LossDetermination-JJAP2009.pdf

Japanese Journal of Applied Physics 48 (2009) 041401 REGULAR PAPER Derivation of Piezoelectric Losses from Admittance Spectra Yuan Zhuang, Seyit O. Ural, Aditya Rajapurkar, Safakcan Tuncdemir, Ahmed Amin1 , and Kenji Uchino International Center for Actuators and Transducers, Materials Research Institute, The Pennsylvania State University, University Park, PA 16802, U.S.A. 1 Naval Undersea Warfare Center, Newport, RI 02841, U.S.A. Received August 11, 2008; accepted January 15, 2009; published online April 20, 2009 High power density piezoelectrics are required to miniaturize devices such as ultrasonic motors, transformers, and sound projectors. The power density is limited by the heat generation in piezoelectrics, therefore, clarification of the loss mechanisms is necessary. This paper provides a methodology to determine the electromechanical losses, i.e., dielectric, elastic and piezoelectric loss factors in piezoelectrics by means of a detailed analysis of the admittance/impedance spectra. This method was applied to determine the piezoelectric losses for lead zirconate titanate ceramics and lead magnesium niobate-lead titanate single crystals. The analytical solution provides a new method for obtaining the piezoelectric loss factor, which is usually neglected in practice by transducer designers. Finite element simulation demonstrated the importance of piezoelectric losses to yield a more accurate fitting to the experimental data. A phenomenological model based on two phase-shifts and the Devonshire theory of a polarizable–deformable insulator is developed to interpret the experimentally observed magnitudes of the mechanical quality factor at resonance and anti-resonance. # 2009 The Japan Society of Applied Physics DOI: 10.1143/JJAP.48.041401 1. "X
¼ "X ð1  j tan 0 Þ; Introduction Piezoelectric ultrasonic motors and transformers can provide 1/10 the volume and weight of equivalent-power-level electromagnetic devices.1–3) The power density is limited by the heat generation in piezoelectric devices. In order to further improve the device miniaturization, it is necessary to understand the origin of losses and loss mechanisms. Losses in piezoelectrics are considered to have three components: dielectric, elastic, and piezoelectric.4) The dielectric loss factor can be measured from the phase delay of dielectric displacement (almost equal to polarization) with respect to the applied electric field. This is easily accomplished by a capacitance meter. The elastic loss factor is determined from the inverse of mechanical quality factor (Qm ) at the resonance point. However, little attention has been paid to the piezoelectric loss factor so far, and sometimes this term is neglected in piezoelectric studies.5) According to our previous study, relatively large piezoelectric losses were reported.6) In the experiments, we measured the hysteresis of strain and electric field (stress free), and also the hysteresis of dielectric displacement and stress (short circuit). Then the piezoelectric loss factor was obtained, which was comparable to dielectric and elastic loss factors. However, this method requires rather sophisticated and bulky instruments, e.g., tensile and compression testing machine (Instron). Thus, it is far from an ‘‘easy and friendly’’ method. On the other hand, a higher quality factor at the antiresonance is usually observed comparing with that at the resonance point,7,8) the reason of which we interpreted from the combination of three loss factors. In this paper we provide an alternative and, more importantly, a simple method to determine piezoelectric losses by analyzing the admittance/ impedance spectra at resonance and anti-resonance. 2. Theoretical Derivation Complex parameters are applied to express the hysteresis losses in piezoelectrics.9) We use tan 0 , tan 0 , and tan 0 to represent ‘‘intensive’’ dielectric, elastic and piezoelectric loss factors, respectively: E
