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Brillouin scattering study on ferroelectric single crystal Sr2Ta2O7
Materials Science and Engineering A 442 (2006) 35–38
Brillouin scattering study on ferroelectric single crystal Sr2Ta2O7 Anwar Hushur ∗ , Seiji Kojima Institute of Materials Science, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan Received 1 August 2005; received in revised form 22 December 2005; accepted 12 January 2006
Abstract Acoustic properties of strontium tantalite (Sr2 Ta2 O7 ) single crystals with perovskite-slab structure were investigated by micro-Brillouin scattering. The two elastic stiffness constants c22 and c33 were determined in the temperature range between −140 ◦ C and 300 ◦ C. The elastic constant c22 shows marked continuous softening on approaching phase transition temperature To around 167 ◦ C from both sides, indicating ferroelastic and nearly second-order nature of this phase transition. The values of c22 can be well reproduced by the Landau theory including bilinear coupling (linear in strain, quadratic in order parameter) between order parameter and the strain. Hypersonic attenuation increases markedly around To , which seems to reflect not only bilinear coupling between the order parameter and the strain in the ferroelastic phase, but also electrostrictive coupling in the paraelectric phase. © 2006 Elsevier B.V. All rights reserved. Keywords: Soft mode; Structural phase transition; Brillouin scattering; Sr2 Ta2 O7 ; Ferroelectrics
1. Introduction Strontium tantalate (Sr2 Ta2 O7 ) and strontium niobate (Sr2 Nb2 O7 ) belong to the A2 B2 O7 -type ferroelectrics with a perovskite-slab structure [1]. Their solid solutions are suitable for use as ferroelectric materials in the field of electronics as a candidate for ferroelectric random access memories (FeRAM), because these materials have a low dielectric constant, low coercive field and high heat-resistance. Recently, its lead-free character has attracted much attention as a green material [2]. Smolenskii et al. [3] first reported the dielectric anomaly in ceramic Sr2 Ta2 O7 samples between −84 ◦ C and −55 ◦ C, and the ferroelectricity was confirmed below this temperature later by Nanamatsu et al. [4]. They observed a large dielectric anomaly along the c-axis at −107 ◦ C (Tc ). Akishige and Ohi [5] have reported a low frequency dielectric dispersion in the temperature range between −223 ◦ C and −163 ◦ C. Difference of Tc was found and was explained by different Nb content in samples. The crystal structure was determined from the three-dimensional Xray diffraction data by Ishizawa et al. [1]. Soft optic phonon responsible for the structural phase transition at 170 ◦ C has been observed by Kojima et al. [6] using Raman scattering study. The high-temperature phase transition was also observed by electron
∗
Corresponding author. Tel.: +81 29 853 5278; fax: +81 29 853 5262. E-mail address: [email protected] (A. Hushur).
