Homogeneity Evaluation of SAW Device Wafers by the LFB Ultrasonic Material Characterization System

Transcription

1 Homogeneity Evaluation of SAW Device Wafers by the LFB Ultrasonic Material Characterization System Jun-ichi Kushibiki and Yuji Ohashi Department of Electrical Engineering, Tohoku University, Sendai , Japan Abstract A line-focus-beam ultrasonic material characterization (LFB-UMC) system was developed to evaluate large-diameter crystals and wafers currently used in electronic devices. The system enables highly accurate detection of slight changes in the physical and chemical properties in and among specimens, as velocity changes of Rayleigh-type leaky surface acoustic waves (LSAWs) excited on the water-loaded specimen surface. A measurement procedure for precisely measuring the LSAW velocities was completed, achieving greater relative accuracy to better than ±0.002% at any single chosen point and absolute accuracy around ±0.01%. The system was applied to evaluate and improve homogeneities of LiNbO 3 and LiTaO 3 crystals and wafers produced in industry for SAW and optoelectronic devices. A guideline for standardizing the specifications in chemical composition for SAW-grade LiTaO 3 wafers was established and the first demonstration of improving the crystal growth conditions for commercially available optical-grade LiTaO 3 crystals, i.e., the starting material composition, was successfully made. I. INTRODUCTION Ferroelectric single crystals of LiNbO 3 [1-4] and LiTaO 3 [1, 5, 6] are widely used as substrate materials not only for surface acoustic wave (SAW) devices but also for optoelectronic devices. Performance of those devices depends upon the quality of materials and the control of device fabrication processes. Therefore, it is of fundamental importance to establish the growth conditions for producing large-diameter crystals with high homogeneity in acoustic and optical properties. We have proposed an ultrasonic method of line-focusbeam (LFB) acoustic microscopy as a new, unique technique for characterizing and evaluating those crystals and wafers [7-14]. Velocity information of leaky SAWs (LSAWs) that are excited using a wedge-shaped focused wave and propagated along one desired direction on the water-loaded specimen surfaces is mainly utilized for evaluation through V(z) curve measurements [7]. This technology enables detecting changes in chemical and physical properties, such as chemical composition, lattice constant, refractive index, and density, as variations in LSAW velocity. We have successfully demonstrated the usefulness and uniqueness for materials characterization through experimental evidence [10, 11]. We have developed various experimental procedures and applications to resolve various kinds of scientific and industrial problems associated with LiNbO 3 and LiTaO 3 crystals and wafers [8-14]. We have developed a new version of the LFB ultrasonic material characterization (UMC) system. It is capable of detecting a very slight velocity change with a resolution better than ±0.002% and is practically applicable to resolve material problems of single crystals and wafers for electronic devices [8]. We have applied the system to investigate homogeneities of LiNbO 3 and LiTaO 3 crystals and wafers produced in industry for optoelectronic device use [11] as well as for SAW device use [13, 14, 33]. In this report, the results are briefly presented. II. LFB-UMC SYSTEM A. System and Accuracy The measurement principle of the LFB-UMC system was described in detail in the literature [7]. The LFB- UMC system enables us to measure the propagation characteristics, viz., phase velocity and attenuation, of LSAWs propagating on the water-loaded specimen surface by analyzing V(z) curves obtained when varying the relative distance z between the LFB ultrasonic device and the specimen, as illustrated in Fig. 1, which represents the principle of forming the V(z) curve. The ultrasonic device consists of a Z-cut sapphire rod with a ZnO piezoelectric film transducer on the top face and a cylindrical acoustic lens on the bottom face. Components #0 and #1 in the figure interfere with each other when the relative distance z is varied. An interference waveform of the V(z) curve is then obtained, as shown in Fig. 2. The velocity information of LSAWs is mainly employed for precise material characterization. The LSAW velocity V LSAW can be obtained from the oscillation interval z in the interference waveform using the following equation: VLSAW = VW. (1) V 2 1 W 1 2 f z Here, V W is the longitudinal velocity in water and f is the ultrasonic frequency.

