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Dielectric polarization and strain behavior of Ba(Ti0.92Zr0.08)O3 single crystals
December 2002
Materials Letters 57 (2002) 349 – 354 www.elsevier.com/locate/matlet
Dielectric polarization and strain behavior of Ba(Ti0.92Zr0.08)O3 single crystals Zhi Yu*, Ruyan Guo, A.S. Bhalla Materials Research Institute, The Pennsylvania State University, University Park, PA 16802, USA Received 23 February 2002; accepted 25 February 2002
Abstract The ferroelectric polarization and strain behavior was studied for Ba(Ti0.92Zr0.08)O3 single crystals with a rhombohedral symmetry at room temperature. The anisotropic dielectric spectra obtained from different orientation cuts showed suppression of permittivity maximum (emax) with increasing frequencies. However, no significant Tc-shift with increasing frequencies was observed. The polarization data obtained from the hysteresis loops measured in a wide temperature range indicated a relaxationlike behavior as the remnant polarization exists much beyond Tc. The unipolar strain level is 0.17% at 55 kV/cm and the piezoelectric strain coefficient d33 f 850 pC/N measured for the h001i oriented Ba(Ti0.92Zr0.08)O3 single crystals are promising values in a lead-free material for the piezoelectric transducer applications. D 2002 Published by Elsevier Science B.V. PACS: 77.84.Dy; 77.65.Bn Keywords: Ferroelectricity; Piezoelectricity; BaTiO3 solid solutions
1. Introduction Since the discovery of the electric field induced ultra high strain (>1%) in lead-based single crystals (e.g., Pb(Zn 1/3 Nb 2/3 )O 3 and Pb(Zn 1/3 Nb 2/3 )O 3 – PbTiO3 [1,2], much work has been focused on the growth and characterization of related lead-based single crystals as well as the understanding of the mechanisms governing their ferroelectric and strain behavior, including its relation to the phase transitions under electric fields and precise phase diagrams in a wide temperature range [3 – 6]. Due to the
*
Corresponding author.
toxicity of lead compounds, there is an increasing desire to develop lead-free materials with high strain capability. One of the most important and promising lead-free piezoelectric materials is barium titanate (BaTiO3) and its solid solutions. The electrostrictive strain levels in BaTiO3 single crystal can reach f 1% at high fields at room temperature [7]. Unfortunately, such high strain is accompanied by large hysteresis, which restricts practical applications. Recent work on BTiO3-based solid solutions has been performed in seeking lead-free transducer materials [8– 12], which show promising strain levels and high piezoelectric activity in Ba(Ti1 xZrx)O3 solid solutions. Further studies are needed to optimize the material perform-
0167-577X/02/$ - see front matter D 2002 Published by Elsevier Science B.V. PII: S 0 1 6 7 - 5 7 7 X ( 0 2 ) 0 0 7 8 9 - 9
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cooling rate of 1 jC/min. The hysteresis loop and strain as a function of electric fields were measured in the temperature range from 10 to 130 jC by a modified Sawyer– Tower circuit and a linear variable differential transducer (LVDT) driven by a lock-in amplifier. The voltage was supplied using a Trek 609C-6 high voltage DC amplifier.
3. Results and discussion 3.1. Dielectric behavior The temperature dependence of dielectric constant (e) and loss (tand) of Ba(Ti0.92Zr0.08)O3 single crystals
Fig. 1. Temperature dependence of e and tand for Ba(Ti0.92Zr0.08)O3 single crystals along h110 i and h 001i orientations at 1, 10, and 100 kHz.
ance and to understand the piezoelectric and electrostrictive mechanism in the BaTiO3-based system. In this paper, we report the dielectric, ferroelectric and strain behavior of Ba(Ti0.92Zr0.08)O3 single crystals with a rhombohedral symmetry at room temperature as a lead-free material for piezoelectric applications.
