Dielectric Properties and Lattice Distortion in Rhombohedral Phase Region and Phase Coexistence Region of PZT Ceramics

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1 Commun. Theor. Phys. (Beijing, China) 43 (2005) pp c International Academic Publishers Vol. 43, No. 5, May 15, 2005 Dielectric Properties and Lattice Distortion in Rhombohedral Phase Region and Phase Coexistence Region of PZT Ceramics ZHANG Duan-Ming, 1,2 ZHONG Zhi-Cheng, 1,3 HAN Xiang-Yun, 1,2 YAN Wen-Sheng, 1,2 SUN Hong-Zhang, 1,2 YANG Feng-Xia, 1,2 ZHENG Ke-Yu, 1,2 WEI Nian, 1,2 and LI Zhi-Hua 1,2, 1 Department of Physics, Huazhong University of Science and Technology, Wuhan , China 2 Faculty of Physics and Electronic Technology, Hubei University, Wuhan , China 3 Department of Physics, Xiangfan College, Xiangfan , China (Received April 16, 2004; Revised November 2, 2004) Abstract In this paper, the relation between the dielectric properties and the lattice distortion in the phase coexistence region is discussed using a phase statistical distribution model, and in the rhombohedral phase region the two connection equations on the dielectric properties and the lattice distortion are established. Particularly, the relation between the dielectric properties and the lattice distortion is investigated in the phase coexistence region of PZT ceramics, and the fitting value of the volume fraction of the tetragonal phase V T to composition x in the equation is determined. Further, the fitting results are well consistent with the related experimental data. It involves more profound physical process than relation between the dielectric properties and composition x. PACS numbers: Fh, q Key words: lattice distortion, dielectric properties, rhombohedral phase, phase coexistence, PZT 1 Introduction Pb(Zr x,ti 1 x )O 3 (generally known as PZT), a continuous solid solution system of perovskite ferroelectric PbTiO 3 and antiferroelectric PbZrO 3 in different Zr/Ti ratios, has been considered as an important material for a wide range of piezoelectric, pyroelectric, and ferroelectric device applications such as transducers, computer memory, and display and pyorelectric sensors. [1,2] The boundary between the tetragonal phase and the rhombohedral phase is called morphotropic phase boundary (MPB), which corresponds to a structural change with the variation in composition. There exist not only the tetragonal phase but also the rhombohedral phase in the coexistence region of PZT ceramics, which results in many particular properties in this region, such as dielectric properties and piezoelectric properties. [3] There are many models interpreting the causes of the occurrence of the phase coexistence, the distribution of the coexisting phase and their chemical properties of PZT ceramics, such as solubility gap model, composition hysterisis model, and Cao and Cross s statistical distribution model. [4 7] But some controversy still exists concerning their real causes. Until now few people have discussed the relation between the dielectric properties and the lattice distortion in the phase coexistence region from the theoretical view. In this paper, we study the relation between the dielectric properties and the lattice distortion in the phase coexistence region using the phase statistical distribution model. [3,8] In Ref. [9], we have studied the relation between the dielectric constant ε r and the lattice distortion c/a for the PZT tetragonal phase region. In this paper, we are going to discuss the relation between the dielectric properties and the lattice distortion in the rhombohedra phase region and the phase coexistence region of PZT ceramics. The two papers will describe completely the change process of the dielectric constant ε r with the lattice distortion from the PZT rhombohedral phase region transitions to tetragonal phase region via the phase coexistence region. 