dot StartStartFraction t OverOver StartFraction normal pi dot d squared Over 4 EndFraction EndEndFraction EndEndEndFraction"/>
where fr is the resonance frequency, fa is anti‐resonance frequency, and fp is parallel resonant frequency.
The frequency method is advantageous since it can determine the complete matrix of the material coefficients, though a complete set of samples needs to be manufactured. The accuracy of the calculated coefficients depends on the overall measurement accuracy of the initial parameters including resonant frequencies, anti‐resonant frequencies, density, and sample dimensions. Besides, it is worth nothing that the measured samples should comply with a minimum aspect ratio stipulated by the IEEE standards.
References
1 1 Coursey, P.R. and Brand, K.G. (1946). Dielectric constants of some titanates. Nature 157: 297–298.
2 2 Roberts, S. (1947). Dielectric and piezoelectric properties of barium titanate. Physical Review 71 (12): 890.
3 3 Shirane, G. and Takeda, A. (1952). Phase transitions in solid solutions of PbZrO3 and PbTiO3: I. Small concentrations of PbTiO3. Journal of the Physical Society of Japan 7: 5–11.
4 4 Shirane, G. and Suzuki, K. (1952). Crystal structure of Pb(Zr–Ti)O3. Journal of the Physical Society of Japan 7: 333–333.
5 5 Sawaguchi, E. (1953). Ferroelectricity versus antiferroelectricity in the solid solutions of PbZrO3 and PbTiO3. Journal of the Physical Society of Japan 8: 615–629.
6 6 Jaffe, B., Roth, R.S., and Marzullo, S. (1954). Piezoelectric properties of lead zirconate–lead titanate solid‐solution ceramics. Journal of Applied Physics 25: 809–810.
7 7 Jaffe, B., Cook, W.R., and Jaffe, H. (1971). Piezoelectric Ceramics. Academic Press.
8 8 Haertling, G.H. (1999). Ferroelectric ceramics: history and technology. Journal of the American Ceramic Society 82: 797–818.
9 9 Uchino, K. (1997). Piezoelectric Actuators and Ultrasonic Motors. Boston, MA: Kluwer Academic Publishers.
10 10 Trolier‐McKinstry, S. and Randall, C.A. (2018). Movers, shakers, and storers of charge: the legacy of ferroelectricians L. Eric Cross and Robert E. Newnham. Journal of the American Ceramic Society 100: 3346–3359.
11 11 EU‐Directive 2002/95/EC (2003). Restriction of the Use of Certain Hazardous Substances in Electrical and Electronic Equipment (RoHS). Official Journal of the European Union 46 (L37): 19.
12 12 Takenaka, T. and Nagata, H. (2005). Current status and prospects of lead‐free piezoelectric ceramics. Journal of the European Ceramic Society 25: 2693–2700.
13 13 Saito, Y., Takao, H., Tani, T. et al. (2004). Lead‐free piezoceramics. Nature 432: 84–87.
14 14 Shrout, T.R. and Zhang, S.J. (2007). Lead‐free piezoelectric ceramics: alternatives for PZT? Journal of Electroceramics 19: 111–124.
15 15 Roedel, J., Jo, W., Seifert, K.T.P. et al. (2009). Perspective on the development of lead‐free piezoceramics. Journal of the American Ceramic Society 92: 1153–1177.
16 16 Li, J.‐F., Wang, K., Zhu, F.‐Y. et al. (2013). (K,Na)NbO3‐based lead‐free piezoceramics: fundamental aspects, processing technologies, and remaining challenges. Journal of the American Ceramic Society 82: 797–818.
17 17 Roedel, J. and Li, J.‐F. (2018). Lead‐free piezoceramics: status and perspectives. MRS Bulletin 43: 576–580.
18 18 Trolier‐McKinstry, S., Zhang, S.J., Bell, A.J., and Tan, X.L. (2018). High‐performance piezoelectric crystals, ceramics, and films. Annual Review of Materials Research 48: 191–217.
19 19 Guo, R., Cross, L.E., Park, S.E. et al. (2000). Origin of the high piezoelectric response in PbZr1−xTixO3. Physical Review Letters 84: 5423–5426.
20 20 Noheda, B., Cox, D.E., Shirane, G. et al. (2001). Stability of the monoclinic phase in the ferroelectric perovskite PbZr1−xTixO3. Physical Review B 63: 014103.
21 21 Noheda, B. (2002). Structure and high‐piezoelectricity in lead oxide solid solutions. Current Opinion in Solid State and Materials Science 6: 27–34.
22 22 IEEE Standard on Piezoelectricity. (1987). ANSI/IEEE, Standard 176‐1987.
23 23 Lines, M.E. and Glass, A.M. (1979). Principles and Applications of Ferroelectrics and Related Materials. Oxford: Clarendon.
24 24 Cady, W.G. (1946). Piezoelectricity: An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals. McGraw‐Hill.
25 25 Sawyer, C.B. and Tower, C.H. (1930). Rochelle salt as a dielectric. Physical Review 35: 269–273.
26 26 Mitsui, T., Tatsuzaki, I., and Nakamura, E. (1976). An Introduction to the Physics of Ferroelectrics. London: Gordon and Breach.
27 27 Damjanovic, D. (1998). Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics. Reports on Progress in Physics 61: 1267–1324.
28 28 Jin, L., Li, F., and Zhang, S.J. (2014). Decoding the fingerprint of ferroelectric loops: comprehension of the material properties and structures. Journal of the American Ceramic Society 97: 1–27.
29 29 Berlincourt, D. and Jaffe, H. (1958). Elastic and piezoelectric coefficients of single‐crystal barium titanate. Physical Review 111: 143–148.
30 30 Jaffe, H. and Berlincourt, D.A. (1965). Piezoelectric transducer materials. Proceedings of the IEEE 53: 1372–1385.
31 31 Cain, M.G. (2014). Characterisation of Ferroelectric Bulk Materials and Thin Films. Springer.
32 32 Yamaguchi, T. and Hamano, K. (1979). Inteferometric method of measuring complex piezoelectric constants of crystals in a frequency range up to about 50 kHz. Japanese Journal of Applied Physics 18: 927–932.
33 33 Zhang, Q.M., Pan, W.Y., and Cross, L.E. (1988). Laser interferometer for the study of piezoelectric and electrostrictive strains. Journal of Applied Physics 63: 2492–2496.
34 34 Li, J.‐F., Moses, P., and Viehland, D. (1995). Simple, high‐resolution interferometer for the measurement of frequency‐dependent complex piezoelectric responses in ferroelectric ceramics. Review of Scientific Instruments 66: 215–221.
35 35 Moilanen, H. and Leppävuori, S. (2001). Laser interferometric measurement of displacement‐field characteristics of piezoelectric actuators and actuator materials. Sensors and Actuators A 92: 326–334.
36 36 Warner, A.W., Onoe, M., and Coquin, G.A. (1966). Determination of elastic and piezoelectric constant in class (3M). Journal of the Acoustical Society of America 42: 1223–1231.
37 37 Fialka, J. and Benes, P. (2013). Comparison of methods for the measurement of piezoelectric coefficients. IEEE Transactions on Instrumentation and Measurement 62: 1047–1057.
2 High‐Performance Lead‐Free Piezoelectrics
2.1 Introduction
Piezoelectric materials are used in a large variety of applications including actuators,
