Numerical modeling of the magnetoelectric effect in magnetostrictive piezoelectric bilayers

Appl. Phys. A (2004) DOI: 10.1007/s00339-004-2983-5 Applied Physics A Materials Science & Processing y. wang 1,2 h. yu 1,2 m. zeng 1,2 j.g. wan 1,2 m.f. zhang 1,2 j.-m. liu 1,2, c.w. nan 3 Numerical modeling of the magnetoelectric effect in magnetostrictive piezoelectric bilayers 1 Laboratory of Solid State Microstructure, Nanjing University, Nanjing 210093, P.R. China 2 International Center for Materials Physics, Chinese Academy of Sciences, Shenyang, P.R. China 3 Department of Material Science and Engineering, Tsinghua University, Beijing, P.R. China Received 13 March 2004; Accepted 5 July 2004 Published online: 20 August 2004 Springer-Verlag 2004 ABSTRACT A numerical modeling of the magnetoelectric (ME) effect in the bilayer structures lead zirconate tiate (PZT)/lanthanum strontium manganite (LSMO), PZT/nickel ferrite (NFO) and PZT/cobalt ferrite (CFO) is investigated for both static and dynamic behaviors. Mainly, this work focuses on the ME coupling of the recgular bilayer structures at the electromechanical resonance (EMR) and predicts a resonance frequency that is found to increase with the decrease of length and the rise of PZT volume fraction. The calculated ME voltage coefficients versus frequency profiles for these three samples show a strong resonance character and the values at the EMR are about 200 times the values far from the EMR state. The estimated resonance frequencies are both at about 120 khz for 15-mm-long NFO/PZT and CFO/PZT bilayers with PZT volume fraction v = 0.25. Furthermore, the relevant experiments were carried out to verify the numerical results. PACS 75.80.+q; 75.50.Gg; 75.60.-d 1 Introduction The magnetoelectric (ME) effect has been greatly studied since the 1990s for the energy conversion between the magnetic and electric fields and the potential use as ME memory elements, smart sensors and transducers [1]. The effect is that the application of either a magnetic field or an electric field induces an electric field or a magnetization, which was first predicted by Landau and Lifshitz in 1957 [2]. So, the ME effect requires the simuleous presence of longrange ordering of both electric dipoles and magnetic moments. There are few such single-phase compounds and the ME effects in these compounds are very weak even at low temperature [3 6]. In order to improve the ME property, van Suchtelen proposed product property composites which were expected to be remarkably magnetoelectric because of mechanical stress-mediated electromagnetic coupling [7]. For insce, the composites may be fabricated by combining a magnetostrictive phase with a piezoelectric phase in differ- Fax: +86-25/8359-5535, E-mail: liujm@nju.edu.cn ent forms. This mechanism can be briefly expressed as [8] magnetoelectric effect = magnetic mechanical mechanical or electric magnetoelectric effect = electric mechanical mechanical magnetic. In the 1970s and 1980s, most interest concentrated on the ferrite lead zirconate tiate (PZT) or ferrite BaTiO 3 bulk composites. Van den Boomgaard et al. synthesized bulk composites of CoFe 2 O 4 (CFO) BaTiO 3 and NiFe 2 O 4 (NFO) BaTiO 3 [9 11]. In general, the ME coefficient in these samples is much smaller than calculated values due to the leakage current associated with the low resistivity of ferrite. Because the layered structure can eliminate the leakage current and the piezoelectric layer can be effectively poled electrically to further strengthen the piezoelectricity, it was predicted that there is a larger ME coefficient in the layered composite than in the bulk [12 16]. In recent years, Srinivasan and Bichurin and others carried out much experimental and theoretical work on the ME effect in bilayer and multilayer composites of ferrite PZT that helps us undersd the essence of the magnetoelectric effect. In particular, they performed the first theoretical modeling of the resonance ME effect in the layered composites [14 22]. In the layered magnetostrictive piezoelectric composites the induced polarization δp is associated with an applied ac magnetic field δh by the expression δp = αδh,whereα is the ME susceptibility. Usually, the parameter measured in the experiment is α E = δe/δh, which is related to α by the expression α = ε 0 ε r α E,whereε 0 is the static dielectric const and ε r is the relative permittivity of the material. For the layered composites, the low-frequency (1 1000 Hz) ME effect based on the mechanical stress-mediated model was much discussed, and we realized that the elastic property relates the strain at the interface to stress and it is in turn converted to an induced electric field or magnetization. This model is helpful for us to undersd the ME effect, but it is static and the effect is still weak. Bichurin et al. first dealt with the ME effect with respect to the frequency response and obtained the properties of thin-disk samples at the electromechanical resonance (EMR) [21].