0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.01.142
microscopy and diffraction [7]. Previous studies revealed that Sr2 Ta2 O7 is orthorhombic structure with space group Cmcm in the high-temperature phase, in result of cooling the paraelectric phase transforms into a non-polar super lattice structure of space group P21 /m, characterized by the tilting of TaO6 octahedra and that the lattice constant along the a-axis doubles at about 170 ◦ C. It is important to note that the monoclinic symmetry is not caused by a distortion of the orthorhombic paraelectric phase but induced by the tilting of TaO6 octahedra. This requires to change into a new unit cell. This transition is displacive-type. Sr2 Ta2 O7 is paraelectric and ferroelastic at room temperature, undergoes a ferroelectric phase transition at −100 ◦ C from P21 /m to the polar phase P21 . Ito et al. [8] observed a soft mode associated with the ferroelectric phase transition by Raman scattering. To understand the phase transition mechanism at To and the elastic property, it is useful to apply Brillouin light scattering technique. Since Brillouin frequency shifts and full-width at half-maximum (FWHM) are very sensitive to phase transitions, it provides fundamental information on interatomic forces, and the structure and dynamics of solids related to phase transitions [9–11]. In the present work, Brillouin scattering studies of Sr2 Ta2 O7 single crystals in the temperature range −140 ◦ C to 300 ◦ C are reported. The temperature dependence of two different longitudinal acoustic (LA) modes has been measured. This allowed for the determination of the temperature changes of two independent elastic-stiffness constant for the both Cmcm and P21 /m phases. A theoretical model using the Landau theory is presented, from
36
A. Hushur, S. Kojima / Materials Science and Engineering A 442 (2006) 35–38
which good agreement with the experimental results is obtained and the nature of the phase transition clearly defined. 2. Experimental The sample used in the Brillouin scattering was a 1.5 mm × 1.7 mm × 3.3 mm rectangular parallelepiped with edges parallel to the a-, b-, and c-axes with respect to the orthorhombic coordinates. A diode-pumped solid-state laser (DPSS532) was used to excite the sample with a wave length of 532 nm and power of 150 mW. The scattered light was analyzed at backward scattering geometry with a six-pass tandem Fabry–Perot interferometer of high contrast. A conventional photon-counting system and a multichannel analyzer were used to detect and average the signal. The sample cell with X–Y adjustment was put on the stage of an optical microscope with a focal point of 1–2 m. The sample was put in a cryostat cell (THMS 600) with temperatures varying from −190 ◦ C to 600 ◦ C and with a stability of ±0.1 K [10,12]. We measured the Brillouin scattering spectra at three different orientations with phonons propagating along the a-, b- and c-axes. 3. Results and discussion Brillouin scattering spectra were measured with a free spectral range (FSR) of 60 GHz at the b(c, c + a)b¯ and c(b, a + b)¯c geometry. Typical Brillouin spectra are shown in Figs. 1 and 2 with a free spectral range of 60 GHz using [0 1 0] and [0 0 1] directions. Only one LA mode is observed at both high and low temperatures, consistent with the selection rule of orthorhombic symmetry at the backward scattering geometry [13]. The intensity of the LA mode increased with the increase of the temperature, it disappeared below −113 ◦ C which is, probably, due to the formation of fine ferroelectric domains. To obtain the temperature dependence of elastic constants and to determine the phase transition temperature, the Brillouin spectra were fitted with an origin-peak-fitting module using the Voigt function. Two
Fig. 1. Temperature dependence of Brillouin spectra of longitudinal acoustic mode propagating along the [0 1 0] direction.
Fig. 2. Temperature dependence of Brillouin spectra of longitudinal acoustic mode propagating along the [0 0 1] direction.
phase transitions are observed for both the frequency shifts and FWHM of the LA mode. The FWHM of the LA mode increases rapidly with the temperature below −100 ◦ C. The changes of the FWHM and the frequency shift of the LA mode propagating along the [0 1 0] direction are more remarkable than those of the LA mode propagating along the [0 0 1] direction. However, we can still observe the broadening of FWHM and the decrease in the frequency shift near To . In both cases, the LA mode becomes soft on approaching the phase transition temperature To from both sides. In the b(c, c + a)b geometry, Brillouin shift of the LA mode is related to the elastic stiffness constant c22 . In c(b, a + b)c geometry, Brillouin shift is related to the c33 . Brillouin shifts of the LA mode and the elastic stiffness coefficient obeys the following relation. 2πν cij = . (1) q ρ In this expression, ρ is the crystal density and q is the scattering wavevector. ν is the Brillouin shift of the LA mode. From Eq. (1) it is clear that elastic constants c22 and c33 also become soft around the To as shown in Fig. 3. This indicates ferroelastic nature of this phase transition. No discontinuity in the elastic constants was detected near To in our measurements and could point to the nearly second-order character of the transition. As shown in Figs. 3 and 4, the anomalies at 167 ◦ C in both frequency and FWHM correspond to the phase transition from orthorhombic Cmcm to monoclinic P21 /m symmetry, as was observed by modulated differential scanning calorimetry (MDSC) at 166.7 ◦ C [14]. On the other hand, the low temperature anomalies around −105 ◦ C correspond to the ferroelectric phase transition. The FWHM increases rapidly while the frequency shift decrease markedly on approaching the transition temperature −105 ◦ C. We can see from Fig. 3 that the elastic constant c22 remains at a finite value of about 154.2 GPa. Kojima et al. also observed the optical soft mode frequency tends to a finite value at To . Since the phase transition is nearly secondorder, some coupling may be considered to explain the fact that both optic and acoustic soft modes remain at finite values at To .
