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The new red-emitting phosphor of oxyfluoride Ca2RF4PO4:Eu3+ (R=Gd, Y) for solid state lighting applications

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Abstract

The novel red-emitting phosphors of Eu3+-activated Ca2 RF4PO4:Eu3+ (R=Gd, Y) prepared by a solid-state reaction have been evaluated as a candidate for white solid state lighting. The detailed luminescence properties, e.g., the excitation spectra, the luminescence spectra and quantum efficiency under the excitation of near-UV, and decay lifetimes were reported. The phosphors can be efficiently excited by near UV light and exhibit a dominant emission peaked at 611 nm (5D07F2) with CIE coordinates of (x=0.661, y=0.333). The thermal stabilities were investigated from the luminescence intensities, color purity and the decay curves by increasing temperature. The luminescence parameters related to white LEDs applications were compared to some red phosphors and discussed in details. The red-emitting Ca2RF4PO4:Eu3+ (R=Gd, Y) may be potentially useful in the fabrication of white LEDs.

©2011 Optical Society of America

1. Introduction

In recent years, white-light emitting diodes (W-LEDs) have attracted much attention because of their special advantages of high color rendering index, long life time and low power consumption [1]. The first W-LED was combined the blue light of an InGaN device (emitting near 460 nm) with an Y3Al5O12:Ce3+ (YAG) phosphor (yellow light) in 1997 [2]. However, it shows lower color rendering index (~80) due to the lack of red color contribution. Consequentially, it yields cool white LEDs with a high color temperature (Tc) above 4000 K, which does not meet the requirements for ambient lighting [3]. To obtain warm white-LEDs (Tc<4000K), red materials, e.g., (Sr1-xCax)S:Eu and (Sr1-xCax)2Si5N8:Eu, are used in the combination with YAG:Ce [4].

In addition to approach using the blue LED plus YAG:Ce, white light can be produced by the excitation of blue, green and red phosphors with a near UV-LED operating in the range 380-400 nm. Since the color is controlled solely by the three phosphors, this approach offers flexibility in designing white LEDs with different color temperatures and color rendering indices. They exhibit a high lumen equivalent, quantum efficiency, and photostability at the same time. Presently used phosphors include red Y2O2S:Eu3+, green ZnS:(Cu+, Al3+) and blue BaMgAl10O17:Eu2+. Unfortunately, Y2O2S:Eu3+ is chemically unstable and its fluorescent efficiency is lower than that of the blue and green-emitting phosphors [5]. The other red-emitting phosphors also have some disadvantages, e.g., sulfide-based phosphor CaS:Eu2+ shows luminescence saturation with an increasing applied current when it is incorporated into phosphor-converted W-LEDs’ devices. Recently, some nitrides and oxynitrides-based compounds have been demonstrated to be good potential as red phosphors due to their good thermal stability [6,7]. However, the very high firing temperatures and high nitrogen pressures are required for their synthesis, resulting in higher production cost.

Therefore, it is necessary to search new red phosphor with low cost and high stability. In recent years, many types of Eu3+-doped compounds e.g. silicates [8], borates [9], phosphates [10,11], molybdates [12], tungstates [13], aluminate [14,15], and sulfates [16], have been reported as interesting candidates for potential red-emitting phosphors in W-LEDs. In particular, rare-earth doped orthophosphates (RPO4) are known to have the high chemical stability and the exceptional optical damage threshold. It is well known that fluorine atoms have the largest electronegative and exhibit strongest attractive electron ability. It has been suggested that after the compound RPO4 is fluoridated into R 3+/F/PO4 3−-containing host, the incorporation of F- ions will draw electronic cloud intensive, which will compete with O2− ions. Therefore, the ionicity of P-O bond is weakened and the covalency of P-O bond is increased in R3+/F/PO4 3−-containing host [17,18]. For example, the P-O bond covalency is increased for PO4 3− anion in NaGdFPO4 relative to that in GdPO4 because all four oxygen ions in PO4 3− ions are bound to Gd3+ ions in GdPO4, but only three oxygen ions of PO4 3− ions are bound to Gd3+ ions in NaGdFPO4 [17]. The more F- ions in crystal lattice of R3+/F-/PO4 3−-containing host might incline to increase covalency of P-O bond for PO4 3- groups. The host absorption edge in Na2GdF2PO4 could move to longer wavelength region (~190 nm [18]) than those in NaGdFPO4 (~180 nm [17]) and GdPO4 (~165 nm [19]). Therefore, R3+/F-/PO4 3−-containing characteristic will have obvious influence on the luminescence properties of the host and the activators.

In this paper, Eu3+-activated oxyfluoride Ca2 RF4PO4 (R=Gd, Y) were prepared by conventional solid state reaction, which were characterized by XRD, photoluminescence excitation and emission spectra. The possible application for near UV InGaN chip-based W-LEDs was evaluated by taking into analysis of the excitation spectra in the near UV region, the luminescence spectra, the quantum efficiency (QE), thermal stability (temperature dependent luminescence, decay and color chromaticity). The luminescence properties were also compared with the references of Eu3+ doped red-emitting materials.

2. Experimental

The polycrystalline samples of Eu3+-doped Ca2 RF4PO4 (R=Gd, Y) were synthesized using a conventional solid-state reaction. The starting material was a stoichiometric mixture of reagent grade CaCO3, NH4H2PO4, CaF2, and RE oxides (Gd2O3, Y2O3 and Eu2O3). Firstly, the mixture was heated up to 350 °C and kept at this temperature for 10h. Then the powder was thoroughly mixed in acetone and heated up to 750 °C and kept at this temperature for 5 h in air. After that, the samples were heated in air at 1100 °C for 10 h for two times.

XRD data were collected on a Rigaku D/Max diffractometer operating at 40 kV, 30 mA with Bragg–Brentano geometry using Cu Kα radiation (λ=1.5405 Å). The UV-excited luminescence spectra were recorded on a Perkin-Elmer LS-50B luminescence spectrometer with Monk–Gillieson type monochromators and a xenon discharge lamp used as an excitation source. To study thermal quenching the same spectrofluorimeter was equipped with a homemade heating cell under the excitation of a 394 nm UV light. The QEs were measured by an Absolute Photoluminescence Quantum Yield Measurement System (C9920-02, Hamamatsu) at room temperature. The excitation was done by changing excitation wavelength of light from 150 W Xe-lamp.

3. Results

3.1 Phase formation

Figure 1 shows the XRD patterns of Eu3+-doped Ca2 RF4PO4 (R=Gd3+, Y3+) 5.0 and 45 mol % phosphors. The patterns were compared with the standard JCPDs Card No. 45-0459 for Ca2YF4PO4. By a comparison between them, the positions and relative intensities of the main peaks are nearly the same. No impurity lines were observed.

 figure: Fig. 1

Fig. 1 XRD patterns of Ca2YF4PO4:Eu3+ 5 mol % (a), 45 mol % (b), and Ca2GdF4PO4:Eu3+ 5 mol % (c), 45 mol % (d), which are compared with JCPDs Card No. 45-0459 (Ca2YF4PO4).

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3.2 The excitation spectra

Figure 2 shows the excitation spectra of Eu3+-doped Ca2 RF4PO4 (R=Gd, Y) and Y2O2S:Eu3+. The excitation spectra by monitoring the 5D07F2 emission (611 nm) of Eu3+-doped Ca2 RF4PO4 (R=Gd, Y) consist of a broad band and some lines. The broad excitation band centered at 280 nm can be attributed to the CT band of Eu3+-O2−. In the range from 350 to 500 nm, Ca2 RF4PO4:Eu3+ (R=Gd, Y) present the intra-configurational 4f–4f transitions of Eu3+:394 nm (7F05L6), 464 nm (7F05D2). This well matches with the output wavelength of near-UV or blue LED chips in phosphor-converted W-LEDs. The excitation peaks at 315 and 311 nm in Ca2GdF4PO4:Eu3+ can be attributed to the ff transition of Gd3+ from the ground state 8S7/2 to the excitation level of 6P(7/2,5/2).

 figure: Fig. 2

Fig. 2 The excitation spectra of the red-emitting phosphors Ca2GdF4PO4:Eu3+ (a), and Ca2YF4PO4:Eu3+ (b) (λem = 611 nm) (b), which are compared with that of Y2O2S:Eu3+em = 627 nm) (c).

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3.3 The luminescence spectra and decay curve

Under the excitation of 394 nm in Ca2 RF4PO4:Eu3+ (R=Gd, Y), the intense and bright red luminescence can be observed. The emission spectra shown in Fig. 3 consist of several emission lines, originating from the transitions of excited states 5D0 to the ground states 7FJ (J = 0, 1, 2, 3, 4) in the 4f6 configuration of Eu3+. The emission spectra are dominated by 611 nm emission attributed to the electric dipole transition of 5D07F2. This implies that the Eu3+ ions are located in the non-inversion centers in this host.

 figure: Fig. 3

Fig. 3 The UV excited (394 nm) luminescence spectra of Ca2GdF4PO4:Eu3+ (a), and Ca2YF4PO4:Eu3+ (b). The inset shows the dependence of the integrated emission intensities of Ca2 RF4PO4:Eu3+ (R=Gd, Y) on the doping levels.

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It is well known that the forced electrical dipole transition 5D07F2 is very sensitive to the local environment, while the magnetic dipole transition 5D07F1 is not much affected by the ligand field around Eu3+. Therefore, the intensity ratio of (5D07F2)/(5D07F1) is a measure of rare-earth ion site symmetry. A lower symmetry of the crystal field around Eu3+ will result in a higher ratio value. The ratio of (5D07F2)/(5D07F1) of Ca2GdF4PO4:Eu3+ and Ca2YF4PO4:Eu3+ are 9.12 and 8.58, respectively. This is favorable to improve the color purity of the red phosphor. The CIE (Commission Internationale de l’Eclairage) chromaticity coordinates are calculated to be x = 0.661, y=0.333.

The inset in Fig. 3 represents concentration dependence of the emission intensity of Ca2 RF4PO4:Eu3+ (R=Gd, Y). The luminescence increases with increasing Eu3+-content until a maximum intensity about 40.0 mol % is reached, and then it decreases because of concentration quenching process.

The decay curve of the 5D07F2 luminescence is shown in Fig. 4 . This can be fitted by a single-exponential function as I=Aexp(−t/τ), and the value of lifetime is 0.94 ms at 300 K. The result shows that the lifetime is short enough for potential applications in displays and lights.

 figure: Fig. 4

Fig. 4 The luminescence decay curve of the 5D0 level (611 nm, 5D07F2) in Ca2GdF4PO4:Eu3+ under the excitation of 355 nm pulsed YAG laser at 300 K.

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3.4 The dependence of luminescence on temperature

The thermal quenching property is one of the important technological parameters for phosphors applied in W-LEDs especially in high-power devices [20]. The emission intensity in the visible should not be drastically temperature dependent over the operating temperature range. Many LEDs that may have the phosphor applied in the LED package in front of the emitting surface reach temperatures of 125 °C or more, and the phosphor must be an efficient emitter at such temperatures. Ideally it will be stable to at least 200 °C [21].

Figure 5 shows the selected temperature dependent emission spectra (a) and integrated intensity (b) in Ca2 RF4PO4:Eu3+ (R=Gd, Y). The intensities keep near constant from 20 to 100 °C and decrease slowly with the increase of temperature to 150 °C (about 92 and 89% of the initial value at 20 °C for Ca2GdF4PO4:Eu3+ and Ca2YF4PO4:Eu3+, respectively). In addition, the intensity ratio of (5D07F2)/(5D07F1) of Ca2GdF4PO4:Eu3+ and Ca2YF4PO4:Eu3+does not change with temperature.

 figure: Fig. 5

Fig. 5 The emission spectra of Ca2GdF4PO4:Eu3+ at the temperature of 20, 135, and 200 °C (a) and the temperature dependence of the integrated emission intensity in Ca2 RF4PO4:Eu3+ (R=Gd, Y) normalized with respect to the value at 20 °C (b). Inset shows the activation energies of the thermal quenching fitted in Eq. (1).

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In Fig. 5, the luminescence intensity curve of heating process is almost the same as that of cooling process, indicating the thermal quenching is recoverable or no change with thermal degradation. The temperature dependence of luminescence intensity can be described by a modified Arrhenius equation as following:

IT=I0[1+cexp(ΔEkT)]1
where I 0 is the initial emission intensity, I T is the intensity at different temperatures, c is a rate constant for the thermally activated escape, ΔE is the activation energy, and k is the Boltzmann constant (8.629 × 10−5eV). Inset in Fig. 5 plots of ln[(I0/IT)-1] versus 1000/T for Ca2GdF4PO4:Eu3+ and Ca2YF4PO4:Eu3+, which are linear with slopes of −4.34 and −3.66, respectively. According to Eq. (1), the activation energy ΔE was calculated to be 0.375 eV for Ca2GdF4PO4:Eu3+ and 0.316 eV for Ca2YF4PO4:Eu3+.

The luminescence decay curves of the 5D0 states at 20 to 200 °C shown in Fig. 6 presenting the single exponential profiles. The lifetime shown on Fig. 6 exhibit a small decrease with increasing temperatures up to 200 °C. The experiments show that Ca2 RF4PO4:Eu3+ (R=Gd, Y) phosphor has the excellent thermal stability and stable color (no emission shift) on the temperature quenching effect.

 figure: Fig. 6

Fig. 6 The luminescence decay curves of the 5D0 states in Ca2GdF4PO4:Eu3+ at 90 to 200°C.

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4. Discussions

In recent years, many kinds of Eu3+ doped compounds have been investigated to develop the red-emitting phosphors for W-LEDs. Compared with the other reported Eu3+-doped red-emitting materials, Ca2 RF4PO4:Eu3+ (R=Gd, Y) phosphors have some characteristics.

Firstly, Ca2 RF4PO4:Eu3+ (R= Gd3+, Y3+) phosphors have efficient absorption in the region of near-UV wavelengths (ff transitions of Eu3+ ions, e.g., 394 nm 7F05L6 and 465 nm 7F05D2), which is very stronger than the absorption in the CT region (Fig. 2). This can match the radiation of near UV-emitting InGaN based LED chips.

Usually, the 4f-4f transitions of Eu3+-doped phosphors can overlap with near violet or blue LED chip radiation. However, the 4f-4f transitions of Eu3+ ions are forbidden transitions with low oscillator strengths (<10−6), so the absorption of InGaN LED radiation by Eu3+ ions will be weak. Generally, in Eu3+-doped phosphors, the intensity of the CT band is much stronger than that of ff transitions in the excitation spectrum. On the excitation spectrum of Y2O2S:Eu3+ (Fig. 2), the strong broad band before 350 nm corresponds to CT transition of Eu3+-O2- and Eu3+-S2− [22]. However, the absorption in the near-ultraviolet or blue region is very weak. The excitation spectra of the other well known red phosphors, for example, Y2O3:Eu3+ [23],RPO4 (R = Y3+, La3+, Gd3+):Eu3+ [24], RVO4:Eu3+ (R = Y, La, Gd) [25], and RBO3 (R = Y, La, Gd):Eu3+ [26], and vanadate garnets [27], the excitation intensities of charge transfer band are much more intense than that of ff transitions.

However, the excitation intensities of ff transition in Ca2RF4PO4:Eu3+ (R=Gd, Y) phosphors are higher than that of CT band. Voort and Blasse [28,29] investigated the influence of an effective charge at Eu3+ ion on the quantum efficiency under the CT excitation (qCT) in the calcium compounds. The Eu3+ ions are subject to a high rate of radiationless processes in its excited charge-transfer state if it is incorporated into calcium compounds. Van der Voort et al. [29] suggested that in some calcium compounds the ions with intraionic transitions seem to show efficient luminescence if they have a positive effective charge. Whereas ions with an interionic transition (CT between Eu3+ and O2-), the luminescence only shows weakly. A positive effective charge gives rise to a large relaxation in the CT states and a low qCT value [28]. For a negative effective charge the relaxation was predicted to be less and qCT was high [29]. In Ca2 RF4PO4:Eu3+ (R=Gd, Y), the low intensity of CT bands of Eu3+ ions could follow this model. This could be ascribed to the effective charge of the Eu3+ influenced by the Ca components.

Secondly, Ca2 RF4PO4:Eu3+ (R=Gd, Y) have better CIE red color coordinates. It is known that the larger intensity ratio of (5D07F2)/(5D07F1), the closer to the optimal value of color chromaticity. The intensity ratios of (5D07F2)/I(5D07F1) of Ca2GdF4PO4:Eu3+ and Ca2YF4PO4:Eu3+ are 9.12 and 8.58, respectively. This value is quite large in comparison with those of the other Eu3+-doped phosphors. For example, the magnitudes of (5D07F2)/(5D07F1) in GdPO4:Eu3+ and Y2O2S:Eu3+ were calculated to be 0.65 and 2.77, respectively (emission spectra omitted). The larger ratio is favorable to improve the color purity. Usually, the orange appearance of the color of the RPO4:Eu3+ (R=La, Gd, Y) is caused by the relative intense line at 594 nm corresponding to a magnetic dipole transition of 5D07F1, while the emission lines from the electric dipole transitions 5D07F2 at 613 and 627 nm are very weak [4]. The CIE of Ca2 RF4PO4:Eu3+ (R=Gd3+, Y3+) phosphors keeps a similar value of about (x = 0.661, y = 0.333), which is closer to the standard of NTSC (x=0.67, y=0.33) than that of Y2O2S:Eu3+ (0.647, 0.343) [30], YVO4:Eu3+ (0.675, 0.301) [31], and Gd(V0.6P0.4)O4:Eu3+ (0.6338, 0.3229) [32].

Thirdly, under the excitation of near UV-light, Ca2 RF4PO4:Eu3+ (R=Gd3+, Y3+) phosphors have high emission intensity. The comparison of emission intensities of the Ca2RF4PO4:Eu3+ (R=Gd3+, Y3+) to that of the red phosphors of Y2O2S:Eu3+, GdPO4:Eu3+ and YPO4:Eu3+ on excitation at 394 nm are listed in Table 1 . All the samples show an increased Eu3+ emission when compared to that of the referred phosphors, for example, Ca2GdF4PO4:Eu3+ is 7.51 times stronger than that of Y2O2S:Eu3+. This corresponds to the sharper and more intense absorption of 5L67F0 Eu3+ line at 394 nm than that of Y2O2S:Eu3+.

Tables Icon

Table 1. Comparison of the Emission Intensities of Ca2RF4PO4:Eu3+ (R= Gd, Y) 5.0 mol % to that of GdPO4:Eu3+ (5.0 mol %), YPO4:Eu3+ (5.0 mol %) and Y2O2S:0.05Eu3+ Red Phosphors Under the Same Measurement Conditions.

Table 2 gives the luminescence QEs of Ca2 RF4PO4:Eu3+ (R=Gd, Y), Y2O2S:Eu3+ and some reported phosphors. For luminescence applications, the QE of a phosphor is often regarded as a measure of its merit. The results demonstrate that QEs of Ca2 RF4PO4:Eu3+ (R=Gd, Y) are higher than that of Y2O2S:Eu3+, and comparable with those by other reports. Certainly, the higher QE can be obtained by further improving the synthesis conditions to reduce the number of defects and impurities and to get high crystallization of the phosphors.

Tables Icon

Table 2. The Luminescence Quantum Efficiencies of Ca2RF4PO4:Eu3+ (R=Gd, Y) and Some Referred Phosphors.

Fourthly, Ca2 RF4PO4:Eu3+ (R=Gd, Y) tolerate high Eu3+ doping. The luminescence intensity enhances with increasing Eu3+-concentration and shows nearly no concentration quenching up to 40 mol %. Usually the concentration quenching is strongly dependent on the structural dimensionality of the host. There is a weak concentration quenching of europium luminescence in the host, where the RE could form a quasi-two-dimensional RE(Eu)3+ sublattice [38]. At present, the detailed structure of Ca2 RF4PO4:Eu3+ (R=Gd, Y) is unknown. However, the high critical values in this host indicate that the energy transfer might be restricted to the quasi-two-dimensional Eu3+ sublattice. The high doping of Eu3+ ions in a phosphor can suffer high power excitation in the applications in W-LEDs. Usually, the quenching concentration of the Eu3+ doping in Y2O2S:Eu3+ and RPO4:Eu3+ (R=La, Gd, Y):Eu3+ is only 5.0 mol % [30,39].

Finally, under high temperature, Ca2 RF4PO4:Eu3+ (R=Gd, Y) has the excellent thermal stability as displayed in Fig. 6. The luminescence intensities of Ca2GdF4PO4:Eu3+ decrease to 92% of the initial value at 20 °C. It has been reported that the luminescence intensities of the red nitride phosphors (Sr0.82Ba0.15Eu0.03)2Si5N8:Eu2+ and (Sr0.75Ca0.25)0.98SiAlN3:Eu2+ at 150 °C are 87% of the value at 25 °C [40]. The luminescence intensity of Y2O2S:Eu3+ or La2O2S:Eu3+ was reported to firstly increase due to its excitation maximum moving to a longer wavelength before reaching its maximum value at 100 °C and dramatically decrease with increasing temperature [9,27]. Compared with these phosphors, the Ca2 RF4PO4:Eu3+ (R= Gd3+, Y3+) phosphors have the excellent thermal stabilities of luminescence and pure color coordinates. In addition, the emission wavelengths show no shifts and the color coordinates have no changes. The CIE keeps the value of about (x=0.661, y=0.333).

5. Conclusions

A new red-emitting phosphor, Ca2 RF4PO4:Eu3+ (R=Gd, Y), was synthesized by solid state reactions for the first time. Compared with the commercial or the reported red-emitting phosphors, Ca2 RF4PO4:Eu3+ (R=Gd, Y) can be effectively excited by near-UV light. Under the excitation of near UV-light, Ca2RF4PO4:Eu3+ (R=Gd3+, Y3+) have high emission intensity, better chrome (x=0.661, y=0.333). The luminescence QEs are 64% and 48% for Ca2GdF4PO4:Eu3+ and Ca2YF4PO4:Eu3+, respectively. Ca2 RF4PO4:Eu3+ (R=Gd, Y) can tolerate high Eu3+ doping concentrations, which can suffer high power excitation in the applications. Under high temperature, Ca2 RF4PO4:Eu3+ (R=Gd, Y) phosphors have the excellent thermal stability and show no emission shifts. The present results indicate that the novel red emitting phosphor is a suitable candidate for the application on near-UV white LEDs.

Acknowledgements

This work was supported by the Industrial Strategic technology development program (Project No: 10037416) funded by the Ministry of Knowledge Economy (MKE, Korea) and by Mid-career Researcher Program through National Research Foundation (NRF) grant funded by the Ministry of Education, Science and Technology (MEST) (Project No. 2009-0078682).

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Figures (6)

Fig. 1
Fig. 1 XRD patterns of Ca2YF4PO4:Eu3+ 5 mol % (a), 45 mol % (b), and Ca2GdF4PO4:Eu3+ 5 mol % (c), 45 mol % (d), which are compared with JCPDs Card No. 45-0459 (Ca2YF4PO4).
Fig. 2
Fig. 2 The excitation spectra of the red-emitting phosphors Ca2GdF4PO4:Eu3+ (a), and Ca2YF4PO4:Eu3+ (b) (λem = 611 nm) (b), which are compared with that of Y2O2S:Eu3+em = 627 nm) (c).
Fig. 3
Fig. 3 The UV excited (394 nm) luminescence spectra of Ca2GdF4PO4:Eu3+ (a), and Ca2YF4PO4:Eu3+ (b). The inset shows the dependence of the integrated emission intensities of Ca2 RF4PO4:Eu3+ (R=Gd, Y) on the doping levels.
Fig. 4
Fig. 4 The luminescence decay curve of the 5D0 level (611 nm, 5D07F2) in Ca2GdF4PO4:Eu3+ under the excitation of 355 nm pulsed YAG laser at 300 K.
Fig. 5
Fig. 5 The emission spectra of Ca2GdF4PO4:Eu3+ at the temperature of 20, 135, and 200 °C (a) and the temperature dependence of the integrated emission intensity in Ca2 RF4PO4:Eu3+ (R=Gd, Y) normalized with respect to the value at 20 °C (b). Inset shows the activation energies of the thermal quenching fitted in Eq. (1).
Fig. 6
Fig. 6 The luminescence decay curves of the 5D0 states in Ca2GdF4PO4:Eu3+ at 90 to 200°C.

Tables (2)

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Table 1 Comparison of the Emission Intensities of Ca2 RF4PO4:Eu3+ (R= Gd, Y) 5.0 mol % to that of GdPO4:Eu3+ (5.0 mol %), YPO4:Eu3+ (5.0 mol %) and Y2O2S:0.05Eu3+ Red Phosphors Under the Same Measurement Conditions.

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Table 2 The Luminescence Quantum Efficiencies of Ca2 RF4PO4:Eu3+ (R=Gd, Y) and Some Referred Phosphors.

Equations (1)

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I T = I 0 [ 1 + c exp ( Δ E k T ) ] 1
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