Applied Physics A Materials Science & Processing FIGURE 2 The estimated magnetoelectric const α as a function of volume fraction v for LSMO-PZT and NFO-PZT bilayers In this paper, we focus on the theoretical calculation of the giant magnetoelectric effect in several recgular bilayered structures. The composites consist of one of the following oxides: cobalt ferrite due to high magnetostriction, nickel ferrite due to strong magneto mechanical coupling or ferromagnetic lanthanum strontium manganite (LSMO) due to structural homogeneity with PZT. The low-frequency magnetoelectric coupling is calculated and compared with the experimental data. We also perform the calculation of the frequency dependence of the ME voltage coefficient for the recgular bilayer based on the one-dimensional approximation. The coupling at the interface is assumed to be ideal in our model. Our calculation confirms that the resonance frequency of the recgular sample increases with decreasing length and augmenting PZT volume fraction. So, we can acquire a suitable resonance frequency through selecting the available parameters. The relevant experiments were also carried out to verify the numerical results. ME coefficients of 5.5 V/cm Oe and 5.0V/cm Oe for NFO-PZT and CFO-PZT bilayers at EMR were obtained in our experiments and the measured resonance frequencies were both at 126 khz. The calculation details and a comparison with experimental results are provided in the following sections. 2 Low-frequency magnetoelectric coupling in bilayer We consider a bilayer ME composite consisting of the magnetostrictive and piezoelectric constituents as shown in Fig. 1. In the mechanical stress-mediated theory, the applied magnetic field leads to a strain in the magnetostrictive phase and the elastic property relates this strain at the interface to stress that induces the electric field in the piezoelectric layer. This process is the energy conversion between the magnetic and electric fields. The assumptions used in this theory are that there is no electric field in the magnetostrictive phase, the top and bottom surfaces of the piezoelectric layer are equipotential surfaces and the magnetic field H is identical throughout the bilayer [12, 15]. Because the wavelength is much larger than the lengths of the samples at the low frequency and the resistivities of the ferromagnetic materials considered in our paper are low, these assumptions are valid. For the magnetostrictive phase with the cubic (m3m)symmetry the relation between the elastic and magnetic variables can be written as the following expression: m T i = m C ij m S j m p ki H k, (1) where H k is the magnetic field intensity and m T i, m S i, m C ij and m p ki are stress, strain, stiffness and piezomagnetic coefficient tensors of the magnetostrictive phase, respectively. Assuming that the magnetic field H is perpendicular to the sample plane, i.e. H 1 = H 2 = 0, H 3 = H, we obtain m T 1 = m C m 11 S 1 + m C 12 ( m S 2 + m S 3 ) m p 31 H 3 = 0, m T 2 = m C m 11 S 2 + m C 12 ( m S 1 + m S 3 ) m p 31 H 3 = 0, m T 3 = m C 11 m S 3 + m C 12 ( m S 1 + m S 2 ) m p 33 H 3. (2a) (2b) (2c) m S 1 and m S 2 can be expressed as a function of m S 3 in terms of (2a) and (2b). Thus, the relation between m T 3 and m S 3 can be determined by taking account of them. For the piezoelectric phase with the m symmetry we propose that the poling axis is parallel to the magnetic field and obtain p T 1 = p C 11 p S 1 + p C 12 p S 2 + p C 13 p S 3 + p e 31 E 3 = 0, p T 2 = p C 12 p S 1 + p