A rate equation model for the energy transfer mechanism of a novel multi-color-emissive phosphor, Ca1.624Sr0.376Si5O3N6:Eu2+

Jin Hee Leea, Satendra Pal Singha, Minseuk Kima, Myoungho Pyob, Woon Bae Park*a and Kee-Sun Sohn*a
aFaculty of Nanotechnology and Advanced Materials Engineering, Sejong University, Seoul 143-747, Republic of Korea
bDepartment of Printed Electronics Engineering, Sunchon National University, Sunchon, Chonnam 540-742, Republic of Korea. E-mail: kssohn@sejong.ac.kr; imjinpp@sejong.ac.kr

Received 9th August 2019 , Accepted 2nd October 2019

First published on 2nd October 2019


Multi-color emissions (or broadband emissions) from a single-phase phosphor with a single activator are an unfamiliar idea compared with those from multi-color-center materials. A single activator that is located in different crystallographic sites of a single-phase phosphor, however, could lead to multimodal emission peaks for multi-color (or broadband) emissions. The discovery of a single-phase-single-activator-broadband-phosphor is rare, and it is regarded as difficult to accomplish. The present investigation introduces a novel single-phase-single-activator-broadband-phosphor (Ca1.624Sr0.376Si5O3N6:Eu2+) and provides an in-depth examination of the energy transfer between different crystallographic sites which is the governing mechanism for the broadband emissions. Structural analysis is backed up by density functional theory (DFT) calculations, which validate the structural model of the discovered novel phosphor. Rate-equation modeling is introduced based on particle swarm optimization (PSO) to provide a complete quantitative analysis for the mechanism of the energy transfer.


Introduction

Multi-color emissions (or broadband emissions) from single-component materials would be favorable for many applications such as in light emitting diodes,1–11 full-color displays,12–15 anti-fake labeling,16–19 cellular imaging,20–22 and fluorescent probes/indicators.23–26 Multi-color emissions from polymer materials based on a single- or multi-chromophore system were not the focus of this study. Instead, we focused on inorganic phosphor materials with broadband emissions. The easiest way to realize multi-color emissions from a single inorganic host material is the introduction of multi-activators.27–38 The most popular multi-activator approach involves up-conversion phosphors.27–33 Another traditional multi-activator approach is referred to as sensitization.34–38 Neither of these approaches address broadband emissions, however, which place them outside the scope of the present investigation.

Our approach targets multi-color emissions from a single-activator system. When a single activator such as Eu2+ or Ce3+ occupies different crystallographic sites in a single host material, the emission energy significantly differs from site to site, which in turn leads to multi-color-emissions and a broadband phosphor. This type of multi-color emission would be greatly affected by inter-site energy transfer (ET). ET from a higher energy site to a lower energy site must be negated in order to achieve broadband emissions since this reaction deteriorates emissions from the higher energy side while emissions from the lower energy side would be boosted. In this regard, ET should be carefully controlled to achieve broadband emissions. Nonetheless, theoretical analyses for inter-site ET have thus far been nonexistent.

A recent discovery of two impressive multi-color-emissive phosphors (Ca 1.5Ba0.5Si5O3N6:Eu2+ and Ca1.62Eu0.38Si5O3N6) based on a single-activator system has attracted a great deal of attention. The former was discovered by our group3,4 and the latter was identified by the group of Prof. Xie.2 These phosphors are composed of two well-known independent phosphor structures: BaSi6N8O:Eu2+ with a 3-D structure and CaSi2O2N2:Eu2+ with a layered structure.39,40 These are the so-called composite structures. Each of the constituent sub-structures provides unique sites for Eu2+ activators. BaSi6N8O:Eu2+ possesses a 3-D site for high-energy emissions (blue) and a 2-D site on the Ca layer for low-energy emissions (yellow).

