**Meet the editor**

Dr. Aimé Peláiz Barranco is a professor in the Physics Faculty, Havana University, Cuba. Her activities range from teaching, to advising undergraduate and graduate students and doing high quality research. She is one of the leaders in the development of ferroelectrics and antiferroelectric materials research in Cuba. She has authored several scientific papers in refereed journals and

presented an important number of research works at national and international conferences. Her results have been recognized in Cuba and worldwide through several awards, such as TWOWS Award for Young Women Scientists in Physics/Mathematics (2010) and TWAS-ROLAC Award for Young Scientists in Physics (2011). She is the international coordinator of the Latin American Network of Ferroelectrics Materials.

Contents

**Preface IX**

**Piezoelectricity 1**

**Transitions 25** Shu-Tao Ai

S. Katiyar

González-Abreu

Martun Hovhannisyan

Chapter 1 **Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and**

Kaoru Miura and Hiroshi Funakubo

**Domain Ferroelectric? 51**

Chapter 2 **Advances in Thermodynamics of Ferroelectric Phase**

Chapter 4 **Self Assembled Nanoscale Relaxor Ferroelectrics 69**

Chapter 5 **Relaxor Behaviour in Ferroelectric Ceramics 85**

Chapter 6 **Electronic Band Structures and Phase Transitions of Ferroelectric and Multiferroic Oxides 109**

Chapter 3 **Pb(Mg1/3Nb2/3)O3 (PMN) Relaxor: Dipole Glass or Nano-**

Desheng Fu, Hiroki Taniguchi, Mitsuru Itoh and Shigeo Mori

Ashok Kumar, Margarita Correa, Nora Ortega, Salini Kumari and R.

A. Peláiz-Barranco, F. Calderón-Piñar, O. García-Zaldívar and Y.

Zhigao Hu, Junhao Chu, Yawei Li, Kai Jiang and Ziqiang Zhu

Chapter 7 **Phase Diagram of The Ternary BaO-Bi2O3-B2O3 System: New Compounds and Glass Ceramics Characterisation 127**

### Contents

### **Preface XIII**



Chapter 18 **Ferroelectrics at the Nanoscale: A First Principle**

Chapter 19 **The Influence of Vanadium Doping on the Physical and**

Chapter 20 **Nanoscale Ferroelectric Films, Strips and Boxes 449**

Chapter 21 **Emerging Applications of Ferroelectric Nanoparticles in**

Chapter 22 **Photorefractive Effect in Ferroelectric Liquid Crystals 499**

**Materials Technologies, Biology and Medicine 475** Yuriy Garbovskiy, Olena Zribi and Anatoliy Glushchenko

**Electrical Properties of Non-Volatile Random Access Memory Using the BTV, BLTV, and BNTV Oxide Thin Films 429**

Kai-Huang Chen, Chien-Min Cheng, Sean Wu, Chin-Hsiung Liao and

Contents **VII**

**Approach 401** Matías Núñez

Jen-Hwan Tsai

Jeffrey F. Webb

Takeo Sasaki


Chapter 16 **Phase Transitions, Dielectric and Ferroelectric Properties of Lead-free NBT-BT Thin Films 351** N. D. Scarisoreanu, R. Birjega, A. Andrei, M. Dinescu, F. Craciun and C. Galassi

Chapter 17 **Thin-Film Process Technology for Ferroelectric Application 369** Koukou Suu

Chapter 18 **Ferroelectrics at the Nanoscale: A First Principle Approach 401** Matías Núñez

Chapter 8 **Optical Properties of Ferroelectrics and Measurement**

Chapter 9 **Ferroelectric Domain Imaging Multiferroic Films Using Piezoresponse Force Microscopy 193**

Hongyang Zhao, Hideo Kimura, Qiwen Yao, Lei Guo, Zhenxiang

Chapter 10 **Gelcasting of Ferroelectric Ceramics: Doping Effect and Further**

Chapter 11 **Electronic Ferroelectricity in II-VI Semiconductor ZnO 231**

Chapter 12 **Doping-Induced Ferroelectric Phase Transition and Ultraviolet-**

Chapter 13 **Raman Scattering Study on the Phase Transition Dynamics of**

Chapter 14 **Electromechanical Coupling Multiaxial Experimental and**

**0.32PbTiO3 Ferroelectric Single Crystal 295** Wan Qiang, Chen Changqing and Shen Yapeng

Chapter 15 **'Universal' Synthesis of PZT (1-X)/X Submicrometric Structures Using Highly Stable Colloidal Dispersions: A Bottom-Up**

Chapter 16 **Phase Transitions, Dielectric and Ferroelectric Properties of**

**Lead-free NBT-BT Thin Films 351**

Chapter 17 **Thin-Film Process Technology for Ferroelectric**

A. Suárez-Gómez, J.M. Saniger-Blesa and F. Calderón-Piñar

N. D. Scarisoreanu, R. Birjega, A. Andrei, M. Dinescu, F. Craciun and

**Illumination Effect in a Quantum Paraelectric Material Studied**

Hiroki Taniguchi, Hiroki Moriwake, Toshirou Yagi and Mitsuru Itoh

**Micro-Constitutive Model Study of Pb(Mg1/3Nb2/3)O3-**

Akira Onodera and Masaki Takesada

**by Coherent Phonon Spectroscopy 257**

**Procedures 163**

**VI** Contents

A.K. Bain and Prem Chand

Cheng and Xiaolin Wang

**Development 209** Dong Guo and Kai Cai

Toshiro Kohmoto

**Approach 323**

C. Galassi

**Application 369** Koukou Suu

**Ferroelectric Oxides 279**


Preface

Since the discovery of the phenomenon of ferroelectricity, the continuous development of materials and technology has provided an important number of applications. The ferroelectricity has been the heart and soul of several industries to the development of piezoelectric transducers, pyroelectric sensors, medical diagnostic transducers, radio and communication filters, ultrasonic motors, electro-optical devices, ferroelectric memories, etc. The most striking properties of the ferroelectric materials are strong coupling effects, strong hysteresis in the field polarization response, and extremely high dielectric permittivity. For the development of new technology by using ferroelectric materials, much research has been carried out from the perspective of theory and of experiment. These have provided revolutionary breakthroughs in the understanding of single crystals, ceramics, thin films and composites. The advances in the studies of these materials at nanoscale have also

The purpose of this book is to offer an up-to-date view of ferroelectricity, covering a wide range of theoretical and experimental topics. Several researches on ceramics, thin films, ferroelectric liquids and ferroelectricity at nanoscale are included. The book is the result of an important number of contributions of researchers from the international scientific

I would like to thank all the authors for their interesting contributions to this book and the

We hope that this book will be of real value to students and researchers moving into the

**Dr. Aimé Peláiz Barranco** Applied Physics Department Physics Faculty, Havana University

Cuba

showed the potential applications of ferroelectric nanoparticles.

community, which work in different topics of ferroelectricity.

InTech team for the outstanding support.

ferroelectric field.

### Preface

Since the discovery of the phenomenon of ferroelectricity, the continuous development of materials and technology has provided an important number of applications. The ferroelectricity has been the heart and soul of several industries to the development of piezoelectric transducers, pyroelectric sensors, medical diagnostic transducers, radio and communication filters, ultrasonic motors, electro-optical devices, ferroelectric memories, etc.

The most striking properties of the ferroelectric materials are strong coupling effects, strong hysteresis in the field polarization response, and extremely high dielectric permittivity. For the development of new technology by using ferroelectric materials, much research has been carried out from the perspective of theory and of experiment. These have provided revolutionary breakthroughs in the understanding of single crystals, ceramics, thin films and composites. The advances in the studies of these materials at nanoscale have also showed the potential applications of ferroelectric nanoparticles.

The purpose of this book is to offer an up-to-date view of ferroelectricity, covering a wide range of theoretical and experimental topics. Several researches on ceramics, thin films, ferroelectric liquids and ferroelectricity at nanoscale are included. The book is the result of an important number of contributions of researchers from the international scientific community, which work in different topics of ferroelectricity.

I would like to thank all the authors for their interesting contributions to this book and the InTech team for the outstanding support.

We hope that this book will be of real value to students and researchers moving into the ferroelectric field.

> **Dr. Aimé Peláiz Barranco** Applied Physics Department Physics Faculty, Havana University Cuba

**Chapter 1**

**Electronic Structures of Tetragonal ABX3:**

Since Cohen proposed an origin for ferroelectricity in perovskites (*ABX*3) [1], investigations of ferroelectric materials using first-principles calculations have been extensively studied [2-20]. Currently, using the pseudopotential (PP) methods, most of the crystal structures in ferroelectric *ABX*3 can be precisely predicted. However, even in BaTiO3, which is a wellknown ferroelectric perovskite oxide with tetragonal structure at room temperature, the op‐ timized structure by the PP methods is strongly dependent on the choice of the Ti PPs as illustrated in Fig. 1; preparation for Ti 3s and 3p semicore states in addition to Ti 3d, 4s, and 4p valence states is essential to the appearance of the tetragonal structure. This is an impor‐ tant problem for ferroelectricity, but it has been generally recognized for a long time that

this problem is within an empirical framework of the calculational techniques [21].

**(FP)**

**Atomic potentials**

and reproduction in any medium, provided the original work is properly cited.

**Unit cell <sup>O</sup> Ti <sup>O</sup> z- axis Wave functions**

**Wave functions (PP)**

© 2013 Miura and Funakubo; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Role of the B-X Coulomb Repulsions for**

**Ferroelectricity and Piezoelectricity**

Additional information is available at the end of the chapter

**Ti 3s, 3p ?**

**Ti 3d, 4s, 4p O 2s, 2p**

**Ti 1s 2p O 1s**

**Valence states (PP)**

**Core states charge density**

**Figure 1.** Illustration of the choice of Ti 3s and 3p states in pseudopotentials.

Kaoru Miura and Hiroshi Funakubo

http://dx.doi.org/10.5772/52187

**1. Introduction**

## **Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity**

Kaoru Miura and Hiroshi Funakubo

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52187

### **1. Introduction**

Since Cohen proposed an origin for ferroelectricity in perovskites (*ABX*3) [1], investigations of ferroelectric materials using first-principles calculations have been extensively studied [2-20]. Currently, using the pseudopotential (PP) methods, most of the crystal structures in ferroelectric *ABX*3 can be precisely predicted. However, even in BaTiO3, which is a wellknown ferroelectric perovskite oxide with tetragonal structure at room temperature, the op‐ timized structure by the PP methods is strongly dependent on the choice of the Ti PPs as illustrated in Fig. 1; preparation for Ti 3s and 3p semicore states in addition to Ti 3d, 4s, and 4p valence states is essential to the appearance of the tetragonal structure. This is an impor‐ tant problem for ferroelectricity, but it has been generally recognized for a long time that this problem is within an empirical framework of the calculational techniques [21].

**Figure 1.** Illustration of the choice of Ti 3s and 3p states in pseudopotentials.

© 2013 Miura and Funakubo; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

It is known that ferroelectric state appears when the long-range forces due to the dipole-di‐ pole interaction overcome the short-range forces due to the Coulomb repulsions. Investiga‐ tions about the relationship between the Ti-O Coulomb repulsions and the appearance of ferroelectricity in *A*TiO3 (*A* = Ba, Pb) were reported both theoretically and experimentally. Theoretically, Cohen first proposed the hybridization between Ti 3d state and O 2p state (Ti 3d-O 2p) as an origin for ferroelectricity in BaTiO3 and PbTiO3 [1]. On the other hand, we investigated [20] the influence of the Ti-O*z* Coulomb repulsions on Ti ion displacement in tetragonal BaTiO3 and PbTiO3, where O*z* denotes the O atom to the *z*-axis (Ti is displaced to the *z*-axis). Whereas the hybridization between Ti 3d state and O*<sup>z</sup>* 2p*<sup>z</sup>* state stabilize Ti ion displacement, the strong Coulomb repulsions between Ti 3s and 3p*z* states and O 2p*<sup>z</sup>* states do not favourably cause Ti ion displacement. Experimentally, on the other hand, Kuroiwa *et al*. [22] showed that the appearance of ferroelectric state is closely related to the total charge density of Ti-O bonding in BaTiO3. As discussed above, investigation about a role of Ti 3s and 3p states is important in the appearance of the ferroelectric state in tetragonal BaTiO3, in addition to the Ti 3d-O 2p hybridization as an origin of ferroelectricity [1].

It seems that the strong *B*-*X* Coulomb repulsions affect the most stable structure of *ABX*3. It has been well known that the most stable structure of *ABX*3 is closely related to the tolerance factor *t*,

$$t = \frac{r\_A + r\_\chi}{\sqrt{2} \left(r\_B + r\_\chi\right)} \quad \text{or} \tag{1}$$

tigated the role of the Zn-O Coulomb repulsions in the appearance of a ferroelectric state in tetragonal BZT [10, 13]. Moreover, we also investigated the structural, ferroelectric, and pie‐ zoelectric properties of tetragonal *ABX*3 and discussed the piezoelectric mechanisms based

Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity

http://dx.doi.org/10.5772/52187

3

In this chapter, based on our recent papers and patents [10-19], we discuss a general role of *B*-*X* Coulomb repulsions for the appearance of the ferroelectric state in *ABX*3. Then, we also discuss the relationship between the *B*-*X* Coulomb repulsions and the piezoelectric proper‐

The calculations for *ABX*3 were performed using the ABINIT code [29], which is one of the normconserving PP (NCPP) methods. Electron-electron interaction was treated in the local-density approximation (LDA) [30]. Pseudopotentials were generated using the OPIUM code [31]:

**i.** In BaTiO3, 5s, 5p and 6s electrons for Ba PP, and 2s and 2p electrons for O PP were

**ii.** In BZT and BMT, 5d, 6s, and 6p electrons for Bi PP, and 2s and 2p electrons for O

rhombohedral (*R3*), monoclinic (*Pm*), and tetragonal (*P4mm*) structures.

PP were treated as semicore or valence electrons, respectively. Moreover, in order to investigate the roles of Zn and Ti 3s and 3p states, and Mg 2s and 2p states, two types of PPs were prepared: the PPs with only Zn and Ti 3d and 4s states, and Mg 3s states, considered as valence electrons (Case I), Zn and Ti 3s, 3p, 3d, and 4s states, and Mg 2s, 2p, and 3s states considered as semicore or valence electrons (Case II). The cutoff energy for plane-wave basis functions was set to be 70 Hr for Case I and 110 Hr for Case II. A 4×4×4 Monkhorst-Pack *k*-point mesh was set in the Brillouin zone of the unit cell. The calculated results can be discussed within 0.02 eV per formula unit (f.u.) using the above conditions. The present calculations were performed for the monoclinic, rhombohedral, and A-, C-, and G-type tetragonal structures. The number of atoms in the unit cell was set to be 10 for the rhombohe‐ dral and monoclinic structures, and 20 for the A-, C-, and G-type tetragonal struc‐ tures. Positions of all the atoms were optimized within the framework of the

nal (*P4mm*) or rhombohedral (*R3m*) structure.

treated as semicore or valence electrons, respectively. Moreover, in order to investi‐ gate the role of Ti 3s and 3p states, two kinds of Ti PPs were prepared: the Ti PP with 3s, 3p, 3d and 4s electrons treated as semicore or valence electrons (Ti3spd4s PP), and that with only 3d and 4s electrons treated as valence electrons (Ti3d4s PP). In both PPs, the differences between the calculated result and experimental one are within 1.5 % of the lattice constant and within 10 % of the bulk modulus in the opti‐ mized calculation of bulk Ti. The cutoff energy for plane-wave basis functions was set to be 50 Hartree (Hr). The number of atoms in the unit cell was set to be five, and a 6×6×6 Monkhorst-Pack *k*-point mesh was set in the Brillouin zone of the unit cell. Positions of all the atoms were optimized within the framework of the tetrago‐

on the *B*-*X* Coulomb repulsions [12, 14, 15, 18, 19].

ties of tetragonal *ABX*3.

**2. Methodology**

where *rA*, *rB*, and *rX* denote the ionic radii of *A*, *B*, and *X* ions, respectively [23]. In general ferroelectric *ABX*3, the most stable structure is tetragonal for *t* ≳ 1, cubic for *t* ≈ 1, and rhom‐ bohedral or orthorhombic for *t* ≲ 1. In fact, BaTiO3 with *t* = 1.062 shows tetragonal structure in room temperature. However, recently, BiZn0.5Ti0.5O3 (BZT) with *t* = 0.935 was experimen‐ tally reported [24] to show a tetragonal PbTiO3-type structure with high *c/a* ratio (1.211). This result is in contrast to that of BiZn0.5Mg0.5O3 (BMT) with *t* = 0.939, *i.e.*, the most stable structure was reported to be the orthorhombic or rhombohedral structure [25, 26]. Several theoretical papers of BZT have been reported [4-6], but the role of the Zn-O Coulomb repul‐ sions in the appearance of the tetragonal structure has not been discussed sufficiently.

Piezoelectric properties in *ABX*3 are also closely related to the crystal structure. Investigations of the relationship between piezoelectric properties and the crystal structure of *ABX*3 by first-prin‐ ciples calculations have been extensively studied [2-19]. Moreover, phenomenological investi‐ gations of the piezoelectric properties have been also performed [27, 28]. However, it seems that the piezoelectric properties in the atomic level have not been sufficiently investigated. There‐ fore, further theoretical investigation of the relationship between piezoelectric properties and the crystal structure of *ABX*3, especially the *B*-*X* Coulomb repulsions, should be needed.

Recently, we investigated the roles of the Ti-O Coulomb repulsions in the appearance of a ferroelectric state in tetragonal BaTiO3 by the analysis of a first-principles PP method [11-15]. We investigated the structural properties of tetragonal and rhombohedral BaTiO3 with two kinds of Ti PPs, and propose the role of Ti 3s and 3p states for ferroelectricity. We also inves‐ tigated the role of the Zn-O Coulomb repulsions in the appearance of a ferroelectric state in tetragonal BZT [10, 13]. Moreover, we also investigated the structural, ferroelectric, and pie‐ zoelectric properties of tetragonal *ABX*3 and discussed the piezoelectric mechanisms based on the *B*-*X* Coulomb repulsions [12, 14, 15, 18, 19].

In this chapter, based on our recent papers and patents [10-19], we discuss a general role of *B*-*X* Coulomb repulsions for the appearance of the ferroelectric state in *ABX*3. Then, we also discuss the relationship between the *B*-*X* Coulomb repulsions and the piezoelectric proper‐ ties of tetragonal *ABX*3.

### **2. Methodology**

It is known that ferroelectric state appears when the long-range forces due to the dipole-di‐ pole interaction overcome the short-range forces due to the Coulomb repulsions. Investiga‐ tions about the relationship between the Ti-O Coulomb repulsions and the appearance of ferroelectricity in *A*TiO3 (*A* = Ba, Pb) were reported both theoretically and experimentally. Theoretically, Cohen first proposed the hybridization between Ti 3d state and O 2p state (Ti 3d-O 2p) as an origin for ferroelectricity in BaTiO3 and PbTiO3 [1]. On the other hand, we investigated [20] the influence of the Ti-O*z* Coulomb repulsions on Ti ion displacement in tetragonal BaTiO3 and PbTiO3, where O*z* denotes the O atom to the *z*-axis (Ti is displaced to the *z*-axis). Whereas the hybridization between Ti 3d state and O*<sup>z</sup>* 2p*<sup>z</sup>* state stabilize Ti ion displacement, the strong Coulomb repulsions between Ti 3s and 3p*z* states and O 2p*<sup>z</sup>* states do not favourably cause Ti ion displacement. Experimentally, on the other hand, Kuroiwa *et al*. [22] showed that the appearance of ferroelectric state is closely related to the total charge density of Ti-O bonding in BaTiO3. As discussed above, investigation about a role of Ti 3s and 3p states is important in the appearance of the ferroelectric state in tetragonal BaTiO3, in

It seems that the strong *B*-*X* Coulomb repulsions affect the most stable structure of *ABX*3. It has been well known that the most stable structure of *ABX*3 is closely related to the tolerance factor *t*,

<sup>+</sup> <sup>=</sup> <sup>+</sup> (1)

 , 2 ( ) *A X B X*

*r r*

where *rA*, *rB*, and *rX* denote the ionic radii of *A*, *B*, and *X* ions, respectively [23]. In general ferroelectric *ABX*3, the most stable structure is tetragonal for *t* ≳ 1, cubic for *t* ≈ 1, and rhom‐ bohedral or orthorhombic for *t* ≲ 1. In fact, BaTiO3 with *t* = 1.062 shows tetragonal structure in room temperature. However, recently, BiZn0.5Ti0.5O3 (BZT) with *t* = 0.935 was experimen‐ tally reported [24] to show a tetragonal PbTiO3-type structure with high *c/a* ratio (1.211). This result is in contrast to that of BiZn0.5Mg0.5O3 (BMT) with *t* = 0.939, *i.e.*, the most stable structure was reported to be the orthorhombic or rhombohedral structure [25, 26]. Several theoretical papers of BZT have been reported [4-6], but the role of the Zn-O Coulomb repul‐ sions in the appearance of the tetragonal structure has not been discussed sufficiently.

Piezoelectric properties in *ABX*3 are also closely related to the crystal structure. Investigations of the relationship between piezoelectric properties and the crystal structure of *ABX*3 by first-prin‐ ciples calculations have been extensively studied [2-19]. Moreover, phenomenological investi‐ gations of the piezoelectric properties have been also performed [27, 28]. However, it seems that the piezoelectric properties in the atomic level have not been sufficiently investigated. There‐ fore, further theoretical investigation of the relationship between piezoelectric properties and

Recently, we investigated the roles of the Ti-O Coulomb repulsions in the appearance of a ferroelectric state in tetragonal BaTiO3 by the analysis of a first-principles PP method [11-15]. We investigated the structural properties of tetragonal and rhombohedral BaTiO3 with two kinds of Ti PPs, and propose the role of Ti 3s and 3p states for ferroelectricity. We also inves‐

the crystal structure of *ABX*3, especially the *B*-*X* Coulomb repulsions, should be needed.

*r r*

addition to the Ti 3d-O 2p hybridization as an origin of ferroelectricity [1].

2 Advances in Ferroelectrics

*t*

The calculations for *ABX*3 were performed using the ABINIT code [29], which is one of the normconserving PP (NCPP) methods. Electron-electron interaction was treated in the local-density approximation (LDA) [30]. Pseudopotentials were generated using the OPIUM code [31]:


**iii.** Relationship between the *B*-*X* Coulomb repulsions and the piezoelectric properties in tetragonal *ABX*3 is investigated. The pseudopotentials were generated using the opium code [31] with semicore and valence electrons (e.g., Ti3spd4s PP), and the virtual crystal approximation [32] were applied to several *ABX*3.

Spontaneous polarizations and piezoelectric constants were also evaluated, due to the Born effective charges [33]. The spontaneous polarization of tetragonal structures along the [001] axis, *P*3, is defined as

$$P\_3 = \sum\_k \frac{ec}{\Omega} Z\_{33}^\* \binom{k}{k} \mu\_3(k) \tag{2}$$

**3. Results and discussion**

The relationship between the piezoelectric *d*33 constant and the *e*33 one is

ferroelectric states in BaTiO3.

1 1.1 1.2 1.3 1.4 1.5

c/a

*3.1.1. Role of Ti 3s and 3p states in ferroelectric BaTiO3*

**3.1.1. Role of Ti 3s and 3p states in ferroelectric BaTiO3**

Figures 2(a) and 2(b)show the optimized results for the ratio *c*/*a* of the lattice parameters and the value of the Ti ion displacement (*δ*Ti) as a function of the *a* lattice parameter in tetragonal BaTiO3, respectively. Results with arrows are the fully optimized results, and the others results are those with the *c* lattice parameters and all the inner coordination optimized for fixed *a*. Note that the fully optimized structure of BaTiO3 is tetragonal with the Ti3spd4s PP, whereas it is cubic with the Ti3d4s PP. This result suggests that the explicit treatment of Ti 3s and 3p semicore states is essential to the appearance of

Figures 2(a) and 2(b) show the optimized results for the ratio *c*/*a* of the lattice parameters and the value of the Ti ion displacement (*δ*Ti) as a function of the *a* lattice parameter in tetragonal BaTiO3, respectively. Results with arrows are the fully optimized results, and the others results are those with the *c* lattice parameters and all the inner coordination optimized for fixed *a*. Note that the fully optimized structure of BaTiO3 is tetragonal with the Ti3spd4s PP, whereas it is cubic with the Ti3d4s PP. This result suggests that

The calculated results shown in Fig. 2 suggest that the explicit treatment of Ti 3s and 3p semicore states is essential to the appearance of ferroelectric states in BaTiO3. In the following, we investigate the role of Ti 3s and 3p states for ferroelectricity from

The calculated results shown in Fig. 2 suggest that the explicit treatment of Ti 3s and 3p semicore states is essential to the appearance of ferroelectric states in BaTiO3. In the follow‐ ing, we investigate the role of Ti 3s and 3p states for ferroelectricity from two viewpoints.

**Figure 2.** Optimized calculated results in tetragonal BaTiO3. Results with arrows are the fully optimized results [11].

0 0.1 0.2 0.3 0.4 0.5 0.6

δTi (Å)

> 3.5 3.6 3.7 3.8 3.9 4 a (Å)

**Ti3d4s**

**Ti3spd4s**

http://dx.doi.org/10.5772/52187

5

One viewpoint concerns hybridizations between Ti 3s and 3p states and other states. Figure 3(a) and 3(b) shows the total density of states (DOS) of tetragonal BaTiO3 with two Ti PPs. Both results are in good agreement with previous calculated results [7] by the full-potential linear augmented plane wave (FLAPW) method. In the DOS with the Ti3spd4s PP, the energy ``levels", not bands, of Ti 3s and 3p states, are located at -2.0 Hr and -1.2 Hr, respectively. This result suggests that the Ti 3s and 3p orbitals do not make any hybridization but only give Coulomb repulsions with the O orbitals as well as the Ba orbitals. In the DOS with the Ti3d4s PP, on the other hand, the energy levels of Ti 3s and 3p states are not shown because Ti 3s and 3p states were treated as the core charges. This result means that the Ti 3s and 3p orbitals cannot even give Coulomb repulsions with the O orbitals as well as the Ba

One viewpoint concerns hybridizations between Ti 3s and 3p states and other states. Fig‐ ure 3(a) and 3(b) shows the total density of states (DOS) of tetragonal BaTiO3 with two Ti PPs. Both results are in good agreement with previous calculated results [7] by the full-potential linear augmented plane wave (FLAPW) method. In the DOS with the Ti3spd4s PP, the energy ``levels", not bands, of Ti 3s and 3p states, are located at -2.0 Hr and -1.2 Hr, respectively. This result suggests that the Ti 3s and 3p orbitals do not make any hybridization but only give Coulomb repulsions with the O orbitals as well as the Ba orbitals. In the DOS with the Ti3d4s PP, on the other hand, the energy levels of Ti 3s and 3p states are not shown because Ti 3s and 3p states were treated as the core charg‐ es. This result means that the Ti 3s and 3p orbitals cannot even give Coulomb repulsions

the explicit treatment of Ti 3s and 3p semicore states is essential to the appearance of ferroelectric states in BaTiO3.

Figure 2. Optimized calculated results in tetragonal BaTiO3. Results with arrows are the fully optimized results [11].

**Ti3spd4s**

3.5 3.6 3.7 3.8 3.9 4

**Ti3d4s**

with the O orbitals as well as the Ba orbitals.

a (Å)

(a) (b)

where *e* and *Ω* denote the charge unit and the volume of the unit cell. *P*3 and *c* denote the spontaneous polarization of tetragonal structures and the lattice parameter of the unit cell along the [001] axis, respectively. *u*3(*k*) denotes the displacement along the [001]

lattice along the [001] axis, which is defined as *η*<sup>3</sup> ≡ (*c* – *c*0)/*c*0; *c*0 denotes the *c* lattice parameter with fully optimized structure. On the other hand, *η*1 denotes the strain of lattice along the [100] axis, which is defined as *η*<sup>1</sup> ≡ (*a* – *a*0)/*a*0; *a*0 denotes the *a* lattice parameter with fully optimized structure. The first term of the right hand in Eq. (3) denotes the clamped term evaluated at

(4)

where *s3jE* denotes the elastic compliance, and ``*T* " denotes the transposition of matrix elements. The suffix *j* denotes the directionindexes of the axis, *i.e.*, 1 along the [100] axis, 2 along the [010] axis, 3 along the [001] axis, and 4 to 6 along the shear directions,

Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity

vanishing internal strain, and the second term denotes the relaxed term that is due to the relative displacements.

6 33 3 3 1 ( ) , *E T j j*

*j d se* 

(*k*) denotes the Born effective charges which contributes to the *P*3 from the *u*3(*k*). *η*3 denotes the strain of

**3.1. Ferroelectricity**

**3. Results and discussion** 

**3.1. Ferroelectricity** 

axis of the *k*th atom, and *Z*33\*

respectively.

two viewpoints.

orbitals.

where *e*, *c*, and *Ω* denote the charge unit, lattice parameter of the unit cell along the [001] axis, and the volume of the unit cell, respectively. *u*3(*k*) denotes the displacement along the [001] axis of the *k*th atom, and *Z*33\* (*k*) denotes the Born effective charges [33] which contrib‐ utes to the *P*3 from the *u*3(*k*).

The piezoelectric *e*33 constant is defined as

$$\varepsilon\_{3j} = \left(\frac{\partial P\_3}{\partial \eta\_3}\right)\_u + \sum\_k \frac{ec}{\Omega} Z\_{33}^\*(k) \frac{\partial u\_3(k)}{\partial \eta\_j} \qquad \{j = 3, 1\}, \tag{3}$$

where *e* and *Ω* denote the charge unit and the volume of the unit cell. *P*3 and *c* denote the spontaneous polarization of tetragonal structures and the lattice parameter of the unit cell along the [001] axis, respectively. *u*3(*k*) denotes the displacement along the [001] axis of the *k*th atom, and *Z*33\* (*k*) denotes the Born effective charges which contributes to the *P*3 from the *u*3(*k*). *η*<sup>3</sup> denotes the strain of lattice along the [001] axis, which is defined as *η*3 ≡ (*c* – *c*0)/*c*0; *c*<sup>0</sup> denotes the *c* lattice parameter with fully optimized structure. On the other hand, *η*1 denotes the strain of lattice along the [100] axis, which is defined as *η*<sup>1</sup> ≡ (*a* – *a*0)/*a*0; *a*<sup>0</sup> denotes the *a* lattice parameter with fully optimized structure. The first term of the right hand in Eq. (3) denotes the clamped term evaluated at vanishing internal strain, and the second term de‐ notes the relaxed term that is due to the relative displacements.

The relationship between the piezoelectric *d*33 constant and the *e*33 one is

$$d\_{33} \equiv \sum\_{j=1}^{6} \mathbf{s}\_{3j}^{E} \times^{T} \{e\_{3j}\} \, \, \, \, \tag{4}$$

where *s3j*E denotes the elastic compliance, and ``*T* " denotes the transposition of matrix ele‐ ments. The suffix *j* denotes the direction-indexes of the axis, *i.e.*, 1 along the [100] axis, 2 along the [010] axis, 3 along the [001] axis, and 4 to 6 along the shear directions, respectively.

where *s3jE* denotes the elastic compliance, and ``*T* " denotes the transposition of matrix elements. The suffix *j* denotes the direction-

6

where *e* and *Ω* denote the charge unit and the volume of the unit cell. *P*3 and *c* denote the spontaneous polarization of tetragonal structures and the lattice parameter of the unit cell along the [001] axis, respectively. *u*3(*k*) denotes the displacement along the [001]

lattice along the [001] axis, which is defined as *η*<sup>3</sup> ≡ (*c* – *c*0)/*c*0; *c*0 denotes the *c* lattice parameter with fully optimized structure. On the other hand, *η*1 denotes the strain of lattice along the [100] axis, which is defined as *η*<sup>1</sup> ≡ (*a* – *a*0)/*a*0; *a*0 denotes the *a* lattice parameter with fully optimized structure. The first term of the right hand in Eq. (3) denotes the clamped term evaluated at

vanishing internal strain, and the second term denotes the relaxed term that is due to the relative displacements.

(*k*) denotes the Born effective charges which contributes to the *P*3 from the *u*3(*k*). *η*3 denotes the strain of

#### **3. Results and discussion** indexes of the axis, *i.e.*, 1 along the [100] axis, 2 along the [010] axis, 3 along the [001] axis, and 4 to 6 along the shear directions,

The relationship between the piezoelectric *d*33 constant and the *e*33 one is

### **3.1. Ferroelectricity**

axis of the *k*th atom, and *Z*33\*

respectively.

orbitals.

**iii.** Relationship between the *B*-*X* Coulomb repulsions and the piezoelectric properties

Spontaneous polarizations and piezoelectric constants were also evaluated, due to the Born effective charges [33]. The spontaneous polarization of tetragonal structures along the [001]

> ( ) \* <sup>3</sup> 33 3 ( ) , *<sup>k</sup>* Ω

where *e*, *c*, and *Ω* denote the charge unit, lattice parameter of the unit cell along the [001] axis, and the volume of the unit cell, respectively. *u*3(*k*) denotes the displacement along the

( ) ( ) 3 3 \*

where *e* and *Ω* denote the charge unit and the volume of the unit cell. *P*3 and *c* denote the spontaneous polarization of tetragonal structures and the lattice parameter of the unit cell along the [001] axis, respectively. *u*3(*k*) denotes the displacement along the [001] axis of the

*u*3(*k*). *η*<sup>3</sup> denotes the strain of lattice along the [001] axis, which is defined as *η*3 ≡ (*c* – *c*0)/*c*0; *c*<sup>0</sup> denotes the *c* lattice parameter with fully optimized structure. On the other hand, *η*1 denotes the strain of lattice along the [100] axis, which is defined as *η*<sup>1</sup> ≡ (*a* – *a*0)/*a*0; *a*<sup>0</sup> denotes the *a* lattice parameter with fully optimized structure. The first term of the right hand in Eq. (3) denotes the clamped term evaluated at vanishing internal strain, and the second term de‐

where *s3j*E denotes the elastic compliance, and ``*T* " denotes the transposition of matrix ele‐ ments. The suffix *j* denotes the direction-indexes of the axis, *i.e.*, 1 along the [100] axis, 2 along the [010] axis, 3 along the [001] axis, and 4 to 6 along the shear directions, respectively.

 h

( ) 3, 1 , <sup>Ω</sup> *<sup>j</sup> u k j P ec u k e Zk j*

virtual crystal approximation [32] were applied to several *ABX*3.

axis, *P*3, is defined as

4 Advances in Ferroelectrics

[001] axis of the *k*th atom, and *Z*33\*

The piezoelectric *e*33 constant is defined as

3 33 3

notes the relaxed term that is due to the relative displacements.

The relationship between the piezoelectric *d*33 constant and the *e*33 one is

6 33 3 3 1 ( ) , *E T j j*

*j d se* =

æ ö ¶ ¶

h

utes to the *P*3 from the *u*3(*k*).

*k*th atom, and *Z*33\*

in tetragonal *ABX*3 is investigated. The pseudopotentials were generated using the opium code [31] with semicore and valence electrons (e.g., Ti3spd4s PP), and the

*ec P Z ku k* <sup>=</sup> å (2)

(*k*) denotes the Born effective charges [33] which contrib‐

º ´ å (4)

= + ç ÷ <sup>=</sup> ç ÷ ¶ ¶ è ø <sup>å</sup> (3)

(*k*) denotes the Born effective charges which contributes to the *P*3 from the

### *3.1.1. Role of Ti 3s and 3p states in ferroelectric BaTiO3* **3. Results and discussion**

Figures 2(a) and 2(b)show the optimized results for the ratio *c*/*a* of the lattice parameters and the value of the Ti ion displacement (*δ*Ti) as a function of the *a* lattice parameter in tetragonal BaTiO3, respectively. Results with arrows are the fully optimized results, and the others results are those with the *c* lattice parameters and all the inner coordination optimized for fixed *a*. Note that the fully optimized structure of BaTiO3 is tetragonal with the Ti3spd4s PP, whereas it is cubic with the Ti3d4s PP. This result suggests that the explicit treatment of Ti 3s and 3p semicore states is essential to the appearance of ferroelectric states in BaTiO3. **3.1. Ferroelectricity 3.1.1. Role of Ti 3s and 3p states in ferroelectric BaTiO3** Figures 2(a) and 2(b) show the optimized results for the ratio *c*/*a* of the lattice parameters and the value of the Ti ion displacement (*δ*Ti) as a function of the *a* lattice parameter in tetragonal BaTiO3, respectively. Results with arrows are the fully optimized results, and the others results are those with the *c* lattice parameters and all the inner coordination optimized for fixed *a*. Note that the fully optimized structure of BaTiO3 is tetragonal with the Ti3spd4s PP, whereas it is cubic with the Ti3d4s PP. This result suggests that

the explicit treatment of Ti 3s and 3p semicore states is essential to the appearance of ferroelectric states in BaTiO3.

Figure 2. Optimized calculated results in tetragonal BaTiO3. Results with arrows are the fully optimized results [11]. **Figure 2.** Optimized calculated results in tetragonal BaTiO3. Results with arrows are the fully optimized results [11].

The calculated results shown in Fig. 2 suggest that the explicit treatment of Ti 3s and 3p semicore states is essential to the appearance of ferroelectric states in BaTiO3. In the following, we investigate the role of Ti 3s and 3p states for ferroelectricity from two viewpoints. The calculated results shown in Fig. 2 suggest that the explicit treatment of Ti 3s and 3p semicore states is essential to the appearance of ferroelectric states in BaTiO3. In the follow‐ ing, we investigate the role of Ti 3s and 3p states for ferroelectricity from two viewpoints.

One viewpoint concerns hybridizations between Ti 3s and 3p states and other states. Figure 3(a) and 3(b) shows the total density of states (DOS) of tetragonal BaTiO3 with two Ti PPs. Both results are in good agreement with previous calculated results [7] by the full-potential linear augmented plane wave (FLAPW) method. In the DOS with the Ti3spd4s PP, the energy ``levels", not bands, of Ti 3s and 3p states, are located at -2.0 Hr and -1.2 Hr, respectively. This result suggests that the Ti 3s and 3p orbitals do not make any hybridization but only give Coulomb repulsions with the O orbitals as well as the Ba orbitals. In the DOS with the Ti3d4s PP, on the other hand, the energy levels of Ti 3s and 3p states are not shown because Ti 3s and 3p states were treated as the core charges. This result means that the Ti 3s and 3p orbitals cannot even give Coulomb repulsions with the O orbitals as well as the Ba One viewpoint concerns hybridizations between Ti 3s and 3p states and other states. Fig‐ ure 3(a) and 3(b) shows the total density of states (DOS) of tetragonal BaTiO3 with two Ti PPs. Both results are in good agreement with previous calculated results [7] by the full-potential linear augmented plane wave (FLAPW) method. In the DOS with the Ti3spd4s PP, the energy ``levels", not bands, of Ti 3s and 3p states, are located at -2.0 Hr and -1.2 Hr, respectively. This result suggests that the Ti 3s and 3p orbitals do not make any hybridization but only give Coulomb repulsions with the O orbitals as well as the Ba orbitals. In the DOS with the Ti3d4s PP, on the other hand, the energy levels of Ti 3s and 3p states are not shown because Ti 3s and 3p states were treated as the core charg‐ es. This result means that the Ti 3s and 3p orbitals cannot even give Coulomb repulsions with the O orbitals as well as the Ba orbitals.

**Ti**

states and O*<sup>x</sup>* (*y*)

with increments of 0.2 *e*/Å3 [11].

tween Ti 3s and 3 p*<sup>x</sup>* (*y*)

**Ox**

**Ti3spd4s Ti3d4s**

Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity

**Figure 4.** Two-dimensional electron-density contour map on the *xz*-plane for tetragonal BaTiO3: (a) with the Ti3spd4s PP, and (b) with the Ti3d4s PP. The optimized calculated results with *a* fixed to be 3.8 Å are shown in both figures. The electron density increases as color changes from blue to red via white. Contour curves are drawn from 0.4 to 2.0 *e*/Å3

Considering the above investigations, we propose the mechanism of Ti ion displacement as follows: Ti ion displacement along the *z*-axis appears when the Coulomb repulsions be‐

2s and 2 p*<sup>x</sup>* (*y*)

interaction, overcome the Coulomb repulsions between Ti 3s and 3pz states and O*z* 2s and 2p*<sup>z</sup>* states. An illustration of the Coulomb repulsions is shown in Fig. 5(a). In fully optimized BaTiO3 with the Ti3spd4s PP, the Ti ion can be displaced due to the above mechanism. In fully optimized BaTiO3 with the Ti3d4sPP, on the other hand, the Ti ion cannot be displaced due to the weaker Coulomb repulsions between Ti and O*<sup>x</sup>* (*y*) ions. However, since the Cou‐ lomb repulsion between Ti and O*z* ions in BaTiO3 with the Ti3d4s PP is also weaker than that in BaTiO3 with the Ti3spd4s PP, the Coulomb repulsions between Ti and O*<sup>x</sup>* (*y*) ions in addi‐ tion to the log-range force become comparable to the Coulomb repulsions between Ti and O*<sup>z</sup>* ions both in Ti PPs, as the *a* lattice parameter becomes smaller. The above discussion sug‐ gests that the hybridization between Ti 3d and O*z* 2s and 2p*z* stabilizes Ti ion displacement,

but contribute little to a driving force for the appearance of Ti ion displacement.

**Ti Ox**

http://dx.doi.org/10.5772/52187

7

states, in addition to the dipole-dipole

**Oz**

**Oz**

**(a) (b)**

**Figure 3.** Total density of states (DOS) of fully optimized tetragonal BaTiO3 with the Ti3spd4s PP (solid line) and cubic BaTiO3 with the Ti3d4s PP (red dashed line) [11].

Another viewpoint is about the Coulomb repulsions between Ti 3s and 3p*<sup>x</sup>* (*y*) states and O*<sup>x</sup>* (*y*) 2s and 2p*<sup>x</sup>* (*y*) states in tetragonal BaTiO3. Figure 4(a) and 4(b) show two-dimensional electron-den‐ sity contour map on the *xz*-plane. These are the optimized calculated results with *a* fixed to be 3.8 Å, and the electron density in Fig. 4(a) is quantitatively in good agreement with the experimen‐ tal result [22]. The electron density between Ti and O*x* ions in Fig. 3(a) is larger than that in Fig. 4(b), which suggests that Ti ion displacement is closely related to the Coulomb repulsions be‐ tween Ti 3s and 3p*<sup>x</sup>* (*y*) states and O*<sup>x</sup>* (*y*) 2s and 2p*<sup>x</sup>* (*y*) states; the Ti-O Coulomb repulsion is an impor‐ tant role in the appearance of the ferroelectric state in BaTiO3.

The present discussion of the Coulomb repulsions is consistent with the previous reports. A recent soft mode investigation [8] of BaTiO3 shows that Ba ions contribute little to the ap‐ pearance of Ti ion displacement along the [001] axis. This result suggests that Ti ion dis‐ placement is closely related to the structural distortion of TiO6 octahedra. In the present calculations, on the other hand, the only difference between BaTiO3 with the Ti3spd4s PP and with the Ti3d4s PP is the difference in the expression for the Ti 3s and 3p states, *i.e.*, the explicit treatment and including core charges. However, our previous calculation [20] shows that the strong Coulomb repulsions between Ti 3s and 3p*<sup>z</sup>* states and O*<sup>z</sup>* 2s and 2p*z* states do not favor Ti ion displacement along the [001] axis. This result suggests that the Coulomb re‐ pulsions between Ti 3s and 3p*<sup>x</sup>* (y) states and O*<sup>x</sup>* (*<sup>y</sup>*) 2s and 2p*<sup>x</sup>* (*<sup>y</sup>*) states would contribute to Ti ion displacement along the [001] axis, and the suggestion is consistent with a recent calcula‐ tion [9] for PbTiO3 indicating that the tetragonal and ferroelectric structure appears more fa‐ vorable as the *a* lattice constant decreases.

Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity http://dx.doi.org/10.5772/52187 7

Density of States (/

and 2p*<sup>x</sup>* (*y*)

tween Ti 3s and 3p*<sup>x</sup>* (*y*)

BaTiO3 with the Ti3d4s PP (red dashed line) [11].

states and O*<sup>x</sup>* (*y*)

vorable as the *a* lattice constant decreases.

tant role in the appearance of the ferroelectric state in BaTiO3.

 **Ti3spd4s Ti3d4s**

**Ti 3s Ti 3p O 2p Ti 3d**

**Figure 3.** Total density of states (DOS) of fully optimized tetragonal BaTiO3 with the Ti3spd4s PP (solid line) and cubic

sity contour map on the *xz*-plane. These are the optimized calculated results with *a* fixed to be 3.8 Å, and the electron density in Fig. 4(a) is quantitatively in good agreement with the experimen‐ tal result [22]. The electron density between Ti and O*x* ions in Fig. 3(a) is larger than that in Fig. 4(b), which suggests that Ti ion displacement is closely related to the Coulomb repulsions be‐

The present discussion of the Coulomb repulsions is consistent with the previous reports. A recent soft mode investigation [8] of BaTiO3 shows that Ba ions contribute little to the ap‐ pearance of Ti ion displacement along the [001] axis. This result suggests that Ti ion dis‐ placement is closely related to the structural distortion of TiO6 octahedra. In the present calculations, on the other hand, the only difference between BaTiO3 with the Ti3spd4s PP and with the Ti3d4s PP is the difference in the expression for the Ti 3s and 3p states, *i.e.*, the explicit treatment and including core charges. However, our previous calculation [20] shows that the strong Coulomb repulsions between Ti 3s and 3p*<sup>z</sup>* states and O*<sup>z</sup>* 2s and 2p*z* states do not favor Ti ion displacement along the [001] axis. This result suggests that the Coulomb re‐ pulsions between Ti 3s and 3p*<sup>x</sup>* (y) states and O*<sup>x</sup>* (*<sup>y</sup>*) 2s and 2p*<sup>x</sup>* (*<sup>y</sup>*) states would contribute to Ti ion displacement along the [001] axis, and the suggestion is consistent with a recent calcula‐ tion [9] for PbTiO3 indicating that the tetragonal and ferroelectric structure appears more fa‐

states in tetragonal BaTiO3. Figure 4(a) and 4(b) show two-dimensional electron-den‐

states and O*<sup>x</sup>* (*y*)

states; the Ti-O Coulomb repulsion is an impor‐

2s

Another viewpoint is about the Coulomb repulsions between Ti 3s and 3p*<sup>x</sup>* (*y*)

2s and 2p*<sup>x</sup>* (*y*)

 Hr)

6 Advances in Ferroelectrics

**Figure 4.** Two-dimensional electron-density contour map on the *xz*-plane for tetragonal BaTiO3: (a) with the Ti3spd4s PP, and (b) with the Ti3d4s PP. The optimized calculated results with *a* fixed to be 3.8 Å are shown in both figures. The electron density increases as color changes from blue to red via white. Contour curves are drawn from 0.4 to 2.0 *e*/Å3 with increments of 0.2 *e*/Å3 [11].