2 Two Connection Equations on Dielectric Properties and Lattice Distortion c/a in Tetragonal Phase Region For the PZT tetragonal phase region, we have attained the equation previously on the dependent relation between the dielectric constant ε r [9] and composition x, C ε 33 = (T T 0 ) + 2ε 0 (6α 11 CP α 111 CP 4 3 ), (1) where ε 0 is the permittivity of free space, C is the Curie constant, α 11 C is the product of the fourth order tetragonal dielectric stiffness and the Curie constant, α 111 C is the product of the sixth order tetragonal dielectric stiffness and the Curie constant, T is the environmental temperature of samples measured, and T 0 is the Curie Weiss temperature. We have also established another equation on the dependent relation between composition x and the lattice The project supported by National Natural Science Foundation of China under Grant No Corresponding author, lily.lzh@263.net

2 856 ZHANG Duan-Ming, ZHONG Zhi-Cheng, HAN Xiang-Yun, et al. Vol. 43 distortion c/a 1 Q 12 1 c/a c/a Q 11 /Q 12 = { [ α 11 C + (α 11 C) 2 3(T T 0)α 111 C ] 1/2} 1 2ε 0 3α 111 C, (2) where Q 11 and Q 12 are the electrostrictive constants for the PZT solid-solution system. All physical parameters in Eqs. (1) and (2), i.e. T 0, α 11 C, α 111 C, Q 11, Q 12, P3 2, and C, have respectively been expressed as functions of composition x by fitting the related experimental data. [9] In this paper, we directly quote their values as follows: P3 2 = ( α 11 C + ((α 11 C) 2 3α 1 Cα 111 C) 1/2 1 ) 3α 111 C, (3) α 11 C = ( x x 2 )10 13, (4) α 111 C = x x 2 )10 13, (5) T 0 = ( x 280.0x x 3 ) C, (6) Q 11 = x , (7) (x 0.5) Q 12 = x , (8) (x 0.5) 2 { α + βx + γx 2 + δx 3 + εx 4 (x 0.5), C = (9) a + bx + cx 2 (x 0.5). In Eq. (9), α β, γ, δ, and ε are , , , , and , respectively; a, b, and c are , , , respectively. We connect Eqs. (1) and (2) and substitute the values of all parameters into the two equations, then we can obtain the relation between the dielectric constant ε r and the lattice distortion c/a for the PZT tetragonal phase region. for high-temperature rhombohedral phase region: χ 2 11 χ 2 12 η 33 = χ χ 11χ (12) 2χ3 12 The relative dielectric constant ε 33 is assumed to be equal to the dielectric susceptibility η 33 for the rhombohedral compositions, [11] thus, ε 33 = η 33. (13) Substituting Eq. (12) into Eq. (13), the results in the following relation: χ 2 11 χ 2 12 ε 33 = χ χ 11χ (14) 2χ3 12 The spontaneous strain χ 4 can be derived in the hightemperature rhombohedral phase region as follows: x 4 = Q 44 P 2 3. (15) In Eq. (15), Q 44 is the electrostrictive constant for the PZT solid-solution system. For the rhombohedral phase region, we can get d R /a R from geometry relation of the rhombohedral cell, d R a R = cos α R, (16) where d R /a R is defined as the PZT rhombohedral lattice distortion, d R the diagonal of the rhombohedro, and a R the side of the rhombohedron. In the above equations, α R is the rhombohedral angle distortion (see Fig. 1). 3 Two Connection Equations on Dielectric Properties and Lattice Distortion d R /a R in Rhombohedral Phase Region We know the relation between the relative dielectric stiffness χ 11, χ 12 and ferroelectric polarization and ferroelectric dielectric stiffness for PZT high-temperature rhombohedral phase region, [10] χ 11 = 2ε 0 [α 1 + (6α α 12 )P 2 3 Fig. 1 The rhombohedra lattice cell structure. + (15α α α 123 )P 4 3 ], (10) χ 12 = 4ε 0 [α 12 P (4α 112 ) + α 123 )P 4 3 ], (11) where ε 0 is the permittivity of free space, α 1, α 11, α 12, α 111, α 112, and α 123 are defined as ferroelectric dielectric stiffness respectively at constant stress, and P 3 is defined as ferroelectric polarization along c axis. The relation for the dielectric susceptibility coefficient η 33 can be derived When the related system lies in the rhombohedral phase region, [12] the spontaneous strains χ 4 is defined as χ 4 = 1 90 (90 α R). (17) Substitute Eq. (15) into Eq. (17), then we can get the following relation: 1 90 (90 α R) = Q 44 P 2 3. (18)