The apparent structures of both the Ca1.5Ba0.5Si5O3N6:Eu2+ (referred to as the B-phase) and Ca1.62Eu0.38Si5O3N6 (referred to as the E-phase) phosphors appear to be very similar with identical symmetry (Cm), but the periodicity in the c-direction differs, i.e., it is doubled in the case of the E-phase. The major issue for the present investigation involved discovering a novel single-activator-multi-color-emissive material based on the composite structure. In this regard, we discovered the Ca1.624Sr0.376Si5O3N6:Eu2+ (referred to as the S-phase) phosphor, the apparent structure of which is more like the E-phase but differs from any of the previously discovered B- and E-phase examples in terms of the details of its SiX4 (X = O or N) networking structure. The discovery of an S-phase was not accomplished via simple ionic substitution, but, rather, by a fine-tuning of the exact structure. The available composition window for the unique structure of the S-phase is very narrow and distinctive from the B- and E-phases (Ca1.5Ba0.5Si5O3N6, Ca1.62Eu0.38Si5O3N6, and Ca1.624Sr0.376Si5O3N6). We investigated the precise structure of the S-phase with the assistance of Rietveld refinement and successive density functional theory (DFT) calculations.

Determining the structure of the S-phase was followed by ET analysis, which is a key to understanding broadband emissions. A rate equation model based on the refined structure of the S-phase, involving every possible non-linear interaction term, was derived, and thereafter a least-square-regression process associated with particle swarm optimization (PSO) was implemented to reproduce the decay curves for every activator site. This allowed us to evaluate all the unknown interaction rate constants (ET rate constants), along with some of the known constants predetermined from the XPS and decay measurements.

Experimental

Sample preparation & characterization

Commercially available starting materials in a solid-powder oxide or nitride state, CaCO3 (Kojundo, 99.99%), SrCO3 (Kojundo, 99.9%), SiO2 (Kojundo, 99.9%), α-Si3N4 (Ube, unreported), and Eu2O3 (Kojundo, 99.9%), were mixed and fired at 1650 °C for 8 h under N2 at a flow rate of 500 cc min−1.

The fired samples were subjected to X-ray diffraction (XRD), X-ray photoemission spectroscopy (XPS), and continuous-wave photoluminescence (CWPL). High-resolution XRD was carried out via synchrotron radiation X-ray sources at the Pohang Accelerator Laboratory (PAL). CWPL was performed using an in-house spectroscope equipped with a xenon lamp at an excitation wavelength of 360 nm with a 380 nm cut-off filter. The time-resolved emission spectra were also recorded 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 1064 nm frequency of the Nd:YAG laser. We used a 400 nm cut-off filter when TRPL was performed to eliminate any undesired influence from the laser pulse. Rietveld refinement was carried out using Fullprof software to determine the true structure of the S-phase.41

DFT calculations

A generalized gradient approximation (GGA) parameterized by Perdew, Burke, and Ernzerhof (PBE)42–45 and a hybrid approach based on the Heyd–Scuseria–Ernzerhof (HSE) functional46,47 were employed as exchange correlation potentials in the Vienna ab initio simulation package (VASP5.4).42–45 We adopted a plane-wave basis set with projector-augmented wave (PAW) potentials.42,43 K-mesh grids (4 × 1×6 and 4 × 1×3) based on the Monkhorst–Pack44 scheme were adopted for the single- (B-phase) and double-cell (E-Phase) models, respectively. The cut-off energy was 500 eV, and the self-consistency field tolerance threshold was 10−5 eV per atom. The positions of all atoms and lattice parameters were fully relaxed until the atomic forces converged to 0.01 eV Å−1.

Results and discussion

Structural determination of the Ca1.624Sr0.376Si5O3N6:Eu2+ phosphor

Structural determination was performed using high-resolution synchrotron X-ray diffraction (SXRD) spectra recorded in the 2θ range 10 to 130.50° at a step size of 0.005° using a wavelength of 1.5226 Å. The SXRD pattern of Ca1.624Sr0.376Si5O3N6:Eu2+ (S-phase) resembles the diffraction patterns of recently discovered compounds Ca1.5Ba0.5Si5O3N6:Eu2+ (B-phase)3,4 and Ca1.62Eu0.38Si5O3N6 (E-phase).2 The crystal structure of B- and E-phases is reported to exist as a monoclinic structure in the Cm space group with similar atomic arrangement and lattice parameters, but with different c-parameters. The c-parameter in the E-phase is double that of the B-phase, which results in a doubling of the unit cell volume with double the number of formula units per unit cell. The unit cell of these compounds consists of a vertex-sharing SiN4 and a SiN3O tetrahedron that forms two types of channels along the c-axis, namely the layer-type and the cage-type, and both of these channels alternately occur along the b-axis. The aim of the present investigation was to determine the true structure of the S-phase and compare the results with previously established structures.

