# Elastic and Dielectric Evaluation of the Piezoelectric Response of Ferroelectrics Using Unpoled Ceramics

## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Piezoelectric Softening

#### 2.2. Temperature Dependence of the Piezoelectric Softening in the Simplest Case

#### 2.3. Anharmonic Stiffening and Electrostrictive Coefficients

#### 2.4. Fluctuations and Thermoelastic Effect

#### 2.5. Flexoelectric and Surface Effects

#### 2.6. Additional Terms in the Expansion of $F\left(P\right)$ and Multiple Ferroelectric Transitions

#### 2.7. Additional Structural Transitions

#### 2.8. Depolarization Field and Influence of the Measurement Frequency

#### 2.9. Extrinsic Domain Wall Contributions

#### 2.10. Polycrystalline Average of Unpoled Ceramic: A Well Defined State

#### 2.11. Experimental Verification and Porosity

#### 2.12. Usefulness of the Elastic Assessment of the Potential Piezoelectric Properties of Unpoled Samples

#### 2.13. Piezoelectric Softening Versus Electromechanical Coupling Factor

## 3. Materials and Methods

## 4. Conclusions

## Abbreviations

C | cubic |

DMA | Dynamic Mechanical Analyzer |

DW | domain wall |

FE | ferroelectric |

O | orthorhombic |

PE | paraelectric |

R | rhombohedral |

RUS | Resonant Ultrasound Spectroscopy |

T | tetragonal |

## References

- Takahashi, H.; Numamoto, Y.; Tani, J.; Tsurekawa, S. Piezoelectric Properties of BaTiO
_{3}Ceramics with High Performance Fabricated by Microwave Sintering. Jpn. J. Appl. Phys.**2006**, 45, 7405. [Google Scholar] [CrossRef] - Karaki, T.; Yan, K.; Adachi, M. Subgrain Microstructure in High-Performance BaTiO3 Piezoelectric Ceramics. Appl. Phys. Express
**2008**, 1, 111402. [Google Scholar] [CrossRef] - Jaffe, B.; Cook, W.R.; Jaffe, H. Piezoelectric Ceramics; Academic Press: London, UK, 1971. [Google Scholar]
- Berlincourt, D.; Krueger, H.A. Domain Processes in Lead Titanate Zirconate and Barium Titanate Ceramics. J. Appl. Phys.
**1959**, 30, 1804–1810. [Google Scholar] [CrossRef] - Cook, W.R.; Berlincourt, D.A.; Scholz, F.J. Thermal Expansion and Pyroelectricity in Lead Titanate Zirconate and Barium Titanate. J. Appl. Phys.
**1963**, 34, 1392. [Google Scholar] [CrossRef] - Zhao, C.; Hou, D.; Chung, C.C.; Zhou, H.; Kynast, A.; Hennig, E.; Liu, W.; Li, S.; Jones, J.L. Deconvolved intrinsic and extrinsic contributions to electrostrain in high performance, Nb-doped Pb(Zr
_{x}Ti_{1−x})O_{3}piezoceramics (0.50 ≤ x ≤ 0.56). Acta Mater.**2018**, 158, 369–380. [Google Scholar] [CrossRef] - IEEE. IEEE Standard on Piezoelectricity ANSI/IEEE Standard No. 176-1987; IEEE: Piscataway Township, NJ, USA, 1987. [Google Scholar]
- Pérez, N.; García, A.; Riera, E.; Pardo, L. Electromechanical Anisotropy at the Ferroelectric to Relaxor Transition of (Bi
_{0.5}Na_{0.5})_{0.94}Ba_{0.06}TiO_{3}Ceramics from the Thermal Evolution of Resonance Curves. Appl. Sci.**2018**, 8, 121. [Google Scholar] [CrossRef] - Zhang, Y.; Tang, L.; Tian, H.; Wang, J.; Cao, W.; Zhang, Z. Determination of temperature dependence of full matrix material constants of PZT-8 piezoceramics using only one sample. J. Alloy. Compd.
**2017**, 714, 20. [Google Scholar] [CrossRef] [PubMed] - Cordero, F. Quantitative evaluation of the piezoelectric response of unpoled ferroelectric ceramics from elastic and dielectric measurements: Tetragonal BaTiO
_{3}. J. Appl. Phys.**2018**, 123, 094103. [Google Scholar] [CrossRef] - Cordero, F.; Craciun, F.; Trequattrini, F.; Galassi, C. Piezoelectric softening in ferroelectrics: Ferroelectric versus antiferroelectric PbZr
_{1−x}Ti_{x}O_{3}. Phys. Rev. B**2016**, 93, 174111. [Google Scholar] [CrossRef] - Damjanovic, D. Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics. Rep. Prog. Phys.