3 No. 5 Dielectric Properties and Lattice Distortion in Rhombohedral Phase Region and Phase Coexistence Region of 857 In the same way we substitute Eq. (16) into Eq. (18) and obtain 90(1 Q 44 P3 2 ) = cos 1[ 1 ( dr ) 2 ] 3. (19) 6 a R When the PZT system lies in the high-temperature rhombohedral phase region, ferroelectric polarization P 3 must make the elastic Gibbs free energy G minimize. At this region, P3 2 can be expressed as [10] P 2 3 = 1 3ξC ( ζc + ((ζc)2 9α 1 CξC) 1/2 ), (20) where C is Curie constant, α 1 C is the product of the second order tetragonal dielectric stiffness and the Curie constant, ζc = 3(α 11 C +α 12 C) and ξc = 3α 111 C +6α 112 C + α 123 C. Substituting Eq. (20) into Eq. (19), the equation between the lattice distortion d R /a R and composition x can be expressed as 90(1 Q 44 ( ζc + ((ζc) 2 9α 1 CξC) 1/2 ) 1 3ξC = cos 1[(( d ) 2 ) R 1 ] 3. (21) a R 6 In Eq. (21), all physical parameters, e.g., Q 44, ζc, ξc, α 1, and C, can be fitted as functions of composition x by fitting the related experimental data (see next paragraph in this paper). Connecting Eqs. (10), (11), (14), and (20), we can get another equation on the dependent equation between the dielectric constant ε r and composition x, ε 33 = {[(2ε 0 ((T T 0 )/2ε 0 C + (6α 11 C/C + 2α 12 C/C)( ζc + ((ζc) 2 9(T T 0 )ξc/2ε 0 ) 1/2 )/(3ξC) + (15α 111 C/C + 14α 112 C/C + α 123 C/C)( ζc + ((ζc) 2 9(T T 0 )ξc/2ε 0 ) 1/2 ) 2 /(3ξC) 2 ] 2 [(4ε 0 (α 12 C/C ( ζc + ((ζc) 2 9(T T 0 )ξc/2ε 0 ) 1/2 )/(3ξC) + (4α 112 C/C + α 123 C/C) ( ζc + ((ζc) 2 9(T T 0 )ξc/2ε 0 ) 1/2 ) 2 /(3ξC) 2 ] 2 }/{[2ε 0 ((T T 0 )/2ε 0 C + (6α 11 C/C + 2α 12 C/C)( ζc + ((ζc) 2 9(T T 0 )ξc/2ε 0 ) 1/2 ) 2 /(3ξC) + (15α 111 C/C + 14α 112 C/C + α 123 C/C)( ζc + ((ζc) 2 9(T T 0 )ξc/2ε 0 ) 1/2 ) 2 /(3ξC) 2 )] 3 3[2ε 0 ((T T 0 )/2ε 0 C + (6α 11 C/C + 2α 12 C/C)( ζc + ((ζc) 2 9(T T 0 )ξc/2ε 0 ) 1/2 /(3ξC)] + (15α 111 C/C + 14α 112 C/C + α 123 C/C)( ζc + ((ζc) 2 9(T T 0 )ξc/2ε 0 ) 1/2 )] [(4ε 0 (α 12 C/C ( ζc + ((ζc) 2 9(T T 0 )ξc/2ε 0 ) 1/2 )/(3ξC) + (4α 112 C/C + α 123 C/C) ( ζc + ((ζc) 2 9(T T 0 ) ξc/2ε 0 ) 1/2 ) 2 /(3ξC) 2 ] 2 + 2[4ε 0 (α 12 C/C ( ζc + ((ζc) 2 9(T T 0 )ξc/2ε 0 ) 1/2 /(3ξC) + (4α 112 C/C + α 123 C/C) ( ζc + ((ζc) 2 9(T T 0 )ξc/2ε 0 ) 1/2 ) 2 /(3ξC) 2 )] 3 }. (22) In Eq. (22), physical parameters, e.g., α 11 C, α 12 C, α 112 C, α 123 C, ζc, and ξc can be fitted as functions of composition x by fitting the related experimental data (see next paragraph in this paper). If we connect Eqs. (21) and (22) and substitute the values of all parameters into the two equations, the dependent relation between the dielectric constant ε r and the lattice distortion d R /a R can be derived. Now we determine the values of all physical parameters in terms of composition x in Eqs. (21) and (22), e.g., T, T 0, C, α 11 C, α 12 C, α 111 C, α 112 C, α 123 C, Q 44, ζc, and ξc, where the values of T 0, C, α 11 C, and α 111 C in terms of composition x are given above. The parameter values ζc and ξc in terms of composition x have been determined from the related experimental data. Equations (34) and (35) in Ref. [13] have given their fitting values, and we directly quote the equations as follows: ζc = [( x) exp( 12.6x) x ]10 14, (23) ξc = [( x) exp( x) x ] (24) The parameter value Q 44 in terms of composition x also can be determined from the related experimental data. [12] It has been given as Q 44 = x (25) (x 0.5) 2 The parameter value α 12 C has been given, [14] α 12 C = 1 3 ζc α 11C. (26) In Eq. (26), the values of ζc and α 11 C have been given by Eqs. (23) and (24). The parameter values α 112 C and α 123 C to composition x have been determined from the

