Visualization of trap distribution in SrAl₂O₄:Eu²⁺,Dy³⁺ revealed by simulations of luminescence changes in charging and emission of persistent luminescence and thermoluminescence based on a random walk model

Abstract

A stochastic analysis of the change in persistent luminescence (PersL) of Sr_0.9Al₂O₄:Eu_0.01x²⁺,Dy_{(0.1−0.01x)}³⁺ (SAOED_x, x = 2, 4, 5, 6, 8, and 10) based on a one-dimensional random walk model (1D RWM) is presented. In the RWM, the excited electron is treated as a particle and can hop stochastically between reservoirs aligned in a straight line. Simulations of changes in PersL and the charging process are performed to determine the hopping probabilities and the electron distributions in reservoirs. We predict potential energies of the electron reservoirs through simulations of the thermoluminescence (TL) spectra of SAOED_x and present potential energy diagrams of the luminescence center and traps involved in PersL.

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1. Introduction

Afterglow or persistent luminescence (PersL) has fascinated researchers involved in science and technology of light since the discovery of SrAl₂O₄:Eu²⁺,Dy³⁺ (SAOED) exhibiting outstanding long PersL over one night.¹ Initially, PersL of SAOED has been employed for clock face as an alternative to radio-active materials or road signs without electrical power in the dark. Recently, it has been applied to photosensitizers and in vivo luminescent materials for photodynamic therapy,² optical imaging,³ thermo-luminescence dosimetry,⁴ and solar cells.⁵ In addition to the application studies, many fundamental studies on PersL materials have been conducted to develop brighter phosphors with much longer lasting luminescence.^6–13 The photoluminescence (PL) of SAOED originates from the 5d–4f electronic transition of Eu²⁺ as a dopant, where the excited electron in the 5d orbital is usually relaxed to the 4f orbital with a lifetime of around 1 μs along with green luminescence.¹⁴ For PersL, in contrast, the excited electron can be transferred to the host material by thermal or photoactivation and then be captured at trapping sites such as oxygen vacancies or defects, and thereafter the electron escaped from the trap by thermal activation returns to the 5d orbital of Eu³⁺ to reproduce the excited *Eu²⁺, resulting in emission of luminescence. The co-dopant Dy³⁺ plays the role of a better electron trap, which contributes to the longer PersL compared to oxygen vacancies or defects. The excited electron may be tightly captured by Dy³⁺ due to the electrostatic attraction, that is, Dy³⁺ might be reduced to Dy²⁺ by analogy to the result that Sm²⁺ and Ce⁴⁺ were detected during the investigation of the electron de-trapping process for an PersL material YPO₄:Ce³⁺,Sm³⁺.¹⁵

Unlike PL, most PersL-decay features do not show a single-exponential curve. Therefore, the PersL decays have been frequently analysed using the double exponential equation,¹⁴

I(t) = I₀ + A₁ exp(−t/τ₁) + A₂ exp(−t/τ₂). (1)

This equation consists of two exponential terms with the fast and slow decay times of τ₁ and τ₂, respectively, where τ₁ may be related to the de-trapping from the oxygen vacancy with a shallow trap depth and τ₂ may originate from the deep trap of Dy³⁺. Occasionally, the PersL decay is analysed using the triple exponential function,

I(t) = I₀ + A₁ exp(−t/τ₁) + A₂ exp(−t/τ₂) + A₃ exp(−t/τ₃). (2)

In this case, the additional component would be attributed to another trap involved in the PersL decay. Thus, it seems hard to find the electron-transfer mechanism satisfying the triple-exponential decay feature.

Some mechanisms for PersL proposed in previous studies are based on a local model composed of a single luminescent centre and a single trap, where an excited electron is assumed to move back-and-forth between the luminescent centre and the trap in the PersL process.¹⁶ In this model, the trap never captures two or more electrons, and the system of luminescence centres and traps is fully isolated and never interacts with others, that is, the electron cannot transfer to another trap in the adjacent system. In this situation, the observed PersL should always show a single-exponential decay curve, unless the electron transfer rates are slow compared with the time scale of PersL observation. On the other hand, a global model consisting of a luminescent centre, a trap, and the conduction band of host crystals has been applied to analyse the PersL decay.¹⁷ The conduction band may be the pathway of electrons between the luminescent centre and the trap, and therefore the electrons involved in PersL would be distributed over the entire crystal in the PersL process. The electron transfer between the excited *Eu²⁺ and the trap through the conduction band, based on the energy diagram with respect to the vacuum referred binding energy (VRBE) level,^18,19 has been taken into account for the PersL process. When electron de-trapping is caused by thermal activation, as per the Boltzmann distribution law, the ratio of the number of electrons in the luminescent centre and those in traps (oxygen vacancies and Dy³⁺) is constant. Even when the excited electrons at the luminescence centre are consumed during the PersL emission, the immediate compensation of electrons from traps within the time scale of PersL observation guarantees that the ratio remains as the constant value. From this consideration, the PersL decay curve should follow a single exponential, even if multi-traps are incorporated in the global model. Therefore, it is better to consider that the electron transfer between the luminescence centre and the trap is slow or comparable to the time scale of PersL observation, when the PersL decay curve can be analysed with a double- or multi-exponential function. Previously, we showed that plural traps contribute to the complicated PersL-decay feature of β-Zn₃(PO₄)₂:Mn²⁺ through the stochastic analysis using a one-dimensional random walk model (1D RWM).²⁰ In the 1D RWM, the excited electron can randomly hop back-and-forth between adjacent traps and occasionally hop into the luminescent centre at the edge of the trap array to emit PersL.²⁰ The simulation for electron migration based on the RWM was found to be simple yet significantly effective to reproduce the multi-exponential PersL decays.