2 The block diagram of the newly developed LFB-UMC system is shown in Fig. 3. The system is designed to provide a stable temperature environment for a longer period than can conventional systems, with the mechanical system containing the ultrasonic device and specimen in a temperature-controlled chamber. In addition, this system is equipped with a specimen transfer system to enable the loading and changing of specimens, a temperaturecontrolled pure water supply/drain system to supply and drain pure water as the water couplant, and an automatic three-axis tilting stage to align the ultrasonic device and specimen while keeping the chamber closed, so that a highly accurate measurement can be obtained efficiently without disturbing the stable temperature environment. Furthermore, the two-dimensional distributions on the substrate are measured with a greater accuracy for the V LSAW measurement by introducing a technique to more precisely measure the temperature of the water couplant and the longitudinal velocity, which directly affect the measurement accuracy [8]. The system [14] operates with the pulse mode measurement system [15] using RF tone burst signals. Experiments in this report are conducted using an LFB ultrasonic device designed for 225-MHz operation with a cylindrical concave surface of 1-mm radius and an aperture half-angle of 60, which can be used in the frequency range from 100 MHz to 300 MHz. The newest system accurately positions the z stage with a resolution of 10 nm using a semiconductor laser interferometer. At the operating ultrasonic frequency of 225 MHz, the relative measurement accuracy of V LSAW in this system is ±0.0013% (±2σ, σ is the standard deviation) at an arbitrary single chosen point on the surface of the specimen and ±0.003% for a two-dimensional continuous scanning area of 75 mm 75 mm. The absolute accuracy is around ±0.01% as determined by the system calibration method using standard specimens [16, 17]. B. Standard Specimens for System Calibration The LFB system must be calibrated when the absolute values of LSAW propagation characteristics need to be obtained, as in determining elastic constants using the LFB-UMC system [18-20]. For this purpose, we proposed a system calibration method using standard specimens [16, 17]. The system is calibrated by measuring acoustical physical constants (elastic constants, piezoelectric constants, dielectric constants and density) for standard specimens, and comparing measured values of LSAW propagation characteristics with the theoretical ones calculated using the constants determined. Therefore, it is important in this method to measure the acoustical physical constants precisely. We developed substrates of LiNbO 3 and LiTaO 3 crystals [21, 22] as standard specimens. ZnO TRANSDUCER θ FOCAL PLANE LSAW y #0 #1 V (z) V (z) 0 1 z x RF PULSE SPECIMEN LINE-FOCUS-BEAM ACOUSTIC LENS WATER defocus Fig. 1. Cross-sectional geometry of the LFB ultrasonic device describing the principle of V(z) curve measurements. RELATIVE OUTPUT [db] f = 225 MHz DISTANCE z [µm] Fig. 2. V(z) curve obtained for a 128 YX-LiNbO 3 specimen. Melt composition: 48.5 Li 2 O-mol%. DIGITAL VOLT METER PURE WATER SUPPLY SYSTEM SAMPLE LOADER/ UNLOADER Directional Bridge Thermocouple MECHANICAL STAGE LFB Ultrasonic Device Water TEMPERATURE-CONTROLLED CHAMBER PULSE MODE MEASUREMENT SYSTEM A/D CONVERTER Specimen V(z) STAGE CONTROLLER COMPUTER Fig. 3. Block diagram of the LFB-UMC system.