2. Experimental procedure Ba(Ti0.92Zr0.08)O3 single crystals were grown by the laser heated pedestal growth (LHPG) technique using properly selected seeds [12]. The crystal orientation was identified by the Laue back reflection technique using a Northstar real-time orientation system. Gold electrodes were sputtered on the crystal specimens. The complex-dielectric constant of the samples was measured using an HP4284A LCR meter in the temperature range 75 to + 160 jC with a
Fig. 2. Polarization ( P) versus electric field (E) for Ba(Ti0.92Zr0.08)O3 single crystals along h110 i and h 001i orientations.
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Although the permittivity maximum (emax) is suppressed from 15,300 to 13,300 with increasing frequency from 1 to 100 kHz seen in Fig. 1. No noticeable Tc shift is observed, i.e., no visible ‘‘ferroelectric relaxor behavior’’ is shown in the dielectric spectra. This is different from Ce-doped BaTiO3 in which a clear ferroelectric ralaxor behavior was observed in samples with 6 at.% Ce doping [13]. In the present work, the suppression of emax could be attributed to domain wall motions in the single crystals in a multidomain state [14]. As shown in the temperature dependence of e, with increasing Zr concentration, Tc of Zr-doped BaTiO3 decreases, but T2 and T3 increase, i.e., Zr-substitution at Ti-sites in BaTiO3 pinches the three phase transition temperatures of pure BaTiO3 [15]. From the dielectric spectra, Ba(Ti0.92Zr0.08)O3 composition shows correspondent characteristics of a rhombohedral point group symmetry at room temperature. It is known that excellent piezoelectric performance can be obtained in a rhombohedral phase in many lead-based relaxor ferroelectrics as mentioned earlier [2 – 4]. However, in our previous study it was shown that a h 001i oriented Ba(Ti1 xZrx)O3 crystal of the orthorhombic phase has a better performance in piezoelectric actuation. In this work, Further details on the polarization and strain behavior of Ba(Ti0.92Zr0.08)O3 with a rhombohedral symmetry are the focus of this study.
Fig. 3. (a,b) Polarization versus electric field in the temperature range from 10 to 120 jC for Ba(Ti0.92Zr0.08)O3 single crystals.
along pseudocubic h110i and h001i orientations is shown in Fig. 1. The dielectric properties show the anisotropic behavior along different orientations. The transition temperature (Tc f 102 jC) from cubic paraelectric to tetragonal ferroelectric and T2 = 71 jC for the transition from tetragonal to orthorhombic were observed. The lowest transition from orthorhombic to rhombohedral is not clearly seen from the curve of e versus temperature, but was evident as seen in tand, which shows T3 = 30 jC.
Fig. 4. Temperature dependence of remnant polarization ( Pr) and coercive field (Ec) for Ba(Ti0.92Zr0.08)O3 single crystals. The values were obtained from Fig. 3.
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dimension restriction of a sample grown by LHPG seeded by h110 i and h001i SrTiO3 seeds. According to the tensor relationships, the spontaneous polarization along the h111i direction can be derived to be about 29 AC/cm2 based on the experimental data measured above. Some hysteresis loops measured in the temperatures range of 10 to + 120 jC are shown in Fig. 3. Obviously at lower temperatures, near saturated loops were obtained, however, at higher temperatures, only slim loops were observed. The temperature dependences of the remnant polarization ( Pr) and the coercive field (Ec) of the B (Ti0.92Zr0.08)O3 single crystals were also studied. It can be seen from Fig. 4 that a maximum for Pr occurs at f 35 jC, which coincides with the third phase transition temperature (T3) obtained from the dielectric spectra as shown in Fig. 1. From the temperature dependence of e, Tc is f 102 jC, corresponding to a sharp decrease in Pr at f 100 jC. However, Pr does not reach zero at 102 jC, but tails to zero near 130 jC (not shown). This behavior is different from that of pure BaTiO3 due to its solidsolution nature. In the Ba(Ti0.92Zr0.08)O3 solid solution, although the shift of Tc with increasing frequency is not evident from the dielectric spectra, the polarization as a function of temperature along with the e dependence on frequencies suggested a relaxation-like behavior. Fig. 5. Bipolar stain versus E for Ba(Ti0.92Zr0.08)O3 single crystals along h110 i and h001i orientations.