Considering the above investigations, we propose the mechanism of Ti ion displacement as follows: Ti ion displacement along the *z*-axis appears when the Coulomb repulsions be‐ tween Ti 3s and 3 p*<sup>x</sup>* (*y*) states and O*<sup>x</sup>* (*y*) 2s and 2 p*<sup>x</sup>* (*y*) states, in addition to the dipole-dipole interaction, overcome the Coulomb repulsions between Ti 3s and 3pz states and O*z* 2s and 2p*<sup>z</sup>* states. An illustration of the Coulomb repulsions is shown in Fig. 5(a). In fully optimized BaTiO3 with the Ti3spd4s PP, the Ti ion can be displaced due to the above mechanism. In fully optimized BaTiO3 with the Ti3d4sPP, on the other hand, the Ti ion cannot be displaced due to the weaker Coulomb repulsions between Ti and O*<sup>x</sup>* (*y*) ions. However, since the Cou‐ lomb repulsion between Ti and O*z* ions in BaTiO3 with the Ti3d4s PP is also weaker than that in BaTiO3 with the Ti3spd4s PP, the Coulomb repulsions between Ti and O*<sup>x</sup>* (*y*) ions in addi‐ tion to the log-range force become comparable to the Coulomb repulsions between Ti and O*<sup>z</sup>* ions both in Ti PPs, as the *a* lattice parameter becomes smaller. The above discussion sug‐ gests that the hybridization between Ti 3d and O*z* 2s and 2p*z* stabilizes Ti ion displacement, but contribute little to a driving force for the appearance of Ti ion displacement.

It seems that the above proposed mechanism for tetragonal BaTiO3 can be applied to the mechanism of Ti ion displacement in rhombohedral BaTiO3, as illustrated in Fig. 5(b). The strong isotropic Coulomb repulsions between Ti 3s and 3px (y, z) states and Ox (y, z) 2s and 2px (y, z) states yield Ti ion displacement along the [111] axis. On the other hand, when the isotropic Coulomb repulsions are weaker or stronger, the Ti ion cannot be displaced and therefore it is favoured for the crystal structure to be cubic.

It seems that the above proposed mechanism for tetragonal BaTiO3 can be applied to the mechanism of Ti ion displacement in rhombohedral BaTiO3, as illustrated in Fig. 5(b). The strong isotropic Coulomb repulsions between Ti 3s and 3px (y, z) states and Ox (y, z) 2s and 2px (y, z) states yield Ti ion displacement along the [111] axis. On the other hand, when the isotropic Coulomb repulsions are weaker or stronger, the Ti ion cannot be displaced and therefore it is favoured for the crystal structure to be cubic.

(a) (b)

Oz

Ox

2s 2 x2s

3px 2px

3s

z

Ti

2pz 3pz

Coulomb repulsion

Figure 5. Illustrations of the proposed mechanisms for the Coulomb repulsions between Ti 3s and 3p states and O 2s and 2p states in BaTiO3: (a) anisotropic Coulomb repulsions between Ti 3s and 3p*<sup>x</sup>* (*<sup>y</sup>*) states and O*<sup>x</sup>* (*<sup>y</sup>*) 2s and 2p*<sup>x</sup>* (*<sup>y</sup>*) states, and between Ti 3s and 3p*z* states and Oz 2s and 2pz states, in the tetragonal structure. (b) isotropic Coulomb repulsions between Ti 3s and 3p*<sup>x</sup>* (*y*, *<sup>z</sup>*) states and O*<sup>x</sup>* (*y*, *<sup>z</sup>*) 2s and 2p*<sup>x</sup>* (*y*, *<sup>z</sup>*) states, in the

x

Let us investigate the structural properties of rhombohedral BaTiO3. Figures 6(a) and 6(b) show the optimized results of the 90-α degree and δTi as a function of fixed volumes of the unit cells in rhombohedral BaTiO3, respectively, where α denotes the angle between two lattice vectors. In these figures, α denotes the angle between two crystal axes of rhombohedral BaTiO3, and δTi denotes the value of the Ti ion displacement along the [111] axis. Results with arrows are the fully optimized results; *V*opt denote the volume of the fully optimized unit cell with the Ti3spd4s PP. The other results are those with all the inner coordination optimized with fixed volumes of the unit cells. The proposal mechanisms about the Coulomb repulsions seem to be consistent with the calculated results shown in Fig.6: For *V*/*V*opt ≲ 0.9 or ≳ 1.3, the isotropic Coulomb repulsions are weaker or stronger, and the Ti ion cannot be displaced along the [111] axis and therefore the crystal structure is cubic for both Ti PPs. For 0.9 ≲ *V*/*V*opt ≲ 1.3, on the other hand, the isotropic Coulomb repulsions are strong enough to yield Ti ion displacement for both Ti PPs. However, since the magnitude of the isotropic Coulomb repulsion is different in the two Ti PPs, the properties of the 90-α degree and δTi are different quantitatively.

Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity

Figure 6. Optimized calculated results as a function of the fixed volumes of the unit cells in rhombohedral BaTiO3: (a) 90-α degree and (b) δTi to the [111] axis. Blue lines correspond to the results with the Ti3spd4s PP, and red lines correspond to those with the Ti3d4s PP. *V*opt denote the volume of the fully optimized unit cell with the Ti3spd4s PP. Results with arrows are the fully optimized results, and the other results are those with all the

**Figure 6.** Optimized calculated results as a function of the fixed volumes of the unit cells in rhombohedral BaTiO3: (a) 90-α degree and (b) δTi to the [111] axis. Blue lines correspond to the results with the Ti3spd4s PP, and red lines corre‐ spond to those with the Ti3d4s PP. *V*opt denote the volume of the fully optimized unit cell with the Ti3spd4s PP. Results with arrows are the fully optimized results, and the other results are those with all the inner coordination optimized

0 0.1 0.2 0.3 0.4

δTi (Å)

 As discussed in Sec. 3.1.1, the Coulomb repulsions between Ti 3s and 3p*<sup>x</sup>* (*<sup>y</sup>*) states and O*<sup>x</sup>* (*<sup>y</sup>*) 2s and 2p*<sup>x</sup>* (*<sup>y</sup>*) states have an important role in the appearance of the ferroelectric state in tetragonal BaTiO3. In this subsection, we discuss the role of Zn 3d (d10) states in

states have an important role in the appearance of the ferroelectric state in tet‐

between the rhombohedral and other structures. Although the lattice constant in each structure except the rhombohedral one

Table 1 shows a summary of the optimized results of BZT in Cases I and II. *ΔE*total denotes the difference in total energy per f.u. between the rhombohedral and other structures. Al‐ though the lattice constant in each structure except the rhombohedral one seems to be quan‐ titatively similar in both cases, properties of *ΔE*total are different. In Case I, the rhombohedral structure is the most stable, which is in disagreement with the experimental result [24]. In Case II, on the other hand, the monoclinic structure, which is the ``pseudo-C-type-tetrago‐ nal'' structure, is the most stable. Unfortunately, this result seems to be in disagreement with the experimental result [24], but is in good agreement with the recent calculated result [6]. Note that the magnitude of *ΔE*total in Case II is markedly smaller than that in Case I. In con‐ trast to BZT, the rhombohedral structure is the most stable structure in both cases in BMT,

Figures 7(a) and 7(b) show two-dimensional electron density contour maps of the C-type tet‐ ragonal BZT in Cases I and II, respectively. The Coulomb repulsion of Zn-O*<sup>x</sup>* in Case II is larger than that in Case I, and the Coulomb repulsion favorably causes Zn ion displacement to O*<sup>z</sup>* in Case II. This result is consistent with Sec. 3.1.1. In contrast to the properties of Zn-O bonding, the inner coordination of the Ti ion is similar in both cases, although the electron densities are markedly different. This result suggests that the Coulomb repulsion magnitude of Ti-O*z* is the same as that of Ti-O*x* in small Ti-O bonding (≈ 1.8 Å), in both Cases I and II. Figures 7(c) and 7(d) show two-dimensional electron density contour maps of the C-type tet‐

ragonal BaTiO3. In this subsection, we discuss the role of Zn 3d (d10) states in addition to 3s

Δ

Δ

*E*total denotes the difference in total energy per f.u.

states and O*<sup>x</sup>* (*y*)

**Ti3d4s**

http://dx.doi.org/10.5772/52187

9

x

Ox

2s

2px

3pz

z

Oz

3px

Coulomb repulsion

3s Ti

*E*total are different. In Case I, the rhombohedral structure is the most

0.8 0.9 1 1.1 1.2 1.3 1.4

**Ti3spd4s**

V/Vopt

rhombohedral structure [11].

inner coordination optimized for fixed volumes of the unit cells [11].

for fixed volumes of the unit cells [11].

0 0.2 0.4 0.6 0.8 1 1.2

90 -

α(deg.)

addition to 3s and 3p states for ferroelectricity in tetragonal BZT.

2s and 2p*<sup>x</sup>* (*y*)

seems to be quantitatively similar in both cases, properties of

Table 1 shows a summary of the optimized results of BZT in Cases I and II.

and 3p states for ferroelectricity in tetragonal BZT.

which is consistent with the experimental result [26].

**3.1.2. Role of Zn 3s, 3p and 3d states in ferroelectric BiZn0.5Ti0.5O3**

*3.1.2. Role of Zn 3s, 3p and 3d states in ferroelectric BiZn0.5Ti0.5O3*

As discussed in Sec. 3.1.1, the Coulomb repulsions between Ti 3s and 3p*<sup>x</sup>* (*y*)

0.8 0.9 1 1.1 1.2 1.3 1.4 V/Vopt

(a) (b)

**Ti3d4s**

**Ti3spd4s**

**Figure 5.** Illustrations of the proposed mechanisms for the Coulomb repulsions between Ti 3s and 3p states and O 2s and 2p states in BaTiO3: (a) anisotropic Coulomb repulsions between Ti 3s and 3p*<sup>x</sup>* (*y*) states and O*<sup>x</sup>* (*y*) 2s and 2p*<sup>x</sup>* (*y*) states, and between Ti 3s and 3p*z* states and Oz 2s and 2pz states, in the tetragonal structure. (b) isotropic Coulomb repulsions between Ti 3s and 3p*x* (*y*, *z*) states and O*x* (*y*, *z*) 2s and 2p*x* (*y*, *z*) states, in the rhombohedral structure [11].

Let us investigate the structural properties of rhombohedral BaTiO3. Figures 6(a) and 6(b) show the optimized results of the 90-α degree and δTi as a function of fixed volumes of the unit cells in rhombohedral BaTiO3, respectively, where α denotes the angle between two lat‐ tice vectors. In these figures, α denotes the angle between two crystal axes of rhombohedral BaTiO3, and δTi denotes the value of the Ti ion displacement along the [111] axis. Results with arrows are the fully optimized results; *V*opt denote the volume of the fully optimized unit cell with the Ti3spd4s PP. The other results are those with all the inner coordination op‐ timized with fixed volumes of the unit cells. The proposal mechanisms about the Coulomb repulsions seem to be consistent with the calculated results shown in Fig.6: For *V*/*V*opt ≲ 0.9 or ≳ 1.3, the isotropic Coulomb repulsions are weaker or stronger, and the Ti ion cannot be displaced along the [111] axis and therefore the crystal structure is cubic for both Ti PPs. For 0.9 ≲ *V*/*V*opt ≲ 1.3, on the other hand, the isotropic Coulomb repulsions are strong enough to yield Ti ion displacement for both Ti PPs. However, since the magnitude of the isotropic Coulomb repulsion is different in the two Ti PPs, the properties of the 90-α degree and δTi are different quantitatively.

the isotropic Coulomb repulsion is different in the two Ti PPs, the properties of the 90-α degree and δTi are different quantitatively.

Figure 5. Illustrations of the proposed mechanisms for the Coulomb repulsions between Ti 3s and 3p states and O 2s and 2p states in BaTiO3: (a) anisotropic Coulomb repulsions between Ti 3s and 3p*<sup>x</sup>* (*<sup>y</sup>*) states and O*<sup>x</sup>* (*<sup>y</sup>*) 2s and 2p*<sup>x</sup>* (*<sup>y</sup>*) states, and between Ti 3s and 3p*z* states and Oz 2s and 2pz states, in the tetragonal structure. (b) isotropic Coulomb repulsions between Ti 3s and 3p*<sup>x</sup>* (*y*, *<sup>z</sup>*) states and O*<sup>x</sup>* (*y*, *<sup>z</sup>*) 2s and 2p*<sup>x</sup>* (*y*, *<sup>z</sup>*) states, in the

x

x

Ox

2s

2px

3pz

z

Oz

3px

Coulomb repulsion

3s Ti

Let us investigate the structural properties of rhombohedral BaTiO3. Figures 6(a) and 6(b) show the optimized results of the 90-α degree and δTi as a function of fixed volumes of the unit cells in rhombohedral BaTiO3, respectively, where α denotes the angle between two lattice vectors. In these figures, α denotes the angle between two crystal axes of rhombohedral BaTiO3, and δTi denotes the value of the Ti ion displacement along the [111] axis. Results with arrows are the fully optimized results; *V*opt denote the volume of the fully optimized unit cell with the Ti3spd4s PP. The other results are those with all the inner coordination optimized with fixed volumes of the unit cells. The proposal mechanisms about the Coulomb repulsions seem to be consistent with the calculated

It seems that the above proposed mechanism for tetragonal BaTiO3 can be applied to the mechanism of Ti ion displacement in rhombohedral BaTiO3, as illustrated in Fig. 5(b). The strong isotropic Coulomb repulsions between Ti 3s and 3px (y, z) states and Ox (y, z) 2s and 2px (y, z) states yield Ti ion displacement along the [111] axis. On the other hand, when the isotropic Coulomb repulsions are weaker or stronger, the Ti ion cannot be displaced and therefore it is favoured for the crystal structure to be cubic.

(a) (b)

Oz

Ox

2s 2 x2s

3px 2px

3s

z

Ti

2pz 3pz

Coulomb repulsion

Figure 6. Optimized calculated results as a function of the fixed volumes of the unit cells in rhombohedral BaTiO3: (a) 90-α degree and (b) δTi to the [111] axis. Blue lines correspond to the results with the Ti3spd4s PP, and red lines correspond to those with the Ti3d4s PP. *V*opt denote the volume of the fully optimized unit cell with the Ti3spd4s PP. Results with arrows are the fully optimized results, and the other results are those with all the inner coordination optimized for fixed volumes of the unit cells [11]. **Figure 6.** Optimized calculated results as a function of the fixed volumes of the unit cells in rhombohedral BaTiO3: (a) 90-α degree and (b) δTi to the [111] axis. Blue lines correspond to the results with the Ti3spd4s PP, and red lines corre‐ spond to those with the Ti3d4s PP. *V*opt denote the volume of the fully optimized unit cell with the Ti3spd4s PP. Results with arrows are the fully optimized results, and the other results are those with all the inner coordination optimized for fixed volumes of the unit cells [11].

#### **3.1.2. Role of Zn 3s, 3p and 3d states in ferroelectric BiZn0.5Ti0.5O3** *3.1.2. Role of Zn 3s, 3p and 3d states in ferroelectric BiZn0.5Ti0.5O3*

rhombohedral structure [11].

It seems that the above proposed mechanism for tetragonal BaTiO3 can be applied to the mechanism of Ti ion displacement in rhombohedral BaTiO3, as illustrated in Fig. 5(b). The strong isotropic Coulomb repulsions between Ti 3s and 3px (y, z) states and Ox (y, z) 2s and 2px (y, z) states yield Ti ion displacement along the [111] axis. On the other hand, when the isotropic Coulomb repulsions are weaker or stronger, the Ti ion cannot be displaced and

**x**

**Figure 5.** Illustrations of the proposed mechanisms for the Coulomb repulsions between Ti 3s and 3p states and O 2s

states, and between Ti 3s and 3p*z* states and Oz 2s and 2pz states, in the tetragonal structure. (b) isotropic Coulomb

2s and 2p*x* (*y*, *z*)

Let us investigate the structural properties of rhombohedral BaTiO3. Figures 6(a) and 6(b) show the optimized results of the 90-α degree and δTi as a function of fixed volumes of the unit cells in rhombohedral BaTiO3, respectively, where α denotes the angle between two lat‐ tice vectors. In these figures, α denotes the angle between two crystal axes of rhombohedral BaTiO3, and δTi denotes the value of the Ti ion displacement along the [111] axis. Results with arrows are the fully optimized results; *V*opt denote the volume of the fully optimized unit cell with the Ti3spd4s PP. The other results are those with all the inner coordination op‐ timized with fixed volumes of the unit cells. The proposal mechanisms about the Coulomb repulsions seem to be consistent with the calculated results shown in Fig.6: For *V*/*V*opt ≲ 0.9 or ≳ 1.3, the isotropic Coulomb repulsions are weaker or stronger, and the Ti ion cannot be displaced along the [111] axis and therefore the crystal structure is cubic for both Ti PPs. For 0.9 ≲ *V*/*V*opt ≲ 1.3, on the other hand, the isotropic Coulomb repulsions are strong enough to yield Ti ion displacement for both Ti PPs. However, since the magnitude of the isotropic Coulomb repulsion is different in the two Ti PPs, the properties of the 90-α degree and δTi

**x**

2s and 2p*<sup>x</sup>* (*y*)

**Ox**

**2s**

**2px**

states and O*<sup>x</sup>* (*y*)

**3pz**

states, in the rhombohedral structure [11].

**z**

**Oz**

**3px**

**Coulomb repulsion**

**3s Ti**

therefore it is favoured for the crystal structure to be cubic.

**(a) (b)**

**Oz**

**Ox**

and 2p states in BaTiO3: (a) anisotropic Coulomb repulsions between Ti 3s and 3p*<sup>x</sup>* (*y*)

states and O*x* (*y*, *z*)

**2s**

**3px 2px**

**3s**

**2 x2s**

**z**

**Ti**

repulsions between Ti 3s and 3p*x* (*y*, *z*)

8 Advances in Ferroelectrics

are different quantitatively.

**2pz 3pz**

**Coulomb repulsion**

 As discussed in Sec. 3.1.1, the Coulomb repulsions between Ti 3s and 3p*<sup>x</sup>* (*<sup>y</sup>*) states and O*<sup>x</sup>* (*<sup>y</sup>*) 2s and 2p*<sup>x</sup>* (*<sup>y</sup>*) states have an important role in the appearance of the ferroelectric state in tetragonal BaTiO3. In this subsection, we discuss the role of Zn 3d (d10) states in addition to 3s and 3p states for ferroelectricity in tetragonal BZT. Table 1 shows a summary of the optimized results of BZT in Cases I and II. Δ*E*total denotes the difference in total energy per f.u. As discussed in Sec. 3.1.1, the Coulomb repulsions between Ti 3s and 3p*<sup>x</sup>* (*y*) states and O*<sup>x</sup>* (*y*) 2s and 2p*<sup>x</sup>* (*y*) states have an important role in the appearance of the ferroelectric state in tet‐ ragonal BaTiO3. In this subsection, we discuss the role of Zn 3d (d10) states in addition to 3s and 3p states for ferroelectricity in tetragonal BZT.

between the rhombohedral and other structures. Although the lattice constant in each structure except the rhombohedral one seems to be quantitatively similar in both cases, properties of Δ*E*total are different. In Case I, the rhombohedral structure is the most Table 1 shows a summary of the optimized results of BZT in Cases I and II. *ΔE*total denotes the difference in total energy per f.u. between the rhombohedral and other structures. Al‐ though the lattice constant in each structure except the rhombohedral one seems to be quan‐ titatively similar in both cases, properties of *ΔE*total are different. In Case I, the rhombohedral structure is the most stable, which is in disagreement with the experimental result [24]. In Case II, on the other hand, the monoclinic structure, which is the ``pseudo-C-type-tetrago‐ nal'' structure, is the most stable. Unfortunately, this result seems to be in disagreement with the experimental result [24], but is in good agreement with the recent calculated result [6]. Note that the magnitude of *ΔE*total in Case II is markedly smaller than that in Case I. In con‐ trast to BZT, the rhombohedral structure is the most stable structure in both cases in BMT, which is consistent with the experimental result [26].

Figures 7(a) and 7(b) show two-dimensional electron density contour maps of the C-type tet‐ ragonal BZT in Cases I and II, respectively. The Coulomb repulsion of Zn-O*<sup>x</sup>* in Case II is larger than that in Case I, and the Coulomb repulsion favorably causes Zn ion displacement to O*<sup>z</sup>* in Case II. This result is consistent with Sec. 3.1.1. In contrast to the properties of Zn-O bonding, the inner coordination of the Ti ion is similar in both cases, although the electron densities are markedly different. This result suggests that the Coulomb repulsion magnitude of Ti-O*z* is the same as that of Ti-O*x* in small Ti-O bonding (≈ 1.8 Å), in both Cases I and II. Figures 7(c) and 7(d) show two-dimensional electron density contour maps of the C-type tet‐ ragonal BMT in Cases I and II, respectively. Although the electron densities in both cases are markedly different, the inner coordination of the Mg ion are similar. This result suggests that the Coulomb repulsion between Mg and O is not strong sufficiently for inducing Mg ion displacement even in Case II.


**Mg Ti**

**Ox**

**Zn Ti Ox**

**Oz**

http://dx.doi.org/10.5772/52187

11

**[110]**

**[001]**

**[110]**

**Oz**

**Zn**

**Oz Oz**

**Figure 8.** Two-dimensional electron density contour maps of BZT in Case II (a) C-type tetragonal and (b) monoclinic. The electron density increases as color changes from blue to red via white. Contour curves are drawn from 0.2 to 2.0

**Bi**

**Bi**

**Oz**

**Oz Oz Oz**

Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity

**Figure 7.** Two-dimensional electron density contour maps of monoclinic (a) BZT in Case I, (b) BZT in Case II, (c) BMT in Case I, and (d) BMT in Case II. The electron density increases as color changes from blue to red via white. Contour

**Zn Ti Ox**

**(a) b**

**(c) d**

**Mg Ti Ox**

**Oz Oz**

curves are drawn from 0.2 to 2.0 *e*/Å3 with increments of 0.2 *e*/Å3 [10].

**Ti Zn**

**(a) b**

**Bi**

**Oz**

**Bi**

**Ti**

**Oz**

*e*/Å3 with increments of 0.2 *e*/Å3 [10].

**[110]**

**[001]**

**Table 1.** Summary of the optimized results of BZT in (a) Case I and (b) Case II. *a* and *c* denote the lattice parameters, and α and β denote angles between two lattice axes. ΔEtotal denotes the difference in total energy per f.u. between the rhombohedral and other structures [10].

Finally in this subsection, we discuss the difference in the electronic structures between the C-type tetragonal and the monoclinic BZT. Figures 8(a) and 8(b) show the electron density contour maps of the C-type tetragonal BZT and that of the monoclinic BZT in Case II, re‐ spectively. This result suggests that the strong Coulomb repulsion between Zn and O*<sup>z</sup>* caus‐ es the small Zn ion displacement in the [110] direction in the monoclinic BZT, which makes the Coulomb repulsion of Zn-O*z* weaker than that in the C-type tetragonal BZT. As a result, this small Zn ion displacement makes the monoclinic BZT more stable than the C-type tet‐ ragonal structure.

Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity http://dx.doi.org/10.5772/52187 11

ragonal BMT in Cases I and II, respectively. Although the electron densities in both cases are markedly different, the inner coordination of the Mg ion are similar. This result suggests that the Coulomb repulsion between Mg and O is not strong sufficiently for inducing Mg

Structure *a* (Å) *c* (Å) *c/a* α (deg.) ΔEtotal (eV/f.u.)

A-type Tetra. 3.748 4.579 1.222 90 0.316

C-type Tetra. 3.681 4.784 1.299 90 0.240

G-type Tetra. 3.725 4.574 1.228 90 0.158

Monoclinic 3.735 4.741 1.269 β = 91.5 0.193

Rhombohedral 5.560 1 59.93 0

Experiment [24] 3.822 4.628 1.211 90 ---

(a)

Structure *a* (Å) *c* (Å) *c/a* α (deg.) ΔEtotal (eV/f.u.)

A-type Tetra. 3.711 4.662 1.256 90 0.135

C-type Tetra. 3.670 4.789 1.305 90 0.091

G-type Tetra. 3.684 4.698 1.275 90 0.047

Monoclinic 3.726 4.740 1.272 β = 91.1 -0.021

Rhombohedral 5.590 1 59.90 0

Experiment [24] 3.822 4.628 1.211 90 ---

(b)

Finally in this subsection, we discuss the difference in the electronic structures between the C-type tetragonal and the monoclinic BZT. Figures 8(a) and 8(b) show the electron density contour maps of the C-type tetragonal BZT and that of the monoclinic BZT in Case II, re‐ spectively. This result suggests that the strong Coulomb repulsion between Zn and O*<sup>z</sup>* caus‐ es the small Zn ion displacement in the [110] direction in the monoclinic BZT, which makes the Coulomb repulsion of Zn-O*z* weaker than that in the C-type tetragonal BZT. As a result, this small Zn ion displacement makes the monoclinic BZT more stable than the C-type tet‐

**Table 1.** Summary of the optimized results of BZT in (a) Case I and (b) Case II. *a* and *c* denote the lattice parameters, and α and β denote angles between two lattice axes. ΔEtotal denotes the difference in total energy per f.u. between the

ion displacement even in Case II.

10 Advances in Ferroelectrics

rhombohedral and other structures [10].

ragonal structure.

**Figure 7.** Two-dimensional electron density contour maps of monoclinic (a) BZT in Case I, (b) BZT in Case II, (c) BMT in Case I, and (d) BMT in Case II. The electron density increases as color changes from blue to red via white. Contour curves are drawn from 0.2 to 2.0 *e*/Å3 with increments of 0.2 *e*/Å3 [10].

**Figure 8.** Two-dimensional electron density contour maps of BZT in Case II (a) C-type tetragonal and (b) monoclinic. The electron density increases as color changes from blue to red via white. Contour curves are drawn from 0.2 to 2.0 *e*/Å3 with increments of 0.2 *e*/Å3 [10].

### **3.2. Piezoelectricity**

the [010] axis) is essential for the appearance of the tetragonal structure.

negative in SrTiO3 while positive in BaTiO3.

products between the *Z*33\*

in the following. Figures 11(a) and 11(b) show the *Z*33\*

0.0 4.0 8.0 12.0 16.0

e33 (C/m^2)

 (C/m2)

differences in the contribution of the ∂*u*3(*k*)/∂*η*3 and ∂*u*3(*k*)/∂*η*1 values, respectively.

*3.2.1. Role of the Ti-O Coulomb repulsions in tetragonal piezoelectric SrTiO3 and BaTiO3*

As discussed in Sec. 3.1, the Coulomb repulsions between Ti 3s and 3p*<sup>x</sup>* (*y*) states and O*<sup>x</sup>* (*y*) 2s and 2p*<sup>x</sup>* (*y*) states have an important role in the appearance of the ferroelectric state in tetrago‐ nal BaTiO3. In this subsection, we discuss the role of the Ti-O Coulomb repulsions for piezo‐ electric SrTiO3 and BaTiO3.

Figures 9(a) shows the optimized results for *c* – *c*cub as a function of the *a* lattice parameters in tetragonal SrTiO3 and BaTiO3, where *c*cub denotes the *c* lattice parameter in cubic SrTiO3 and BaTiO3, respectively. These results are the fully optimized results and the results with the *c* lattice parameters and all the inner coordination optimized for fixed *a*. The fully opti‐ mized parameters of SrTiO3 (*a* = 3.84 Å: cubic) and BaTiO3 (*a* = 3.91 Å and *c* = 4.00 Å: tetrago‐ nal) are within 2.0 % in agreement with the experimental results in room temperature. Figure 9(b) shows the evaluated results for *P*3 as a function of the *a* lattice parameters in tet‐ ragonal SrTiO3 and BaTiO3, where *P*3, which is evaluated by Eq. (2), denotes the spontane‐ ous polarization along the [001] axis. Note that the tetragonal and ferroelectric structures appear even in SrTiO3 when the fixed *a* lattice parameter is compressed to be smaller than the fully-optimized *a* lattice parameter. As shown in Figs. 9(a) and 9(b), the tetragonal and ferroelectric structure appears more favorable as the fixed *a* lattice parameter decreases, which is consistent with previous calculated results [9, 11]. The results would be due to the suggestion discussed in the previous section that the large Coulomb repulsion of Ti-O bond‐ ing along the [100] axis (and the [010] axis) is a driving force of the displacement of Ti ions along the [001] axis, *i.e.*, the large Coulomb repulsion along the [100] axis (and the [010] axis) is essential for the appearance of the tetragonal structure. suggestion discussed in the previous section that the large Coulomb repulsion of Ti-O bonding along the [100] axis (and the [010] axis) is a driving force of the displacement of Ti ions along the [001] axis, *i.e.*, the large Coulomb repulsion along the [100] axis (and

Figure 9. Optimized calculated results as a function of *a* lattice parameters in compressive tetragonal SrTiO3 and BaTiO3: (a) *c* – *c*cub and (b) *P*3, *i.e*., spontaneous polarization along the [001] axis [12]. **Figure 9.** Optimized calculated results as a function of *a* lattice parameters in compressive tetragonal SrTiO3 and Ba‐ TiO3: (a) *c* – *c*cub and (b) *P*3, *i.e*., spontaneous polarization along the [001] axis [12].

In the following, we use *c* – *c*cub as a functional parameter, because *c* – *c*cub is closely related to *η*3. Figures 10(a) and 10(b) shows the piezoelectric properties of *e*33 and *e*31 as a function of *c* – *c*cub in tetragonal SrTiO3 and BaTiO3. The value *c* – *c*cub is optimized value as shown in Fig. 9(a) and *e*33 and *e*31 are evaluated values in their optimized structures. Note that *e*33 become larger at *c* – *c*cub ≈ 0, especially in SrTiO3. These properties seem to be similar to the properties arond the Curie temperatures in piezoelectric *AB*O3; Damjanovic emphasized the importance of the polarization extension as a mechanism of larger piezoelectric constants in a recent In the following, we use *c* – *c*cub as a functional parameter, because *c* – *c*cub is closely related to *η*3. Figures 10(a) and 10(b) shows the piezoelectric properties of *e*33 and *e*<sup>31</sup> as a function of *c* – *c*cub in tetragonal SrTiO3 and BaTiO3. The value *c* – *c*cub is optimized value as shown in Fig. 9(a) and *e*33 and *e*31 are evaluated values in their optimized structures. Note that *e*<sup>33</sup> become

paper [28]. Contrary to *e*33, on the other hand, the changes in *e*31 are much smaller than the changes in *e*33, but note that *e*31 shows

Figure 10.Evaluated piezoelectric constants as a function of *c* – *c*cub in optimized tetragonal SrTiO3 and BaTiO3: (a) *e*33 and (b) *e*31 [12].

0.00 0.20 0.40 0.60 0.80 1.00

BaTiO3

(a) (b)

SrTiO3

c - c\_cub (Å)

As expressed in Eq. (3), *e*3*<sup>j</sup>* is the sum of the contributions from the clamped term and the relaxed term. However, it has been generally known that the contribution to *e*3*<sup>j</sup>* from the clamped term is much smaller than that from the relaxed term; in fact, the absolute values of the *e*33 clamped terms are less than 1 C/m2 in both SrTiO3 and BaTiO3. We therefore investigate the contributions to the relaxed term of *e*33 and *e*31 in detail. As expressed in Eq. (3), the relaxed terms of *e*3*<sup>j</sup>* are proportional to the sum of the


e31 (C/m^2)

e31 (C/m2)

c - ccub (Å) c - ccub (Å)

(*k*) and ∂*u*3 (*k*)/∂*ηj* (*j* = 3 or 1) values. Let us show the evaluated results of *Z*33\*

are quantitatively similar in both SrTiO3 and BaTiO3. Therefore, the difference in the properties of *e*33 and *e*31 between SrTiO3 and BaTiO3 must be due to the difference in the properties of ∂*u*3(*k*)/∂*ηj*. Figures 12(a) and 12(b) show the ∂*u*3(*k*)/∂*η*3 values in SrTiO3 and BaTiO3, respectively. In these figures, O*x* and O*z* denote oxygen atoms along the [100] and [001] axes, respectively. Clearly, the absolute values of ∂*u*3(*k*)/∂*η*3 are different in between SrTiO3 and BaTiO3. On the other hand, Figs. 13(a) and 13(b) show the ∂*u*3(*k*)/∂*η*1 values in SrTiO3 and BaTiO3, respectively. The absolute values of ∂*u*3(*k*)/∂*η*1, especially for Ti, O*x*, and O*z* are different in between SrTiO3 and BaTiO3. As a result, the quantitative differences in *e*33 and *e*31 between SrTiO3 and BaTiO3 are due to the

(*k*) values in SrTiO3 and BaTiO3, respectively. Properties of the *Z*33\*

0.00 0.20 0.40 0.60 0.80 1.00

BaTiO3

SrTiO3

c - c\_cub (Å)

(*k*), ∂*u*3(*k*)/∂*η*3, and ∂*u*3(*k*)/∂*η*<sup>1</sup>

(*k*) values

larger at *c* – *c*cub ≈ 0, especially in SrTiO3. These properties seem to be similar to the proper‐ ties arond the Curie temperatures in piezoelectric *AB*O3; Damjanovic emphasized the impor‐ tance of the polarization extension as a mechanism of larger piezoelectric constants in a recent paper [28]. Contrary to *e*33, on the other hand, the changes in *e*31 are much smaller than

Figure 9. Optimized calculated results as a function of *a* lattice parameters in compressive tetragonal SrTiO3 and BaTiO3: (a) *c* – *c*cub and (b) *P*3, *i.e*.,

Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity

a (Å) a (Å)

0.0 0.2 0.4 0.6 0.8 1.0

P3 (C/m^2)

P3 (C/m2)

3.6 3.7 3.8 3.9 4.0

BaTiO3

http://dx.doi.org/10.5772/52187

13

SrTiO3

a (A)

In the following, we use *c* – *c*cub as a functional parameter, because *c* – *c*cub is closely related to *η*3. Figures 10(a) and 10(b) shows the piezoelectric properties of *e*33 and *e*31 as a function of *c* – *c*cub in tetragonal SrTiO3 and BaTiO3. The value *c* – *c*cub is optimized value as shown in Fig. 9(a) and *e*33 and *e*31 are evaluated values in their optimized structures. Note that *e*33 become larger at *c* – *c*cub ≈ 0, especially in SrTiO3. These properties seem to be similar to the properties arond the Curie temperatures in piezoelectric *AB*O3; Damjanovic emphasized the importance of the polarization extension as a mechanism of larger piezoelectric constants in a recent paper [28]. Contrary to *e*33, on the other hand, the changes in *e*31 are much smaller than the changes in *e*33, but note that *e*31 shows

suggestion discussed in the previous section that the large Coulomb repulsion of Ti-O bonding along the [100] axis (and the [010] axis) is a driving force of the displacement of Ti ions along the [001] axis, *i.e.*, the large Coulomb repulsion along the [100] axis (and

the [010] axis) is essential for the appearance of the tetragonal structure.

SrTiO3

3.6 3.7 3.8 3.9 4.0

(a) (b)

BaTiO3

a (A)

spontaneous polarization along the [001] axis [12].

0.0 0.2 0.4 0.6 0.8 1.0

c - c\_cub (A)

c - ccub (Å)

negative in SrTiO3 while positive in BaTiO3.

products between the *Z*33\*

and (b) *e*31 [12].

in the following. Figures 11(a) and 11(b) show the *Z*33\*

respectively. Properties of the *Z*33\*

relaxed terms of *e*3*<sup>j</sup>*

As expressed in Eq. (3), *e*3*<sup>j</sup>*

0.0 4.0 8.0 12.0 16.0

e33 (C/m^2)

e33 (C/m2)

differences in the contribution of the ∂*u*3(*k*)/∂*η*3 and ∂*u*3(*k*)/∂*η*1 values, respectively.

must be due to the difference in the properties of ∂*u*3(*k*)/∂*η<sup>j</sup>*

in the contribution of the ∂*u*3(*k*)/∂*η*3 and ∂*u*3(*k*)/∂*η*1 values, respectively.

the *e*33 clamped terms are less than 1 C/m2

the changes in *e*33, but note that *e*31 shows negative in SrTiO3 while positive in BaTiO3.

Figure 10.Evaluated piezoelectric constants as a function of *c* – *c*cub in optimized tetragonal SrTiO3 and BaTiO3: (a) *e*33 and (b) *e*31 [12].

relaxed term. However, it has been generally known that the contribution to *e*3*<sup>j</sup>*

0.00 0.20 0.40 0.60 0.80 1.00

BaTiO3

(a) (b)

SrTiO3

c - c\_cub (Å)

As expressed in Eq. (3), *e*3*<sup>j</sup>* is the sum of the contributions from the clamped term and the relaxed term. However, it has been generally known that the contribution to *e*3*<sup>j</sup>* from the clamped term is much smaller than that from the relaxed term; in fact, the absolute values of the *e*33 clamped terms are less than 1 C/m2 in both SrTiO3 and BaTiO3. We therefore investigate the contributions to the relaxed term of *e*33 and *e*31 in detail. As expressed in Eq. (3), the relaxed terms of *e*3*<sup>j</sup>* are proportional to the sum of the

**Figure 10.** Evaluated piezoelectric constants as a function of *c* – *c*cub in optimized tetragonal SrTiO3 and BaTiO3: (a) *e*<sup>33</sup>


e31 (C/m^2)

e31 (C/m2)

c - ccub (Å) c - ccub (Å)

(*k*) and ∂*u*3 (*k*)/∂*ηj* (*j* = 3 or 1) values. Let us show the evaluated results of *Z*33\*

are quantitatively similar in both SrTiO3 and BaTiO3. Therefore, the difference in the properties of *e*33 and *e*31 between SrTiO3 and BaTiO3 must be due to the difference in the properties of ∂*u*3(*k*)/∂*ηj*. Figures 12(a) and 12(b) show the ∂*u*3(*k*)/∂*η*3 values in SrTiO3 and BaTiO3, respectively. In these figures, O*x* and O*z* denote oxygen atoms along the [100] and [001] axes, respectively. Clearly, the absolute values of ∂*u*3(*k*)/∂*η*3 are different in between SrTiO3 and BaTiO3. On the other hand, Figs. 13(a) and 13(b) show the ∂*u*3(*k*)/∂*η*1 values in SrTiO3 and BaTiO3, respectively. The absolute values of ∂*u*3(*k*)/∂*η*1, especially for Ti, O*x*, and O*z* are different in between SrTiO3 and BaTiO3. As a result, the quantitative differences in *e*33 and *e*31 between SrTiO3 and BaTiO3 are due to the

clamped term is much smaller than that from the relaxed term; in fact, the absolute values of

gate the contributions to the relaxed term of *e*33 and *e*31 in detail. As expressed in Eq. (3), the

(*k*)/∂*η<sup>j</sup>* (*j* = 3 or 1) values. Let us show the evaluated results of *Z*33\* (*k*), ∂*u*3(*k*)/∂*η*3, and ∂*u*3(*k*)/ ∂*η*1 in the following. Figures 11(a) and 11(b) show the *Z*33\* (*k*) values in SrTiO3 and BaTiO3,

TiO3. Therefore, the difference in the properties of *e*33 and *e*<sup>31</sup> between SrTiO3 and BaTiO3

∂*u*3(*k*)/∂*η*3 values in SrTiO3 and BaTiO3, respectively. In these figures, O*x* and O*z* denote oxy‐ gen atoms along the [100] and [001] axes, respectively. Clearly, the absolute values of ∂*u*3(*k*)/ ∂*η*3 are different in between SrTiO3 and BaTiO3. On the other hand, Figs. 13(a) and 13(b) show the ∂*u*3(*k*)/∂*η*1 values in SrTiO3 and BaTiO3, respectively. The absolute values of ∂*u*3(*k*)/ ∂*η*1, especially for Ti, O*x*, and O*z* are different in between SrTiO3 and BaTiO3. As a result, the quantitative differences in *e*33 and *e*31 between SrTiO3 and BaTiO3 are due to the differences

(*k*) values in SrTiO3 and BaTiO3, respectively. Properties of the *Z*33\*

(*k*) values are quantitatively similar in both SrTiO3 and Ba‐

in both SrTiO3 and BaTiO3. We therefore investi‐

. Figures 12(a) and 12(b) show the

is the sum of the contributions from the clamped term and the

are proportional to the sum of the products between the *Z*33\* (*k*) and ∂*u*<sup>3</sup>

0.00 0.20 0.40 0.60 0.80 1.00

BaTiO3

SrTiO3

c - c\_cub (Å)

(*k*), ∂*u*3(*k*)/∂*η*3, and ∂*u*3(*k*)/∂*η*<sup>1</sup>

from the

(*k*) values

0.0 0.2 0.4 0.6 0.8 1.0

SrTiO3

BaTiO3

P3 (C/m^2)

P3 (C/m2)

suggestion discussed in the previous section that the large Coulomb repulsion of Ti-O bonding along the [100] axis (and the [010] axis) is a driving force of the displacement of Ti ions along the [001] axis, *i.e.*, the large Coulomb repulsion along the [100] axis (and

the [010] axis) is essential for the appearance of the tetragonal structure.

SrTiO3

(a) (b)

BaTiO3

0.2 0.4 0.6 0.8 1.0

c - c\_cub (A)

c - ccub (Å)

negative in SrTiO3 while positive in BaTiO3.

larger at *c* – *c*cub ≈ 0, especially in SrTiO3. These properties seem to be similar to the proper‐ ties arond the Curie temperatures in piezoelectric *AB*O3; Damjanovic emphasized the impor‐ tance of the polarization extension as a mechanism of larger piezoelectric constants in a recent paper [28]. Contrary to *e*33, on the other hand, the changes in *e*31 are much smaller than the changes in *e*33, but note that *e*31 shows negative in SrTiO3 while positive in BaTiO3. Figure 9. Optimized calculated results as a function of *a* lattice parameters in compressive tetragonal SrTiO3 and BaTiO3: (a) *c* – *c*cub and (b) *P*3, *i.e*., spontaneous polarization along the [001] axis [12]. In the following, we use *c* – *c*cub as a functional parameter, because *c* – *c*cub is closely related to *η*3. Figures 10(a) and 10(b) shows the piezoelectric properties of *e*33 and *e*31 as a function of *c* – *c*cub in tetragonal SrTiO3 and BaTiO3. The value *c* – *c*cub is optimized value as shown in Fig. 9(a) and *e*33 and *e*31 are evaluated values in their optimized structures. Note that *e*33 become larger at *c* – *c*cub ≈ 0,

especially in SrTiO3. These properties seem to be similar to the properties arond the Curie temperatures in piezoelectric *AB*O3; Damjanovic emphasized the importance of the polarization extension as a mechanism of larger piezoelectric constants in a recent paper [28]. Contrary to *e*33, on the other hand, the changes in *e*31 are much smaller than the changes in *e*33, but note that *e*31 shows

**3.2. Piezoelectricity**

12 Advances in Ferroelectrics

electric SrTiO3 and BaTiO3.

the [010] axis) is essential for the appearance of the tetragonal structure.

SrTiO3

3.6 3.7 3.8 3.9 4.0

(a) (b)

BaTiO3

a (A)

TiO3: (a) *c* – *c*cub and (b) *P*3, *i.e*., spontaneous polarization along the [001] axis [12].

spontaneous polarization along the [001] axis [12].

0.0 0.2 0.4 0.6 0.8 1.0

c - c\_cub (A)

c - ccub (Å)

negative in SrTiO3 while positive in BaTiO3.

products between the *Z*33\*

in the following. Figures 11(a) and 11(b) show the *Z*33\*

0.0 4.0 8.0 12.0 16.0

e33 (C/m^2)

e33 (C/m2)

differences in the contribution of the ∂*u*3(*k*)/∂*η*3 and ∂*u*3(*k*)/∂*η*1 values, respectively.

*3.2.1. Role of the Ti-O Coulomb repulsions in tetragonal piezoelectric SrTiO3 and BaTiO3*

As discussed in Sec. 3.1, the Coulomb repulsions between Ti 3s and 3p*<sup>x</sup>* (*y*) states and O*<sup>x</sup>* (*y*) 2s and 2p*<sup>x</sup>* (*y*) states have an important role in the appearance of the ferroelectric state in tetrago‐ nal BaTiO3. In this subsection, we discuss the role of the Ti-O Coulomb repulsions for piezo‐

Figures 9(a) shows the optimized results for *c* – *c*cub as a function of the *a* lattice parameters in tetragonal SrTiO3 and BaTiO3, where *c*cub denotes the *c* lattice parameter in cubic SrTiO3 and BaTiO3, respectively. These results are the fully optimized results and the results with the *c* lattice parameters and all the inner coordination optimized for fixed *a*. The fully opti‐ mized parameters of SrTiO3 (*a* = 3.84 Å: cubic) and BaTiO3 (*a* = 3.91 Å and *c* = 4.00 Å: tetrago‐ nal) are within 2.0 % in agreement with the experimental results in room temperature. Figure 9(b) shows the evaluated results for *P*3 as a function of the *a* lattice parameters in tet‐ ragonal SrTiO3 and BaTiO3, where *P*3, which is evaluated by Eq. (2), denotes the spontane‐ ous polarization along the [001] axis. Note that the tetragonal and ferroelectric structures appear even in SrTiO3 when the fixed *a* lattice parameter is compressed to be smaller than the fully-optimized *a* lattice parameter. As shown in Figs. 9(a) and 9(b), the tetragonal and ferroelectric structure appears more favorable as the fixed *a* lattice parameter decreases, which is consistent with previous calculated results [9, 11]. The results would be due to the suggestion discussed in the previous section that the large Coulomb repulsion of Ti-O bond‐ ing along the [100] axis (and the [010] axis) is a driving force of the displacement of Ti ions along the [001] axis, *i.e.*, the large Coulomb repulsion along the [100] axis (and the [010] axis) is essential for the appearance of the tetragonal structure. suggestion discussed in the previous section that the large Coulomb repulsion of Ti-O bonding along the [100] axis (and the [010] axis) is a driving force of the displacement of Ti ions along the [001] axis, *i.e.*, the large Coulomb repulsion along the [100] axis (and

Figure 9. Optimized calculated results as a function of *a* lattice parameters in compressive tetragonal SrTiO3 and BaTiO3: (a) *c* – *c*cub and (b) *P*3, *i.e*.,

**Figure 9.** Optimized calculated results as a function of *a* lattice parameters in compressive tetragonal SrTiO3 and Ba‐

a (Å) a (Å)

0.0 0.2 0.4 0.6 0.8 1.0

P3 (C/m^2)

P3 (C/m2)

3.6 3.7 3.8 3.9 4.0

BaTiO3

SrTiO3

a (A)

In the following, we use *c* – *c*cub as a functional parameter, because *c* – *c*cub is closely related to *η*3. Figures 10(a) and 10(b) shows the piezoelectric properties of *e*33 and *e*31 as a function of *c* – *c*cub in tetragonal SrTiO3 and BaTiO3. The value *c* – *c*cub is optimized value as shown in Fig. 9(a) and *e*33 and *e*31 are evaluated values in their optimized structures. Note that *e*33 become larger at *c* – *c*cub ≈ 0, especially in SrTiO3. These properties seem to be similar to the properties arond the Curie temperatures in piezoelectric *AB*O3; Damjanovic emphasized the importance of the polarization extension as a mechanism of larger piezoelectric constants in a recent paper [28]. Contrary to *e*33, on the other hand, the changes in *e*31 are much smaller than the changes in *e*33, but note that *e*31 shows

In the following, we use *c* – *c*cub as a functional parameter, because *c* – *c*cub is closely related to *η*3. Figures 10(a) and 10(b) shows the piezoelectric properties of *e*33 and *e*<sup>31</sup> as a function of *c* – *c*cub in tetragonal SrTiO3 and BaTiO3. The value *c* – *c*cub is optimized value as shown in Fig. 9(a) and *e*33 and *e*31 are evaluated values in their optimized structures. Note that *e*<sup>33</sup> become

Figure 10.Evaluated piezoelectric constants as a function of *c* – *c*cub in optimized tetragonal SrTiO3 and BaTiO3: (a) *e*33 and (b) *e*31 [12].