E ð1Þ 0 ð2Þ d ¼ dð1  j tan  Þ: ð3Þ s ¼ s ð1  j tan  Þ;
0 Here j is the imaginary notation, "X is the relative permittivity under constant stress, sE is the elastic compliance under constant electric field, and d is the piezoelectric constant. Practically 0 is the phase delay of dielectric displacement D, under an applied electric field E; 0 is the phase delay of strain x, under an applied stress X; 0 is the phase delay of x for a given E, or D for a given X. This is schematically depicted in Fig. 1, where we , wm , and wem are used to denote dielectric (electrical) loss, elastic (mechanical) loss, and electromechanical loss, respectively. The loss for each process can be derived by the hysteresis loop. It should be noted that in Fig. 1(c) the product x0 E0 must be multiplied by the scale factor (d=s) to convert to energy density units (N/m2 ), where d is the piezoelectric coefficient (m/V) and s the elastic compliance (m2 /N), and in Fig. 1(d) the product D0 X0 must be multiplied by (d="0 ") with " being the dielectric constant. This is the principle for our previous methodology to determine the loss factors.10) It should be noted that these three loss factors are depicting the ‘‘intensive’’ losses in terms of ‘‘intensive’’ parameters X and E, which are externally controllable. On the other hand, to consider the real physical meaning of losses ‘‘extensive’’ loss factors must be used, which can be derived from intensive losses.10) In this paper, we derive the necessary equations for a simplest rectangular piezoelectric sample in k31 fundamental mode as shown in the Fig. 2. Similar derivations can be extended to a disk and a rod mode. For the piezoelectric plate, the length l, width w, and thickness b should satisfy l  w  b. In this case the electromechanical coupling factor k31 is given by 2 k31 ¼ 2 d31 : sE11 "0 "X 33 ð4Þ The electric admittance for the ceramic (Y) can be represented as 041401-1 # 2009 The Japan Society of Applied Physics Jpn. J. Appl. Phys. 48 (2009) 041401 Y. Zhuang et al. Admittance Y max 3dB Y max 2 ω a1 ω a ω a2 Frequency (a) Fig. 1. Schematic representation of experimentally determined hysteresis curves: (a) D–E (stress-free conditions), (b) x–X (short-circuit conditions), (c) x –E (stress-free conditions), and (d) D–X (shortcircuit conditions). Admittance 2Y min Electrode 3dB P b Y min w z l y x Fig. 2. Longitudinal vibration through the transverse piezoelectric effect in a rectangular plate (k31 mode). ω b1 ωb ω b2 Frequency (b)
 "0 "33 wl 2 2 tanð!l=2vÞ 1  k31 þ k31 ; Y ¼ j! b !l=2v Fig. 3. Admittance schematics: (a) resonance peak and (b) antiresonance peak. ð5Þ and the complete expression with loss factors is given by10) Y ¼ j!Cd ð1  j tan Þ   tanð!l=2v
Þ ð6Þ 2 ; ½1  jð2 tan 0  tan 0 Þ þ j!C0 k31 !l=2v
where ! is the driving frequency in radians, Cd the damped capacitance, C0 the motional capacitance, tan  the ‘‘extensive’’ dielectric loss factor, and v
the sound velocity in the material with loss effect: wl "0 "X 33 ; b 2 Cd ¼ ð1  k31 ÞC0 ; 1 2 ½tan 0  k31 ð2 tan 0  tan 0 Þ; tan  ¼ 2 1  k31   1 tan 0 : v
¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ v 1 þ j 2 sE11 ð1  j tan 0 Þ C0 ¼ ð7Þ ð8Þ and anti-resonance (B-type resonance), respectively. According to the definition, the quality factors are given by: !a ; ð11Þ QA ¼ !a2  !a1 !b ; ð12Þ QB ¼ !b2  !b1 where !a is the resonance frequency, !b is the antiresonance frequency, and (!a2  !a1 ) or (!b2  !b1 ) corresponds to the 3 dB bandwidth in the admittance curves around the resonance or anti-resonance peaks, respectively. As is derived in our previous paper,10) QA is only related to the intensive elastic loss: ð9Þ ð10Þ In eq. (10),  is the mass density, and the result is derived with the first-order approximation considering that the loss factor tan 0 is a small number (typically less than 0.1). Figure 3 shows schematic illustrations of the admittance and impedance spectra near the piezoelectric resonance and anti-resonance modes. Then QA and QB are used to represent the quality factor under the resonance (A-type resonance) QA ¼ 1 : tan 0 ð13Þ Next, the expression for QB will be derived in terms of loss factors and coupling coefficient. We will introduce a new parameter for analysis: ¼ !l : 2v ð14Þ is called a ‘‘normalized frequency’’ here, which is a real number and proportional to !, considering v is a real number as well. Accordingly, 041401-2 # 2009 The Japan Society of Applied Physics Jpn. J. Appl. Phys. 48 (2009) 041401 Y. Zhuang et al.  1 0 1  j tan  ; ð15Þ 2   tan 0 tanð!l=2v
Þ ¼ tan j 2 tan 0 : ð16Þ ¼ tan  j 2 cos2 Since first-order approximation is utilized in eq. (16), it can only be applied under the assumption that the first-order term is much larger than the second-order in the expansion. In this paper, we specify this requirement as the first-order term must be at least 10 times larger than the second-order portion. Putting eqs. (15) and (16) into the admittance expression eq. (6), we can get the following equation: From eq. (23), we can also get:  !l ¼ 2v
l Y ¼ Cr þ jCi ; 2vC0 ð17Þ cos2 ¼ b ð1  Smin ¼ A2 2 þ tan 0 k31 2 Ci ¼ ð1  k31 Þ 2 cos2 ; ð18Þ 2 þ k31 tan 2  k31 M tan 0 2 cos2 ; Ci ¼ ð1  þ tan : 0 0 b 2 þ k31 tan b ¼ 0; tan 0 ¼ tan 0 2 cos2 ð21Þ 0 ¼ b b b þ
: ð30Þ 1
; cos2 b 1 þ 2 b tan þ 2 cos2 b b þ ð31Þ b
: ð32Þ where 1 cos2 b 1 þ 2 b tan b 2 þ k31 tan 0 ; 2 cos2 b 2 k31 2 C ¼ ð1  k31 Þþ : cos2 b Since C  B, the solution can be obtained as 2 2 B ¼ ð1  k31 Þ tan  þ k31 M ð23Þ b !b l : 2v ð29Þ Putting eqs. (31) and (32) into eq. (29), we can get:  2   b b 2 A þ 2AB  ðB2 þ C 2 Þ ¼ 0; ð33Þ


¼ ð28Þ : Þ ¼ 2Smin : ¼ 2 cos2 where b # Accordingly, Consequently, the anti-resonance corresponds to the minimum value of Sð Þ. Since each term in the Cr part includes loss factors (small parameters), the value of Sð Þ is mainly determined by Ci , while Cr is still essential for deriving the 3 dB bandwidth. Therefore, at the anti-resonance point we can have the following approximation: 2 Ci ¼ ð1  k31 Þ 2 b It is notable that the 3 dB bandwidth corresponds to 2Smin here, because Sð Þ is proportional to jYj2 . Assuming that the difference between b and 0 is
, we obtain: ð19Þ To study the admittance magnitude, we define the function Sð Þ, which is proportional to jYj2 : 2    L ð22Þ Y  ¼ Cr2 þ Ci2 : Sð Þ ¼  2vC0 ð27Þ Note that tan  in eq. (25) has been replaced by tan 0 using eq. (9). For the 3 dB bandwidth, we take into account the following equation: 3 tan 0 : ð20Þ 2 Since loss factors are typically quite small, tan , tan 0 , and M can be treated as perturbations and some terms in the equation can be neglected so that: 2 k31 ð26Þ 2 b: A ¼ ðtan 0 þ tan 0  2 tan 0 Þ "  2 tan 0 1 1þ  k31 þ k31 2 M ¼ 2 tan 0  2 k31 Þ : where Sð 2 þ Mk31 tan 4 þ k31 Then, considering eqs. (9), (20), (23), and (26), eq. (25) can be rewritten as where 2 Þ Cr ¼ tan ð1  k31 4 k31 2 2 2 k31 Þ b ¼ C : A ð34Þ ð35Þ ð36Þ According to the definition of quality factor, QB is given ð24Þ by Then the minimum value of Sð Þ is given by
2 0 2 b k31 tan  2 2 : Þ b þ Mk31 tan b þ Smin ¼ tan ð1  k31 2 cos2 b ð25Þ QB ¼ b 2j
j ¼ C ; 2A ð37Þ i.e., 2 k31 cos2 b QB ¼