Rietveld refinement was performed to determine the exact lattice parameters and atomic position coordinates of the S-phase. During the Rietveld refinement, a pseudo-Voigt function and a linear interpolation between the set background points with refinable heights were used to define the profile shape and the background, respectively. Parameters such as the scale factor, background, half-width parameters, mixing parameters, lattice parameters, positional coordinates, and thermal parameters were varied in the course of refinement. Occupancy parameters were refined and kept at optimum to maintain the charge neutrality of the compound, and it was necessary to use anisotropic peak broadening in the course of refinement. The Rietveld refinement was first carried out using lattice parameters (a = 7.07033, b = 23.86709(6), c = 4.825304 Å and β = 109.0647°) and position coordinates of the B-phase as the initial, so-called single-cell model. Fig. 1(a) depicts the full-pattern Rietveld refinement fit showing the observed and calculated profiles. This figure shows that some of the peaks with very low intensity occurring around 2θ = 17.41, 22.88, 26.45, 29.97, 31.89, 34.48, 47.83, and 48.46° (marked with ‘∇’ in the inset of Fig. 1(a)) were not indexed in the refinement and do not belong to any known impurity phase. This signifies that the structure of the S-phase does not fit the structural model of the B-phase. It should be noted that the unaccounted-for peaks occurring in this compound do not appear in the XRD pattern of a pure B-phase. We then refined the structural model of the E-phase with cell doubling along the c-axis (a = 7.0595, b = 23.750, c = 9.6345 Å and β = 109.0393°), which is referred to as a so-called double-cell model. This model resulted in a very good fit between the observed and calculated profiles shown in Fig. 1(b) with almost a flat difference profile and very good values for the agreement factors (Rp = 6.36, Rwp = 8.24, Rexp = 4.62 and χ2 = 3.16). All the peaks that were not indexed in the single-cell model were fully accounted for using the double-cell model. In addition to this phase, a very small amount of an impurity phase of Ca2Si5N8 with a monoclinic structure in the Cc space group,48,49 marked with an asterisk (*) in the inset of Fig. 1, was also detected and considered a secondary phase during the Rietveld refinement. The phase fraction for the impurity phase (Ca2Si5N8), as calculated from Rietveld refinement, was less than 1.5 wt%.


image file: c9qi01002e-f1.tif
Fig. 1 Rietveld refinement fit on synchrotron X-ray diffraction data of Ca1.624Sr0.376Si5O3N6:Eu2+ using a monoclinic structure in the Cm space group along with an impurity phase (Ca2Si5N8) in the Cc space group for (a) a single-cell model and (b) a double-cell model. In the figure, the black dots, red line, and blue line represent the observed, calculated, and difference profiles, respectively. The inset depicts the zoom portion of the Rietveld fit in the various 2θ range. The vertical tick marks above the difference profile in the first and second lines from the top denote the positions of the Bragg reflections for the Ca1.624Sr0.376Si5O3N6:Eu2+ and Ca2Si5N8 phases, respectively.

The Ca ions in the S-phase partially share positions with the Sr ions at Wyckoff site 2a in the crystal structure, and it is important to note that this is in contrast to a single-cell model where Ca ions lie solely at Wyckoff site 4b and are surrounded by 6 oxygen ions that form a trigonal prism that does not share positions with the Ba ions. The values of the structural parameters such as the atomic position, the thermal displacement parameter, and the site occupancy factor obtained after the Rietveld refinement are presented in the ESI (Table S1). A schematic of the crystal structure of the S-phase viewed along the c- and b-axes is shown in Fig. 2. Most of the Ca ions constitute a layer between Si tetrahedron networks, and each of the Ca ions in the layer is surrounded by six O ions to form a distorted trigonal prism (as shown in Fig. 2). The Ca–O bond distance ranges from 2.021 to 2.889 Å with an average bond distance of 2.405 Å. The hole created by the vertex sharing of the SiX4 (X = N, O) tetrahedra is mainly occupied by Sr ions with partial sharing by Ca ions. The Sr/Ca–N bond distance ranges from 2.30 to 3.646 with an average bond distance of 3.1965 Å in the S-phase structure.


image file: c9qi01002e-f2.tif
Fig. 2 Schematic of the crystal structure Ca1.624Sr0.376Si5O3N6:Eu2+ viewed along (a) the c-axis and (b) the b-axis.