C 11 p S 2 + p C 13 p S 3 + p e 31 E 3 = 0, p T 3 = p C 13 ( p S 1 + p S 2 ) + p C 33 p S 3 + p e 33 E 3. (3) Here p T i, p S i and p C ij are stress, strain and stiffness tensors of the piezoelectric phase, respectively. Similarly, p T 3 can be expressed as a function of p S 3. Next, using the boundary conditions p T 3 = m T 3 and p S 3 = m S 3 (1 v)/v [13], where v = p v/( p v + m v) and p v and m v are the volumes of piezoelectric and magnetostrictive phases, respectively, from (2) and (3) we obtain = p T 3 1 + v 1 v 2 m C 12 m p 31 m C 11 + m C 12 m p 33 m C 11 ( m C 11 + m C 12 ) 2 m C 12 2 m C 11 + m C 12 p C 11 + p C 12 p C 33 ( p C 11 + p C 12 ) 2 p C 13 2 H 3. (4) In the right-hand side of (4) the term concerning E 3 is neglected because it is much smaller than the listed term. According to the principle of the energy conversion, we obtain δe H = 3vλ 001 p T 3 /D s = αh 3, (5) FIGURE 1 Schematic diagram showing a magnetostrictive piezoelectric bilayer. Fields are all along the z direction where α and D s are the magnetoelectric const and the saturation electric displacement, respectively. According to (5) we can estimate the magnetoelectric const α = δe H /H 3

WANG et al. Numerical modeling of the magnetoelectric effect in magnetostrictive piezoelectric bilayers by compliance coefficients of two components, the piezomagnetic coefficient, and the electrostriction and saturation electric displacement of the piezoelectric phase. The calculated ME consts of bilayer composites of LSMO-PZT and NFO-PZT versus v are displayed in Fig. 2. The physical parameters of these components are as follows [17]: PZT: p C 11 = 12.6 10 10 N/m 2, p C 12 = 7.95 10 10 N/m 2, p C 11 = 8.4 10 10 N/m 2, p C 33 = 11.7 10 10 N/m 2, λ 001 = 600 10 6, D s = 32 10 2 C/m 2, LSMO: m C 11 = 10 10 10 N/m 2, p C 12 = 5 10 10 N/m 2, m p 31 = 5.5N/Am, m p 33 = 13 N/Am, NiFe 2 O 4 : m C 11 = 21.99 10 10 N/m 2, p C 12 = 10.77 10 10 N/m 2, m p 31 = 33.55 N/Am, m p 33 = 122.1N/Am. In Fig. 2 we can see that the composite NFO-PZT shows a stronger magnetoelectric effect than LSMO-PZT for the larger magnetostriction. At low frequency, much work indicated that the largest ME consts were at approximately the volume ratio 1 : 1 of magnetostrictive and piezoelectric phases as shown in Fig. 2 [20]. The calculated values for v = 0.5 are given by α = 6 and 20 mv/cm Oe for LSMO-PZT and NFO-PZT, respectively. In their experiments, Srinivasan et al. pointed out that longitudinal ME voltage coefficients of two composites were 13 and 60 mv/cm Oe [14, 15], which are both larger than our calculated values. This model tells us that to obtain a large ME const the giant magnetostriction, strong piezoelectric effect and small saturation polarization are necessary. 3 Frequency dependence of ME effect in recgular bilayers Although many studies on bulk and layered composites have been made, most of the interest concentrated on the low-frequency ME effect and the frequency-dispersive ME effect was not investigated sufficiently. We pay attention to the frequency dependence of the ME effect, especially the behavior at the electromechanical resonance (EMR) ranging between 10 Hz and 500 khz, and are interested in predicting the resonance frequency accurately. Recently, Bichurin et al. first studied the theoretical modeling of the resonance ME coupling of thin-disk samples [21]. This dynamic procedure in the bilayer composite can be described by the equation of motion in the elastic medium. In Bichurin s work, for the axial symmetry of