A. Hushur, S. Kojima / Materials Science and Engineering A 442 (2006) 35–38
37
Fig. 5. Elastic stiffness coefficient c22 below To . Solid line is the fitted result.
of Sr2 Ta2 O7 is modulated by q = a ∗ /2, linear coupling to uniform strain is forbidden, where a* is the reciprocal lattice vector in the high-temperature phase. The free-energy expansion of this crystal is written as follows [15,16]: 1 1 1 1 F = Fo + αη2 + βη4 + γη6 + co s2 + dη2 s. 2 4 6 2
(2)
where η is an order parameter, s is strain, co is a bare elastic constant, α = λ(T − To ) and other coefficients do not depend on temperature. By considering a complex elastic constant of a crystal in a case of coupling of elastic strain and the Landau–Khalatnikov mechanism of motion, we have Fig. 3. Temperature dependence of elastic stiffness coefficients (a) c22 (b) c33 .
The changes of elastic stiffness constants in the neighborhood of phase transitions can be determined using the Landau theory. In the ferroelastic phase, the Landau theory predicted that the bilinear coupling (linear in strain, quadratic in order parameter) between order parameter and the strain causes the softening below To . Since the order parameter in the high-symmetry phase
c22 = 0 or c22 = co c22 = co − β + β1
(T > To ) 2d 2
(3) (T < To )
1 + (4γλ(To − T )/β12 ) − 1 (4)
where β1 = β − 2d2 /co . The temperature dependence of elastic stiffness constants c22 in the ferroelastic phase was fitted by Eq. (4) as shown in Fig. 5. The fitted result is shown by the solid line in the same figure. The bilinear coupling gives reasonable values for the softening of the c22 below To . Above To the Landau–Khalatnikov mechanism gives temperature independent values for the elastic stiffness constants. However, the additional curvature in the paraelastic phase, indicating the existence of an additional contribution. The situation becomes clear if we look at the FWHM in Fig. 4, which is related to the hypersonic damping. The order parameter fluctuation may be the origin of this damping. 4. Conclusion
Fig. 4. Temperature dependence of the full-width at half-maximum of the longitudinal acoustic mode propagating along the [0 1 0] direction.
The micro-Brillouin technique was applied to examining the acoustic properties of the Sr2 Ta2 O7 single crystals in the temperature range from −140 ◦ C to 300 ◦ C. Two phase transitions were detected by Brillouin scattering method at 167 ◦ C and around
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A. Hushur, S. Kojima / Materials Science and Engineering A 442 (2006) 35–38
−105 ◦ C. Elastic stiffness constant of c22 , c33 and the FWHM of LA mode propagating along the [0 1 0] and [0 0 1] directions showed remarkable anomalies at To and Tc . c22 showed continuous softening on approaching To from both sides, indicating the nearly second-order and ferroelastic nature of these phase transition. This may reflect not only the Landau–Khalatnikov mechanism (T < To ) but also the electrostrictive coupling (fluctuation damping), which is consistent with the observation of the second harmonic generation above To . Acknowledgement This work was supported in part by the 21st century COE program under the Japanese Ministry of Education, Culture, Sports, Science and Technology. References [1] N. Ishizawa, F. Marumo, T. Kawamura, M. Kimura, Acta Crystallogr. B32 (1976) 2564–2566. [2] Y. Fujimori, N. Izumi, T. Nakamura, A. Kamasiwa, Jpn. J. Appl. Phys. 37 (1998) 5207–5210.