For SAOED, we can observe the luminescence enhancement during continuous excitation corresponding to the charge of PresL.¹⁴ In this charging process, the luminescence intensity keeps increasing and eventually reaches the maximum value when the increase rate of excited electrons by photoexcitation is balanced with the depletion rate of electrons due to PL and PersL.¹⁶ The excited electrons seem to be continuously transferred to/from the host material and accumulated in traps until filling up. When the photoirradiation ceased, we can observe PersL alone from SAOED. Both PersL and charging processes should be accounted for by the electron transfer between the excited state of *Eu²⁺ and the trap state of (Eu³⁺–Trap⁻),¹¹ where the electrons can move between the potential surfaces of these states by overwhelming the activation state by thermal energy. Even when PersL disappeared due to the depletion of trapped electrons, luminescence can be recovered by other excited electrons squeezed out from deep traps by heating. The thermoluminescence (TL) spectrum, which is the luminescence observed at a variety of temperatures, provides information of trap depths in SAOED. A peak temperature in the TL spectrum is related to the trap depth, and the peak intensity is proportional to the number of escaped electrons from the trap.^21,22 Detailed analyses of the charging and PersL decay processes and the TL spectrum reveal the characteristics of dopants in SAOED and the mechanism of electron transfer between the luminescence centre and the trap involved in PersL.²³

In this study, we show the effectiveness of the 1D RWM in simulating not only the non-exponential PersL decay curve of SAOED but also the charging process of PersL and TL. For the 1D RWM, we consider a train of multiple traps aligned in a straight line, where electrons can move back-and-forth between adjacent traps, and a luminescence centre is attached to the end of the trap line, where excitation of electrons and emission of luminescence take place. The simulation of the charging and PersL processes provides hopping probabilities among the luminescence centre and traps, corresponding to the electron transfer rates. Also, it provides evidence of electron population in traps, which helps to unveil the distribution of trap states in the SAOED crystal. In addition, the simulation of the TL spectrum provides the activation energy for each back-and-forth electron transfer process, which contributes to creating the energy diagram for traps involved in PersL for the SAOED crystal.

2. Simulation

2.1. A proposed mechanism of PersL emission

We propose a scheme for the PersL emission process based on the potential curves of Eu, *Eu, and trapping sites (Ts) in SAOED as shown in Fig. 1. The horizontal axis corresponds to a reaction coordinate along the electron transfer.¹¹ We denote that the excitation rate of Eu is k₀ and the relaxation rate of *Eu is k_e. The electron transfer rates between potential curves are k_m (m = ±1, ±2, …, ±M), and the capturing rate at deep trap or quenching rate is k_q. Although the tunnelling between potential curves might be necessary to consider the PersL emission mechanism, we do not treat the tunnelling explicitly in the RWM because all electrons are postulated to be particles. When focused on a single electron, a ground state configuration can be represented as G,

G: (Eu)^e − (*Eu) − (T₁) − (T₂) ⋯ − (T_M)

When *Eu is produced by photoexcitation, the electronic configuration can be represented as E₀,

E₀: (Eu) − (*Eu)^e − (T₁) − (T₂) ⋯ − (T_M)

The excited electron in the 5d orbital of *Eu can either relax to the 4f orbital to emit luminescence resulting in the G configuration or get trapped at closest trap T₁ which is represented as the E₁ configuration. The electron at T₁ (E₁) can move to the next trap T₂ (E₂) or return to the luminescence centre *Eu (E₁). The electron at T₂ can move to T₃ and so on,

E₁: (Eu) − (*Eu) − (T₁)^e − (T₂) ⋯ − (T_M)

E₂: (Eu) − (*Eu) − (T₁) − (T₂)^e ⋯ − (T_M)

⋮

E_M: (Eu) − (*Eu) − (T₁) − (T₂) ⋯ − (T_M)^e

Since there must be considerably many Eu ions in the bulk SAOED crystal, we should consider many sets of configurations {n₀E₀, n₁E₁, n₂E₂, …, n_ME_M} for the PersL process, where n₀, n₁, n₂, …, n_M are the numbers of respective excited configurations. We assume that this situation can be statistically equivalent to the following configuration with large numbers (n₀, n₁, n₂, …, n_M) of electrons in each trap:

(*Eu)^n₀e − (T₁)^n₁e − (T₂)^n₂e ⋯ − (T_M)^n_Me

When n₀ is decreased by luminescence emission, it will be compensated by the electron transfer from T₁ to *Eu, which contributes to the elongation of the luminescence lifetime resulting in PersL. Since the emission intensity is proportional to the product of n₀ and k_e, the simulation of PersL decay is equivalent to the determination of n₀k_e as a function of t. The simulation will provide not only n₀ and k_e but also the other rate constants k_±m and the number of electrons (n₁, n₂, …, n_M) in each trap. For the charging process, since n_m = 0 (m = 0, 1, 2, …, M) at the beginning, the photoexcitation contributes to an increase in n₀ alone. After that, while n₀ gradually decreases by the electron transfer to T₁ and the relaxation to Eu, the continuous photoexcitation from Eu and the backward electron transfer from T₁ contribute to an increase in n₀. Thus, n₀ initially increases and eventually will reach the maximum value when the increase and decrease rates of n₀ are balanced. This charging process should also be simulated from n₀k_e as a function of t using the rate constants determined by the PersL simulation. Since the electron transfer rate k_m corresponds to the de-trapping rate of electrons from the potential well, it would strongly depend on temperature. Therefore, for simulations of the TL spectrum, the temperature dependence in the rate constant k_±m can be incorporated as

k_±m = k_0,±m exp(−ΔE_±m/k_BT), (3)

Fig. 1 A schematic energy diagram for Eu, *Eu, and T_m (m = 1, 2, …, M) for PersL of SAOED. k_m (m = ±1, ±2, …, ±M) are rate constants for electron transfer between potential wells.

where k_0,±m (m = 1, 2, …, M) are pre-exponential factors, ΔE_±m is trap depth or activation energy for each electron transfer process, k_B is Boltzmann's constant, and T is the absolute temperature. The simulation of the TL spectrum is equivalent to the prediction of n₀ as a function of T and contributes to the determination of the activation energies for electron transfer (ΔE_±m).

2.2. A 1D random walk model (1D RWM) for the PersL

Based on the mechanism described above, we stochastically simulate the change in luminescence of the charging process, the non-exponential decay of PersL, and the TL spectrum using the 1D RWM as displayed in Fig. 2. The rectangle reservoirs being able to store electrons are aligned in a straight line, where each reservoir represents Eu, *Eu, or Ts. Eu is fixed at the left end, and *Eu is next to it. An array of Ts is set next to *Eu. We treat Dy³⁺ and other traps (oxygen vacancies or defects) as the same T_m. We assume that the electrons can move forward and backward between the very next reservoirs with certain probabilities, respectively, or can remain at the same reservoir. We introduce an electron-extinction path from T_M to represent the dropping in deep traps or the quenching. In this model, we introduce a probability p_m for each hopping between reservoirs instead of the electron-transfer rate constant k_m. Since the electron-transfer rate (k_m) to overwhelm the activation barrier (ΔE_m) is represented as k_m = k_m,0 exp(−ΔE_m/k_BT), we can describe the hopping probability as

p_m = k_m/k_m,0 = exp(−ΔE_m/k_BT). (4)

Fig. 2 Hopping probabilities (p₀, p_e, p_±m, and p_q) and numbers of electrons (N, N₀, and N_m) at reservoirs (Eu, *Eu, and T_m) aligned in a straight line (m = 1 − M). Boxes are electron reservoirs, and small circles are excited electrons involved in the PersL.

For example, the electron at Eu can move to *Eu with a probability p₀ (0 ≤ p₀ ≤ 1) upon excitation. For the electron at *Eu, the probabilities of moving to Eu and T₁ are p_e (0 ≤ p_e ≤ 1) and p₁ (0 ≤ p₁ ≤ 1), respectively, and probability of remaining at *Eu is 1 − p_e − p₁.