0.00 0.20 0.40 0.60 0.80 1.00

BaTiO3

(a) (b)

SrTiO3

c - c\_cub (Å)

As expressed in Eq. (3), *e*3*<sup>j</sup>* is the sum of the contributions from the clamped term and the relaxed term. However, it has been generally known that the contribution to *e*3*<sup>j</sup>* from the clamped term is much smaller than that from the relaxed term; in fact, the absolute values of the *e*33 clamped terms are less than 1 C/m2 in both SrTiO3 and BaTiO3. We therefore investigate the contributions to the relaxed term of *e*33 and *e*31 in detail. As expressed in Eq. (3), the relaxed terms of *e*3*<sup>j</sup>* are proportional to the sum of the


e31 (C/m^2)

e31 (C/m2)

c - ccub (Å) c - ccub (Å)

(*k*) and ∂*u*3 (*k*)/∂*ηj* (*j* = 3 or 1) values. Let us show the evaluated results of *Z*33\*

are quantitatively similar in both SrTiO3 and BaTiO3. Therefore, the difference in the properties of *e*33 and *e*31 between SrTiO3 and BaTiO3 must be due to the difference in the properties of ∂*u*3(*k*)/∂*ηj*. Figures 12(a) and 12(b) show the ∂*u*3(*k*)/∂*η*3 values in SrTiO3 and BaTiO3, respectively. In these figures, O*x* and O*z* denote oxygen atoms along the [100] and [001] axes, respectively. Clearly, the absolute values of ∂*u*3(*k*)/∂*η*3 are different in between SrTiO3 and BaTiO3. On the other hand, Figs. 13(a) and 13(b) show the ∂*u*3(*k*)/∂*η*1 values in SrTiO3 and BaTiO3, respectively. The absolute values of ∂*u*3(*k*)/∂*η*1, especially for Ti, O*x*, and O*z* are different in between SrTiO3 and BaTiO3. As a result, the quantitative differences in *e*33 and *e*31 between SrTiO3 and BaTiO3 are due to the

(*k*) values in SrTiO3 and BaTiO3, respectively. Properties of the *Z*33\*

0.00 0.20 0.40 0.60 0.80 1.00

BaTiO3

SrTiO3

c - c\_cub (Å)

(*k*), ∂*u*3(*k*)/∂*η*3, and ∂*u*3(*k*)/∂*η*<sup>1</sup>

(*k*) values

As expressed in Eq. (3), *e*3*<sup>j</sup>* is the sum of the contributions from the clamped term and the relaxed term. However, it has been generally known that the contribution to *e*3*<sup>j</sup>* from the clamped term is much smaller than that from the relaxed term; in fact, the absolute values of the *e*33 clamped terms are less than 1 C/m2 in both SrTiO3 and BaTiO3. We therefore investigate the contributions **Figure 10.** Evaluated piezoelectric constants as a function of *c* – *c*cub in optimized tetragonal SrTiO3 and BaTiO3: (a) *e*<sup>33</sup> and (b) *e*31 [12].

(*k*) values

Figure 10.Evaluated piezoelectric constants as a function of *c* – *c*cub in optimized tetragonal SrTiO3 and BaTiO3: (a) *e*33 and (b) *e*31 [12].

to the relaxed term of *e*33 and *e*31 in detail. As expressed in Eq. (3), the relaxed terms of *e*3*<sup>j</sup>* are proportional to the sum of the products between the *Z*33\* (*k*) and ∂*u*3 (*k*)/∂*ηj* (*j* = 3 or 1) values. Let us show the evaluated results of *Z*33\* (*k*), ∂*u*3(*k*)/∂*η*3, and ∂*u*3(*k*)/∂*η*<sup>1</sup> in the following. Figures 11(a) and 11(b) show the *Z*33\* (*k*) values in SrTiO3 and BaTiO3, respectively. Properties of the *Z*33\* are quantitatively similar in both SrTiO3 and BaTiO3. Therefore, the difference in the properties of *e*33 and *e*31 between SrTiO3 and BaTiO3 must be due to the difference in the properties of ∂*u*3(*k*)/∂*ηj*. Figures 12(a) and 12(b) show the ∂*u*3(*k*)/∂*η*3 values in SrTiO3 and BaTiO3, respectively. In these figures, O*x* and O*z* denote oxygen atoms along the [100] and [001] axes, respectively. Clearly, the absolute values of ∂*u*3(*k*)/∂*η*3 are different in between SrTiO3 and BaTiO3. On the other hand, Figs. 13(a) and 13(b) show the ∂*u*3(*k*)/∂*η*1 values in SrTiO3 and BaTiO3, respectively. The absolute values of ∂*u*3(*k*)/∂*η*1, especially for Ti, O*x*, and O*z* are different in between SrTiO3 and BaTiO3. As a result, the quantitative differences in *e*33 and *e*31 between SrTiO3 and BaTiO3 are due to the differences in the contribution of the ∂*u*3(*k*)/∂*η*3 and ∂*u*3(*k*)/∂*η*1 values, respectively. As expressed in Eq. (3), *e*3*<sup>j</sup>* is the sum of the contributions from the clamped term and the relaxed term. However, it has been generally known that the contribution to *e*3*<sup>j</sup>* from the clamped term is much smaller than that from the relaxed term; in fact, the absolute values of the *e*33 clamped terms are less than 1 C/m2 in both SrTiO3 and BaTiO3. We therefore investi‐ gate the contributions to the relaxed term of *e*33 and *e*31 in detail. As expressed in Eq. (3), the relaxed terms of *e*3*<sup>j</sup>* are proportional to the sum of the products between the *Z*33\* (*k*) and ∂*u*<sup>3</sup> (*k*)/∂*η<sup>j</sup>* (*j* = 3 or 1) values. Let us show the evaluated results of *Z*33\* (*k*), ∂*u*3(*k*)/∂*η*3, and ∂*u*3(*k*)/ ∂*η*1 in the following. Figures 11(a) and 11(b) show the *Z*33\* (*k*) values in SrTiO3 and BaTiO3, respectively. Properties of the *Z*33\* (*k*) values are quantitatively similar in both SrTiO3 and Ba‐ TiO3. Therefore, the difference in the properties of *e*33 and *e*<sup>31</sup> between SrTiO3 and BaTiO3 must be due to the difference in the properties of ∂*u*3(*k*)/∂*η<sup>j</sup>* . Figures 12(a) and 12(b) show the ∂*u*3(*k*)/∂*η*3 values in SrTiO3 and BaTiO3, respectively. In these figures, O*x* and O*z* denote oxy‐ gen atoms along the [100] and [001] axes, respectively. Clearly, the absolute values of ∂*u*3(*k*)/ ∂*η*3 are different in between SrTiO3 and BaTiO3. On the other hand, Figs. 13(a) and 13(b) show the ∂*u*3(*k*)/∂*η*1 values in SrTiO3 and BaTiO3, respectively. The absolute values of ∂*u*3(*k*)/ ∂*η*1, especially for Ti, O*x*, and O*z* are different in between SrTiO3 and BaTiO3. As a result, the quantitative differences in *e*33 and *e*31 between SrTiO3 and BaTiO3 are due to the differences in the contribution of the ∂*u*3(*k*)/∂*η*3 and ∂*u*3(*k*)/∂*η*1 values, respectively.

axis and the [001] axis, respectively [12].

Figure 11.Evaluated Born effective charges *Z*33\*


Figure 11.Evaluated Born effective charges *Z*33\* (*k*) as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3. O*x* and O*z* denote oxygen atoms along the [100] axis and the [001] axis, respectively [12]. **Figure 11.** Evaluated Born effective charges *Z*33\* (*k*) as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3. O*x* and O*z* denote oxygen atoms along the [100] axis and the [001] axis, respectively [12]. Figure 11.Evaluated Born effective charges *Z*33\* (*k*) as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3. O*x* and O*z* denote oxygen atoms along the [100] c - c\_cub (A) c - c\_cub (A) c - ccub (Å) c - ccub (Å)

c - ccub (Å) c - ccub (Å)


(a) (b) BTO

STO (a) (b) BTO Figure 12.Evaluated values of ∂*u*3(*k*)/∂*η*3 as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3 [12]. **Figure 12.** Evaluated values of ∂*u*3(*k*)/∂η3 as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3 [12].

(a) (b) BTO

STO

c - c\_cub (A)

Figure 12.Evaluated values of ∂*u*3(*k*)/∂*η*3 as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3 [12].

STO

c - c\_cub (A)

0.00 0.20 0.40 0.60 0.80 1.00

Sr Ti

0.0 0.1

c - ccub (Å) c - ccub (Å)

Ba

c - c\_cub (A)

c - c\_cub (A)

0.00 0.20 0.40 0.60 0.80 1.00

and *r*O*<sup>z</sup>* (*r*Ti + *r*O*<sup>z</sup>*) along the [001] axis as a function of *c* – *c*cub. Note that *R*Ti-O*<sup>z</sup>* is smaller than *r*Ti + *r*O*<sup>z</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *R*Ti-O*<sup>z</sup>* and *r*Ti + *r*O*<sup>z</sup>* in SrTiO3 is smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Ti-Oz Coulomb repulsion along the [001] axis in SrTiO3 is smaller than that in BaTiO3 and that therefore the Ti ion of SrTiO3 can be displaced more easily along the [001] axis than that of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Ti and O*z* ions in SrTiO3 are larger than that in BaTiO3. Figure 14(b) shows the difference between the *A*-O*x* distance (*RA*-O*<sup>x</sup>*) and the sum of *rA* and *r*O*<sup>x</sup>* (*rA* + *r*O*<sup>x</sup>*) on the (100) plane as a function of *c* – *c*cub, where the values of the ionic radii are defined as Shannon's ones [23]. Note that *RA*-O*<sup>x</sup>* is smaller than *rA* + *r*O*<sup>x</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *RA*-O*<sup>x</sup>* and *r*A + *r*O*<sup>x</sup>* in SrTiO3 is much smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Sr-O*x* Coulomb re‐ pulsion on the (100) plane in SrTiO3 is much smaller than the Ba-O*x* Coulomb repulsion in BaTiO3 and that therefore Sr and O*x* ions of SrTiO3 can be displaced more easily along the [001] axis than Ba and O*<sup>x</sup>* ions of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Sr and O*x* ions in SrTiO3 are larger than those of Ba and O*x* ions in BaTiO3.

Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity

Figure 14.Evaluated values as a function of *c* – *c*cub in optimized tetragonal SrTiO3 and BaTiO3: (a) difference between the Ti-O*<sup>z</sup>* distance (*R*Ti-O*<sup>z</sup>*) and *r*Ti + *r*O*<sup>z</sup>*. (b) difference between the *A*-O*<sup>x</sup>* distance (*RA*-Ox) and *rA* + *r*O*<sup>x</sup>*. *RA*-O*<sup>x</sup>* and *R*Ti-O*<sup>z</sup>* in *A*TiO3 are also illustrated. Note that all the ionic radii are

**Figure 14.** Evaluated values as a function of *c* – *c*cub in optimized tetragonal SrTiO3 and BaTiO3: (a) difference between the Ti-O*z* distance (*R*Ti-O*z*) and *r*Ti + *r*O*z*. (b) difference between the *A*-O*<sup>x</sup>* distance (*RA*-Ox) and *rA* + *r*O*x*. *RA*-O*<sup>x</sup>* and *R*Ti-O*<sup>z</sup>* in *A*TiO3 are also illustrated. Note that all the ionic radii are much larger and that *A* and Ti ions are displaced along the

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Finally, in this subsection, we discuss the relationship between ∂*u*3(*k*)/∂*η*<sup>3</sup> and *c* – *c*cub in detail. Figure 15(a) shows the differences in the total energy (*ΔE*total ) as a function of *u*3(Ti). In this figure, the properties of SrTiO3 with *η*<sup>3</sup> = 0.011, SrTiO3 with *η*<sup>3</sup> = 0.053 and fully optimized BaTiO3 as a reference, are shown. Calculations of *E*total were performed with the fixed crystal structures of previously optimized structures except Ti ions. Clearly, the magnitude of *u*3(Ti) at the minimum points of the *ΔE*total and the depth of the potential are closely related to the spontaneous polarization *P*<sup>3</sup> and the Curie temperature (*T*C), respectively. On the other hand, *e*<sup>33</sup> seems to be closely related to the deviation at the minimum points of the *ΔE*total. Figure 15(b) shows illustrations of *ΔE*total curves with deviations at the minimum points of the *ΔE*total values, corresponding to the *ΔE*total curves of SrTiO3 in Fig. 15(a). Clearly, as *η*<sup>3</sup> becomes smaller, the deviated value at the minimum point of the *ΔE*total values becomes smaller, *i.e.*, the Ti ion can be displaced more favourably. On the other hand, as shown in Fig. 12(a), the absolute value of ∂*u*3(Ti)/∂*η*<sup>3</sup> becomes larger as *η*<sup>3</sup>

Finally, in this subsection, we discuss the relationship between ∂*u*3(*k*)/∂*η*3 and *c* – *c*cub in de‐ tail. Figure 15(a) shows the differences in the total energy (*ΔE*total ) as a function of *u*3(Ti). In

Therefore, the Ti ion can be displaced more favourably as the deviated value at the minimum point of the *ΔE*total values becomes

Figure 15.(a) *ΔE*total as a function of *u*3(Ti) in tetragonal SrTiO3 and BaTiO3. (b) Illustration of the *ΔE*total curves in tetragonal SrTiO3 with *η<sup>3</sup>* = 0.011

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The previous discussion in Sec. 3.2.1 suggests that the piezoelectric properties of *e*<sup>33</sup> are closely related to the *B*-*X* Coulomb repulsions in tegtragonal *ABX*3. In the viewpoint of the change of the *B*-*X* Coulomb repulsions, we recently proposed new

It has been known that BaNiO3 shows the 2H hexagonal structure as the most stable structure in room temperature. Moreover, the ionic radius of Ni4+ (d6) with the low-spin state in 2H BaNiO3 is 0.48 Å, which is much smaller than that of Ti4+ (d0), 0.605 Å, in

much larger and that *A* and Ti ions are displaced along the [001] axis in actual *A*TiO3 [12].

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��Ti��

��Ti��

(�) (�)

� � ����� (Å)

����

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[001] axis in actual *A*TiO3 [12].

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smaller. The relationship between *e*<sup>33</sup> and ∂*u*3(Ti)/∂*η*<sup>3</sup> is discussed in Sec. 3.2.3.

and SrTiO3 with *η*<sup>3</sup> = 0.053 with deviations at the minimum point of *ΔE*total [14].

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(�) (�) ⊿������ ���� ⊿�Ti

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��Ti (�)

piezoelectric materials [16, 17], *i.e.*, BaTi1*-x*Ni*x*O3 and Ba(Ti1*-3z*Nb*3z*)(O1-zNz)3.

**3.2.2. Proposal of new piezoelectric materials**

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⊿���� (��)

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becomes smaller.

(*k*) as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3. O*x* and O*z* denote oxygen atoms along the [100]

is smaller than *r*Ti + *r*O*<sup>z</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *R*Ti-O*<sup>z</sup>* and *r*Ti + *r*O*<sup>z</sup>* in SrTiO3 is Let us discuss the reasons of the quantitative differences in *e*33 between SrTiO3 and BaTiO3. Figure 14(a) shows the difference Figure 13.Evaluated values of ∂*u*3(*k*)/∂*η*1 as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3 [12]. **Figure 13.** Evaluated values of ∂*u*3(*k*)/∂η1 as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3 [12].

Figure 13.Evaluated values of ∂*u*3(*k*)/∂*η*1 as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3 [12].

SrTiO3 are larger than those of Ba and O*x* ions in BaTiO3.

SrTiO3 are larger than those of Ba and O*x* ions in BaTiO3.

SrTiO3 are larger than those of Ba and O*x* ions in BaTiO3.

0.0 0.1

smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Ti-Oz Coulomb repulsion along the [001] axis in SrTiO3 is smaller than that in BaTiO3 and that therefore the Ti ion of SrTiO3 can be displaced more easily along the [001] axis than that of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Ti and O*z* ions in SrTiO3 are larger than that in BaTiO3. Figure 14(b) shows the difference between the *A*-O*x* distance (*RA*-O*<sup>x</sup>*) and the sum of *rA* and *r*O*<sup>x</sup>* (*rA* + *r*O*<sup>x</sup>*) on the (100) plane as between the Ti-O*z* distance (*R*Ti-O*<sup>z</sup>*) and the sum of the *r*Ti and *r*O*<sup>z</sup>* (*r*Ti + *r*O*<sup>z</sup>*) along the [001] axis as a function of *c* – *c*cub. Note that *R*Ti-O*<sup>z</sup>* is smaller than *r*Ti + *r*O*<sup>z</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *R*Ti-O*<sup>z</sup>* and *r*Ti + *r*O*<sup>z</sup>* in SrTiO3 is smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Ti-Oz Coulomb repulsion along the [001] axis in SrTiO3 is smaller than that in BaTiO3 and that therefore the Ti ion of SrTiO3 can be displaced more easily along the [001] axis Let us discuss the reasons of the quantitative differences in *e*33 between SrTiO3 and BaTiO3. Figure 14(a) shows the difference between the Ti-O*z* distance (*R*Ti-O*<sup>z</sup>*) and the sum of the *r*Ti and *r*O*<sup>z</sup>* (*r*Ti + *r*O*<sup>z</sup>*) along the [001] axis as a function of *c* – *c*cub. Note that *R*Ti-O*<sup>z</sup>* is smaller than *r*Ti + *r*O*<sup>z</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *R*Ti-O*<sup>z</sup>* and *r*Ti + *r*O*<sup>z</sup>* in SrTiO3 is Let us discuss the reasons of the quantitative differences in *e*<sup>33</sup> between SrTiO3 and BaTiO3. Figure 14(a) shows the difference between the Ti-O*z* distance (*R*Ti-O*<sup>z</sup>*) and the sum of the *r*Ti

a function of *c* – *c*cub, where the values of the ionic radii are defined as Shannon's ones [23]. Note that *RA*-O*<sup>x</sup>* is smaller than *rA* + *r*O*<sup>x</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *RA*-O*<sup>x</sup>* and *r*A + *r*O*<sup>x</sup>* in SrTiO3 is much smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Sr-O*x* Coulomb repulsion on the (100) plane in SrTiO3 is much smaller than the Ba-O*x* Coulomb repulsion in BaTiO3 and that therefore Sr and O*x* ions of SrTiO3 can be displaced more easily along the [001] axis than Ba and O*x* ions of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Sr and O*x* ions in

smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Ti-Oz Coulomb repulsion along the [001] axis in SrTiO3 is smaller than that in BaTiO3 and that therefore the Ti ion of SrTiO3 can be displaced more easily along the [001] axis than that of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Ti and O*z* ions in SrTiO3 are larger than that in BaTiO3. Figure 14(b) shows the difference between the *A*-O*x* distance (*RA*-O*<sup>x</sup>*) and the sum of *rA* and *r*O*<sup>x</sup>* (*rA* + *r*O*<sup>x</sup>*) on the (100) plane as a function of *c* – *c*cub, where the values of the ionic radii are defined as Shannon's ones [23]. Note that *RA*-O*<sup>x</sup>* is smaller than *rA* + *r*O*<sup>x</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *RA*-O*<sup>x</sup>* and *r*A + *r*O*<sup>x</sup>* in SrTiO3 is much smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Sr-O*x* Coulomb repulsion on the (100) plane in SrTiO3 is much smaller than the Ba-O*x* Coulomb repulsion in BaTiO3 and that therefore Sr and O*x* ions of SrTiO3 can be displaced more easily along the [001] axis than Ba and O*x* ions of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Sr and O*x* ions in

than that of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Ti and O*z* ions in SrTiO3 are larger than that in BaTiO3. Figure 14(b) shows the difference between the *A*-O*x* distance (*RA*-O*<sup>x</sup>*) and the sum of *rA* and *r*O*<sup>x</sup>* (*rA* + *r*O*<sup>x</sup>*) on the (100) plane as a function of *c* – *c*cub, where the values of the ionic radii are defined as Shannon's ones [23]. Note that *RA*-O*<sup>x</sup>* is smaller than *rA* + *r*O*<sup>x</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *RA*-O*<sup>x</sup>* and *r*A + *r*O*<sup>x</sup>* in SrTiO3 is much smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Sr-O*x* Coulomb repulsion on the (100) plane in SrTiO3 is much smaller than the Ba-O*x* Coulomb repulsion in BaTiO3 and that therefore Sr and O*x* ions of SrTiO3 can be displaced more easily along the [001] axis than Ba and O*x* ions of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Sr and O*x* ions in

Let us discuss the reasons of the quantitative differences in *e*33 between SrTiO3 and BaTiO3. Figure 14(a) shows the difference between the Ti-O*z* distance (*R*Ti-O*<sup>z</sup>*) and the sum of the *r*Ti and *r*O*<sup>z</sup>* (*r*Ti + *r*O*<sup>z</sup>*) along the [001] axis as a function of *c* – *c*cub. Note that *R*Ti-O*<sup>z</sup>*

and *r*O*<sup>z</sup>* (*r*Ti + *r*O*<sup>z</sup>*) along the [001] axis as a function of *c* – *c*cub. Note that *R*Ti-O*<sup>z</sup>* is smaller than *r*Ti + *r*O*<sup>z</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *R*Ti-O*<sup>z</sup>* and *r*Ti + *r*O*<sup>z</sup>* in SrTiO3 is smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Ti-Oz Coulomb repulsion along the [001] axis in SrTiO3 is smaller than that in BaTiO3 and that therefore the Ti ion of SrTiO3 can be displaced more easily along the [001] axis than that of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Ti and O*z* ions in SrTiO3 are larger than that in BaTiO3. Figure 14(b) shows the difference between the *A*-O*x* distance (*RA*-O*<sup>x</sup>*) and the sum of *rA* and *r*O*<sup>x</sup>* (*rA* + *r*O*<sup>x</sup>*) on the (100) plane as a function of *c* – *c*cub, where the values of the ionic radii are defined as Shannon's ones [23]. Note that *RA*-O*<sup>x</sup>* is smaller than *rA* + *r*O*<sup>x</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *RA*-O*<sup>x</sup>* and *r*A + *r*O*<sup>x</sup>* in SrTiO3 is much smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Sr-O*x* Coulomb re‐ pulsion on the (100) plane in SrTiO3 is much smaller than the Ba-O*x* Coulomb repulsion in BaTiO3 and that therefore Sr and O*x* ions of SrTiO3 can be displaced more easily along the [001] axis than Ba and O*<sup>x</sup>* ions of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Sr and O*x* ions in SrTiO3 are larger than those of Ba and O*x* ions in BaTiO3.

Figure 11.Evaluated Born effective charges *Z*33\*

Figure 11.Evaluated Born effective charges *Z*33\*




**Figure 11.** Evaluated Born effective charges *Z*33\*







∂u3/∂η1

∂u3/∂η1

∂u3/∂η1

∂u3/∂η1

∂u3/∂η1

∂u3/∂η1

∂u3/∂η3

∂u3/∂η3

∂u3/∂η3

∂u3/∂η3

∂u3/∂η3

∂u3/∂η3

Z\*33

Z\*33

Z33

Z\*33

Z33

\*

Z33

\*

\*

14 Advances in Ferroelectrics

Figure 12.Evaluated values of ∂*u*3(*k*)/∂*η*3 as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3 [12].

Figure 12.Evaluated values of ∂*u*3(*k*)/∂*η*3 as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3 [12].

Oz

Figure 12.Evaluated values of ∂*u*3(*k*)/∂*η*3 as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3 [12].

Ti

Ox

Ti

Ti

Oz

Ox

Ox

Oz

Ox Oz

Ox Oz

Ox Oz

Ti

Ti

Ti

oxygen atoms along the [100] axis and the [001] axis, respectively [12]. Figure 11.Evaluated Born effective charges *Z*33\*

0.00 0.20 0.40 0.60 0.80 1.00

0.00 0.20 0.40 0.60 0.80 1.00

c - c\_cub (A)

c - c\_cub (A)

0.00 0.20 0.40 0.60 0.80 1.00

0.00 0.20 0.40 0.60 0.80 1.00

c - c\_cub (A)

0.00 0.20 0.40 0.60 0.80 1.00

c - c\_cub (A)

c - c\_cub (A)

**Figure 13.** Evaluated values of ∂*u*3(*k*)/∂η1 as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3 [12].

0.00 0.20 0.40 0.60 0.80 1.00

c - c\_cub (A)

STO

STO

STO

0.00 0.20 0.40 0.60 0.80 1.00

Ti

Ti

Sr

Ti

Sr

Oz Ox

Oz Ox

Oz Ox

Sr

(a) STO (b)

(a) STO (b)

(a) STO (b)

c - c\_cub (A)

0.00 0.20 0.40 0.60 0.80 1.00

c - c\_cub (A)

0.00 0.20 0.40 0.60 0.80 1.00

c - c\_cub (A)

STO

STO

STO

(a) (b) BTO

(a) (b) BTO

(a) (b) BTO

(a) (b) BTO

**Figure 12.** Evaluated values of ∂*u*3(*k*)/∂η3 as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3 [12].

(a) (b) BTO

(a) (b) BTO

Sr Ti

Sr Ti

Sr Ti

Sr Ti

Sr Ti

Sr Ti

Figure 13.Evaluated values of ∂*u*3(*k*)/∂*η*1 as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3 [12].

Figure 13.Evaluated values of ∂*u*3(*k*)/∂*η*1 as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3 [12].

Figure 13.Evaluated values of ∂*u*3(*k*)/∂*η*1 as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3 [12].

SrTiO3 are larger than those of Ba and O*x* ions in BaTiO3.

SrTiO3 are larger than those of Ba and O*x* ions in BaTiO3.

SrTiO3 are larger than those of Ba and O*x* ions in BaTiO3.

Let us discuss the reasons of the quantitative differences in *e*33 between SrTiO3 and BaTiO3. Figure 14(a) shows the difference between the Ti-O*z* distance (*R*Ti-O*<sup>z</sup>*) and the sum of the *r*Ti and *r*O*<sup>z</sup>* (*r*Ti + *r*O*<sup>z</sup>*) along the [001] axis as a function of *c* – *c*cub. Note that *R*Ti-O*<sup>z</sup>* is smaller than *r*Ti + *r*O*<sup>z</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *R*Ti-O*<sup>z</sup>* and *r*Ti + *r*O*<sup>z</sup>* in SrTiO3 is smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Ti-Oz Coulomb repulsion along the [001] axis in SrTiO3 is smaller than that in BaTiO3 and that therefore the Ti ion of SrTiO3 can be displaced more easily along the [001] axis than that of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Ti and O*z* ions in SrTiO3 are larger than that in BaTiO3. Figure 14(b) shows the difference between the *A*-O*x* distance (*RA*-O*<sup>x</sup>*) and the sum of *rA* and *r*O*<sup>x</sup>* (*rA* + *r*O*<sup>x</sup>*) on the (100) plane as a function of *c* – *c*cub, where the values of the ionic radii are defined as Shannon's ones [23]. Note that *RA*-O*<sup>x</sup>* is smaller than *rA* + *r*O*<sup>x</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *RA*-O*<sup>x</sup>* and *r*A + *r*O*<sup>x</sup>* in SrTiO3 is much smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Sr-O*x* Coulomb repulsion on the (100) plane in SrTiO3 is much smaller than the Ba-O*x* Coulomb repulsion in BaTiO3 and that therefore Sr and O*x* ions of SrTiO3 can be displaced more easily along the [001] axis than Ba and O*x* ions of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Sr and O*x* ions in

Let us discuss the reasons of the quantitative differences in *e*33 between SrTiO3 and BaTiO3. Figure 14(a) shows the difference between the Ti-O*z* distance (*R*Ti-O*<sup>z</sup>*) and the sum of the *r*Ti and *r*O*<sup>z</sup>* (*r*Ti + *r*O*<sup>z</sup>*) along the [001] axis as a function of *c* – *c*cub. Note that *R*Ti-O*<sup>z</sup>* is smaller than *r*Ti + *r*O*<sup>z</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *R*Ti-O*<sup>z</sup>* and *r*Ti + *r*O*<sup>z</sup>* in SrTiO3 is smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Ti-Oz Coulomb repulsion along the [001] axis in SrTiO3 is smaller than that in BaTiO3 and that therefore the Ti ion of SrTiO3 can be displaced more easily along the [001] axis than that of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Ti and O*z* ions in SrTiO3 are larger than that in BaTiO3. Figure 14(b) shows the difference between the *A*-O*x* distance (*RA*-O*<sup>x</sup>*) and the sum of *rA* and *r*O*<sup>x</sup>* (*rA* + *r*O*<sup>x</sup>*) on the (100) plane as a function of *c* – *c*cub, where the values of the ionic radii are defined as Shannon's ones [23]. Note that *RA*-O*<sup>x</sup>* is smaller than *rA* + *r*O*<sup>x</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *RA*-O*<sup>x</sup>* and *r*A + *r*O*<sup>x</sup>* in SrTiO3 is much smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Sr-O*x* Coulomb repulsion on the (100) plane in SrTiO3 is much smaller than the Ba-O*x* Coulomb repulsion in BaTiO3 and that therefore Sr and O*x* ions of SrTiO3 can be displaced more easily along the [001] axis than Ba and O*x* ions of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Sr and O*x* ions in

Let us discuss the reasons of the quantitative differences in *e*33 between SrTiO3 and BaTiO3. Figure 14(a) shows the difference between the Ti-O*z* distance (*R*Ti-O*<sup>z</sup>*) and the sum of the *r*Ti and *r*O*<sup>z</sup>* (*r*Ti + *r*O*<sup>z</sup>*) along the [001] axis as a function of *c* – *c*cub. Note that *R*Ti-O*<sup>z</sup>* is smaller than *r*Ti + *r*O*<sup>z</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *R*Ti-O*<sup>z</sup>* and *r*Ti + *r*O*<sup>z</sup>* in SrTiO3 is smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Ti-Oz Coulomb repulsion along the [001] axis in SrTiO3 is smaller than that in BaTiO3 and that therefore the Ti ion of SrTiO3 can be displaced more easily along the [001] axis than that of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Ti and O*z* ions in SrTiO3 are larger than that in BaTiO3. Figure 14(b) shows the difference between the *A*-O*x* distance (*RA*-O*<sup>x</sup>*) and the sum of *rA* and *r*O*<sup>x</sup>* (*rA* + *r*O*<sup>x</sup>*) on the (100) plane as a function of *c* – *c*cub, where the values of the ionic radii are defined as Shannon's ones [23]. Note that *RA*-O*<sup>x</sup>* is smaller than *rA* + *r*O*<sup>x</sup>* in both SrTiO3 and BaTiO3. However, the difference in absolute value between *RA*-O*<sup>x</sup>* and *r*A + *r*O*<sup>x</sup>* in SrTiO3 is much smaller than the difference in BaTiO3 for 0 ≲ *c* – *c*cub ≲ 0.20. This result suggests that the Sr-O*x* Coulomb repulsion on the (100) plane in SrTiO3 is much smaller than the Ba-O*x* Coulomb repulsion in BaTiO3 and that therefore Sr and O*x* ions of SrTiO3 can be displaced more easily along the [001] axis than Ba and O*x* ions of BaTiO3. This would be a reason why the absolute values of ∂*u*3(*k*)/∂*η*3 of Sr and O*x* ions in

Let us discuss the reasons of the quantitative differences in *e*<sup>33</sup> between SrTiO3 and BaTiO3. Figure 14(a) shows the difference between the Ti-O*z* distance (*R*Ti-O*<sup>z</sup>*) and the sum of the *r*Ti

axis and the [001] axis, respectively [12].

axis and the [001] axis, respectively [12].

axis and the [001] axis, respectively [12].

(*k*) as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3. O*x* and O*z* denote oxygen atoms along the [100]

(*k*) as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3. O*x* and O*z* denote

(*k*) as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3. O*x* and O*z* denote oxygen atoms along the [100]

(*k*) as a function of *c* – *c*cub: (a) SrTiO3 and (b) BaTiO3. O*x* and O*z* denote oxygen atoms along the [100]

0.00 0.20 0.40 0.60 0.80 1.00

c - c\_cub (A)

c - c\_cub (A)

0.00 0.20 0.40 0.60 0.80 1.00

0.00 0.20 0.40 0.60 0.80 1.00

0.00 0.20 0.40 0.60 0.80 1.00

c - c\_cub (A)

c - c\_cub (A)

0.00 0.20 0.40 0.60 0.80 1.00

0.00 0.20 0.40 0.60 0.80 1.00

0.00 0.20 0.40 0.60 0.80 1.00

c - c\_cub (A)

c - c\_cub (A)

0.00 0.20 0.40 0.60 0.80 1.00

0.00 0.20 0.40 0.60 0.80 1.00

c - c\_cub (A)

c - c\_cub (A)

BTO

BTO

BTO

Ba Ox Oz

Ti

Ba Ox Oz

Ti

Ba Ox Oz

Ba

Ox Oz

Ba

Ox Oz

Ba

Ox Oz

Ba

Oz

Ba

Oz

Ox

Oz

Ba

Ox

Ox

becomes smaller.

Ti

c - c\_cub (A)










∂u3/∂η1

c - ccub (Å) c - ccub (Å)

c - ccub (Å) c - ccub (Å)

c - ccub (Å) c - ccub (Å)

∂u3/∂η1

∂u3/∂η1

∂u3/∂η1

∂u3/∂η1

∂u3/∂η1

∂u3/∂η3

∂u3/∂η3

∂u3/∂η3

∂u3/∂η3

∂u3/∂η3

∂u3/∂η3

c - ccub (Å) c - ccub (Å)

c - ccub (Å) c - ccub (Å)

c - ccub (Å) c - ccub (Å)

Z\*33

\*

Z\*33

Z33

Z\*33

Z33

\*

Z33

\*

c - ccub (Å) c - ccub (Å)

c - ccub (Å) c - ccub (Å)

c - ccub (Å) c - ccub (Å)

Figure 14.Evaluated values as a function of *c* – *c*cub in optimized tetragonal SrTiO3 and BaTiO3: (a) difference between the Ti-O*<sup>z</sup>* distance (*R*Ti-O*<sup>z</sup>*) and *r*Ti + *r*O*<sup>z</sup>*. (b) difference between the *A*-O*<sup>x</sup>* distance (*RA*-Ox) and *rA* + *r*O*<sup>x</sup>*. *RA*-O*<sup>x</sup>* and *R*Ti-O*<sup>z</sup>* in *A*TiO3 are also illustrated. Note that all the ionic radii are much larger and that *A* and Ti ions are displaced along the [001] axis in actual *A*TiO3 [12]. Finally, in this subsection, we discuss the relationship between ∂*u*3(*k*)/∂*η*<sup>3</sup> and *c* – *c*cub in detail. Figure 15(a) shows the differences in **Figure 14.** Evaluated values as a function of *c* – *c*cub in optimized tetragonal SrTiO3 and BaTiO3: (a) difference between the Ti-O*z* distance (*R*Ti-O*z*) and *r*Ti + *r*O*z*. (b) difference between the *A*-O*<sup>x</sup>* distance (*RA*-Ox) and *rA* + *r*O*x*. *RA*-O*<sup>x</sup>* and *R*Ti-O*<sup>z</sup>* in *A*TiO3 are also illustrated. Note that all the ionic radii are much larger and that *A* and Ti ions are displaced along the [001] axis in actual *A*TiO3 [12].

the total energy (*ΔE*total ) as a function of *u*3(Ti). In this figure, the properties of SrTiO3 with *η*<sup>3</sup> = 0.011, SrTiO3 with *η*<sup>3</sup> = 0.053 and fully optimized BaTiO3 as a reference, are shown. Calculations of *E*total were performed with the fixed crystal structures of previously optimized structures except Ti ions. Clearly, the magnitude of *u*3(Ti) at the minimum points of the *ΔE*total and the depth Finally, in this subsection, we discuss the relationship between ∂*u*3(*k*)/∂*η*3 and *c* – *c*cub in de‐ tail. Figure 15(a) shows the differences in the total energy (*ΔE*total ) as a function of *u*3(Ti). In

of the potential are closely related to the spontaneous polarization *P*<sup>3</sup> and the Curie temperature (*T*C), respectively. On the other hand, *e*<sup>33</sup> seems to be closely related to the deviation at the minimum points of the *ΔE*total. Figure 15(b) shows illustrations of *ΔE*total curves with deviations at the minimum points of the *ΔE*total values, corresponding to the *ΔE*total curves of SrTiO3 in Fig. 15(a). Clearly, as *η*<sup>3</sup> becomes smaller, the deviated value at the minimum point of the *ΔE*total values becomes smaller, *i.e.*, the Ti ion can be displaced more favourably. On the other hand, as shown in Fig. 12(a), the absolute value of ∂*u*3(Ti)/∂*η*<sup>3</sup> becomes larger as *η*<sup>3</sup>

Therefore, the Ti ion can be displaced more favourably as the deviated value at the minimum point of the *ΔE*total values becomes

Figure 15.(a) *ΔE*total as a function of *u*3(Ti) in tetragonal SrTiO3 and BaTiO3. (b) Illustration of the *ΔE*total curves in tetragonal SrTiO3 with *η<sup>3</sup>* = 0.011

������

⊿�� (Ti ) (Å) ⊿�� (Ti ) (���� ��i�)

⊿������ (���� ��i�)

�

⊿���� (��)

⊿������ ���� ⊿�� (Ti) ⊿������ ���� ⊿�� (Ti)

���

�

��Ti (�)

The previous discussion in Sec. 3.2.1 suggests that the piezoelectric properties of *e*<sup>33</sup> are closely related to the *B*-*X* Coulomb repulsions in tegtragonal *ABX*3. In the viewpoint of the change of the *B*-*X* Coulomb repulsions, we recently proposed new

It has been known that BaNiO3 shows the 2H hexagonal structure as the most stable structure in room temperature. Moreover, the ionic radius of Ni4+ (d6) with the low-spin state in 2H BaNiO3 is 0.48 Å, which is much smaller than that of Ti4+ (d0), 0.605 Å, in

smaller. The relationship between *e*<sup>33</sup> and ∂*u*3(Ti)/∂*η*<sup>3</sup> is discussed in Sec. 3.2.3.

and SrTiO3 with *η*<sup>3</sup> = 0.053 with deviations at the minimum point of *ΔE*total [14].

��� ��� ��� ���

(�) (�) ⊿������ ���� ⊿�Ti

��Ti�� (η� � �����)

��Ti��

��Ti�� (η� � �����)

��Ti (�)

piezoelectric materials [16, 17], *i.e.*, BaTi1*-x*Ni*x*O3 and Ba(Ti1*-3z*Nb*3z*)(O1-zNz)3.

**3.2.2. Proposal of new piezoelectric materials**

������ ������ ������ ����� ����� ����� �����

⊿���� (��)

⊿������ (��)

this figure, the properties of SrTiO3 with *η*3 = 0.011, SrTiO3 with *η*<sup>3</sup> = 0.053 and fully opti‐ mized BaTiO3 as a reference, are shown. Calculations of *E*total were performed with the fixed crystal structures of previously optimized structures except Ti ions. Clearly, the magnitude of *u*3(Ti) at the minimum points of the *ΔE*total and the depth of the potential are closely relat‐ ed to the spontaneous polarization *P*3 and the Curie temperature (*T*C), respectively. On the other hand, *e*33 seems to be closely related to the deviation at the minimum points of the *ΔE*total. Figure 15(b) shows illustrations of *ΔE*total curves with deviations at the minimum points of the *ΔE*total values, corresponding to the *ΔE*total curves of SrTiO3 in Fig. 15(a). Clearly, as *η*3 becomes smaller, the deviated value at the minimum point of the *ΔE*total values becomes smaller, *i.e.*, the Ti ion can be displaced more favourably. On the other hand, as shown in Fig. 12(a), the absolute value of ∂*u*3(Ti)/∂*η*3 becomes larger as *η*3 becomes smaller. Figure 14.Evaluated values as a function of *c* – *c*cub in optimized tetragonal SrTiO3 and BaTiO3: (a) difference between the Ti-O*z* distance (*R*Ti-O*<sup>z</sup>*) and *r*Ti + *r*O*<sup>z</sup>*. (b) difference between the *A*-O*x* distance (*RA*-Ox) and *rA* + *r*O*<sup>x</sup>*. *RA*-O*<sup>x</sup>* and *R*Ti-O*<sup>z</sup>* in *A*TiO3 are also illustrated. Note that all the ionic radii are much larger and that *A* and Ti ions are displaced along the [001] axis in actual *A*TiO3 [12]. Finally, in this subsection, we discuss the relationship between ∂*u*3(*k*)/∂*η*3 and *c* – *c*cub in detail. Figure 15(a) shows the differences in the total energy (*ΔE*total ) as a function of *u*3(Ti). In this figure, the properties of SrTiO3 with *η*3 = 0.011, SrTiO3 with *η*3 = 0.053 and fully optimized BaTiO3 as a reference, are shown. Calculations of *E*total were performed with the fixed crystal structures of previously optimized structures except Ti ions. Clearly, the magnitude of *u*3(Ti) at the minimum points of the *ΔE*total and the depth of the potential are closely related to the spontaneous polarization *P*3 and the Curie temperature (*T*C), respectively. On the other hand, *e*33 seems to be closely related to the deviation at the minimum points of the *ΔE*total. Figure 15(b) shows illustrations of *ΔE*total curves with deviations at the minimum points of the *ΔE*total values, corresponding to the *ΔE*total curves of SrTiO3 in Fig. 15(a). Clearly, as *η*3 becomes smaller, the deviated value at the minimum point of the *ΔE*total values becomes smaller, *i.e.*, the Ti ion can be

ergy difference *ΔE*total between 2H and tetragonal structures of BaTi1*-x*Ni*x*O3 as a function of *x*. The most stable structure changes at *x* ≈ 0.26. Figure 16(b) shows *c* – *c*cub as a function of *x*. The *c* – *c*cub value shows 0 around *x* = 0.26, which suggests the appearance of the MPB, *i.e.*,

Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity

http://dx.doi.org/10.5772/52187

17

**Figure 16.** a) ΔEtotal (total-energy difference between 2H and tetragonal structures), and (b) *c* – *c*cub of the tetragonal structure, as a function of *x* in BaTi1*-x*Ni*x*O3 [16]. For or 0.26 ≤ *x* ≦ 1, the tetragonal structure is not the most stable one.

**Figure 17.** (a) ΔEtotal (total-energy difference between cubic and tetragonal structures), and (b) *c* – *c*cub, as a function of

Another proposal is tetragonal Ba(Ti1*-*3*<sup>z</sup>*Nb3*z*)(O1-*z*N*z*)3, which consists of BaTiO3 and BaN‐ bO2N [17]. Due to the change of (Ti1*-*3*<sup>z</sup>*Nb3*z*)-(O1-*z*N*z*) Coulomb repulsions, the *e*33 piezoelectric values are expected to be larger than that in tetragonal BaTiO3. Recent experimental paper reported that the most stable structure of BaNbO2N is cubic in room temperature [34]. Con‐ trary to the experimental result, however, our calculations suggest that the tetragonal struc‐ ture will be more stable than the cubic one, as shown in Fig. 17(a). Figure 17(b) shows *c* – *c*cub

*x* in Ba(Ti1*-*3*z*Nb3*z*)(O1-*z*N*z*)3 [17].

the *e*33 piezoelectric value shows a maximum at *x* ≈ 0.26.

**(a) b**

Therefore, the Ti ion can be displaced more favourably as the deviated value at the mini‐ mum point of the *ΔE*total values becomes smaller. The relationship between *e*33 and ∂*u*3(Ti)/∂*η*<sup>3</sup> is discussed in Sec. 3.2.3. displaced more favourably. On the other hand, as shown in Fig. 12(a), the absolute value of ∂*u*3(Ti)/∂*η*3 becomes larger as *η*<sup>3</sup> becomes smaller. Therefore, the Ti ion can be displaced more favourably as the deviated value at the minimum point of the *ΔE*total values becomes

Figure 15.(a) *ΔE*total as a function of *u*3(Ti) in tetragonal SrTiO3 and BaTiO3. (b) Illustration of the *ΔE*total curves in tetragonal SrTiO3 with *η<sup>3</sup>* = 0.011 and SrTiO3 with *η*3 = 0.053 with deviations at the minimum point of *ΔE*total [14]. **Figure 15.** a) ΔEtotal as a function of *u*3(Ti) in tetragonal SrTiO3 and BaTiO3. (b) Illustration of the ΔEtotal curves in tetrago‐ nal SrTiO3 with η*3* = 0.011 and SrTiO3 with η3 = 0.053 with deviations at the minimum point of ΔEtotal [14].

#### **3.2.2. Proposal of new piezoelectric materials**  *3.2.2. Proposal of new piezoelectric materials*

smaller. The relationship between *e*33 and ∂*u*3(Ti)/∂*η*3 is discussed in Sec. 3.2.3.

The previous discussion in Sec. 3.2.1 suggests that the piezoelectric properties of *e*33 are closely related to the *B*-*X* Coulomb repulsions in tegtragonal *ABX*3. In the viewpoint of the change of the *B*-*X* Coulomb repulsions, we recently proposed new piezoelectric materials [16, 17], *i.e.*, BaTi1*-x*Ni*x*O3 and Ba(Ti1*-3z*Nb*3z*)(O1-zNz)3. It has been known that BaNiO3 shows the 2H hexagonal structure as the most stable structure in room temperature. Moreover, the The previous discussion in Sec. 3.2.1 suggests that the piezoelectric properties of *e*<sup>33</sup> are closely related to the *B*-*X* Coulomb repulsions in tegtragonal *ABX*3. In the viewpoint of the change of the *B*-*X* Coulomb repulsions, we recently proposed new piezoelectric materials [16, 17], *i.e.*, BaTi1*-x*Ni*x*O3 and Ba(Ti1*-3z*Nb*3z*)(O1-zNz)3.

ionic radius of Ni4+ (d6) with the low-spin state in 2H BaNiO3 is 0.48 Å, which is much smaller than that of Ti4+ (d0), 0.605 Å, in

It has been known that BaNiO3 shows the 2H hexagonal structure as the most stable struc‐ ture in room temperature. Moreover, the ionic radius of Ni4+ (d6 ) with the low-spin state in 2H BaNiO3 is 0.48 Å, which is much smaller than that of Ti4+ (d0 ), 0.605 Å, in BaTiO3. There‐ fore, due to the drastic change in the (Ti1-*x*Ni*x*)-O Coulomb repulsions in tetragonal BaTi1 *<sup>x</sup>*Ni*x*O3, the *e*33 piezoelectric values are expected to be larger than that in tetragonal BaTiO3, especially around the morphotropic phase boundary (MPB). Figure 16(a) shows the total-en‐ ergy difference *ΔE*total between 2H and tetragonal structures of BaTi1*-x*Ni*x*O3 as a function of *x*. The most stable structure changes at *x* ≈ 0.26. Figure 16(b) shows *c* – *c*cub as a function of *x*. The *c* – *c*cub value shows 0 around *x* = 0.26, which suggests the appearance of the MPB, *i.e.*, the *e*33 piezoelectric value shows a maximum at *x* ≈ 0.26.

this figure, the properties of SrTiO3 with *η*3 = 0.011, SrTiO3 with *η*<sup>3</sup> = 0.053 and fully opti‐ mized BaTiO3 as a reference, are shown. Calculations of *E*total were performed with the fixed crystal structures of previously optimized structures except Ti ions. Clearly, the magnitude of *u*3(Ti) at the minimum points of the *ΔE*total and the depth of the potential are closely relat‐ ed to the spontaneous polarization *P*3 and the Curie temperature (*T*C), respectively. On the other hand, *e*33 seems to be closely related to the deviation at the minimum points of the *ΔE*total. Figure 15(b) shows illustrations of *ΔE*total curves with deviations at the minimum points of the *ΔE*total values, corresponding to the *ΔE*total curves of SrTiO3 in Fig. 15(a). Clearly, as *η*3 becomes smaller, the deviated value at the minimum point of the *ΔE*total values becomes smaller, *i.e.*, the Ti ion can be displaced more favourably. On the other hand, as shown in

Figure 14.Evaluated values as a function of *c* – *c*cub in optimized tetragonal SrTiO3 and BaTiO3: (a) difference between the Ti-O*z* distance (*R*Ti-O*<sup>z</sup>*) and *r*Ti + *r*O*<sup>z</sup>*. (b) difference between the *A*-O*x* distance (*RA*-Ox) and *rA* + *r*O*<sup>x</sup>*. *RA*-O*<sup>x</sup>* and *R*Ti-O*<sup>z</sup>* in *A*TiO3 are also illustrated. Note that all the ionic radii are

Finally, in this subsection, we discuss the relationship between ∂*u*3(*k*)/∂*η*3 and *c* – *c*cub in detail. Figure 15(a) shows the differences in the total energy (*ΔE*total ) as a function of *u*3(Ti). In this figure, the properties of SrTiO3 with *η*3 = 0.011, SrTiO3 with *η*3 = 0.053 and fully optimized BaTiO3 as a reference, are shown. Calculations of *E*total were performed with the fixed crystal structures of previously optimized structures except Ti ions. Clearly, the magnitude of *u*3(Ti) at the minimum points of the *ΔE*total and the depth of the potential are closely related to the spontaneous polarization *P*3 and the Curie temperature (*T*C), respectively. On the other hand, *e*33 seems to be closely related to the deviation at the minimum points of the *ΔE*total. Figure 15(b) shows illustrations of *ΔE*total curves with deviations at the minimum points of the *ΔE*total values, corresponding to the *ΔE*total curves of SrTiO3 in Fig. 15(a). Clearly, as *η*3 becomes smaller, the deviated value at the minimum point of the *ΔE*total values becomes smaller, *i.e.*, the Ti ion can be displaced more favourably. On the other hand, as shown in Fig. 12(a), the absolute value of ∂*u*3(Ti)/∂*η*3 becomes larger as *η*<sup>3</sup>

Fig. 12(a), the absolute value of ∂*u*3(Ti)/∂*η*3 becomes larger as *η*3 becomes smaller.