DFT calculations for the Ca1.624Sr0.376Si5O3N6:Eu2+ phosphor

The first issue for the DFT calculations of the S-phase is to determine whether the structure resembles a B-phase or an E-phase in terms of the c axis periodicity. Rietveld refinement already confirmed that the structure of the S-phase is refined as a double-cell model, which is similar to the E-phase. We calculated the formation energy, band gap, density of states, and band structure based on both single- and double-cell models using GGA and HSE06 exchange correlation functionals. Although the DFT-calculated formation energy at zero Kelvin was not generally known to be predictive of relative phase stability, it could be so when the same elemental substances are dealt with just like the present case wherein the single and double cell models for the S-phase were compared. A list of reference materials (elemental substances) used for the formation energy calculation is presented along with their structural information and the reference energy in the ESI (Table S2). Every elemental substance has several different reference energies for their various structures appearing in the ICSD, and we adopted the lowest. The S-phase in the double-cell model can be regarded as more stable than that in the single-cell, because it has a more negative formation enthalpy relative to the elemental substances, as shown in Table 1. This result capably verifies the Rietveld refinement results wherein the double-cell model was more suitable for the S phase.
Table 1 DFT-calculated formation energy and band gap for the single- and double-cell models of Ca1.624Sr0.376Si5O3N6
    Single Double
Formation energy (eV per atom) GGA −1.77573 −1.80301
HSE06 −1.99209 −2.02175
Band gap (eV) GGA 3.285 3.852
HSE06 4.749 5.208


The DFT-calculated band gap was compared with the experimentally measured optical band gap energy to validate the calculation. The diffuse reflectance is a general method which can be used to measure an experimental optical band gap. To avoid luminescence interference from reflected light, we synthesized a Eu2+-free S-phase sample and used it for the diffuse reflectance measurement leading to the measurement of the exact optical band gap (5.24 eV). Fig. 3 shows the diffuse reflectance spectra and the measurement details for the band gap of the S phase. The optical band gap should be obtained from the undoped S phase since the Eu2+ emission interfered with the optical band gap measurement as evidenced by the red line representing the 2 mol% Eu2+-doped S-phase. The GGA-based band gap was underestimated far below the experimental band gap for both the single- and double-cell models. On the other hand, the hybrid exchange correlation functional (HSE06) gave a significantly enhanced band gap energy comparable to the experimental value, as shown in Table 1. Fig. 4 shows the density of states and the band structure (electronic dispersion) calculated for both the single- and double-cell models based on the GGA and HSE06 exchange correlation functional schemes. The HSE06 calculated band gap energy for the double-cell model (5.208 eV) is greater than that for the single-cell model (4.749 eV) and closer to the experimental value. This verifies that the structure of the S-phase is that of a double-cell.


image file: c9qi01002e-f3.tif
Fig. 3 Diffuse reflectance spectra for both the Eu2+-activated and undoped Ca1.624Sr0.376Si5O3N6 samples and a graphical description for the optical band gap evaluation from the {F(R)}2 vs. energy () plot; the straight line in the plot intersects the energy axis at 5.24 eV.

image file: c9qi01002e-f4.tif
Fig. 4 GGA (top) and HSE06 (bottom) band structures along with DOS for both the single- and double-cell models of Ca1.624Sr0.376Si5O3N6:Eu2+.