**1998**, 61, 1267. [Google Scholar] [CrossRef] - Cordero, F. Piezoelectricity from Elastic and Dielectric Measurements on Unpoled Ferroelectrics. Mater. Res.
**2018**, 21, e20170852. [Google Scholar] [CrossRef] - Lines, M.E.; Glass, A.M. Principles and Applications of Ferroelectrics and Related Materials; Oxford University Press: Oxford, UK, 1977. [Google Scholar]
- Strukov, B.A.; Levanyuk, A.P. Ferroelectric Phenomena in Crystals; Springer: Heidelberg, Germany, 1998. [Google Scholar]
- Uchino, K.; Cross, L.E. Electrostriction and Its Interrelation with Other Anharmonic Properties of Materials. Jpn. J. Appl. Phys.
**1980**, 19, L171. [Google Scholar] [CrossRef] - Han, H.S.; Jo, W.; Kang, J.K.; Ahn, C.W.; Kim, I.W.; Ahn, K.K.; Lee, J.S. Incipient piezoelectrics and electrostriction behavior in Sn-doped Bi
_{1/2}(Na_{0.82}K_{0.18})_{1/2}TiO_{3}lead-free ceramics. J. Appl. Phys.**2013**, 113, 154102. [Google Scholar] [CrossRef] - Li, F.; Jin, L.; Xu, Z.; Zhang, S. Electrostrictive effect in ferroelectrics: An alternative approach to improve piezoelectricity. Appl. Phys. Rev.
**2014**, 1, 011103. [Google Scholar] [CrossRef][Green Version] - Weaver, P.M.; Cain, M.G.; Stewart, M. Temperature dependence of strain-polarization coupling in ferroelectric ceramics. Appl. Phys. Lett.
**2010**, 96, 142905. [Google Scholar] [CrossRef] - Devonshire, A.F. Theory of barium titanate—Part II. Philos. Mag.
**1951**, 42, 1065. [Google Scholar] [CrossRef] - Amin, A.; Haun, M.J.; Badger, B.; McKinstry, H.; Cross, L.E. A phenomenological Gibbs function for the single cell region of the PbZrO
_{3}:PbTiO_{3}solid solution system. Ferroelectrics**1985**, 65, 107. [Google Scholar] [CrossRef] - Tröster, A.; Schranz, W. General theory of heat diffusion dynamics. Phys. Rev. B
**2002**, 66, 184110. [Google Scholar] [CrossRef] - Tagantsev, A.K.; Meunier, V.; Sharma, P. Novel Electromechanical Phenomena at the Nanoscale: Phenomenological Theory and Atomistic Modeling. MRS Bull.
**2009**, 34, 643. [Google Scholar] [CrossRef] - Zhang, X.; Pan, Q.; Tian, D.; Zhou, W.; Chen, P.; Zhang, H.; Chu, B. Large Flexoelectriclike Response from the Spontaneously Polarized Surfaces in Ferroelectric Ceramics. Phys. Rev. Lett.
**2018**, 121, 057602. [Google Scholar] [CrossRef] [PubMed] - Ishibashi, Y.; Iwata, M. Theory of Morphotropic Phase Boundary in Solid-Solution Systems of Perovskite-Type Oxide Ferroelectrics: Elastic Properties. Jpn. J. Appl. Phys.
**1999**, 38, 1454–1458. [Google Scholar] [CrossRef] - Heitmann, A.A.; Rossetti, G.A., Jr. Thermodynamics of Ferroelectric Solid Solutions with Morphotropic Phase Boundaries. J. Am. Ceram. Soc.
**2014**, 97, 1661. [Google Scholar] [CrossRef] - Cordero, F. Elastic Properties and Enhanced Piezoelectric Response at Morphotropic Phase Boundaries. Materials
**2015**, 8, 8195. [Google Scholar] [CrossRef] [PubMed] - Cordero, F.; Craciun, F.; Galassi, C. Low-temperature phase transformations of PbZr
_{1−x}Ti_{x}O_{3}in the morphotropic phase-boundary region. Phys. Rev. Lett.**2007**, 98, 255701. [Google Scholar] [CrossRef] [PubMed] - Carpenter, M.A. Elastic anomalies accompanying phase transitions in (Ca,SrTiO
_{3}perovskites: Part I. Landau theory and a calibration for SrTiO_{3}. Am. Mineral.**2007**, 92, 309–327. [Google Scholar] [CrossRef] - Cordero, F.; Craciun, F.; Trequattrini, F.; Galassi, C. Effects of coupling between octahedral tilting and polar modes on the phase diagram of the ferroelectric perovskites PbZr
_{1−x}Ti_{x}O_{3}and (Na_{1/2}Bi_{1/2})_{1−x}Ba_{x}TiO_{3}. Phase Transit.**2014**, 87, 255. [Google Scholar] [CrossRef] - Cordero, F.; Trequattrini, F.; Craciun, F.; Galassi, C. Merging of the polar and tilt instability lines near the respective morphotropic phase boundaries of PbZr
_{1−x}Ti_{x}O_{3}. Phys. Rev. B**2013**, 87, 094108. [Google Scholar] [CrossRef] - Jones, G.O.; Thomas, P.A. Investigation of the structure and phase transitions in the novel A-site substituted distorted perovskite compound Na
_{0.5}Bi_{0.5}TiO_{3}. Acta Cryst. B**2002**, 58, 168–178. [Google Scholar] [CrossRef] - Cordero, F.; Craciun, F.; Trequattrini, F.; Mercadelli, E.; Galassi, C. Phase transitions and phase diagram of the ferroelectric perovskite (Na
