 (2026). https://doi.org/10.1038/s41467-026-68549-3
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DOI: https://doi.org/10.1038/s41467-026-68549-3
