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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 17 — Aug. 16, 2010
  • pp: 17805–17818
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Particle-swarm-optimization-assisted rate equation modeling of the two-peak emission behavior of non-stoichiometric CaAlxSi(7-3x)/4N3:Eu2+ phosphors

Yoon Won Jung, Bonghyun Lee, Satendra Pal Singh, and Kee-Sun Sohn  »View Author Affiliations


Optics Express, Vol. 18, Issue 17, pp. 17805-17818 (2010)
http://dx.doi.org/10.1364/OE.18.017805


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Abstract

We examined non-stoichiometric CaAlxSi(7-3x)/4N3:Eu2+ phosphors that were intentionally prepared with x = 0.7 ~1.3 to identify the origin of the deconvoluted Gaussian components that constitute the emission spectra of stoichiometric CaAlSiN3:Eu2+ phosphors. The Al/Si molar ratio around the Eu2+ activator caused the deconvoluted Gaussian peaks. The Eu2+ activator sites in Al-rich environments gave rise to the lower-energy emission peak, while those in Si-rich environments were related to the higher-energy emission peaks. Active energy transfer from the Eu2+ activator site in the Si-rich environment to the Eu2+ activator site in the Al-rich environment was confirmed. Particle swarm optimization was employed to estimate the nine unknown decision parameters that control the energy transfer process. All of the decision parameters were estimated within the range of reasonable values.

© 2010 OSA

1. Introduction

Recently, considerable effort has been devoted to the development of nitride-based phosphors for use in light-emitting diodes (LED) [1

1. C. Kulshreshtha, J. H. Kwak, Y.-J. Park, and K.-S. Sohn, “Photoluminescent and decay behaviors of Mn2+ and Ce2+ co-activated MgSiN2 phosphors for use in LED applications,” Opt. Lett. 34(6), 794–796 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-6-794. [CrossRef] [PubMed]

-12]. CaAlSiN3:Eu2+, which will hereafter be denoted as CASIN, is one of the most successfully commercialized nitride phosphors for use in LED applications. Broad band-type emission in the red-light wavelength range enhances the color rendering index of white LEDs. Consequently, many investigations have aimed to develop a promising synthesis method for CASIN phosphors along with detailed structural analysis[4

4. X. Piao, K. Machida, T. Horikawa, H. Hanzawa, Y. Shimomura, and N. Kijima, “Preparation of CaAlSiN3:Eu2+ phosphors by the Self-Propagating High-Temperature Synthesis and Their Luminescent Properties,” Chem. Mater. 19(18), 4592–4599 (2007), http://pubs.acs.org/doi/abs/10.1021/cm070623c?prevSearch=%255Bauthor%253A%2BX.%2BPiao%255D&searchHistoryKey=. [CrossRef]

~12]; however, in-depth luminescence studies of CASIN phosphors are lacking. We have recently observed an interesting photoluminescence behavior: two-peak emission behavior and variation in decay behavior with respect to the emission wavelength [13

13. S. Lee and K.-S. Sohn, “Effect of inhomogeneous broadening on time-resolved photoluminescence in CaAlSiN3:Eu2+,” Optic. Lett 35, 1004–1006 (2010), http://www.opticsinfobase.org/ol/search.cfm.

]. However, the origin of this two-peak emission behavior remains to be clarified. Likewise, the origin of each deconvoluted emission peak remains ambiguous. Because the host structure of CASIN provides Eu2+ activators that have only a Wyckoff site, it is difficult to envisage local structures that could be assigned to the deconvoluted emission peaks. In addition, decay curve variation with respect to the emission wavelength is of particular concern in relation to energy transfer between Eu2+ activators. However, detailed investigations regarding energy-transfer behavior and the ensuing quantitative analyses remain to be performed.

The inexplicable result that was reported previously [13

13. S. Lee and K.-S. Sohn, “Effect of inhomogeneous broadening on time-resolved photoluminescence in CaAlSiN3:Eu2+,” Optic. Lett 35, 1004–1006 (2010), http://www.opticsinfobase.org/ol/search.cfm.

] originated from failure to identify the local structures around the Eu2+ activator that caused the variation in decay behavior with respect to the emission wavelength. Thus, one of the most important purposes of the present investigation was to assign deconvoluted emission components to their corresponding local structures. The CASIN structure (Cmc21) exhibited random distribution of Al or Si ions around the Eu2+ activator at the Ca site (m). In addition, the CASIN structure might have inhomogeneous distribution of both oxygen impurities and cation vacancies around the Eu2+ activators. This unidentified diversity in the CASIN structure inevitably gives rise to a certain degree of inhomogeneous broadening. From a practical point of view, it is essential to examine every possible local structure around the Eu2+ activator and its consequence on PL behavior, because correct identification of local structures would make it possible to predict the exact shape and width of the emission (excitation or absorption) spectra of CASIN phosphors.

Although precise identification of all possible local structures was not achieved in the present study, the variation in both the shape and the width of the emission spectra could be explained in terms of local structures by examining artificially manipulated model phosphors. Non-stoichiometric CaAlxSi(7-3x)/4N3:Eu2+ phosphors with x = 0.7 ~1.3 were adopted as our model system, and their spectral distribution, structural refinement, and time-resolved emission spectra were extensively investigated. This model system was specially designed to monitor the effect of the Al/Si molar ratio around the Eu2+ activator site on the steady-state and time-resolved PL behaviors. Although slight deviations from the stoichiometric CASIN structure might induce unexpected defects or defect impurities, such as oxygen or cation vacancy, we succeeded in monitoring the PL behavior of non-stoichiometric CaAlxSi(7-3x)/4N3:Eu2+ phosphors as a function of the Al/Si molar ratio alone.

2. Experimental procedures

CaAlxSi(7-3x)/4N3:Eu2+ phosphor samples with x = 0.7 ~1.3, which were adopted as our model system in the present investigation, were prepared by a typical solid-state reaction method. Raw materials such as Ca3N2, AlN, α-Si3N4, and EuN were dry mixed in a glove box with oxygen and a moisture content maintained below 1 ppm. The Eu2+ concentration was fixed at 0.02 in each sample. The mixed raw materials were fired at 1800 °C under a pressurized N2 gas environment (10 atm). The fired samples were ground and subjected to X-ray diffraction (XRD), energy dispersive spectroscopy (EDS), continuous wave photoluminescence (CWPL), and time-resolved photoluminescence (TRPL). CWPL was measured using an in-house spectroscope equipped with a xenon lamp with an excitation wavelength of 390 nm. The time-resolved emission spectra of the CaAlxSi(7-3x)/4N3:Eu2+ phosphors were also measured using an in-house photoluminescence system that included a picosecond Nd:yttrium aluminum garnet (YAG) laser with a pulse repetition frequency of 10 Hz and a charge-coupled device sensor with a time resolution of 10 ns. An excitation wavelength of 355 nm was produced by tripling the 1066 nm frequency of the Nd:YAG laser.