the circular sample the strain and stress are independent of the transverse variable θ. The boundary conditions and the symmetry of the recgular sample are both different from those of the circular one, so we discuss them in this section. The bilayer composite can be viewed as a homogeneous medium. We have [17 20] S i = s ij T j + d ki E k + q ki H k, D k = d ki T i + ε kn E n + α kn H k, (6) B k = q ki T i + α kn E n + µ kn H n, where S i and T i are the strain and stress tensors, E k, D k, H k and B k are the effective electric field, electric displacement, magnetic field and magnetic induction, s ij, d ki and q ki are the effective compliance, piezoelectric and piezomagnetic coefficients, and ε kn, µ kn and α kn are the effective permittivity, permeability and ME coefficient. All the effective parameters discussed in [13] and [20] can be determined from the parameters of individual phases. The bilayer geometry is a recgle in the (x, y) plane with thickness t, width w and length L. The thickness of the electrode is supposed to be negligible. Applied transverse magnetic fields are along the x axis, namely the length direction, and an ac magnetic field δh induces the harmonic waves. Because t and w L, the resonance frequencies for both width and thickness modes are much higher than that for the length mode considering the case of a sding wave, and in the measured frequency range these two modes hardly influence the bilayer s vibration and are not observed. Moreover, it is very difficult to solve the partial differential equation set with two variables. So, the one-dimensional approximation is valid and only the harmonic wave in the length direction is considered. Thus the generalized Hooke s law and the corresponding equations have the following form: S 1 = s 11 T 1 + d 31 E 3 + q 11 H 1, D 3 = d 31 T 1 + ε 33 E 3 + α 31 H 1, (7) where S 1 = u 1 / x; u 1 is the displacement coordinate of medium. The equation of elastic dynamics is expressed as dt 1 dx + ϱω2 u 1 = 0, (8) where ϱ is the density and ω is the angular frequency. Taking into account (7) and using the boundary conditions T 1 = 0 at x = 0 and x = L, from (2) we get the expression for T 1 : T 1 = 1 s 11 [ sin ω(l x) V sin V + sin ωx V ] 1 (d 31 E 3 + q 11 H 1 ), (9) where V = 1/ ϱs 11 is the acoustic speed and L is the length of the bilayer. Obviously, s 11 > 0 is required. According to the open-circuit condition, i.e. s D 3 ds = 0, the electric field is E 3 = d ( 2V 31q 11 d 2 31 ( 2V 2V 1) + s 11 α 31 2V 1) + s 11 ε 33 H 1. (10) The transverse ME voltage coefficient α T,31 is ( 2V ( 2 2V 2V 1) + s 11 α 31 α T,31 = E 3 = d 31q 11 H 1 2V 1). (11) + s 11 ε 33 We next carry out some analysis of (11). First, the resonance frequency is determined by ( ) 2V 2 2V 1 + s 11 ε 33 = 0, ( 2V = 1 s ) 11ε 33 2 2V. (12)

Applied Physics A Materials Science & Processing Because the function x is periodic, the number of the solutions of the equation is infinite. On the other hand, for the acoustic speed V being const for a composite, the wavelength becomes shorter with the frequency increase. Therefore, the length of the composite is larger than the wavelength for the higher frequency and the reflected wave interferes with the original wave that makes the resonance character disappear. Thus, only the resonance frequency corresponding to the first solution of (12) can be observed in the experiment. Second, according to mathematical knowledge, we know that x/x 1 0 when x 1. So, at low frequencies the terms in respect of the frequency