[3] G.A. Smolenskii, V.A. Isupov, A.I. Agranovskaia, Dokl. Akad. Nauk. (USSR). 108 (1956) 232–235 (in Russian). [4] S. Nanamatsu, M. Kimura, T. Kawamura, J. Phys. Soc. Jpn. 38 (1975) 817–824. [5] Y. Akishige, K. Ohi, J. Phys. Soc. Jpn. 61 (1992) 1351–1356. [6] S. Kojima, K. Ohi, T. Nakamura, Solid State Commun. 35 (1980) 79–81. [7] N. Yamamoto, K. Yagi, G. Honjo, M. Kimura, T. Kawamura, J. Phys. Soc. Jpn. 48 (1980) 185–191. [8] K. Ito, T. Toya, K. Ohi, Ferroelectrics 168 (1995) 161–167. [9] A. Hushur, H. Shigematsu, Y. Akishige, S. Kojima, Appl. Phys. Lett. 86 (2005) 2903–2905. [10] A. Hushur, H. Shigematsu, Y. Akishige, S. Kojima, Jpn. J. Appl. Phys. 43 (2004) 6825–6828. [11] J.-H. Ko, A. Hushur, S. Kojima, B.C. Sih, Z.G. Ye, Appl. Phys. Lett. 81 (2002) 4043–4045. [12] S. Kojima, Ultrasonic Technol. 15 (2003) 55–59 (in Japanese). [13] R. Vacher, L. Voyer, Phys. Rev. B 6 (1972) 639–673. [14] A. Hushur, G. Shabbir, J.-H. Ko, S. Kojima, J. Phys. D: Appl. Phys. 37 (2004) 1127–1131. [15] M. Borissov (Ed.), Proceedings of the 2nd International School on Condensed Matter Physics, Optical and Acoustic Waves in Solids, World Scientific, Singapore, 1983. [16] M.E. Lines, A.M. Glass, Principles and Applications of Ferroelectrics and Related Materials, Clarendon Press, Oxford, 1977.
Brillouin scattering study on ferroelectric single crystal Sr2Ta2O7 Anwar Hushur ∗ , Seiji Kojima Institute of Materials Science, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan Received 1 August 2005; received in revised form 22 December 2005; accepted 12 January 2006
Abstract Acoustic properties of strontium tantalite (Sr2 Ta2 O7 ) single crystals with perovskite-slab structure were investigated by micro-Brillouin scattering. The two elastic stiffness constants c22 and c33 were determined in the temperature range between −140 ◦ C and 300 ◦ C. The elastic constant c22 shows marked continuous softening on approaching phase transition temperature To around 167 ◦ C from both sides, indicating ferroelastic and nearly second-order nature of this phase transition. The values of c22 can be well reproduced by the Landau theory including bilinear coupling (linear in strain, quadratic in order parameter) between order parameter and the strain. Hypersonic attenuation increases markedly around To , which seems to reflect not only bilinear coupling between the order parameter and the strain in the ferroelastic phase, but also electrostrictive coupling in the paraelectric phase. © 2006 Elsevier B.V. All rights reserved. Keywords: Soft mode; Structural phase transition; Brillouin scattering; Sr2 Ta2 O7 ; Ferroelectrics
1. Introduction Strontium tantalate (Sr2 Ta2 O7 ) and strontium niobate (Sr2 Nb2 O7 ) belong to the A2 B2 O7 -type ferroelectrics with a perovskite-slab structure [1]. Their solid solutions are suitable for use as ferroelectric materials in the field of electronics as a candidate for ferroelectric random access memories (FeRAM), because these materials have a low dielectric constant, low coercive field and high heat-resistance. Recently, its lead-free character has attracted much attention as a green material [2]. Smolenskii et al. [3] first reported the dielectric anomaly in ceramic Sr2 Ta2 O7 samples between −84 ◦ C and −55 ◦ C, and the ferroelectricity was confirmed below this temperature later by Nanamatsu et al. [4]. They observed a large dielectric anomaly along the c-axis at −107 ◦ C (Tc ). Akishige and Ohi [5] have reported a low frequency dielectric dispersion in the temperature range between −223 ◦ C and −163 ◦ C. Difference of Tc was found and was explained by different Nb content in samples. The crystal structure was determined from the three-dimensional Xray diffraction data by Ishizawa et al. [1]. Soft optic phonon responsible for the structural phase transition at 170 ◦ C has been observed by Kojima et al. [6] using Raman scattering study. The high-temperature phase transition was also observed by electron
∗
Corresponding author. Tel.: +81 29 853 5278; fax: +81 29 853 5262. E-mail address: [email protected] (A. Hushur).