We define the number of electrons at reservoirs as N₀ at *Eu and N_m at T_m (m = 1, 2, …, M). For the random walk of electrons in the charging process, we initially prepare Eu with N electrons and the empty *Eu and Ts, that is, N⁰_m = 0 (m = 0, 1, 2, …, M). After the first step, Np₀ electrons move to *Eu from Eu, but N(1 − p₀) electrons remain in Eu, that is, N₀¹ = Np₀ and N_m¹ = 0 (m = 1, 2, …, M). After the second step, we can find N₀² = N(1 − p₀)p₀ + Np₀(1 − p_e) + Np₀(1 − p₁), N₁² = Np₀p₁, and N_m² = 0 (m = 2, …, M). The number of electrons Nⁿ_m (m = 0, 1, 2, …, M) at every reservoir after many steps (n) can be obtained in similar ways, which are represented as functions of step number. By converting the step number (n) into time (t), we can predict the luminescence change by time from p_eNⁿ₀. For PersL, since the photoirradiation is ceased, p₀ alone is needed to be p₀ = 0. After the first step, the numbers of electrons at *Eu and T₁ are N₀¹ = N₀(1 − p_e − p₁) + N₁p₋₁ and N₁¹ = N₀p₁ + N₁(1 − p₋₁ − p₂) + N₂p₋₂, respectively. After the second step, the number of electrons at *Eu is determined as N₀² = N₀¹(1 − p_e − p₁) + N₁¹p₋₁. After n steps, we can analogously obtain the number of electrons at *Eu as Nⁿ₀ = Nⁿ⁻¹₀(1 − p_e − p₁) + Nⁿ⁻¹₁p₋₁. For both charging and PersL processes, we perform a curve-fitting of luminescence intensity of SAOED as a function of step number n to determine p_e, p_±m (m = 1, 2, …, M), p_q and N_m (m = 0, 1, 2, …, M). Since the observation of charging and PersL processes was conducted at room temperature T₃₀₀ = 300 K, the hopping probability can be described as p³⁰⁰_±. Therefore, the probability at T can be represented as eqn (5) as a function of T,

p^T_±m = p³⁰⁰_± exp(−ΔE_±m/k_B)(1/T − 1/T₃₀₀)). (5)

Using p^T_±m, we performed a simulation of the TL spectrum for SAOED, which contributes to the determination of ΔE_±m.

3. Results

3.1. Charging and PersL

We observed luminescence changes in the charging process for SAOED_x (x = 2, 4, 5, 6, 8, and 10) which had been heated at 500 K for 1 h to de-trap electrons prior to the luminescence measurement. The luminescence intensities (I) at 520 nm were recorded at time intervals of 0.5 s during the continuous photoirradiation with 370 nm light, and log(I) was plotted with grey circles against log(t) as shown in Fig. 3. As can be seen from the plots, the log(I) values in the charging process exhibit complicated curves. For SAOED₂, SAOED₄, and SAOED₅, each log(I) increases, reaches the maximum value log(I_max), and then decreases. The increase in log(I) at the early stage becomes reduced with increasing x and disappeared for SAOED₆, SAOED₈, and SAOED₁₀. Still, the simulation, red lines, based on 1D RWM successfully reproduces the complex decay features of the charging processes for the entire samples. The SAOED_x that underwent the charging process was employed for the PersL measurement. As shown in Fig. 3, the plot of each log(I) against log(t) for PersL shows an upward convex curve, suggesting that the PersL decays non-exponentially or multi-exponentially. The times when each PersL intensity becomes I_max/100 are t = 470 s (x = 2), 195 s (x = 4), 505 s (x = 5), 116 s (x = 6), 82 s (x = 8), and 70 s (x = 10). Among them, the SAOED₅ likely shows the best performance for PersL. The red lines are simulated decay curves for the PersL and consistent with the experimental results.

Fig. 3 (left) Changes in luminescence of SAOED_x (x = 2, 4, 5, 6, 8, and 10) monitored at 520 nm in the charging process with the continuous irradiation of 370 nm at 300 K. (right) PersL of SAOED_x (x = 2, 4, 5, 6, 8, and 10) monitored at 520 nm after the photoexcitation with 370 nm light for 1 min at 300 K. The relative integrated intensities of PersL for entire samples were 5.10 × 10³ (x = 2), 2.06 × 10³ (x = 4), 1.54 × 10³ (x = 5), 4.30 × 10² (x = 6), 1.15 × 10² (x = 8), and 1.00 (x = 10).

Each hopping probability p_m (m = e, 1, −1, 2, −2, …, or q) between reservoirs was restricted to be the same for the charging and PersL processes in the simulation and determined through curve fitting. Listed in Table 1 are p_m values converted to the hopping probabilities per second. It is noticed that the probabilities (p₅, p₋₅, p₆, p₋₆, p₇, and p₋₇) for the entire samples are almost zero, which suggests the very small contribution of reservoirs T₅ T₆, and T₇ to the PersL of SAOED_x. In other words, we can focus on *Eu, T₁, T₂, T₃, and T₄ to analyse the charging and PersL processes. The averaged hopping-probability from *Eu to T₁ (p₁) is as half as the average probability p_e, corresponding to the intrinsic relaxation probability from excited *Eu²⁺ in SAOED_x. This result means that the 5d electron in the luminescence center easily transfers to the adjacent trap, leading to PersL for the SAOED_x system. The simulation can provide the number of electrons N_m at reservoirs. Fig. S3 (ESI†) shows N_m (m = 0–6) for SAOED_x (x = 2, 4, 5, 6, 8, and 10), where N₀ is normalized to unity. It is obvious that N₅ and N₆ are almost zero for the entire samples; there is no population of excited electrons. N₂, N₃, and N₄ for SAOED₅ are larger than those for the other samples, respectively. This result seems to relate to the best performance in PersL for SAOED₅.