4 858 ZHANG Duan-Ming, ZHONG Zhi-Cheng, HAN Xiang-Yun, et al. Vol. 43 related experimental data. [11] We can directly quote the equations α 112 C = ( exp( x) x )10 14, (27) α 123 C = ξc 3α 111 C 6α 112 C. (28) In Eq. (28), the parameter values of ξc, α 111 C, and α 112 C have been given by Eqs. (24), (5), and (27). 4 Phase Statistical Distribution Model and Determination of Parameter V T In the coexistence phase region, there are two phases, i.e. the tetragonal phase and the rhombohedral phase. The dielectric properties of the PZT in the phase coexistence region can be calculated considering that the phase coexistence corresponds to a statistical distribution of phases with the same composition. [3] The authors consider that the two phases meet a statistical distribution in the phase coexistence region in this model, and there are two distinct structure phase to the same chemical composition. In the phase coexistence region, the relation between the dielectric properties and the crystal structure of each phase are considered to fit that of each phase outside the coexistence range. The dielectric properties of the PZT material result from the contribution of the two phases. For the dielectric constant of the ceramics in the phase coexistence region, we adopt the following equation, [14] [ ln{vt ε (V T 0.65) T + V T ε (V T 0.65) ε = exp V T 0.65 R } ], (29) where ε is the dielectric constant of the compound with the two phases, V T is the volume fraction of the tetragonal phase, ε T is the tetragonal phase dielectric constant, and ε R is the rhombohedral phase dielectric constant. In Eq. (29), for the same composition in the phase coexistence region, it corresponds to the two distinct structure phases. The contribution of the tetragonal phase to the dielectric constant of the phase coexistence region can be given by Eqs. (1) and (2). The contribution of the rhombohedral phase to the dielectric constant of the phase coexistence region can be given by Eqs. (21) and (22). The parameter V T is the volume fraction of the tetragonal phase. The volume fractions of the phases that coexist in a PZT ceramics can be directly determined by X-ray diffraction. Their calculation and results were presented. [8] We can directly use the results of the relation between f T and composition x. Figure 4 in Ref. [8] is about molar fractions of the tetragonal phase and the rhombohedral phase respectively in PZT solid solutions. In this paper, we only quote the part of molar fractions of the tetragonal phase sintered at 1250 C for 2 h from Fig. 4 in Ref. [8]. Its result is shown in Fig. 2 in this paper. We fit the curve about molar fractions of the tetragonal phase f T sintered at 1250 C for 2 h. The fitting value f T to composition x is as f T = ( ) 2(x ) / (30) π/2 We know the relation between f T and V T, [8] Thence V T f T. (31) V T + V R f T + f R V T f T. (32) We obtain the relation between parameter V T and composition x, V T = ( ) 2(x ) / (33) π/2 Fig. 2 Molar fractions of the tetragonal phases, f T, in PZT solid solutions. The points correspond to experimental results and the curve is fitted using the statistical model: sintered ceramics at 1250 C for 2 h. 5 Comparing Theoretical Results with Related Experimental Data In Eq. (29), for the same composition x in the coexistence phase region, contribution of each phase to the dielectric constant of the compound in the phase coexistence region has been expressed as a function of the lattice distortion. The values of V T also have been fitted as a function of composition x. Thence, connecting Eqs. (1), (2), (21), (22), and (29) and substituting the values of all parameters into the five equations, the relation between the dielectric constant of the compound with the two phases and the lattice distortion (c/a, d R /a R ) can be attained.