BaTiO3

SrTiO3 (η3 = 0.053)

much larger and that *A* and Ti ions are displaced along the [001] axis in actual *A*TiO3 [12].

smaller. The relationship between *e*33 and ∂*u*3(Ti)/∂*η*3 is discussed in Sec. 3.2.3.

and SrTiO3 with *η*3 = 0.053 with deviations at the minimum point of *ΔE*total [14].

*3.2.2. Proposal of new piezoelectric materials*

0.0 0.1 0.2 0.3

(a) (b) ⊿Etotal v.s.⊿uTi

SrTiO3 (η3 = 0.011)

u\_Ti (A)

piezoelectric materials [16, 17], *i.e.*, BaTi1*-x*Ni*x*O3 and Ba(Ti1*-3z*Nb*3z*)(O1-zNz)3.

[16, 17], *i.e.*, BaTi1*-x*Ni*x*O3 and Ba(Ti1*-3z*Nb*3z*)(O1-zNz)3.

ture in room temperature. Moreover, the ionic radius of Ni4+ (d6

2H BaNiO3 is 0.48 Å, which is much smaller than that of Ti4+ (d0

**3.2.2. Proposal of new piezoelectric materials** 


⊿Etot (eV)

⊿Etotal (eV)

is discussed in Sec. 3.2.3.

16 Advances in Ferroelectrics

becomes smaller.

Therefore, the Ti ion can be displaced more favourably as the deviated value at the mini‐ mum point of the *ΔE*total values becomes smaller. The relationship between *e*33 and ∂*u*3(Ti)/∂*η*<sup>3</sup>

Therefore, the Ti ion can be displaced more favourably as the deviated value at the minimum point of the *ΔE*total values becomes

Figure 15.(a) *ΔE*total as a function of *u*3(Ti) in tetragonal SrTiO3 and BaTiO3. (b) Illustration of the *ΔE*total curves in tetragonal SrTiO3 with *η<sup>3</sup>* = 0.011

**Figure 15.** a) ΔEtotal as a function of *u*3(Ti) in tetragonal SrTiO3 and BaTiO3. (b) Illustration of the ΔEtotal curves in tetrago‐

nal SrTiO3 with η*3* = 0.011 and SrTiO3 with η3 = 0.053 with deviations at the minimum point of ΔEtotal [14].


⊿u3 (Ti ) (Å) ⊿u3 (Ti ) (arb. unit)

⊿Etotal (arb. unit)

0

0

u\_Ti (A)

) with the low-spin state in

), 0.605 Å, in BaTiO3. There‐

⊿Etot (eV)

⊿Etotal v.s.⊿u3 (Ti) ⊿Etotal v.s.⊿u3 (Ti)

The previous discussion in Sec. 3.2.1 suggests that the piezoelectric properties of *e*33 are closely related to the *B*-*X* Coulomb repulsions in tegtragonal *ABX*3. In the viewpoint of the change of the *B*-*X* Coulomb repulsions, we recently proposed new

The previous discussion in Sec. 3.2.1 suggests that the piezoelectric properties of *e*<sup>33</sup> are closely related to the *B*-*X* Coulomb repulsions in tegtragonal *ABX*3. In the viewpoint of the change of the *B*-*X* Coulomb repulsions, we recently proposed new piezoelectric materials

It has been known that BaNiO3 shows the 2H hexagonal structure as the most stable structure in room temperature. Moreover, the ionic radius of Ni4+ (d6) with the low-spin state in 2H BaNiO3 is 0.48 Å, which is much smaller than that of Ti4+ (d0), 0.605 Å, in

It has been known that BaNiO3 shows the 2H hexagonal structure as the most stable struc‐

fore, due to the drastic change in the (Ti1-*x*Ni*x*)-O Coulomb repulsions in tetragonal BaTi1 *<sup>x</sup>*Ni*x*O3, the *e*33 piezoelectric values are expected to be larger than that in tetragonal BaTiO3, especially around the morphotropic phase boundary (MPB). Figure 16(a) shows the total-en‐

**Figure 16.** a) ΔEtotal (total-energy difference between 2H and tetragonal structures), and (b) *c* – *c*cub of the tetragonal structure, as a function of *x* in BaTi1*-x*Ni*x*O3 [16]. For or 0.26 ≤ *x* ≦ 1, the tetragonal structure is not the most stable one.

**Figure 17.** (a) ΔEtotal (total-energy difference between cubic and tetragonal structures), and (b) *c* – *c*cub, as a function of *x* in Ba(Ti1*-*3*z*Nb3*z*)(O1-*z*N*z*)3 [17].

Another proposal is tetragonal Ba(Ti1*-*3*<sup>z</sup>*Nb3*z*)(O1-*z*N*z*)3, which consists of BaTiO3 and BaN‐ bO2N [17]. Due to the change of (Ti1*-*3*<sup>z</sup>*Nb3*z*)-(O1-*z*N*z*) Coulomb repulsions, the *e*33 piezoelectric values are expected to be larger than that in tetragonal BaTiO3. Recent experimental paper reported that the most stable structure of BaNbO2N is cubic in room temperature [34]. Con‐ trary to the experimental result, however, our calculations suggest that the tetragonal struc‐ ture will be more stable than the cubic one, as shown in Fig. 17(a). Figure 17(b) shows *c* – *c*cub as a function of *x*. The *c* – *c*cub value shows almost 0 at *x* ≈ 0.12. Although the MPB does not appear in tetragonal Ba(Ti1*-*3*<sup>z</sup>*Nb3*z*)(O1-*z*N*z*)3, the *e*33 piezoelectric values are expected to show a maximum at *x* ≈ 0.12.

### *3.2.3. Piezoelectric properties of in tetragonal ABX3*

In the following, we discuss the role of the *B*-*X* Coulomb repulsions in piezoelectric *ABX*3.

Figures 18(a) and 18(b) show the piezoelectric properties of *e*33 as a function of the value *c* – *c*cub in tetragonal *ABX*3, where *c*cub denotes the *c* lattice parameter in cubic *ABX*3; *c* – *c*cub is a closely related parameter to *η*3. For *ABX*3, SrTiO3, BaTiO3 and PbTiO3 with the *c* lattice pa‐ rameter and all the inner coordination optimized for fixed *a*, and BaTi1*-x*Ni*x*O3 (0 ≦ *x* ≦ 0.05), Ba(Ti1*-*3*<sup>z</sup>*Nb3*z*)(O1-*z*N*z*)3 (0 ≦ *z* ≦ 0.125), Ba1*-y*Sr*y*TiO3 (0 ≦ *y* ≦ 0.5), BaTi1*-x*Zr*x*O3 (0 ≦ *x* ≦ 0.06), and Bi*M'*O3 (*M'* = Al, Sc) with fully optimized, were prepared [15]. Note that *e*<sup>33</sup> becomes larger as *c* – *c*cub becomes smaller and that the trend of *e*33 is almost independent of the kind of *A* ions. Moreover, note also that *e*33 of BaTi1*-x*Ni*x*O3 and that of Ba(Ti1*-*3*<sup>z</sup>*Nb3*z*)(O1-*z*N*z*)3 show much larger values than the other *ABX*3.

**Figure 18.** *e*33 as a function of *c* – *c*cub for different scales [15].

Figure 19. *∂u*3(*k*)/*∂η*3 as a function of *c* – *c*cub : (a) BaTi1*-x*Ni*x*O3 and (b) BaTiO3 [15].

Ba

x = 0

x = 0.04

Ox

Oz

0.00 0.20 0.40 0.60 0.80 1.00

(a) (b)


∂u3

(k)/∂η3

Figure 18.*e*33 as a function of *c* – *c*cub for different scales [15].

in BaTiO3.

Let us discuss the relationship between *∂u*3(*k*)/*∂η3* and *c* – *c*cub in BaTi1*-x*Ni*x*O3 and BaTiO3 in the following. Figures 19(a) and 19(b) show the *∂u*3(*k*)/*∂η3* values. In these figures, O*x* and O*z* denote oxygen atoms along the [100] and [001] axes, respectively. Clearly, the absolute values of *∂u*3(*k*)/*∂η3* in BaTi1*-x*Ni*x*O3 are much larger than those in BaTiO3. Moreover, in comparison with Fig.18, properties of *e*33 are closely related to those of *∂u*3(*k*)/*∂η3*. Figure 20(a) shows the difference between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* along the [001] axis, and Fig. 20(b) shows the difference between *RA*-O*<sup>x</sup>* and *rA* + *r*O*<sup>x</sup>* on the (100) plane for several *AB*O3, as a function of *c* – *c*cub. Clearly, the difference between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* is closely related to *e*33 shown in Fig. 18, rather than the difference between *RA*-O*<sup>x</sup>* and *rA* + *r*O*<sup>x</sup>*. Moreover, note that the difference in absolute value between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* in BaTi1*-x*Ni*x*O3 is much smaller than that in BaTiO3. This result suggests that the (Ti1*-x*Ni*x*)-O*z* Coulomb repulsion along the [001] axis in BaTi1*-x*Ni*x*O3 is much smaller than the Ti-O*<sup>z</sup>* Coulomb repulsion in BaTiO3 and that therefore Ti1*-x*Ni*x* ion of BaTi1*-x*Ni*x*O3 can be displaced more easily along the [001] axis than Ti ion of BaTiO3. This must be a reason why the absolute value of *∂u*3(*k*)/*∂η3* of Ti1*-x*Ni*x* and O*z* ions in BaTi1*-x*Ni*x*O3 is larger than those Let us discuss the relationship between *∂u*3(*k*)/*∂η3* and *c* – *c*cub in BaTi1*-x*Ni*x*O3 and BaTiO3 in the following. Figures 19(a) and 19(b) show the *∂u*3(*k*)/*∂η3* values. In these figures, O*x* and O*<sup>z</sup>* denote oxygen atoms along the [100] and [001] axes, respectively. Clearly, the absolute val‐ ues of *∂u*3(*k*)/*∂η3* in BaTi1*-x*Ni*x*O3 are much larger than those in BaTiO3. Moreover, in compari‐ son with Fig.18, properties of *e*<sup>33</sup> are closely related to those of *∂u*3(*k*)/*∂η3*. Figure 20(a) shows the difference between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* along the [001] axis, and Fig. 20(b) shows the differ‐ ence between *RA*-O*<sup>x</sup>* and *rA* + *r*O*<sup>x</sup>* on the (100) plane for several *AB*O3, as a function of *c* – *c*cub. Clearly, the difference between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* is closely related to *e*<sup>33</sup> shown in Fig. 18, rather than the difference between *RA*-O*<sup>x</sup>* and *rA* + *r*O*<sup>x</sup>*. Moreover, note that the difference in absolute value between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* in BaTi1*-x*Ni*x*O3 is much smaller than that in BaTiO3. This result suggests that the (Ti1*-x*Ni*x*)-O*z* Coulomb repulsion along the [001] axis in BaTi1 *<sup>x</sup>*Ni*x*O3 is much smaller than the Ti-O*z* Coulomb repulsion in BaTiO3 and that therefore Ti1 *<sup>x</sup>*Ni*x* ion of BaTi1*-x*Ni*x*O3 can be displaced more easily along the [001] axis than Ti ion of

> -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

∂u3

c - ccub (Å) c - ccub (Å)

(k)/∂η3

Ti1-xNix Ba

0.00 0.20 0.40 0.60 0.80 1.00

Oz

Ti

BaTiO3. This must be a reason why the absolute value of *∂u*3(*k*)/*∂η3* of Ti1*-x*Ni*x* and O*z* ions in

Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity


∂u3

c - ccub (Å) c - ccub (Å)

Figure 20.Evaluated values of in optimized tetragonal BaTi1*-x*Ni*x*O3, BaTiO3, and several *AB*O3, as a function of *c* – *c*cub: (a) *RB*-O*<sup>z</sup>* - (*rB* + *r*O*<sup>z</sup>*), and (b) *RA*-

**Figure 20.** Evaluated values of in optimized tetragonal BaTi1*-x*Ni*x*O3, BaTiO3, and several *AB*O3, as a function of *c* – *c*cub:

���� ���� ���3 ���� ����

�����3

��������� ��3

�����3

������ � (��� � ���) (�)

� ��) (Å)

�����3

�����3

���� � (��

�

Figure 21(a) shows Δ*E*total as a function of the displacement of the Ti1*-x*Ni*<sup>x</sup>* ions with fixed crystal structures of fully-optimized BaTi1 *<sup>x</sup>*Ni*x*O3. Calculations of *E*total were performed with the fixed crystal structures of previously optimized structures except Ti1*-x*Ni*<sup>x</sup>* ions. The deviated value at the minimum point of Δ*E*total, *i.e.*, *∂*(Δ*E*total)/*∂u*3(Ti1*-x*Ni*x*), becomes smaller as *x* becomes larger. Moreover, both *e*<sup>33</sup> and *∂u*3(Ti1*-x*Ni*x*)/*∂η<sup>3</sup>* become larger as *x* becomes larger, as shown in Figs. 18 and 19. This result is consistent with the result of

Figure 21(a) shows *ΔE*total as a function of the displacement of the Ti1*-x*Ni*x* ions with fixed crystal structures of fully-optimized BaTi1*-x*Ni*x*O3. Calculations of *E*total were performed with the fixed crystal structures of previously optimized structures except Ti1*-x*Ni*x* ions. The deviated value at the minimum point of *ΔE*total, *i.e.*, *∂*(*ΔE*total)/*∂u*3(Ti1*-x*Ni*x*), becomes

> total total 3 1 3 1 3 3 (Ti Ni ) . (Ti Ni )

total 3 1 3 1 3

 

*E u*

larger as *∂*(*ΔE*total)/*∂u*3(Ti1*-x*Ni*x*) becomes smaller. This result is consistent with the result of SrTiO3 discussed in Sec. 3.2.1.

**x=0**

**smaller e33**

*x x*

**x=0.04 x=0.05**

��3

(�) (�)

As shown in Fig. 21(b), *∂*(Δ*E*total)/ *∂η<sup>3</sup>* is almost constant, and therefore, *∂*(Δ*E*total)/*∂u*3(Ti1*-x*Ni*x*) is almost proportional to (*∂u*3(Ti1*-*

Ti Ni . (Ti Ni )

On the other hand, according to Eq. (3), *e*<sup>33</sup> becomes larger as *∂u*3(Ti1*-x*Ni*x*)/*∂η<sup>3</sup>* becomes larger. This is a reason why *e*<sup>33</sup> becomes

Figure 21. (a) Δ*E*total as a function of *u*3(Ti1-*x*Ni*x*) in BaTi1*-x*Ni*x*O3. Results with *x* = 0.05, 0.04, and 0 are shown. Dashed lines denote guidelines of

Finally, we comment on the difference in the properties between *e*<sup>33</sup> and *d*<sup>33</sup> in tetragonal *ABX*3. Figures 22(a) and 22(b) show the piezoelectric properties of *d*<sup>33</sup> as a function of *c* – *c*cub. Note that the trend of *d*<sup>33</sup> is closely dependent on the kind of *A* ions. This

*E Eu*

1

(5)

����� ���� ���� ���� ���� ����

**∂(**⊿**E)/∂η3** ≒**(const.)** 

**BaTi1-xNixO3**

⊿**E v.s. η3**

**BaTiO3**

η3

**η3**

���� ���� ���� ���� ���� ����

�����3

�����3

�����3

� � ����� (Å)

� � ���� (Å)

*x x*

(6)

 

<sup>1</sup>

*x x*

����� ����� ����� �3��� ����� ����� ���

⊿���� (����3 ��)

⊿**Etotal**

**(**×**10-3Hr)** 

(k)/∂η3

Ti1-xNix Ba

Let us discuss the relationship between *∂u*3(*k*)/*∂η3* and *c* – *c*cub in BaTi1*-x*Ni*x*O3 and BaTiO3 in the following. Figures 19(a) and 19(b) show the *∂u*3(*k*)/*∂η3* values. In these figures, O*x* and O*z* denote oxygen atoms along the [100] and [001] axes, respectively. Clearly, the absolute values of *∂u*3(*k*)/*∂η3* in BaTi1*-x*Ni*x*O3 are much larger than those in BaTiO3. Moreover, in comparison with Fig.18, properties of *e*33 are closely related to those of *∂u*3(*k*)/*∂η3*. Figure 20(a) shows the difference between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* along the [001] axis, and Fig. 20(b) shows the difference between *RA*-O*<sup>x</sup>* and *rA* + *r*O*<sup>x</sup>* on the (100) plane for several *AB*O3, as a function of *c* – *c*cub. Clearly, the difference between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* is closely related to *e*33 shown in Fig. 18, rather than the difference between *RA*-O*<sup>x</sup>* and *rA* + *r*O*<sup>x</sup>*. Moreover, note that the difference in absolute value between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* in BaTi1*-x*Ni*x*O3 is much smaller than that in BaTiO3. This result suggests that the (Ti1*-x*Ni*x*)-O*z* Coulomb repulsion along the [001] axis in BaTi1*-x*Ni*x*O3 is much smaller than the Ti-O*<sup>z</sup>* Coulomb repulsion in BaTiO3 and that therefore Ti1*-x*Ni*x* ion of BaTi1*-x*Ni*x*O3 can be displaced more easily along the [001] axis than Ti ion of BaTiO3. This must be a reason why the absolute value of *∂u*3(*k*)/*∂η3* of Ti1*-x*Ni*x* and O*z* ions in BaTi1*-x*Ni*x*O3 is larger than those

0

0.00 0.05 0.10 0.15 0.20

BaTiO3

PbTiO3

BiMO3

c - c\_cub (Å)

SrTiO3

Ba(Ti1-3zNb3z)(O1-zNz)3

c - ccub (Å)

0.00 0.20 0.40 0.60 0.80 1.00

Oz

Ti

http://dx.doi.org/10.5772/52187

19

Ox

c- ccub (Å)

5

10

e33 (C/m^2)

e33 (C/m2)

e33 (C/m2)

15

20

BaTi 1-xNi xO3

x = 0.04

<sup>x</sup> <sup>=</sup> <sup>0</sup> BaTi 1-xZrxO3 Ba1-ySryTiO3

BaTi1*-x*Ni*x*O3 is larger than those in BaTiO3.

Figure 19. *∂u*3(*k*)/*∂η*3 as a function of *c* – *c*cub : (a) BaTi1*-x*Ni*x*O3 and (b) BaTiO3 [15].

Ba

x = 0.04 x = 0

Ox

Oz

0.00 0.20 0.40 0.60 0.80 1.00

**Figure 19.** ∂u3(*k*)/∂η3 as a function of *c* – *c*cub : (a) BaTi1*-x*Ni*x*O3 and (b) BaTiO3 [15].

Let us discuss the above reasons in the following. *∂*(⊿*E*total)/*∂u*3(Ti1*-x*Ni*x*) can be written as

���� ���� ���� ���� ���� ����

� � ���� (Å)

� � ����� (Å)

�����3

��������� � � � ��3 ���

(�) (�)

�����3

� � ����

�����3

*u*

*∂*(*ΔE*total)/*∂u*3(Ti1*-x*Ni*x*) for each *x*. (b) *ΔE*total as a function of *η<sup>3</sup>* for BaTi1*-x*Ni*x*O3 and BaTiO3 [15].

��� ��� ��� ��3

���� ���������� �(( �) )

���������

**larger e33**

���� ���� ��� ��� ��� 3��

⊿���� (����3 ��)

������

(���3 ��)

⊿ *x x*

*u*

(a) (b)


���3

(a) *RB*-O*z* - (*rB* + *r*O*z*), and (b) *RA*-O*x* – (*rA* + *r*O*x*) [15].

����

� � �

�����3 � (��� � ���) (�)

� ��) (Å)

���� � (��

�

����

���

∂u3

(k)/∂η3

Figure 18.*e*33 as a function of *c* – *c*cub for different scales [15].

x = 0.05

BaTi 1-xNi xO3

x = 0.04

Ba(Ti

x = 0

e33 (C/m^2)

e33 (C/m2)

0.00 0.20 0.40 0.60 0.80 1.00

c - c\_cub (Å)

1-3zNb3z)(O1-zNz)3

(a) (b)

c - ccub (Å)

in BaTiO3.

<sup>O</sup>*<sup>x</sup>* – (*rA* + *r*O*<sup>x</sup>*) [15].

*<sup>x</sup>*Ni*x*)/*∂η3*)-1, *i.e.*,

SrTiO3 shown in Fig. 15(a).

Ox

Let us discuss the relationship between *∂u*3(*k*)/*∂η3* and *c* – *c*cub in BaTi1*-x*Ni*x*O3 and BaTiO3 in the following. Figures 19(a) and 19(b) show the *∂u*3(*k*)/*∂η3* values. In these figures, O*x* and O*z* denote oxygen atoms along the [100] and [001] axes, respectively. Clearly, the absolute values of *∂u*3(*k*)/*∂η3* in BaTi1*-x*Ni*x*O3 are much larger than those in BaTiO3. Moreover, in comparison with Fig.18, properties of *e*33 are closely related to those of *∂u*3(*k*)/*∂η3*. Figure 20(a) shows the difference between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* along the [001] axis, and

0

0.00 0.05 0.10 0.15 0.20

BaTiO3

PbTiO3

BiMO3

c - c\_cub (Å)

SrTiO3

Ba(Ti1-3zNb3z)(O1-zNz)3

c - ccub (Å)

c-ccub(Å)

5

10

e33 (C/m^2)

e33 (C/m2)

e33 (C/m2)

15

20

BaTi 1-xNi xO3

x = 0.04

<sup>x</sup> <sup>=</sup> <sup>0</sup> BaTi 1-xZrxO3 Ba1-ySryTiO3

BaTiO3. This must be a reason why the absolute value of *∂u*3(*k*)/*∂η3* of Ti1*-x*Ni*x* and O*z* ions in BaTi1*-x*Ni*x*O3 is larger than those in BaTiO3. Coulomb repulsion in BaTiO3 and that therefore Ti1*-x*Ni*x* ion of BaTi1*-x*Ni*x*O3 can be displaced more easily along the [001] axis than Ti ion of BaTiO3. This must be a reason why the absolute value of *∂u*3(*k*)/*∂η3* of Ti1*-x*Ni*x* and O*z* ions in BaTi1*-x*Ni*x*O3 is larger than those in BaTiO3.

**Figure 19.** ∂u3(*k*)/∂η3 as a function of *c* – *c*cub : (a) BaTi1*-x*Ni*x*O3 and (b) BaTiO3 [15].

Figure 19. *∂u*3(*k*)/*∂η*3 as a function of *c* – *c*cub : (a) BaTi1*-x*Ni*x*O3 and (b) BaTiO3 [15].

Figure 18.*e*33 as a function of *c* – *c*cub for different scales [15].

x = 0.05

BaTi 1-xNi xO3

x = 0.04

Ba(Ti

x = 0

e33 (C/m^2)

e33 (C/m2)

0.00 0.20 0.40 0.60 0.80 1.00

c - c\_cub (Å)

1-3zNb3z)(O1-zNz)3

(a) (b)

c - ccub (Å)

as a function of *x*. The *c* – *c*cub value shows almost 0 at *x* ≈ 0.12. Although the MPB does not appear in tetragonal Ba(Ti1*-*3*<sup>z</sup>*Nb3*z*)(O1-*z*N*z*)3, the *e*33 piezoelectric values are expected to show a

In the following, we discuss the role of the *B*-*X* Coulomb repulsions in piezoelectric *ABX*3.

Figures 18(a) and 18(b) show the piezoelectric properties of *e*33 as a function of the value *c* – *c*cub in tetragonal *ABX*3, where *c*cub denotes the *c* lattice parameter in cubic *ABX*3; *c* – *c*cub is a closely related parameter to *η*3. For *ABX*3, SrTiO3, BaTiO3 and PbTiO3 with the *c* lattice pa‐ rameter and all the inner coordination optimized for fixed *a*, and BaTi1*-x*Ni*x*O3 (0 ≦ *x* ≦ 0.05), Ba(Ti1*-*3*<sup>z</sup>*Nb3*z*)(O1-*z*N*z*)3 (0 ≦ *z* ≦ 0.125), Ba1*-y*Sr*y*TiO3 (0 ≦ *y* ≦ 0.5), BaTi1*-x*Zr*x*O3 (0 ≦ *x* ≦ 0.06), and Bi*M'*O3 (*M'* = Al, Sc) with fully optimized, were prepared [15]. Note that *e*<sup>33</sup> becomes larger as *c* – *c*cub becomes smaller and that the trend of *e*33 is almost independent of the kind of *A* ions. Moreover, note also that *e*33 of BaTi1*-x*Ni*x*O3 and that of Ba(Ti1*-*3*<sup>z</sup>*Nb3*z*)(O1-*z*N*z*)3 show

Let us discuss the relationship between *∂u*3(*k*)/*∂η3* and *c* – *c*cub in BaTi1*-x*Ni*x*O3 and BaTiO3 in the following. Figures 19(a) and 19(b) show the *∂u*3(*k*)/*∂η3* values. In these figures, O*x* and O*z* denote oxygen atoms along the [100] and [001] axes, respectively. Clearly, the absolute values of *∂u*3(*k*)/*∂η3* in BaTi1*-x*Ni*x*O3 are much larger than those in BaTiO3. Moreover, in comparison with Fig.18, properties of *e*33 are closely related to those of *∂u*3(*k*)/*∂η3*. Figure 20(a) shows the difference between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* along the [001] axis, and Fig. 20(b) shows the difference between *RA*-O*<sup>x</sup>* and *rA* + *r*O*<sup>x</sup>* on the (100) plane for several *AB*O3, as a function of *c* – *c*cub. Clearly, the difference between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* is closely related to *e*33 shown in Fig. 18, rather than the difference between *RA*-O*<sup>x</sup>* and *rA* + *r*O*<sup>x</sup>*. Moreover, note that the difference in absolute value between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* in BaTi1*-x*Ni*x*O3 is much smaller than that in BaTiO3. This result suggests that the (Ti1*-x*Ni*x*)-O*z* Coulomb repulsion along the [001] axis in BaTi1*-x*Ni*x*O3 is much smaller than the Ti-O*<sup>z</sup>* Coulomb repulsion in BaTiO3 and that therefore Ti1*-x*Ni*x* ion of BaTi1*-x*Ni*x*O3 can be displaced more easily along the [001] axis than Ti ion of BaTiO3. This must be a reason why the absolute value of *∂u*3(*k*)/*∂η3* of Ti1*-x*Ni*x* and O*z* ions in BaTi1*-x*Ni*x*O3 is larger than those

Let us discuss the relationship between *∂u*3(*k*)/*∂η3* and *c* – *c*cub in BaTi1*-x*Ni*x*O3 and BaTiO3 in the following. Figures 19(a) and 19(b) show the *∂u*3(*k*)/*∂η3* values. In these figures, O*x* and O*<sup>z</sup>* denote oxygen atoms along the [100] and [001] axes, respectively. Clearly, the absolute val‐ ues of *∂u*3(*k*)/*∂η3* in BaTi1*-x*Ni*x*O3 are much larger than those in BaTiO3. Moreover, in compari‐ son with Fig.18, properties of *e*<sup>33</sup> are closely related to those of *∂u*3(*k*)/*∂η3*. Figure 20(a) shows the difference between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* along the [001] axis, and Fig. 20(b) shows the differ‐ ence between *RA*-O*<sup>x</sup>* and *rA* + *r*O*<sup>x</sup>* on the (100) plane for several *AB*O3, as a function of *c* – *c*cub. Clearly, the difference between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* is closely related to *e*<sup>33</sup> shown in Fig. 18, rather than the difference between *RA*-O*<sup>x</sup>* and *rA* + *r*O*<sup>x</sup>*. Moreover, note that the difference in absolute value between *RB*-O*<sup>z</sup>* and *rB* + *r*O*<sup>z</sup>* in BaTi1*-x*Ni*x*O3 is much smaller than that in BaTiO3. This result suggests that the (Ti1*-x*Ni*x*)-O*z* Coulomb repulsion along the [001] axis in BaTi1 *<sup>x</sup>*Ni*x*O3 is much smaller than the Ti-O*z* Coulomb repulsion in BaTiO3 and that therefore Ti1 *<sup>x</sup>*Ni*x* ion of BaTi1*-x*Ni*x*O3 can be displaced more easily along the [001] axis than Ti ion of

0


∂u3

c - ccub (Å) c - ccub (Å)

(k)/∂η3

Ti1-xNix Ba

0.00 0.05 0.10 0.15 0.20

BaTiO3

PbTiO3

BiMO3

c - c\_cub (Å)

SrTiO3

Ba(Ti1-3zNb3z)(O1-zNz)3

c - ccub (Å)

0.00 0.20 0.40 0.60 0.80 1.00

Oz

Ti

Ox

c -ccub (Å)

5

Ba1-ySryTiO3

10

e33 (C/m^2)

e33 (C/m2)

e33 (C/m2)

15

20

BaTi 1-xNi xO3

x = 0.04

<sup>x</sup> <sup>=</sup> <sup>0</sup> BaTi 1-xZrxO3

maximum at *x* ≈ 0.12.

18 Advances in Ferroelectrics

*3.2.3. Piezoelectric properties of in tetragonal ABX3*

much larger values than the other *ABX*3.

x = 0.05

BaTi 1-xNi xO3

x = 0.04

**Figure 18.** *e*33 as a function of *c* – *c*cub for different scales [15].

Ba(Ti

x = 0

Figure 18.*e*33 as a function of *c* – *c*cub for different scales [15].

e33 (C/m^2)

e33 (C/m2)

0.00 0.20 0.40 0.60 0.80 1.00

c - c\_cub (Å)

1-3zNb3z)(O1-zNz)3

(a) (b)

c - ccub (Å)

Figure 19. *∂u*3(*k*)/*∂η*3 as a function of *c* – *c*cub : (a) BaTi1*-x*Ni*x*O3 and (b) BaTiO3 [15].

Ba

x = 0

x = 0.04

Ox

Oz

0.00 0.20 0.40 0.60 0.80 1.00

(a) (b)


∂u3

(k)/∂η3

in BaTiO3.

Figure 20.Evaluated values of in optimized tetragonal BaTi1*-x*Ni*x*O3, BaTiO3, and several *AB*O3, as a function of *c* – *c*cub: (a) *RB*-O*<sup>z</sup>* - (*rB* + *r*O*<sup>z</sup>*), and (b) *RA*-<sup>O</sup>*<sup>x</sup>* – (*rA* + *r*O*<sup>x</sup>*) [15]. **Figure 20.** Evaluated values of in optimized tetragonal BaTi1*-x*Ni*x*O3, BaTiO3, and several *AB*O3, as a function of *c* – *c*cub: (a) *RB*-O*z* - (*rB* + *r*O*z*), and (b) *RA*-O*x* – (*rA* + *r*O*x*) [15].

Figure 21(a) shows Δ*E*total as a function of the displacement of the Ti1*-x*Ni*<sup>x</sup>* ions with fixed crystal structures of fully-optimized BaTi1 *<sup>x</sup>*Ni*x*O3. Calculations of *E*total were performed with the fixed crystal structures of previously optimized structures except Ti1*-x*Ni*<sup>x</sup>* ions. The deviated value at the minimum point of Δ*E*total, *i.e.*, *∂*(Δ*E*total)/*∂u*3(Ti1*-x*Ni*x*), becomes smaller as *x* becomes larger. Moreover, both *e*<sup>33</sup> and *∂u*3(Ti1*-x*Ni*x*)/*∂η<sup>3</sup>* become larger as *x* becomes larger, as shown in Figs. 18 and 19. This result is consistent with the result of SrTiO3 shown in Fig. 15(a). Figure 21(a) shows *ΔE*total as a function of the displacement of the Ti1*-x*Ni*x* ions with fixed crystal structures of fully-optimized BaTi1*-x*Ni*x*O3. Calculations of *E*total were performed with the fixed crystal structures of previously optimized structures except Ti1*-x*Ni*x* ions. The deviated value at the minimum point of *ΔE*total, *i.e.*, *∂*(*ΔE*total)/*∂u*3(Ti1*-x*Ni*x*), becomes

> total total 3 1 3 1 3 3 (Ti Ni ) . (Ti Ni )

total 3 1 3 1 3

 

*E u*

larger as *∂*(*ΔE*total)/*∂u*3(Ti1*-x*Ni*x*) becomes smaller. This result is consistent with the result of SrTiO3 discussed in Sec. 3.2.1.

**x=0**

**smaller e33**

*x x*

**x=0.04 x=0.05**

��3

(�) (�)

As shown in Fig. 21(b), *∂*(Δ*E*total)/ *∂η<sup>3</sup>* is almost constant, and therefore, *∂*(Δ*E*total)/*∂u*3(Ti1*-x*Ni*x*) is almost proportional to (*∂u*3(Ti1*-*

Ti Ni . (Ti Ni )

On the other hand, according to Eq. (3), *e*<sup>33</sup> becomes larger as *∂u*3(Ti1*-x*Ni*x*)/*∂η<sup>3</sup>* becomes larger. This is a reason why *e*<sup>33</sup> becomes

Figure 21. (a) Δ*E*total as a function of *u*3(Ti1-*x*Ni*x*) in BaTi1*-x*Ni*x*O3. Results with *x* = 0.05, 0.04, and 0 are shown. Dashed lines denote guidelines of

Finally, we comment on the difference in the properties between *e*<sup>33</sup> and *d*<sup>33</sup> in tetragonal *ABX*3. Figures 22(a) and 22(b) show the piezoelectric properties of *d*<sup>33</sup> as a function of *c* – *c*cub. Note that the trend of *d*<sup>33</sup> is closely dependent on the kind of *A* ions. This

*E Eu*

1

(5)

����� ���� ���� ���� ���� ����

**∂(**⊿**E)/∂η3** ≒**(const.)** 

**BaTi1-xNixO3**

⊿**E v.s. η3**

**BaTiO3**

η3

**η3**

*x x*

(6)

 

<sup>1</sup>

*x x*

����� ����� ����� �3��� ����� ����� ���

⊿���� (����3 ��)

⊿**Etotal**

**(**×**10-3Hr)** 

Let us discuss the above reasons in the following. *∂*(⊿*E*total)/*∂u*3(Ti1*-x*Ni*x*) can be written as

*u*

*∂*(*ΔE*total)/*∂u*3(Ti1*-x*Ni*x*) for each *x*. (b) *ΔE*total as a function of *η<sup>3</sup>* for BaTi1*-x*Ni*x*O3 and BaTiO3 [15].

��� ��� ��� ��3

���� ���������� �(( �) )

���������

**larger e33**

���� ���� ��� ��� ��� 3��

⊿���� (����3 ��)

������

(���3 ��)

⊿

*<sup>x</sup>*Ni*x*)/*∂η3*)-1, *i.e.*,

*x x*

*u*

SrTiO3 shown in Fig. 15(a).

*<sup>x</sup>*Ni*x*)/*∂η3*)-1, *i.e.*,

smaller as *x* becomes larger. Moreover, both *e*33 and *∂u*3(Ti1*-x*Ni*x*)/*∂η3* become larger as *x* becomes larger, as shown in Figs. 18 and 19. This result is consistent with the result of SrTiO3 shown in Fig. 15(a). Figure 20.Evaluated values of in optimized tetragonal BaTi1*-x*Ni*x*O3, BaTiO3, and several *AB*O3, as a function of *c* – *c*cub: (a) *RB*-O*z* - (*rB* + *r*O*z*), and (b) *RA*-<sup>O</sup>*<sup>x</sup>* – (*rA* + *r*O*<sup>x</sup>*) [15].

*<sup>x</sup>*Ni*x*O3. Calculations of *E*total were performed with the fixed crystal structures of previously optimized structures except Ti1*-x*Ni*x* ions. The deviated value at the minimum point of Δ*E*total, *i.e.*, *∂*(Δ*E*total)/*∂u*3(Ti1*-x*Ni*x*), becomes smaller as *x* becomes larger. Moreover, both

Let us discuss the above reasons in the following. *∂*(*⊿E*total)/*∂u*3(Ti1*-x*Ni*x*) can be written as Figure 21(a) shows Δ*E*total as a function of the displacement of the Ti1*-x*Ni*x* ions with fixed crystal structures of fully-optimized BaTi1*-*

Let us discuss the above reasons in the following. *∂*(⊿*E*total)/*∂u*3(Ti1*-x*Ni*x*) can be written as

*x x*

⊿


$$
\left(\frac{\partial \Delta \mathcal{E}\_{\text{total}}}{\partial u\_3(\text{Ti}\_{1-x}\text{Ni}\_x)}\right) = \left(\frac{\partial \Delta \mathcal{E}\_{\text{total}}}{\partial \eta\_3}\right) \times \left(\frac{\partial u\_3(\text{Ti}\_{1-x}\text{Ni}\_x)}{\partial \eta\_3}\right)^{-1}.\tag{5}
$$

*s3jE* as well as *e*3*<sup>j</sup>*. In fact, the absolute value of *s3jE* in Bi*BX*3 or Pb*BX*3 is generally larger than that in *ABX*3 with alkaline-earth *A* ions.

Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity

result is in contrast with the trend of *e*33 as shown in Fig. 18. As expressed in Eq. (4), *d*33 is closely related to the elastic compliance *s3jE* as well as *e*3*<sup>j</sup>*. In fact, the absolute value of *s3jE* in Bi*BX*3 or Pb*BX*3 is generally larger than that in *ABX*3 with alkaline-earth *A* ions.

0.0

0.0 40.0 80.0 120.0 160.0

0.00 0.05 0.10 0.15 0.20

BaTiO3

0.00 0.05 0.10 0.15 0.20

BaTiO3

1-3zNb3z)(O1-zNz)3

1-3zNb3z)(O1-zNz)3

PbTiO3

http://dx.doi.org/10.5772/52187

21

PbTiO3

BiMO3

> BiMO3

c - c\_cub (Å)

c - ccub (Å)

c - c\_cub (Å)

c - ccub (Å)

40.0 80.0

d33 (pC/N)

d33 (pC/N)

d33 (pC/N)

d33 (pC/N)

120.0 160.0

BaTi1-xNixO3

Ba(Ti

Ba1-ySryTiO3

Ba1-ySryTiO3

BaTi1-xNixO3

BaTi1-xZrxO3

Ba(Ti

BaTi1-xZrxO3

Figure 23. (a) Illustration of the relationship between the *B*-*X* Coulomb repulsions and the ferroelectric and piezoelectric states in tetragonal *ABX*3. (b) Illustration of the relationship between *e*33 and the deviation. *P*3 and *T*C denote the spontaneous polarization and the Curie temperature,

**Figure 23.** (a) Illustration of the relationship between the B-X Coulomb repulsions and the ferroelectric and piezoelec‐ tric states in tetragonal *ABX3*. (b) Illustration of the relationship between *e33* and the deviation. *P3* and *TC* denote the

Figure 23. (a) Illustration of the relationship between the *B*-*X* Coulomb repulsions and the ferroelectric and piezoelectric states in tetragonal *ABX*3. (b) Illustration of the relationship between *e*33 and the deviation. *P*3 and *T*C denote the spontaneous polarization and the Curie temperature,

We have discussed a general role of the *B*-*X* Coulomb repulsions for ferroelectric and piezoelectric properties of tetragonal *ABX*3, based on our recent papers and patents. We have found that both ferroelectric state and piezoelectric state are closely related to the *B*-*Xz* Coulomb repulsions as well as the *B*-*Xx* ones, as illustrated in Fig. 23(a). Moreover, as illustrated in Fig. 23(b), we have also

We have discussed a general role of the *B*-*X* Coulomb repulsions for ferroelectric and piezo‐

found that both ferroelectric state and piezoelectric state are closely related to the *B*-*Xz* Cou‐

We have discussed a general role of the *B*-*X* Coulomb repulsions for ferroelectric and piezoelectric properties of tetragonal *ABX*3, based on our recent papers and patents. We have found that both ferroelectric state and piezoelectric state are closely related to the *B*-*Xz* Coulomb repulsions as well as the *B*-*Xx* ones, as illustrated in Fig. 23(a). Moreover, as illustrated in Fig. 23(b), we have also

, based on our recent papers and patents. We have

ones, as illustrated in Fig. 23(a). Moreover, as illustrated

We thank Professor M. Azuma in Tokyo Institute of Technology for useful discussion. We also thank T. Furuta, M. Kubota, H. Yabuta, and T. Watanabe for useful discussion and computational support. The present work was partly supported by the

in Fig. 23(b), we have also found that *e*33 is closely related to the deviation at the minimum

We thank Professor M. Azuma in Tokyo Institute of Technology for useful discussion. We also thank T. Furuta, M. Kubota, H. Yabuta, and T. Watanabe for useful discussion and computational support. The present work was partly supported by the

[2] Wu Z, Cohen R E. Pressure-induced anomalous phase transitions and colossal enhancement of piezoelectricity in PbTiO3.

[2] Wu Z, Cohen R E. Pressure-induced anomalous phase transitions and colossal enhancement of piezoelectricity in PbTiO3.

[3] Deguez O, Rabe K M, Vanderbilt D. First-principles study of epitaxial strain in perovskites. Phys. Rev. B 2005; 72 (14): 144101,

[3] Deguez O, Rabe K M, Vanderbilt D. First-principles study of epitaxial strain in perovskites. Phys. Rev. B 2005; 72 (14): 144101,

Elements Science and Technology Project from the Ministry of Education, Culture, Sports, Science and Technology.

Elements Science and Technology Project from the Ministry of Education, Culture, Sports, Science and Technology.

found that *e*33 is closely related to the deviation at the minimum point of the *ΔE*total.

electric properties of tetragonal *ABX*<sup>3</sup>

lomb repulsions as well as the *B*-*Xx*

spontaneous polarization and the Curie temperature, respectively.

found that *e*33 is closely related to the deviation at the minimum point of the *ΔE*total.

[1] Cohen R E. Origin of ferroelectricity in perovskite oxides. Nature 1992; 451 (6382) : 136-8.

[1] Cohen R E. Origin of ferroelectricity in perovskite oxides. Nature 1992; 451 (6382) : 136-8.

Phys. Rev. Lett. 2005; 95 (3): 037601, and related references therein.

Phys. Rev. Lett. 2005; 95 (3): 037601, and related references therein.

This result must be due to the larger Coulomb repulsion of Bi-*X* or Pb-*X* derived from 6s electrons in Bi (Pb) ion.

This result must be due to the larger Coulomb repulsion of Bi-*X* or Pb-*X* derived from 6s electrons in Bi (Pb) ion.

0.00 0.20 0.40 0.60 0.80 1.00

<sup>c</sup> - c\_cub (Å) c - ccub (Å)

0.00 0.20 0.40 0.60 0.80 1.00

B BiScO3 iAlO3

B BiScO3 iAlO3

(a) (b)

(a) (b)

<sup>c</sup> - c\_cub (Å) c - ccub (Å)

Figure 22. *d*33 as a function of *c* – *c*cub for different scales [15].

Figure 22. *d*33 as a function of *c* – *c*cub for different scales [15].

0.0

0.0 80.0 160.0 240.0 320.0

80.0

PbTiO3

PbTiO3

**Figure 22.** *d33* as a function of *c – ccub* for different scales [15].

160.0

d33 (pC/N)

d33 (pC/N)

d33 (pC/N)

d33 (pC/N)

240.0

320.0

respectively.

**4. Conclusion** 

**4. Conclusion** 

respectively.

**Acknowledgements** 

**Acknowledgements** 

**4. Conclusion**

point of the *ΔE*total.

and related references therein.

and related references therein.

**5. References** 

**5. References** 

As shown in Fig. 21(b), *∂*(Δ*E*total)/ *∂η*3 is almost constant, and therefore, *∂*(Δ*E*total)/*∂u*3(Ti1-xNix) is almost proportional to (*∂u*3(Ti1-xNix)/*∂η*3) -1, *i.e.*, 1 total total 3 1 (Ti Ni ) . (Ti Ni ) *x x E Eu u* - æ ö ¶ ¶¶ æ ö æ ö ç ÷ ç ÷ ç ÷ ¶ ¶¶ è ø è ø è ø ⊿ ⊿(5)

3 1 3 3

h

total 3 1

*E u*

larger as *∂*(*ΔE*total)/*∂u*3(Ti1*-x*Ni*x*) becomes smaller. This result is consistent with the result of SrTiO3 discussed in Sec. 3.2.1.

$$
\left(\frac{\partial \Lambda \mathbf{E}\_{\mathrm{total}}}{\partial u\_3 (\mathbf{T} \mathbf{i}\_{1-x} \mathbf{N} \mathbf{i}\_x)}\right) \propto \left(\frac{\partial u\_3 \left(\mathbf{T} \mathbf{i}\_{1-x} \mathbf{N} \mathbf{i}\_x\right)}{\partial \eta\_3}\right)^{-1}.\tag{6}
$$


 h

<sup>1</sup>


*x x*

On the other hand, according to Eq. (3), *e33* becomes larger as *∂u*3(Ti1-xNix)/*∂η*3 becomes larg‐ er. This is a reason why *e33* becomes larger as *∂*(*ΔE*total)/*∂u*3(Ti1-xNix) becomes smaller. This re‐ sult is consistent with the result of SrTiO3 discussed in Sec. 3.2.1. 3 1 3 Ti Ni . (Ti Ni ) *x x u* h æ ö ¶ æ ö ¶ ç ÷ <sup>µ</sup> ç ÷ ¶ ¶ è ø è ø (6) On the other hand, according to Eq. (3), *e*33 becomes larger as *∂u*3(Ti1*-x*Ni*x*)/*∂η3* becomes larger. This is a reason why *e*33 becomes

Figure 21. (a) Δ*E*total as a function of *u*3(Ti1-*x*Ni*x*) in BaTi1*-x*Ni*x*O3. Results with *x* = 0.05, 0.04, and 0 are shown. Dashed lines denote guidelines of *∂*(*ΔE*total)/*∂u*3(Ti1*-x*Ni*x*) for each *x*. (b) *ΔE*total as a function of *η3* for BaTi1*-x*Ni*x*O3 and BaTiO3 [15]. **Figure 21.** (a) ΔEtotal as a function of *u*3(Ti1-xNix) in BaTi1-xNixO3. Results with *x* = 0.05, 0.04, and 0 are shown. Dashed lines denote guidelines of ∂(ΔEtotal)/∂u3(Ti1-xNix) for each *x*. (b) ΔEtotal as a function of η3 for BaTi1-xNixO3 and BaTiO3 [15].