Table 1 Hopping probabilities between reservoirs predicted through the simulation of charging and PersL processes for SAOED_x (x = 2, 4, 5, 6, 8, and 10)

x	pe	p1	p2	p3	p4	p5	p6	p7	pq
		p−1	p−2	p−3	p−4	p−5	p−6	p−7
2	0.312	0.218	0.166	0.009	0.002	0.000	0.074	0.083	0.068
		0.410	0.061	0.012	0.002	0.039	0.026	0.019
4	0.338	0.157	0.042	0.007	0.003	0.001	0.083	0.032	0.186
		0.154	0.034	0.007	0.001	0.000	0.000	0.000
5	0.233	0.115	0.369	0.012	0.002	0.000	0.099	0.032	0.067
		0.200	0.135	0.010	0.002	0.000	0.000	0.000
6	0.280	0.064	0.109	0.006	0.001	0.000	0.099	0.032	0.059
		0.152	0.076	0.012	0.002	0.000	0.000	0.000
8	0.328	0.082	0.294	0.007	0.001	0.000	0.100	0.032	0.188
		0.236	0.165	0.015	0.002	0.000	0.000	0.000
10	0.469	0.164	0.077	0.019	0.001	0.000	0.099	0.032	0.000
		0.380	0.067	0.014	0.003	0.000	0.000	0.000
Ave	0.330	0.116	0.178	0.010	0.001	0.001	0.096	0.032	0.111
		0.224	0.096	0.016	0.002	0.000	0.000	0.000

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3.2. Thermoluminescence

In Fig. 4, the thermoluminescence (TL) spectrum of each sample is plotted with grey circles from 200 to 400 K. We notice that the TL intensities of the entire samples increase from 0 at 200 K to the maximum value in the range of 250–300 K and decrease to 0 at 400 K. The peak of each spectrum would correspond to the trap depth in the corresponding sample. With increasing x, the peak temperature likely shifts to the low temperature region. It is suggested that the less the Eu²⁺ content (x) in SAOED_x, the deeper the trap depth. In addition, each spectral profile strongly depends on x. For example, we can see two peaks in the TL spectra of SAOED₂ and SAOED₄ but one for the others. We performed simulations of the TL spectra based on the RWM. In this simulation, the temperature dependence in hopping probability was incorporated using the following relationship:

p_m,T = p_m,300 exp(−(ΔE_m/k_B)(1/T − 1/300)), (6)

where p_m,300 (m = 1, −1, 2, −2, …, q) is the hopping probability at 300 K determined through the simulation of charging and PersL processes, k_B is Boltzmann's constant, and ΔE_m is the activation energy for hopping between reservoirs. As shown in Fig. 4, the simulated TL spectra plotted with red lines are consistent with the experimental spectra of the entire samples. Specifically, the twin peaks for SAOED₂ are also well reproduced through the simulation. For each sample, we can prepare a potential energy diagram for the activation states and reservoirs from the resultant ΔE_m as shown in Fig. 5, where the potential energies of them are relatively depicted with respect to that of *Eu. For the entire samples, the energies of T₁ are lower than those of *Eu, and the activation energies ΔE₁ from *Eu to T₁ are 12 kJ mol⁻¹ or less. These results strongly suggest that the 5d electron in the excited state of Eu²⁺ spontaneously transfers to the next trap T₁. The ΔE₋₁ values for the backward hopping from T₁ to *Eu are 35 kJ mol⁻¹ (x = 2), 34 kJ mol⁻¹ (x = 4), 27 kJ mol⁻¹ (x = 5), 29 kJ mol⁻¹ (x = 6), 24 kJ mol⁻¹ (x = 8), and 20 kJ mol⁻¹ (x = 10). With an increase in the Eu²⁺ content x, the electrons at T₁ seem to become easier to move back to *Eu. This can reasonably account for the low temperature shift of peak temperature in TL when x is increased. For SAOED₂ and SAOED₄, both ΔE₂ and ΔE₋₂ are large, that is, the activation states between T₁ and T₂ are energetically high compared with the others. Most electrons are suggested to be accommodated in the relatively deep traps T₁ and/or T₂, which might be related to the twin peaks in each TL spectrum of SAOED₂ and SAOED₄. For SAOED₅, T₁, T₂, and T₃ sink together, and the activation energy ΔE₄ for hopping from T₃ to T₄ is remarkably large. These results suggest that the most excited electrons are populated in traps of T₁ through T₃. Although a high activation barrier of ΔE₄ is found for SAOED₆, the excited electrons would be mostly accommodated in T₁ because T₂ and T₃ are higher in energy than T₁. For SAOED₈, it is considered that the electron populations at T₁ and T₃ are larger than the others. For SAOED₁₀, it is characteristic that the reservoirs T₁ through T₄ act as sinks compared with *Eu. This result may suggest that the excited electrons are widely distributed over T₁ through T₆.