3. Results and discussion

3.1 Spectral and structural analyses

As shown in Figs. 2(a)
Fig. 2 XRD patterns and corresponding Rietveld refining results of (a) N-CASIN-0.7 and (b) N-CASIN-1.2.
and 2(b), we refined the XRD patterns for two representative samples, CaAl0.7Si1.225N3 (x = 0.7) and CaAl1.2Si0.85N3 (x = 1.2); hereafter, the former will be denoted as N-CASIN-0.7 and the latter as N-CASIN-1.2. A major constituent phase retained the CASIN structure despite detection of impurities. Both samples contained an AlN phase as an impurity, and only N-CASIN-1.2 included both an AIN and a Ca3AlN3 phase.

The refined atomic position and the occupancy data are presented in Table 1

Table 1. Atomic coordination and Ca-N bond length data

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along with Ca-N bond length data. In particular, the Al/Si occupancy ratio and the average Ca-N bond length in Table 1 are of interest. The occupancy of the other constituent atoms was fixed during refinement, so that we could monitor only the Al/Si occupancy ratio in the CASIN structure. The refined Al/Si occupancy ratio was similar to the starting composition as well as the energy dispersive spectroscopy (EDS) composition analysis, which is given in Table 2

Table 2. EDS compositional analysis (atomic %) of eight different spots

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. The average Ca-N bond length for N-CASIN-0.7 was slightly greater than that for N-CASIN-1.2, which also coincided with the emission spectrum measurement. These results indicate that a higher energy component was observed only for N-CASIN-0.7. It is well known that the smaller the polyhedron around the activator, the higher the emission energy[17

17. P. Dorenbos, L. Pierron, L. Dinca, C. W. E. Eijk, A. Kahn-Harari, and B. Viana, “C.W.E van Eijk, A. Kahn-Harari, B. Viana, “4f-5d spectroscopy of Ce3+ in CaBPO5, LiCaPO4 and Li2CaSiO4,” J. Phys. Condens. Matter 15(3), 511–520 (2003), http://iopscience.iop.org/0953-8984/15/3/315?fromSearchPage=true. [CrossRef]

-21

21. P. Dorenbos, “5d-level energies of Ce3+ and the crystalline environment. III. Oxides containing ionic complexes,” Phys. Rev. B 64(12), 125117 (2001), http://prb.aps.org/abstract/PRB/v64/i12/e125117. [CrossRef]

], which will be discussed in greater detail later in this subsection. It seems to be impractical to distinguish aluminum and silicon in the Rietveld refinement of powder x-ray diffraction patterns due to similar electron density.

The Rietveld refinement process involving different Al/Si ratio was implemented just to see the effect on Ca-N polyhedron size and eventually to check if the resultant Ca-N polyhedron size coincides with the PL data. In this regard, a reasonable result was obtained with an acceptable fitting quality (Rwp = 12.1, Rp = 11.0). The EDS data in Table 2 also provided useful information about the oxygen content. However, the EDS results cannot be used for correct quantitative analyses but for rough measures because of the presence of impurity phases. If the greater oxygen content was induced by a greater Al/Si ratio to meet the electrical neutrality requirement, the high-energy emission component would have been activated for the higher Al/Si ratios, causing a blue-shift in emission. However, the observed result was the opposite of this hypothetical result. That is, the higher Al/Si ratio deactivated the high-energy component and a red-shift was observed. Namely, in contrast to our prior expectation, which was based on electrical neutrality, the EDS measurement showed that the oxygen contents of N-CASIN-0.7 and N-CASIN-1.2 were similar, and the emission spectra showed a clear red shift as the Al/Si ratio increased. Consequently, factors other than oxygen contamination might explain the experimental data.

The above-described assignment of the deconvoluted Gaussian peaks can be explained by Dorenbos’ analysis[17

17. P. Dorenbos, L. Pierron, L. Dinca, C. W. E. Eijk, A. Kahn-Harari, and B. Viana, “C.W.E van Eijk, A. Kahn-Harari, B. Viana, “4f-5d spectroscopy of Ce3+ in CaBPO5, LiCaPO4 and Li2CaSiO4,” J. Phys. Condens. Matter 15(3), 511–520 (2003), http://iopscience.iop.org/0953-8984/15/3/315?fromSearchPage=true. [CrossRef]

-21

21. P. Dorenbos, “5d-level energies of Ce3+ and the crystalline environment. III. Oxides containing ionic complexes,” Phys. Rev. B 64(12), 125117 (2001), http://prb.aps.org/abstract/PRB/v64/i12/e125117. [CrossRef]

], wherein the centroid shift and crystal field splitting was monitored with respect to neighboring ions for the Ce3+ doped system. Although the exact Stokes shift was not estimated, the Dorenbos’ analysis on centroid shift and crystal field splitting could explain approximately the observed emission spectra. Dorenbos reported that Al3+ gave rise to a larger centroid shift than Si4+ for a Ce3+-doped oxide system [21

21. P. Dorenbos, “5d-level energies of Ce3+ and the crystalline environment. III. Oxides containing ionic complexes,” Phys. Rev. B 64(12), 125117 (2001), http://prb.aps.org/abstract/PRB/v64/i12/e125117. [CrossRef]

]. The same result should be observed for the present non-stoichiometric CASIN system, because Dorenbos’ analysis of Ce3+-doped systems can be applied to Eu2+-doped systems [20

20. P. Dorenbos, “Relation between Eu2+ and Ce3+ f ↔ d-transition energies in inorganic compounds,” J. Phys. Condens. Matter 15(27), 4797–4807 (2003), http://iopscience.iop.org/0953-8984/15/27/311?fromSearchPage=true. [CrossRef]

]. In addition to Dorenbros’ analysis of centroid shift, that of crystal field splitting was also in good agreement with our experimental data. The crystal field splitting of the 5d level was controlled by the distance to the nearest anion ligands. Based on Rietvelt refinement, the average Ca-N bond length for N-CASIN-1.2 was slightly shorter than that for N-CASIN-0.7, as can be seen in Table 1. Although a clear separation between the local environments, which leads to the higher and lower energy emissions, was practically impossible, the average Ca-N bond length might be a good indicator of the plausible assignment of deconvoluted emission spectra to their corresponding local environments. For example, an Al-rich local environment induced a smaller polyhedron around the Eu2+ activator, which was assigned to the lower energy emission, while a Si-rich local environment led to a larger polyhedron and a higher energy emission. Both the centroid shift and crystal field splitting analyses were in good agreement with our experimental data and provided a plausible explanation of the two-peak emission behavior.