are negligible and the dynamic ME coefficient coincides with the static value of α 31 /ε 33. In Fig. 3 the solid lines show the calculated values versus frequency based on the current theory for parameters in Tables 1 and 2. The measured results are shown as the triangles and will be discussed in Sect. 4. The bias field H is relevant to the maximal piezomagnetic coefficient, and the lengths and PZT volume fractions are the same for the three samples. For the calculated results the resonance character is remarkable and the peak values of the three samples at EMR are 2.8 15 V/cm Oe that are all extremely large compared with the low-frequency values of 14 70 mv/cm Oe. The resonance values are about 200 times the low-frequency values far from the resonance state. The resonance line width is small, which is determined by d 31 and q 11 in (11). It is clear that the ME coefficient of the CFO-PZT bilayer is largest at EMR, due to the strongest magnetostriction, and that that of the LSMO-PZT bilayer is lowest. We have discussed the situation that the volume fraction v of the piezoelectric phase is relatively small and the effective compliance coefficient s 11 > 0.Whenv is large and s 11 is negative, the transverse ME voltage coefficient α T,31 has the following expression: ( 2V ) h + s11 α 31 ( 2 2V α T,31 = E 3 = d 31q 11 H 1 2V h 2V ) + s11 ε 33, (13) where V = 1/ ϱs 11 is the acoustic speed. Estimates indicate that the resonance character disappears and the ME Material s 11 s 12 q 31 q 11 d 31 ε 33 /ε 0 (10 12 (10 12 (10 12 (10 12 (10 12 m 2 /N) m 2 /N) m/a) m/a) m/v) LSMO 15 5 120 300 NFO 6.5 2.4 125 840 CFO 6.5 2.4 556 1880 PZT 15.3 5 175 1750 TABLE 1 Compliance coefficient s, piezomagnetic coefficient q, piezoelectric coefficient d and permittivity ε for LSMO, NFO, CFO and PZT LSMO-PZT NFO-PZT CFO-PZT v 0.25 0.25 0.25 L (mm) 15 15 15 TABLE 2 PZT volume fraction v and length L for LSMO-PZT, NFO-PZT and CFO-PZT composites FIGURE 3 Comparison of the calculated ME voltage coefficient with respect to the frequency response with the experimental data. The solid lines are the calculated results of the frequency dependence of the transverse ME voltage coefficient α T,31 = E 3 / H 1 for NFO-PZT, CFO-PZT and LSMO- PZT bilayers. Estimates are for the thickness and width much smaller than the length. The resonance character is noticeable. Parameters used for the estimates are given in Tables 1 and 2. The triangles are data points for NFO- PZT and CFO-PZT bilayers with 4-mm width and 2-mm thickness. The lines are guides to the eye voltage coefficient is saturated at the high frequency. These properties are apparent in Fig. 4. The maximal slope of h x lies at the point x = 0 and its value is 1. So the equation 2 ( 2V h 2V ) + s 11ε 33 = 0 has no solution for s 11 ε 33 / 2 > 1. That is, the composite has no resonance frequency. Furthermore, we calculate the length dependence of the resonance frequency for a LSMO-PZT bilayer shown in Fig. 5. The curves are fitted and coincide with the function f = a/l,wherea is a function of v. It is well known that V = 2L f at the mechanical resonance frequency for a recgular sample. Using V = 1/ ϱs 11 and s 11 =[ v m s 11 + (1 v) p s 11 ]/[4v(1 v) 2 ] in the bilayer composite, approximately we can undersd that the electromechanical resonance frequency increases with the decrease of length and the rise of PZT volume fraction. This diagram can instruct us to select the available v and L to acquire the needed resonance frequency in practical applications.