 2 1 2ðtan 0 þ tan 0  2 tan 0 Þ þ 1 þ  k31 k31 2 ð1  k31 Þþ 041401-3 2 b  ; ð38Þ tan 0 # 2009 The Japan Society of Applied Physics Jpn. J. Appl. Phys. 48 (2009) 041401 Y. Zhuang et al. where b ¼ !b l=2v (=2 < b < ). Considering eqs. (13) and (26), we can finally get the following equation: 1 1 ¼ þ QB QA 2  1  k31 1þ k31 2 ðtan 0 þ tan 0  2 tan 0 Þ: 2 b ð39Þ 0 Therefore, the piezoelectric loss factor tan  can be calculated by QA , QB , k31 , and other loss factors as tan 0 þ tan 0 tan 0 ¼ 2  "  2 # ð40Þ 1 1 1 1 2 þ   k31 1þ b : 4 QA QB k31 Then, let us discuss the prerequisite for eqs. (39) and (40). As mentioned before, final results are based on eq. (16) with the first-order approximation:  0 b tan 
tanð!b l=2v Þ ¼ tan b  j 2 ð41Þ ¼ tan b þ t1 þ t2 þ     tan b þ t1 ; (a) 10 10 r t 10 3 2 1 where tan 0 ; ð42Þ 2 cos2 b 2 sin b tan2 0 t2 ¼  b : ð43Þ 2 cos3 b In order to guarantee the feasibility of first-order approximation, we claim in this paper that the first-order term in the expansion t1 should be 10 times larger than the second-order t2 . Therefore, if we define the ratio of the two terms as rt , it must satisfy t1 ¼ j rt ¼ b jt1 j > 10: jt2 j 10 10 QA=100 QA=500 QA=1000 QA=1500 0 -1 0.1 0.2 0.3 0.4 0.5 k 0.6 0.7 0.8 0.9 31 (b) Fig. 4. rt values determined by QA and k31 : (a) general trend and (b) k31 dependence for materials with different QA . ð44Þ (5) Finally obtain tan 0 by: Considering eq. (23) as well, the ratio of jt1 j and jt2 j is basically determined by k31 and tan 0 . Figure 4 shows the dependence of rt on k31 and QA . In Fig. 4(b), rt increases with k31 for a given QA , and the theory is only effective for the points above the straight line (rt ¼ 10). Therefore, there is a lower boundary of k31 for the first-order approximation, represented as kmin here. This is also the application condition of eqs. (39) and (40). Further, the result shows that a higher QA gives a smaller kmin . In summary, piezoelectric loss factor tan 0 in k31 mode can be derived through the admittance/impedance analysis, whose value is finally determined by QA , QB , tan 0 , tan 0 , and k31 . The derivation process is as follows: (1) Obtain tan 0 from a capacitance meter at a frequency far off from the resonance or anti-resonance range. (2) Obtain the following parameters experimentally from an impedance spectrum analyzer: !a , !b , QA , QB (from the 3 dB bandwidth method) and the normalized frequency b ¼ !b l=2v. (3) Obtain tan 0 from the inverse value of QA (quality factor at the resonance). (4) Calculate k31 from the !a and !b with the following equation:11)
 2 k31  !b ð!b  !a Þ tan ¼ : 2 1  k31 2 !a 2!a tan 0 ¼ 3. tan 0 þ tan 0 2  "  2 1 1 1 1 þ   k31 1þ 4 QA QB k31 2 b # : Simulation and Experiment Results 3.1 Finite element method simulation The necessity of the value tan 0 was demonstrated in a finite element method (FEM) computer simulation. We employed the software ATILA (ver. 5.2.4) commercialized by Institute Superieure de l’Electronique et du Numerique, Lille, France (ISEN) and distributed by Micromechatronics, which has the capability to apply three losses (dielectric, elastic, and piezoelectric losses) in the simulation. So far, none of the studies using ATILA in the literature reported the computer simulation with the piezoelectric loss. In our simulation, a typical ‘‘hard’’ lead zirconate titanate ceramic (PZT-8) plate was used with a dimension of 20  3  1 mm3 . For this material, QA ¼ 1000, i.e., tan 0 ¼ 0:001; tan 0 ¼ 0:004; k31 ¼ 0:3. Several tan 0 values were selected as shown in Table I, and the admittance spectra were plotted in each case. Figure 5(a) gives the mesh condition of FEM, and Fig. 5(b) shows the results for different tan 0 . Then, QA or QB can be obtained around resonance or anti-resonance peak. It is important to note that when 2 tan 0 ¼ tan 0 þ 041401-4 # 2009 The Japan Society of Applied Physics Jpn. J. Appl. Phys. 48 (2009) 041401 Y. Zhuang et al. Table I. Quality factors obtained by ATILA FEM and analytical calculation. tan 0 QA by simulation QB by simulation QB by analytical calculation Table II. Material properties of APC 841 and Mn–PMN–PT (k31 mode). 