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Single phase tunable warm white-light-emitting Sr3La(PO4)3:Eu2+, Sm3+ phosphor for white LEDs

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Abstract

A novel single phased white emitting phosphor Sr3La(PO4)3:Eu2+, Sm3+ was synthesized by the solid-state method, and the crystal structure and luminescence properties of phosphors were investigated in detail by the X-ray diffraction, photoluminescence spectra and decay curves. Through the energy transfer, the warm white light can be realized with superior chromaticity coordinates of (0.3824, 0.3381), low correlated color temperature (CCT = 3598 K), and Ra of 78.4 by varying the molar ratios of Eu2+ and Sm3+ in the Sr3La(PO4)3:Eu2+, Sm3+ phosphor. The results indicate that the developed phosphor can be used as a potential warm white emitting phosphor for white light emitting diodes.

© 2015 Optical Society of America

1. Introduction

Nowadays, as one of the hot spots in solid-state lighting areas, warm white-light-emitting diodes (WLEDs), with low correlated color temperature (CCT) of ≤5000 K or even≤3500 K, have attracted considerable attention for its great significance in the indoor illumination applications because it is more favorable for human sight [1–3]. Currently, the most common commercial WLED fabricated with a blue chip and a yellow phosphor YAG:Ce3+ is un-optimized for indoor use for having emission spectrum deficient in the red region with a result of high correlated color temperature (CCT~7750 K) and low color-rendering index (CRI~70-80) [4–6]. Therefore, as an alternative, a combination of near-ultraviolet (n-UV) LED or UV LED with red, green and blue emitting phosphors has been developed to improve the CRI, color stability and to tune CCT value [7, 8]. However, the luminescence efficiency is low in this system because the blue emission is reabsorbed by green and red phosphors [9]. Therefore, single-phased white emitting phosphors for UV or n-UV excitations have drawn much attention for solid state lighting due to the merits of excellent Ra and color stability. Up to now, several single-component warm white emitting phosphors have been reported, such as NaGd(WO4)2:Tm3+, Dy3+, Eu3+ [10], Sr10[(PO4)5.5(BO4)0.5](BO2):Eu2+, Mn2+, Tb3+ [11], Sr9Gd(PO4)3:Eu2+, Mn2+ [12], Ca9La(PO4)7:Eu2+, Mn2+ [13], Na0.34Ca0.66Al1.66Si2.34O8:Eu2+, Mn2+ [14], Sr3Lu(PO4)3:Eu2+, Mn2+ [15], and Sr2Ca0.995MoO6:Sm3+ [16]. Although some warm white light achieved by tuning the ratios of activators with the above hosts, yet overall there are few satisfactory warm white light suitable for indoor application. Hence, it is still to be considered as the state of urgency for the development of single-phase warm white emitting phosphor that has lower CCT and excellent luminescence.

Recently, eulytite-type orthophosphates with the general formula M3IMII(PO4)3 (MI = Ca, Sr, Ba and Pb; MII = La, Y, Sc, Bi, Tb and In) have attracted extensive attention as host materials for lanthanide activators because of their excellent thermal stability and optical property [15,17–19]. Hence, there is an increasing interest in the synthesis of novel efficient single-component warm white phosphors having structures derived from the Eulytite-type orthophosphate family. As a eulytine-type phosphate, Sr3La(PO4)3 has been studied as a host for photo-luminescent material due to its good chemical/physical stability, such as Sr3La(PO4)3:Ce3+, Tb3+ [20], Sr3La(PO4)3:Eu3+, Sm3+ [21], Sr3La(PO4)3:Ce3+, Mn2+ [22] and Sr3La(PO4)3:Eu2+, Mn2+ [23]. However, to the best of our knowledge, there are no warm white emissions producing by double doping in these reports. Our previous work has described that Eu2+ ions occupy two different Sr2+ sites in Sr3La(PO4)3 lattice and form two kinds of luminescent centers. Nine-coordinated Eu2+ luminescent centers (Eu2+ (I)) emit blue light and six-coordinated Eu2+ luminescent centers (Eu2+ (II)) emit cyan light [24]. Thus it can be easily inferred that a white light emission might be obtained by doping a red emission activator into the Sr3La(PO4)3:Eu2+. It is well known that Mn2+, Eu3+ and Sm3+ are generally used as red emission activators. However, the reported Sr3La(PO4)3:Eu2+, Mn2+ emits red light. In addition, the ratio of Eu3+ and Eu2+ in a single-phased phosphor is very difficult to control. As reported in the Ca9Y(PO4)7:Eu2+, Sm3+ [25], the coexistence of Eu2+ and Sm3+ was achieved by annealing the phosphor in a reducing atmosphere, and Sm3+ emits red light in both the H2/N2 and the air atmosphere. Therefore, it is reasonable to expect that a warm white light emission might be achieved by doping Sm3+ into Sr3La(PO4)3:Eu2+.