3 C. Bulk-Wave Velocity Measurements In the LFB-UMC system developed above, we can replace the LFB ultrasonic devices with plane-wave (PW) ultrasonic devices and can precisely measure the velocity of bulk waves in the composite ultrasonic transmission line using the complex-mode measurement method. The measurement method and system are described in detail elsewhere [23]. The experimental arrangement is illustrated in Fig. 4. The PW ultrasonic device employed here consists of a cylindrical buffer rod of synthetic silica (SiO 2 ) glass with a transducer fabricated on one end of the rod. A ZnO piezoelectric film transducer was used for the longitudinal wave measurement, and an X-cut LiNbO 3 transducer bonded to the rod, for the shear wave measurement. Coupling materials are pure water for longitudinal waves and salol (phenyl salicylate) for shear waves. Salol bonds the buffer rod and a specimen with a thickness of less than 1 µm, typically 0.2 to 0.5 µm. In the longitudinal wave velocity measurements, a pure water layer is established with a certain distance, here typically mm, between the SiO 2 buffer rod and the specimen, so the reflected pulse signals from the buffer rod end, V 1, can be separated from the reflected pulse signals from the front surface (V 2 ) and back surface (V 3 ) of the specimen in the time domain. By measuring the phase shift of V 3 /V 2, φ, and the thickness of the specimen h, we obtain the bulk wave velocity of the specimen, V l, by the following equation: h Vl = 2ω φ π θ, (2) where ω is the angular frequency, and θ is the difference between the phase advances of signals V 2 and V 3 caused by diffraction. RF pulse COUPLING MATERIAL SiO 2ROD TRANSDUCER V 1 V 2 SPECIMEN Fig. 4. Experimental arrangement for bulk velocity measurements of solid specimens using bulk ultrasonic RF pulses. V3 h AIR In the shear wave velocity measurements, additional phase shift occurs when shear waves transmit through or reflect on the bonding layer since V 1 signals cannot be separated from V 2 signals in the time domain. The effect of this phase shift, θ BL, on the velocity measurement must be corrected. θ BL is contained in the phase shift φ to be measured. It can be calculated using the acoustic parameters (velocity, attenuation coefficients, and density) of the bonding layer and by estimating the layer thickness by comparing measured and calculated frequency dependences of the reflection coefficient at the boundary between the SiO 2 buffer rod and the bonding layer. This system was used for developing the piezoelectric standard specimens. III. SENSITIVITY TO CHEMICAL COMPOSITION VARIATIONS Acoustic inhomogeneities observed as velocity variations might be caused primarily by chemical composition ratio changes, secondarily by residual multidomains due to an incomplete poling process, and thirdly by surface damage during the slicing and polishing processes. The first cause must still be considered to be the main problem. The second problem causes great changes in velocity but can be easily removed by paying careful attention to the proper poling operation. The third is a future problem that is more serious for SAW devices operating in the SHF range, as well as for surface waveguide-type optical devices, which utilize the material properties within one micron beneath the surface. To obtain basic data for evaluating homogeneities of LiNbO 3 and LiTaO 3 single crystals, we experimentally investigated the relationships among LSAW velocities, chemical compositions, Curie temperatures, densities, and lattice constants as the calibration lines for crystal evaluation using X-, Y-, and Z-cut substrates prepared from three LiNbO 3 and LiTaO 3 single crystals grown with different Li 2 O contents ranging from 48 to 49 mol% [11, 24]. For example, it is shown for LiTaO 3 that, as the Li 2 O content increases around the congruent composition, the LSAW velocities linearly increase for all specimen surfaces and all propagation directions, and the increase rate is maximum for Z-cut, Y-axis propagating (ZY) LiTaO 3 [11]. We successfully demonstrated that this ultrasonic method and system have much higher sensitivities and resolutions, as determined by velocity measurements, to the chemical and physical properties than those of conventional methods, such as differential thermal analysis [5, 6, 25], X-ray diffractometry by the Bond method [26], the prism coupler method [27], and the Archimedes method [28].