Luminescence and decay behaviors of the Ca1.624Sr0.376Si5O3N6:Eu2+ phosphor

Fig. 5 shows the photoluminescence (PL) and photoluminescence excitation (PLE) spectra for the S-phase. The PL spectrum clearly shows multi-color emissions, one for blue color light emission (referred to as P1) at around 462 nm and the other for yellowish color light emission (referred to as P2) at around 584 nm. As the Eu2+ concentration increased, the P1 emission dramatically decreased, and the P2 emission grew with an expanse of P1 since the ET from the P1 emission to the P2 excitation took place briskly. The PLE spectra were monitored at both the P1 and P2 emission peak positions. The main absorption (excitation) peak for P1 at around 322 appeared as a conspicuous shoulder on the P2 excitation. The P1 excitation peak at 322 nm also decreased as the Eu2+ concentration increased, which coincided with the degradation of the P1 emission peak. Most of the energy absorbed by the Eu2+ ions located at the Sr site, did not lead to light emission but was transferred in a non-radiative manner to the Eu2+ ions located at the Ca site when the Eu2+ concentration exceeded a certain threshold. The PL and PLE spectra show that the P1 emission peak at 462 nm and the P1 excitation peak at 322 nm never shifted with the Eu2+ concentration, but the P2 emission peak at 584 nm was slightly red shifted as the Eu2+ concentration increased. This sort of redshift occurs with almost all Eu2+-doped phosphors, the reason for which is well described elsewhere.49–52
image file: c9qi01002e-f5.tif
Fig. 5 (a) Photoluminescence (PL) at 355 and 460 nm excitations and (b) photoluminescence excitation (PLE) spectra detected at 462 nm (P1 emission) and 584 nm (P2 emission) for Ca1.624Sr0.376Si5O3N6:Eu2+.

To examine the ET between Eu2+ ions located at different crystallographic sites, the time resolved photoluminescence (TRPL) spectra were recorded and the decay curve at both P1 and P2 emission peak locations was extracted from the TRPL, as shown in Fig. 6. The TRPL and decay results exhibit a typical finding that the P1 emission (donor emission) is much faster than the P2 emission (acceptor emission) and both emissions also become faster as the activator concentration increases. The evaluation of the exact Eu2+ content should be a prerequisite for quantitative ET analysis. The XPS measurement revealed the exact Eu2+ and Eu3+ contents at both the Ca and Sr sites for three different Eu concentrations as shown in Fig. 7. The Eu3+ content was higher than had been expected and the relative portions of Eu2+ occupation in the Sr and Ca sites were similar. The Eu3+ content could be disadvantageous in that it reduces the Eu2+ activators and causes a non-radiative ET from Eu2+ to Eu3+. No typical red emissions induced by the intra-4f transition for Eu3+ were detected. Although no inversion symmetry was found for either the Sr or Ca sites, the relaxation of the forbidden intra-4f transition was negated. Nonetheless, the ET should be active from the Eu2+ excited state to the Eu3+ ground state, resulting in an Eu3+ excited state, which could be normally thought of as an Eu3+–O charge transfer state that is always located in a lower energy side than the 4f–5d state in contrast to other lanthanides.53 The Eu3+–O charge transfer state energy range has a considerable overlap with the 5d → 4f emission energy for Eu2+. The energy transferred to the Eu3+–O charge transfer state should be eventually quenched in a non-radiative manner via both the phonon and the nearby killer sites such as defects and defect impurities that have not yet been identified. To more precisely examine the Eu2+ → Eu3+ ET, we derived appropriate rate equations.

image file: c9qi01002e-t1.tif

image file: c9qi01002e-t2.tif

image file: c9qi01002e-t3.tif

image file: c9qi01002e-t4.tif

image file: c9qi01002e-t5.tif

image file: c9qi01002e-t6.tif

image file: c9qi01002e-t7.tif
(X stands for the Eu population at each state.)


image file: c9qi01002e-f6.tif
Fig. 6 (a)–(c) TRPL spectra and (d)–(e) decay curves at 462 and 584 nm for Ca1.624Sr0.376Si5O3N6:Eu2+ as a function of the Eu concentration. The concentration presented in the figure stands for the starting Eu concentration rather than the true Eu2+ concentration.

image file: c9qi01002e-f7.tif
Fig. 7 The Eu 3d core-level X-ray photoelectron spectrum together with fitted lines for Ca1.624Sr0.376Si5O3N6:xEu2+ with x = 2, 5, and 10 mol%.