Finally, we comment on the difference in the properties between *e*33 and *d*33 in tetragonal *ABX*3. Figures 22(a) and 22(b) show the piezoelectric properties of *d*33 as a function of *c* – *c*cub. Note that the trend of *d*33 is closely dependent on the kind of *A* ions. This Finally, we comment on the difference in the properties between *e*33 and *d*33 in tetragonal *ABX*3. Figures 22(a) and 22(b) show the piezoelectric properties of *d*<sup>33</sup> as a function of *c* – *c*cub. Note that the trend of *d*<sup>33</sup> is closely dependent on the kind of *A* ions. This result is in contrast with the trend of *e*33 as shown in Fig. 18. As expressed in Eq. (4), *d*33 is closely related to the elastic compliance *s3j <sup>E</sup>* as well as *e*<sup>3</sup>*<sup>j</sup>* . In fact, the absolute value of *s3j* E in Bi*BX*3 or Pb*BX*3 is gen‐ erally larger than that in *ABX*3 with alkaline-earth *A* ions. This result must be due to the larger Coulomb repulsion of Bi-*X* or Pb-*X* derived from 6s electrons in Bi (Pb) ion.

20 Advances in Ferroelectrics result is in contrast with the trend of *e*33 as shown in Fig. 18. As expressed in Eq. (4), *d*33 is closely related to the elastic compliance *s3jE* as well as *e*3*<sup>j</sup>*. In fact, the absolute value of *s3jE* in Bi*BX*3 or Pb*BX*3 is generally larger than that in *ABX*3 with alkaline-earth *A* ions. result is in contrast with the trend of *e*33 as shown in Fig. 18. As expressed in Eq. (4), *d*33 is closely related to the elastic compliance Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity http://dx.doi.org/10.5772/52187 21

*s3jE* as well as *e*3*<sup>j</sup>*. In fact, the absolute value of *s3jE* in Bi*BX*3 or Pb*BX*3 is generally larger than that in *ABX*3 with alkaline-earth *A* ions.

This result must be due to the larger Coulomb repulsion of Bi-*X* or Pb-*X* derived from 6s electrons in Bi (Pb) ion.

c - c\_cub (Å)

c - ccub (Å)

Figure 22. *d*33 as a function of *c* – *c*cub for different scales [15]. **Figure 22.** *d33* as a function of *c – ccub* for different scales [15].

<sup>c</sup> - c\_cub (Å) c - ccub (Å)

Figure 22. *d*33 as a function of *c* – *c*cub for different scales [15].

respectively. **4. Conclusion**  (b) Illustration of the relationship between *e*33 and the deviation. *P*3 and *T*C denote the spontaneous polarization and the Curie temperature, respectively. **Figure 23.** (a) Illustration of the relationship between the B-X Coulomb repulsions and the ferroelectric and piezoelec‐ tric states in tetragonal *ABX3*. (b) Illustration of the relationship between *e33* and the deviation. *P3* and *TC* denote the spontaneous polarization and the Curie temperature, respectively.

Figure 23. (a) Illustration of the relationship between the *B*-*X* Coulomb repulsions and the ferroelectric and piezoelectric states in tetragonal *ABX*3. (b) Illustration of the relationship between *e*33 and the deviation. *P*3 and *T*C denote the spontaneous polarization and the Curie temperature,

Figure 23. (a) Illustration of the relationship between the *B*-*X* Coulomb repulsions and the ferroelectric and piezoelectric states in tetragonal *ABX*3.

#### We have discussed a general role of the *B*-*X* Coulomb repulsions for ferroelectric and piezoelectric properties of tetragonal *ABX*3, based on our recent papers and patents. We have found that both ferroelectric state and piezoelectric state are closely related to the We have discussed a general role of the *B*-*X* Coulomb repulsions for ferroelectric and piezoelectric properties of tetragonal *ABX*3, **4. Conclusion**

**4. Conclusion** 

and related references therein.

and related references therein.

**5. References** 

**5. References** 

smaller as *x* becomes larger. Moreover, both *e*33 and *∂u*3(Ti1*-x*Ni*x*)/*∂η3* become larger as *x* becomes larger, as shown in Figs. 18 and 19. This result is consistent with the result of

Figure 20.Evaluated values of in optimized tetragonal BaTi1*-x*Ni*x*O3, BaTiO3, and several *AB*O3, as a function of *c* – *c*cub: (a) *RB*-O*z* - (*rB* + *r*O*z*), and (b) *RA*-

Figure 21(a) shows Δ*E*total as a function of the displacement of the Ti1*-x*Ni*x* ions with fixed crystal structures of fully-optimized BaTi1 *<sup>x</sup>*Ni*x*O3. Calculations of *E*total were performed with the fixed crystal structures of previously optimized structures except Ti1*-x*Ni*x* ions. The deviated value at the minimum point of Δ*E*total, *i.e.*, *∂*(Δ*E*total)/*∂u*3(Ti1*-x*Ni*x*), becomes smaller as *x* becomes larger. Moreover, both *e*33 and *∂u*3(Ti1*-x*Ni*x*)/*∂η3* become larger as *x* becomes larger, as shown in Figs. 18 and 19. This result is consistent with the result of

Let us discuss the above reasons in the following. *∂*(*⊿E*total)/*∂u*3(Ti1*-x*Ni*x*) can be written as

total total 3 1 3 1 3 3 (Ti Ni ) . (Ti Ni )

<sup>æ</sup> ¶ ¶¶ ö æ öæ <sup>ö</sup> <sup>ç</sup> ÷ ç ÷ç = ´ <sup>÷</sup> <sup>ç</sup> <sup>÷</sup> ¶ ¶¶ <sup>è</sup> ø è øè <sup>ø</sup>

h

*E Eu*

 ⊿

total 3 1 3 1 3

æ ö ¶ æ ö ¶ ç ÷ <sup>µ</sup> ç ÷ ¶ ¶ è ø è ø

*E u*

larger as *∂*(*ΔE*total)/*∂u*3(Ti1*-x*Ni*x*) becomes smaller. This result is consistent with the result of SrTiO3 discussed in Sec. 3.2.1.

**x=0**

**smaller e33**

*x x*

⊿


*x x*


sult is consistent with the result of SrTiO3 discussed in Sec. 3.2.1.

1-xNixO3

(a) (b)

**x=0.04 x=0.05**

*u*

*ΔE u*

*x x*


As shown in Fig. 21(b), *∂*(Δ*E*total)/ *∂η*3 is almost constant, and therefore, *∂*(Δ*E*total)/*∂u*3(Ti1-xNix)


total total 3 1 3 1 3 3 (Ti Ni ) . (Ti Ni )

h

æ ö ¶ ¶¶ æ ö æ ö ç ÷ ç ÷ ç ÷ ¶ ¶¶ è ø è ø è ø

Ti Ni . (Ti Ni )

total 3 1 3 1 3

æ ö ¶ æ ö ¶ ç ÷ <sup>µ</sup> ç ÷ ¶ ¶ è ø è ø

As shown in Fig. 21(b), *∂*(Δ*E*total)/ *∂η3* is almost constant, and therefore, *∂*(Δ*E*total)/*∂u*3(Ti1*-x*Ni*x*) is almost proportional to (*∂u*3(Ti1*-*

On the other hand, according to Eq. (3), *e33* becomes larger as *∂u*3(Ti1-xNix)/*∂η*3 becomes larg‐ er. This is a reason why *e33* becomes larger as *∂*(*ΔE*total)/*∂u*3(Ti1-xNix) becomes smaller. This re‐

On the other hand, according to Eq. (3), *e*33 becomes larger as *∂u*3(Ti1*-x*Ni*x*)/*∂η3* becomes larger. This is a reason why *e*33 becomes

Figure 21. (a) Δ*E*total as a function of *u*3(Ti1-*x*Ni*x*) in BaTi1*-x*Ni*x*O3. Results with *x* = 0.05, 0.04, and 0 are shown. Dashed lines denote guidelines of

**Figure 21.** (a) ΔEtotal as a function of *u*3(Ti1-xNix) in BaTi1-xNixO3. Results with *x* = 0.05, 0.04, and 0 are shown. Dashed lines denote guidelines of ∂(ΔEtotal)/∂u3(Ti1-xNix) for each *x*. (b) ΔEtotal as a function of η3 for BaTi1-xNixO3 and BaTiO3 [15].

Finally, we comment on the difference in the properties between *e*33 and *d*33 in tetragonal *ABX*3. Figures 22(a) and 22(b) show the piezoelectric properties of *d*33 as a function of *c* – *c*cub. Note that the trend of *d*33 is closely dependent on the kind of *A* ions. This

Finally, we comment on the difference in the properties between *e*33 and *d*33 in tetragonal *ABX*3. Figures 22(a) and 22(b) show the piezoelectric properties of *d*<sup>33</sup> as a function of *c* – *c*cub. Note that the trend of *d*<sup>33</sup> is closely dependent on the kind of *A* ions. This result is in contrast with the trend of *e*33 as shown in Fig. 18. As expressed in Eq. (4), *d*33 is closely related to the

erally larger than that in *ABX*3 with alkaline-earth *A* ions. This result must be due to the

larger Coulomb repulsion of Bi-*X* or Pb-*X* derived from 6s electrons in Bi (Pb) ion.

. In fact, the absolute value of *s3j*

Ti Ni . (Ti Ni )

*ΔE ΔE u*

*x x*

⊿


Let us discuss the above reasons in the following. *∂*(⊿*E*total)/*∂u*3(Ti1*-x*Ni*x*) can be written as

*u*

BaTi

**larger e33**

*∂*(*ΔE*total)/*∂u*3(Ti1*-x*Ni*x*) for each *x*. (b) *ΔE*total as a function of *η3* for BaTi1*-x*Ni*x*O3 and BaTiO3 [15].

0.0 0.1 0.2 0.3

u\_Ti 1u-TxNiNii x(( ÅA) )

*<sup>E</sup>* as well as *e*<sup>3</sup>*<sup>j</sup>*


⊿Etot (×10^-3 Hr)

elastic compliance *s3j*

⊿ Etotal (×10-3 Hr)

*u*

*u*

is almost proportional to (*∂u*3(Ti1-xNix)/*∂η*3)

1

(5)

(6)


1

(5)


**∂(**⊿**E)/∂η3** ≒**(const.)** 

**BaTi1-xNixO3**

⊿**E v.s. η3**

**BaTiO3**

E in Bi*BX*3 or Pb*BX*3 is gen‐

η3

**η3**


*x x*



(6)

*x x*

 h

> h


( ) <sup>1</sup>

h


h


*x x*

<sup>1</sup>

*x x*


⊿Etot (×10^-3 Hr)

⊿**Etotal**

**(**×**10-3Hr)** 


SrTiO3 shown in Fig. 15(a).

<sup>O</sup>*<sup>x</sup>* – (*rA* + *r*O*<sup>x</sup>*) [15].

*<sup>x</sup>*Ni*x*)/*∂η3*)-1, *i.e.*,

SrTiO3 shown in Fig. 15(a).

found that *e*33 is closely related to the deviation at the minimum point of the *ΔE*total. **Acknowledgements**  We thank Professor M. Azuma in Tokyo Institute of Technology for useful discussion. We also thank T. Furuta, M. Kubota, H. Yabuta, and T. Watanabe for useful discussion and computational support. The present work was partly supported by the *B*-*Xz* Coulomb repulsions as well as the *B*-*Xx* ones, as illustrated in Fig. 23(a). Moreover, as illustrated in Fig. 23(b), we have also found that *e*33 is closely related to the deviation at the minimum point of the *ΔE*total. **Acknowledgements**  We thank Professor M. Azuma in Tokyo Institute of Technology for useful discussion. We also thank T. Furuta, M. Kubota, H. Yabuta, and T. Watanabe for useful discussion and computational support. The present work was partly supported by the We have discussed a general role of the *B*-*X* Coulomb repulsions for ferroelectric and piezo‐ electric properties of tetragonal *ABX*<sup>3</sup> , based on our recent papers and patents. We have found that both ferroelectric state and piezoelectric state are closely related to the *B*-*Xz* Cou‐ lomb repulsions as well as the *B*-*Xx* ones, as illustrated in Fig. 23(a). Moreover, as illustrated in Fig. 23(b), we have also found that *e*33 is closely related to the deviation at the minimum point of the *ΔE*total.

[2] Wu Z, Cohen R E. Pressure-induced anomalous phase transitions and colossal enhancement of piezoelectricity in PbTiO3.

[2] Wu Z, Cohen R E. Pressure-induced anomalous phase transitions and colossal enhancement of piezoelectricity in PbTiO3.

[3] Deguez O, Rabe K M, Vanderbilt D. First-principles study of epitaxial strain in perovskites. Phys. Rev. B 2005; 72 (14): 144101,

[3] Deguez O, Rabe K M, Vanderbilt D. First-principles study of epitaxial strain in perovskites. Phys. Rev. B 2005; 72 (14): 144101,

Elements Science and Technology Project from the Ministry of Education, Culture, Sports, Science and Technology.

Elements Science and Technology Project from the Ministry of Education, Culture, Sports, Science and Technology.

[1] Cohen R E. Origin of ferroelectricity in perovskite oxides. Nature 1992; 451 (6382) : 136-8.

[1] Cohen R E. Origin of ferroelectricity in perovskite oxides. Nature 1992; 451 (6382) : 136-8.

Phys. Rev. Lett. 2005; 95 (3): 037601, and related references therein.

Phys. Rev. Lett. 2005; 95 (3): 037601, and related references therein.

*B*-*Xz* Coulomb repulsions as well as the *B*-*Xx* ones, as illustrated in Fig. 23(a). Moreover, as illustrated in Fig. 23(b), we have also

based on our recent papers and patents. We have found that both ferroelectric state and piezoelectric state are closely related to the

### **Acknowledgements**

We thank Professor M. Azuma in Tokyo Institute of Technology for useful discussion. We also thank T. Furuta, M. Kubota, H. Yabuta, and T. Watanabe for useful discussion and computational support. The present work was partly supported by the Elements Sci‐ ence and Technology Project from the Ministry of Education, Culture, Sports, Science and Technology.

[8] Oguchi T, Ishii F, Uratani Y. New method for calculating physical properties from first principles-piezoelectric and multiferroics. Butsuri 2009; 64 (4): 270-6 [in Japa‐

Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity

http://dx.doi.org/10.5772/52187

23

[9] Uratani Y, Shishidou T, Oguchi T. First-principles calculations of colossal piezoelec‐ tric response in thin film PbTiO3. Jpn. Soc. Appl. Phys. 2008: conference proceedings,

[10] Miura K, Kubota M, Azuma M, and Funakubo H. Electronic and structural proper‐

[11] Miura K, Furuta T, Funakubo H. Electronic and structural properties of BaTiO3: A proposal about the role of Ti 3s and 3p states for ferroelectricity. Solid State Com‐

[12] Furuta T, Miura K. First-principles study of ferroelectric and piezoelectric properties of tetragonal SrTiO3 and BaTiO3 with in-plane compressive structures. Solid State

[13] Miura K, Azuma M, Funakubo H. Electronic and structural properties of ABO3: Role of the B-O Coulomb repulsions for ferroelectricity. Materials 2011; 4 (1): 260-73.

[14] Miura K. First-principles study of ABO3: Role of the B-O Coulomb repulsions for fer‐ roelectricity and piezoelectricity. Lallart M. (ed.) Ferroelectrics - Characterization and Modeling. Rijeka: InTech; 2012. p395-410. Available from http://www.intechop‐ en.com/books/ferroelectrics-characterization-and-modeling/first-principles-study-ofabo3-role-of-the-b-o-coulomb-repulsions-for-ferroelectricity-and-piezoelec (accessed

[15] Miura K, Funakubo H. First-principles analysis of tetragonal ABO3: Role of the B-O Coulomb repulsions for ferroelectricity and piezoelectricity. Proceedings of 15th US-Japan Seminar on Dielectric and Piezoelectric Ceramics, 6-9 November 2011, Kagosh‐

[16] Miura K, Ifuku T, Kubota M, Hayashi J. Piezoelectric Material. Japan Patent

[17] Miura K, Kubota M, Hayashi J, Watanabe T. Piezoelectric Material. submitted to Ja‐

[18] Miura K, Furuta T. First-principles study of structural trend of BiMO3 and BaMO3: Relationship between tetragonal and rhombohedral structure and the tolerance fac‐

[19] Miura K, Kubota M, Azuma M, and Funakubo H. Electronic, structural, and piezo‐ electric properties of BiFe1-xCoxO3. Jpn. J. Appl. Phys. 2010; 49 (9): 09ME07.

[20] Miura K, Tanaka M. Electronic structures of PbTiO3: I. Covalent interaction between

http://www.mdpi.com/1996-1944/4/1/260 (accessed 20 July 2012).

27-30 March 2008, Funabashi, Japan [in Japanese].

mun. 2010; 150 (3-4): 205-8.

20 July 2012).

ima, Japan.

2011-001257 [in Japanese].

pan Patent [in Japanese].

tors. Jpn. J. Appl. Phys. 2010; 49 (3): 031501 (2010).

Ti and O ions. Jpn. J. Appl. Phys. 1998; 37 (12A): 6451-9.

Commun. 2010; 150 (47-48): 2350-3.

ties of BiZn0.5Ti0.5O3. Jpn. J. Appl. Phys. 2009; 48 (9): 09KF05.

nese].

### **Author details**

Kaoru Miura1 and Hiroshi Funakubo2


### **References**


[8] Oguchi T, Ishii F, Uratani Y. New method for calculating physical properties from first principles-piezoelectric and multiferroics. Butsuri 2009; 64 (4): 270-6 [in Japa‐ nese].

**Acknowledgements**

and Technology.

22 Advances in Ferroelectrics

**Author details**

1 Canon Inc., Tokyo, Japan

and Hiroshi Funakubo2

2 Tokyo Institute of Technology, Yokohama, Japan

Kaoru Miura1

**References**

136-8.

lated references therein.

Appl. Phys. 2009; 105 (5): 053713.

ences therein.

We thank Professor M. Azuma in Tokyo Institute of Technology for useful discussion. We also thank T. Furuta, M. Kubota, H. Yabuta, and T. Watanabe for useful discussion and computational support. The present work was partly supported by the Elements Sci‐ ence and Technology Project from the Ministry of Education, Culture, Sports, Science

[1] Cohen R E. Origin of ferroelectricity in perovskite oxides. Nature 1992; 451 (6382) :

[2] Wu Z, Cohen R E. Pressure-induced anomalous phase transitions and colossal en‐ hancement of piezoelectricity in PbTiO3. Phys. Rev. Lett. 2005; 95 (3): 037601, and re‐

[3] Deguez O, Rabe K M, Vanderbilt D. First-principles study of epitaxial strain in per‐

[4] Qi T, Grinberg I, Rappe A M. First-principles investigation of the high tetragonal fer‐ roelectric material BiZn1/2Ti1/2O3. Phys. Rev. B 2010; 79 (13): 134113, and related refer‐

[5] Wang H, Huang H, Lu W, Chan H L W, Wang B, Woo C H. Theoretical prediction on the structural, electronic, and polarization properties of tetragonal Bi2ZnTiO6. J.

[6] Dai J Q, Fang Z. Structural, electronic, and polarization properties of Bi2ZnTiO6 su‐

[7] Khenata R, Sahnoun M, Baltache H, Rerat M, Rashek A H, Illes N, Bouhafs B. Firstprinciple calculations of structural, electronic and optical properties of BaTiO3 and BaZrO3 under hydrostatic pressure. Solid State Commun. 2005; 136 (2): 120-125.

percell from first-principles. J. Appl. Phys. 2012; 111 (11): 114101.

ovskites. Phys. Rev. B 2005; 72 (14): 144101, and related references therein.


[21] LDA\_TM\_psp1\_data: ABINIT. http://www.abinit.org/downloads/psp-links/ lda\_tm\_psp1\_data/ (accessed 20 July 2012).

**Chapter 2**

**Advances in Thermodynamics**

Shu-Tao Ai

**1. Introduction**

http://dx.doi.org/10.5772/52089

**of Ferroelectric Phase Transitions**

Additional information is available at the end of the chapter

are a kind of technical but not fundamental progress.

The thermodynamics of ferroelectric phase transitions is an important constituent part of the phenomenological theories of them, as well as the interface dynamics of them. In particular, if we confine the thermodynamics to the equilibrium range, we can say that the Landau-Devonshire theory is a milestone in the process of the development of ferroelectric phase transition theories. This can be found in many classical books such as [1,2]. Many studies centering on it, especially the size-effects and surface-effects of ferroelectric phase transitions, have been carried out. For the reason of simplicity, we just cite a few [3,4]. But we think these

Why we think so is based on that the Landau-Devonshire theory is confined to the equilibrium range in essene so it can't deal with the outstanding irreversible phenomonon of first-order ferroelectric phase transitions strictly, which is the "thermal hysteresis". The Landau-Devon‐ shire theory attributes the phenomenon to a series of metastable states existing around the Curie temperature *TC*. In principle, the metastable states are not the equilibrium ones and can not be processed by using the equilibrium thermodynamics. Therefore, we believe the Landau-Devonshire theory is problematic though it is successful in mathematics. The real processes of phase transition were distorted. In this contribution, the Landau-Devonshire theory will be reviewed critically, then the latest phenomenological theory of ferroelectric phase transitions

will be established on the basis of non-equilibrium or irreversible thermodynamics.

This contribution are organized as the follows. In Section 2, we will show the unpleasant consequence caused by the metastable states hypothesis, and the evidence for the nonexistence of metastable states, i.e. the logical conflict. Then in Section 3 and 4, we will give the non-equilibrium ( or irreversible ) thermodynamic description of ferroelectric phase transi‐

and reproduction in any medium, provided the original work is properly cited.

© 2013 Ai; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.


**Chapter 2**

### **Advances in Thermodynamics of Ferroelectric Phase Transitions**

Shu-Tao Ai

[21] LDA\_TM\_psp1\_data: ABINIT. http://www.abinit.org/downloads/psp-links/

[22] Kuroiwa Y, Aoyagi S, Sawada A, Harada J, Nishibori E, Tanaka M, and Sakata M. Evidence for Pb-O covalency in tetragonal PbTiO3. Phys. Rev. Lett. 2001; 87 (21):

[23] Shannon R D. Revised effective ionic radii and systematic studies of interatomic dis‐ tances in halides and chalcogenides. Acta Crystallogr. Sect. A. 1976; 32 (5): 751-67.

[24] Suchomel M R, Fogg A M, Allix M, Niu H, Claridge J B, Rosseinsky M J. Bi2ZnTiO6: A lead-free closed-shell polar perovskite with a calculated ionic polarization of 150

[25] Khalyavin D D, Salak A N, Vyshatko N P, Lopes A B, Olekhnovich N M, Pushkarev A V, Maroz I I, Radyush T V. Crystal structure of metastable perovskite Bi(Mg1/2Ti1/2)O3: Bi-based structural analogue of antiferroelectric PbZrO3. Chem. Ma‐

[26] Randall C A, Eitel R, Jones B, Shrout T R, Woodward D I, Reaney I M, Investigation of a high TC piezoelectric system: (1-x)Bi(Mg1/2Ti1/2)O3-(x)PbTiO3. J. Appl. Phys. 2004;

[27] Rossetti Jr G A, Khachaturyan A G. Concepts of morphotropism in ferroelectric solid solutions. Proceedings of 13th US-Japan Seminar on Dielectric and Piezoelectric Ce‐

[28] Damjanovic D. A morphotropic phase boundary system based on polarization rota‐ tion and polarization extension. Appl. Phys. Lett. 2010; 97 (6): 062906, and related ref‐

[29] Gonze X, Beuken J-M, Caracas R, Detraux F, Fuchs M, Rignanese G-M, Sindic L, Ver‐ straete M, Zerah G, Jollet F, Torrent M, Roy A, Mikami M, Ghosez P, Raty J-Y, Allan D C. First-principles computation of material properties: the ABINIT software

[30] Hohenberg P, Kohn W. Inhomogeneous electron gas. Phys. Rev. 1964; 136 (3B):

[31] Opium - pseudopotential generation project. http://opium.sourceforge.net/

[32] Ramer N J, Rappe A M. Application of a new virtual crystal approach for the study

[33] Resta R. Macroscopic polarization in crystalline dielectrics: the geometric phase ap‐

[34] Kim Y-I, Lee E. Constant-wavelength neutron diffraction study of cubic perovskites

of disordered perovskites. J. Phys. Chem. Solids. 2000; 61 (2): 315-20.

BaTaO2N and BaNbO2N. J. Ceram. Soc. Jpn. 2011; 119 (5): 371-4.

ramics, 4-7 November 2007, Awaji, Japan, and related references therein.

lda\_tm\_psp1\_data/ (accessed 20 July 2012).

μC cm-2. Chem. Mater. 2006; 18 (21), 4987-9.

project. Comput. Mater. Sci. 2002; 25 (3), 478-92.

proach. Rev. Mod. Phys. 1994; 66 (3): 899-915.

index.html (accessed 20 July 2012).

ter. 2006; 18 (21), 5104-10.

95 (7): 3633-9.

erences therein.

B864-71.

217601.

24 Advances in Ferroelectrics

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52089

### **1. Introduction**

The thermodynamics of ferroelectric phase transitions is an important constituent part of the phenomenological theories of them, as well as the interface dynamics of them. In particular, if we confine the thermodynamics to the equilibrium range, we can say that the Landau-Devonshire theory is a milestone in the process of the development of ferroelectric phase transition theories. This can be found in many classical books such as [1,2]. Many studies centering on it, especially the size-effects and surface-effects of ferroelectric phase transitions, have been carried out. For the reason of simplicity, we just cite a few [3,4]. But we think these are a kind of technical but not fundamental progress.

Why we think so is based on that the Landau-Devonshire theory is confined to the equilibrium range in essene so it can't deal with the outstanding irreversible phenomonon of first-order ferroelectric phase transitions strictly, which is the "thermal hysteresis". The Landau-Devon‐ shire theory attributes the phenomenon to a series of metastable states existing around the Curie temperature *TC*. In principle, the metastable states are not the equilibrium ones and can not be processed by using the equilibrium thermodynamics. Therefore, we believe the Landau-Devonshire theory is problematic though it is successful in mathematics. The real processes of phase transition were distorted. In this contribution, the Landau-Devonshire theory will be reviewed critically, then the latest phenomenological theory of ferroelectric phase transitions will be established on the basis of non-equilibrium or irreversible thermodynamics.

This contribution are organized as the follows. In Section 2, we will show the unpleasant consequence caused by the metastable states hypothesis, and the evidence for the nonexistence of metastable states, i.e. the logical conflict. Then in Section 3 and 4, we will give the non-equilibrium ( or irreversible ) thermodynamic description of ferroelectric phase transi‐

© 2013 Ai; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

tions, which eliminates the unpleasant consequence caused by the metastable states hypoth‐ esis. In Section 5, we will give the non-equilibrium thermodynamic explanation of the irreversibility of ferroelectric phase transitions, i.e. the thermal hysteresis and the domain occurrences in ferroelectrics. At last, in Section 6 we will make some concluding remarks and look forward to some possible developments.

( ) 1 1 *G g TD* = , (1)

Advances in Thermodynamics of Ferroelectric Phase Transitions

http://dx.doi.org/10.5772/52089

27

( ) ( ) 11 1 1 *f G TD G g TD* , , =- = , 0 (2)

¶ ¶ -- = ¶ ¶ (3)

¶ <sup>=</sup> ¶ (5)

*h DT* ( , 0 ) = (6)

, 0. After all, if our discussion are limited in the

(4)

, and the electric displace‐

The long-standing, close correlation between analytical dynamics and thermodynamics

where *G*1, *T* , *D* are the generalized displacements. The possible displacements d*G*1, d*T* and

In the Landau-Devonshire theory, the scleronomic constraint equation, i.e. Equation (1) is expressed in the form of the power series of *D* ( For simplicity, only the powers whose orders

> 111 246 *GG D D D* =+ + +

where *α*, *β*, *γ* are the functions of *T* , and *G*<sup>10</sup> is the elastic Gibbs energy of paraelectric phase. The relation between *G*1 and *D* at various temperature, which belongs to first-order phase transition ferroelectrics is represented graphically in Figure 1. The electric displacements

ment which corresponds to the middle minimum of *G*1 equals zero. The possible electric

Equivalently, imposed on the generalized displacements *G*1, *T* , *D* is a constraint, which is

<sup>1</sup> <sup>0</sup> *<sup>G</sup> D*

So, the possible displacement d*D* should be the follows

equilibrium thermodynamics strictly, there must be the third constraint, i.e. the equilibrium

displacements should be the above ones which correspond to the minima of *G*1.

−0= ± *D* \*

abg

2 46

1 1 <sup>1</sup> d d d0 *g g G TD T D*

implies that Equation (1) can be taken as a scleronomic constraint equation

d*D* satisfy the following equation

are not more than six are considered )

d*D* \*

, −d*D* \*

*D* and *T* should satisfy

, 0−(±*D* \*) = ∓ *D* \*

1 10

which correspond to the bilateral minima of *G*1 are identified as ±*<sup>D</sup>* \*

, ± *D* \*

### **2. Limitations of Landau-Devonshire theory and demonstration of new approach**

The most outstanding merit of Landau-Devonshire theory is that the Curie temperature and the spontaneous polarization at Curie temperature can be determined simply. However, in the Landau-Devonshire theory, the path of a first-order ferroelectric phase transition is believed to consist of a series of metastable states existing around the Curie temperature. This is too difficult to believe because of the difficulties encounted ( just see the follows) *TC*.

### **2.1. Unpleasant consequence caused by metastable states hypothesis**

Basing on the Landau-Devonshire theory, we make the following inference. Because of the thermal hysteresis, a first-order ferroelectric phase transition must occur at another tempera‐ ture, which is different from the Curie temperature [5]. The state corresponding to the mentioned temperature ( i.e. actural phase transition temperature ) is a metastable one. Since the unified temperature and spontanous polarization can be said about the metastable state, we neglect the heterogeneity of system actually. In other words, every part of the system, i.e. either the surface or the inner part, is of equal value physically. When the phase transition occurs at the certain temperature, every part of the system absorbs or releases the latent heat simultaneously by a kind of action at a distance. ( The concept arose in the electromagnetism first. Here it maybe a kind of heat transfer. ) Otherwise, the heat transfer in system, with a finite rate, must destroy the homogeneity of system and lead to a non-equilibrium thermodynamic approach. The unpleasant consequence, i.e. the action at a distance should be eliminated and the lifeforce should be bestowed on the non-equilibrium thermodynamic approach.

In fact, a first-order phase transition process is always accompanied with the fundamental characteristics, called the co-existence of phases and the moving interface ( i.e. phase boun‐ dary ). The fact reveals that the phase transition at various sites can not occur at the same time. Yet, the phase transition is induced by the external actions ( i.e. absorption or release of latent heat ). It conflicts sharply with the action at a distance.

### **2.2. Evidence for non-existence of metastable states: logical conflict**

In the Landau-Devonshire theory, if we neglect the influence of stress, the elastic Gibbs energy *G*1 can be expressed with a binary function of variables, namely the temperature *T* and the electric displacement *D* ( As *G*1 is independent of the orientation of *D*, here we are interested in the magnitude of *D* only )

$$G\_1 = \mathcal{g}\_1(T, D) \tag{1}$$

The long-standing, close correlation between analytical dynamics and thermodynamics implies that Equation (1) can be taken as a scleronomic constraint equation

tions, which eliminates the unpleasant consequence caused by the metastable states hypoth‐ esis. In Section 5, we will give the non-equilibrium thermodynamic explanation of the irreversibility of ferroelectric phase transitions, i.e. the thermal hysteresis and the domain occurrences in ferroelectrics. At last, in Section 6 we will make some concluding remarks and

**2. Limitations of Landau-Devonshire theory and demonstration of new**

The most outstanding merit of Landau-Devonshire theory is that the Curie temperature and the spontaneous polarization at Curie temperature can be determined simply. However, in the Landau-Devonshire theory, the path of a first-order ferroelectric phase transition is believed to consist of a series of metastable states existing around the Curie temperature. This is too difficult to believe because of the difficulties encounted ( just see the follows) *TC*.

Basing on the Landau-Devonshire theory, we make the following inference. Because of the thermal hysteresis, a first-order ferroelectric phase transition must occur at another tempera‐ ture, which is different from the Curie temperature [5]. The state corresponding to the mentioned temperature ( i.e. actural phase transition temperature ) is a metastable one. Since the unified temperature and spontanous polarization can be said about the metastable state, we neglect the heterogeneity of system actually. In other words, every part of the system, i.e. either the surface or the inner part, is of equal value physically. When the phase transition occurs at the certain temperature, every part of the system absorbs or releases the latent heat simultaneously by a kind of action at a distance. ( The concept arose in the electromagnetism first. Here it maybe a kind of heat transfer. ) Otherwise, the heat transfer in system, with a finite rate, must destroy the homogeneity of system and lead to a non-equilibrium thermodynamic approach. The unpleasant consequence, i.e. the action at a distance should be eliminated and

the lifeforce should be bestowed on the non-equilibrium thermodynamic approach.

heat ). It conflicts sharply with the action at a distance.

in the magnitude of *D* only )

**2.2. Evidence for non-existence of metastable states: logical conflict**

In fact, a first-order phase transition process is always accompanied with the fundamental characteristics, called the co-existence of phases and the moving interface ( i.e. phase boun‐ dary ). The fact reveals that the phase transition at various sites can not occur at the same time. Yet, the phase transition is induced by the external actions ( i.e. absorption or release of latent

In the Landau-Devonshire theory, if we neglect the influence of stress, the elastic Gibbs energy *G*1 can be expressed with a binary function of variables, namely the temperature *T* and the electric displacement *D* ( As *G*1 is independent of the orientation of *D*, here we are interested

**2.1. Unpleasant consequence caused by metastable states hypothesis**

look forward to some possible developments.

**approach**

26 Advances in Ferroelectrics

$$f\_1(\mathbb{G}\_{1'}T, D) = \mathbb{G}\_1 - \mathbb{g}\_1(T, D) = 0 \tag{2}$$

where *G*1, *T* , *D* are the generalized displacements. The possible displacements d*G*1, d*T* and d*D* satisfy the following equation

$$\mathbf{d}\mathbf{G}\_1 - \frac{\partial \mathbf{g}\_1}{\partial T} \mathbf{d}T - \frac{\partial \mathbf{g}\_1}{\partial D} \mathbf{d}D = \mathbf{0} \tag{3}$$

In the Landau-Devonshire theory, the scleronomic constraint equation, i.e. Equation (1) is expressed in the form of the power series of *D* ( For simplicity, only the powers whose orders are not more than six are considered )

$$\mathcal{G}\_1 = \mathcal{G}\_{10} + \frac{1}{2} a D^2 + \frac{1}{4} \beta D^4 + \frac{1}{6} \gamma D^6 \tag{4}$$

where *α*, *β*, *γ* are the functions of *T* , and *G*<sup>10</sup> is the elastic Gibbs energy of paraelectric phase. The relation between *G*1 and *D* at various temperature, which belongs to first-order phase transition ferroelectrics is represented graphically in Figure 1. The electric displacements which correspond to the bilateral minima of *G*1 are identified as ±*<sup>D</sup>* \* , and the electric displace‐ ment which corresponds to the middle minimum of *G*1 equals zero. The possible electric displacements should be the above ones which correspond to the minima of *G*1.

Equivalently, imposed on the generalized displacements *G*1, *T* , *D* is a constraint, which is

$$\frac{\partial \mathbf{G}\_1}{\partial D} = 0 \tag{5}$$

So, the possible displacement d*D* should be the follows d*D* \* , −d*D* \* , 0−(±*D* \*) = ∓ *D* \* , ± *D* \* −0= ± *D* \* , 0. After all, if our discussion are limited in the equilibrium thermodynamics strictly, there must be the third constraint, i.e. the equilibrium *D* and *T* should satisfy

$$h(D, T) = 0\tag{6}$$

**Figure 1.** The relation between elastic Gibbs energy *G*1 and zero field electric displacement *D* belongs to the ferroelec‐ trics at various temperature, which undergoes a first-order phase transition [5]. *T*<sup>0</sup> :the lowest temperature at which the paraelectric phase may exist, i.e. the Curie-Weiss temperature; *TC* :the phase transition temperature, i.e. the Curie temperature; *T*1 : the highest temperature at which the ferroelectric phase may exist; *T*<sup>2</sup> : the highest temperature at which the ferroelectric phase may be induced by the external electric field.

where *h* is a binary function of the variables *D* and *T* . It can be determined by the principle of minimum energy

$$G\_1 = \min \tag{7}$$

( ) ( ) 2 1 1 2 *f G TDt G g TDt* ¢ ¢ ,, , =- = ,, 0 (9)

¶¶¶ ¢ - - -= ¶¶ ¶ (10)

22 2 <sup>1</sup> d d d d0 *ggg G TDt TD t*

self-contradiction arises. So the metastable states can not come into being.

which do not vary with the time but may be not metastable.

velocity of interface ( where the phase transition is occurring ), *Jq*

**2.3. Real path: Existence of stationary states**

±*lρv* ′ = *Jq*

stationary.

Comparing Equation (3) with Equation (10), we may find that the possible displacements here are not the same as those in the former case which characterize the metastable states for they satisfy the different constraint equations, respectively. ( In the latter case, the possible dis‐ placements are time-dependent, whereas in the former case they are not. ) Yet, the integral of possible displacement d*D* is the possible electric displacement in every case. The possible electric displacements which characterize one certain metastable state vary with the cases. A

What are the real states among a phase transition process? In fact, both the evolution with time and the spatial heterogeneity need to be considered when the system is out of equilibrium [6-9]. Just as what will be shown in Section 2.3, the real states should be the stationary ones,

The real path of a first-order ferroelectric phase transition is believed by us to consist of a series of stationary states. At first, this was conjectured according to the experimental results, then

Because in the experiments the ferroelectric phase transitions are often achieved by the quasistatic heating or cooling, we conjetured that they are stationary states processes [8]. The results on the motion of interface in ferroelectrics and antiferroelectrics support our opinion [10-12]. From Figure 2, we may find that the motion of interface is jerky especially when the average velocity *va* is small. A sequence of segments of time corresponding to the states of rest may be found. The experimental results about other materials such as PbTiO3 are alike [11]. This reveals that in these segments of time ( i.e. characteristic time of phase transition ) the stationary distributions of temperature, heat flux, stress, etc. may be established. Otherwise, if the motion of interface is continuous and smooth, with the unceasing moving of interface ( where the temperature is *TC* ) to the inner part, the local temperature of outer part must change to keep the temperature gradient ∇*T* of this region unchanged for it is determined by

*diff* = −*κ*∇*T* , where *l* is the latent heat ( per unit mass ), *ρ* is the mass density, *v* ′

heat conduction, *κ* is the thermal conductivity ( and maybe a tensor. ) Then, the states are not

is the

*diff* is the diffusion of heat, i.e.

was demonstrated reliable with the aid of non-equilibrium variational principles.

1, d*T* and d*D* satisfy the following equation

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In this case, the possible displacements d*G* ′

for certain *T* . Then, the metastable states are excluded. Thus, the thermal hysteresis does not come into being. The corollary conflicts with the fact sharply. This reveals that the first-order ferroelectric phase transition processes must not be reversible at all so as not to be dealt with by using the equilibrium thermodynamics.

How can this difficulty be overcome? An expedient measure adopted by Devonshire is that the metastable states are considered. However, do they really exist?

Because the metastable states are not the equilibrium ones, the relevant thermodynamic variables or functions should be dependent on the time *t*. In addition, the metastable states are close to equilibrium, so the heterogeneity of system can be neglected. Here, the elastic Gibbs energy *G* ′ 1 should be

$$\mathbf{G}'\_1 = \mathbf{g}\_2 \left( T\_\prime D\_\prime t \right) \tag{8}$$

For the same reason as was mentioned above, Equation (8) can be regarded as a rheonomic constraint on the generalized displacements *G* ′ 1, *T* , *D*

$$\mathcal{G}\_2\left(\mathbf{G}\_1', T, D, t\right) = \mathbf{G}\_1' - \mathbf{g}\_2\left(T, D, t\right) = 0\tag{9}$$

In this case, the possible displacements d*G* ′ 1, d*T* and d*D* satisfy the following equation

$$\mathbf{d}G\_1' - \frac{\partial \mathbf{g}\_2}{\partial T} \mathbf{d}T - \frac{\partial \mathbf{g}\_2}{\partial D} \mathbf{d}D - \frac{\partial \mathbf{g}\_2}{\partial t} \mathbf{d}t = \mathbf{0} \tag{10}$$

Comparing Equation (3) with Equation (10), we may find that the possible displacements here are not the same as those in the former case which characterize the metastable states for they satisfy the different constraint equations, respectively. ( In the latter case, the possible dis‐ placements are time-dependent, whereas in the former case they are not. ) Yet, the integral of possible displacement d*D* is the possible electric displacement in every case. The possible electric displacements which characterize one certain metastable state vary with the cases. A self-contradiction arises. So the metastable states can not come into being.

What are the real states among a phase transition process? In fact, both the evolution with time and the spatial heterogeneity need to be considered when the system is out of equilibrium [6-9]. Just as what will be shown in Section 2.3, the real states should be the stationary ones, which do not vary with the time but may be not metastable.

### **2.3. Real path: Existence of stationary states**

where *h* is a binary function of the variables *D* and *T* . It can be determined by the principle

**Figure 1.** The relation between elastic Gibbs energy *G*1 and zero field electric displacement *D* belongs to the ferroelec‐ trics at various temperature, which undergoes a first-order phase transition [5]. *T*<sup>0</sup> :the lowest temperature at which the paraelectric phase may exist, i.e. the Curie-Weiss temperature; *TC* :the phase transition temperature, i.e. the Curie temperature; *T*1 : the highest temperature at which the ferroelectric phase may exist; *T*<sup>2</sup> : the highest temperature at

for certain *T* . Then, the metastable states are excluded. Thus, the thermal hysteresis does not come into being. The corollary conflicts with the fact sharply. This reveals that the first-order ferroelectric phase transition processes must not be reversible at all so as not to be dealt with

How can this difficulty be overcome? An expedient measure adopted by Devonshire is that

Because the metastable states are not the equilibrium ones, the relevant thermodynamic variables or functions should be dependent on the time *t*. In addition, the metastable states are close to equilibrium, so the heterogeneity of system can be neglected. Here, the elastic Gibbs

For the same reason as was mentioned above, Equation (8) can be regarded as a rheonomic

1, *T* , *D*

the metastable states are considered. However, do they really exist?

which the ferroelectric phase may be induced by the external electric field.

*G*<sup>1</sup> = min (7)

( ) 1 2 *G g TDt* ¢ = , , (8)

of minimum energy

28 Advances in Ferroelectrics

energy *G* ′

by using the equilibrium thermodynamics.

constraint on the generalized displacements *G* ′

1 should be

The real path of a first-order ferroelectric phase transition is believed by us to consist of a series of stationary states. At first, this was conjectured according to the experimental results, then was demonstrated reliable with the aid of non-equilibrium variational principles.

Because in the experiments the ferroelectric phase transitions are often achieved by the quasistatic heating or cooling, we conjetured that they are stationary states processes [8]. The results on the motion of interface in ferroelectrics and antiferroelectrics support our opinion [10-12]. From Figure 2, we may find that the motion of interface is jerky especially when the average velocity *va* is small. A sequence of segments of time corresponding to the states of rest may be found. The experimental results about other materials such as PbTiO3 are alike [11]. This reveals that in these segments of time ( i.e. characteristic time of phase transition ) the stationary distributions of temperature, heat flux, stress, etc. may be established. Otherwise, if the motion of interface is continuous and smooth, with the unceasing moving of interface ( where the temperature is *TC* ) to the inner part, the local temperature of outer part must change to keep the temperature gradient ∇*T* of this region unchanged for it is determined by ±*lρv* ′ = *Jq diff* = −*κ*∇*T* , where *l* is the latent heat ( per unit mass ), *ρ* is the mass density, *v* ′ is the velocity of interface ( where the phase transition is occurring ), *Jq diff* is the diffusion of heat, i.e. heat conduction, *κ* is the thermal conductivity ( and maybe a tensor. ) Then, the states are not stationary.

**Figure 2.** The position of phase boundary as a function of time for NaNbO3 single crystal for the various values of aver‐ age velocity *va* (the values in μm/s given against the curves) [12].

The non-equilibrium variational principles are just the analogue and generalization of the variational principles in analytical dynamics. The principle of least dissipation of energy, the Gauss's principle of least constraint and the Hamiltonian principle etc., in non-equilibrium thermodynamics play the fundamental roles as those in analytical dynamics. They describe the characteristics of stationary states or determine the real path of a non-equilibrium process.

For the basic characteristics of non-equilibrium processes is the dissipation of energy, the dissipation function *φ* is defined as

$$
\varphi = \sigma\_s - \pi \tag{11}
$$

( ) ( )

 x

*i i* x

(1)

the external power supply exists, similarly the evolution of deviation *ξ<sup>i</sup>*

x

( ) ( )

 x

*i i t e*

. If the coefficients *Vi*

0

where *ξ<sup>i</sup>*

where *Vi*

*χEi* (1)

thermodynamic force *χTi*

equilibrium stationary state *ξ<sup>i</sup>*

the system choose a real path.

phase transition is occurring [13].

(0) is the initial value of *ξ<sup>i</sup>*

variation in the dissipative force *χDi*

(1)

variable from a given non-equilibrium stationary state *ξ<sup>i</sup>*

Equation (12) define the real path with the addition that *Ri*

2 0

extensive pseudo-thermodynamic variable from a given non-equilibrium stationary state *ξ*˙*<sup>i</sup>*

2( )


*i i i S V <sup>t</sup> <sup>R</sup>*

is the coefficient between the linear variation in the force related to the external power

, *Ri*

and the deviation of the extensive pseudo-thermodynamic variable from a given non-

**Figure 3.** Three types of regions and their interfaces in the ferroelectric-paraelectric system in which a first-order

*i i S t R*

*t e* <sup>=</sup> (12)

Advances in Thermodynamics of Ferroelectric Phase Transitions

(*t*), *Si* is the coefficient between the linear variation in the

(*ξ*˙) and the time-derivative of the deviation of the

<sup>=</sup> (13)

assume the suitable values *Vi*

is the coefficient between the linear

should be a suitable value *Ri*

can be obtained

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.