3.2. Ferroelectric polarization Fig. 2 shows hysteresis loops of Ba(Ti0.92Zr0.08)O3 single crystals with different orientations measured at room temperature. The saturation polarization ( Ps) 22 and 17 AC/cm2 along h110 i and h001i directions, respectively, and the remnant polarization ( Pr) 17 and 12 AC/cm2 along h110 i and h001i directions, respectively, were measured. Both the saturation and remnant polarization values are larger in h110 i direction than those in the h001i direction. Polarization data in a sample with h111i orientation would be very meaningful, since it is a polar direction of a rhombohedral phase. Unfortunately, a h111i oriented sample is not easy to obtain due to the
Fig. 6. Unipolar strain versus E for Ba(Ti0.92Zr0.08)O3 single crystals along h110 i and h001i orientations.
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Fig. 7. Piezoelectric coefficient d33 versus E for Ba(Ti0.92Zr0.08)O3 single crystals along h110 i and h001i orientations.
3.3. Strain and piezoelectric properties Bipolar strain as a function of electric field (E) for Ba (Ti0.92Zr0.08)O3 single crystal is shown in Fig. 5. Butterfly-shaped strain versus electric fields can be observed for different orientations. The hysteresis behavior is different for the samples along h110 i and h001i directions. The difference in the strain behavior might be attributed to different domain configurations [2]. The h001i oriented crystal exhibits less hysteresis and higher strain levels. This was also observed by the unipolar strain behavior under higher electric fields, as shown in Fig. 6. The strain levels 0.17% at 55 kV/cm in the h001i direction and 0.10% in h110 i direction were measured. The piezoelectric strain coefficient d33 as a function of electric field can be obtained by calculating the slope of the strain versus E curve. The d33 coefficients along h001i and h110 i orientations are plotted in Fig. 7. It can be seen that d33 is higher than 800 pC/N for h001i oriented crystal below 10 kV/cm and higher than 250 pC/N up to 20 kV/cm. For h110 i orientation, d33 is about 440 pC/N at low fields ( < 5 kV/cm) and then decreases to a lower value of about 200 pC/N at 20 kV/cm. The ferroelectric and strain properties show that the remnant polarization is smaller, but strain level is higher for h001i orientation than those for h110 i ori-
ented Ba(Ti0.92Zr0.08)O3 crystal with a rhombohedral symmetry.
4. Conclusions The dielectric, ferroelectric and piezoelectric behavior of Ba(Ti0.92Zr0.08)O3 single crystals of different orientations h001i and h110 i with rhombohedral symmetry was studied. The anisotropic dielectric spectra revealed details of the pinched phase evolution in this crystal. The suppression of emax values with increasing frequencies and the residual polarization above Tc implies a relaxation behavior. The lack of significant Tc shift with increasing frequencies, however, indicates a regular ferroelectric crystal with long range order preserved. The respectable strain level with small hysteresis and high piezoelectric coefficient found in Ba(Ti0.92Zr0.08)O3 single crystals are reported to be of potential for practical lead-free piezoelectric transducer applications.
Acknowledgements One of the authors (Zhi Yu) would like to thank Dr. Chen Ang for the stimulating discussion during the
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course of this work. This work is supported by the Office of Naval Research under grant No. N00014-981-0527.