The four different Eu sites are Eu2+ in the Sr site, Eu2+ in the Ca site, Eu3+ in the Sr site, and Eu3+ in the Ca site, and there are five energy-transfer constants (kxy) that designate the interactions between these sites. Ksc refers to the interaction between Eu2+ in the Sr site and Eu2+ in the Ca site; Ksc refers to the interaction between Eu2+ in the Sr site and Eu3+ in the Ca site; kss refers to the interaction between Eu2+ in the Sr site and Eu3+ in the Sr site; kcs refers to the interaction between Eu2+ in the Ca site and Eu3+ in the Sr site; and kcc refers to the interaction between Eu2+ in the Ca site and Eu3+ in the Ca site. The backward Eu3+ → Eu2+ ET was not included in the rate equations since it would not take place. Kn designates the total quenching rate for the Eu3+–O charge transfer state. Kr1 and kr2 represent the radiative rates for P1 (Eu2+ in the Sr site) and P2 emissions (Eu2+ in the Ca site), respectively. We used a simple linear regression model to obtain both radiative rates by using samples with nominal activator concentrations.54 Detailed procedures for the radiative rate evaluation are well described in the ESI. Because we used normalized donor and acceptor decay curves, it was reasonable to exclude the excitation rate constants (G and Q) from the interaction rate constant evaluation process based on the least-squares-fitting. In fact, excluding G and Q revealed that the excitation rate constant never affected the decay profile but the overall population level in our previous reports.55,56 In other words, the excitation coefficients G and Q were fixed at a predetermined value since it was clearly proven that the excitation coefficient never affects the regression fitting procedures for normalized decay curves. Consequently, we had six unknown rate constants to be evaluated from the rate equation computation, five ET rate constants (kxys), and a quenching rate constant (kn).

The initial condition was clearly obtained from the XPS results, as shown in the ESI (Table S3). A typical Runge–Kutta method57 was used to numerically treat the rate equations, and a particle swarm optimization (PSO) method58 was successively employed to perform the least squares regression and eventually to achieve the best-fit constants. Fig. 8 exhibits the calculated results, wherein a good match between the calculated and experimental decay curves for the acceptor emission (P2) can be seen, and although the match for the donor emission (P1) was slightly biased for the highest Eu concentration (10 mol%), the overall feature should not be seen as a mismatch. The PSO is a well-established metaheuristic algorithm which has been successfully used for various optimization problems. It enabled us to evaluate six interaction rate constants in a very efficient manner. The typical gradient method would never have achieved such a brilliant result. There are two notable points in the present decay analysis; the first is the use of PSO, and the second is the simultaneous use of both the donor and acceptor decay data for the least squares fitting process, which was possible thanks to the adoption of the PSO algorithm. The detailed computation process is well described in our previous reports.55,56 Table 2 shows the best-fit kxy values from the PSO process. All the kxy values are in a plausible range. As the activator concentration increased, all the kxy values also increased, which implies that the shortened inter-ionic distance caused by the increased Eu concentration more briskly activated the ET in every possible route.


image file: c9qi01002e-f8.tif
Fig. 8 Experimental decay data for an acceptor detected at 584 nm and for a donor detected at 462 nm, and the calculated rate-equation model (green and purple solid lines indicate the acceptor and the donor, respectively).
Table 2 Best-fitted kxy values from the PSO process and relative GF values for Ca1.624Sr0.376Si5O3N6:xEu2+ with x = 2, 5, and 10 mol%. The relative SO values deduced from both the kxy and relative GF values are also given
2 mol%
  Kxy Relative G. F. Relative S. O.
ksc 2.91 × 10−16 cm3 s−1 0.00673 1
ksc 1.70 × 10−16 cm3 s−1   0.585526
kss 2.35 × 10−17 cm3 s−1 0.00512 0.106151
kcs 1.83 × 10−17 cm3 s−1 0.00112 0.378523
kcc 9.27 × 10−17 cm3 s−1 1 0.002147
Kn 3.09 × 106 1 s−1 N/A