31

\* . If

\*\* , *Ri* \*\* ,

(*ξ*) and the deviation of the extensive pseudo-thermodynamic

, *Ri*

where *σs* is the rate of local entropy production and *π* is the external power supply ( per unit volume and temperature ). After the rather lengthy deducing and utilizing the thermodynamic Gauss's principle of least constraint which makes the system choose a real path [13], the evolution with time *t* of the deviation from a given non-equilibrium stationary state, *ξ*(*t*)={*ξ<sup>i</sup>* (*t*)}, was obtained in two cases. If no external power supply,

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$$
\xi\_i(t) = \xi\_i(0)e^{\frac{2S\_i}{R\_i}t} \tag{12}
$$

where *ξ<sup>i</sup>* (0) is the initial value of *ξ<sup>i</sup>* (*t*), *Si* is the coefficient between the linear variation in the thermodynamic force *χTi* (1) (*ξ*) and the deviation of the extensive pseudo-thermodynamic variable from a given non-equilibrium stationary state *ξ<sup>i</sup>* , *Ri* is the coefficient between the linear variation in the dissipative force *χDi* (1) (*ξ*˙) and the time-derivative of the deviation of the extensive pseudo-thermodynamic variable from a given non-equilibrium stationary state *ξ*˙*<sup>i</sup>* . Equation (12) define the real path with the addition that *Ri* should be a suitable value *Ri* \* . If the external power supply exists, similarly the evolution of deviation *ξ<sup>i</sup>* can be obtained

$$\xi\_i(t) = \xi\_i(0)e^{\frac{2\left(S\_i - V\_i\right)}{R\_i}t} \tag{13}$$

where *Vi* is the coefficient between the linear variation in the force related to the external power *χEi* (1) and the deviation of the extensive pseudo-thermodynamic variable from a given nonequilibrium stationary state *ξ<sup>i</sup>* . If the coefficients *Vi* , *Ri* assume the suitable values *Vi* \*\* , *Ri* \*\* , the system choose a real path.

**Figure 2.** The position of phase boundary as a function of time for NaNbO3 single crystal for the various values of aver‐

The non-equilibrium variational principles are just the analogue and generalization of the variational principles in analytical dynamics. The principle of least dissipation of energy, the Gauss's principle of least constraint and the Hamiltonian principle etc., in non-equilibrium thermodynamics play the fundamental roles as those in analytical dynamics. They describe the characteristics of stationary states or determine the real path of a non-equilibrium process. For the basic characteristics of non-equilibrium processes is the dissipation of energy, the

> *s* js

(*t*)}, was obtained in two cases. If no external power supply,

 p

where *σs* is the rate of local entropy production and *π* is the external power supply ( per unit volume and temperature ). After the rather lengthy deducing and utilizing the thermodynamic Gauss's principle of least constraint which makes the system choose a real path [13], the evolution with time *t* of the deviation from a given non-equilibrium stationary state,

= - (11)

age velocity *va* (the values in μm/s given against the curves) [12].

dissipation function *φ* is defined as

30 Advances in Ferroelectrics

*ξ*(*t*)={*ξ<sup>i</sup>*

**Figure 3.** Three types of regions and their interfaces in the ferroelectric-paraelectric system in which a first-order phase transition is occurring [13].

Both the real paths in the two cases reveal that the deviations decrease exponentially when the system regresses to the stationary states. Stationary states are a king of attractors to nonequilibrium states. The decreases are steep. So the regressions are quick. It should be noted that we are interested in calculating the change in the generalized displacements during a macroscopically small time interval. In other words, we are concerned with the determination of the path of an irreversible process which is described in terms of a finite difference equation. In the limit as the time interval is allowed to approach zero, we obtain the variational equation of thermodynamic path.

dd d d*i i*

*Ts u* = -× - *E D* å

within a random small volume, respectively; *s*, *u*, *μi*

*s*

we have

where *Ju*, *JP*, *Jni*

Then we deduce the following

transport

*i*

where *T* , *E*, *D* is the temperature, the electric field intensity and the electric displacement

energy density, the chemical potential and the molar quantity density in the small volume, respectively. And there, it was assumed that the crystal system is mechanically-free ( i.e. no

force is exerted on it ). Differentiating Equation (15) and using the following relations

<sup>0</sup> *<sup>u</sup>*

( ) <sup>0</sup>

<sup>0</sup> *<sup>i</sup> <sup>n</sup>*

1 *<sup>i</sup>*

*tT T T* ¶ =- Ñ× - × + Ñ× ¶ å

> *uq P* =+ + j

> > *E* = -Ñj

where *Jq* is the heat flux, *φ* is the electrical potential and satisfies

*i*

*i i*

 m

*u P ni*

polarization. *Ju* should consist of three parts: the energy flux caused by the heat conduction, the energy flux caused by the charge transport and the energy flux caused by the matter

m

is the energy flux, the polarization current and the matter flux; *P* is the

+Ñ× =

¶ +Ñ× =

*t*

*t t* ¶ ¶ + = = ¶ ¶ e

> *t* ¶

*D E P*

m

, *ni*

*P*

¶ *<sup>u</sup> <sup>J</sup>* (16)

¶ *ni <sup>J</sup>* (18)

*<sup>E</sup> JJ J* (19)

å *ni JJ J J* (20)

(21)

*<sup>n</sup>* (15)

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*J* (17)

is the entropy density, the internal

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33

So, if the irreversible process is not quick enough, it can be regarded as the one that consists of a series of stationary states. The ferroelectric phase transitions are usually achieved by the quasi-static heating or cooling in the experiments. So, the processes are not quick enough to make the states deviate from the corresponding stationary states in all the time. In Figure 3, three types of regions and their interfaces are marked with I, II, III, 1, 2 respectively. The region III where the phase transition will occur is in equilibrium and has no dissipation. In the region I where the phase transition has occurred, there is no external power supply, and in the region II ( i.e. the paraelectric-ferroelectric interface as a region with finite thickness instead of a geometrical plane ) where the phase transition is occurring, there exists the external power supply, i.e. the latent heat ( per unit volume and temperature ). According to the former analysis in two cases, we may conclude that they are in stationary states except for the very narrow intervals of time after the sudden lose of phase stability.

### **3. Thermo-electric coupling**

In the paraelectric-ferroelectric interface dynamics induced by the latent heat transfer [6,7], the normal velocity of interface *vn* was obtained

$$\left. \psi\_n = \frac{1}{l\rho} \right[k\_{fer} \left( \nabla T \right)\_{fer} - k\_{par} \left( \nabla T \right)\_{pur} \right] \cdot \mathbf{n} \tag{14}$$

where *l* is the latent heat ( per unit mass ), *ρ* is the density of metastable phase (paraelectric phase ), *k fer* is the thermal conductivity coefficient of ferroelectric phase, *k par* is the thermal conductivity coefficient of paraelectric phase, (∇*T* ) *fer* is the temperature gradient in ferro‐ electric phase part, (∇*T* ) *par* is the temperature gradient in paraelectric phase part, *n* is the unit vector in normal direction and directs from the ferroelectric phase part to the paraelectric phase part. The temperature gradients can be studied from the point of view that a ferroelectric phase transition is a stationary, thermo-electric coupled transport process [8].

### **3.1. Local entropy production**

In the thermo-electric coupling case, the Gibbs equation was given as the following [8]

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$$\mathbf{Tds} = \mathbf{d}u - \mathbf{E} \cdot \mathbf{d}\mathbf{D} - \sum\_{i} \mu\_{i} \mathbf{d}n\_{i} \tag{15}$$

where *T* , *E*, *D* is the temperature, the electric field intensity and the electric displacement within a random small volume, respectively; *s*, *u*, *μi* , *ni* is the entropy density, the internal energy density, the chemical potential and the molar quantity density in the small volume, respectively. And there, it was assumed that the crystal system is mechanically-free ( i.e. no force is exerted on it ). Differentiating Equation (15) and using the following relations

$$\frac{\partial \boldsymbol{\mu}}{\partial t} + \nabla \cdot \mathbf{J}\_{\boldsymbol{u}} = \mathbf{0} \tag{16}$$

$$\frac{\partial \mathbf{D}}{\partial t} = \frac{\partial \left(\varepsilon\_0 \mathbf{E} + \mathbf{P}\right)}{\partial t} = \mathbf{J}\_P \tag{17}$$

$$\frac{\partial \mathbf{u}\_i}{\partial t} + \nabla \cdot \mathbf{J}\_{ni} = \mathbf{0} \tag{18}$$

we have

Both the real paths in the two cases reveal that the deviations decrease exponentially when the system regresses to the stationary states. Stationary states are a king of attractors to nonequilibrium states. The decreases are steep. So the regressions are quick. It should be noted that we are interested in calculating the change in the generalized displacements during a macroscopically small time interval. In other words, we are concerned with the determination of the path of an irreversible process which is described in terms of a finite difference equation. In the limit as the time interval is allowed to approach zero, we obtain the variational equation

So, if the irreversible process is not quick enough, it can be regarded as the one that consists of a series of stationary states. The ferroelectric phase transitions are usually achieved by the quasi-static heating or cooling in the experiments. So, the processes are not quick enough to make the states deviate from the corresponding stationary states in all the time. In Figure 3, three types of regions and their interfaces are marked with I, II, III, 1, 2 respectively. The region III where the phase transition will occur is in equilibrium and has no dissipation. In the region I where the phase transition has occurred, there is no external power supply, and in the region II ( i.e. the paraelectric-ferroelectric interface as a region with finite thickness instead of a geometrical plane ) where the phase transition is occurring, there exists the external power supply, i.e. the latent heat ( per unit volume and temperature ). According to the former analysis in two cases, we may conclude that they are in stationary states except for the very

In the paraelectric-ferroelectric interface dynamics induced by the latent heat transfer [6,7], the

where *l* is the latent heat ( per unit mass ), *ρ* is the density of metastable phase (paraelectric phase ), *k fer* is the thermal conductivity coefficient of ferroelectric phase, *k par* is the thermal conductivity coefficient of paraelectric phase, (∇*T* ) *fer* is the temperature gradient in ferro‐ electric phase part, (∇*T* ) *par* is the temperature gradient in paraelectric phase part, *n* is the unit vector in normal direction and directs from the ferroelectric phase part to the paraelectric phase part. The temperature gradients can be studied from the point of view that a ferroelectric phase

ê ú ë û *<sup>n</sup>* (14)

( ) ( ) <sup>1</sup> *n fer par fer par v kT kT*

é ù = Ñ-Ñ ×

In the thermo-electric coupling case, the Gibbs equation was given as the following [8]

narrow intervals of time after the sudden lose of phase stability.

*l*

r

transition is a stationary, thermo-electric coupled transport process [8].

of thermodynamic path.

32 Advances in Ferroelectrics

**3. Thermo-electric coupling**

**3.1. Local entropy production**

normal velocity of interface *vn* was obtained

$$\frac{\partial \mathbf{\hat{s}}}{\partial t} = -\frac{1}{T} \nabla \cdot \mathbf{J}\_u - \frac{E}{T} \cdot \mathbf{J}\_p + \sum\_i \frac{\mu\_i}{T} \nabla \cdot \mathbf{J}\_{ni} \tag{19}$$

where *Ju*, *JP*, *Jni* is the energy flux, the polarization current and the matter flux; *P* is the polarization. *Ju* should consist of three parts: the energy flux caused by the heat conduction, the energy flux caused by the charge transport and the energy flux caused by the matter transport

$$\mathbf{J}\_u = \mathbf{J}\_q + q\mathbf{J}\_p + \sum\_i \mu\_i \mathbf{J}\_{ni} \tag{20}$$

where *Jq* is the heat flux, *φ* is the electrical potential and satisfies

$$E = -\nabla \varphi \tag{21}$$

Then we deduce the following

$$\frac{\partial \mathbf{\hat{s}}}{\partial t} = -\nabla \cdot \left(\frac{\mathbf{J}\_q + q \mathbf{J}\_P}{T}\right) + \mathbf{J}\_q \cdot \nabla \left(\frac{1}{T}\right) + \mathbf{J}\_p \cdot \nabla \left(\frac{\rho}{T}\right) - \sum\_i \frac{\mathbf{J}\_{ni}}{T} \cdot \nabla \mu\_i \tag{22}$$

If we define a entropy flux *Js* and a rate of local entropy production *σs* as

$$\mathbf{J}\_s = \frac{\mathbf{J}\_q + \rho \mathbf{J}\_p}{T} \tag{23}$$

If the external electric field is not applied, *φe* can be a random constant. There is no harm in

1 *<sup>i</sup>*

According to the crystal structures of ferroelectrics [2], we know the polarization current *JP* originates from the displacement or ordering of ions in ferroelectrics. We may consider it as

Assume the external electric field is not applied. Here are the thermodynamic fluxex *Jq*, *JP* and

1 *T* æ ö = Ñç ÷ è ø

*i T* æ ö = Ñç ÷ è ø j

where *L qq*, *L qP*, *L Pq* and *L PP* are the transport coefficients, which are four second-order

*T*

Because to a first-order ferroelectric phase transition the electric displacement changes suddenly and so does the internal electrical potential, the force *XP* of the region where the phase transition is occurring can be regarded as a large constant roughly in the characteristic

(*i*, *j* =*q*, *P*)

tensors. They should satisfy the generalized Onsager relations [14]

j

*q P J J* (30)

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*X<sup>q</sup>* (31)

*X<sup>P</sup>* (32)

= ×+ × *q qq q qP P J LX LX* (33)

= ×+ × *P Pq q PP P J LXLX* (34)

*PP PP L L* = (35)

*<sup>s</sup> T T* æ ö æ ö = ×Ñ + ×Ñ ç ÷ ç ÷ è ø è ø

letting the constant equal zero. Then the entropy production equals

s

the transport of charges influenced by the internal electric field.

the corresponding thermodynamic forces *Xq*, *XP*

can be expanded linearly with *Xj*

*Ji*

*L qq* = *L qq*

*<sup>T</sup>* , *<sup>L</sup> qP* <sup>=</sup> *<sup>L</sup> Pq*

*T* ,

**3.2. Description of phase transitions and verification of interface dynamics**

$$\boldsymbol{\sigma}\_{s} = \mathbf{J}\_{q} \cdot \nabla \left(\frac{1}{T}\right) + \mathbf{J}\_{P} \cdot \nabla \left(\frac{\boldsymbol{\rho}}{T}\right) - \sum\_{i} \frac{\mathbf{J}\_{ni}}{T} \cdot \nabla \mu\_{i} \tag{24}$$

Equation (22) can be written as

$$\frac{\partial \mathbf{\hat{s}}}{\partial t} + \nabla \cdot \mathbf{J}\_s = \sigma\_s \tag{25}$$

This is the local entropy balance equation. We know, the system is in the crystalline states before and after a phase transition so that there is no diffusion of any kind of particles in the system. So, *Jni* =0. The local entropy production can be reduced as

$$\boldsymbol{\sigma}\_{s} = \mathbf{J}\_{q} \cdot \nabla \left(\frac{1}{T}\right) + \mathbf{J}\_{p} \cdot \nabla \left(\frac{\boldsymbol{\varrho}}{T}\right) \tag{26}$$

We know the existence of ferroics is due to the molecular field. It is an internal field. So we must take it into account. Here, the electric field should be the sum of the external electric field *Ee* and the internal electric field *Ei*

$$E = E\_e + E\_i \tag{27}$$

Correspondingly, there are the external electrical potential *φ<sup>e</sup>* and the internal electrical potential *φ<sup>i</sup>* and they satisfy

$$E\_e = -\nabla \varphi\_e \tag{28}$$

$$E\_i = -\nabla \varphi\_i \tag{29}$$

If the external electric field is not applied, *φe* can be a random constant. There is no harm in letting the constant equal zero. Then the entropy production equals

$$
\sigma\_s = \mathbf{J}\_q \cdot \nabla \left(\frac{1}{T}\right) + \mathbf{J}\_p \cdot \nabla \left(\frac{\varphi\_i}{T}\right) \tag{30}
$$

According to the crystal structures of ferroelectrics [2], we know the polarization current *JP* originates from the displacement or ordering of ions in ferroelectrics. We may consider it as the transport of charges influenced by the internal electric field.

### **3.2. Description of phase transitions and verification of interface dynamics**

Assume the external electric field is not applied. Here are the thermodynamic fluxex *Jq*, *JP* and the corresponding thermodynamic forces *Xq*, *XP*

$$X\_q = \nabla \left(\frac{1}{T}\right) \tag{31}$$

$$X\_P = \nabla \left(\frac{\rho\_i}{T}\right) \tag{32}$$

*Ji* can be expanded linearly with *Xj* (*i*, *j* =*q*, *P*)

1

*T* <sup>+</sup> <sup>=</sup> j*q P*

*s i T TT <sup>i</sup>* æö æö = ×Ñ + ×Ñ - ×Ñ ç÷ ç÷ èø èø <sup>å</sup> j

*s*

j

*q P J J* (26)

*<sup>e</sup>* = + *EE E<sup>i</sup>* (27)

*E<sup>e</sup>* (28)

*E<sup>i</sup>* (29)

s

This is the local entropy balance equation. We know, the system is in the crystalline states before and after a phase transition so that there is no diffusion of any kind of particles in the

We know the existence of ferroics is due to the molecular field. It is an internal field. So we must take it into account. Here, the electric field should be the sum of the external electric field

Correspondingly, there are the external electrical potential *φ<sup>e</sup>* and the internal electrical

*<sup>e</sup>* = -Ñj

*<sup>i</sup>* = -Ñj

*J J*

 *q P ni q P J J J*

*t T T TT* ¶ æ ö <sup>+</sup> æö æö = -Ñ ×ç ÷ + ×Ñ + ×Ñ - ×Ñ ç÷ ç÷ ¶ èø èø è ø <sup>å</sup>

If we define a entropy flux *Js* and a rate of local entropy production *σs* as

*s*

1

*q P*

*s t*

¶

system. So, *Jni* =0. The local entropy production can be reduced as

s

¶ +Ñ× =

1 *<sup>s</sup> T T* æö æö = ×Ñ + ×Ñ ç÷ ç÷ èø èø

s

j

*s*

34 Advances in Ferroelectrics

Equation (22) can be written as

*Ee* and the internal electric field *Ei*

and they satisfy

potential *φ<sup>i</sup>*

*i*

m

*i*

*J J* (22)

*J* (23)

*<sup>s</sup> J* (25)

 m*ni*

*J J* (24)

*J*

j

$$\mathbf{J}\_q = \mathbf{L}\_{qq} \cdot \mathbf{X}\_q + \mathbf{L}\_{qP} \cdot \mathbf{X}\_P \tag{33}$$

$$J\_P = L\_{pq} \cdot X\_q + L\_{pp} \cdot X\_p \tag{34}$$

where *L qq*, *L qP*, *L Pq* and *L PP* are the transport coefficients, which are four second-order tensors. They should satisfy the generalized Onsager relations [14] *L qq* = *L qq <sup>T</sup>* , *<sup>L</sup> qP* <sup>=</sup> *<sup>L</sup> Pq T* ,

$$\mathbf{L}\_{\rm PP} = \mathbf{L}\_{\rm PP}^T \tag{35}$$

Because to a first-order ferroelectric phase transition the electric displacement changes suddenly and so does the internal electrical potential, the force *XP* of the region where the phase transition is occurring can be regarded as a large constant roughly in the characteristic times of phase transition ( i.e. the times in which the interface keeps rest ). For the ferroelectric phase transition may be regarded as a stationary states process, the principle of minimum entropy production must be satisfied [15].

According to Equations (30)-(35), we have

$$
\sigma\_s = \mathbf{L}\_{qq} : \mathbf{X}\_q \mathbf{X}\_q + 2\mathbf{L}\_{qP} : \mathbf{X}\_q \mathbf{X}\_P + \mathbf{L}\_{pp} : \mathbf{X}\_P \mathbf{X}\_P \tag{36}
$$

( ) *fer fer*

Ñ ×

In order to compare it with the experiments, we make use of the following values which are

value of the velocity of the interface's fast motion, which has been measured in the experiments, is 0.5mm/s [11]. According to Equation (39), we calculate the corresponding temperature gradient to be 57.35K/cm. However, in [19] it is reported that the experimental temperature gradient varies from 1.5 to 3.5K/mm while the experimental velocity of the interface's motion varies from 732 to 843μm/s. Considering the model is rather rough, we may conclude that the

In the experiments, the latent heat and the spotaneous polarization are measured often for first-order ferroelectric phase transitions. So in the follows, we will establish the relation between latent heat and spontaneous polarization in the realm of non-equilibrium thermody‐

All the quantities of the region where the phase transition has occurred are marked with the superscript "I"; all the quantities of the region where the phase transition is occurring are marked with the superscript "II"; and all the quantities of the region where the phase transition will occur are marked with the superscript "III". Let's consider the heating processes of phase

where we have ignored the difference between the mass density of ferroelectric phase and that of paraelectric phase ( almost the same ) and denote them as *ρ*, *va* is the average velocity of

> 0 *II II II II q qq q qP* =× +× = *II*

*II II II II II*

The heat which is transferred to the region where the phase transition is occurring is absorbed as the latent heat because the pure heat conduction and the heat conduction induced by the

r

*<sup>P</sup> J LX LX* (41)

*P Pq q PP P J LX LX* =×+× (42)

*<sup>n</sup>* (39)

Advances in Thermodynamics of Ferroelectric Phase Transitions

(40)

erg/cmsK [18]. The

http://dx.doi.org/10.5772/52089

37

*l*

r

*k T*

*n*

=

about PbTiO3 crystal: *ρ* =7.1g/cm3 [16], *l* =900cal/mol [17], *k fer* =8.8 × 105

**3.3. Relation between latent heat and spontaneous polarization**

transition firstly. In the region where the phase transition has occurred,

interface. In the region where the phase transition is occurring,

thermo-electric coupling cancel out each other. So,

*I para I para I para q qq q <sup>a</sup> <sup>l</sup>* - -- *J LX v* =× =

theory coincides with the experiments.

namics.

*v*

If there is no any restriction on *Xq* and *XP*, according to the conditions on which the entropy production is a minimum

$$\left(\frac{\partial \sigma\_s}{\partial \mathbf{X}\_q}\right)\_{\mathbf{X}\_p} = \mathbf{2L}\_{qq} \cdot \mathbf{X}\_q + \mathbf{2L}\_{qp} \cdot \mathbf{X}\_p = \mathbf{2J}\_q = \mathbf{0} \tag{37}$$

$$\left(\frac{\partial \sigma\_s}{\partial \mathbf{X}\_P}\right)\_{\mathbf{X}\_q} = \mathbf{2L}\_{pq} \cdot \mathbf{X}\_q + \mathbf{2L}\_{pp} \cdot \mathbf{X}\_P = \mathbf{2J}\_P = \mathbf{0} \tag{38}$$

We know the stationary states are equilibrium ones actually. If we let *Xq* ( or *XP* ) be a constant, according to Equation (38) ( or (37) ) we know *JP* ( or *Jq* ) which is corresponded to another force *XP* ( or *Xq* ) should be zero.

Then, a first-order ferroelectric phase transition can be described by the second paradigm. Since the force *XP* of the region where the phase transition is occurring is a large constant, the flux *Jq* of this region should be zero ( but *JP* ≠0 ). This states clearly that the pure heat conduction and the heat conduction induced by the thermo-electric coupling cancel out each other so as to release or absorb the latent heat. It is certain that the latent heat passes through the region where the phase transition has occurred ( at the outside of the region where the phase transition is occurring ) and exchanges itself with the thermal bath. Accompanied with the change of the surface's temperature and the unceasing jerky moving of the region where the phase transition is occurring, a constant temperature gradient is kept in the region where the phase transition has occurred, i.e. the force *Xq* is a constant. So, the flux *JP* =0 ( but *Jq* ≠0 ). This states clearly that the electric displacement of the region where the phase transition has occurred will not change but keep the value at Curie temperature or zero until the phase transition finishes. Differently, the region where the phase transition will occur should be described by the first paradigm for there is no restriction on the two forces *XP*, *Xq*. The states of this region are equilibrium ones. So the temperature gradient ∇*T* should be zero.

Considering that (∇*T* ) *par* ≈0 for the region where the phase transition will occur ( i.e. the paraelectric phase part ) can be regarded as an equilibrium system, we modify Equation (14) as Advances in Thermodynamics of Ferroelectric Phase Transitions http://dx.doi.org/10.5772/52089 37

$$\upsilon\_n = \frac{k\_{fer} \left(\nabla T\right)\_{fer} \cdot n}{l \,\rho} \tag{39}$$

In order to compare it with the experiments, we make use of the following values which are about PbTiO3 crystal: *ρ* =7.1g/cm3 [16], *l* =900cal/mol [17], *k fer* =8.8 × 105 erg/cmsK [18]. The value of the velocity of the interface's fast motion, which has been measured in the experiments, is 0.5mm/s [11]. According to Equation (39), we calculate the corresponding temperature gradient to be 57.35K/cm. However, in [19] it is reported that the experimental temperature gradient varies from 1.5 to 3.5K/mm while the experimental velocity of the interface's motion varies from 732 to 843μm/s. Considering the model is rather rough, we may conclude that the theory coincides with the experiments.

#### **3.3. Relation between latent heat and spontaneous polarization**

times of phase transition ( i.e. the times in which the interface keeps rest ). For the ferroelectric phase transition may be regarded as a stationary states process, the principle of minimum

If there is no any restriction on *Xq* and *XP*, according to the conditions on which the entropy

*qq q qP P q*

*Pq q PP P P*

*2L X 2L X 2J*

We know the stationary states are equilibrium ones actually. If we let *Xq* ( or *XP* ) be a constant, according to Equation (38) ( or (37) ) we know *JP* ( or *Jq* ) which is corresponded to another

Then, a first-order ferroelectric phase transition can be described by the second paradigm. Since the force *XP* of the region where the phase transition is occurring is a large constant, the flux *Jq* of this region should be zero ( but *JP* ≠0 ). This states clearly that the pure heat conduction and the heat conduction induced by the thermo-electric coupling cancel out each other so as to release or absorb the latent heat. It is certain that the latent heat passes through the region where the phase transition has occurred ( at the outside of the region where the phase transition is occurring ) and exchanges itself with the thermal bath. Accompanied with the change of the surface's temperature and the unceasing jerky moving of the region where the phase transition is occurring, a constant temperature gradient is kept in the region where the phase transition has occurred, i.e. the force *Xq* is a constant. So, the flux *JP* =0 ( but *Jq* ≠0 ). This states clearly that the electric displacement of the region where the phase transition has occurred will not change but keep the value at Curie temperature or zero until the phase transition finishes. Differently, the region where the phase transition will occur should be described by the first paradigm for there is no restriction on the two forces *XP*, *Xq*. The states of this region are

Considering that (∇*T* ) *par* ≈0 for the region where the phase transition will occur ( i.e. the paraelectric phase part ) can be regarded as an equilibrium system, we modify Equation (14) as

*2L X 2L X 2J*

<sup>0</sup> *<sup>s</sup>* æ ö ¶ ç ÷ = ×+ × = = ç ÷ ¶è ø

<sup>0</sup> *<sup>s</sup>* æ ö ¶ ç ÷ = ×+ × = = ç ÷ ¶è ø

=+ + *qq q q qP q P PP P P L X X 2L X X L X X* (36)

*<sup>X</sup>* (37)

*<sup>X</sup>* (38)

: :: *<sup>s</sup>*

entropy production must be satisfied [15].

According to Equations (30)-(35), we have

s

s

s

*P*

*q*

equilibrium ones. So the temperature gradient ∇*T* should be zero.

*P X*

*<sup>q</sup> <sup>X</sup>*

production is a minimum

36 Advances in Ferroelectrics

force *XP* ( or *Xq* ) should be zero.

In the experiments, the latent heat and the spotaneous polarization are measured often for first-order ferroelectric phase transitions. So in the follows, we will establish the relation between latent heat and spontaneous polarization in the realm of non-equilibrium thermody‐ namics.

All the quantities of the region where the phase transition has occurred are marked with the superscript "I"; all the quantities of the region where the phase transition is occurring are marked with the superscript "II"; and all the quantities of the region where the phase transition will occur are marked with the superscript "III". Let's consider the heating processes of phase transition firstly. In the region where the phase transition has occurred,

$$\mathbf{J}\_q^{I-para} = \mathbf{L}\_{qq}^{I-para} \cdot \mathbf{X}\_q^{I-para} = l \, \rho \mathbf{v}\_a \tag{40}$$

where we have ignored the difference between the mass density of ferroelectric phase and that of paraelectric phase ( almost the same ) and denote them as *ρ*, *va* is the average velocity of interface. In the region where the phase transition is occurring,

$$\mathbf{J}\_{\boldsymbol{q}}^{\rm II} = \mathbf{L}\_{\boldsymbol{q}q}^{\rm II} \cdot \mathbf{X}\_{\boldsymbol{q}}^{\rm II} + \mathbf{L}\_{\boldsymbol{q}p}^{\rm II} \cdot \mathbf{X}\_{\boldsymbol{P}}^{\rm II} = \mathbf{0} \tag{41}$$

$$\mathbf{J}\_P^{\text{II}} = \mathbf{L}\_{pq}^{\text{II}} \cdot \mathbf{X}\_q^{\text{II}} + \mathbf{L}\_{pp}^{\text{II}} \cdot \mathbf{X}\_p^{\text{II}} \tag{42}$$

The heat which is transferred to the region where the phase transition is occurring is absorbed as the latent heat because the pure heat conduction and the heat conduction induced by the thermo-electric coupling cancel out each other. So,

$$\mathbf{l}\,\rho\mathbf{v}\_a = \mathbf{L}\_{qq}^{\mathrm{II}} \cdot \mathbf{X}\_q^{\mathrm{II}} \tag{43}$$

stressed [9], there is some difference between *P spon*( = *P spon*−*III* ) and the equilibrium spontane‐

The comprehensive thermo-electro-mechanical coupling may be found in ferroelectric phase transition processes. Because there exists not only the change of polarization but also the changes of the system's volume and shape when a ferroelectric phase transition occurs in it, the mechanics can not be ignored even if it is mechanically-free, i.e. no external force is exerted on it. To a first-order ferroelectric phase transition, it occurs at the surface layer of system firstly,

Since one aspect of the nature of ferroelectric phase transitions is the thermo-electro-mechan‐ ical coupling, we take the mechanics into account on the basis of Section 3, where only the thermo-electric coupling has been considered. This may lead to a complete description in the

For a continuum, the momentum equation in differential form can be written as

<sup>2</sup> *<sup>v</sup>* <sup>2</sup>

d d *k t*

Ñ× + = *σfa* r

 r

where *σ*, *f* , *a*, *ρ* is the stress, the volume force exerted on unit mass, the acceleration and the

we can deduce the following balance equation of mechanical energy basing on Equations

(51)

Advances in Thermodynamics of Ferroelectric Phase Transitions

http://dx.doi.org/10.5772/52089

39

be the local kinetic energy density (per unit mass), with

= × *v a* (52)

∇ ⋅ (*v* ⋅*σ*)=*v* ⋅ (∇ ⋅*σ*) + (∇*v*):*σ* (53)

*σ v σd* : : (Ñ =) (54)

ous polarization without the affects of stress *P spon*′ because of the piezoelectric effect.

**4. Thermo-electro-mechanical coupling**

sense of continuum physics.

**4.1. Deformation mechanics**

mass density, respectively. Let *<sup>k</sup>* <sup>=</sup> <sup>1</sup>

where *t* is the time. In terms of

*v* is the velocity. Then

(51) (52)

then in the inner part. So, the stress may be found in the system.

According to Eqations (41)-(43), we work out

$$\mathbf{J}\_{P}^{\mathrm{II}} = \mathbf{l} \rho \left[ \mathbf{L}\_{pq}^{\mathrm{II}} \cdot \left( \mathbf{L}\_{qq}^{\mathrm{II}} \right)^{-1} - \mathbf{L}\_{PP}^{\mathrm{II}} \cdot \left( \mathbf{L}\_{qP}^{\mathrm{II}} \right)^{-1} \right] \cdot \mathbf{v}\_{a} \tag{44}$$

where the superscript "-1" means reverse. While

$$\int \mathbf{J}\_P^{II} \mathbf{d}t = -\mathbf{D}^{\text{spom}} = \varepsilon\_0 \nabla \varphi\_i^{III} - \mathbf{P}^{\text{spom}-III} = -\mathbf{P}^{\text{spom}-III} \tag{45}$$

where we utilized the boundary condition of *D* and considered the region where the phase transition will occur is in equilibrium, the superscript "spon" means spontaneous. So,

$$\mathbf{P}^{\text{spon}} = \mathbf{P}^{\text{spon}-\text{III}} = -\int \mathbf{J}\_P^{\text{II}} \mathbf{d}t \tag{46}$$

The relation between latent heat and spontaneous polarization are obtained. In the cooling processes of phase transition,

$$-l\rho \mathbf{v}\_a = \mathbf{L}\_{qq}^{1-ferr} \cdot \mathbf{X}\_q^{1-ferr} + \mathbf{L}\_{qP}^{1-ferr} \cdot \mathbf{X}\_P^{1-ferr} = \mathbf{L}\_{qq}^{II} \cdot \mathbf{X}\_q^{II} \tag{47}$$

$$\int \mathbf{J}\_p^{II'} \mathbf{d}t = \mathbf{D}^{spom-I} = -\varepsilon\_0 \nabla \phi\_i^I + \mathbf{P}^{spom-I} \tag{48}$$

Repeating the above steps, we obtain

$$\mathbf{P}^{\text{spon}-I} = \int \mathbf{J}\_P^{II'} \mathbf{d}t + \varepsilon\_0 \nabla \varphi\_i^I \tag{49}$$

With

$$\mathbf{J}\_{P}^{\text{II}'} = -\mathbf{l}\rho \left[ \mathbf{L}\_{p\_{\text{P}}}^{\text{II}} \cdot \left( \mathbf{L}\_{q\text{q}}^{\text{II}} \right)^{-1} - \mathbf{L}\_{\text{PP}}^{\text{II}} \cdot \left( \mathbf{L}\_{q\text{P}}^{\text{II}} \right)^{-1} \right] \cdot \mathbf{v}\_{a} = -\mathbf{J}\_{P}^{\text{II}} \tag{50}$$

Then we find that *P spon*−*III* ( serves as the equilibrium polarization ) is not equal *P spon*−*<sup>I</sup>* ( serves as the non-equilibrium polarization ). For the region where the phase transition will occur is stressed [9], there is some difference between *P spon*( = *P spon*−*III* ) and the equilibrium spontane‐ ous polarization without the affects of stress *P spon*′ because of the piezoelectric effect.

### **4. Thermo-electro-mechanical coupling**

The comprehensive thermo-electro-mechanical coupling may be found in ferroelectric phase transition processes. Because there exists not only the change of polarization but also the changes of the system's volume and shape when a ferroelectric phase transition occurs in it, the mechanics can not be ignored even if it is mechanically-free, i.e. no external force is exerted on it. To a first-order ferroelectric phase transition, it occurs at the surface layer of system firstly, then in the inner part. So, the stress may be found in the system.

Since one aspect of the nature of ferroelectric phase transitions is the thermo-electro-mechan‐ ical coupling, we take the mechanics into account on the basis of Section 3, where only the thermo-electric coupling has been considered. This may lead to a complete description in the sense of continuum physics.

### **4.1. Deformation mechanics**

*II II a qq q l*

*v LX* = × (43)

(45)

(48)

(49)

( serves

(44)

*<sup>P</sup> <sup>t</sup>* - = = -ò *PP J* (46)

*v L X L X LX* (47)

(50)

( serves as the equilibrium polarization ) is not equal *P spon*−*<sup>I</sup>*

r

( ) ( ) 1 1 *II II II II II P Pq qq PP qP a <sup>l</sup>* - - é ù = × -× × ê ú ë û *J LL LL v*

<sup>0</sup> <sup>d</sup>*II spon III spon III spon III P i <sup>t</sup>* - - =- = Ñ - =- ò *JD P P* e j

transition will occur is in equilibrium, the superscript "spon" means spontaneous. So,

d *spon spon III II*

*a qq q qP P qq q <sup>l</sup>* ---- - = × + × =×

<sup>0</sup> <sup>d</sup>*II spon I <sup>I</sup> spon I P i <sup>t</sup>* ¢ - - = =- Ñ + ò *JD P* e j

> <sup>0</sup> <sup>d</sup> *spon I II <sup>I</sup> P i <sup>t</sup>* - ¢ = +Ñ ò *P J*

( ) ( ) 1 1 *II II II II II II <sup>P</sup> Pq qq PP qP a P <sup>l</sup>* - - ¢ é ù =- × - × × =- ê ú ë û *J LL LL v J*

as the non-equilibrium polarization ). For the region where the phase transition will occur is

e j

where we utilized the boundary condition of *D* and considered the region where the phase

The relation between latent heat and spontaneous polarization are obtained. In the cooling

*I ferr I ferr I ferr I ferr II II*

According to Eqations (41)-(43), we work out

38 Advances in Ferroelectrics

where the superscript "-1" means reverse. While

processes of phase transition,

r

Repeating the above steps, we obtain

Then we find that *P spon*−*III*

r

With

r

For a continuum, the momentum equation in differential form can be written as

$$
\nabla \cdot \boldsymbol{\sigma} + \rho \, \boldsymbol{f} = \rho \mathbf{a} \tag{51}
$$

where *σ*, *f* , *a*, *ρ* is the stress, the volume force exerted on unit mass, the acceleration and the mass density, respectively. Let *<sup>k</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> *<sup>v</sup>* <sup>2</sup> be the local kinetic energy density (per unit mass), with *v* is the velocity. Then

$$\frac{\mathbf{d}k}{\mathbf{d}t} = \mathbf{v} \cdot \mathbf{a} \tag{52}$$

where *t* is the time. In terms of

$$\nabla \cdot (v \cdot \sigma) = v \cdot (\nabla \cdot \sigma) + (\nabla v) \colon \sigma \tag{53}$$

$$
\sigma : \left( \nabla \nu \right) = \sigma : d \tag{54}
$$

we can deduce the following balance equation of mechanical energy basing on Equations (51) (52)

$$
\rho \frac{\mathbf{d}k}{\mathbf{d}t} - \nabla \cdot (\mathbf{v} \cdot \boldsymbol{\sigma}) = \rho v \cdot f - \boldsymbol{\sigma} \cdot \mathbf{d} \tag{55}
$$

where *e* is the total energy with *e* =*u* + *k*. We know, the system is in the crystalline states before and after a phase transition so that there is no diffusion of any kind of particles in the system.

After the lengthy and troublesome deduction [9], the local entropy balance equation in

 s

*<sup>T</sup>* ) <sup>−</sup>*<sup>σ</sup>* : *<sup>v</sup>*∇( <sup>1</sup>

1 *<sup>q</sup> T* æ ö = Ñç ÷ è ø

*i <sup>P</sup> T* æ ö = Ñç ÷ è ø j

1 *T* æ ö =- Ñç ÷ è ø *X v* s

d *s s*

*diff* <sup>+</sup> *<sup>φ</sup>iJP* <sup>−</sup>*<sup>v</sup>* <sup>⋅</sup>*<sup>σ</sup>*

*dt* stands for the velocity *v*, *Ei* <sup>=</sup> −∇*φ<sup>i</sup>*

= -Ñ × + *J* (61)

*<sup>T</sup>* (62)

*<sup>T</sup>* (63)

*<sup>T</sup>* ) <sup>−</sup>*ρ<sup>f</sup>* <sup>⋅</sup> *<sup>v</sup>*

*diff* ( <sup>=</sup> *Jq*), *JP*, *<sup>σ</sup>*( <sup>=</sup> *<sup>J</sup>σ*), *<sup>ρ</sup><sup>f</sup>* ( <sup>=</sup> *Jf* ) and the corresponding

*X* (64)

*X* (65)

(66)

*dni dt* =0.

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Advances in Thermodynamics of Ferroelectric Phase Transitions

*dD dt* stands 41

( *φ<sup>i</sup>* stands for the

as the local molar quantity ( per unit mass ) does not change with *t*, i.e.

d

*Js* = *Jq*

*<sup>T</sup>* ) <sup>+</sup> *JP* ⋅∇( *<sup>φ</sup><sup>i</sup>*

*s t* r

So, *ni*

for the polarization current *JP*, while *dr*

and the rate of local entropy production *σ<sup>s</sup>*

*σ<sup>s</sup>* = *Jq*

Here are the thermodynamic fluxes *Jq*

thermodynamic forces *Xq*, *XP*, *Xσ*, *Xf*

*diff* ⋅∇( <sup>1</sup>

*diff* is the diffusion of heat, i.e. heat conduction.

internal electric potential ).

with the entropy flux *Js*

where *Jq*

Lagrangian form can be obtained

which is in differential form and in Lagrangian form. Or in Eulerian form

$$\frac{\partial(\rho k)}{\partial t} + \nabla \cdot (\rho k v - v \cdot \sigma) = \rho v \cdot f - \sigma : d \tag{56}$$

where *<sup>d</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> (∇*<sup>v</sup>* <sup>+</sup> *<sup>v</sup>*∇)= <sup>1</sup> <sup>2</sup> <sup>∇</sup>*<sup>v</sup>* <sup>+</sup> (∇*v*)*T* is the rate of deformation or strain rate ( the super‐ script "T" means transposition ).

To a ferroelectric phase transition, *f* and *σ* may be the nominal volume force and stress, which are the embodiments of the actions of thermo-electro-mechanical coupling and are two internal fields. Generally, they are the sums of real and nominal volume force or stress

$$f = f^{real} + f^{nom} \tag{57}$$

$$
\sigma = \sigma^{real} + \sigma^{non} \tag{58}
$$

The nominal volume force and stress are not zero until the eigen ( or free ) deformation of system finishes in phase transitions. If they are zero, the eigen ( or free ) deformation finishes.

### **4.2. Local entropy production and description of phase transitions**

The Gibbs equation was given as the following [9]

$$T\,\mathrm{ds} + \sum\_{i} \mu\_{i}\mathrm{d}n\_{i} + \frac{1}{\rho}E \cdot \mathrm{d}D + f \cdot \mathrm{d}r + \frac{1}{\rho}\nabla \cdot (\mathbf{v} \cdot \boldsymbol{\sigma})\mathrm{d}t = \mathrm{d}u + \mathrm{d}k \tag{59}$$

where *T* , *E*, *D* is the local temperature, the local electric field intensity and the local electric displacement, respectively; *s*, *u*, *ni* , *μi* is the local entropy density ( per unit mass ), the local internal energy density ( per unit mass ), the local molar quantity density ( per unit mass ) and the chemical potential, respectively; *r* is the displacement vector. If the external electric field is not applied, the quantity *E* is the internal electric field *Ei* only [8].

Make the material derivative of Equation (59) with *t*, then obtain

$$
\rho \frac{\mathbf{d}s}{\mathbf{d}t} + \frac{\rho}{T} \sum\_{i} \mu\_{i} \frac{\mathbf{d}n\_{i}}{\mathbf{d}t} + \frac{1}{T} E\_{i} \cdot \frac{\mathbf{d}D}{\mathbf{d}t} + \frac{\rho}{T} f \cdot \frac{\mathbf{d}r}{\mathbf{d}t} + \frac{1}{T} \nabla \cdot (\mathbf{v} \cdot \boldsymbol{\sigma}) = \frac{\rho}{T} \frac{\mathbf{d}e}{\mathbf{d}t} \tag{60}
$$

where *e* is the total energy with *e* =*u* + *k*. We know, the system is in the crystalline states before and after a phase transition so that there is no diffusion of any kind of particles in the system.

So, *ni* as the local molar quantity ( per unit mass ) does not change with *t*, i.e. *dni dt* =0. *dD dt* stands

for the polarization current *JP*, while *dr dt* stands for the velocity *v*, *Ei* <sup>=</sup> −∇*φ<sup>i</sup>* ( *φ<sup>i</sup>* stands for the internal electric potential ).

After the lengthy and troublesome deduction [9], the local entropy balance equation in Lagrangian form can be obtained

$$
\rho \frac{\text{ds}}{\text{dt}} = -\nabla \cdot \mathbf{J}\_s + \sigma\_s \tag{61}
$$

with the entropy flux *Js*

*ρ* d*k*

∂(*ρk*)

<sup>2</sup> (∇*<sup>v</sup>* <sup>+</sup> *<sup>v</sup>*∇)= <sup>1</sup>

script "T" means transposition ).

where *<sup>d</sup>* <sup>=</sup> <sup>1</sup>

40 Advances in Ferroelectrics

which is in differential form and in Lagrangian form. Or in Eulerian form

<sup>d</sup>*<sup>t</sup>* −∇ <sup>⋅</sup> (*<sup>v</sup>* <sup>⋅</sup>*σ*)=*ρ<sup>v</sup>* <sup>⋅</sup> *<sup>f</sup>* <sup>−</sup>*<sup>σ</sup>* :*<sup>d</sup>* (55)

<sup>∂</sup>*<sup>t</sup>* <sup>+</sup> <sup>∇</sup> <sup>⋅</sup> (*ρkv* <sup>−</sup>*<sup>v</sup>* <sup>⋅</sup>*σ*)=*ρ<sup>v</sup>* <sup>⋅</sup> *<sup>f</sup>* <sup>−</sup>*<sup>σ</sup>* :*<sup>d</sup>* (56)

<sup>2</sup> <sup>∇</sup>*<sup>v</sup>* <sup>+</sup> (∇*v*)*T* is the rate of deformation or strain rate ( the super‐

*real nom ff f* = + (57)

*real nom σσ σ* = + (58)

∇ ⋅ (*v* ⋅*σ*)d*t* =d*u* + d*k* (59)

is the local entropy density ( per unit mass ), the local

only [8].

*<sup>T</sup>* <sup>∇</sup> <sup>⋅</sup> (*<sup>v</sup>* <sup>⋅</sup>*σ*)= *<sup>ρ</sup>*

*T* d*e*

<sup>d</sup>*<sup>t</sup>* (60)

To a ferroelectric phase transition, *f* and *σ* may be the nominal volume force and stress, which are the embodiments of the actions of thermo-electro-mechanical coupling and are two internal

The nominal volume force and stress are not zero until the eigen ( or free ) deformation of system finishes in phase transitions. If they are zero, the eigen ( or free ) deformation finishes.

> 1 *ρ*

where *T* , *E*, *D* is the local temperature, the local electric field intensity and the local electric

internal energy density ( per unit mass ), the local molar quantity density ( per unit mass ) and the chemical potential, respectively; *r* is the displacement vector. If the external electric field

> *<sup>T</sup> <sup>f</sup>* <sup>⋅</sup> <sup>d</sup>*<sup>r</sup>* <sup>d</sup>*<sup>t</sup>* <sup>+</sup> 1

<sup>d</sup>*<sup>t</sup>* <sup>+</sup> *<sup>ρ</sup>*

fields. Generally, they are the sums of real and nominal volume force or stress

**4.2. Local entropy production and description of phase transitions**

*<sup>ρ</sup> <sup>E</sup>* <sup>⋅</sup>d*<sup>D</sup>* <sup>+</sup> *<sup>f</sup>* <sup>⋅</sup>d*<sup>r</sup>* <sup>+</sup>

, *μi*

is not applied, the quantity *E* is the internal electric field *Ei*

Make the material derivative of Equation (59) with *t*, then obtain

1 *<sup>T</sup> Ei* <sup>⋅</sup> <sup>d</sup>*<sup>D</sup>*

The Gibbs equation was given as the following [9]

*T* d*s* + ∑ *i μi* d*ni* + 1

displacement, respectively; *s*, *u*, *ni*

*ρ* d*s* <sup>d</sup>*<sup>t</sup>* <sup>+</sup> *<sup>ρ</sup> <sup>T</sup>* ∑ *i μi* d*ni* <sup>d</sup>*<sup>t</sup>* <sup>+</sup>

$$J\_s = \frac{J\_q^{diff} + qjJ\_P - \upsilon \cdot \sigma}{T} \tag{62}$$

and the rate of local entropy production *σ<sup>s</sup>*

$$\boldsymbol{\sigma}\_{s} = \boldsymbol{I}\_{q}^{\text{diff}} \cdot \nabla \left( \frac{1}{T} \right) + \boldsymbol{I}\_{P} \cdot \nabla \left( \frac{\varphi\_{i}}{T} \right) - \boldsymbol{\sigma} \cdot \left[ \boldsymbol{\upsilon} \nabla \left( \frac{1}{T} \right) \right] - \rho \boldsymbol{f} \cdot \frac{\boldsymbol{\upsilon}}{T} \tag{63}$$

where *Jq diff* is the diffusion of heat, i.e. heat conduction.