References [1] J. Kuwata, K. Uchino, S. Nomura, Jpn. J. Phys. Part 1 21 (1982) 1298. [2] S.E. Park, T.R. Shrout, J. Appl. Phys. 82 (1997) 1804. [3] D.S. Paik, S.E. Park, S. Wada, S.F. Liu, T.R. Shrout, J. Appl. Phys. 85 (1999) 1080. [4] S.F. Liu, S.E. Park, T.R. Shrout, L.E. Cross, J. Appl. Phys. 85 (1999) 2810. [5] M.K. Durbin, E.W. Jacobs, J.C. Hicks, S.E. Park, Appl. Phys. Lett. 74 (1999) 2848.
[6] D.E. Cox, B. Noheda, G. Shirane, Y. Uesu, K. Fujishiro, Y. Yamada, arXiv: cond-mat/0102457, v1 25 Feb. 2001. [7] S.E. Park, S. Wada, L.E. Cross, T.R. Shrout, J. Appl. Phys. 86 (1999) 2746. [8] P.W. Rherig, S.E. Park, S.T. McKinstry, G.L. Messing, B. Jones, T.M. Shrout, J. Appl. Phys. 86 (1999) 1657. [9] Z. Yu, R. Guo, A.S. Bhalla, J. Appl. Phys. 88 (2000) 410. [10] Z. Yu, R. Guo, A.S. Bhalla, Appl. Phys. Lett. 77 (2000) 1535. [11] Z. Yu, R. Guo, A.S. Bhalla, Ferroelectr. Lett. 27 (5/6) (2000) 113. [12] Z. Yu, R. Guo, A.S. Bhalla, J. Cryst. Growth 233 (2001) 460. [13] C. Ang, Z. Yu, Z. Jing, Phys. Rev., B61 (2000) 957 – 961. [14] Y.M. Poplavko, V.G. Tsykalov, V.I. Molchanov, Sov. Phys.Solid State 10 (1969) 2708. [15] B. Jaffe, W. Cook, H. Jaffe, Piezoelectric Ceramics, Academic Press, London, 1971.
Materials Letters 57 (2002) 349 – 354 www.elsevier.com/locate/matlet
Dielectric polarization and strain behavior of Ba(Ti0.92Zr0.08)O3 single crystals Zhi Yu*, Ruyan Guo, A.S. Bhalla Materials Research Institute, The Pennsylvania State University, University Park, PA 16802, USA Received 23 February 2002; accepted 25 February 2002
Abstract The ferroelectric polarization and strain behavior was studied for Ba(Ti0.92Zr0.08)O3 single crystals with a rhombohedral symmetry at room temperature. The anisotropic dielectric spectra obtained from different orientation cuts showed suppression of permittivity maximum (emax) with increasing frequencies. However, no significant Tc-shift with increasing frequencies was observed. The polarization data obtained from the hysteresis loops measured in a wide temperature range indicated a relaxationlike behavior as the remnant polarization exists much beyond Tc. The unipolar strain level is 0.17% at 55 kV/cm and the piezoelectric strain coefficient d33 f 850 pC/N measured for the h001i oriented Ba(Ti0.92Zr0.08)O3 single crystals are promising values in a lead-free material for the piezoelectric transducer applications. D 2002 Published by Elsevier Science B.V. PACS: 77.84.Dy; 77.65.Bn Keywords: Ferroelectricity; Piezoelectricity; BaTiO3 solid solutions
1. Introduction Since the discovery of the electric field induced ultra high strain (>1%) in lead-based single crystals (e.g., Pb(Zn 1/3 Nb 2/3 )O 3 and Pb(Zn 1/3 Nb 2/3 )O 3 – PbTiO3 [1,2], much work has been focused on the growth and characterization of related lead-based single crystals as well as the understanding of the mechanisms governing their ferroelectric and strain behavior, including its relation to the phase transitions under electric fields and precise phase diagrams in a wide temperature range [3 – 6]. Due to the
*
Corresponding author.