3.2 Rate equations for decay analysis

Figures 5(a)
Fig. 5 Decay curves monitored at (a) 525 and 700 nm for N-CASIN-0.7, and (b) 525 and 725 nm for N-CASIN-1.2. The solid lines represent the rate equation model.
and 5(b) show typical donor and acceptor decay curves measured at different emission wavelengths for N-CASIN-0.7 and N-CASIN-1.2, respectively. In an attempt to provide a quantitative interpretation of the decay behavior based on energy transfer, rate equations were set up, wherein the Eu2+ activators were categorized into two types. The first category was for regular Eu2+ ions located at the characteristic site of the host lattice with no adjacent quenching (or killer) sites. The criterion that we used to identify regular Eu2+ ions was based on whether the primary energy transfer occurs with other Eu2+ ions or killer sites. If the energy transfer rate from Eu2+ ions to other Eu2+ ions is much faster than the rate of energy transfer to killer sites, then we can regard them as regular Eu2+ ions. The number density of Eu2+ ions in this category was designated as ρx   orye   or   g. The superscript of ρ indicates whether Eu2+ ions are in a ground (4f7) or an excited state (4f65d1), and the subscript indicates whether Eu2+ ions are located at either the Ca1 or the Ca2 site. The regular Eu2+ ions are not susceptible to direct quenching by killer sites, but are apt to interact with their neighboring Eu2+ ions. Consequently, inter-activator energy transfer takes place.

When a killer site is located near an excited Eu2+ ion, inter-activator energy transfer does not occur, but, rather, direct quenching is dominant. This category of Eu2+ ions, which are located near killer sites, are defined as defective Eu2+ ions. The killer sites were not identified, but were presumed to be defects or impurities in the host lattice. Eu2+ ions at perturbed, non-emitting Eu2+ sites, the symmetry of which deviated from that of the characteristic Ca1 or Ca2 sites of the host lattice due to defects, were also killer sites [25

25. S. O. Vásquez, “Energy transfer processes in organized media. III. A two-center model for nonhomogeneous crystals,” J. Chem. Phys. 108(2), 723–728 (1998), http://jcp.aip.org/jcpsa6/v108/i2/p723_s1. [CrossRef]

]. The number density of defective Eu2+ ions was designated as ρ¯x   orye   or   g. It should be noted that categorization of the activator into regular and defective ions was required to explain the early stage of the acceptor decay curve. Without this classification, a conspicuous rising would occur during the initial stage of the calculated acceptor decay curve, which would be inconsistent with the actual measured decay curve. In fact, no dramatic rising was observed in the actual measured decay curve, as shown in Figs. 5(a) and 5(b). This issue will be discussed in more detail later in this section.

The rate equations consisting of ρx   orye   or   g and ρ¯x   orye   or   g were derived as follows:
dρCa1edt   =   GρCa1gkrCa1ρCa1e   k12ρSr1eρSr2g        k11ρCa1eρ¯Ca1g   k12ρCa1eρ¯Ca2g   dρCa2edt   =   GρCa2gkrCa2ρCa2e+   k12ρCa1eρCa2g        k22ρCa2eρ¯Ca2g   dρCa1gdt   =   GρCa1g+krCa1ρCa1e+   k12ρCa1eρCa2g   +     k11ρCa1eρ¯Ca1g+   k12ρCa1eρ¯Ca2gdρCa2gdt   =   GρCa2g+krCa2ρCa2e   k12ρCa1eρCa2g   +     k22ρCa2eρ¯Ca2g                                                                                                           (1)dρ¯Ca1edt   =   Gρ¯Ca1g   (krCa1+Kn)ρ¯Ca1e+   k11ρCa1eρ¯Ca1g   dρ¯Ca2edt   =   Gρ¯Ca2g   (krCa2+Kn)ρ¯Ca2e   +   k12ρCa1eρ¯Ca2g   +     k22ρCa2eρ¯Ca2g     dρ¯Ca1gdt   =   Gρ¯Ca1g+   (krCa1+Kn)ρ¯Ca1e   k11ρCa1eρ¯Ca1g   dρ¯Ca2gdt   =   Gρ¯Ca2g+   (krCa2+Kn)ρ¯Ca2e      k12ρCa1eρ¯Ca2g        k22ρCa2eρ¯Ca2g   ρCa1e   +   ρCa2e   +ρCa1g+ρCa2g+ρ¯Ca1e+ρ¯Ca2e+ρ¯Ca1g+   ρ¯Ca2g=   totalEu2+   numberper   unit   volume     (ρTotal)
G is the excitation rate of Eu2+ ions, which was estimated to be 100 sec−1 using the so-called Einstein’s coefficients [26

26. R. Loudon, In The Quantum Theory of Light (Oxford Univ, Oxford, 1973)

], and the oscillator strength of the 4f-5d transition for Eu2+ in some oxide hosts [27

27. R. Reisfeld, E. Greenberg, R. Velapodi, and B. Barnett, “Luminescence Quantum Efficiency of Gd and Tb in Borate Glasses and the Mechanism of Energy Transfer between Them,” J. Chem. Phys. 56(4), 1698–1705 (1972), http://jcp.aip.org/jcpsa6/v56/i4/p1698_s1. [CrossRef]

,28]. Although we failed to determine the exact G value for N-CASIN-0.7 and N-CASIN-1.2, our estimate is reasonable because the variation in G does not affect the shape of the decay curve, but, rather, has a dramatic influence on the overall number density. Considering our use of normalized decay curves for the least-square fitting process, a rough estimation of G would not affect the determination of other more important parameters. KrCa1(1/τCa1) and KrCa2(1/τCa2) are the radiative decay rates for P1 and P2, which are presumed to vary with respect to the activator site environment. Kn is the non-radiative decay rate of defective Eu2+ ions, which represents direct quenching by killer sites. kxy is the energy transfer constant of interactions between Eu2+ ions, wherein the subscripts x and y stand for the Ca1 and Ca2 sites in the host lattice, respectively. The energy transfer constant (kxy) should never be misunderstood to be identical to the energy transfer rate. In fact, the non-radiative energy transfer rate is kxyρygand kxyρ¯yg.