4 Table I. Chemical composition dependences of acoustical physical constants of LiNbO 3 and LiTaO 3 crystals. LiNbO 3 LiTaO 3 Elastic constant ( N/m 2 ) Piezoelectric constant (C/m 2 ) Dielectric constant Density (kg/m 3 ) Congruent c11 E ± c12 E ± c13 E ± c14 E ± c33 E ± c44 E ± c66 E ± e ±0.022 e ±0.015 e ±0.032 e ±0.054 ε11 ε0 ± ε33 ε0 ± ρ ±0.2 Gradient (/mol%) Congruent ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.4 Gradient (/mol%) IV. REFERENCES FOR CHARACTERIZATION To evaluate and analyze LiNbO 3 and LiTaO 3 single crystals using the LFB-UMC system, we must prepare calibration lines of the acoustic properties to be measured against the chemical composition to be evaluated. For example, when implementing an evaluation with the LSAW velocities measured by the LFB system, we must obtain the relationship between the chemical compositions and LSAW velocities by preparing several substrate specimens with different chemical compositions for a required cut surface and then measuring the LSAW velocities on each specimen. It is very time-consuming. Chemical compositions of rotated Y-cut X- propagating (128 YX)-LiNbO 3 [29], X-cut rotated Y-propagating (X-112 Y)-LiTaO 3 [30], and 36 YX-LiTaO 3 [31] substrates have been so far evaluated according to such experimental procedures [10, 13]. In addition, we are unable to directly evaluate the SAW velocities, which are closely related to the characteristics of the SAW devices, and, in particular, the pseudo surface acoustic wave (PSAW) velocities of SH type, since the LFB system enables measuring only the Rayleigh propagation mode on the water-loaded specimen surfaces from its principle of operation. If we could have all independent components of the acoustical physical constants of these crystals as a function of the chemical composition ratio, we could obtain chemical composition dependences of the acoustic properties for an arbitrarily cut specimen surface, wave propagation direction, and mode by numerical calculations, enabling a very efficient evaluation.

5 Furthermore, the calculated relationships between LSAW velocities and SAW velocities can be easily obtained. We could then evaluate SAW velocities from measured LSAW velocities. So, we have determined the chemical composition dependences of the constants for LiNbO 3 and LiTaO 3 crystals around the congruent composition, as shown in Table I [21, 22]. V. STANDARDIZED EVALUATION FOR LiTaO 3 CRYSTALS In this study of evaluating the chemical compositions of LiTaO 3 crystals, we implemented a standardized comparison and evaluation of the tolerances for T C and a, which are determined independently by individual manufacturers, through the Rayleigh-type V LSAW measured by the LFB-UMC system [14]. We first measured V LSAW for 42 YX-LiTaO 3 wafers obtained from three crystal manufacturers, and then experimentally obtained the relationship between T C and a measured by the individual manufacturers. Using the relative accuracy of V LSAW measurements, ±0.0013% (±0.04 m/s), we can estimate the resolutions of this ultrasonic method for the chemical and physical properties of 42 YX-LiTaO 3 crystals, as shown in Table II. The relationship between V LSAW and V SAW for 42 YX-LiTaO 3 wafers was obtained through numerical calculations based on the chemical composition dependences of the acoustical physical constants of LiTaO 3 crystals, as shown in Fig. 5. T C and a can be converted into values of the same parameter (V LSAW or V SAW ) using these relationships. As a result, we found that the tolerance for a of ± nm is 1.6 times larger than the tolerance for T C of ±3 C, thus clarifying one problem in the current evaluation methods, as presented in Table III. However, we suggested a serious problem, associated with the measuring instruments and conditions for T C, that the experimental results revealed a difference of 2 C between the T C values measured by two individual manufacturers. A standardized tolerance of ±0.04% for V SAW [32] corresponds to the