WANG et al. Numerical modeling of the magnetoelectric effect in magnetostrictive piezoelectric bilayers FIGURE 4 Estimates of α T,31 versus f for large PZT volume fraction v.effective compliance coefficients s 11 of three composites are negative with the chosen v values. The EMR property does not remain in this situation FIGURE 5 Resonance frequency as a function of length L and PZT volume fraction v. The resonance frequency becomes higher with shortening the length and increasing the PZT volume fraction 4 Relevant experiments The theory and calculated results of the recgular bilayer composites were obtained, which displayed a strong resonance ME effect at EMR. To verify the calculation the relevant experiments were carried out. Our estimate and other works [15] indicated that the ME effect of a LSMO-PZT bilayer was weak even at low temperature; therefore, we paid attention to NFO-PZT and CFO-PZT bilayer composites. In the experiments, bilayer composite samples of NFO-PZT and CFO-PZT were synthesized from two individual phases. Submicron NFO (CFO) powder and commercial PZT were used. The magnetostrictive phases of CFO and NFO were prepared by a conventional solid state reaction method. The powders of high-purity NiO or Co 2 O 3 and Fe 2 O 3 weremixedinaball mill, presintered at 1270 K for 10 h and then pressed into pellets and sintered at 1470 K for 10 h. PZT-502 powder (from PKI, USA) used for the piezoelectric phase was sintered at 1520 K for 1hin a lead-atmosphere environment. After sintering it was sliced, covered with silver electrodes and poled with a 40-kV/cm electric field in 395 K silicone oil for 15 min. The sliced NFO (CFO) and PZT layers were of 1.5- and 0.5-mm thicknesses, respectively, 15-mm length and 4-mm width, and bonded with an epoxy. Here the very thin and elastic epoxy is used to relate the strain at the interface of the two individual phases to stress and is supposed to have no effect on the ME voltage coefficient as an ideal medium. For the two bilayer samples ME voltage coefficients α E = d E/d H were measured by the induced electric field under an applied small ac magnetic field less than 5Oe (generated by a solenoid) superposed on a dc magnetic bias field (generated by an electromagnet). The measured results are shown as the triangles in Fig. 3 and the most significant realization is the occurrence of the predicted giant ME effect at EMR. The resonance frequencies of the two composites are both at 126 khz. Measured ME voltage coefficients at EMR are 5.5 and 5.0V/cm Oe for NFO-PZT and CFO-PZT composites and contrasted with the low-frequency values of 40 70 mv/cm Oe, respectively. Because the one-dimensional approximation is used in the model and the harmonic wave in the width direction is neglected, there is a small departure between the calculated resonance frequency of 120 khz and the measured values. The features of the strong ME coupling at resonance and the small line width of the experimental results are in good agreement with the estimated results. It is noted in Fig. 3 that the measured ME voltage coefficient at EMR of a CFO-PZT bilayer is almost equal to that of a NFO-PZT bilayer; however, the calculated ME voltage coefficient of the former is larger than that of the latter. In our model the coupling at the interface is assumed to be perfect, so that the ME coefficient is proportional to the magnetostriction. In a practical system, loss at the interface is unavoidable. We assume that although the magnetostriction of CFO is stronger than that of NFO, the coupling at the interface of NFO-PZT is better; therefore, the common influence of two factors produces the observed results. On the other hand, the measured ME coefficients at EMR of CFO-PZTand NFO-PZTare 33% and 64% of their calculated values, respectively. This proves that the interface coupling of the NFO-PZT bilayer is better. Clearly, the experimental and calculated results coincide with each other excellently. Unfortunately, the ME effect of the LSMO-PZT bilayer is very small at room temperature and our experimental data are unrepeatable. So, there are no data points of the LSMO-PZT bilayer in Fig. 3. 