Fig. 4 TL spectra (grey circles) and their simulations (red lines) for SAOED_x (x = 2, 4, 5, 6, 8, and 10). The samples were treated with the de-trapping at 500 K and the charging with 370 nm light for 10 min at 200 K. The luminescence of each sample was measured at 520 nm from 200 to 500 K with the increasing rate of 1.03 K s⁻¹ after 10 min in the dark at 200 K.

Fig. 5 Potential energy diagrams of *Eu and T_m (m = 1–4) for SAOEDx (x = 2, 4, 5, 6, 8, and 10) predicted by the simulation based on the RWM. The numbers are ΔE values in kJ mol⁻¹ between the electron reservoir (*Eu or T_m) and the activation state, respectively.

4. Discussion

4.1. Multiple traps

X-ray crystallography revealed that there are two Sr²⁺ sites (Sr1 and Sr2) in host crystal SrAl₂O₄, where both Sr²⁺ sites are crystallographically nonequivalent to each other yet very similar in coordination structure.²⁴ These Sr²⁺ sites in the SrAl₂O₄:Eu²⁺ crystal can be exchanged with dopant Eu²⁺ due to their similar ionic radii of 1.21 Å (Sr²⁺) and 1.20 Å (Eu²⁺)²⁵ and the same electric charges. Since the electronic configuration of excited Eu²⁺ is represented as 4f⁶5d¹ involving the 5d orbital, the excited energy is sensitive to the difference in coordination geometry. Therefore, the modulation of coordination number of Eu²⁺ by O atoms by varying the SrO/Al₂O₃ ratio of host material results in the drastic change in the emission peak from 275 nm for SrAl₁₂O₁₉:Eu²⁺ to 467 nm for Sr₃Al₂O₆:Eu²⁺.²⁶ SrAl₂O₄:Eu²⁺ emits blue and green luminescence peaking at 455 and 520 nm, respectively, which are assigned to the emission from *Eu²⁺ at Sr1 and Sr2.^26,27 The DFT calculation predicted that co-dopant Dy³⁺, as well as Eu²⁺, occupies the most energetically stable Sr²⁺ site.²⁷ This is consistent with the result of field-emission scanning electron microscopy (FE-SEM) that Eu²⁺ and Dy³⁺ are evenly distributed in the SrAl₂O₄ host.²⁸ On the other hand, another SEM measurement suggests the agglomeration of Dy³⁺ at grain boundaries in SrAl₂O₄.¹¹ Given these reports for the SAOED structure, we should consider that there are a variety of coordination structures around Eu²⁺ and Dy³⁺ for SAOED depending on its crystal structure and the contents of Eu²⁺ and Dy³⁺. Especially, the variation in the coordination structure around Dy³⁺ will result in a variety of trap depths contributing to the multi-exponential PersL decay. Despite the complex trap system in SAOED, the simple 1D RWM is effective for the simulation of the charging and PersL processes and the TL spectra and contributes to the prediction of the traps and activation states in the SAOED crystals. The validity of the 1D trap array assumption would be rationally accounted for by the electron hopping path which can be uniquely determined when the hopping preferentially occurs between the closest traps in the crystal.

4.2. Analysis using the triple-exponential function

For SAOED₂, we observed the rise-and-decay curve for the change in luminescence intensity during the charging process as mentioned above. In accordance with the global system consisting of a single *Eu–trap system with a conduction band, the luminescence rise at the early stage is accounted for by the increase in the number of excited electrons in the *Eu–trap system due to the continuous photoirradiation. When a curve fitting using an equation incorporating the rise component was carried out (Fig. S4, ESI†), we needed two additional decay components to match the simulation with the experimental result. This strongly suggests that the number of excited electrons at the luminescence centre *Eu does not remain constant even after filling them up. In this simulation, we employed eqn (7),

I_c = I_c1 exp(−t/τ_c1) + I_c2 exp(−t/τ_c2) + I_c3 exp(−t/τ_c3) + I_c0 (7)

where τ_c1, τ_c2, and τ_c3 are lifetimes, I_c2, I_c2, and I_c3 are pre-exponential factors, and I_c0 is a constant. From the best fit, we determined τ_c1 = 44 s, τ_c2 = 5.1 × 10⁴ s, and τ_c3 = 1.3 s, I_c1 = 0.025, I_c2 = 0.26, and I_c3 = −1.1, and I_c0 = 0.72, as listed in Table S2 (ESI†). The electron filling process represented by the rise component is likely completed within τ_c3 = 1.3 s in the present case. The decay components would correspond to electron transfer from the luminescence centre to traps and/or leaking electrons from the *Eu–trap system along with no luminescence such as quenching or dropping into deep traps. It might be essential that the electron transfer rate from *Eu to traps involved in PersL of SAOED₂ is slow. At the later stage showing the constant intensity I_c0 = 0.72, the electron filling rate is likely balanced with the electron consuming rates due to the leaking and the luminescence emission. For PersL of SAOED₂, Fig. S4 (ESI†) shows that a curve fitting using eqn (8) provides the best result and affords τ_p1 = 5.0 s (89%), τ_p2 = 65 s (10%), and τ_p3 = 430 s (1%),