5 No. 5 Dielectric Properties and Lattice Distortion in Rhombohedral Phase Region and Phase Coexistence Region of 859 In Fig. 2 of Ref. [3] it showed the evolution of the lattice distortions of the rhombohedral phase, d R /a R, and of the tetragonal phase, c/a, with the composition of PZT ceramics sintered at 1250 C for 2 h. That is Fig. 3 in this paper, where the dots are the experimental results measured in ceramics sintered at 1250 C for 2 h. Thence, we can attain the experimental results between dielectric permittivity of PZT and the lattice distortion (c/a, d R /a R measured in ceramics sintered at 1250 C for 2 h from Figs. 3 and 4. The experimental data measured in the ceramics sintered at 1250 C for 2 h are listed in Table 1. Table 1 The experimental data measured in ceramics sintered at 1250 C for 2 h. Dielectric permittivity ε r c/a d R/a R Fig. 3 Evolution of the unit cells distortions of the rhombohedral phase, d R/a R, and of the tetragonal phase c/a, with the composition of PZT ceramics sintered at 1250 C for 2 h. For the PZT samples in phase coexistence region in Ref. [3], compositions x of PZT samples prepared all belong to the second order phase transition, then, T 0 = T C. Because the dielectric permittivity was measured at room temperature, we can reasonably assume T = 25 C. We can fit results on the relation between the dielectric constant ε r and the lattice distortion (c/a, d R /a R ) in phase coexistence region based on Eqs. (1), (2), (21), (22), and (29). The asterisks in Fig. 5 are experimental data measured in ceramics sintered at 1250 C for 2 h. The successive curve is the fitted results. The relative warp is at range from 0.02 to It indicates that the phase statistical distribution model is suitable to describe the relation between the dielectric properties and the lattice distortion in the phase coexistence region of PZT ceramics. Fig. 4 Dielectric permittivity of PZT ceramics, measured at 100 khz at room temperature and sintered at 1250 C for 2 h. Figure 9 in Ref. [3] is about the dielectric permittivity of PZT, measured at 100 khz at room temperature and, with values calculated by supposing that both phases have the same composition, but different structures inside the phase coexistence interval. The dots are the experimental results measured in ceramics sintered at 1250 C for 2 h. In this paper, as shown in Fig. 4, we only quote the experimental results measured in ceramics sintered at 1250 C for 2 h from Fig. 9 in Ref. [3]. Fig. 5 The dielectric constant ε r versus the lattice distortion c/a and d R/a R: the curve is the calculated results; the asterisks are the experimental results measured in ceramics sintered at 1250 C for 2 h. It can be found that the theoretical results are in good agreement with the related experimental results.

6 860 ZHANG Duan-Ming, ZHONG Zhi-Cheng, HAN Xiang-Yun, et al. Vol Conclusion In summary, we have investigated in detail the relation between the dielectric properties and the lattice distortion for PZT ceramics from the lattice structure point of view that few people have discussed. Our theoretical results agree with the related experimental one perfectly in the phase coexistence region. We have discussed the relation between the dielectric constant ε r and the lattice distortion for the PZT in the rhombohedral phase region and phase coexistence region. Associating our previous work, [9] this picture clearly describes overall changing process of the dielectric constant ε r with the lattice distortion from the PZT rhombohedral phase region transition to the tetragonal phase region, via the phase coexistence region. References [1] S.L. Fu, S.Y. Cheng, and C.C. Wei, Ferroelectrics 67 (1986) 93. [2] B. Jaffc, R. Cook, and H. Jaffc, Piezoelectric Ceramics, Academic Press, London (1971). [3] M.R. Senos and P.Q. Mantas, J. European Ceramic Society 20 (2000) 321. [4] L. Benguigui, Solid State Commun. 11 (1972) 825. [5] V.A. Isupov, Sov. Phys. Solid State 22 (1980) 98. [6] W. Cao and L. Cross, Phys. Rev. B47 (1993) [7] W. Cao and L. Cross, J. Appl. Phys. 73 (1993) [8] M.R. Soars, A.M.R. Senos, and P.Q. Mantas, J. European Ceramic Society 19 (1999) [9] D.M. Zhang, W.S. Yan, et al., Acta Phys. Sin. (in Chinese) 53 (2004) [10] M.J. Haun, E. Furman, S.J. Jang, and L.E. Cross, Ferroelectrics 99 (1989) 13. [11] M.J. Haun, E. Furman, S.J. Jang, and L.E. Cross, Ferroelectrics 99 (1989) 45. [12] M.J. Haun, Z.Q. Zhuang, E. Furman, S.J. Jang, and L.E. Cross, J. Am. Ceram. Soc. 72 (1989) [13] M.J. Haun, E. Furman, S.J. Jang, and L.E. Cross, Ferroelectrics 99 (1989) 27. [14] K. Wakino, J. Am. Ceram. Soc. 76 (1993) 2588.