Our rate equations involved four non-linear terms that describe the energy transfer between the Eu2+ activators of various types: regular Ca1 to regular Ca2, regular Ca1 to defective Ca1 (and Ca2), and regular Ca2 to defective Ca2. Energy transfer from defective to regular or to defective sites was not considered here because the defective activators are apt to be directly quenched by the killer site. Backward energy transfer from Ca2 to Ca1 was ignored because of the negligible spectral overlap, and migration between the same type of activators was also ignored. We categorized the donors into four different types and possible energy transfer routes between them are included in our rate equations. This means that a certain degree of migration effect was already taken into account partly in the present rate equation model, even though we failed to accommodate a complete consideration of migration. We only adopted three independent energy transfer constants that represented the following energy transfers: Ca1 → Ca1 (k11), Ca2 → Ca2 (k22) and Ca1 → Ca2 (k12) interactions (energy transfer routes). Because the energy transfer that occurred between Eu2+ activators of particular types was examined in the present investigation, the inhomogeneous distribution of Eu2+ activators in the crystalline host was an important issue. However, in CASIN hosts, the four characteristic activator sites, i.e., regular Ca1, defective Ca1, regular Ca2 and defective Ca2, all belong to an identical Wyckoff site, so that the acceptor distribution around a donor is identical, irrespective of which activator sites were considered the donor sites. Accordingly, all four energy transfer constants should involve the same acceptor distribution function. As a distribution function, we adopted the angular-class model proposed by Vásquez[25

25. S. O. Vásquez, “Energy transfer processes in organized media. III. A two-center model for nonhomogeneous crystals,” J. Chem. Phys. 108(2), 723–728 (1998), http://jcp.aip.org/jcpsa6/v108/i2/p723_s1. [CrossRef]

,29

29. S. O. Vásquez, “Energy transfer processes in organized media. I. A crystal model for cubic sites,” J. Chem. Phys. 104(19), 7652–7657 (1996), http://jcp.aip.org/jcpsa6/v104/i19/p7652_s1. [CrossRef]

-32

32. S. O. Vásquez, “Energy-transfer processes in quasi-bidimensional crystal arrays,” Phys. Rev. B 64(12), 125103 (2001), http://prb.aps.org/abstract/PRB/v64/i12/e125103. [CrossRef]

], which predicts a non-radiative energy-transfer rate based on an inhomogeneous distribution of activators in the crystalline host. The primary determinant of the energy transfer constant is not the distribution function, but some other factor included in kxy because the distribution function included in every kxy is identical. kxy is comprised of the product of the radiative decay rate of the donor, the spectral overlap, the oscillator strength of the acceptor, and the acceptor distribution function.
kxy   =   3e2c3h5πε2m   ×   krx   ×​  SO   ×fy  ​×lnlmax(1Rlg)6.
(2)
where e is the electron charge, m is the electron mass, c is the speed of light, h is the Plank constant, ε is the dielectric constant of the host, Krx is the radiative rate of donors (x stands for regular Ca1 or defective Ca1), SO is the spectral overlap between donor emission and acceptor absorption, fy is the oscillator strength of acceptor absorption, nlmax is the maximum number of Eu2+ ions in the l th angular class of the CASIN host, and Rlg is the possible donor-acceptor distance (all possible Ca-Ca distances in the CASIN host structure) for the pointing-generating vectors of the l th angular class. Since the distribution function (lnlmax(1Rlg)6) included in kxy was assumed to be identical for each activator site, kxy should be strongly affected by SO and fy. It is reasonable to assume k12 to be greater than k11 and k22, because the spectral overlap between the emission of Eu2+ activators at Ca1 and the absorption of Eu2+ activators at Ca2 would be greater than the spectral overlap between Ca1 and Ca1 or between Ca2 and Ca2. In addition, k11 and k22 were assumed to be identical in the same context.

Reproduction of the decay curves was accomplished using the well-known Runge-Kutta method [33

33. W. E. Boyce and R. C. Diprima, In Elementry Differential Equations and Boundary Value Problems (John Wiley & Sons, 1986)

], followed by least-square-fitting (LSF). The donor (P1) and acceptor (P2) decay curves were simultaneously subjected to LSF by use of the following relationships:
m(ρCa1e   +   ρ¯Ca1e)*   +       (1m)(ρCa2e   +ρ¯Ca2e)*=   I525
n(ρCa2e   +   ρ¯Ca2e)*   +       (1n)(ρCa1e   +ρ¯Ca1e)*   =   I700for  N_CASIN_0.7 n(ρCa2e   +   ρ¯Ca2e)*   +       (1n)(ρCa1e   +ρ¯Ca1e)*   =   I725   for    N_CASIN_1.2
(4)
Where I525, I700 and I725 are the normalized decay curves monitored at 525, 700 and 725 nm, representing donor and acceptor decay curves, respectively, m is the P2 contribution to the 525-nm decay curve, and n is the P1 contribution to the 700-nm (or 725-nm) decay curve, and the superscript asterisk represents normalized values. The wavelengths, at which the donor and acceptor decay curves were detected, were selected to minimize the influence of one on the other. Namely, we used the largest possible separation between the wavelengths within the emission spectrum. So, both m and n values should be close to 1. The distribution of Eu2+ activators in both sites was considered unequal, because the Ca1/Ca2 ratio was intentionally altered by introducing different Al/Si ratios. In this regard, RCa1 was defined as the ratio of Ca1 sites to total Ca sites in the CASIN host. Thus, the initial condition for the Runge-Kutta process is given as follows:
ρCa1e=0,   ρCa2e=0,   ρCa1g=RCa1(1q)2ρTotal,   ρCa2g=(1RCa1)(1q)2ρTotal,​ ​  ρ¯Ca1e=0,   ρ¯Ca2e=0,   ρ¯Ca1g=RCa1q2ρTotal,   ρ¯Ca2g=(1RCa1)q2ρTotal
(5)
where q is the fraction of defective Eu2+ activators. The pulse duration of the laser light source was 20 picoseconds, which allowed the excitation rate (G) to take effect only at t = 0 and then vanish immediately.