I_p = I_p1 exp(−t/τ_p1) + I_p2 exp(−t/τ_p2) + I_p3 exp(−t/τ_p3). (8)

This triple exponential function strongly suggests that there are at least three electron-trapping sites or reservoirs involving the PersL decay of SAOED₂. For the other samples, the simulation using eqn (7) and (8) successfully reproduced the luminescence changes in the charging and PersL as shown in Fig. S4 (ESI†). The parameters determined by the simulation are listed in Table S2 (ESI†). The resultant τ_p1, τ_p2, and τ_p3 for PersL of each sample are roughly the same: τ_p1 = 3–7 s (ca. 90%), τ_p2 = 27–71 s (ca. 10%), and τ_p3 = 170–450 s (ca. 1%). No remarkable change in lifetimes is observed. On the other hand, for the charging process, although the result for SAOED₄ is similar to that for SAOED₂, the others are different. For SAOED₅, two rise components, I_c2 = −0.025 (τ_c2 = 3.0 s) and I_c3 = −0.0083 (τ_c3 = 70 s), are needed to fit the experimental luminescence change. For SAOED₆ and SAOED₈, the luminescence change in the charging process can be reproduced with two decay components, since the third component is negligibly small. For SAOED₁₀, three decay components are needed to reproduce the luminescence change. Although we can estimate lifetimes for the charging and PersL processes based on eqn (7) and (8), it seems hard to clarify the role of traps in the luminescence behaviours in the entire SAOED_x, even when we take multi-traps in a SAOED crystal into account for the PersL decay and the charging process. In contrast, it is significant that the stochastic analysis using the RWM can predict the energy diagram of *Eu and traps, the electron hopping probabilities between traps, and the electron distribution in traps for the entire samples.

4.3. Trap depth from TL spectra

According to the literature,^29–31 we can determine the trap depth (E) for each SAOED_x from the corresponding TL spectrum, assuming the single luminescence centre-single trap system. To do so, we created a plot of ln(T²/β) vs. 1/k_BT in accordance with Hoogenstraaten's equation:²⁹where T is the peak temperature resulting from a TL spectrum, k_B is Boltzmann's constant, β is the heating rate, and s is the oscillating factor. The slope of the plot provides a trap depth E₁ for each sample, which is listed in Table 2. Alternatively, we can also determine the trap depth from eqn (10) derived from the fact that the change in β brings the shift of the peak temperature of the TL spectrum:^30,31

Table 2 Trap depths (E₁ and E₂ in kJ mol⁻¹) for SAOED_x (x = 2, 4, 5, 6, 8, and 10) evaluated from the TL spectra, obtained from eqn (3) and (4), respectively. ΔE_d values in kJ mol⁻¹ are determined from the potential energies of deepest reservoirs for SAOED_x, respectively, with respect to that for *Eu

x	2	4	5	6	8	10
E1 (kJ mol^−1)	53.3	41.3	32.6	38.6	73.5	84.4
E2 (kJ mol^−1)	63.5	40.5	68.5	64.4	93.6	99.8
ΔEd (kJ mol^−1)	35	26	27	17	24	34

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The trap depths (E₂) determined from this equation are listed in Table 2. The resultant E₁ and E₂ are in the range of 30–100 kJ mol⁻¹, which are consistent with the values reported by other researchers.²² However, it seems that there is no relationship between the trap depth and the Eu²⁺ content x in SAOED_x. Especially, the deepest E₁ and E₂ are obtained from both methods for SAOED₁₀, which is inconsistent with the physical intuition that the lower the peak temperature in the TL spectrum, the shallower the trap depth. Table 2 lists energy gaps ΔE_d between *Eu and the deepest trap in SAOED_x obtained through the analyses for TL spectra using the RWM. ΔE_d, the largest being 35 kJ mol⁻¹ for SAOED₂ among them, decreases and increases with increasing x. For SAOED₁₀, ΔE_d is comparable to that for SAOED₂. It is interesting that this trend in ΔE_d against x is similar to that in E₁.

5. Conclusions

We employed SAOED_x (x = 2, 4, 5, 6, 8, and 10) containing 10% dopants (Eu²⁺ and Dy³⁺) in host crystal SrAl₂O₄ to produce a large amount of electron traps owing to Eu²⁺, Dy³⁺, oxygen vacancies, and defects. It should be noted that although the 1D trap array must be far apart from the actual arrangement of electron traps of Eu²⁺, Dy³⁺, oxygen vacancies, and defects in SAOED_x crystals, the simulation based on the 1D RWM successfully reproduced the complicated luminescence changes in charging and PersL and the TL spectrum for each SAOED_x. The resultant energy diagram of the luminescence centre and traps for each sample is simple and helpful for understanding the PersL of SAOED_x with a complicated trap system. The RWM is simple yet effective to visualize the picture that electrons migrate in the SAOED crystal with a variety of traps.