following tolerances for each characteristic value: ±1.6 C for T C, ± nm for a, and ±0.9 m/s for V LSAW. Using the statistical results of T C and a for LiTaO 3 crystals manufactured by two manufacturers from 1999 to 2000, we evaluated the crystals between the two manufacturers on the same scale of V LSAW. This clarified the differences in V LSAW or chemical composition for average crystal ingots from the two manufacturers as well as the differences in the distributions among crystals that correspond to the tolerances of individual manufacturers. As described above, some problems in growing and evaluating crystals were clarified by examining the interrelationships among various chemical and physical Table II. Resolutions of Li 2 O content, Curie temperature, lattice constant, and SAW velocity for 42 YX-LiTaO 3 by LSAW velocity measurements. LSAW velocity Resolution ±0.04 m/s Li 2 O content ±0.002 mol% Curie temperature ±0.07 C Lattice constant a ± nm SAW velocity ±0.08 m/s (LSAW-m/s)/mol% 42.4 (SAW-m/s)/mol% Li O CONCENTRATION [mol%] 2 Fig. 5. Calculated Li 2 O concentration dependences of LSAW and SAW velocities for 42 YX-LiTaO 3 single crystals. characteristics using the LFB-UMC system and by evaluating the crystals using the standardized scale of V LSAW. It is at least necessary to perform evaluations at the top and bottom sides of crystals to identify the variations and distributions within the crystals. Furthermore, it is important to feed the evaluation results back to the growth conditions to obtain more homogeneous crystals, and to provide such results for users who fabricate SAW devices. Grown crystals should also be evaluated under standardized criteria to supply crystal substrates with small variations in the physical and chemical characteristics. To that end we believe that this study will provide an important guideline for standardizing the specifications for wafers. In addition, since the LFB-UMC system can be used to evaluate both Rayleigh-type SAW device materials and SH-type SAW device materials, we consider that this SAW VELOCITY [m/s]

6 Table III. Tolerances for lattice constant a and Curie temperature T C of LiTaO 3 single crystals and their corresponding variations of LSAW and SAW velocities. Manufacturer Property Tolerance Corresponding distribution V LSAW [m/s] V SAW [m/s] A, B T C [ C] ±3 ±1.76 ±3.27 C a [nm] ± ±2.85 ±5.27 system is an important material evaluation technology that enables superior accuracy and a standardized evaluation of SAW device materials. VI. EVALUATION OF SAW-GRADE LiTaO 3 CRYSTALS In this study, we applied the LFB-UMC system to evaluate a mass-production line of LiTaO 3 single crystals for SAW devices in a manufacturer that uses a maximum charge number of 60 to grow crystals [33]. We measured Rayleigh-type V LSAW for 36 YX-LiTaO 3 and 42 YX-LiTaO 3 wafers, as well as T C, and then compared them. This study revealed that the average V LSAW was m/s with a maximum difference of 2.2 m/s, that each crystal ingot had a velocity increase of about 1 m/s from the top to the bottom, and that there was no dependence on the charge number, as shown in Fig. 6. Because the average velocity corresponded to the chemical composition of Li 2 O-mol%, we could understand that the manufacturer was producing the crystals grown from the melt composition controlled around that value through T C measurements. We also estimated the congruent composition to be Li 2 O-mol% from V LSAW distributions along the crystal pulling-axis direction obtained for a series of the 42 YX-LiTaO 3 crystal ingots. The congruent composition was about 0.3 Li 2 O-mol% less than the melt composition controlled for the current mass production by the manufacturer. A more accurate estimate of the congruent composition will be made by further experiments using V LSAW measurements in a similar way. From this investigation, we found that the measurement conditions associated with T C measurements mainly used for chemical composition analysis in