Figure 5(a) and 5(b) show the measured and calculated decay curves with acceptable fitting quality. The best-fitted parameters obtained from the PSO process were all reasonable, as shown in Table 3

Table 3. The best-fitted parameters from the PSO process

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. The radiative decay rates for the two distinct local environments, KrCa1 and KrCa2, were determined to be similar to one another within the same order of magnitude, though we set them as independent free parameters in the PSO process. The non-linear energy-transfer terms in the rate equation, which were of particular interest, were coupled with two energy-transfer constants (k12,k11 = k22).k12 for N-CASIN-1.2 was much greater than that for N-CASIN-0.7 while k11 ( = k22) did not differ significantly between N-CASIN-0.7 and N-CASIN-1.2. This result indicates that the energy transfer from Ca1 to Ca2 was enhanced as the Al/Si ratio around the activator increased. The estimated RCa1 was also noteworthy because N-CASIN-1.2 had a lower RCa1 value than N-CASIN-0.7. RCa1 showed an inverse relationship with the Al/Si ratio, i.e., the higher the Al/Si ratio, the lower the RCa1 value. This result coincided with the spectral measurement, which showed a reduction in the high-energy emission (P1) at higher Al/Si ratios that ultimately caused a red shift. The fraction of defective Eu2+ activators (q) was indicative of the killer concentration (defects or defect impurities). The m and n values were estimated to be close to 1, which were quite reasonable considering the spectral deconvolution shown in Figs. 4(a) and 4(b). Thus, the detection wavelengths used for donor and acceptor decay were far enough from each other to allow adequate separation. The time evolution of the donors in the four different local environments, i.e., ρCa1e,ρ¯Ca1e,ρCa2e and ρ¯Ca2e curves, was calculated based on the fitting results for N-CASIN-0.7 and N-CASIN-1.2 (Fig. 6
Fig. 6 Calculated time evolution (decay curve) of normalized ρCa1e,     ρ¯Ca1e,     ρCa2e and    ρ¯Ca2e for (a) N-CASIN-0.7 and (b) N-CASIN-1.2. All curves were normalized by ρCa1e(0).
). It should be noted that separate experimental measurements for the ρCa1e,     ρ¯Ca1e,     ρCa2e and    ρ¯Ca2e curves were unavailable.

It was readily apparent that active energy transfer between Ca1 and Ca2 occurred. This type of energy transfer, which has been called site-to-site energy transfer [13

13. S. Lee and K.-S. Sohn, “Effect of inhomogeneous broadening on time-resolved photoluminescence in CaAlSiN3:Eu2+,” Optic. Lett 35, 1004–1006 (2010), http://www.opticsinfobase.org/ol/search.cfm.

], was clearly evidenced by the variation in decay rate with respect to the detection wavelength. Site-to-site energy transfer might involve a variety of local environments, so that precise classification of the local environment might lead to a more complicated rate equation model along with more terms and more decision parameters. In the present investigation, however, we adopted only two local environments, which were simplified as a Si-rich environment (Ca1) and an Al-rich environment (Ca2). Accordingly, only two Gaussian emission components (P1 and P2) were deconvoluted in an attempt to implement our quantitative analysis of the decay curves. In fact, this simplification was based on the fact that the two-peak deconvolution yielded the best fit compared with other deconvolution cases that involved more Gaussian components. Furthermore, the analysis based on the two-peak simplification could be generalized and applied to more complicated systems that involved more Gaussian components, resulting in a more refined classification of the local environments.

4. Conclusion

We examined non-stoichiometric CASIN phosphors to assign the deconvoluted Gaussian components that comprise the emission spectrum to their corresponding local environments. The Al/Si molar ratio around the Eu2+ activator was a key parameter in assigning the deconvoluted Gaussian peaks. The high Al/Si ratio near the Eu2+ activator abolished the high energy peak. The Eu2+ activator site in an Al-rich environment with a smaller average Ca-N distance was assigned to the lower energy emission peak, while that in a Si-rich environment with a longer average Ca-N distance was assigned to the higher energy emission peak.

The energy transfer between Eu2+ activators in Si- and Al-rich environments was also taken into account. Decay curve analysis along with rate equation modeling confirmed active energy transfer from the Eu2+ activator site in the Si-rich environment (donor) to the Eu2+ activator site in the Al-rich environment (acceptor). In particular, numerical computation based on particle swarm optimization was implemented to solve the rate equations with nine unknown decision parameters and to correctly evaluate the decision parameters. Use of this strategy gave precise donor and acceptor decay curves with outstanding fitting quality along with plausible best-fitted parameters. The estimated energy-transfer rate, radiative decay rate, defect concentration, and relative contribution of each local environment were all within a reasonable range of values. Decay curve fitting with the quality reported in the present study is extremely rare in the published literature, especially for simultaneous treatment of the donor and acceptor decay curves for Eu2+ activator systems. In this context, our energy-transfer modeling and the ensuing PSO process resulted in excellent agreement between the calculated and experimental results.

Acknowledgements

This work was supported by the IT R&D Program of MKE/IITA (2009-F-020-01) and partly supported by the WCU (World Class University) program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology.

References and links

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C. Kulshreshtha, J. H. Kwak, Y.-J. Park, and K.-S. Sohn, “Photoluminescent and decay behaviors of Mn2+ and Ce2+ co-activated MgSiN2 phosphors for use in LED applications,” Opt. Lett. 34(6), 794–796 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-6-794. [CrossRef] [PubMed]

2.

T. Suehiro, H. Onuma, N. Hirosaki, R.-J. Xie, T. Sato, and A. Miyamoto, “Powder Synthesis of Y-α-SiAlON and Its Potential as a Phosphor Host,” J. Phys. Chem. C 114(2), 1337–1342 (2010). [CrossRef]

3.

C. Zhang, H. Lian, D. Kong, S. Huang, and J. Lin, “Stuctural and Bluish-White Luminescent Properties of Li+-Doped BPO4 as a Potential Environmentally Friendly Phosphor Material,” J. Phys. Chem. C 113(4), 1580–1588 (2009). [CrossRef]

4.

X. Piao, K. Machida, T. Horikawa, H. Hanzawa, Y. Shimomura, and N. Kijima, “Preparation of CaAlSiN3:Eu2+ phosphors by the Self-Propagating High-Temperature Synthesis and Their Luminescent Properties,” Chem. Mater. 19(18), 4592–4599 (2007), http://pubs.acs.org/doi/abs/10.1021/cm070623c?prevSearch=%255Bauthor%253A%2BX.%2BPiao%255D&searchHistoryKey=. [CrossRef]

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Y. Q. Li, N. Hirosaki, R.-J. Xie, T. Takeda, and M. Mitomo, “Yellow-Orange-Emitting CaAlSiN3:Ce3+ phosphor: Structure, Photoluminescence, and Application in White LEDs,” Chem. Mater. 20(21), 6704–6714 (2008), http://pubs.acs.org/doi/abs/10.1021/cm801669x?prevSearch=%255Bauthor%253A%2BM.%2BMitomo%255D&searchHistoryKey=. [CrossRef]

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K. Uheda, N. Hirosaki, Y. Yamamoto, A. Naito, T. Nakajima, and H. Tamamoto, “Luminescence Properties of a Red Phosphor, CaAlSiN3:Eu2+, for White Light-Emitting Diodes,” Electrochem. Soc 9, H22–H25 (2006), http://www.ecsdl.org/vsearch/servlet/VerityServlet?KEY=JESOAN&ONLINE=YES&smode=strresults&sort=rel&maxdisp=25&threshold=0&pjournals=ESLEF6%2CJESOAN%2CMAECES%2CECSTF8%2CTESOAV&possible1=K.+Uheda&possible1zone=author&OUTLOG=NO&viewabs=ESLEF6&key=DISPLAY&docID=6&page=1&chapter=0.