toxicity of lead compounds, there is an increasing desire to develop lead-free materials with high strain capability. One of the most important and promising lead-free piezoelectric materials is barium titanate (BaTiO3) and its solid solutions. The electrostrictive strain levels in BaTiO3 single crystal can reach f 1% at high fields at room temperature [7]. Unfortunately, such high strain is accompanied by large hysteresis, which restricts practical applications. Recent work on BTiO3-based solid solutions has been performed in seeking lead-free transducer materials [8– 12], which show promising strain levels and high piezoelectric activity in Ba(Ti1 xZrx)O3 solid solutions. Further studies are needed to optimize the material perform-
0167-577X/02/$ - see front matter D 2002 Published by Elsevier Science B.V. PII: S 0 1 6 7 - 5 7 7 X ( 0 2 ) 0 0 7 8 9 - 9
350
Z. Yu et al. / Materials Letters 57 (2002) 349–354
cooling rate of 1 jC/min. The hysteresis loop and strain as a function of electric fields were measured in the temperature range from 10 to 130 jC by a modified Sawyer– Tower circuit and a linear variable differential transducer (LVDT) driven by a lock-in amplifier. The voltage was supplied using a Trek 609C-6 high voltage DC amplifier.
3. Results and discussion 3.1. Dielectric behavior The temperature dependence of dielectric constant (e) and loss (tand) of Ba(Ti0.92Zr0.08)O3 single crystals
Fig. 1. Temperature dependence of e and tand for Ba(Ti0.92Zr0.08)O3 single crystals along h110 i and h 001i orientations at 1, 10, and 100 kHz.
ance and to understand the piezoelectric and electrostrictive mechanism in the BaTiO3-based system. In this paper, we report the dielectric, ferroelectric and strain behavior of Ba(Ti0.92Zr0.08)O3 single crystals with a rhombohedral symmetry at room temperature as a lead-free material for piezoelectric applications.
2. Experimental procedure Ba(Ti0.92Zr0.08)O3 single crystals were grown by the laser heated pedestal growth (LHPG) technique using properly selected seeds [12]. The crystal orientation was identified by the Laue back reflection technique using a Northstar real-time orientation system. Gold electrodes were sputtered on the crystal specimens. The complex-dielectric constant of the samples was measured using an HP4284A LCR meter in the temperature range 75 to + 160 jC with a
Fig. 2. Polarization ( P) versus electric field (E) for Ba(Ti0.92Zr0.08)O3 single crystals along h110 i and h 001i orientations.
Z. Yu et al. / Materials Letters 57 (2002) 349–354
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Although the permittivity maximum (emax) is suppressed from 15,300 to 13,300 with increasing frequency from 1 to 100 kHz seen in Fig. 1. No noticeable Tc shift is observed, i.e., no visible ‘‘ferroelectric relaxor behavior’’ is shown in the dielectric spectra. This is different from Ce-doped BaTiO3 in which a clear ferroelectric ralaxor behavior was observed in samples with 6 at.% Ce doping [13]. In the present work, the suppression of emax could be attributed to domain wall motions in the single crystals in a multidomain state [14]. As shown in the temperature dependence of e, with increasing Zr concentration, Tc of Zr-doped BaTiO3 decreases, but T2 and T3 increase, i.e., Zr-substitution at Ti-sites in BaTiO3 pinches the three phase transition temperatures of pure BaTiO3 [15]. From the dielectric spectra, Ba(Ti0.92Zr0.08)O3 composition shows correspondent characteristics of a rhombohedral point group symmetry at room temperature. It is known that excellent piezoelectric performance can be obtained in a rhombohedral phase in many lead-based relaxor ferroelectrics as mentioned earlier [2 – 4]. However, in our previous study it was shown that a h 001i oriented Ba(Ti1 xZrx)O3 crystal of the orthorhombic phase has a better performance in piezoelectric actuation. In this work, Further details on the polarization and strain behavior of Ba(Ti0.92Zr0.08)O3 with a rhombohedral symmetry are the focus of this study.