0.001 0.0025 0.004 997.8 810.4 997.8 999.1 997.8 1287.7 APC 841 1298.5 Mn–PMN–PT 813.1 1000.0 Material QA (1= tan 0 ) QB 1183 1986 0.0035 0.31 0.0042 240 445 0.007 0.50 0.0094 ð45Þ D ¼ dX: ð46Þ Admittance Magnitude (S) Let us consider eq. (45) first. On the whole, the hysteresis loss in this case is the piezoelectric loss tan 0 . However, the process from E to x (E ! x) can be viewed as two successive processes: (1) the applied electric field E generates an induced dielectric displacement D (almost the same as the polarization P), and (2) this P results in strain x. The delay time from the first process may be derived from -2 10 -3 10 -4 10 tanθ'=0.001 tanθ'=0.0025 tanθ'=0.004 P ¼ "0 "E; -6 10 85 86 87 88 89 90 91 Frequency (kHz) (b) Fig. 5. Simulation by ATILA FEM: (a) mesh structure and (b) simulated admittance curves. tan 0 ¼ 0:005, QA ¼ QB ; when 2 tan 0 < tan 0 þ tan 0 , QA > QB ; when 2 tan 0 > tan 0 þ tan 0 , QA < QB . On the other hand, analytical solutions were calculated using eq. (39). Both simulation and analytical results are listed in Table I, which shows an excellent match. 3.2 Application to practical piezoelectric materials The QB values were measured using an HP Impedance Analyzer (HP4192) under an electric voltage level of 0.5 Vrms. In the experiment, we used a hard PZT based ceramic sample (17  3  1 mm3 ) APC 841 (American Piezo Ceramic International), and a manganese-doped Pb(Mg1=3 Nb2=3 )O3 –PbTiO3 (Mn–PMN–PT) single crystal (Ceracomp) with a size of 17  3  1 mm3 . The details on the experiments will be reported in successive papers.12,13) Table II provides the properties of APC 841 and Mn– PMN–PT single crystal. By applying eq. (40), the piezoelectric loss factor tan 0 was obtained. It is notable that the piezoelectric loss is comparable to, and even larger than other two losses. 4. tan 0 x ¼ dE; -1 10 -5 k31 by the term (tan 0 þ tan 0  2 tan 0 ). According to eq. (39), if 2 tan 0 > ðtan 0 þ tan 0 Þ, then QB > QA , which is consistent with experimental observations. Otherwise, if 2 tan 0 < ðtan 0 þ tan 0 Þ, then QA > QB which has not been experimentally observed so far. The condition 2 tan 0 ¼ ðtan 0 þ tan 0 Þ, sets QB ¼ QA . In what follows, we present a possible explanation of these observations using Devonshire theory for a polarizable–deformable insulator. Let’s start from the piezoelectric equations: (a) 10 tan 0 Phenomenological Model The difference between QB and QA is mainly determined ð47Þ leading to tan 0 . The loss factor for P ! x is denoted as tan  0 . Now, we can represent the intensive piezoelectric loss tan 0 as a linear combination of the intensive dielectric loss tan 0 and electrostrictive loss tan  0 : tan 0 ¼ tan 0 þ tan  0 : ð48Þ In order to identify the meaning of tan  0 , we further consider the Devonshire phenomenology.14) Taking into account the electrostrictive coupling in perovskite-type ferroelectrics in a ferroelectric phase, we obtain x ¼ QðPs þ
PÞ2 ¼ QP2s þ 2QPs
P þ ðhigh order termsÞ: ð49Þ Here, Q is the electrostriction coefficient (this is not the mechanical quality factor in this section), and Ps is the spontaneous polarization. The term QP2s corresponds to a huge spontaneous strain, and the second term 2QPs
P corresponds to the smaller piezoelectrically induced strain. Considering