7.

H. Watanabe, H. Yamane, and N. Kijima, “Crystal structure and luminescence of Sr0.99Eu0.01AlSiN3,” J. Solution Chem. 181, 1848–1852 (2008), http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WM2-4SDX30C-4&_user=404030&_coverDate=08%2F31%2F2008&_alid=1371026111&_rdoc=1&_fmt=high&_orig=search&_cdi=6922&_sort=r&_docanchor=&view=c&_ct=1&_acct=C000019098&_version=1&_urlVersion=0&_userid=404030&md5=40b1daf98bdb00f8e87fc9f993dbc649.

8.

J. Li, T. Watanabe, N. Sakamoto, H. Wada, T. Setoyama, and M. Yoshimura, “Synthesis of a Multinary Nitride, Eu-Doped CaAlSiN3, from Alloy at Low Temperatures,” Chem. Mater. 20(6), 2095–2105 (2008), http://pubs.acs.org/doi/abs/10.1021/cm071612m?prevSearch=%255Bauthor%253A%2BT.%2BWatanabe%255D&searchHistoryKey=. [CrossRef]

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K. Uheda, N. Hirosaki, and H. Yamamoto, “Host lattice materials in the system Ca3N2-AlN-Si3N4 for white light emitting diode,” Phys. Status Solidi 203(11), 2712–2717 (2006), http://www3.interscience.wiley.com/search/allsearch?mode=quicksearch&products=journal&WISsearch2=%281862-6319%2C1521-396X%29&WISindexid2=issn&contentTitle=physica+status+solidi+%28a%29&contextLink=blah&groupTitleOID=122649431&restrict=groupTitle&WISsearch1=K.+Uheda&WISindexid1=WISauthor&articleGo.x=8&articleGo.y=11. [CrossRef]

10.

J. Li, T. Watanabe, H. Wada, T. Setoyama, and M. Yoshimura, “Synthesis of Eu-Doped CaAlSiN3 from Ammonometallates: Effects of Sodium Content and Pressure,” J. Am. Ceram. Soc. 92(2), 344–349 (2009), http://www3.interscience.wiley.com/search/allsearch?mode=citation&contextLink=blah&issn=1551-2916&volume=92&issue=&pages=344. [CrossRef]

11.

H. Watanabe, H. Wada, K. Seki, M. Itou, and N. Kijima, “Synthetic Method and Luminescence Properties of SrxCa1-xAlSiN3:Eu2+ Mixed Nitride Phosphors,” J. Electrochem. Soc. 155(3), F31–F36 (2008), http://www.ecsdl.org/vsearch/servlet/VerityServlet?KEY=JESOAN&ONLINE=YES&smode=strresults&sort=rel&maxdisp=25&threshold=0&possible1=H.+Watanabe&possible1zone=author&OUTLOG=NO&viewabs=JESOAN&key=DISPLAY&docID=11&page=1&chapter=0. [CrossRef]

12.

H. Watanabe and N. Kijima, “Crystal structure and luminescence properties of SrxCa1-xAlSiN3:Eu2+ mixed nitride phosphor,” J. Alloy. Comp. 475(1-2), 434–439 (2009), http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TWY-4T8R1PM-5&_user=404030&_coverDate=05%2F05%2F2009&_alid=1371055120&_rdoc=1&_fmt=high&_orig=search&_cdi=5575&_sort=r&_docanchor=&view=c&_ct=1&_acct=C000019098&_version=1&_urlVersion=0&_userid=404030&md5=145691746bab9f3ac3ccc3fd9a5528c1. [CrossRef]

13.

S. Lee and K.-S. Sohn, “Effect of inhomogeneous broadening on time-resolved photoluminescence in CaAlSiN3:Eu2+,” Optic. Lett 35, 1004–1006 (2010), http://www.opticsinfobase.org/ol/search.cfm.

14.

R. C. Eberhart and J. Kennedy, In A new optimizer using particle swarm theory (proc IEEE Int Conf on Neural networks, Nagoya, 1995) p. 39–43.

15.

J. Kennedy and R. C. Eberhart, In Particle swarm optimization (proc IEEE Int Conf on Neural networks, Perth, 1995) p. 1942–1948.

16.

J. Kennedy, In The particle swarm: Social adaption of knowledge (Proc IEEE Int Conf on Evolutionary Computation, Indianapolis, 1997) p. 303–308.

17.

P. Dorenbos, L. Pierron, L. Dinca, C. W. E. Eijk, A. Kahn-Harari, and B. Viana, “C.W.E van Eijk, A. Kahn-Harari, B. Viana, “4f-5d spectroscopy of Ce3+ in CaBPO5, LiCaPO4 and Li2CaSiO4,” J. Phys. Condens. Matter 15(3), 511–520 (2003), http://iopscience.iop.org/0953-8984/15/3/315?fromSearchPage=true. [CrossRef]

18.

P. Dorenbos, “5d-level energies of Ce3+ and the crystalline environment. II. Chloride, bromide, and iodide compounds,” Phys. Rev. B 62(23), 15650–15659 (2000), http://prb.aps.org/abstract/PRB/v62/i23/p15650_1. [CrossRef]

19.

P. Dorenbos, “5d-level energies of Ce3+ and the crystalline environment. I. Fluoride compounds,” Phys. Rev. B 62(23), 15640–15649 (2000), http://prb.aps.org/abstract/PRB/v62/i23/p15640_1. [CrossRef]

20.

P. Dorenbos, “Relation between Eu2+ and Ce3+ f ↔ d-transition energies in inorganic compounds,” J. Phys. Condens. Matter 15(27), 4797–4807 (2003), http://iopscience.iop.org/0953-8984/15/27/311?fromSearchPage=true. [CrossRef]

21.