industry, including the measurement environment of room temperatures, must be reconsidered for reliable production control. VII. EVALUATION AND IMPROVEMENT OF OPTICAL-GRADE LiTaO 3 CRYSTALS We made the first demonstration of improving the crystal growth conditions for optical-grade LiTaO : Top : Middle : Bottom CHARGE NUMBER Fig. 6. LSAW velocity distributions with charge number for 42 YX-LiTaO 3 wafers prepared from five crystals of a series of 39 crystals grown successively in one furnace. crystals, i.e., the starting material composition, to obtain higher homogeneity of chemical compositions using the LFB-UMC system [34]. We evaluated a commercial LiTaO 3 single crystal (Sample Old-C) for optical use grown with a nominally congruent composition along the Y-axis direction by the CZ method, and then attempted to improve the crystal growth conditions. By investigating the chemical composition distributions in the crystal through measuring the LSAW velocity distributions on the ZY-LiTaO 3 substrate specimens, we found that changes in chemical composition were particularly significant along the pulling axis direction ( (Li 2 O- mol%)/mm), as seen from Fig. 7. We developed a method of estimating the proper starting material composition that leads to a more homogeneous crystal. A new crystal (Sample New-C) with Li 2 O content in the starting material of mol% was grown using the same growth furnace and crucible under the same growth conditions as the old crystal. The crystal grown had a smaller compositional distribution, approximately half

7 40 M 50 Y B 80 Z X seed (a) 10 T B (b) M T (c) (m/s)/mm B X 65 Z seed (a) T M Y (b) B M T (c) (m/s)/mm Fig. 7. LSAW velocity distributions for a ZY-LiTaO 3 specimen prepared from Old-C crystal ingot. (a) Twodimensional scanning area. (b) Diameter direction. (c) Pulling axis direction. Fig. 8. LSAW velocity distributions for ZY-LiTaO 3 specimen prepared from New-C crystal ingot. (a) Twodimensional scanning area. (b) Diameter direction. (c) Pulling axis direction. that before the improvement, as shown in Fig. 8. We thus succeeded in growing a more homogeneous crystal. Furthermore, similar velocity (or chemical composition) profiles were observed below the shoulder portion of the crystal boules along the pulling axis direction before and after the improvement. This might suggest problems in the growth conditions other than the starting material composition [11, 35, 36]. This is a change that cannot be detected by conventional methods, such as the Curie temperature measurements and the lattice constant measurements by the Bond method at certain positions. This gives a good example of the advantage of the LFB- UMC system in that the distributions in crystals can be measured to much greater details. As described above, once we apply the system with such excellent resolution to complicated and unsolved problems associated with crystal growth, we could detect and understand the true meanings and interpretations of the problems. A supply of homogeneous high-quality substrates is a very important point for developing future high-performance SAW devices or optical devices and for manufacturing those devices with better yield. VIII. CONCLUSION Thus, we have completed the LFB-UMC system and method for evaluating inhomogeneities in acoustic properties among crystal ingots and wafers of LiNbO 3 and LiTaO 3 single crystals due to the chemical composition changes, based on measuring the velocities, such as the