_{0.5}Bi_{0.5})_{1−x}Ba_{x}TiO_{3}by anelastic and dielectric measurements. Phys. Rev. B**2010**, 81, 144124. [Google Scholar] [CrossRef] - Yang, J. An Introduction to the Theory of Piezoelectricity; Springer: Singapore, 2005. [Google Scholar]
- Pramanick, A.; Damjanovic, D.; Daniels, J.E.; Nino, J.C.; Jones, J.L. Origins of Electro-Mechanical Coupling in Polycrystalline Ferroelectrics During Subcoercive Electrical Loading. J. Am. Ceram. Soc.
**2011**, 94, 293. [Google Scholar] [CrossRef] - Nowick, A.S. Dielectric and anelastic relaxation of crystals containing point defects. II. Adv. Phys.
**1967**, 16, 1. [Google Scholar] [CrossRef] - Damjanovic, D.G.; Mayergoyz, I.D. Hysteresis in Piezoelectric and Ferroelectric Materials. In The Science of Hysteresis: Hysteresis in Materials; Bertotti, G., Mayergoyz, I.D., Eds.; Academic Press: Oxford, UK, 2006; Chapter 4; Volume 3, pp. 337–465. [Google Scholar]
- Li, F.X.; Rajapakse, R.K.N.D. Analytical saturated domain orientation textures and electromechanical properties of ferroelectric ceramics due to electric/mechanical poling. J. Appl. Phys.
**2007**, 101, 054110. [Google Scholar] [CrossRef] - Cheng, B.L.; Gabbay, M.; Duffy, W., Jr.; Fantozzi, G. Mechanical loss and Young’s modulus associated with phase transitions in barium titanate based ceramics. J. Mater. Sci.
**1996**, 31, 4951. [Google Scholar] [CrossRef] - Schaefer, A.; Schmitt, H.; Dörr, A. Elastic and piezoelectric coefficients of TSSG barium titanate single crystals. Ferroelectrics
**1986**, 69, 253. [Google Scholar] [CrossRef] - Budimir, M.; Damjanovic, D.; Setter, N. Piezoelectric anisotropy-phase transition relations in perovskite single crystals. J. Appl. Phys.
**2003**, 94, 6753. [Google Scholar] [CrossRef] - Cordero, F.; Langhammer, H.T.; Müller, T.; Buscaglia, V.; Nanni, P. Rotational instability of the electric polarization and divergence of the shear elastic compliance. Phys. Rev. B
**2016**, 93, 064111. [Google Scholar] [CrossRef] - Cordero, F.; Craciun, F.; Pardo, L.; Galassi, C. Comparison between the piezoelectric coefficients of ceramic Ca-modofied PbTiO
_{3}and those deduced from the piezoelectric softening of the unpoled material. Unpublished work. - Tang, L.; Cao, W. Temperature dependence of self-consistent full matrix material constants of lead zirconate titanate ceramics. Appl. Phys. Lett.
**2015**, 106, 052902. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dunn, M.L. Effects of grain shape anisotropy, porosity, and microcracks on the elastic and dielectric constants of polycrystalline piezoelectric ceramics. J. Appl. Phys.
**1995**, 78, 1533. [Google Scholar] [CrossRef] - Cordero, F.; Castellano, C.; Cantelli, R.; Ferretti, M. Glassy dynamics of the inhomogeneous metallic phase in La
_{1−x}Ca_{x}MnO_{3}. Phys. Rev. B**2002**, 65, 012403. [Google Scholar] [CrossRef] - Cordero, F.; Trequattrini, F.; Barbeta, V.B.; Jardim, R.F.; Torikachvili, M.S. Anelastic spectroscopy study of the metal-insulator transition of Nd
_{1−x}Eu_{x}NiO_{3}. Phys. Rev. B**2011**, 84, 125127. [Google Scholar] [CrossRef] - Cordero, F.; Bella, L.D.; Corvasce, F.; Latino, P.M.; Morbidini, A. An insert for anelastic spectroscopy measurements from 80 K to 1100 K. Meas. Sci. Technol.
**2009**, 20, 015702. [Google Scholar] [CrossRef] - Cordero, F.; Craciun, F.; Dinescu, M.; Scarisoreanu, N.; Galassi, C.; Schranz, W.; Soprunyuk, V. Elastic response of (1-x)Ba(Ti0.8Zr0.2)O3 - x(Ba0.7Ca0.3)TiO3 (x = 0.45-0.55) and the role of the intermediate orthorhombic phase in enhancing the piezoelectric coupling. Appl. Phys. Lett.
**2014**, 105, 232904. [Google Scholar] [CrossRef] - Jimenez, B.; Olmo, L.D.; Mendiola, J.; Calzada, L.; Pardo, L.; Alemany, C. Preparation and Properties of Ferroelectric Ceramic Obtained from Colloidal Phases. Mater. Sci. Forum
**1991**, 62–64, 295. [Google Scholar] [CrossRef]