P. Dorenbos, “5d-level energies of Ce3+ and the crystalline environment. III. Oxides containing ionic complexes,” Phys. Rev. B 64(12), 125117 (2001), http://prb.aps.org/abstract/PRB/v64/i12/e125117. [CrossRef]

22.

D. Ahn, N. Shin, K. D. Park, and K.-S. Sohn, “Energy Transfer Between Activators at Different Crystallographic Sites,” J. Electrochem. Soc. 156(9), J242–J248 (2009), http://www.ecsdl.org/vsearch/servlet/VerityServlet?KEY=JESOAN&ONLINE=YES&smode=strresults&sort=rel&maxdisp=25&threshold=0&possible1=D.+Ahn&possible1zone=author&OUTLOG=NO&viewabs=JESOAN&key=DISPLAY&docID=1&page=1&chapter=0. [CrossRef]

23.

K.-S. Sohn, S. Lee, R.-J. Xie, and N. Hirosaki, “Time-resolved photoluminescence analysis of two-peak emission behavior in Sr2Si5N8:Eu2+,” Appl. Phys. Lett. 95(12), 121903 (2009), http://apl.aip.org/applab/v95/i12/p121903_s1. [CrossRef]

24.

K.-S. Sohn, B. Lee, R.-J. Xie, and N. Hirosaki, “Rate-equation model for energy transfer between activators at different crystallographic sites in Sr2Si5N8:Eu(2+),” Opt. Lett. 34(21), 3427–3429 (2009). [CrossRef] [PubMed]

25.

S. O. Vásquez, “Energy transfer processes in organized media. III. A two-center model for nonhomogeneous crystals,” J. Chem. Phys. 108(2), 723–728 (1998), http://jcp.aip.org/jcpsa6/v108/i2/p723_s1. [CrossRef]

26.

R. Loudon, In The Quantum Theory of Light (Oxford Univ, Oxford, 1973)

27.

R. Reisfeld, E. Greenberg, R. Velapodi, and B. Barnett, “Luminescence Quantum Efficiency of Gd and Tb in Borate Glasses and the Mechanism of Energy Transfer between Them,” J. Chem. Phys. 56(4), 1698–1705 (1972), http://jcp.aip.org/jcpsa6/v56/i4/p1698_s1. [CrossRef]

28.

H. Ebendorff-Heidepriem and D. Ehrt, “Formation and UV absorption of cerium, europium and terbium ions in different valencies in glasses,” Opt. Mater. 15(1), 7–25 (2000), http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TXP-40NFV77-2&_user=404030&_coverDate=09%2F30%2F2000&_alid=1371320381&_rdoc=1&_fmt=high&_orig=search&_cdi=5596&_sort=r&_docanchor=&view=c&_ct=1&_acct=C000019098&_version=1&_urlVersion=0&_userid=404030&md5=ec854d1d4a17443404bf23c32bfbec52. [CrossRef]

29.

S. O. Vásquez, “Energy transfer processes in organized media. I. A crystal model for cubic sites,” J. Chem. Phys. 104(19), 7652–7657 (1996), http://jcp.aip.org/jcpsa6/v104/i19/p7652_s1. [CrossRef]

30.

S. O. Vásquez, “Energy transfer processes in organized media. II. Generalization of the crystal model for dipole-dipole interactions in cubic sites,” J. Chem. Phys. 106(21), 8664–8671 (1997), http://jcp.aip.org/jcpsa6/v106/i21/p8664_s1. [CrossRef]

31.

S. O. Vásquez, “Crystal model for energy-transfer processes in organized media: Higher-order electric multipolar interations,” Phys. Rev. B 60(12), 8575–8585 (1999), http://prb.aps.org/abstract/PRB/v60/i12/p8575_1. [CrossRef]

32.

S. O. Vásquez, “Energy-transfer processes in quasi-bidimensional crystal arrays,” Phys. Rev. B 64(12), 125103 (2001), http://prb.aps.org/abstract/PRB/v64/i12/e125103. [CrossRef]

33.

W. E. Boyce and R. C. Diprima, In Elementry Differential Equations and Boundary Value Problems (John Wiley & Sons, 1986)

OCIS Codes
(160.2540) Materials : Fluorescent and luminescent materials
(250.5230) Optoelectronics : Photoluminescence
(300.6280) Spectroscopy : Spectroscopy, fluorescence and luminescence

ToC Category:
Materials

History
Original Manuscript: June 16, 2010
Revised Manuscript: July 8, 2010
Manuscript Accepted: July 20, 2010
Published: August 3, 2010

Citation
Yoon Won Jung, Bonghyun Lee, Satendra Pal Singh, and Kee-Sun Sohn, "Particle-swarm-optimization-assisted rate equation modeling of the two-peak emission behavior of non-stoichiometric CaAlxSi(7-3x)/4N3:Eu2+ phosphors," Opt. Express 18, 17805-17818 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-17805