0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.01.142
microscopy and diffraction [7]. Previous studies revealed that Sr2 Ta2 O7 is orthorhombic structure with space group Cmcm in the high-temperature phase, in result of cooling the paraelectric phase transforms into a non-polar super lattice structure of space group P21 /m, characterized by the tilting of TaO6 octahedra and that the lattice constant along the a-axis doubles at about 170 ◦ C. It is important to note that the monoclinic symmetry is not caused by a distortion of the orthorhombic paraelectric phase but induced by the tilting of TaO6 octahedra. This requires to change into a new unit cell. This transition is displacive-type. Sr2 Ta2 O7 is paraelectric and ferroelastic at room temperature, undergoes a ferroelectric phase transition at −100 ◦ C from P21 /m to the polar phase P21 . Ito et al. [8] observed a soft mode associated with the ferroelectric phase transition by Raman scattering. To understand the phase transition mechanism at To and the elastic property, it is useful to apply Brillouin light scattering technique. Since Brillouin frequency shifts and full-width at half-maximum (FWHM) are very sensitive to phase transitions, it provides fundamental information on interatomic forces, and the structure and dynamics of solids related to phase transitions [9–11]. In the present work, Brillouin scattering studies of Sr2 Ta2 O7 single crystals in the temperature range −140 ◦ C to 300 ◦ C are reported. The temperature dependence of two different longitudinal acoustic (LA) modes has been measured. This allowed for the determination of the temperature changes of two independent elastic-stiffness constant for the both Cmcm and P21 /m phases. A theoretical model using the Landau theory is presented, from
36
A. Hushur, S. Kojima / Materials Science and Engineering A 442 (2006) 35–38
which good agreement with the experimental results is obtained and the nature of the phase transition clearly defined. 2. Experimental The sample used in the Brillouin scattering was a 1.5 mm × 1.7 mm × 3.3 mm rectangular parallelepiped with edges parallel to the a-, b-, and c-axes with respect to the orthorhombic coordinates. A diode-pumped solid-state laser (DPSS532) was used to excite the sample with a wave length of 532 nm and power of 150 mW. The scattered light was analyzed at backward scattering geometry with a six-pass tandem Fabry–Perot interferometer of high contrast. A conventional photon-counting system and a multichannel analyzer were used to detect and average the signal. The sample cell with X–Y adjustment was put on the stage of an optical microscope with a focal point of 1–2 m. The sample was put in a cryostat cell (THMS 600) with temperatures varying from −190 ◦ C to 600 ◦ C and with a stability of ±0.1 K [10,12]. We measured the Brillouin scattering spectra at three different orientations with phonons propagating along the a-, b- and c-axes. 3. Results and discussion Brillouin scattering spectra were measured with a free spectral range (FSR) of 60 GHz at the b(c, c + a)b¯ and c(b, a + b)¯c geometry. Typical Brillouin spectra are shown in Figs. 1 and 2 with a free spectral range of 60 GHz using [0 1 0] and [0 0 1] directions. Only one LA mode is observed at both high and low temperatures, consistent with the selection rule of orthorhombic symmetry at the backward scattering geometry [13]. The intensity of the LA mode increased with the increase of the temperature, it disappeared below −113 ◦ C which is, probably, due to the formation of fine ferroelectric domains. To obtain the temperature dependence of elastic constants and to determine the phase transition temperature, the Brillouin spectra were fitted with an origin-peak-fitting module using the Voigt function. Two
Fig. 1. Temperature dependence of Brillouin spectra of longitudinal acoustic mode propagating along the [0 1 0] direction.
