# 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 |

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**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