In this paper, we have firstly demonstrated a series of single-component color-tunable white-light-emitting Sr3La(PO4)3:Eu2+, Sm3+ phosphors by energy transfer Eu2+-Sm3+, the color can be tuned from deep blue to pale-blue, warm-white, and eventually to orange-red. A warm white emitting phosphor, with low CCT, is achieved by varying the relative doping content of Eu2+ and Sm3+. The fluorescence decay curves have been fitted and analyzed. The corresponding photoluminescence and energy transfer of Eu2+ and Sm3+ codoped Sr3La(PO4)3 phosphors are investigated in detail.

2. Sample synthesis and measurement

According to our previous work [24], Sr3La(PO4)3 (SLP):Eu2+ shows the strongest blue emission when Eu2+ content is 3% . Therefore, the optimal Eu2+ content for SLP:Eu2+, Sm3+ is set to be 3%. The powder samples with the general formula Sr2.97La1-y(PO4)3:3%Eu2+, ySm3+ (y = 0-15%) were prepared by the conventional solid-state reaction. Initial materials, SrCO3, La2O3, NH4H2PO4 (all materials are of analytical grade), Sm2O3 (99.99%) and Eu2O3 (99.99%) are weighed in stoichiometric proportion, thoroughly mixed and ground by an agate mortar and pestle for more than 30 min till they are uniformly distributed. The mixed powders are calcined in corundum crucibles at 1300 °C for 3 h in a reducing atmosphere (5%H2/95%N2). Finally they are cooled down to room temperature and ground thoroughly again into powders.

Phase formation of sample is carefully checked by powder X-ray diffraction (XRD) analysis (Bruker AXS D8 advanced automatic diffractometer (Bruker Co., German)), with Ni-filtered Cu Kα1 radiation (λ = 0.15406 nm) operating at 40 kV and 40 mA, and a scan rate of 0.02°/s is applied to record the patterns in the 2θ range from 20° to 70°. Luminescence spectra, quantum efficiency and luminescence decay curves are detected by a FLS920 fluorescence spectrometer, and exciting source is a 450 W Xe lamp. The Commission International de I’Eclairage (CIE) chromaticity coordinates of the samples are measured by using a PMS-80 spectra analysis system. All measurements are carried out at room temperature.

3. Results and discussion

3.1 Phase formation

Figure 1 shows the x-ray powder diffraction (XRD) patterns of the series of samples of SLP:3%Eu2+, ySm3+ (y = 0, 1%, 3%, 5%, 7%, 10%, and 15%). It is can be seen from Fig. 1 that the diffraction peaks of low Sm3+ concentration samples (y<10%) can be exactly assigned to pure eulytite-type structure of Sr3La(PO4)3 with JCPDS card No. 29-1306, meanwhile with the larger quantity of impurity, such as 3%Eu2+/10%Sm3+ and 3%Eu2+/15%Sm3+, the XRD pattern peaks slightly shift to the bigger angles. This attributes to the larger La3+ (r = 103 pm) is substituted by smaller Sm3+ (r = 96 pm) in terms of Vegard’s law. The result indicates that the Eu2+ and Sm3+ ions have been doped into the lattice and thus solid solutions were formed in SLP:3%Eu2+, ySm3+ samples. However, as an exceptional case, one additional weak diffraction peak near 26° for the samples should be ascribed to a small amount of La2O3 remained. The crystal structures of the eulytite-type materials M3IMII(PO4)3 are well known to be cubic (space group number 220) and isomorphous with eulytine mineral (Bi4Si3O12). The Sr3La(PO4)3 structure is described in the cubic I43d space group with lattice constant a = 10.192Å, and Sr and La occupy the same site with a C3 symmetry. The general feature of Sr3La(PO4)3 structure could be regarded as a three-dimensional packing of [PO4]3- anionic tetrahedra and La/Sr octahedra, arranged in a manner to share common apices. It is interesting that all the [PO4]3- tetrahedral are totally independent while the La/Sr octahedra share edges with each other and form a three-dimensional network, as shown in Fig. 2.

 figure: Fig. 1

Fig. 1 XRD patterns of SLP:3%Eu2+, ySm3+ (y = 0-15%) with the standard data of Sr3La(PO4)3 (JCPDS) No. 29-1306.

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 figure: Fig. 2

Fig. 2 Three-dimensional view of Sr3La(PO4)3 eulytite structure.

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3.2 Luminescence properties of SLP:Eu2+ and SLP:Sm3+

The excitation and emission spectrum of SLP:3%Eu2+ is shown in Fig. 3(a). When excited at 325 nm, the phosphor shows a strong blue emission around 421 nm and a shoulder centered at 500 nm. Both the intense broad peaks are originated from 5d→4f transition of Eu2+ ions. Monitored at 421 nm, the excitation spectrum exhibits a broad band in the range from 220 to 400 nm, which is assigned to the transitions between the ground state 4f7 and the crystal-field split 4f 65d configuration. SLP: Eu2+ can be excited by ultraviolet light ranged from 300 nm to 380 nm effectively.

 figure: Fig. 3

Fig. 3 Emission and excitation spectra of SLP:3%Eu2+ (a), and SLP:3%Sm3+ (b), and SLP:3%Eu2+, 3%Sm3+ (c).

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As shown in Fig. 3(b), the emission spectrum of SLP:Sm3+ has three orange-red emission bands, which are attributed to the transitions from 4G5/2 excited state to 6HJ (J = 5/2, 7/2, 9/2) ground states. The strongest emission locates at 600 nm which is originated from the 4G5/26H7/2 typical transition of Sm3+. The other emission peaks locate at 560 nm and 645 nm which are attributed to the 4G5/26H5/2 and 4G5/26H9/2 transitions of Sm3+, respectively. Moreover, all emission peaks which correspond to the transitions of 4G5/26H7/2, 4G5/26H5/2 and 4G5/26H9/2 of Sm3+ are split, and the energy level splitting which is induced by the crystal interaction [26]. The excitation spectrum of SLP: 3%Sm3+ consists of several narrow excitation lines due to the f-f characteristics transition of Sm3+. The strongest excitation line (401 nm) corresponds to the 6H5/24K11/2 transition of Sm3+, which matches well with the emission wavelength of n-UV LED chip.

Comparing the excitation spectrum of the Sm3+ ions with emission spectrum of the Eu2+ ions, a significant spectral overlap was observed. Thus, a resonance-type energy transfer from the Eu2+ to Sm3+ ions in a co-doped sample should be possible.

3.3 Luminescent properties and energy transfer of SLP:Eu2+, Sm3+

As expected in Fig. 3(c), it is clearly seen that the excitation spectrum (λem = 600nm) of SLP: 3%Eu2+, 3%Sm3+ is much stronger than that of SLP:3%Sm3+, especially in the range of 220~400nm wave band, which is consistent with the excitation spectrum of Eu2+. This can be explained that the excitation spectrum of the doubly doped sample consists of both excitation spectra of Eu2+ and Sm3+, which indicates the energy transfer from Eu2+ to Sm3+. As a result, the excitation into the excitation band of the Eu2+ ions (for example 375nm) yields emission from both Eu2+ and Sm3+ ions, and consists of a blue band which corresponds to the f–d transitions of the Eu2+ ions and a reddish-orange band attributed to the 4G5/26H7/2 transitions of the Sm3+ ions. Thus, white-light might be achieved by combining the blue emission of the Eu2+ ions and reddish-orange emission of the Sm3+ ions in a single host lattice by varying the ratio of Eu2+ to Sm3+ ions through the principle of energy transfer.

In order to investigate the energy transfer from Eu2+ to Sm3+, the emission spectra of SLP: 3%Eu2+, ySm3+ (y = 0-15%) under the 375nm radiation excitation are measured, and displayed in Fig. 4. One can see that the emission intensity for Eu2+ decreases monotonically with an increase in Sm3+ content. Meanwhile, the emission of Sm3+ increases gradually until the Sm3+ concentration is above 5% and concentration quenching occurs. This indicates that the energy transfer from Eu2+ to Sm3+ clearly exists, unfortunately quenching concentration of Sm3+ is relatively low. In addition, it can be noticed that there are clear absorption dips on Eu2+ emission, which indicates the radiation reabsorption of Eu2+ also plays a role. And moreover there is an overlap (20 nm) between the excitation and emission spectrum of SLP:3%Eu2+ in Fig. 3(a), which will result in the radiation reabsorption of Eu2+.

 figure: Fig. 4

Fig. 4 Emission spectra of SLP:3%Eu2+, ySm3+ (y = 0-15%) (λex = 375 nm).

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In order to further prove the energy transfer from Eu2+ to Sm3+, the decay curves of the Eu2+ ions in the SLP:3%Eu2+, ySm3+ (y = 0-15%) samples were measured by monitoring the emission of the Eu2+ ions at 421 nm and are presented in Fig. 5. The decay curves are well fitted with a second-order exponential decay mode by the following equation [27]

I=I0+A1exp(t/τ1)+A2exp(t/τ2)
where I is the luminescent intensity; A1 and A2 are constants; t is the time; τ1 and τ2 are rapid and slow lifetimes for exponential components, respectively. Using the above parameters, the average decay times (τ*) can be determined by the formula as follows [27]

 figure: Fig. 5

Fig. 5 Decay curves of Eu2+ emission monitored at 421 nm for SLP:3%Eu2+, ySm3+ (y = 0-15%) phosphors under excitation at 320 nm.

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τ*=(A1τ12+A2τ22)/(A1τ1+A2τ2)

For SLP:3%Eu2+, ySm3+ samples, as the value of y increases from 0 to 15%, the lifetime for Eu2+ is found to decline monotonically from 285.5 ns to 10.2 ns, implying the introduction of the extra decay pathway, i.e. energy transfer from Eu2+ to Sm3+. The total decay rate (Ktot) of Eu2+ 5d levels in single doped SLP:3%Eu2+ is given by

Ktot=KF+Kq=τEu1
where KF is the radiative rate, Kq is the quenching rate, and τEu is the lifetime of the 5d level in SLP:3%Eu2+. In double doped SLP:3%Eu2+, ySm3+ (y = 1%-15%), the extra decay pathway from the 5d level of Eu2+ to the 4G5/2 level of Sm3+ was generated. Therefore, the total decay rate is given by
Ktot=KF+Kq+KET=τEu,Sm1
Where KET is the non-radiative energy transfer rate (Eu2+→Sm3+) and τEu,Sm is the lifetime of the 5d level in SLP:3%Eu2+, ySm3+ (y = 1%-15%).

In order to further clarify the energy transfer process from Eu2+ to Sm3+, the decay curves of the Sm3+ ions in the SLP:3%Eu2+, ySm3+ (y = 1%-15%) samples were also measured under 401nm excitation, as shown in Fig. 6. The decay curves could be well fitted using a single exponential equation [28]

I=I0exp(t/τ)
where I and I0 are the luminescence intensities at time t and 0, respectively, and τ is the decay lifetime. The lifetime τ values for SLP:3%Eu2+, ySm3+ (y = 1%-15%) are determined to 7.14 ms, 6.91ms, 6.40ms, 5.58ms, 4.28ms, and 2.83ms, respectively. Obviously, with the value of y increasing from 1% to 15%, the lifetime for Sm3+ is found to decrease monotonically. Since the decay rate of the Eu2+ emission is much faster compared to the Sm3+ emission, so here the energy transfer rate KET should be neglected, and the total decay rate of Sm3+ 4G5/2 level in SLP:3%Eu2+, ySm3+ should be given by
Ktot=KF+Kq=τSm1
where τSm is the lifetime of the 4G5/2 level in SLP:3%Eu2+, ySm3+. With the increase of Sm3+ concentration, the quenching rate Kq increases, consequently the lifetime τSm decreases.

 figure: Fig. 6

Fig. 6 Decay curves of Sm3+ emission monitored at 600 nm for SLP:3%Eu2+, ySm3+ (y = 1-15%) phosphors under excitation at 401 nm.

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In order to clarify the energy transfer efficiency from Eu2+ to Sm3+, the external quantum efficiency (EQE) for SLP:3%Eu2+, ySm3+ (y = 0-15%) samples are measured and shown in Table 1. As seen from the Table 1, EQE is relatively low, which indicates that there is a low energy transfer efficiency from Eu2+ to Sm3+ due to the energy transferring to quenching centers. Therefore, the EQE should be improved through further optimization of the processing conditions and composition.

Tables Icon

Table 1. A comparison of CIE Chromaticity Coordinates (x, y), CCT (k) and EQE for SLP:3%Eu2+, ySm3+ phosphors (λex = 365 nm).

It is also known that the dominant interaction is strongly dependent on the separation between the donors and the acceptors. As we know that the non-radiative energy transfer often occurs as a result of an exchange interaction or a multipole–multipole interaction [29]. The exchange interaction is generally responsible for the energy transfer of forbidden transition and the typical critical distance is about 5 Å. In order to confirm which interaction dominates the energy transfer between the Eu2+ and Sm3+ ions, we estimated the average separation REu-Sm between Eu2+ and Sm3+ ions using the formula suggested by Blasse [30]

REuSm=2[3V/(4πCN)]13
where N is number of Z ions in the unit cell, and V is volume of unit cell (for Sr3La(PO4)3, N = 4, V = 1.0587 nm3). C is the sum concentration of Eu2+ and Sm3+. we calculate REu-Sm at different Sm3+ concentrations, and they are 23.3 Å, 20.3 Å, 18.5 Å, 17.2 Å, 15.7Å and 14.1Å at y = 1%, 3%, 5%, 7%, 10% and 15%, respectively. Since the above calculated REu-Sm values at different Sm3+ concentrations are much larger than the typical critical distance for exchange interaction (5Å) [31], thus the exchange interaction plays no role in the energy transfer process. Therefore, the energy transfer process should be controlled by the electric multipole–multipole interaction [32].

In order to further confirm the energy transfer from Eu2+ and Sm3+, the comparison of emission intensity (λem = 600nm) for samples of SLP:ySm3+ and SLP:3%Eu2+, ySm3+ (y = 0-15%) under the 375nm excitation are shown in Fig. 7. It is obvious to see that the emission intensity of SLP:3%Eu2+, ySm3+ are stronger than that of SLP:ySm3+ at different Sm3+ concentration, which strongly confirms the continuous energy transfer from Eu2+ to Sm3+.

 figure: Fig. 7

Fig. 7 Dependence of emission intensity of SLP:ySm3+ and SLP:3%Eu2+, ySm3+ (y = 0-15%) under the 375nm excitation on Sm3+ concentration (y).

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Figure 8 shows the energy level diagrams of Eu2+ and Sm3+, the energy gap between the 5d level of Eu2+ and the 4G5/2 level of Sm3+ is small, therefore, the emission spectra of Sr3La(PO4)3: Eu2+, Sm3+ indicates the energy transfer from Eu2+ to Sm3+.

 figure: Fig. 8

Fig. 8 Energy level diagrams of Eu2+ and Sm3+ showing the energy transfer process from Eu2+ to Sm3+.

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3.4 CIE of SLP:Eu2+, SLP:Sm3+ and SLP:Eu2+, Sm3+

The Commission Internationale de L’Eclairage (CIE) chromaticity coordinates and CCT of SLP:3%Eu2+, ySm3+ samples were measured, and the results are shown in Fig. 9 and Table 1. For SLP:3%Eu2+, ySm3+, as the value of y increases from 0 to 15%, the luminescence color can be easily modulated from deep blue to pale-blue, warm-white, and eventually to orange-red, which is due to the different emission composition of the Eu2+ and Sm3+ ions. Moreover, the CCT of warm white light can be tuned from 4240K (the fifth Sample) to 3598 K (the seventh Sample), which indicates that a warm-white-light with a different CCT can be produced by varying the Sm3+ content in the SLP:Eu2+, Sm3+ phosphor system. For example, when the value of y is 10%, a warm-white-light emission with CIE chromaticity coordinates of (0.3824, 0.3381), CCT of 3598 K, Ra of 78.4, and EQE of 18.0% can be realized in a single host, as shown in Fig. 9 for the seventh sample.

 figure: Fig. 9

Fig. 9 The CIE chromaticity coordinates of SLP:3%Eu2+,ySm3+ (y = 0-15%) (λex = 365 nm). The corresponding luminescence photos of SLP:3%Eu2+, ySm3+ (y = 0-15%) and SLP: 5%Sm3+ex = 365 nm).

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The insets in Fig. 9 also demonstrate the phosphor color images of different Eu2+/Sm3+ molar ratios excited at 365 nm in an UV box, the results also indicate that there are the energy transfer of Eu2+-Sm3+ in SLP:Eu2+, Sm3+. From the above results, we can clearly see that the emission of Eu2+ and Sm3+ ions occurs simultaneously and yields a warm white light emission as a result of energy transfer. Hence, this material would be potentially used as a warm white light emitting source to meet the needs of indoor illumination application.

4. Conclusion

A series of single-component warm-white-light SLP:Eu2+, Sm3+ phosphors for white LEDs have been synthesized. The investigation reveals that the absorption band of the obtained phosphors perfectly match the characteristic emission of n-UV-light emitting diode chip. The energy transfer from Eu2+ to Sm3+ in SLP:Eu2+, Sm3+ has been validated The luminescence color can be easily modulated from deep blue to pale-blue, warm-white, and eventually to orange-red by simply adjusting the content of Sm3+. More importantly, a warm-white-light emission from the SLP:3%Eu2+, 10%Sm3+ phosphor with a CCT of 3598 K, CIE coordinates of (0.3824, 0.3381), Ra of 78.4, and EQE of 18.0% was realized. There results indicate that the developed phosphor exhibits a potential to act as a single-component warm-white-emitting phosphor for white LEDs.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC) (No. 61475016), the Natural Science Foundation of Hebei Province, China (No.E2014201037) and the Education Office Research Foundation of Hebei Province, China (No.QN2014085).

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Figures (9)

Fig. 1
Fig. 1 XRD patterns of SLP:3%Eu2+, ySm3+ (y = 0-15%) with the standard data of Sr3La(PO4)3 (JCPDS) No. 29-1306.
Fig. 2
Fig. 2 Three-dimensional view of Sr3La(PO4)3 eulytite structure.
Fig. 3
Fig. 3 Emission and excitation spectra of SLP:3%Eu2+ (a), and SLP:3%Sm3+ (b), and SLP:3%Eu2+, 3%Sm3+ (c).
Fig. 4
Fig. 4 Emission spectra of SLP:3%Eu2+, ySm3+ (y = 0-15%) (λex = 375 nm).
Fig. 5
Fig. 5 Decay curves of Eu2+ emission monitored at 421 nm for SLP:3%Eu2+, ySm3+ (y = 0-15%) phosphors under excitation at 320 nm.
Fig. 6
Fig. 6 Decay curves of Sm3+ emission monitored at 600 nm for SLP:3%Eu2+, ySm3+ (y = 1-15%) phosphors under excitation at 401 nm.
Fig. 7
Fig. 7 Dependence of emission intensity of SLP:ySm3+ and SLP:3%Eu2+, ySm3+ (y = 0-15%) under the 375nm excitation on Sm3+ concentration (y).
Fig. 8
Fig. 8 Energy level diagrams of Eu2+ and Sm3+ showing the energy transfer process from Eu2+ to Sm3+.
Fig. 9
Fig. 9 The CIE chromaticity coordinates of SLP:3%Eu2+,ySm3+ (y = 0-15%) (λex = 365 nm). The corresponding luminescence photos of SLP:3%Eu2+, ySm3+ (y = 0-15%) and SLP: 5%Sm3+ex = 365 nm).

Tables (1)

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Table 1 A comparison of CIE Chromaticity Coordinates (x, y), CCT (k) and EQE for SLP:3%Eu2+, ySm3+ phosphors (λex = 365 nm).

Equations (7)

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I= I 0 + A 1 exp( t/ τ 1 )+ A 2 exp( t/ τ 2 )
τ * =( A 1 τ 1 2 + A 2 τ 2 2 )/( A 1 τ 1 + A 2 τ 2 )
K tot = K F + K q = τ Eu 1
K tot = K F + K q + K ET = τ Eu,Sm 1
I= I 0 exp( t/τ )
K tot = K F + K q = τ Sm 1
R EuSm =2 [ 3V/( 4πCN ) ] 1 3
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