Author contributions

R. Koizumi and R. Takahashi conducted the preparation and identifications of materials, the luminescence measurements, and the simulation. S. Hattori and S. Yamada contributed to the experimental work and participated in discussions. K. Shinozaki supervised this project and wrote the manuscript.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting this article have been included as part of the ESI.†

References

1. T. Matsuzawa, Y. Aoki, N. Takeuchi and Y. Murayama, J. Electrochem. Soc., 1993, 143, 2670.
2. A. Bessière, J.-O. Durand and C. Noûs, Nanophotonics, 2021, 10, 2999.
3. S. K. Sharma, D. Gourier, E. Teston, D. Scherman, C. Richard and B. Viana, Opt. Mater., 2017, 63, 51.
4. X. Tang, E. D. Ehler, E. Brost and D. C. Mathew, J. Appl. Clin. Med. Phys., 2021, 22, 191.
5. S. Singh, V. Tanwar, A. P. Simantilleke and D. Singh, Nano-Struct. Nano-Objects, 2020, 21, 100427.
6. J. Xu and S. Tanabe, J. Lumin., 2019, 205, 581.
7. H. F. Brito, J. Hölsä, T. Laamanen, M. Lastusaari, M. Malkamäki and L. C. V. Rodrigues, Opt. Mater. Express, 2012, 2, 371.
8. C. Chiatti, C. Fabiani and A. L. Pisello, Annu. Rev. Mater. Res., 2021, 51, 409.
9. K. Van den Eeckhout, P. F. Smet and D. Poelman, Materials, 2010, 3, 2536.
10. R. E. Rojas-Hernandez, F. Rubio-Marcos, M. Á. Rodriguez and J. F. Fernandez, Renewable Sustainable Energy Rev., 2018, 81, 2759.
11. D. Van der Heggen, J. J. Joos, A. Feng, V. Fritz, T. Delgado, N. Gartmann, B. Walfort, D. Rytz, H. Hagemann, D. Poelman, B. Viana and P. F. Smet, Adv. Funct. Mater., 2022, 32, 2208809.
12. C. Jia, D. Gao, Z. Wang, X. Zhang, S. Yun, J. Z. Zhang and X. Wang, Mater. Today Chem., 2024, 39, 102182.
13. H. Li, R. Li and C. Chang, Ceram. Int., 2021, 47, 30156.
14. T. Delgado, N. Gartmann, B. Walfort, F. LaMattina, M. Pollnau, A. Rosspeintner, J. Afshani, J. Olchowka and H. Hagemann, Adv. Photonics Res., 2022, 3, 2100199.
15. A. J. J. Bos, N. R. J. Poolton, J. Wallinga, A. Bessière and P. Dorenbos, Radiat. Meas., 2010, 45, 343.
16. J. Botterman, J. J. Joos and P. F. Smet, Phys. Rev. B:Condens. Matter Mater. Phys., 2014, 90, 085147.
17. C. Tydtgat, K. W. Meert, D. Poelman and P. F. Smet, Opt. Mater. Express, 2018, 6, 844.
18. P. Dorenbos, J. Lumin., 2000, 91, 155.
19. P. Dorenbos, J. Lumin., 2003, 104, 239.
20. K. Ikegaya, S. Yamada and K. Shinozaki, Dalton Trans., 2019, 48, 6746.
21. M. L. Chithambo, A. H. Wako and A. A. Finch, Radiat. Meas., 2019, 97, 1.
22. P. Zeng, X. Wei, M. Yin and Y. Chen, J. Lumin., 2018, 199, 400.
23. K. Van den Eeckhout, A. J. J. Bos, D. Poelman and P. F. Smet, Phys. Rev. B:Condens. Matter Mater. Phys., 2013, 87, 045126.
24. V. Vitola, D. Millers, I. Bite, K. Smits and A. Spustaka, Mater. Sci. Tech. Ser., 2019, 35, 1.
25. F. Clabau, X. Rocquefelte, S. Jobic, P. Deniard, M.-H. Whangbo, A. Garcia and T. Le Mercier, Chem. Mater., 2005, 19, 3904.
26. D. Dutczak, T. Jüstel, C. Ronda and A. Meijerink, Phys. Chem. Chem. Phys., 2015, 19, 15236.
27. K. Hoang, Phys. Rev. Appl., 2023, 19, 024060.
28. M. Zheng, X. Li, Y. Bai, S. Tang, P. Li and Q. Zhu, Coatings, 2023, 13, 1050.
29. W. Hoogenstraaten, Philips Res. Rep., 1958, 13, 515.
30. R. Chen, J. L. Lawless and V. Pagonis, Radiat. Meas., 2022, 150, 106692.
31. M. Maghrabi, J. Lumin., 2018, 198, 54.

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Footnote

† Electronic supplementary information (ESI) available: Experimental procedures and supplementary figures and tables. See DOI: https://doi.org/10.1039/d5cp01232e

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