Fig. 3. (a,b) Polarization versus electric field in the temperature range from 10 to 120 jC for Ba(Ti0.92Zr0.08)O3 single crystals.
along pseudocubic h110i and h001i orientations is shown in Fig. 1. The dielectric properties show the anisotropic behavior along different orientations. The transition temperature (Tc f 102 jC) from cubic paraelectric to tetragonal ferroelectric and T2 = 71 jC for the transition from tetragonal to orthorhombic were observed. The lowest transition from orthorhombic to rhombohedral is not clearly seen from the curve of e versus temperature, but was evident as seen in tand, which shows T3 = 30 jC.
Fig. 4. Temperature dependence of remnant polarization ( Pr) and coercive field (Ec) for Ba(Ti0.92Zr0.08)O3 single crystals. The values were obtained from Fig. 3.
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Z. Yu et al. / Materials Letters 57 (2002) 349–354
dimension restriction of a sample grown by LHPG seeded by h110 i and h001i SrTiO3 seeds. According to the tensor relationships, the spontaneous polarization along the h111i direction can be derived to be about 29 AC/cm2 based on the experimental data measured above. Some hysteresis loops measured in the temperatures range of 10 to + 120 jC are shown in Fig. 3. Obviously at lower temperatures, near saturated loops were obtained, however, at higher temperatures, only slim loops were observed. The temperature dependences of the remnant polarization ( Pr) and the coercive field (Ec) of the B (Ti0.92Zr0.08)O3 single crystals were also studied. It can be seen from Fig. 4 that a maximum for Pr occurs at f 35 jC, which coincides with the third phase transition temperature (T3) obtained from the dielectric spectra as shown in Fig. 1. From the temperature dependence of e, Tc is f 102 jC, corresponding to a sharp decrease in Pr at f 100 jC. However, Pr does not reach zero at 102 jC, but tails to zero near 130 jC (not shown). This behavior is different from that of pure BaTiO3 due to its solidsolution nature. In the Ba(Ti0.92Zr0.08)O3 solid solution, although the shift of Tc with increasing frequency is not evident from the dielectric spectra, the polarization as a function of temperature along with the e dependence on frequencies suggested a relaxation-like behavior. Fig. 5. Bipolar stain versus E for Ba(Ti0.92Zr0.08)O3 single crystals along h110 i and h001i orientations.
3.2. Ferroelectric polarization Fig. 2 shows hysteresis loops of Ba(Ti0.92Zr0.08)O3 single crystals with different orientations measured at room temperature. The saturation polarization ( Ps) 22 and 17 AC/cm2 along h110 i and h001i directions, respectively, and the remnant polarization ( Pr) 17 and 12 AC/cm2 along h110 i and h001i directions, respectively, were measured. Both the saturation and remnant polarization values are larger in h110 i direction than those in the h001i direction. Polarization data in a sample with h111i orientation would be very meaningful, since it is a polar direction of a rhombohedral phase. Unfortunately, a h111i oriented sample is not easy to obtain due to the
Fig. 6. Unipolar strain versus E for Ba(Ti0.92Zr0.08)O3 single crystals along h110 i and h001i orientations.
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Fig. 7. Piezoelectric coefficient d33 versus E for Ba(Ti0.92Zr0.08)O3 single crystals along h110 i and h001i orientations.