Here are the thermodynamic fluxes *Jq diff* ( <sup>=</sup> *Jq*), *JP*, *<sup>σ</sup>*( <sup>=</sup> *<sup>J</sup>σ*), *<sup>ρ</sup><sup>f</sup>* ( <sup>=</sup> *Jf* ) and the corresponding thermodynamic forces *Xq*, *XP*, *Xσ*, *Xf*

$$\mathbf{X}\_q = \nabla \left(\frac{1}{T}\right) \tag{64}$$

$$\mathbf{X}\_P = \nabla \left(\frac{\varphi\_i}{T}\right) \tag{65}$$

$$X\_{\sigma} = -\nu \nabla \left(\frac{1}{T}\right) \tag{66}$$

$$X\_f = -\frac{\nu}{T} \tag{67}$$

( )

*LX LX L X LX*

*qq q qP P q qf f*

( )

*LXLX L X LX*

*Pq q PP P P Pf f*

( )

*LXLX L X LX*

( )

This reveals that if the *k* forces among those are kept constant, i.e. *Xi* =*const*(*i* =1, 2, ⋅ ⋅ ⋅ , *k*, *k* <4), the fluxes corresponding to the left 4−*k* forces are zero. Of

*LX LX L X LX*

*fq q fP P f ff f*

s s

= ×+ × + + ×

*T ff f f qq PP f f*

s

ss s s

:

 s

(*i* =1, 2, 3, 4), all the flues are zero. ( For convenience,

*qf q Pf P f fq q*

*LX LX L X LX*

s

ss

*LXLX LXLX*

s s

*qP q Pq q PP P P*

*LXLX LX L X*

s s

*q q P P qq PP*

2 :

æ ö ¶ ç ÷ = ×+ × + +× ç ÷ ¶è ø

= ×+ × + +×

*T T Pf f P fP f*

 s

ç ÷ = ×+ × + ×+ × ç ÷ ¶è ø

*LXLXLX*

+ + ×+ ×

*s TT T*

+×+ + ×

s s

*LX L X LX*

*fP P f ff f*

= ×+ × + +×

æ ö ¶ ç ÷ =×+ × + +× ç ÷ ¶è ø

s

s

= ×+ × + + ×

ç ÷ = ×+ ×+ × + ç ÷ ¶è ø

*LX L X LX*

+×+ +×

s

 s

*qq q qP P q qf f*

*LX LX L X LX*

s s

2 :

s s

Advances in Thermodynamics of Ferroelectric Phase Transitions

http://dx.doi.org/10.5772/52089

 s (75)

43

(76)

(77)

(78)

, ,

*XXX*

*P f*

s

 2 0

 2 0

æ ö ¶

*X*

s

s

= =

*P*

æ ö ¶

*X*

s

= =

*s*

s

*X*

*q*

:

*q*

*J*

, ,

*XX X*

*q f s T*

s

:

*P*

*J*

, ,

*XXX*

*qP f*

ss s s

> s

s

*J*

, ,

*XXX*

*q P*

: 2

s

*f*

we have modified the superscripts *q*, *P*, *σ*, *f* to be 1,2,3,4 ).

*J*

2:

2:

= =

 2 0

 2 0

course, if there are no restrictions on *Xi*

= =

*f*

s

*X*

2:

2:

2:

*s T T*

*TT T Pq P q fq f*

s

*LX L X LX*

+×+ +×

*Ji* (*i* =*q*, *P*, *σ*, *f* ) can be expanded linearly with *Xj* ( *j* =*q*, *P*, *σ*, *f* )

$$\mathbf{J}\_q = \mathbf{L}\_{qq} \cdot \mathbf{X}\_q + \mathbf{L}\_{qP} \cdot \mathbf{X}\_P + \mathbf{L}\_{q\sigma} : \mathbf{X}\_\sigma + \mathbf{L}\_{q\circ} \cdot \mathbf{X}\_f \tag{68}$$

$$\mathbf{J}\_{\rm p} = \mathbf{L}\_{\rm p\_{\rm q}} \cdot \mathbf{X}\_{\rm q} + \mathbf{L}\_{\rm pp} \cdot \mathbf{X}\_{\rm p} + \mathbf{L}\_{\rm p\_{\sigma}} : \mathbf{X}\_{\sigma} + \mathbf{L}\_{\rm p\_{\uparrow}} \cdot \mathbf{X}\_{\rm f} \tag{69}$$

$$\mathbf{J}\_{\sigma} = \mathbf{L}\_{\sigma\eta} \cdot \mathbf{X}\_{\eta} + \mathbf{L}\_{\sigma\mathbf{P}} \cdot \mathbf{X}\_{\mathbf{P}} + \mathbf{L}\_{\sigma\sigma} : \mathbf{X}\_{\sigma} + \mathbf{L}\_{\sigma f} \cdot \mathbf{X}\_{f} \tag{70}$$

$$\mathbf{J}\_f = \mathbf{L}\_{f\mathbf{q}} \cdot \mathbf{X}\_q + \mathbf{L}\_{f\mathbf{p}} \cdot \mathbf{X}\_p + \mathbf{L}\_{f\sigma} : \mathbf{X}\_\sigma + \mathbf{L}\_{f\mathbf{f}} \cdot \mathbf{X}\_f \tag{71}$$

where *L qq*, *L PP*, *L ff* , *L qP*, *L Pq*, *L qf* , *L fq*, *L Pf* , *L fP* are nine second-order tensors, *L <sup>q</sup>σ*, *L <sup>σ</sup>q*, *L <sup>P</sup>σ*, *L <sup>σ</sup>P*, *L <sup>σ</sup><sup>f</sup>* , *L <sup>f</sup><sup>σ</sup>* are six third-order tensors, *L σσ* is a fourth-order tensor. They should satisfy the generalized Onsager relations [14] *L αα* = *L αα T*

$$\left(\alpha = q, \ P, \ \sigma, f\right) \tag{72}$$

*L αβ* = *L βα T*

$$\left(\alpha,\,\beta=q,\,P,\,\sigma\_{\,\,\,t} \,\,\text{and}\,\,a\neq\beta\right)\tag{73}$$

So the rate of local entropy production can be written as

$$\begin{aligned} \sigma\_s &= \mathbf{L}\_{qq} : \mathbf{X}\_q \mathbf{X}\_q + \mathbf{L}\_{q\overline{p}} \cdots \mathbf{X}\_p \mathbf{X}\_q + \mathbf{L}\_{q\sigma} \cdots \mathbf{X}\_{\sigma}^T \mathbf{X}\_q + \mathbf{L}\_{q\overline{f}} \cdots \mathbf{X}\_{\overline{f}} \mathbf{X}\_q \\ &+ \mathbf{L}\_{pq} \cdots \mathbf{X}\_q \mathbf{X}\_p + \mathbf{L}\_{pp} : \mathbf{X}\_p \mathbf{X}\_p + \mathbf{L}\_{p\sigma} \cdots \mathbf{X}\_{\sigma}^T \mathbf{X}\_p + \mathbf{L}\_{p\overline{f}} \cdots \mathbf{X}\_{\overline{f}} \mathbf{X}\_p \\ &+ \mathbf{L}\_{\sigma q} \cdots \mathbf{X}\_q \mathbf{X}\_{\sigma}^T + \mathbf{L}\_{\sigma p} \cdots \mathbf{X}\_p \mathbf{X}\_{\sigma}^T + \mathbf{L}\_{\sigma \sigma} \cdots \cdots \mathbf{X}\_{\sigma}^T \mathbf{X}\_{\sigma}^T + \mathbf{L}\_{\sigma f} \cdots \mathbf{X}\_f \mathbf{X}\_{\sigma}^T \\ &+ \mathbf{L}\_{fq} \cdots \mathbf{X}\_q \mathbf{X}\_f + \mathbf{L}\_{f\overline{p}} \cdots \mathbf{X}\_p \mathbf{X}\_f + \mathbf{L}\_{f\sigma} \cdots \mathbf{X}\_{\sigma}^T \mathbf{X}\_f + \mathbf{L}\_{f\overline{f}} : \mathbf{X}\_f \mathbf{X}\_f \end{aligned} \tag{74}$$

According to the condition on which the local entropy production is a minimum, from Equation (74) we can deduce the following

$$\begin{aligned} \left(\frac{\partial \sigma\_s}{\partial \mathbf{X}\_q}\right)\_{\mathbf{X}\_p, \mathbf{X}\_\sigma, \mathbf{X}\_f} &= 2\mathbf{L}\_{qq} \cdot \mathbf{X}\_q + \mathbf{L}\_{q\mathbf{P}} \cdot \mathbf{X}\_\mathbf{P} + \mathbf{L}\_{q\sigma} : \mathbf{X}\_\sigma + \mathbf{L}\_{q\mathbf{f}} \cdot \mathbf{X}\_f \\ &+ \mathbf{L}\_{pq}^T \cdot \mathbf{X}\_\mathbf{P} + \mathbf{L}\_{\sigma q}^T : \mathbf{X}\_\sigma + \mathbf{L}\_{fq}^T \cdot \mathbf{X}\_f \\ &= 2\left(\mathbf{L}\_{qq} \cdot \mathbf{X}\_q + \mathbf{L}\_{q\mathbf{P}} \cdot \mathbf{X}\_\mathbf{P} + \mathbf{L}\_{q\sigma} : \mathbf{X}\_\sigma + \mathbf{L}\_{q\mathbf{f}} \cdot \mathbf{X}\_f\right) \\ &= 2\mathbf{J}\_q \\ &= 0 \end{aligned} \tag{75}$$

*<sup>f</sup> <sup>T</sup>* = - *<sup>v</sup> <sup>X</sup>* (67)

= *qP f* , , , ) (72)

 s (74)

, , , , , and = *qP f* ¹ ) (73)

( *j* =*q*, *P*, *σ*, *f* )

: *q qq q qP P q qf f* = ×+ × + +×

: *P Pq q PP P P Pf f* = ×+ × + + ×

: *qq PP f f* = ×+ × + + ×

: *f fq q fP P f ff f* = ×+ × + +×

s

(a

:

+ ×× + ×× + +

s s*J LX LX L X LX* (68)

s s*J LXLX L X LX* (69)

ss s s*J LXLX L X LX* (70)

s s*J LX LX L X LX* (71)

where *L qq*, *L PP*, *L ff* , *L qP*, *L Pq*, *L qf* , *L fq*, *L Pf* , *L fP* are nine second-order tensors, *L <sup>q</sup>σ*, *L <sup>σ</sup>q*, *L <sup>P</sup>σ*, *L <sup>σ</sup>P*, *L <sup>σ</sup><sup>f</sup>* , *L <sup>f</sup><sup>σ</sup>* are six third-order tensors, *L σσ* is a fourth-order tensor. They

> s

> > a b

*T*

 s

*T T T T T*

L

L

*T*

 s  s s s

*T*

 s

 s

*s qq q q qP P q q q qf f q*

= + ×× + + ×× + ×× + + + ×× + + +× +

*L XX L XX L X X L XX*

*Pq q P PP P P P P Pf f P*

*L XX L XX L X X L XX*

s

LLL L

*fq q f fP P f f f ff f f*

 s ss

*L XX L XX L X X L XX*

*qq PP f f*

*L XX L XX L X X L XX*

s

According to the condition on which the local entropy production is a minimum, from

L

s

*Ji*

42 Advances in Ferroelectrics

*L αα* = *L αα T*

*L αβ* = *L βα T*

(*i* =*q*, *P*, *σ*, *f* ) can be expanded linearly with *Xj*

s s

should satisfy the generalized Onsager relations [14]

(ab

So the rate of local entropy production can be written as

 s s

:

s

Equation (74) we can deduce the following

s

:

$$\begin{aligned} \left(\frac{\partial \sigma\_s}{\partial \mathbf{X}\_p}\right)\_{\mathbf{X}\_q, \mathbf{X}\_{\sigma}, \mathbf{X}\_f} &= \mathbf{L}\_{q\mathbf{P}}^T \cdot \mathbf{X}\_q + \mathbf{L}\_{\mathbf{P}q} \cdot \mathbf{X}\_q + 2\mathbf{L}\_{\mathbf{P}\mathbf{P}} \cdot \mathbf{X}\_{\mathbf{P}} + \mathbf{L}\_{\mathbf{P}\sigma} : \mathbf{X}\_{\sigma} \\ &+ \mathbf{L}\_{\mathbf{P}f} \cdot \mathbf{X}\_f + \mathbf{L}\_{\sigma\mathbf{P}}^T : \mathbf{X}\_{\sigma} + \mathbf{L}\_{f\mathbf{P}}^T \cdot \mathbf{X}\_f \\ &= 2\left(\mathbf{L}\_{\mathbf{P}q} \cdot \mathbf{X}\_q + \mathbf{L}\_{\mathbf{P}\mathbf{P}} \cdot \mathbf{X}\_p + \mathbf{L}\_{\mathbf{P}\sigma} : \mathbf{X}\_{\sigma} + \mathbf{L}\_{\mathbf{P}f} \cdot \mathbf{X}\_f\right) \\ &= 2\mathbf{J}\_p \\ &= 0 \end{aligned} \tag{76}$$

$$\begin{aligned} \left(\frac{\partial \sigma\_s}{\partial \mathbf{X}\_\sigma}\right)\_{\mathbf{X}\_q, \mathbf{X}\_p, \mathbf{X}\_f} &= \mathbf{L}\_{q\sigma}^T \cdot \mathbf{X}\_q + \mathbf{L}\_{p\sigma}^T \cdot \mathbf{X}\_p + \mathbf{L}\_{\sigma q} \cdot \mathbf{X}\_q + \mathbf{L}\_{\sigma p} \cdot \mathbf{X}\_p \\ &+ 2\mathbf{L}\_{\sigma\sigma} : \mathbf{X}\_\sigma + \mathbf{L}\_{\sigma f} \cdot \mathbf{X}\_f + \mathbf{L}\_{f\sigma}^T \cdot \mathbf{X}\_f \\ &= 2\left(\mathbf{L}\_{\sigma q} \cdot \mathbf{X}\_q + \mathbf{L}\_{\sigma P} \cdot \mathbf{X}\_p + \mathbf{L}\_{\sigma\sigma} : \mathbf{X}\_\sigma + \mathbf{L}\_{\sigma f} \cdot \mathbf{X}\_f\right) \\ &= 2\mathbf{J}\_\sigma \\ &= 0 \end{aligned} \tag{77}$$

$$\begin{aligned} \left(\frac{\partial \sigma\_s}{\partial \mathbf{X}\_f}\right)\_{\mathbf{X}\_q, \mathbf{X}\_p, \mathbf{X}\_\sigma} &= \mathbf{L}\_{\mathbf{q}'}^T \cdot \mathbf{X}\_q + \mathbf{L}\_{\mathbf{p}'}^T \cdot \mathbf{X}\_p + \mathbf{L}\_{\sigma f}^T : \mathbf{X}\_\sigma + \mathbf{L}\_{f\mathbf{q}} \cdot \mathbf{X}\_q \\ &+ \mathbf{L}\_{\mathbf{p}'} \cdot \mathbf{X}\_p + \mathbf{L}\_{f\sigma} : \mathbf{X}\_\sigma + 2\mathbf{L}\_{f\mathbf{f}'} \cdot \mathbf{X}\_f \\ &= 2\left(\mathbf{L}\_{f\mathbf{q}} \cdot \mathbf{X}\_q + \mathbf{L}\_{f\mathbf{p}} \cdot \mathbf{X}\_p + \mathbf{L}\_{f\sigma} : \mathbf{X}\_\sigma + \mathbf{L}\_{f\mathbf{f}'} \cdot \mathbf{X}\_f\right) \\ &= 2\mathbf{J}\_f \\ &= 0 \end{aligned} \tag{78}$$

This reveals that if the *k* forces among those are kept constant, i.e. *Xi* =*const*(*i* =1, 2, ⋅ ⋅ ⋅ , *k*, *k* <4), the fluxes corresponding to the left 4−*k* forces are zero. Of course, if there are no restrictions on *Xi* (*i* =1, 2, 3, 4), all the flues are zero. ( For convenience, we have modified the superscripts *q*, *P*, *σ*, *f* to be 1,2,3,4 ).

We may describe a ferroelectric phase transition by using the two paradigms above similarly as we have done in Section 3. To a first-order ferroelectric phase transition, the forces *XP*, *Xσ*, *Xf* of the region where the phase transition is occurring can be regarded as three large constants roughly in the characteristic times of phase transition ( i.e. the times in which the interface keeps rest ) because the electric displacement, the volume and the shape change suddenly. So, the flux *Jq diff* of the region should be zero (but *JP* <sup>≠</sup>0, *<sup>σ</sup>* <sup>≠</sup>0, *<sup>ρ</sup><sup>f</sup>* <sup>≠</sup><sup>0</sup> ). This states clearly that the pure heat conduction and the heat conduction induced by the thermo-electric coupling and the thermo-mechanical coupling cancel out each other so as to release or absorb the latent heat. The phase transition occurs at the surface layer firstly, which is mechanicallyfree. So, when the phase transition occurs in this region, the flux *σ* maybe the nominal stress *σ nom* only, which does work to realize the transformation from internal energy to kinetic energy. When the phase transition occurs in the inner part, the flux *σ* should be the sum of *σ real* and *σ nom* because the sudden changes of the inner part's volume and shape have to overcome the bound of outer part then *σ real* arises. The region where the phase transition is occurring, i.e. the phase boundary is accompanied with the real stress *σ real* usually, which does work to realize the transformation from kinetic energy to internal energy. This has been predicted and described with a propagating stress wave [20].

latent heat is transferred within a finite time so the occurrence of phase transition in the inner part is delayed. ( Of course, another cause is the stress, just see Section 5 ) In other words, the various parts absorb or release the latent heat at the various times. The action at a distance

Advances in Thermodynamics of Ferroelectric Phase Transitions

http://dx.doi.org/10.5772/52089

45

**5. Irreversibility: Thermal hysteresis and occurrences of domain structure**

The "thermal hysteresis" of first-order ferroelectric phase transitions is an irreversible phenomenon obviously. But it was treated by using the equilibrium thermodynamics for ferroelectric phase transitions, the well-known Landau-Devonshire theory [2]. So, there is an inherent contradiction in this case. The system in which a first-order ferroelectric phase transition occurs is heterogeneous. The occurrences of phase transition in different parts are not at the same time. The phase transition occurs at the surface layer then in the inner part of system. According to the description above, we know a constant temperature gradient is kept in the region where the phase transition has occurred. The temperature of surface layer, which is usually regarded as the temperature of the whole system in experiments, must be higher

No doubt that the shape and the area of surface can greatly affect the above processes. We may conclude that the thermal hysteresis can be reduced if the system has a larger specific surface and, the thermal hysteresis can be neglected if a finite system has an extremely-large specific

The region where the phase transition will occur can be regarded as an equilibrium system for there are no restrictions on the forces *Xq*, *XP*, *Xσ*, *Xf* . In other words, the forces and the corresponding fluxes are zero in this region. To a system where a second-order ferroelectric phase transition occurs, the case is somewhat like that of the region where a first-order ferroelectric phase transition will occur. The spontaneous polarization, the volume and the shape of system are continuous at the Curie temperature and change with the infinitesimal

magnitudes. The second-order phase transition occurs in every part of the system simultane‐ ously, i.e. there is no the co-existence of two phases ( ferroelectric and paraelectric ). So, there

The region where a first-order ferroelectric phase transition will occur is stressed. This reveals that the occurrences of phase transition in the inner part have to overcome the bound of outer part, where the phase transition occurs earlier. This may lead to the delay of phase transition

Though the rationalization of the existence of domain structures can be explained by the equilibrium thermodynamics, the evolving characteristics of domain occurrences in

*diff* , *JP*, *<sup>σ</sup>*, *<sup>ρ</sup><sup>f</sup>* can be arbitrary infinitesimal

( or lower ) than the Curie temperature. This may lead to the thermal hysteresis.

surface. So, the thermal hysteresis is not an intrinsic property of the system.

magnitudes. This means *Xq*, *XP*, *Xσ*, *Xf* and *Jq*

**5.2. Occurrences of domain structure**

in the inner part.

is no the latent heat and stress. The thermal hysteresis disappears.

does not affect the phase transition necessarily.

**5.1. Thermal hysteresis**

It is certain that the latent heat passes through the region where the phase transition has occurred ( at the outside of the region where the phase transition is occurring ) and exchange itself with the thermal bath. For ±*lρva* = *Jq diff* = −*κ* ⋅∇*T* , a constant temperature gradient ∇*T* is kept in the region where the phase transition has occurrd, i.e. the force *Xq* at every site is a constant ( which does not change with the time but may vary with the position ). So, the fluxes *JP* =*σ* =*ρf* =0 ( but the flux *Jq diff* ≠0 ). This states clearly that the electric displacement *D* will not change but keep the value at Curie temperature or zero until the phase transition finishes and *σ real* = −*σ nom* in this region. Because the electric displacement *D* and the strain ( or deformation ) are all determined by the crystal structure of system, *JP* =0 reveals that *D* of this region does not change so does not the crystal structure then does not the strain ( or deformation ). According to [20], we know the region where the phase transition has occurred is unstressed, i.e. *σ real* =0, then *σ nom* =0. This reveals that the eigen ( or free ) strain ( or deformation ) of system induced by the thermo-electro-mechanical coupling of phase transition is complete and the change of it terminates before the phase transition finishes. The two deductions coincide with each other. *σ real* may relaxes via the free surface.

The region where the phase transition will occur should be in equilibrium because there are no restrictions on the forces *Xq*, *XP*, *Xσ*, *Xf* . Whereas, according to [20], the region is stressed, i.e. *σ real* ≠0. To the heating process of phase transition, this may lead to a change of the spontaneous polarization of this region because of the electro-mechanical coupling ( piezo‐ electric effect ).

An immediate result of the above irreversible thermodynamic description is that the action at a distance, which is the kind of heat transfer at phase transitions, is removed absolutely. The latent heat is transferred within a finite time so the occurrence of phase transition in the inner part is delayed. ( Of course, another cause is the stress, just see Section 5 ) In other words, the various parts absorb or release the latent heat at the various times. The action at a distance does not affect the phase transition necessarily.

### **5. Irreversibility: Thermal hysteresis and occurrences of domain structure**

### **5.1. Thermal hysteresis**

We may describe a ferroelectric phase transition by using the two paradigms above similarly as we have done in Section 3. To a first-order ferroelectric phase transition, the forces *XP*, *Xσ*, *Xf* of the region where the phase transition is occurring can be regarded as three large constants roughly in the characteristic times of phase transition ( i.e. the times in which the interface keeps rest ) because the electric displacement, the volume and the shape change

clearly that the pure heat conduction and the heat conduction induced by the thermo-electric coupling and the thermo-mechanical coupling cancel out each other so as to release or absorb the latent heat. The phase transition occurs at the surface layer firstly, which is mechanicallyfree. So, when the phase transition occurs in this region, the flux *σ* maybe the nominal stress *σ nom* only, which does work to realize the transformation from internal energy to kinetic energy. When the phase transition occurs in the inner part, the flux *σ* should be the sum of

and *σ nom* because the sudden changes of the inner part's volume and shape have to

work to realize the transformation from kinetic energy to internal energy. This has been

It is certain that the latent heat passes through the region where the phase transition has occurred ( at the outside of the region where the phase transition is occurring ) and exchange

kept in the region where the phase transition has occurrd, i.e. the force *Xq* at every site is a constant ( which does not change with the time but may vary with the position ). So, the fluxes

change but keep the value at Curie temperature or zero until the phase transition finishes and

induced by the thermo-electro-mechanical coupling of phase transition is complete and the change of it terminates before the phase transition finishes. The two deductions coincide with

The region where the phase transition will occur should be in equilibrium because there are no restrictions on the forces *Xq*, *XP*, *Xσ*, *Xf* . Whereas, according to [20], the region is stressed,

spontaneous polarization of this region because of the electro-mechanical coupling ( piezo‐

An immediate result of the above irreversible thermodynamic description is that the action at a distance, which is the kind of heat transfer at phase transitions, is removed absolutely. The

= −*σ nom* in this region. Because the electric displacement *D* and the strain ( or deformation ) are all determined by the crystal structure of system, *JP* =0 reveals that *D* of this region does not change so does not the crystal structure then does not the strain ( or deformation ). According to [20], we know the region where the phase transition has occurred is unstressed,

=0, then *σ nom* =0. This reveals that the eigen ( or free ) strain ( or deformation ) of system

≠0. To the heating process of phase transition, this may lead to a change of the

occurring, i.e. the phase boundary is accompanied with the real stress *σ real*

predicted and described with a propagating stress wave [20].

may relaxes via the free surface.

*diff* of the region should be zero (but *JP* <sup>≠</sup>0, *<sup>σ</sup>* <sup>≠</sup>0, *<sup>ρ</sup><sup>f</sup>* <sup>≠</sup><sup>0</sup> ). This states

arises. The region where the phase transition is

*diff* = −*κ* ⋅∇*T* , a constant temperature gradient ∇*T* is

*diff* ≠0 ). This states clearly that the electric displacement *D* will not

usually, which does

suddenly. So, the flux *Jq*

44 Advances in Ferroelectrics

overcome the bound of outer part then *σ real*

itself with the thermal bath. For ±*lρva* = *Jq*

*JP* =*σ* =*ρf* =0 ( but the flux *Jq*

*σ real*

*σ real*

i.e. *σ real*

i.e. *σ real*

each other. *σ real*

electric effect ).

The "thermal hysteresis" of first-order ferroelectric phase transitions is an irreversible phenomenon obviously. But it was treated by using the equilibrium thermodynamics for ferroelectric phase transitions, the well-known Landau-Devonshire theory [2]. So, there is an inherent contradiction in this case. The system in which a first-order ferroelectric phase transition occurs is heterogeneous. The occurrences of phase transition in different parts are not at the same time. The phase transition occurs at the surface layer then in the inner part of system. According to the description above, we know a constant temperature gradient is kept in the region where the phase transition has occurred. The temperature of surface layer, which is usually regarded as the temperature of the whole system in experiments, must be higher ( or lower ) than the Curie temperature. This may lead to the thermal hysteresis.

No doubt that the shape and the area of surface can greatly affect the above processes. We may conclude that the thermal hysteresis can be reduced if the system has a larger specific surface and, the thermal hysteresis can be neglected if a finite system has an extremely-large specific surface. So, the thermal hysteresis is not an intrinsic property of the system.

The region where the phase transition will occur can be regarded as an equilibrium system for there are no restrictions on the forces *Xq*, *XP*, *Xσ*, *Xf* . In other words, the forces and the corresponding fluxes are zero in this region. To a system where a second-order ferroelectric phase transition occurs, the case is somewhat like that of the region where a first-order ferroelectric phase transition will occur. The spontaneous polarization, the volume and the shape of system are continuous at the Curie temperature and change with the infinitesimal magnitudes. This means *Xq*, *XP*, *Xσ*, *Xf* and *Jq diff* , *JP*, *<sup>σ</sup>*, *<sup>ρ</sup><sup>f</sup>* can be arbitrary infinitesimal magnitudes. The second-order phase transition occurs in every part of the system simultane‐ ously, i.e. there is no the co-existence of two phases ( ferroelectric and paraelectric ). So, there is no the latent heat and stress. The thermal hysteresis disappears.

The region where a first-order ferroelectric phase transition will occur is stressed. This reveals that the occurrences of phase transition in the inner part have to overcome the bound of outer part, where the phase transition occurs earlier. This may lead to the delay of phase transition in the inner part.

### **5.2. Occurrences of domain structure**

Though the rationalization of the existence of domain structures can be explained by the equilibrium thermodynamics, the evolving characteristics of domain occurrences in ferroelectrics can not be explained by it, but can be explained by the non-equilibrium thermodynamics.

In the region where the phase transition is occurring, the thermodynamic forces *XP* (= ∇( *<sup>φ</sup><sup>i</sup> <sup>T</sup>* ) ), *<sup>X</sup>σ* (= −*v*∇( <sup>1</sup> *<sup>T</sup>* ) ), *Xf* (= <sup>−</sup> *<sup>v</sup> <sup>T</sup>* ) can be regarded as three large constants in the characteristic times of transition and the thermodynamic flux *Jq diff* =0 (but *JP* <sup>≠</sup>0, *<sup>σ</sup>* <sup>≠</sup>0, *ρ<sup>f</sup>* <sup>≠</sup>0 ). The local entropy production (cf. Equation (63)) reduces to

$$\boldsymbol{\sigma}\_{s} = \mathbf{J}\_{\mathbf{P}} \cdot \nabla \left( \frac{\boldsymbol{\sigma}\_{i}}{T} \right) - \quad : \left[ \mathbf{v} \nabla \left( \frac{1}{T} \right) \right] - \boldsymbol{\rho} \mathbf{f} \cdot \frac{\mathbf{v}}{T} \tag{79}$$

It seems that the picture of domain occurrences for first-order ferroelectric phase transition systems should disappear when we face second-order ferroelectric phase transition systems. This is true if the transition processes proceed infinitely slowly as expounded by the equili‐ brium thermodynamics. But any actual process proceeds with finite rate, so it is irreversible.

In [21], the domain occurrences in ferromagnetics can be described parallelly by analogy. And the case of ferroelastic domain occurrences is a reduced, simpler one compared with that of

It is well known that the Landau theory or the Curie principle tells us how to determine the symmetry change at a phase transition. A concise statement is as follows [22]: for a crystal undergoing a phase transition with a space-group symmetry reduction from *G*0 to *G*, whereas *G* determines the symmetry of transition parameter ( or vice versa ), it is the symmetry operations lost in going from *G*0 to *G* that determine the domain structure in the low-symmetry phase. The ferroic phase transitions are the ones accompanied by a change of point group symmetry [23]. Therefore, the substitution of "point group" for the "space group" in the above statement will be adequate for ferroic phase transitions. From the above statement, the domain structure is a manifestation of the symmetry operations lost at the phase transition. In our treatment of the domain occurrences in ferroics, we took into account the finiteness of system ( i.e. existence of surface ) and the irreversibility of process ( asymmetry of time ). The finiteness

space symmetry. The infinite space symmetry, combined with the asymmetry of time, reproduces the symmetry operations lost at the phase transition in the ferroic phase. It can be

After all, for the domain structures can exist in equilibrium systems, they are the equilibrium structures but not the dissipative ones, for the latter can only exist in systems far from

In order to overcome the shortcoming of Landau-Devonshire theory, the non-equilibrium thermodynamics was applied to study the ferroelectric phase transitions. The essence of transitions is the thermo-electro-mechanical coupling. Moreover, the irreversibility, namely thermal hysteresis and domain occurrences can be explained well in the realm of non-

The non-equilibrium thermodynamic approach utilized here is the linear thermodynamic one actully. In order to get the more adequate approaches, we should pay attention to the new developments of non-equilibrium thermodynamics. The thermodynamics with internal variables [25] and the extended ( irreversible ) thermodynamics [26] are two current ones. They all expand the fundamental variables spaces to describe the irreversible processes more

*<sup>T</sup>* ), <sup>−</sup>*v*∇( <sup>1</sup>

Advances in Thermodynamics of Ferroelectric Phase Transitions

http://dx.doi.org/10.5772/52089

47

*<sup>T</sup>* ), <sup>−</sup> *<sup>v</sup>*

*<sup>T</sup>* have infinite

Then the above picture revives.

ferroelectrics or ferromagnetics.

of system make the thermodynamic forces such as ∇( *<sup>φ</sup><sup>i</sup>*

viewed as an embodiment of time-space symmetry.

equilibrium [24].

**6. Concluding remarks**

equilibrium thermodynamics.

Now, we are facing a set of complicated fields of *T* , *v* and *φ<sup>i</sup>* , respectively. Assume that the phase transition front is denoted by *S*. The points included in *S* stand for the locations where the transition is occurring. Because the transition occurs along all directions from the outer part to the inner part, we may infer that the orientations of ∇( *<sup>φ</sup><sup>i</sup> <sup>T</sup>* ), or of −*v*∇( <sup>1</sup> *<sup>T</sup>* ) and or of − *v <sup>T</sup>* vary continuously such that they are differently oriented at different locations.

There are always several ( at least two ) symmetry equivalent orientations in the prototype phase ( in most cases it is the high temperature phase ), which are the possible orientations for spontaneous polarization ( or spontaneous deformation or spontaneous displacement ). Therefore, the spontaneous polarization, the spontaneous deformation and the spontaneous displacement must take an appropriate orientation respectively to ensure *σ<sup>s</sup>* is a positive minimum when the system transforms from the prototype ( paraelectric ) phase to the ferroelectric ( low temperature ) phase. The underlying reasons are that

$$\mathbf{J}\_{\mathbf{P}} = \frac{\mathbf{d}\mathbf{D}}{\mathbf{d}t} = \frac{\mathbf{d}\left(\varepsilon\_{0}E\_{i} + \mathbf{P}\right)}{\mathbf{d}t} \tag{80}$$

$$\mathbf{r} = \mathsf{L}^\* \mathbf{r} \tag{81}$$

$$
\rho \,\rho \mathbf{f} = \rho \frac{\mathbf{d} \mathbf{v}}{\mathbf{d}t} - \nabla \cdot \tag{82}
$$

where *L* , *ε*, and *ε*0 are the modulus of rigidity, the strain and the permittivity of vacuum, respectively. Therefore, *P*, *ε* at different locations will be differently oriented. The domain structures in ferroelectrics thus occur.

It seems that the picture of domain occurrences for first-order ferroelectric phase transition systems should disappear when we face second-order ferroelectric phase transition systems. This is true if the transition processes proceed infinitely slowly as expounded by the equili‐ brium thermodynamics. But any actual process proceeds with finite rate, so it is irreversible. Then the above picture revives.

In [21], the domain occurrences in ferromagnetics can be described parallelly by analogy. And the case of ferroelastic domain occurrences is a reduced, simpler one compared with that of ferroelectrics or ferromagnetics.

It is well known that the Landau theory or the Curie principle tells us how to determine the symmetry change at a phase transition. A concise statement is as follows [22]: for a crystal undergoing a phase transition with a space-group symmetry reduction from *G*0 to *G*, whereas *G* determines the symmetry of transition parameter ( or vice versa ), it is the symmetry operations lost in going from *G*0 to *G* that determine the domain structure in the low-symmetry phase. The ferroic phase transitions are the ones accompanied by a change of point group symmetry [23]. Therefore, the substitution of "point group" for the "space group" in the above statement will be adequate for ferroic phase transitions. From the above statement, the domain structure is a manifestation of the symmetry operations lost at the phase transition. In our treatment of the domain occurrences in ferroics, we took into account the finiteness of system ( i.e. existence of surface ) and the irreversibility of process ( asymmetry of time ). The finiteness

of system make the thermodynamic forces such as ∇( *<sup>φ</sup><sup>i</sup> <sup>T</sup>* ), <sup>−</sup>*v*∇( <sup>1</sup> *<sup>T</sup>* ), <sup>−</sup> *<sup>v</sup> <sup>T</sup>* have infinite space symmetry. The infinite space symmetry, combined with the asymmetry of time, reproduces the symmetry operations lost at the phase transition in the ferroic phase. It can be viewed as an embodiment of time-space symmetry.

After all, for the domain structures can exist in equilibrium systems, they are the equilibrium structures but not the dissipative ones, for the latter can only exist in systems far from equilibrium [24].

### **6. Concluding remarks**

ferroelectrics can not be explained by it, but can be explained by the non-equilibrium

In the region where the phase transition is occurring, the thermodynamic forces *XP* (= ∇( *<sup>φ</sup><sup>i</sup>*

1 *<sup>i</sup> <sup>s</sup> T TT* æ ö é ù æ ö = ×Ñ - Ñ - × ç ÷ ê ú ç ÷ è ø ë û è ø

phase transition front is denoted by *S*. The points included in *S* stand for the locations where the transition is occurring. Because the transition occurs along all directions from the outer

There are always several ( at least two ) symmetry equivalent orientations in the prototype phase ( in most cases it is the high temperature phase ), which are the possible orientations for spontaneous polarization ( or spontaneous deformation or spontaneous displacement ). Therefore, the spontaneous polarization, the spontaneous deformation and the spontaneous displacement must take an appropriate orientation respectively to ensure *σ<sup>s</sup>* is a positive minimum when the system transforms from the prototype ( paraelectric ) phase to the

> ( ) <sup>0</sup> d d d d *t t* <sup>+</sup> = = e*i*

> > d d*t*

where *L* , *ε*, and *ε*0 are the modulus of rigidity, the strain and the permittivity of vacuum, respectively. Therefore, *P*, *ε* at different locations will be differently oriented. The domain

 r

*D E P*

*P*

*<sup>T</sup>* vary continuously such that they are differently oriented at different locations.

j

Now, we are facing a set of complicated fields of *T* , *v* and *φ<sup>i</sup>*

part to the inner part, we may infer that the orientations of ∇( *<sup>φ</sup><sup>i</sup>*

ferroelectric ( low temperature ) phase. The underlying reasons are that

*P*

r

structures in ferroelectrics thus occur.

*<sup>T</sup>* ) can be regarded as three large constants in the characteristic

*<sup>v</sup> J vf* : (79)

*J* (80)

 *L* = : *ε* (81)

= -Ñ× *<sup>v</sup> <sup>f</sup>* (82)

 r

*diff* =0 (but *JP* <sup>≠</sup>0, *<sup>σ</sup>* <sup>≠</sup>0, *ρ<sup>f</sup>* <sup>≠</sup>0 ). The local

, respectively. Assume that the

*<sup>T</sup>* ) and or of

*<sup>T</sup>* ), or of −*v*∇( <sup>1</sup>

*<sup>T</sup>* ) ),

thermodynamics.

46 Advances in Ferroelectrics

*<sup>X</sup>σ* (= −*v*∇( <sup>1</sup>

− *v*

*<sup>T</sup>* ) ), *Xf* (= <sup>−</sup> *<sup>v</sup>*

times of transition and the thermodynamic flux *Jq*

entropy production (cf. Equation (63)) reduces to

s

In order to overcome the shortcoming of Landau-Devonshire theory, the non-equilibrium thermodynamics was applied to study the ferroelectric phase transitions. The essence of transitions is the thermo-electro-mechanical coupling. Moreover, the irreversibility, namely thermal hysteresis and domain occurrences can be explained well in the realm of nonequilibrium thermodynamics.

The non-equilibrium thermodynamic approach utilized here is the linear thermodynamic one actully. In order to get the more adequate approaches, we should pay attention to the new developments of non-equilibrium thermodynamics. The thermodynamics with internal variables [25] and the extended ( irreversible ) thermodynamics [26] are two current ones. They all expand the fundamental variables spaces to describe the irreversible processes more adequately. Whereas, the relevant theoretical processing must be more complicated undoubt‐ ly. This situation needs very much effort.

[10] Yufatova, S M, Sindeyev, Y G, Garilyatchenko, V G, & Fesenko, E G. Different Kinetics Types of Phase Transformations in Lead Titanate. Ferroelectrics (1980). , 26(1), 809-812.

Advances in Thermodynamics of Ferroelectric Phase Transitions

http://dx.doi.org/10.5772/52089

49

[11] Dec, J. Jerky Phase Front Motion in PbTiO3 Crystal. Journal of Physics C (1988). , 21(7),

[12] Dec, J, & Yurkevich, J. The Antiferroelectric Phase Boundary as a Kink. Ferroelectrics

[13] Ai, S T, Xu, C T, Wang, Y L, Zhang, S Y, Ning, X F, & Noll, E. Comparison of and Comments on Two Thermodynamic Approaches ( Reversible and Irreversible ) to

[14] Ai, S T, Zhang, S Y, Jiang, J S, & Wang, C C. On Transport Coefficients of Ferroelectrics

[15] Lavenda, B H. Thermodynamics of Irreversible Processes. London and Basingstoke:

[16] Chewasatn, S, & Milne, S T. Synthesis and Characterization of PbTiO3 and Ca and Mn Modified PbTiO3 Fibres Produced by Extrusion of Diol Based Gels. Journal of Material

[17] Nomura, S, & Sawada, S. Dielectric Properties of Lead-Strontium Titanate. Journal of

[18] Mante, A, & Volger, H. J. The Thermal Conductivity of PbTiO3 in the Neighbourhood of Its Ferroelectric Transition Temperature. Physics Letters A (1967). , 24(3), 139-140.

[20] Gordon, A. Propagation of Solitary Stress Wave at First-Order Ferroelectric Phase

[21] Ai, S T, Zhang, S Y, Ning, X F, Wang, Y L, & Xu, C T. Non-Equilibrium Thermodynamic Explanation of Domain Occurrences in Ferroics. Ferroelectrics Letters Section (2010). ,

[22] Janovec, V. A Symmetry Approach to Domain Structures. Ferroelectrics (1976). , 12(1),

[23] Wadhawan, V K. Ferroelasticity and Related Properties of Crystals. Phase Transitions

[24] Glansdoff, P, & Prigogine, I. Thermodynamic Theory of Structure, Stability and

[25] Maugin, G A, & Muschik, W. Thermodynamics with Internal Variables (I) General Concepts, (II) Applications. Journal of Non-Equilibrium Thermodynamics (1994).

[26] Jou, D, Casas-vázquez, J, & Lebon, G. Extended Irreversible Thermodynamics ( 3rd

[19] Dec, J. The Phase Boundary as a Kink. Ferroelectrics (1989). , 89(1), 193-200.

Ferroelectric Phase Transitions. Phase Transitions (2008). , 81(5), 479-490.

and Onsager Reciprocal Relations. Ferroelectrics (2011). , 423(1), 54-62.

1257-1263.

(1990). , 110(1), 77-83.

Macmillan; (1978).

37(2), 30-34.

(1982). , 3(1), 3-103.

edition ). Berlin: Springer; (2001).

43-53.

Science (1994). , 29(14), 3621-3629.

Transitions. Physics Letters A (1991).

the Physical Society of Japan (1955). , 10(2), 108-111.

Fluctuation. New York: Wiley-Interscience; (1971).

### **Acknowledgements**

The work is supported by the Natural Science Foundation Program of Shandong Province, China ( Grant No. ZR2011AM019 ).

### **Author details**

Shu-Tao Ai\*

School of Science, Linyi University, Linyi, People's Republic of China

### **References**


[10] Yufatova, S M, Sindeyev, Y G, Garilyatchenko, V G, & Fesenko, E G. Different Kinetics Types of Phase Transformations in Lead Titanate. Ferroelectrics (1980). , 26(1), 809-812.

adequately. Whereas, the relevant theoretical processing must be more complicated undoubt‐

The work is supported by the Natural Science Foundation Program of Shandong Province,

[1] Grindlay, I. An Introduction to the Phenomenological Theory of Ferroelectricity.

[2] Lines, M E, & Glass, A M. Principles and Applications of Ferroelectrics and Related

[3] Tilley, D R, & Zeks, B. Landau Theory of Phase Transitions in Thick Films. Solid State

[4] Scott, J F, Duiker, H M, Beale, P D, Pouligny, B, Dimmler, K, Parris, M, Butler, D, & Eaton, S. Properties of Ceramic KNO3 Thin-Film Memories. Physica B+C (1988).

[6] Gordon, A. Finite-Size Effects in Dynamics of Paraelectric-Ferroelectric Interfaces

[7] Gordon, A, Dorfman, S, & Fuks, D. Temperature-Induced Motion of Interface Boun‐ daries in Confined Ferroelectrics. Philosophical Magazine B (2002). , 82(1), 63-71.

[8] Ai, S T. Paraelectric-Ferroelectric Interface Dynamics Induced by Latent Heat Transfer and Irreversible Thermodynamics of Ferroelectric Phase Transitions. Ferroelectrics

[9] Ai, S T. Mechanical-Thermal-Electric Coupling and Irreversibility of Ferroelectric Phase

[5] Zhong, W L. Physics of Ferroelectrics ( in Chinese ). Beijing: Science Press; (1996).

Induced by Latent Heat Transfer. Physics Letters A (2001).

Transitions. Ferroelectrics (2007). , 350(1), 81-92.

School of Science, Linyi University, Linyi, People's Republic of China

ly. This situation needs very much effort.

**Acknowledgements**

48 Advances in Ferroelectrics

**Author details**

Shu-Tao Ai\*

**References**

China ( Grant No. ZR2011AM019 ).

Oxford: Pergamon; (1970).

(2006). , 345(1), 59-66.

Materials. Oxford: Clarendon; (1977).

Communications (1984). , 49(8), 823-828.


**Chapter 3**

4 will

**Pb(Mg1/3Nb2/3)O3 (PMN) Relaxor: Dipole Glass or Nano-**

In sharp contrast to normal ferroelectric (for example BaTiO3), relaxors show unusually large dielectric constant over a large temperature range (~100 K) (Fig. 1 (a)) [1,2]. Such large dielectric response is strongly dependent on the frequency. Its origin has been the focus of interest in the solid-state physics. Unlike the dielectric anomaly in BaTiO3, which is associat‐ ed with a ferroelectric phase transition, the maximum of dielectric response in relaxor does not indicate the occurrence of a ferroelectric phase transition. Such huge dielectric response suggests that local polarization might occur in the crystal. This was envisioned by Burns and Dacol from the deviation from linearity of refractive index *n*(*T*) (Fig. 1(c)) around the socalled Burns temperature (*T*Burns) [3] because the deviation Δ*n* is proportional to polarization *P*s. The local polarization is suggested to occur in a nano-region, and is generally called as polar nano-region (PNR). The existence of PNR in relaxor is well confirmed from the neu‐ tron scattering measurements [4] and transition electron microscopy (TEM) observation. However, it is still unknown how PNRs contribute to the large dielectric response. More re‐ cently, it is also suggested that a strong coupling between zone-center and zone-boundary soft-modes may play a key role in understanding the relaxor behaviors [5]. Clearly, the question "what is the origin of the giant dielectric constant over a broad temperature

Another longstanding issue on relaxors is how PNRs interact at low temperature. There are two acceptable models: (1) dipole glass model [2,8-11], and (2) random-field model [12,13]. In spherical random-bond–random-field (SRBRF) model, Pirc and Blinc assumed that PNRs are spherical and interact randomly and proposed a frozen dipole glass state for relaxor (Fig. 2(a)) [10,11]. It predicts that the scaled third-order nonlinear susceptibility *a*3=-*ε*3/*ε*<sup>1</sup>

> © 2013 Fu et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

**Domain Ferroelectric?**

http://dx.doi.org/10.5772/52139

Shigeo Mori

**1. Introduction**

range?"is still unclear [6,7].