Fig. 2. Temperature dependence of Brillouin spectra of longitudinal acoustic mode propagating along the [0 0 1] direction.
phase transitions are observed for both the frequency shifts and FWHM of the LA mode. The FWHM of the LA mode increases rapidly with the temperature below −100 ◦ C. The changes of the FWHM and the frequency shift of the LA mode propagating along the [0 1 0] direction are more remarkable than those of the LA mode propagating along the [0 0 1] direction. However, we can still observe the broadening of FWHM and the decrease in the frequency shift near To . In both cases, the LA mode becomes soft on approaching the phase transition temperature To from both sides. In the b(c, c + a)b geometry, Brillouin shift of the LA mode is related to the elastic stiffness constant c22 . In c(b, a + b)c geometry, Brillouin shift is related to the c33 . Brillouin shifts of the LA mode and the elastic stiffness coefficient obeys the following relation. 2πν cij = . (1) q ρ In this expression, ρ is the crystal density and q is the scattering wavevector. ν is the Brillouin shift of the LA mode. From Eq. (1) it is clear that elastic constants c22 and c33 also become soft around the To as shown in Fig. 3. This indicates ferroelastic nature of this phase transition. No discontinuity in the elastic constants was detected near To in our measurements and could point to the nearly second-order character of the transition. As shown in Figs. 3 and 4, the anomalies at 167 ◦ C in both frequency and FWHM correspond to the phase transition from orthorhombic Cmcm to monoclinic P21 /m symmetry, as was observed by modulated differential scanning calorimetry (MDSC) at 166.7 ◦ C [14]. On the other hand, the low temperature anomalies around −105 ◦ C correspond to the ferroelectric phase transition. The FWHM increases rapidly while the frequency shift decrease markedly on approaching the transition temperature −105 ◦ C. We can see from Fig. 3 that the elastic constant c22 remains at a finite value of about 154.2 GPa. Kojima et al. also observed the optical soft mode frequency tends to a finite value at To . Since the phase transition is nearly secondorder, some coupling may be considered to explain the fact that both optic and acoustic soft modes remain at finite values at To .
A. Hushur, S. Kojima / Materials Science and Engineering A 442 (2006) 35–38
37
Fig. 5. Elastic stiffness coefficient c22 below To . Solid line is the fitted result.
of Sr2 Ta2 O7 is modulated by q = a ∗ /2, linear coupling to uniform strain is forbidden, where a* is the reciprocal lattice vector in the high-temperature phase. The free-energy expansion of this crystal is written as follows [15,16]: 1 1 1 1 F = Fo + αη2 + βη4 + γη6 + co s2 + dη2 s. 2 4 6 2
(2)
where η is an order parameter, s is strain, co is a bare elastic constant, α = λ(T − To ) and other coefficients do not depend on temperature. By considering a complex elastic constant of a crystal in a case of coupling of elastic strain and the Landau–Khalatnikov mechanism of motion, we have Fig. 3. Temperature dependence of elastic stiffness coefficients (a) c22 (b) c33 .