5 Conclusions The low-frequency ME const in the mechanical stress-mediated model was obtained. The calculated values were compared with the experimental results. We also discussed the frequency-dispersive ME effect of the recgular bilayer composite and obtained the expression for the ME voltage coefficient based on the one-dimensional approximation. The estimated resonance frequencies for NFO-PZT and CFO-PZT bilayers coincide with the measured results excellently. The ME voltage coefficient at EMR is giant and much larger than the low-frequency value. The influence of the volume fraction v and the length L on the resonance frequency is evident. ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China (10021001 and 50372020), the National Key Projects for Basic Research of China (2002CB613303), the Provincial Natural Science Foundation of Jiangsu (BK2003412) and LSSMS of Nanjing University. REFERENCES 1 V.E. Wood, A.E. Austin: in Proc. Symp. Magnetoelectric Interaction Phenomena in Crystals, Seattle, 1973, ed. by A.J. Freeman, H. Schmid (Gordon and Breach, New York 1975) p. 181

Applied Physics A Materials Science & Processing 2 L.D. Landau, E.M. Lifshitz: Electrodynamics of Continuous Media (Gostekhizdat, Moscow 1957) 3 D.N. Astrov: Zh. Eksp. Teor. Fiz. 40, 1035 (1961) [Sov. Phys. JETP 13, 729 (1961)]; G.T. Rado, V.J. Folen: Phys. Rev. Lett. 7, 310 (1961); S. Foner, M. Hanabusa: J. Appl. Phys. 34, 1246 (1963) 4 I. Kornev, M. Bichurin, J.-P. Rivera, S. Gentil, A.G.M. Jansen, H. Schmid, P. Wyder: Phys. Rev. B 62, 12 247 (2000) 5 M. Bichurin: Ferroelectrics 204, 356 (1997) 6 H. Schmid: Proc. SPIE 12, 4097 (2000) 7 J. van Suchtelen: Philips Res. Rep. 27, 28 (1972) 8 C.-W. Nan: Phys. Rev. B 50, 6082 (1994) 9 J. Van den Boomgaard, D.R. Terrell, R.A.J. Born: J. Mater. Sci. 9, 1705 (1974) 10 J. Van den Boomgaard, A.M.J.G. van Run, J. van Suchtelen: Ferroelectrics 14, 727 (1976) 11 J. Van den Boomgaard, R.A.J. Born: J. Mater. Sci. 13, 1538 (1978) 12 G. Harshe, J.P. Dougherty, R.E. Newnham: Int. J. Appl. Electromagn. Mater. 4, 145 (1993); M. Avellaneda, G. Harshe: J. Intell. Mater. Syst. Struct. 5, 501 (1994) 13 G. Harshe, J.P. Dougherty, R.E. Newnham: Int. J. Appl. Electromagn. Mater. 4, 161 (1993) 14 G. Srinivasan, E.T. Rasmussen, J. Gallegos, R. Srinivasan, Yu.I. Bokhan, V.M. Laletin: Phys. Rev. B 64, 214 408 (2001) 15 G. Srinivasan, E.T. Rasmussen, B.J. Levin, R. Hayes: Phys. Rev. B 65, 134 402 (2002) 16 G. Srinivasan, E.T. Rasmussen, R. Hayes: Phys. Rev. B 67, 014 418 (2003) 17 M.I. Bichurin, I.A. Kornev, V.M. Petrov, A.S. Tatarenko, Yu.V. Kiliba, G. Srinivasan: Phys. Rev. B 64, 094 409 (2001) 18 M.I. Bichurin, V.M. Petrov, G. Srinivasan: J. Appl. Phys. 92, 7681 (2002) 19 M.I. Bichurin, V.M. Petrov, Yu.V. Kiliba, G. Srinivasan: Phys. Rev. B 66, 134 404 (2002) 20 M.I. Bichurin, V.M. Petrov, G. Srinivasan: Phys. Rev. B 68, 054 402 (2003) 21 M.I. Bichurin, D.A. Filippov, V.M. Petrov, V.M. Laletsin, N. Paddubnaya, G. Srinivasan: Phys. Rev. B 68, 132 408 (2003) 22 D.A. Filippov, M.I. Bichurin, V.M. Petrov, V.M. Laletin, N.N. Paddubnaya, G. Srinivasan: Tech. Phys. Lett. 30, 6 (2004)