8 LSAW and bulk-wave velocities obtained by the LFB/PW-UMC system. Once we obtain a true congruent composition using this ultrasonic method, it is easy to grow homogeneous crystals from the starting material with a true congruent composition and their recycled material (such as failed crystals and material remaining at the top and bottom parts of the crystal ingots), easily satisfying the required condition of a tolerance of ±0.04% in SAW velocity [32] because the corresponding tolerance for T C is ±1.6 C [14]. We suggest that in the near future it will be possible to simplify the evaluation procedure after the establishment of crystal growth conditions for each furnace, probably eliminating the need to measure V LSAW and T C. Also, from the technical point of view in the acoustic properties, it may be possible to grow homogeneous crystals with charge numbers of over 60, leading to lower cost and improvement in productivity. In general, these results also suggest to other manufacturers that the industrial conditions (especially chemical composition) for growth of LiTaO 3 and LiNbO 3 crystals should be re-examined for more efficient production of crystals with better homogeneity. REFERENCES [1] A. A. Ballman, J. Am. Ceram. Soc. 48, 112 (1965). [2] K. Nassau, H. J. Levinstein, and G. M. Loiacono, J. Phys. Chem. Solids 27, 983 (1966). [3] K. Nassau, H. J. Levinstein, and G. M. Loiacono, J. Phys. Chem. Solids 27, 989 (1966). [4] J. R. Carruthers, G. E. Peterson, M. Grasso, and P. M. Bridenbaugh, J. Appl. Phys. 42, 1846 (1971). [5] R. L. Barns and J. R. Carruthers, J. Appl. Crystallogr. 3, 395 (1970). [6] S. Miyazawa and H. Iwasaki, J. Cryst. Growth 10, 276 (1971). [7] J. Kushibiki and N. Chubachi, IEEE Trans. Sonics Ultrason. SU-32, 189 (1985). [8] J. Kushibiki, Y. Ono, Y. Ohashi, and M. Arakawa, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 49, 99 (2002). [9] J. Kushibiki, H. Takahashi, T. Kobayashi, and N. Chubachi, Appl. Phys. Lett. 58, 893 (1991). [10] J. Kushibiki, H. Takahashi, T. Kobayashi, and N. Chubachi, Appl. Phys. Lett. 58, 2622 (1991). [11] J. Kushibiki, T. Okuzawa, J. Hirohashi, and Y. Ohashi, J. Appl. Phys. 87, 4395 (2000). [12] J. Kushibiki, T. Kobayashi, H. Ishiji, and C. K. Jen, J. Appl. Phys. 85, 7863 (1999). [13] J. Kushibiki, Y. Ohashi, and Y. Ono, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 47, 1068 (2000). [14] J. Kushibiki, Y. Ohashi, and T. Ujiie, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 49, 454 (2002). [15] J. Kushibiki, T. Sannomiya, and N. Chubachi, IEEE Trans. Sonics Ultrason. SU-29, 338 (1982). [16] J. Kushibiki and M. Arakawa, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 45, 421 (1998). [17] J. Kushibiki, M. Arakawa, and R. Okabe, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 49, 827 (2002). [18] J. Kushibiki, T. Ueda, and N. Chubachi, IEEE Ultrason. Symp. Proc. (Denver, CO, 1987), pp [19] J. Kushibiki, T. Ishikawa, and N. Chubachi, Appl. Phys. Lett. 57, 1967 (1990). [20] I. Takanaga and J. Kushibiki, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 49, 893 (2002). [21] J. Kushibiki, I. Takanaga, M. Arakawa, and T. Sannomiya, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 46, 1315 (1999). [22] J. Kushibiki, I. Takanaga, S. Komatsuzaki, and T. Ujiie, J. Appl. Phys. 91, 6341 (2002). [23] J. Kushibiki and M. Arakawa, J. Acoust. Soc. Amer. 108, 564 (2000). [24] J. Kushibiki, J. Hirohashi, and Y. Ohashi (unpublished). [25] M. Sato, A. Iwata, J. Yamada, M. Hikita, and Y. Furukawa, Jpn. J. Appl. Phys. 28, 111 (1989). [26] W. L. Bond, Acta Crystallogr. 13, 814 (1960). [27] V. V. Atuchin, Opt. Spectrosc. 67, 771 (1989). [28] H. A. Bowman and R. M. Schoonover, J. Res. Natl. Bur. Stand., Sect. C 71, 179 (1967). [29] K. Shibayama, K. Yamanouchi, H. Sato, and T. Meguro, Proc. IEEE 64, 595 (1976). [30] H. Hirano, T. Fukuda, S. Matsumura, and S. Takahashi, Proc. 1st Meeting on Ferroelectric Materials and Their Applications (Kyoto, Japan, 1978), pp [31] K. Nakamura, M. Kazumi, and H. Shimizu, IEEE Ultrason. Symp. Proc. (Phoenix, AZ, 1977), pp [32] K. Yamada, T. Omi, S. Matsumura, and T. Nishimura, IEEE Ultrason. Symp. Proc. (Dallas, TX, 1984), pp [33] J. Kushibiki, Y. Ohashi, and M. Mochizuki, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. (in press). [34] J. Kushibiki, Y. Ohashi, Y. Ono, and T. Sasamata, IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 49, 905 (2002). [35] S. Matsumura, J. Cryst. Growth 51, 41 (1981). [36] J. Trauth and B. C. Grabmaier, J. Cryst. Growth 112, 451 (1991).