Desheng Fu, Hiroki Taniguchi, Mitsuru Itoh and

Additional information is available at the end of the chapter

### **Pb(Mg1/3Nb2/3)O3 (PMN) Relaxor: Dipole Glass or Nano-Domain Ferroelectric?**

Desheng Fu, Hiroki Taniguchi, Mitsuru Itoh and Shigeo Mori

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52139

### **1. Introduction**

In sharp contrast to normal ferroelectric (for example BaTiO3), relaxors show unusually large dielectric constant over a large temperature range (~100 K) (Fig. 1 (a)) [1,2]. Such large dielectric response is strongly dependent on the frequency. Its origin has been the focus of interest in the solid-state physics. Unlike the dielectric anomaly in BaTiO3, which is associat‐ ed with a ferroelectric phase transition, the maximum of dielectric response in relaxor does not indicate the occurrence of a ferroelectric phase transition. Such huge dielectric response suggests that local polarization might occur in the crystal. This was envisioned by Burns and Dacol from the deviation from linearity of refractive index *n*(*T*) (Fig. 1(c)) around the socalled Burns temperature (*T*Burns) [3] because the deviation Δ*n* is proportional to polarization *P*s. The local polarization is suggested to occur in a nano-region, and is generally called as polar nano-region (PNR). The existence of PNR in relaxor is well confirmed from the neu‐ tron scattering measurements [4] and transition electron microscopy (TEM) observation. However, it is still unknown how PNRs contribute to the large dielectric response. More re‐ cently, it is also suggested that a strong coupling between zone-center and zone-boundary soft-modes may play a key role in understanding the relaxor behaviors [5]. Clearly, the question "what is the origin of the giant dielectric constant over a broad temperature range?"is still unclear [6,7].

Another longstanding issue on relaxors is how PNRs interact at low temperature. There are two acceptable models: (1) dipole glass model [2,8-11], and (2) random-field model [12,13]. In spherical random-bond–random-field (SRBRF) model, Pirc and Blinc assumed that PNRs are spherical and interact randomly and proposed a frozen dipole glass state for relaxor (Fig. 2(a)) [10,11]. It predicts that the scaled third-order nonlinear susceptibility *a*3=-*ε*3/*ε*<sup>1</sup> 4 will

© 2013 Fu et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

shows a nearly divergent behavior at the freezing temperature *T*<sup>f</sup> of the spherical glass phase. In sharp contrast, the random field model of Fisch [13], which assumes non-random two-spin exchange, predicts a ferroelectric or ferroelectric domain state in relaxor (Fig. 2(b)). This random Potts field model also predicts a broadening specific heat peak for the glass phase and shows that the latent heat at ferroelectric transition *T*c is so small that it may be difficult to be detected, which reasonably explains the data reported by Moriya et al. [14].

eral nm has been observed in PMN crystal by TEM. It should be noticed that PNR and COR belong to different symmetry groups and are considered to have non-centrosymmetry of*R*3*m*

along the <111>c direction of pseudocubic structure in PNRs [20,21], but none of *P*s exists in CORs. In addition to chemical and structural inhomogeneities, we will show that PMN relax‐ or also has inhomogeneity of ferroelectric domain structure. Multiple inhomogeneities are

> 0.0 0.5 1.0 1.5 2.0 2.5 3.0

**Figure 2.** Two phase diagrams proposed for relaxor by (a) a dipole glass model [10], and (b) a random field model [13].

**Figure 3.** (a) ABO3perovskite structure. (b) Model for relaxor structure. PNR and COR represent the polar nano-region and chemically order region, respectively. (c) &(d) show two models of atom arrangement for COR. To maintain the

T/J

ferroelectric

thus considered to play a crucial role in inducing the intriguing behaviors in relaxors.

/ ~ D=0 ferroelectric

¯*m*, respectively. Therefore, there is spontaneous polarization *P*<sup>s</sup>

Pb(Mg1/3Nb2/3)O3 (PMN) Relaxor: Dipole Glass or Nano-Domain Ferroelectric?

http://dx.doi.org/10.5772/52139

53

(b) random field model

0 2 4 6 8 10

h/J

[1,1] domain ferroelectric [1,0]

?

paraelectric

paraelectric glass

and centrosymmetryof*Fm*3

0 1

spherical dipole glass

> J 0 /J

(a) dipole glass model

~ D=0

paraelectric

0

electric neutrality, a Nb-rich layer is required for case (c).

1

T/J

Combining our recent results [15] from the electrical polarization, Raman scattering, TEM measurements and those reported in the literature, here, we propose a physical picture to understand the dielectric behaviors of Pb(Mg1/3Nb2/3)O3 (PMN) relaxor.

**Figure 1.** Temperature dependence of (a) linear dielectric constant [9], (b) cold [35] & thermal [4] neutrons diffuse intensities, and (c) refractive index [3] for PMN relaxor crystal.

### **2. Multiple inhomogeneities in relaxors**

PMN is a prototypical relaxor with A(B',B")O3 perovskite structure (Fig. 3), in which B-sites are occupied by two kinds of heterovalent cations. Such chemical inhomogeneity is a com‐ mon feature of relaxor crystals. Although it remains an average centrosymmetric cubic struc‐ ture down to 5 K [16], local structural inhomogeneity has been detected in PMN relaxor. In addition to PNR mentioned above, chemically ordering region (COR) [17-19] with size of sev‐ eral nm has been observed in PMN crystal by TEM. It should be noticed that PNR and COR belong to different symmetry groups and are considered to have non-centrosymmetry of*R*3*m* and centrosymmetryof*Fm*3 ¯*m*, respectively. Therefore, there is spontaneous polarization *P*<sup>s</sup> along the <111>c direction of pseudocubic structure in PNRs [20,21], but none of *P*s exists in CORs. In addition to chemical and structural inhomogeneities, we will show that PMN relax‐ or also has inhomogeneity of ferroelectric domain structure. Multiple inhomogeneities are thus considered to play a crucial role in inducing the intriguing behaviors in relaxors.

shows a nearly divergent behavior at the freezing temperature *T*<sup>f</sup> of the spherical glass phase. In sharp contrast, the random field model of Fisch [13], which assumes non-random two-spin exchange, predicts a ferroelectric or ferroelectric domain state in relaxor (Fig. 2(b)). This random Potts field model also predicts a broadening specific heat peak for the glass phase and shows that the latent heat at ferroelectric transition *T*c is so small that it may be difficult to be detected, which reasonably explains the data reported by Moriya et al. [14]. Combining our recent results [15] from the electrical polarization, Raman scattering, TEM measurements and those reported in the literature, here, we propose a physical picture to

**Figure 1.** Temperature dependence of (a) linear dielectric constant [9], (b) cold [35] & thermal [4] neutrons diffuse

PMN is a prototypical relaxor with A(B',B")O3 perovskite structure (Fig. 3), in which B-sites are occupied by two kinds of heterovalent cations. Such chemical inhomogeneity is a com‐ mon feature of relaxor crystals. Although it remains an average centrosymmetric cubic struc‐ ture down to 5 K [16], local structural inhomogeneity has been detected in PMN relaxor. In addition to PNR mentioned above, chemically ordering region (COR) [17-19] with size of sev‐

intensities, and (c) refractive index [3] for PMN relaxor crystal.

**2. Multiple inhomogeneities in relaxors**

understand the dielectric behaviors of Pb(Mg1/3Nb2/3)O3 (PMN) relaxor.

52 Advances in Ferroelectrics

**Figure 2.** Two phase diagrams proposed for relaxor by (a) a dipole glass model [10], and (b) a random field model [13].

**Figure 3.** (a) ABO3perovskite structure. (b) Model for relaxor structure. PNR and COR represent the polar nano-region and chemically order region, respectively. (c) &(d) show two models of atom arrangement for COR. To maintain the electric neutrality, a Nb-rich layer is required for case (c).

### **3. Evolution of the electrical polarization and origin of the huge dielectric responses**

In order to understand the nature of the huge dielectric response and the ground state of the electrical polarization in PMN, it is essential to know the polarization hysteresis of all states including virgin state in PMN crystal. Although there are many reports on the polarization hysteresis of PMN crystal, there is a lack of understanding of the polarization hysteresis of the virgin state. In our polarization measurements, in order to access the virgin state of the crystal at a temperature, it was firstly annealed at 360 K and then cooled to the desired tem‐ perature for the measurements.

Figure 4 shows the *D-E* hysteresis at three typical temperatures observed for (110)c-cut PMN crystal A*t T*=360 K that is greatly higher than the freezing temperature *T*<sup>f</sup> =224 K assumed for PMN crystal [9],there is no remnant polarization within the experimental time scale of *τ*~10 ms (one cycle of the *D–E* loop) and the polarization is history-independent, indicating that the crystal is macroscopically paraelectric at this temperature. When temperature is lower than room temperature (for example, *T*=250 K), remnant polarization was observed but it generally disappears after removing the electric field.

Upon further cooling to temperatures lower than ~220 K,PMN shows polarization hysteresis similar to that of normal ferroelectric [22]. Fig. 4(a) shows an example of the characteristic hysteresis loop in this temperature range. In the virgin state as indicated by the thick red line, it appears that there is no remnant polarization in the crystal at zero electric field. How‐ ever, as increasing the electric field, we can see the gradual growth of the polarization. This is a characteristic behavior of the polarization reversal (switching) in ferroelectric. When the applied field is larger than the coercive field *E*c, ferroelectric domains are aligned along the direction of the electric field, leading to a stable macroscopic polarization in the crystal. This is evident from the fact that the remnant polarization is identical to the saturation polariza‐ tion. These results clearly indicate that PMN is a ferroelectric rather than a dipole glass at temperature lower than 220 K.

**Figure 4.** (a) Polarization hysteresis in PMN crystal at 190 K, 250 K, 360K. (b) Frequency dependence of polarization hysteresis. (c) Relationship between frequency and coercive field determined from the peak of switching currents. Su‐

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As discussed in following, ferroelectric micro-domain and soft-mode behaviors have also been observed at zero field in PMN in our measurements. Also, lowering of symmetry of local structure at zero field was also revealed around 210 K by a NMR study [21]. All these results direct to the fact of occurrence of a ferroelectric state at zero field in PMN crystal. We therefore consider that it is more rational to attribute *E*<sup>c</sup> to the coercive field required for the domain switching rather than the critical field for a field-induced phase transition. In fact, the exponential relationship of coercive field with frequency is well-known as Merz's law (*f*=1/*τ*∝exp(-*α*/*E*c), *τ*=switching time, *α*=activation field) [23-28] in the normal ferroelectrics (for example, BaTiO3,TGS) [23,24]. Using this relationship, the activation field is estimated to be 83. 5 kV/cm for PMN crystal at 200 K (Fig. 4c), which is one order of magnitude greater than that of the BaTiO3 crystal. This indicates that the domain switching in PMN become more and more difficult as lowering temperatures. For example, when an electric field of 1 kV/cm is applied to the crystal at 200 K, an unpractical time of 2. 3×1029s (~7. 3×1021 years) is required for the domain switching. Actually, it is impossible to observe the spontaneous po‐

perscript *i* denotes the value observed for the virgin state.

larization by this weak field within a limit time at this temperature.

There are many reports on the electric-field induced phase transition in PMN. On the basis of the change of dielectric constant under the application of a DC electric field, Colla et al. proposed an *E-T* phase diagram for PMN [8], which suggests a phase transition from a glass phase to a ferroelectric phase at a critical field of *E*<sup>t</sup> =1. 5kV/cm in the temperature range of 160 K - 200 K. However, from our polarization results, we cannot find such a critical field except the coercive field *E*c. If we assume that *E*<sup>c</sup> is the critical field, then its value completely disagrees with the reported value. We observed that *E*<sup>c</sup> increases rapidly with lowering the temperature, for example, *E*<sup>c</sup> can reach a value of ~11 kV/cm at 180 K, which is about 7 times of *E*<sup>t</sup> . Moreover, *E*c is strongly dependent on frequency (Fig. 4b), and has an exponential form(1/*f*∝ exp (α/*E*c) (Fig. 4c), which will result in an undefined critical field if we assume *E*c=*E*<sup>t</sup> because *E*c is dependent on the measurement time-scale.

**3. Evolution of the electrical polarization and origin of the huge dielectric**

In order to understand the nature of the huge dielectric response and the ground state of the electrical polarization in PMN, it is essential to know the polarization hysteresis of all states including virgin state in PMN crystal. Although there are many reports on the polarization hysteresis of PMN crystal, there is a lack of understanding of the polarization hysteresis of the virgin state. In our polarization measurements, in order to access the virgin state of the crystal at a temperature, it was firstly annealed at 360 K and then cooled to the desired tem‐

Figure 4 shows the *D-E* hysteresis at three typical temperatures observed for (110)c-cut PMN

PMN crystal [9],there is no remnant polarization within the experimental time scale of *τ*~10 ms (one cycle of the *D–E* loop) and the polarization is history-independent, indicating that the crystal is macroscopically paraelectric at this temperature. When temperature is lower than room temperature (for example, *T*=250 K), remnant polarization was observed but it

Upon further cooling to temperatures lower than ~220 K,PMN shows polarization hysteresis similar to that of normal ferroelectric [22]. Fig. 4(a) shows an example of the characteristic hysteresis loop in this temperature range. In the virgin state as indicated by the thick red line, it appears that there is no remnant polarization in the crystal at zero electric field. How‐ ever, as increasing the electric field, we can see the gradual growth of the polarization. This is a characteristic behavior of the polarization reversal (switching) in ferroelectric. When the applied field is larger than the coercive field *E*c, ferroelectric domains are aligned along the direction of the electric field, leading to a stable macroscopic polarization in the crystal. This is evident from the fact that the remnant polarization is identical to the saturation polariza‐ tion. These results clearly indicate that PMN is a ferroelectric rather than a dipole glass at

There are many reports on the electric-field induced phase transition in PMN. On the basis of the change of dielectric constant under the application of a DC electric field, Colla et al. proposed an *E-T* phase diagram for PMN [8], which suggests a phase transition from a glass

160 K - 200 K. However, from our polarization results, we cannot find such a critical field except the coercive field *E*c. If we assume that *E*<sup>c</sup> is the critical field, then its value completely disagrees with the reported value. We observed that *E*<sup>c</sup> increases rapidly with lowering the temperature, for example, *E*<sup>c</sup> can reach a value of ~11 kV/cm at 180 K, which is about 7 times

. Moreover, *E*c is strongly dependent on frequency (Fig. 4b), and has an exponential form(1/*f*∝ exp (α/*E*c) (Fig. 4c), which will result in an undefined critical field if we assume

=224 K assumed for

=1. 5kV/cm in the temperature range of

crystal A*t T*=360 K that is greatly higher than the freezing temperature *T*<sup>f</sup>

generally disappears after removing the electric field.

phase to a ferroelectric phase at a critical field of *E*<sup>t</sup>

because *E*c is dependent on the measurement time-scale.

**responses**

54 Advances in Ferroelectrics

perature for the measurements.

temperature lower than 220 K.

of *E*<sup>t</sup>

*E*c=*E*<sup>t</sup>

**Figure 4.** (a) Polarization hysteresis in PMN crystal at 190 K, 250 K, 360K. (b) Frequency dependence of polarization hysteresis. (c) Relationship between frequency and coercive field determined from the peak of switching currents. Su‐ perscript *i* denotes the value observed for the virgin state.

As discussed in following, ferroelectric micro-domain and soft-mode behaviors have also been observed at zero field in PMN in our measurements. Also, lowering of symmetry of local structure at zero field was also revealed around 210 K by a NMR study [21]. All these results direct to the fact of occurrence of a ferroelectric state at zero field in PMN crystal. We therefore consider that it is more rational to attribute *E*<sup>c</sup> to the coercive field required for the domain switching rather than the critical field for a field-induced phase transition. In fact, the exponential relationship of coercive field with frequency is well-known as Merz's law (*f*=1/*τ*∝exp(-*α*/*E*c), *τ*=switching time, *α*=activation field) [23-28] in the normal ferroelectrics (for example, BaTiO3,TGS) [23,24]. Using this relationship, the activation field is estimated to be 83. 5 kV/cm for PMN crystal at 200 K (Fig. 4c), which is one order of magnitude greater than that of the BaTiO3 crystal. This indicates that the domain switching in PMN become more and more difficult as lowering temperatures. For example, when an electric field of 1 kV/cm is applied to the crystal at 200 K, an unpractical time of 2. 3×1029s (~7. 3×1021 years) is required for the domain switching. Actually, it is impossible to observe the spontaneous po‐ larization by this weak field within a limit time at this temperature.

The high-resolution data of the polarization obtained by a 14-bit oscilloscope allow us to cal‐ culate the linear and nonlinear dielectric susceptibilities (defined by the expansion *P*=*ε*0(χ1*E* +χ3*E*<sup>3</sup> + )) directly from the *D-E* hysteresis by differentiating the polarization with respective to the electric field. The calculated results are summarized in Fig. 5 for various electric fields in the virgin state and the zero electric field after the polarization reversal. These results al‐ low us to have a deep insight into the nature of abnormal dielectric behaviors and the phase transition in PMN relaxor. Fig. 5(a) shows the linear dielectric response for the virgin state in various electric fields. When comparing the response obtained at zero field with that ob‐ tained by LCR meter at the frequency corresponding to the sampling rate used in *D-E* hyste‐ resis measurements, one might find that they both behavior in the same way with the temperature. As mentioned above that the polarization response under the electric field in the virgin state is essentially due to the polarization reorientation, we therefore can reasona‐ bly attribute the dielectric anomaly usually observed in PMN relaxor to the polarization re‐ orientation. This indicates that the reorientation of the PNRs dominates the huge dielectric response in PMN relaxor. Fig. 5 (a) also shows that the peak of dielectric response shifts to lower temperature at a higher electric field. This means that the activation field required for domain switching increases with lowering the temperature.

The nonlinear dielectric susceptibility *ε*3v and its scaled value *a*3v for the virgin state are given in Fig. 5(c) and (d), respectively. *ε*3vshows a broad peak around 255 K, which is in good agree‐ ment with those obtained by Levstik et al. using a lock-in to wave analyzer technique for vari‐ ous frequencies [9]. Levstik et al. attributed this behavior to the freezing of dipole glass at *T*f =224 K in PMN. However, this picture is inconsistent with the results shown in the above polarization measurements, which indicates that PMN relaxor is ferroelectric but not glass at *T*<*T*<sup>f</sup> . Such dipole glass picture is also excluded by the results of the scaled susceptibility *a*<sup>3</sup> shown in Fig. 5(d), and those reported in the previous studies [29, 30]. We can see that there is no divergent behavior of *a*3 in PMN relaxor. This result again suggests that there is no freez‐ ing of dipole glass in PMN as predicted for the dipole glass model. The observed anomaly of nonlinear dielectric susceptibility around 255 K is more consistent with the phase diagram of the random field model proposed by Fisch [13], in which a glass phase occurs between the paraelectric phase and the ferroelectric phase in relaxors. Such anomaly of nonlinear dielec‐ tric susceptibility may be a manifestation of spherical dipole glass with random interaction used in SRBRF model, and its nature requires further theoretical investigations.

In the random field model, the ferroelectric phase transition is suggested to be smeared due to the quenched random fields, but it may be visible if the random fields are overcome by an external electric field [12]. This ferroelectric phase transition has been convincingly shown by the sharp peak of the linear dielectric susceptibility *ε*1p (Fig. 5(b)) and the anomaly of its nonlinear components *ε*3p and *a*3p (solid circles in Fig. 5(c) and (d)) when ferroelectric do‐ mains are aligned by an external field. The sharp peak of linear susceptibility indicates that the ferroelectric phase transition occurs at *T*=*T*c=225 K, around which Curie-Weise law was observed. The value of Curie constant is estimated to be *C* = 2. 05 \*105 K, which is character‐ istic of that of the displacive-type phase transition, suggesting a soft mode-driven phase transition in this system. This conclusion is supported by the occurrence of soft–mode in the crystal observed by neutron [31] and Raman scattering measurements [32].

**Figure 5.** Dielectric responses in PMN crystal. (a) Chang of linear dielectric responses (ε1=∂*D*/∂*E*|*E*) with the electric field for the virgin state. Inset shows an example of the polarization and dielectric responses, in which the thick lines indicate the virgin state. Dielectric constant obtained by LCR impedance measurements at an ac level of 1 V/cm (red solid line) is also shown for comparison. (b) Linear dielectric response (ε1p=∂*D*/∂*E*|E=0) at zero field after the polarization reversal (indi‐ cated by the square in inset of (a)), and its inverse. (c) Nonlinear dielectric constant(ε3=∂<sup>3</sup>*D*/∂<sup>3</sup>*E*|*E*=0) at zero field for the vir‐

4|E=0 for these two states.

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gin state, and the state after the polarization reversal. (d) The corresponding *a*3=ε3/ε<sup>1</sup>

The high-resolution data of the polarization obtained by a 14-bit oscilloscope allow us to cal‐ culate the linear and nonlinear dielectric susceptibilities (defined by the expansion *P*=*ε*0(χ1*E*

The nonlinear dielectric susceptibility *ε*3v and its scaled value *a*3v for the virgin state are given in Fig. 5(c) and (d), respectively. *ε*3vshows a broad peak around 255 K, which is in good agree‐ ment with those obtained by Levstik et al. using a lock-in to wave analyzer technique for vari‐ ous frequencies [9]. Levstik et al. attributed this behavior to the freezing of dipole glass at

=224 K in PMN. However, this picture is inconsistent with the results shown in the above polarization measurements, which indicates that PMN relaxor is ferroelectric but not glass at

In the random field model, the ferroelectric phase transition is suggested to be smeared due to the quenched random fields, but it may be visible if the random fields are overcome by an external electric field [12]. This ferroelectric phase transition has been convincingly shown by the sharp peak of the linear dielectric susceptibility *ε*1p (Fig. 5(b)) and the anomaly of its nonlinear components *ε*3p and *a*3p (solid circles in Fig. 5(c) and (d)) when ferroelectric do‐ mains are aligned by an external field. The sharp peak of linear susceptibility indicates that the ferroelectric phase transition occurs at *T*=*T*c=225 K, around which Curie-Weise law was observed. The value of Curie constant is estimated to be *C* = 2. 05 \*105 K, which is character‐ istic of that of the displacive-type phase transition, suggesting a soft mode-driven phase transition in this system. This conclusion is supported by the occurrence of soft–mode in the

used in SRBRF model, and its nature requires further theoretical investigations.

crystal observed by neutron [31] and Raman scattering measurements [32].

. Such dipole glass picture is also excluded by the results of the scaled susceptibility *a*<sup>3</sup> shown in Fig. 5(d), and those reported in the previous studies [29, 30]. We can see that there is no divergent behavior of *a*3 in PMN relaxor. This result again suggests that there is no freez‐ ing of dipole glass in PMN as predicted for the dipole glass model. The observed anomaly of nonlinear dielectric susceptibility around 255 K is more consistent with the phase diagram of the random field model proposed by Fisch [13], in which a glass phase occurs between the paraelectric phase and the ferroelectric phase in relaxors. Such anomaly of nonlinear dielec‐ tric susceptibility may be a manifestation of spherical dipole glass with random interaction

domain switching increases with lowering the temperature.

+ )) directly from the *D-E* hysteresis by differentiating the polarization with respective to the electric field. The calculated results are summarized in Fig. 5 for various electric fields in the virgin state and the zero electric field after the polarization reversal. These results al‐ low us to have a deep insight into the nature of abnormal dielectric behaviors and the phase transition in PMN relaxor. Fig. 5(a) shows the linear dielectric response for the virgin state in various electric fields. When comparing the response obtained at zero field with that ob‐ tained by LCR meter at the frequency corresponding to the sampling rate used in *D-E* hyste‐ resis measurements, one might find that they both behavior in the same way with the temperature. As mentioned above that the polarization response under the electric field in the virgin state is essentially due to the polarization reorientation, we therefore can reasona‐ bly attribute the dielectric anomaly usually observed in PMN relaxor to the polarization re‐ orientation. This indicates that the reorientation of the PNRs dominates the huge dielectric response in PMN relaxor. Fig. 5 (a) also shows that the peak of dielectric response shifts to lower temperature at a higher electric field. This means that the activation field required for

+χ3*E*<sup>3</sup>

56 Advances in Ferroelectrics

*T*f

*T*<*T*<sup>f</sup>

**Figure 5.** Dielectric responses in PMN crystal. (a) Chang of linear dielectric responses (ε1=∂*D*/∂*E*|*E*) with the electric field for the virgin state. Inset shows an example of the polarization and dielectric responses, in which the thick lines indicate the virgin state. Dielectric constant obtained by LCR impedance measurements at an ac level of 1 V/cm (red solid line) is also shown for comparison. (b) Linear dielectric response (ε1p=∂*D*/∂*E*|E=0) at zero field after the polarization reversal (indi‐ cated by the square in inset of (a)), and its inverse. (c) Nonlinear dielectric constant(ε3=∂<sup>3</sup>*D*/∂<sup>3</sup>*E*|*E*=0) at zero field for the vir‐ gin state, and the state after the polarization reversal. (d) The corresponding *a*3=ε3/ε<sup>1</sup> 4|E=0 for these two states.

Here, we can see that there are two characteristic temperatures in relaxors: Burns tempera‐ ture *T*Burns and ferroelectric phase transition temperature *T*c. A*tT*Burns, local polarizations (PNRs) begin to occur. Before the ferroelectric transition occurs, PNRs are dynamic, and more importantly, interactions among them are random. Consequently, the existence of PNRs be‐ tween *T*c and *T*Burns can be considered as precursor phenomenon in a phase transition [33]. Ac‐ tually, such precursor phenomenon has also been observed in the normal ferroelectric BaTiO3at temperatures far above *T*c[34]. A difference between BaTiO3 and PMN relaxor is that the temperate region of precursor existence in PMN is greatly lager than in BaTiO3. To probe the precursor behavior, one has to consider the time scale used. For example, we cannot de‐ tect a spontaneous polarization at ms scale by *D-E* measurements for PMN at room tempera‐ ture, but at the probe time scale of 2 ns, cold neutron high-flux backscattering spectrometer can detect PNRs up to ~400K [35] (Fig. 1b). In contrast, due to a shorter probe time scale of ~6 ps [4], thermal neutron scattering can probe PNRs up to 600 K, which is close to *T*Burns deter‐ mined by the optical measurements that have the shortest probe time scale.

### **4. Soft mode behaviors in PMN relaxor**

In the displacive-type ferroelectrics, soft-modes should occur in the lattice dynamics of the crystal. Actually, in a study of neutron inelastic scattering, a FE soft mode was revealed to re‐ covers, i. e., becomes underdamped, below 220 K, and from there its energy squared (*ħω*s) 2 in‐ creases linearly with decreasing *T* as for normal FEs below *T*c(see also Fig. 6(e)) [31]. This has long been a puzzle for PMN relaxor: how can this be, since it has been thought that PMN re‐ mains cubic to at least 5 K [16]? However, such soft-mode behavior is exactly consistent with our results from polarization measurements shown in previous section, which show a ferro‐ electric phase transition at *T*c~225 K. Our Raman scattering measurements also support the oc‐ currence of FE softmode in PMN relaxor [32].

In Raman scattering studies for relaxors, the multiple inhomogeneities due to the coexistence of different symmetry regions such as the PNR and COR has been a tremendous barrier to clarify the dynamical aspect of relaxor behavior in PMN. In particular, the intense tempera‐ ture-independent peak at 45 cm−1 (indicated by↓in Fig. 6(a)), which stems from the COR with *Fm*3 ¯*m* [32], always precludes a detailed investigation of low-wave number spectra of PMN crystal. Our angular dependence of the Raman spectra together with the results from the Raman tensor calculations clearly indicate that the strong *F*2*<sup>g</sup>*mode located at 45 cm−1 can be eliminated by choosing a crossed Nicols configuration with the polarization direction of the incident laser along <110>c direction (see right panel in Fig. 6(c)).

Such special configuration allows us to observe the other low-wave number modes easily. Fig. 6 (d) shows the spectra obtained by this configuration. At the lowest temperature, a well-de‐ fined mode can be seen from the spectrum, which softens as increasing the temperature, indi‐ cating the occurrence of FE soft mode in PMN relaxor. Due to the multiple inhomogeneities of the system, the shape of soft-mode of PMN relaxor is not as sharp as that observed in normal displacive-type ferroelectrics. However, we still can estimate its frequency reliably from the careful spectrum analysis. Its temperature dependence is shown in Fig. 6(e) (indicated by ●) in comparison with the results obtained by neutron inelastic scattering (○) [31].

**Figure 6.** a) Room-temperature Raman spectra in PMN observed by the parallel (upper panel) and crossed (bottom panel) Nicols configurations, with the polarization direction of the incident laser parallel to <100>c(*P* // <100>c). (b) Scattering configurations used in measurements on the angular dependence of the Raman spectra. (c) Angular de‐ pendence of the low-wave number Raman spectra obtained at room temperature. (d) Temperature dependence of Raman spectra observed by the crossed Nicols configuration with the polarization direction of the incident laser along <110>c direction (*P* // <110>c). (e) Soft mode wave numbers obtained by Raman scattering (●) and by neutron inelas‐

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tic scattering by Wakimoto et al. (○) [31].

Pb(Mg1/3Nb2/3)O3 (PMN) Relaxor: Dipole Glass or Nano-Domain Ferroelectric? http://dx.doi.org/10.5772/52139 59

Here, we can see that there are two characteristic temperatures in relaxors: Burns tempera‐ ture *T*Burns and ferroelectric phase transition temperature *T*c. A*tT*Burns, local polarizations (PNRs) begin to occur. Before the ferroelectric transition occurs, PNRs are dynamic, and more importantly, interactions among them are random. Consequently, the existence of PNRs be‐ tween *T*c and *T*Burns can be considered as precursor phenomenon in a phase transition [33]. Ac‐ tually, such precursor phenomenon has also been observed in the normal ferroelectric BaTiO3at temperatures far above *T*c[34]. A difference between BaTiO3 and PMN relaxor is that the temperate region of precursor existence in PMN is greatly lager than in BaTiO3. To probe the precursor behavior, one has to consider the time scale used. For example, we cannot de‐ tect a spontaneous polarization at ms scale by *D-E* measurements for PMN at room tempera‐ ture, but at the probe time scale of 2 ns, cold neutron high-flux backscattering spectrometer can detect PNRs up to ~400K [35] (Fig. 1b). In contrast, due to a shorter probe time scale of ~6 ps [4], thermal neutron scattering can probe PNRs up to 600 K, which is close to *T*Burns deter‐

In the displacive-type ferroelectrics, soft-modes should occur in the lattice dynamics of the crystal. Actually, in a study of neutron inelastic scattering, a FE soft mode was revealed to re‐ covers, i. e., becomes underdamped, below 220 K, and from there its energy squared (*ħω*s)

creases linearly with decreasing *T* as for normal FEs below *T*c(see also Fig. 6(e)) [31]. This has long been a puzzle for PMN relaxor: how can this be, since it has been thought that PMN re‐ mains cubic to at least 5 K [16]? However, such soft-mode behavior is exactly consistent with our results from polarization measurements shown in previous section, which show a ferro‐ electric phase transition at *T*c~225 K. Our Raman scattering measurements also support the oc‐

In Raman scattering studies for relaxors, the multiple inhomogeneities due to the coexistence of different symmetry regions such as the PNR and COR has been a tremendous barrier to clarify the dynamical aspect of relaxor behavior in PMN. In particular, the intense tempera‐ ture-independent peak at 45 cm−1 (indicated by↓in Fig. 6(a)), which stems from the COR with

¯*m* [32], always precludes a detailed investigation of low-wave number spectra of PMN crystal. Our angular dependence of the Raman spectra together with the results from the Raman tensor calculations clearly indicate that the strong *F*2*<sup>g</sup>*mode located at 45 cm−1 can be eliminated by choosing a crossed Nicols configuration with the polarization direction of the

Such special configuration allows us to observe the other low-wave number modes easily. Fig. 6 (d) shows the spectra obtained by this configuration. At the lowest temperature, a well-de‐ fined mode can be seen from the spectrum, which softens as increasing the temperature, indi‐ cating the occurrence of FE soft mode in PMN relaxor. Due to the multiple inhomogeneities of the system, the shape of soft-mode of PMN relaxor is not as sharp as that observed in normal displacive-type ferroelectrics. However, we still can estimate its frequency reliably from the careful spectrum analysis. Its temperature dependence is shown in Fig. 6(e) (indicated by ●)

in comparison with the results obtained by neutron inelastic scattering (○) [31].

2 in‐

mined by the optical measurements that have the shortest probe time scale.

**4. Soft mode behaviors in PMN relaxor**

currence of FE softmode in PMN relaxor [32].

incident laser along <110>c direction (see right panel in Fig. 6(c)).

*Fm*3

58 Advances in Ferroelectrics

**Figure 6.** a) Room-temperature Raman spectra in PMN observed by the parallel (upper panel) and crossed (bottom panel) Nicols configurations, with the polarization direction of the incident laser parallel to <100>c(*P* // <100>c). (b) Scattering configurations used in measurements on the angular dependence of the Raman spectra. (c) Angular de‐ pendence of the low-wave number Raman spectra obtained at room temperature. (d) Temperature dependence of Raman spectra observed by the crossed Nicols configuration with the polarization direction of the incident laser along <110>c direction (*P* // <110>c). (e) Soft mode wave numbers obtained by Raman scattering (●) and by neutron inelas‐ tic scattering by Wakimoto et al. (○) [31].

demonstrated in the NMR study [21]. According to previous results, the local symmetry in the PNR changes from cubic to rhombohedral. Therefore, the soft mode can be assumed to split from the *F*1u mode to the *A*1 and *E* modes. Generally, the *A*1 mode is higher in wave

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In a short summary, we may say that the polarization in PMN is induced by the soft mode. This interpretation is essentially consistent with the results described in the polarization measurements, and the results obtained in previous neutron studies [36, 37], in which the crystallographic structure of PNR is attributed to the displacement pattern of the soft mode. The results of the Raman study also support that a ferroelectric state exists in PMN even at

In order to understand the microstructures of COR and PNR together with the domain structures and its evolution with temperature in the ferroelectric phase of PMN relaxor, we have carried out a detailed TEM observation. The typical results are summarized in Fig. 7. As shown in Fig. 7 (a")-(c"), COR was found to be spherical shape and has size less than 5 nm. It is very stable and remains unchanged within the temperature range of 130 K-675K. In the TEM observation, large amount of CORs were found to distribute in the PMN crystal. In a previous HRTEM study, its volume fraction has been estimated to be ~1/3 of the crystal

[18]. CORs are thus considered to be the intense sources of the strong random fields.

0 100 200 300 400 500 600

0 100 200 300 400 500 600

*T* (K)

**Figure 8.** Temperature variation of PNR observed by (a) TEM [38]and (b) Neutron scattering [4]. In (b), ξ is the correla‐ tion length, and *I*0 is the integrated diffuse scattering intensity and can be written to Nξ*<sup>3</sup>*|Qδ|*<sup>2</sup>*, where *N* is the total

*N* |*Q*d| 2

Indensity

**Sperical** TEM

Yohsida et al.

Size

**~Number of PNR** Tc

Neutrons Xu. et al. PRB**69** 064112

0

50

*I*

*/*x

~*N* 

*Q*| d|2

Indensity

3

*0*

100

0

5

Correlation length x (nm)

10

0

10

Size (nm)

20

30

(b)

(a)

x

number of PNR, δ is the average displacement of atoms within the PNR, and *Q* is the wave vector.

**Elliptical**

number than the *E* mode due to the depolarization field effect.

**5. Ferroelectric domain structures observed by TEM**

the zero-bias field.

**Figure 7.** a)-(c) Temperature variation of TEM images observed for PMN relaxor. (a')-(c') & (a")-(c") show images of PNR and COR derived from (a)-(c). (d)Micrometric domain structure observed in the ferroelectric phase at 130K. (e) Schematic domain patterns shown in (d). Arrows and lines indicate the polarization directions and domain bounda‐ ries, respectively.

We find that the soft-mode exhibits softening towards *T*<sup>c</sup> on heating and follows the conven‐ tional Curie–Weiss law (solid line) over a large temperature region. However, upon further heating, the soft mode becomes over damped in a temperature region extending over ∼200 K, which does not allow us to estimate the frequency of the mode. At temperatures above 480 K, the soft mode recovers the under damped oscillation and hardens as the temperature increases. These phenomena are very similar to those revealed by neutron inelastic scatter‐ ing [31]. A major difference is that the wave number of the soft mode in the present study is significantly lower than that observed by the neutron inelastic scattering. This can be rea‐ sonably understood by the splitting of the soft mode due to the lowering of symmetry as demonstrated in the NMR study [21]. According to previous results, the local symmetry in the PNR changes from cubic to rhombohedral. Therefore, the soft mode can be assumed to split from the *F*1u mode to the *A*1 and *E* modes. Generally, the *A*1 mode is higher in wave number than the *E* mode due to the depolarization field effect.

In a short summary, we may say that the polarization in PMN is induced by the soft mode. This interpretation is essentially consistent with the results described in the polarization measurements, and the results obtained in previous neutron studies [36, 37], in which the crystallographic structure of PNR is attributed to the displacement pattern of the soft mode. The results of the Raman study also support that a ferroelectric state exists in PMN even at the zero-bias field.

### **5. Ferroelectric domain structures observed by TEM**

**Figure 7.** a)-(c) Temperature variation of TEM images observed for PMN relaxor. (a')-(c') & (a")-(c") show images of PNR and COR derived from (a)-(c). (d)Micrometric domain structure observed in the ferroelectric phase at 130K. (e) Schematic domain patterns shown in (d). Arrows and lines indicate the polarization directions and domain bounda‐

We find that the soft-mode exhibits softening towards *T*<sup>c</sup> on heating and follows the conven‐ tional Curie–Weiss law (solid line) over a large temperature region. However, upon further heating, the soft mode becomes over damped in a temperature region extending over ∼200 K, which does not allow us to estimate the frequency of the mode. At temperatures above 480 K, the soft mode recovers the under damped oscillation and hardens as the temperature increases. These phenomena are very similar to those revealed by neutron inelastic scatter‐ ing [31]. A major difference is that the wave number of the soft mode in the present study is significantly lower than that observed by the neutron inelastic scattering. This can be rea‐ sonably understood by the splitting of the soft mode due to the lowering of symmetry as

ries, respectively.

60 Advances in Ferroelectrics

In order to understand the microstructures of COR and PNR together with the domain structures and its evolution with temperature in the ferroelectric phase of PMN relaxor, we have carried out a detailed TEM observation. The typical results are summarized in Fig. 7. As shown in Fig. 7 (a")-(c"), COR was found to be spherical shape and has size less than 5 nm. It is very stable and remains unchanged within the temperature range of 130 K-675K. In the TEM observation, large amount of CORs were found to distribute in the PMN crystal. In a previous HRTEM study, its volume fraction has been estimated to be ~1/3 of the crystal [18]. CORs are thus considered to be the intense sources of the strong random fields.

**Figure 8.** Temperature variation of PNR observed by (a) TEM [38]and (b) Neutron scattering [4]. In (b), ξ is the correla‐ tion length, and *I*0 is the integrated diffuse scattering intensity and can be written to Nξ*<sup>3</sup>*|Qδ|*<sup>2</sup>*, where *N* is the total number of PNR, δ is the average displacement of atoms within the PNR, and *Q* is the wave vector.

In contrast to CORs, PNRs exhibit remarkable change with temperature. As shown in Fig. 7(a')-(c') and Fig. 8, PNRs with size of several nm were found to occur in the crystal for *T*<*T*Burns. These spherical PNRs show continuous growth as lowering the temperature. How‐ ever, PNRs change from spherical shape to elliptical shape around *T*c. Associating with the change in shape, its intensity was also found to drop rapidly. These results are consistent with those derived from neutron scattering (Fig. 8(b)) [4]. Neutron scattering study by Xu et al. [4] shows that the ''correlation length'' *ξ*, which is a direct measure of the length scale of the PNR, increases on cooling and changes remarkably around *T*c. At the same time, the number of PNR increases on cooling from high temperatures and then drops dramatically at around *T*c, remaining roughly constant below *T*c.

together to form micrometric domain in the FE phase. Thus, our TEM observations support the random field model suggested by Fisch [13]. It should be emphasized that PNRs cannot merge together completely due to the blocking of the intense CORs in the crystal (Fig. 10).

Pb(Mg1/3Nb2/3)O3 (PMN) Relaxor: Dipole Glass or Nano-Domain Ferroelectric?

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63

Here, we made a discussion on the volume fraction of PNRs in PMN crystal. Neutron scatter‐ ing technique has been used to estimate the volume fraction of PNRs. Fig. 9 replots two re‐ sults reported by Jeong et al. [37] and Uesu et al. [39], respectively. Both studies indicate that PNRs occupy a volume fraction > 25% at the lowest measurement temperature. This volume fraction is larger than the threshold of 22% to form a percolated ferroelectric state with an el‐ lipsoidal-shape [40], supporting again the picture of a ferroelectric state in PMN relaxor.

In summary, we propose a physics picture for relaxors. Figure 10 schematically shows a model of structure evolution in PMN relaxor. Since COR has been observed at *T*>*T*Burns, it can

ing the temperature. Around room temperature, PNRs grow to a size of about 10 nm. For *T*<*T*<sup>c</sup> (~225K), neighboring PNRs merge together to form elliptical shape with anisotropy, as‐ sociating with the reduction of its number. Due to the blocking by the intense CORs, indi‐ vidual PNR merely grows to a size of about 20 nm at *T*<*T*c. However, PNRs with elliptical

¯*m*phasein this high temperature. Upon cooling, spherical PNRs occur

¯*m* phase for *T*<*T*Burns. Both number and size of PNR increase as lower‐

¯*m* symmetry

be considered that there is a coexistence of paraelectric phase of COR with *Fm*3

**Figure 10.** A physics picture of structure evolution in PMN relaxor.

**6. A physics picture of relaxor**

and paraelectric *Pm*3

from paraelectric *Pm*3

Associating with the change in the number and the shape of PNR, micrometric ferroelectric domains were found to occur for *T*<*T*c. It is because of growing into macroscopic domain that the number of PNR drop sharply in the ferroelectric phase. Figs. 7(d) and (e) show the struc‐ ture of a FE micrometric domain in the FE phase of 130 Kand its schematic patterns, respec‐ tively. The micrometric domains are formed in the crystal with the spontaneous *P*s along the <111>c direction. In comparison with the domain structure of normal ferroelectric such as Ba‐ TiO3, domain size is relatively small and the domain boundaries blur in PMN relaxor.

**Figure 9.** Volume fraction of PNR estimated from neutron scattering measurements [37,39]. Solid line denotes the threshold of percolation for elliptical shape [40].

The occurrence of micrometric domains, the soft-mode observed by Raman scattering, and the macroscopic polarization all direct to the same conclusion: PMN is essentially ferroelec‐ tric but not dipole-glass at *T*<*T*c although it exhibits some unique characteristic properties including broadening soft-mode, smearing domain wall, and very large activation field re‐ quired for domain switching. Our TEM measurements clearly indicate that interactions among PNRs for *T*<*T*c are not random, but cooperative, which is different from the picture expected in the SRBRF model [10]. It is due to such non-random interaction, PNRs team up together to form micrometric domain in the FE phase. Thus, our TEM observations support the random field model suggested by Fisch [13]. It should be emphasized that PNRs cannot merge together completely due to the blocking of the intense CORs in the crystal (Fig. 10).

Here, we made a discussion on the volume fraction of PNRs in PMN crystal. Neutron scatter‐ ing technique has been used to estimate the volume fraction of PNRs. Fig. 9 replots two re‐ sults reported by Jeong et al. [37] and Uesu et al. [39], respectively. Both studies indicate that PNRs occupy a volume fraction > 25% at the lowest measurement temperature. This volume fraction is larger than the threshold of 22% to form a percolated ferroelectric state with an el‐ lipsoidal-shape [40], supporting again the picture of a ferroelectric state in PMN relaxor.

**Figure 10.** A physics picture of structure evolution in PMN relaxor.

### **6. A physics picture of relaxor**

In contrast to CORs, PNRs exhibit remarkable change with temperature. As shown in Fig. 7(a')-(c') and Fig. 8, PNRs with size of several nm were found to occur in the crystal for *T*<*T*Burns. These spherical PNRs show continuous growth as lowering the temperature. How‐ ever, PNRs change from spherical shape to elliptical shape around *T*c. Associating with the change in shape, its intensity was also found to drop rapidly. These results are consistent with those derived from neutron scattering (Fig. 8(b)) [4]. Neutron scattering study by Xu et al. [4] shows that the ''correlation length'' *ξ*, which is a direct measure of the length scale of the PNR, increases on cooling and changes remarkably around *T*c. At the same time, the number of PNR increases on cooling from high temperatures and then drops dramatically at

Associating with the change in the number and the shape of PNR, micrometric ferroelectric domains were found to occur for *T*<*T*c. It is because of growing into macroscopic domain that the number of PNR drop sharply in the ferroelectric phase. Figs. 7(d) and (e) show the struc‐ ture of a FE micrometric domain in the FE phase of 130 Kand its schematic patterns, respec‐ tively. The micrometric domains are formed in the crystal with the spontaneous *P*s along the <111>c direction. In comparison with the domain structure of normal ferroelectric such as Ba‐

0 100 200 300 400 500 600 700

 Uesu et al.JKPS **29**( S703) Jeong et al PRL94(147602)

> *P*C (a/b=1:3)

 *T*C

T (K)

**Figure 9.** Volume fraction of PNR estimated from neutron scattering measurements [37,39]. Solid line denotes the

The occurrence of micrometric domains, the soft-mode observed by Raman scattering, and the macroscopic polarization all direct to the same conclusion: PMN is essentially ferroelec‐ tric but not dipole-glass at *T*<*T*c although it exhibits some unique characteristic properties including broadening soft-mode, smearing domain wall, and very large activation field re‐ quired for domain switching. Our TEM measurements clearly indicate that interactions among PNRs for *T*<*T*c are not random, but cooperative, which is different from the picture expected in the SRBRF model [10]. It is due to such non-random interaction, PNRs team up

TiO3, domain size is relatively small and the domain boundaries blur in PMN relaxor.

around *T*c, remaining roughly constant below *T*c.

62 Advances in Ferroelectrics

0

20

Volume fraction (%)

threshold of percolation for elliptical shape [40].

40

In summary, we propose a physics picture for relaxors. Figure 10 schematically shows a model of structure evolution in PMN relaxor. Since COR has been observed at *T*>*T*Burns, it can be considered that there is a coexistence of paraelectric phase of COR with *Fm*3 ¯*m* symmetry and paraelectric *Pm*3 ¯*m*phasein this high temperature. Upon cooling, spherical PNRs occur from paraelectric *Pm*3 ¯*m* phase for *T*<*T*Burns. Both number and size of PNR increase as lower‐ ing the temperature. Around room temperature, PNRs grow to a size of about 10 nm. For *T*<*T*<sup>c</sup> (~225K), neighboring PNRs merge together to form elliptical shape with anisotropy, as‐ sociating with the reduction of its number. Due to the blocking by the intense CORs, indi‐ vidual PNR merely grows to a size of about 20 nm at *T*<*T*c. However, PNRs with elliptical shape tend to team up together to form a larger domain at *T*<*T*<sup>c</sup> at which a ferroelectric state occurs. The existence of a multi-scale inhomogeneity of ferroelectric domain structure pro‐ vides a key point to understand the huge electromechanical coupling effects in relaxors and piezoelectrics with morphotropic phase boundary (MPB). This also gives idea to design new material with domain structure to have large elastic deformation with the application of an electric or magnetic field [41].

[6] Vugmeister, B. E. Polarization dynamics and formation of polar nanoregions in relax‐

Pb(Mg1/3Nb2/3)O3 (PMN) Relaxor: Dipole Glass or Nano-Domain Ferroelectric?

http://dx.doi.org/10.5772/52139

65

[7] Grinberg, I., Juhas, P., Davies, P. K., Rappe, A. M. Relationship between local struc‐ ture and relaxor behavior in perovskite oxides. *Phys. Rev. Lett.* (2007), 99, 267603. [8] Colla, E.V., Koroleva, E. Yu., Okuneva, N. M., Vakhrushev, S. B. Long-time relaxa‐ tion of the dielectric response in lead magnoniobate. *Phys. