Journal of Alloys and Compounds 559 (2013) 92–96
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Photoluminescence and piezoelectric properties of Pr-doped NBT–xBZT ceramics: Sensitive to structure transition Peng Du a, Laihui Luo a,⇑, Weiping Li a, Yueping Zhang a,b, Hongbing Chen b a b
Department of Physics, Ningbo University, 315211 Ningbo, China Institute of Materials Science and Engineering, Ningbo University, 315211 Ningbo, China
a r t i c l e
i n f o
Article history: Received 1 January 2013 Received in revised form 19 January 2013 Accepted 21 January 2013 Available online 4 February 2013 Keywords: Lead-free ceramic Piezoelectric Ferroelectric Photoluminescence
a b s t r a c t Lead-free (1 x)Na0.5Bi0.5TiO3–xBa(Zr0.05Ti0.95)O3:Pr (NBT–xBZT:Pr, x = 0–0.12) were synthesized by conventional solid state method. The influence of Ba(Zr0.05Ti0.95)O3 (BZT) content on the piezoelectric and photoluminescence performance was investigated. The results show that all ceramics exhibit strong red emission at 607 nm, and the NBT–0.07BZT:Pr ceramic shows the optimal photoluminescence property. In addition, piezoelectric performance is also enhanced with the increasing of BZT fraction, and reaches the maximum value at x = 0.07. The results show that the Pr3+ ions emission may be used as a probe for the phase transition of the ferroelectric materials. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction In Recent years, luminescence properties of rare-earth (RE)doped materials have attracted considerable attention due to their widely technology applications in photonic devices and next-generation flat-panel display. Ferroelectrics are multifunction materials; they show excellent piezoelectric, ferroelectric and pyroelectric performance. Researchers have revealed that the ferroelectric materials exhibit enhanced piezoelectric, ferroelectric, and optimal photoluminescence (PL) properties by rare-earth doping [1–6]. RE doping in ferroelectrics can induce some interesting performances, which is essential as a smart material. Wang et al. [1] reported that electro-mechano, mechano-optic and electro-optic coupling, can be realized in the Pr doped BaTiO3-based ceramics. Our previous investigation also demonstrates the PL performance in ferroelectrics can be improved by the poling [7]. It has been reported that the dielectric, ferroelectric, and piezoelectric properties of Na1/2Bi1/2TiO3 (NBT) ceramics are improved by the addition of Er2O3 [2]. Sun et al. [3] revealed that Pr-doped ferroelectric NBT ceramics show a strong photoluminescence performance, and enhanced ferroelectric performance. In the Pr-doped Sr1 xCaxTiO3 system the blue emission is enhanced when tetragonal–orthorhombic phase transition happens [8]. Zhang et al. [9] reported that there are little mechanoluminescence (ML) in the pure tetragonal or orthorhombic phase in the (Ba, Ca)TiO3:Pr ceramics,
⇑ Corresponding author. Tel.: +86 574 87600953; fax: +86 574 87600744. E-mail address:
[email protected] (L. Luo). 0925-8388/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jallcom.2013.01.096
strong ML only occurs in diphase region. Therefore, it seems that the optical properties of the RE doped ferroelectric materials are sensitive to the structure and phase transition. With the growing demand for global environmental protection, lead-free materials have attracted an increasing attention because of the high toxicity of lead oxide and its high volatility during sintering. Therefore, it is necessary to search for a new lead-free piezoelectric ceramic to replace the lead-based materials. Bismuth sodium titanate Na0.5Bi0.5TiO3 is one of the most popular lead-free piezoelectric material with a rhombohedral perovskite structure discovered by Smolenkii et al. in 1960 [9]. NBT ceramics have strong ferroelectric properties, a large remanent polarization (Pr = 38 lC/cm2) and a high coercive field (Ec = 73 kV/cm). Many researchers have been devoted to improving the piezoelectric properties of the NBT ceramics by adding other perovskite ferroelectrics, such as NBT–BaTiO3, [11,12] (1 x)NBT–xBiAlO3, [13] and NBT–BaTiO3–Ba0.77Ca0.23TiO3 [14]. The (1 x)NBT–xBaTiO3 ceramics have optimal properties compared with NBT ceramics, and reach the extreme values near the MPB with x = 0.06 [11]. Ba(Zr1 xTix)O3 (BZT) ceramics also have received much attention for its excellent dielectric properties [15–18]. Ba(Zr1 xTix)O3 ceramics exhibit fairly satisfactory piezoelectric constant d33 = 236 pC/N, and optimal electromechanical coupling coefficient K33 = 56.5% when x = 0.05 [15]. However, the large temperature dependence of the electrical performances and a low Curie temperature (Tc = 102 °C) for Ba(Ti0.95Zr0.05)O3 ceramic limits it’s practical applications [16]. Considering the NBT ceramics have a higher Curie temperature, we choose the (1 x)Na0.5Bi0.5TiO3–xBa(Zr0.05Ti0.95)O3 (abbreviated as NBT–xBZT) system as a host material.
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It was reported that the emission intensity of Pr doped materials reached the optimal value when the doped concentration was 0.3 mol% [3]. Thus in this work we present the influence of BZT content on the piezoelectric and photoluminescence performance of the 0.3 mol% Pr-doped (1 x)Na0.5Bi0.5TiO3-xBaZr0.05Ti0.95O3 (abbreviated as NBT–xNBT:Pr, x = 0, 0.02, 0.04, 0.06, 0.07, 0.1, 0.12) ceramics. 2. Experimental High-purity powders TiO2, Bi2O3, Na2CO3, Pr2O3, BaCO3 and ZrO2 were used as starting materials to synthesize the 0.3 mol% Pr-doped NBT–xBZT ceramics by a conventional solid-state reaction technique. The NBT–xBZT:Pr ceramics were prepared by using a modified two-step sintering approach, in which NBT:Pr and BZT source powders were calcined separately and then remilled together. To prepare the NBT:Pr source power, TiO2, Bi2O3, Na2CO3 and Pr2O3 powders in the stoichiometric ratio were mixed in alcohol by agate balls for 10 h, and then dried and calcined at 850 °C for 2 h. For Ba(Zr0.05Ti0.95)O3 power, TiO2, BaCO3 and ZrO2 powders in the stoichiometric ratio were mixed in alcohol by agate balls for 10 h, and then dried and calcined at 1200 °C for 2 h. The NBT:Pr and BZT source powders were then weighted and mixed according to the formulas of (1 x)NBT–xBZT:Pr, where x is the content of BZT, ranging from 0 to 0.12. After remilled, mixed thoroughly with a PVA binder solution and pressed into pellets; the pellets were finally sintered at 1150 °C for 2 h in air. For carrying out the electrical measurement, all the sintered pellets were ground to 1 mm in thickness and silver electrode was coated on both sides. The samples were subjected to a 40 kV/cm electric field at 80 °C in silicone oil bath for 20 min, and then cooled to room temperature under field. The crystal structure of the ceramics was checked using X-ray diffraction (XRD) analyzer (Bruker D8 Advance) with Cu Ka radiation. The piezoelectric coefficient d33 of the samples was measured by a quasistatic piezoelectric meter (ZJ-3AN, China). Dielectric constant e33 and dielectric loss tan d of the sintered samples as a function of temperature were measured at 100 Hz, 1 kHz, 10 kHz, 100 kHz, 1 MHz using a computer-controlled impedance analyzer Agilent 4294A. The polarization vs. electric field (P–E) hysteresis loops were measured at 1 Hz by the RT Premier II ferroelectric workstation. The morphology of calcined powders was investigated by a scanning electron microscope (SEM) (Hitachi SU-700). The photoluminescence (PL) and photoluminescence excitation (PLE) spectra at room temperature were recorded using a spectrofluorometer (Hitachi F-4500). The density was measured using Archimedes method.
3. Results and discussion The XRD patterns of the NBT–xBZT:Pr ceramics with 0 6 x 6 0.12 are shown in Fig. 1. The XRD patterns of samples in the 2h ranging from 45° to 48.5° are zoomed in Fig. 1b. The results suggest that all the ceramics possess a pure perovskite phase and no any secondary impurity phase can be observed. Without BZT addition, the NBT:Pr ceramic shows a rhombohedral phase. However, the peaks are shifted slightly to the lower-angle with the increase in BZT, as shown in Fig. 1b. It is attributed to the lattice enlarging. The larger size Ba2+ ion (1.61 Å) occupies the A site in the ceramics by replacing Na+ (1.39 Å) and Bi3+ (1.30 Å), and Zr4+
Fig. 1. (a) X-ray diffraction patterns of the sintered NBT–xBZT:Pr ceramics (x = 0, 0.02, 0.04, 0.06, 0.07, 0.1, 0.12); (b) the zoomed XRD patterns between 45° and 48.5°.
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(0.72 Å) ion substitutes the Ti4+ (0.605 Å) ion. However, as the content of BZT exceeds 0.1, the XRD peaks shift slightly to the largerangle sides, and the rhombohedral to tetragonal phase transition should be responsible for this phenomenon. NBT is rhombohedral phase at room temperature [10], BZT is tetragonal phase at ambient temperature [16]. We can see that the peak (2 0 2) finally splits into two peaks with the increment of the BZT. It is believed that the (2 0 2) peak is assigned to the rhombohedral phase, and the splitting of peaks (0 0 2) and (2 0 0) is related to the tetragonal phase transition. It means that a phase transition from rhombohedral to tetragonal happened with the increasing BZT fraction. As shown in Fig. 1b only the (2 0 2) peak is observed when x < 0.07, implying that NBT–xBZT:Pr ceramics are rhombohedral phase. With the increment of BZT to x = 0.1, (2 0 2) peak splits to (0 0 2) and (2 0 0) peaks, suggesting that the ceramics transform from rhombohedral to tetragonal phase. These results indicate that the rhombohedral– tetragonal morphotropic phase boundary (MPB) of NBT–xBZT:Pr ceramics is near to x = 0.07 at room temperature, which is consistent with earlier report [19]. Typical microstructure of the composition with x = 0, 0.07, and 0.12 are shown in Fig. 2a–c. It is seen that all NBT–xBZT:Pr ceramics are densely sintered. The grain size increases with the increasing of BZT fraction as shown in Fig. 2a and b. However, as the BZT content increased further, abnormal grain growth happens, and some large grains appear as shown in Fig. 2c. Furthermore, all NBT–xBZT:Pr ceramics have a density of 5.66–5.83 g/ cm3 as shown in Fig. 2d. It is seen that the relative density enhances with the increment of BZT content when x 6 0.07. However, the relative density starts to decrease when x P 0.1. The reduced volatilization of Na, Bi ions and abnormal grain growth should be responsible for this phenomenon. It is well known that the Na, Bi ions evaporates during the sintering process, resulting in lowly dense samples. With BZT addition, Na, Bi ions can be substituted by Ba ions, and the volatility of Na, Bi ions is reduced. Therefore, the relative density is enhanced. Nevertheless, the abnormal grain growth may be responsible for the decreased relative density when x P 0.1. The dielectric constant e33 and dielectric loss tan d for the poled NBT–xBZT:Pr ceramics as a function of temperature are measured with frequencies 100 Hz, 1 kHz, 10 kHz, 100 kHz and 1 MHz. Several typical curves are shown in Fig. 3. From Fig. 3a–c, we clearly see that there are three dielectric anomaly peaks at Td, Tp and Tm, as marked in the Figs. The first dielectric anomaly peak occurs at the depolarization temperature Td, which is corresponding to the temperature of the first tan d peak. Near Td, lots of interesting results are observed in BNT-based ceramics, such as large electricfield induced strain, and constricted P–E loop [20,21]. The second dielectric anomaly peak appears at temperature Tp, the dielectric anomaly might be attributed to the polarization of space charges and dipolar defects [22]. From Fig. 3a–c, we can observe that Tp is dependent on the measurement frequency, and the dielectric anomaly nearly disappear when the test frequency increases to 1 MHz. The third dielectric anomaly appears at temperature Tm, at which maximal dielectric constant is achieved. As shown in Fig. 3, the dielectric anomaly peaks at Tm for all the ceramics are relatively broad. The researchers have different opinions on the dielectric anomaly at temperature Tm. Zhang et al. [23] attributed the dielectric anomaly at Tm to the antiferroelectric–paraelectric phase transition. However, Ma et al. [24] revealed that the dielectric anomaly at Tm does not identify a phase transition, and in the NBT–BT system, the cubic paraelectric phase appears far above Tm. However, some researchers reported that the dielectric anomaly at Tm originates from the contribution of the rhombohedra to tetragonal phase transition and the thermal evolution of tetragonal polar nanoregions [25,26]. Therefore, further studies are still needed to better clarify the origin of the anomaly at Tm. Fig. 3d shows the var-
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Fig. 2. SEM micrograph of NBT–xBZT:Pr ceramic sintered at 1150 °C for 2 h. (a) x = 0, (b) x = 0.07, (c) x = 0.12; (d) the dependence of density and relative density on BZT content.
Fig. 3. Temperature dependences of e33 and tan d for poled NBT–xBZT:Pr ceramics at 100 Hz, 1 k Hz, 100 k Hz, and 1 M Hz: (a) x = 0, (b) x = 0.07, (c) x = 0.12; (d) the dependence of Td on BZT content.
iation of Td with BZT content. It is seen that the value of Td decreases with the increment of BZT content when x 6 0.07. However, the value of the Td starts to increase when x P 0.1. We note that similar result was found in NBT–BT ceramics [11]. The incre-
ment of Td may be ascribed to the enhancement of the ferroelectric order induced by the introduction of BZT. Fig 4a shows the P–E hystersis loops of NBT–xBZT:Pr ceramics at room temperature. It is found that well saturated and square-like
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Fig. 4. (a) P–E hysteresis loops of NBT–xBZT:Pr ceramics at room temperature; (b) Variations of Pr and Ec with the BZT content x for the NBT–xBZT:Pr ceramics.
loops are obtained for all the studied compositions. Fig 4b presents the variations of the remnant polarization Pr and coercive filed Ec with the BZT content x. We can see that the maximum remanent polarization Pr (33.5 lC/cm2) is found at x = 0.04, and the NBT– 0.07BZT:Pr ceramic owns the minimum coercive field Ec (17.7 kV/ cm). The variation of Pr and Ec can be attributed to the change of the structure parameters [19]. The composition of NBT–0.07BZT:Pr ceramic shows a ferroelectric properties with higher remanent polarization Pr (31.3 lC/cm2) and minimum coercive field Ec (17.7 kV/cm). The dielectric constant e33 and dielectric loss tan d of NBT– xBZT:Pr ceramics measured at 1 kHz are shown in Fig. 5a. The maximum values of e33 = 1332 and tan d = 0.056 are obtained at x = 0.07, which is near the MPB. At the MPB region, the polarization vector can easily switch between all the allowed polarization orientations. Therefore, the dielectric properties increase [27]. Fig. 5b illustrates the variations of the piezoelectric coefficient d33 with x. It can be seen that the d33 increases with x, and achieves its maximum when x = 0.07. Two possible reasons are given to the
phenomenon. One reason is that the number of possible spontaneous polarization directions for compositions nears the rhombohedral–tetragonal MPB are increased [28]. Another reason is that the depolarization temperature may also be related to the piezoelectric constant. It is known to us that macro–micro domain transition at depolarization temperature [14]. Therefore, depolarization temperature can be seen as the indication of the stability of ferroelectric domains. The decreased depolarization temperature suggests that the addition BZT makes the ferroelectric domains less stable, then the domains reorientate easily [11,13]. Therefore, the piezoelectric properties are enhanced. However, the value of d33 jumps to a low value as x increases to 0.12. In order to investigate the influence of BZT addition on the luminescence properties, the photoluminescence excitation spectra and emission spectra of the NBT–xBZT:Pr ceramics were shown in Fig. 6. There are no shifts about the excitation and emission peaks for all the NBT–xBZT:Pr ceramics. The excitation spectra of NBT–xBZT:Pr ceramics monitored at kem = 607 nm is shown in Fig. 6a. From the excitation spectra, we can see three strong sharp
Fig. 5. (a) Dielectric constant e33 and dielectric loss tan d of the NBT–xBZT:Pr ceramics, (b) Piezoelectric constant d33 of the NBT–xBZT:Pr ceramics.
Fig. 6. (a) Photoluminescence excitation spectra of NBT–xBZT:Pr (x = 0, 0.02, 0.04, 0.06, 0.07, 0.1, 0.12) ceramics monitoring at 607 nm, (b) photoluminescence spectra of NBT–xBZT:Pr (x = 0, 0.02, 0.04, 0.06, 0.07, 0.1, 0.12) ceramics excitation at 452 nm. Inset: the dependence of emission intensity on BZT content.
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excitation peaks between 440 and 500 nm. The broad excitation band is attributed to the Pr3+ (4f) ? Ti4+ (3d) charge transition. The excitation peaks between 440 and 500 nm are due to the typical 4f–4f transition from the 3H4 ground state to the excited states 3 PJ (J = 0, 1, 2) of Pr3+ [3,6,7]. The intensely sharp peaks around 452, 478, and 494 nm are related to the 3H4 ? 3P2, 3H4 ? 3P1, and 3 H4 ? 3P0 transitions. Fig. 6b shows the photoluminescence emission of NBT–xBZT:Pr ceramics. By 452 nm excitation, the NBT– xBZT:Pr ceramics show two emission bands located at 607 nm and 655 nm. When electrons are excited to 3P2 levels, they may either relax to 3P0 state and recombine to 3F4 state or nonradiatively relax to 1D2 state and recombine to lower levels following selection rule. The strong red emission band at 607 nm is due to the inter-4f transition from the excited 1D2 to the ground state 3H4 [1,3], and the weak red emission located at 655 nm is due to the 3 P0 ? 3F2 transition [29,30]. The inset in Fig. 6b illustrates the intensity of the 1D2 ? 3H4 transition of NBT–xBZT:Pr ceramics as a function of the content of BZT. From the Fig. 6b we find that the intensity of the NBT–xBZT:Pr ceramics increase with the adding of BZT content and reach their maximum at x = 0.07. As the content of BZT exceeds 0.07, the PL intensity decreases rapidly, this coincides well with the piezoelectric behavior of the ceramics. The phase transition should be responsible for the increment of the PL intensity [8]. It is clearly to us that the rare-earth ions have a sharp line output due to the transition within the 4f shell, which is well shielded by outer 5s- and 5p-electron shells and shows a negligible interaction with the host [31]. Strictly speaking, these transition are forbidden, but become allowed due to the surrounding crystal field relaxing the selection rules [32]. The peak at 607 nm is due to the forced electric dipole transition (1D2 ? 3H4), which is allowed in this case as the Pr does not occupy a center of symmetry in NBT–xBZT ceramics. It is also hypersensitive to environment effects. With the increment of the BZT content, the crystal field around the Pr3+ ions would be changed, which is induced by the rhombohedral to tetragonal phase transition [33]. The presence of a crystal field in most crystalline hosts would shift 5d energy level of Pr3+ and allows stronger 4f ? 4f emission occurring [31]. Therefore, the red emission is enhanced with rhombohedral to tetragonal transition. The results suggest that the rare-earth Pr3+ ions emission may be used as a structure probe for the ferroelectric materials. 4. Conclusions The NBT–xBZT:Pr ceramics were prepared by conventional solid state method. The piezoelectric, ferroelectric and photoluminescence properties were investigated. With the increment of BZT content, both the piezoelectric and photoluminescence properties are increased, and reach the optimal value when x = 0.07. From the structure and phase analysis, phase transition from rhombohedral to tetragonal was observed with the increasing of BZT content, indicating the existence of MPB in the NBT–xBZT:Pr system. In addition, upon the excitation of 452 nm, the NBT–xBZT:Pr ceramics exhibit a strong red emission at 607 nm, and the NBT–0.07BZT:Pr
ceramic shows the optimum photoluminescence performance at rhombohedral–tetragonal transition. The structure transition can be attributed to this phenomenon, and the results imply that the emission of the Pr3+ ions may be used as a structure transition probe for the ferroelectric. Acknowledgments This work was supported by the National Natural Science Foundation of China (51002082, 11004113), The Prior Project in Key Science & Technology Program of Zhejiang Province of China (Grant No. 2009C11144), Natural Science Foundation of Ningbo (2012A610118), and the K.C. Wong Magna Foundation in Ningbo University (XYL10012, xkz11203). References [1] X. Wang, C. Xu, H. Yamada, K. Nishikubo, X. Zheng, Adv. Mater. 17 (2005) 1254. [2] M. Wu, Y. Lu, Y. Li, J. Am. Ceram. Soc. 90 (11) (2007) 3642. [3] H. Sun, D. Peng, X. Wang, M. Tang, Q. Zhang, X. Yao, J. Appl. Phys. 110 (2011) 016102. [4] J. Hao, Y. Zhang, X. Wei, Angew. Chem. 123 (2011) 7008. [5] P. Zhang, M. Shen, L. Fang, F. Zheng, X. Wu, J. Shen, H. Chen, Appl. Phys. Lett. 92 (2008) 222908. [6] H. Sun, D. Peng, X. Wang, M. Tang, Q. Zhang, X. Yao, J. Appl. Phys. 111 (2012) 046102. [7] P. Du, L. Luo, W. Li, Y. Zhang, H. Chen, J. Alloys Comp. 551 (2013) 219. [8] W. Jia, W. Xu, I. RiVera, A. Perez, F. Fernandez, Solid State Commun. 126 (2003) 153. [9] J.C. Zhang, X. Wang, X. Yao, C.N. Xu, H. Yamada, J. Electrochem. Soc. 157 (2010) 269. [10] G.A. Sommerdijk, V.A. Isupov, A.I. Agranovskaya, N.N. Krainik, Sov. Phys. Solid State 2 (1961) 2651. [11] B. Chu, D. Chen, G. Li, Q. Yin, J. Eur. Ceram. Soc. 22 (2002) 2115. [12] K. Moon, D. Rout, H. Lee, S. Kang, J. Cryst. Growth. 317 (2011) 28. [13] H. Yu, Z. Ye, Appl. Phys. Lett. 93 (2008) 112902. [14] L. Luo, F. Ni, H. Zhang, H. Chen, J. Alloys Comp. 536 (2012) 113. [15] Z. Yu, C. Ang, R. Guo, A.S. Bhalla, J. Appl. Phys. 92 (2002) 1489. [16] . N, Nanakorn, P. Jalupoom, N. Vaneesorn, A. Thanaboonmbut, Ceram. Int. 34 (2008) 779. [17] T. Maiti, R. Guo, A.S. Bhalla, Appl. Phys. Lett. 90 (2007) 182801. [18] H. Chen, C. Yang, C. Fu, J. Shi, J. Zhang, W. Leng, J. Mater. Sci.: Mater. Electron. 19 (2008) 379. [19] W. Lee, Y. Lee, M. Tseng, C. Huang, Y. Wu, J. Am. Ceram. Soc. 95 (2009) 1069. [20] S. Zhang, A.B. Kounga, E. Aulbach, H. Ehrenberg, J. Rodel, Appl. Phys. Lett. 91 (2007) 112906. [21] Y. Guo, M. Gu, H. Luo, Y. Liu, R.L. Withers, Phys. Rev. B 83 (5) (2011) 054118. [22] C.R. Zhou, X.Y. Liu, W.Z. Liu, C.L. Yuan, Solid State Commun. 149 (2009) 481. [23] S.T. Zhang, A.B. Kounga, E. Aulbach, T. Granzow, W. Jo, H.J. Kleebe, J. Rodel, J. Appl. Phys. 103 (2008) 03108. [24] C. Ma, X. Tan, J. Am. Ceram. Soc. 94 (2011) 4040. [25] W. Jo, S. Schaab, E. Sapper, L.A. Schmitt, H.J. Kleebe, A.J. Bell, J. Rodel, J. Appl. Phys. 110 (2011) 074106. [26] G. Viola, H. Ning, M.J. Reece, R. Wilison, T.M. Correia, P. Weaver, M.G. Cain, H. Yan, J. Phys. D: Appl. Phys. 45 (2012) 355302. [27] C. Peng, J. Li, W. Gong, Mater. Lett. 59 (2005) 1576. [28] Q. Xu, X. Chen, W. Chen, S. Chen, B. Kim, J. Lee, Mater. Lett. 59 (2005) 2437. [29] D. Peng, H. Sun, X. Wang, J. Zhang, M. Tang, X. Yao, J. Alloys Comp. 511 (2012) 159. [30] Q.J. Chen, W.J. Zhang, X.Y. Huang, G.P. Dong, M.Y. Peng, Q.Y. Zhang, J. Alloys Comp. 513 (2012) 139. [31] S.Y. Kang, Y.H. Kim, J. Moon, K.S. Kang, Jpn. J. Appl. Phys. 48 (2009) 052301. [32] B.R. Judd, Phys. Rev. 127 (1962) 750. [33] Y. Zhang, J. Hao, C.L. Mak, X. Wei, Opt. Express 19 (3) (2011) 1824.