The changes of elastic stiffness constants in the neighborhood of phase transitions can be determined using the Landau theory. In the ferroelastic phase, the Landau theory predicted that the bilinear coupling (linear in strain, quadratic in order parameter) between order parameter and the strain causes the softening below To . Since the order parameter in the high-symmetry phase
c22 = 0 or c22 = co c22 = co − β + β1
(T > To ) 2d 2
(3) (T < To )
1 + (4γλ(To − T )/β12 ) − 1 (4)
where β1 = β − 2d2 /co . The temperature dependence of elastic stiffness constants c22 in the ferroelastic phase was fitted by Eq. (4) as shown in Fig. 5. The fitted result is shown by the solid line in the same figure. The bilinear coupling gives reasonable values for the softening of the c22 below To . Above To the Landau–Khalatnikov mechanism gives temperature independent values for the elastic stiffness constants. However, the additional curvature in the paraelastic phase, indicating the existence of an additional contribution. The situation becomes clear if we look at the FWHM in Fig. 4, which is related to the hypersonic damping. The order parameter fluctuation may be the origin of this damping. 4. Conclusion
Fig. 4. Temperature dependence of the full-width at half-maximum of the longitudinal acoustic mode propagating along the [0 1 0] direction.
The micro-Brillouin technique was applied to examining the acoustic properties of the Sr2 Ta2 O7 single crystals in the temperature range from −140 ◦ C to 300 ◦ C. Two phase transitions were detected by Brillouin scattering method at 167 ◦ C and around
38
A. Hushur, S. Kojima / Materials Science and Engineering A 442 (2006) 35–38
−105 ◦ C. Elastic stiffness constant of c22 , c33 and the FWHM of LA mode propagating along the [0 1 0] and [0 0 1] directions showed remarkable anomalies at To and Tc . c22 showed continuous softening on approaching To from both sides, indicating the nearly second-order and ferroelastic nature of these phase transition. This may reflect not only the Landau–Khalatnikov mechanism (T < To ) but also the electrostrictive coupling (fluctuation damping), which is consistent with the observation of the second harmonic generation above To . Acknowledgement This work was supported in part by the 21st century COE program under the Japanese Ministry of Education, Culture, Sports, Science and Technology. References [1] N. Ishizawa, F. Marumo, T. Kawamura, M. Kimura, Acta Crystallogr. B32 (1976) 2564–2566. [2] Y. Fujimori, N. Izumi, T. Nakamura, A. Kamasiwa, Jpn. J. Appl. Phys. 37 (1998) 5207–5210.
[3] G.A. Smolenskii, V.A. Isupov, A.I. Agranovskaia, Dokl. Akad. Nauk. (USSR). 108 (1956) 232–235 (in Russian). [4] S. Nanamatsu, M. Kimura, T. Kawamura, J. Phys. Soc. Jpn. 38 (1975) 817–824. [5] Y. Akishige, K. Ohi, J. Phys. Soc. Jpn. 61 (1992) 1351–1356. [6] S. Kojima, K. Ohi, T. Nakamura, Solid State Commun. 35 (1980) 79–81. [7] N. Yamamoto, K. Yagi, G. Honjo, M. Kimura, T. Kawamura, J. Phys. Soc. Jpn. 48 (1980) 185–191. [8] K. Ito, T. Toya, K. Ohi, Ferroelectrics 168 (1995) 161–167. [9] A. Hushur, H. Shigematsu, Y. Akishige, S. Kojima, Appl. Phys. Lett. 86 (2005) 2903–2905. [10] A. Hushur, H. Shigematsu, Y. Akishige, S. Kojima, Jpn. J. Appl. Phys. 43 (2004) 6825–6828. [11] J.-H. Ko, A. Hushur, S. Kojima, B.C. Sih, Z.G. Ye, Appl. Phys. Lett. 81 (2002) 4043–4045. [12] S. Kojima, Ultrasonic Technol. 15 (2003) 55–59 (in Japanese). [13] R. Vacher, L. Voyer, Phys. Rev. B 6 (1972) 639–673. [14] A. Hushur, G. Shabbir, J.-H. Ko, S. Kojima, J. Phys. D: Appl. Phys. 37 (2004) 1127–1131. [15] M. Borissov (Ed.), Proceedings of the 2nd International School on Condensed Matter Physics, Optical and Acoustic Waves in Solids, World Scientific, Singapore, 1983. [16] M.E. Lines, A.M. Glass, Principles and Applications of Ferroelectrics and Related Materials, Clarendon Press, Oxford, 1977.