P ¼ "0 "E, the piezoelectric coefficient d is represented as: d ¼ 2"0 "QPs : ð50Þ Now, we can introduce the complex numbers for d, ", and Q, leading to the loss factors tan 0 , tan 0 , and tan  0 . tan  0 is denoted as the ‘‘electrostrictive loss factor’’. Similarly, we can start the analysis from eq. (46). Initially, stress X induces strain x and provides the elastic loss tan 0 . Then the strain generates polarization (or dielectric displacement) through the converse electrostrictive effect (x ! P), whose loss factor is denoted as tan . The accumulated loss is therefore represented as: 041401-5 # 2009 The Japan Society of Applied Physics Jpn. J. Appl. Phys. 48 (2009) 041401 Y. Zhuang et al. tan 0 ¼ tan 0  tan : ð51Þ Summing up eqs. (48) and (51), we finally get an important expression: 2 tan 0  tan 0  tan 0 ¼ tan  0  tan : ð52Þ 0 According to eq. (52), only if tan  is larger than tan  (we expect that both factors tan  0 and tan  are positive), tan 0 can be larger than ðtan 0 þ tan 0 Þ=2, which explains consistently the experimental results: the anti-resonance quality factor QB is higher than the resonance quality factor QA , in consideration of eq. (39). 5. Conclusions In this paper, we proposed a simple, easy and friendly method to determine the piezoelectric loss factor tan 0 in k31 mode through admittance/impedance spectrum analysis. The derivation process is as follows: (1) Obtain tan 0 from a capacitance meter at a frequency far off from the resonance or anti-resonance range; (2) Obtain the following parameters experimentally from an impedance spectrum analyzer: !a , !b , QA , QB (from the 3 dB bandwidth method), and the normalized frequency b ¼ !b l=2v; (3) Obtain tan 0 from the inverse value of QA (quality factor at the resonance); (4) Calculate k31 from the !a and !b with the following equation:11)
 2 k31  !b ð!b  !a Þ ¼ tan ; 2 1  k31 2 !a 2!a (5) Finally obtain tan 0 by: tan 0 þ tan 0 tan 0 ¼ 2  "  2 1 1 1 1   k31 þ 1þ 4 QA QB k31 2 b # : The necessity of tan 0 has been demonstrated in the FEM computer simulation, particularly in the quality factor analysis. In addition one physical explanation is introduced, based on loss mechanisms in the piezoelectric effect. From experiments, anti-resonance mechanical quality factor QB always shows a higher value than resonance mechanical quality factor QA , leading to a larger tan 0 than the value of ðtan 0 þ tan 0 Þ=2. In order to explain the physical origin, we introduced the new ‘‘electrostrictive’’ loss factors tan  0 and tan  for the electrostrictive coefficients. Acknowledgements This work was supported by the Office of Naval Research, U.S.A. through the Contract N00014-99-1-0754. The authors also would like to thank Professor Ho-Yong Lee, Ceracomp Co., Ltd., Korea, for providing single crystals. 1) T. Sashida: Mech. Autom. 15 (1983) 31 [in Japanese]. 2) S. Cagatay, B. Koc, P. Moses, and K. Uchino: Jpn. J. Appl. Phys. 43 (2004) 1429. 3) C. A. Rosen: Proc. Electronic Component Symp., 1957, p. 205. 4) G. E. Martin: Proc. Ultrasonics Symp., 1974, p. 613. 5) T. Ikeda: Fundamentals of Piezoelectric Materials Science (Ohmsha, Tokyo, 1984) p. 83 [in Japanese]. 6) K. Uchino, J. H. Zheng, Y. H. Chen, X. H. Du, J. Ryu, Y. Gao, and S. Ural: J. Mater. Sci. 41 (2006) 217. 7) S. Hirose, M. Aoyagi, Y. Tomikawa, S. Takahashi, and K. Uchino: Ultrasonics 34 (1996) 213. 8) A. V. Mezheritsky: IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49 (2002) 484. 9) G. Arlt and H. Dederichs: Ferroelectrics 29 (1980) 47. 10) K. Uchino and S. Hirose: IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48 (2001) 307. 11) IEEE Std. 176 (1987) p. 54. 12) A. Rajapurkar, Y. Zhuang, S. O. Ural, S.-H. Park, and K. Uchino: submitted to Jpn. J. Appl. Phys. 13) A. Rajapurkar, S. O. Ural, H.-Y. Lee, A. Amin, and K. Uchino: submitted to J. Electroceram. 14) A. F. Devonshire: Adv. Phys. 3 (1954) 85. 041401-6 # 2009 The Japan Society of Applied Physics