3.3. Strain and piezoelectric properties Bipolar strain as a function of electric field (E) for Ba (Ti0.92Zr0.08)O3 single crystal is shown in Fig. 5. Butterfly-shaped strain versus electric fields can be observed for different orientations. The hysteresis behavior is different for the samples along h110 i and h001i directions. The difference in the strain behavior might be attributed to different domain configurations [2]. The h001i oriented crystal exhibits less hysteresis and higher strain levels. This was also observed by the unipolar strain behavior under higher electric fields, as shown in Fig. 6. The strain levels 0.17% at 55 kV/cm in the h001i direction and 0.10% in h110 i direction were measured. The piezoelectric strain coefficient d33 as a function of electric field can be obtained by calculating the slope of the strain versus E curve. The d33 coefficients along h001i and h110 i orientations are plotted in Fig. 7. It can be seen that d33 is higher than 800 pC/N for h001i oriented crystal below 10 kV/cm and higher than 250 pC/N up to 20 kV/cm. For h110 i orientation, d33 is about 440 pC/N at low fields ( < 5 kV/cm) and then decreases to a lower value of about 200 pC/N at 20 kV/cm. The ferroelectric and strain properties show that the remnant polarization is smaller, but strain level is higher for h001i orientation than those for h110 i ori-
ented Ba(Ti0.92Zr0.08)O3 crystal with a rhombohedral symmetry.
4. Conclusions The dielectric, ferroelectric and piezoelectric behavior of Ba(Ti0.92Zr0.08)O3 single crystals of different orientations h001i and h110 i with rhombohedral symmetry was studied. The anisotropic dielectric spectra revealed details of the pinched phase evolution in this crystal. The suppression of emax values with increasing frequencies and the residual polarization above Tc implies a relaxation behavior. The lack of significant Tc shift with increasing frequencies, however, indicates a regular ferroelectric crystal with long range order preserved. The respectable strain level with small hysteresis and high piezoelectric coefficient found in Ba(Ti0.92Zr0.08)O3 single crystals are reported to be of potential for practical lead-free piezoelectric transducer applications.
Acknowledgements One of the authors (Zhi Yu) would like to thank Dr. Chen Ang for the stimulating discussion during the
354
Z. Yu et al. / Materials Letters 57 (2002) 349–354
course of this work. This work is supported by the Office of Naval Research under grant No. N00014-981-0527.
References [1] J. Kuwata, K. Uchino, S. Nomura, Jpn. J. Phys. Part 1 21 (1982) 1298. [2] S.E. Park, T.R. Shrout, J. Appl. Phys. 82 (1997) 1804. [3] D.S. Paik, S.E. Park, S. Wada, S.F. Liu, T.R. Shrout, J. Appl. Phys. 85 (1999) 1080. [4] S.F. Liu, S.E. Park, T.R. Shrout, L.E. Cross, J. Appl. Phys. 85 (1999) 2810. [5] M.K. Durbin, E.W. Jacobs, J.C. Hicks, S.E. Park, Appl. Phys. Lett. 74 (1999) 2848.
[6] D.E. Cox, B. Noheda, G. Shirane, Y. Uesu, K. Fujishiro, Y. Yamada, arXiv: cond-mat/0102457, v1 25 Feb. 2001. [7] S.E. Park, S. Wada, L.E. Cross, T.R. Shrout, J. Appl. Phys. 86 (1999) 2746. [8] P.W. Rherig, S.E. Park, S.T. McKinstry, G.L. Messing, B. Jones, T.M. Shrout, J. Appl. Phys. 86 (1999) 1657. [9] Z. Yu, R. Guo, A.S. Bhalla, J. Appl. Phys. 88 (2000) 410. [10] Z. Yu, R. Guo, A.S. Bhalla, Appl. Phys. Lett. 77 (2000) 1535. [11] Z. Yu, R. Guo, A.S. Bhalla, Ferroelectr. Lett. 27 (5/6) (2000) 113. [12] Z. Yu, R. Guo, A.S. Bhalla, J. Cryst. Growth 233 (2001) 460. [13] C. Ang, Z. Yu, Z. Jing, Phys. Rev., B61 (2000) 957 – 961. [14] Y.M. Poplavko, V.G. Tsykalov, V.I. Molchanov, Sov. Phys.Solid State 10 (1969) 2708. [15] B. Jaffe, W. Cook, H. Jaffe, Piezoelectric Ceramics, Academic Press, London, 1971.