5 mol%
  Kxy Relative G. F. Relative S. O.
ksc 7.27 × 10−15 cm3 s−1 0.00673 1
ksc 2.35 × 10−15 cm3 s−1   0.323684
kss 3.04 × 10−17 cm3 s−1 0.00512 0.0055
kcs 1.68 × 10−17 cm3 s−1 0.00112 0.013852
kcc 3.38 × 10−15 cm3 s−1 1 0.003135
Kn 5.13 × 106 1 s−1 N/A

10 mol%
  Kxy Relative G. F. Relative S. O.
ksc 7.41 × 10−15 cm3 s−1 0.00673 1
ksc 3.89 × 10−15 cm3 s−1   0.525161
kss 3.66 × 10−17 cm3 s−1 0.00512 0.006496
kcs 2.86 × 10−17 cm3 s−1 0.00112 0.023183
kcc 3.29 × 10−15 cm3 s−1 1 0.002987
Kn 7.31 × 106 1 s−1 N/A


According to the ET theory presented in the ESI, the rate constant kxy connotes several important factors such as spectral overlap (SO) between donor emission and acceptor absorption, the geometry factor (GF), oscillator strength for acceptor absorption, and the radiative rate for donors.59 Among these factors involved in kxys, the SO and GF should be the most influential to constitute kxy. According to reasonable speculations on every emission wavelength for Eu2+ in Sr and Ca sites and its Stokes shift, the spectral overlap for ksc should be greater than for any other since ksc stands for the interaction between divalent Eu ions while all the others represent the interaction between divalent and trivalent Eu ions. The SO for ksc ranks next to ksc, and the SO for kss and kcc should be greater than that for kcc. The geometry factor stands for the probability of potential ET routes in the crystal structure of the S-phase, which is very important since the ET never occurs in a homogeneous media but, rather, in a certain crystal structure that provides only deterministic site-to-site paths. We computed the GF by referring to the principle reported in our previous approaches.4,60 It is our opinion that the GF plays a significant role in the ET taking place in a crystalline host. Although most theoretical models for ET have been based on the averaged ET route in homogeneous media, in the real world the ET takes place in a crystalline host for every phosphor. A strict crystal structure would never allow such a smeared-out-medium-approximation but would instead create discrete (deterministic) ET routes. The GF refers to a rough measure of such inhomogeneous ET routes. Assuming that the interaction scheme is wholly multipolar (dipole–dipole), we considered every possible ET route (Ris) and summed up the 1/Ri6 values within 15 Å around an Sr (or Ca) site of concern. The relative SO should be deduced from the evaluated kxy values by dividing them by the calculated GF value. These deduced SO values were reasonably well-matched with the pre-speculation described above, as shown in Table 2. The details on the GF computation protocol are described in the ESI.

Conclusions

A novel broadband phosphor, Ca1.624Sr0.376Si5O3N6:Eu2+, with the Cm symmetry of a double-cell structure was identified with the lattice parameters a = 7.06115(2) Å, b = 23.78568(5) Å, and c = 9.63830(2) Å. DFT calculations further verified the discovered double-cell structure by comparing the formation energy and band gap between single- and double-cell structural models. The valence state of Eu ions and the occupation at both Sr and Ca sites were precisely evaluated via XPS, and a comprehensive rate equation model was set up which involved the interaction terms between all possible Eu ion states (i.e., Eu2+ and Eu3+ in Sr and Ca sites). The ET rate constant for the non-linear interaction terms in the rate equation model was estimated to be within a reasonable range and the spectral overlap was deduced from the evaluated rate constant and the geometry factor.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This research was supported by the Creative Materials Discovery Program through the National Research Foundation of Korea (NRF) funded by the the Ministry of Science, ICT, and Future Planning (2015M3D1A1069705) and partly by a NRF grant (2018R1C1B6006943).

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Footnotes

Electronic supplementary information (ESI) available: Rietveld refinement results, formation energy calculation details, initial condition for the Runge–Kutta method, radiative rate evaluation procedures, energy transfer theory and geometry factor computation details. See DOI: 10.1039/c9qi01002e
These authors contributed equally.

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