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References

  1. C. Kulshreshtha, J. H. Kwak, Y.-J. Park, and K.-S. Sohn, “Photoluminescent and decay behaviors of Mn2+ and Ce2+ co-activated MgSiN2 phosphors for use in LED applications,” Opt. Lett. 34(6), 794–796 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-6-794 . [CrossRef] [PubMed]
  2. T. Suehiro, H. Onuma, N. Hirosaki, R.-J. Xie, T. Sato, and A. Miyamoto, “Powder Synthesis of Y-α-SiAlON and Its Potential as a Phosphor Host,” J. Phys. Chem. C 114(2), 1337–1342 (2010). [CrossRef]
  3. C. Zhang, H. Lian, D. Kong, S. Huang, and J. Lin, “Stuctural and Bluish-White Luminescent Properties of Li+-Doped BPO4 as a Potential Environmentally Friendly Phosphor Material,” J. Phys. Chem. C 113(4), 1580–1588 (2009). [CrossRef]
  4. X. Piao, K. Machida, T. Horikawa, H. Hanzawa, Y. Shimomura, and N. Kijima, “Preparation of CaAlSiN3:Eu2+ phosphors by the Self-Propagating High-Temperature Synthesis and Their Luminescent Properties,” Chem. Mater. 19(18), 4592–4599 (2007). [CrossRef]
  5. Y. Q. Li, N. Hirosaki, R.-J. Xie, T. Takeda, and M. Mitomo, “Yellow-Orange-Emitting CaAlSiN3:Ce3+ phosphor: Structure, Photoluminescence, and Application in White LEDs,” Chem. Mater. 20(21), 6704–6714 (2008). [CrossRef]
  6. K. Uheda, N. Hirosaki, Y. Yamamoto, A. Naito, T. Nakajima, and H. Tamamoto, “Luminescence Properties of a Red Phosphor, CaAlSiN3:Eu2+, for White Light-Emitting Diodes,” Electrochem. Soc 9, H22–H25 (2006).
  7. H. Watanabe, H. Yamane, and N. Kijima, “Crystal structure and luminescence of Sr0.99Eu0.01AlSiN3,” J. Solution Chem. 181, 1848–1852 (2008).
  8. J. Li, T. Watanabe, N. Sakamoto, H. Wada, T. Setoyama, and M. Yoshimura, “Synthesis of a Multinary Nitride, Eu-Doped CaAlSiN3, from Alloy at Low Temperatures,” Chem. Mater. 20(6), 2095–2105 (2008). [CrossRef]
  9. K. Uheda, N. Hirosaki, and H. Yamamoto, “Host lattice materials in the system Ca3N2-AlN-Si3N4 for white light emitting diode,” Phys. Status Solidi 203(11), 2712–2717 (2006). [CrossRef]
  10. J. Li, T. Watanabe, H. Wada, T. Setoyama, and M. Yoshimura, “Synthesis of Eu-Doped CaAlSiN3 from Ammonometallates: Effects of Sodium Content and Pressure,” J. Am. Ceram. Soc. 92(2), 344–349 (2009). [CrossRef]
  11. H. Watanabe, H. Wada, K. Seki, M. Itou, and N. Kijima, “Synthetic Method and Luminescence Properties of SrxCa1-xAlSiN3:Eu2+ Mixed Nitride Phosphors,” J. Electrochem. Soc. 155(3), F31–F36 (2008). [CrossRef]
  12. H. Watanabe and N. Kijima, “Crystal structure and luminescence properties of SrxCa1-xAlSiN3:Eu2+ mixed nitride phosphor,” J. Alloy. Comp. 475(1-2), 434–439 (2009). [CrossRef]
  13. S. Lee, and K.-S. Sohn, “Effect of inhomogeneous broadening on time-resolved photoluminescence in CaAlSiN3:Eu2+,” Opt. Lett. 35, 1004–1006 (2010).
  14. R. C. Eberhart, and J. Kennedy, In A new optimizer using particle swarm theory (Proc IEEE Int Conf on Neural networks, Nagoya, 1995) p. 39–43.
  15. J. Kennedy and R. C. Eberhart, In Particle swarm optimization (Proc IEEE Int Conf on Neural Networks, Perth, 1995) p. 1942–1948.
  16. J. Kennedy, In The particle swarm: Social adaption of knowledge (Proc. IEEE Int. Conf. on Evolutionary Computation, Indianapolis, 1997) p. 303–308.
  17. P. Dorenbos, L. Pierron, L. Dinca, C. W. E. van Eijk, A. Kahn-Harari, and B. Viana, "4f-5d spectroscopy of Ce3+ in CaBPO5, LiCaPO4 and Li2CaSiO4," J. Phys. Condens. Matter 15(3), 511–520 (2003). [CrossRef]
  18. P. Dorenbos, “5d-level energies of Ce3+ and the crystalline environment. II. Chloride, bromide, and iodide compounds,” Phys. Rev. B 62(23), 15650–15659 (2000). [CrossRef]
  19. P. Dorenbos, “5d-level energies of Ce3+ and the crystalline environment. I. Fluoride compounds,” Phys. Rev. B 62(23), 15640–15649 (2000). [CrossRef]
  20. P. Dorenbos, “Relation between Eu2+ and Ce3+ f ↔ d-transition energies in inorganic compounds,” J. Phys. Condens. Matter 15(27), 4797–4807 (2003). [CrossRef]
  21. P. Dorenbos, “5d-level energies of Ce3+ and the crystalline environment. III. Oxides containing ionic complexes,” Phys. Rev. B 64(12), 125117 (2001). [CrossRef]
  22. D. Ahn, N. Shin, K. D. Park, and K.-S. Sohn, “Energy Transfer Between Activators at Different Crystallographic Sites,” J. Electrochem. Soc. 156(9), J242–J248 (2009). [CrossRef]
  23. K.-S. Sohn, S. Lee, R.-J. Xie, and N. Hirosaki, “Time-resolved photoluminescence analysis of two-peak emission behavior in Sr2Si5N8:Eu2+,” Appl. Phys. Lett. 95(12), 121903 (2009). [CrossRef]
  24. K.-S. Sohn, B. Lee, R.-J. Xie, and N. Hirosaki, “Rate-equation model for energy transfer between activators at different crystallographic sites in Sr2Si5N8:Eu2+,” Opt. Lett 34, 3427–3429 (2009). [CrossRef] [PubMed]
  25. S. O. Vásquez, “Energy transfer processes in organized media. III. A two-center model for nonhomogeneous crystals,” J. Chem. Phys. 108(2), 723–728 (1998). [CrossRef]
  26. R. Loudon, In The Quantum Theory of Light (Oxford Univ, Oxford, 1973)
  27. R. Reisfeld, E. Greenberg, R. Velapodi, and B. Barnett, “Luminescence Quantum Efficiency of Gd and Tb in Borate Glasses and the Mechanism of Energy Transfer between Them,” J. Chem. Phys. 56(4), 1698–1705 (1972). [CrossRef]
  28. H. Ebendorff-Heidepriem and Ebendorff-Heidepriem and D. Ehrt, “Formation and UV absorption of cerium, europium and terbium ions in different valencies in glasses,” Opt. Mater. 15(1), 7–25 (2000). . [CrossRef]
  29. S. O. Vásquez, “Energy transfer processes in organized media. I. A crystal model for cubic sites,” J. Chem. Phys. 104(19), 7652–7657 (1996). [CrossRef]
  30. S. O. Vásquez, “Energy transfer processes in organized media. II. Generalization of the crystal model for dipole-dipole interactions in cubic sites,” J. Chem. Phys. 106(21), 8664–8671 (1997). [CrossRef]
  31. S. O. Vásquez, “Crystal model for energy-transfer processes in organized media: Higher-order electric multipolar interations,” Phys. Rev. B 60(12), 8575–8585 (1999). [CrossRef]
  32. S. O. Vásquez, “Energy-transfer processes in quasi-bidimensional crystal arrays,” Phys. Rev. B 64(12), 125103 (2001). [CrossRef]
  33. W. E. Boyce and R. C. Diprima, In Elementry Differential Equations and Boundary Value Problems (John Wiley & Sons, 1986)

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