**Figure 1.**The two mechanisms causing the piezoelectric softening: direct and converse piezoelectric effects arising from the electrostrictive coupling with spontaneous polarization ${P}_{0}.$

**Figure 2.**Various factors that determine the actual softening in the FE phase: (

**a**) piezoelectric softening within the FE phase; (

**b**) linear anharmonic stiffening of the background compliance; (

**c**) fluctuations and thermoelastic effect; (

**d**) additional terms in the FE free energy with respect to the simplest expansion (11).

**Figure 3.**Compliance and dielectric permittivity of PbZr${}_{0.86}$Ti${}_{0.14}$O${}_{3}$, with anomalies at ${T}_{\mathrm{C}}$ and the octahedral tilting transition at ${T}_{\mathrm{T}}$ (data from Ref. [31])

**Figure 4.**(

**a**) Compliance of three samples of BaTiO${}_{3}$ from different laboratories and with different porosities; (

**b**) After rescaling to the stiffer sample, with smaller porosity, in the PE phase; the dashed line is ${s}^{0}\left(T\right)$ extrapolated from >800 K; (

**c**) After subtraction of the extrapolated ${s}^{0}\left(T\right);$ curves 4 and 5 are $\Delta {s}^{\mathrm{piezo}}$ calculated from different sets of the $\u03f5$ and d tensors from the literature; (

**d**) Change of the magnitude of the polarization vector $\mathbf{P}$ at the PE/FE transition, with the corresponding longitudinal fluctuations in red, and change of the direction of $\mathbf{P}$ at the transition between tetragonal and orthorhombic FE phases, with transverse fluctuations.

**Figure 6.**(

**a**) Hypothetical compliance curves obtained by varying a material parameter, for example doping; (

**b**) The same curves after normalization to ${s}_{0}\left(T\right)$ in the PE phase, in order to remove the dependence on porosity; (

**c**) Dielectric permittivity measured of the same samples or compositions; (

**d**) Effective piezoelectric coefficient, with the effect of different porosities removed.

**Figure 7.**Hypothetical compliance curves obtained by doping in a manner that changes the FE transition into an orbital/charge order transition.

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**MDPI and ACS Style**

Cordero, F. Elastic and Dielectric Evaluation of the Piezoelectric Response of Ferroelectrics Using Unpoled Ceramics. *Ceramics* **2018**, *1*, 211-228.
https://doi.org/10.3390/ceramics1020018

**AMA Style**

Cordero F. Elastic and Dielectric Evaluation of the Piezoelectric Response of Ferroelectrics Using Unpoled Ceramics. *Ceramics*. 2018; 1(2):211-228.
https://doi.org/10.3390/ceramics1020018

**Chicago/Turabian Style**

Cordero, Francesco. 2018. "Elastic and Dielectric Evaluation of the Piezoelectric Response of Ferroelectrics Using Unpoled Ceramics" *Ceramics* 1, no. 2: 211-228.
https://doi.org/10.3390/ceramics1020018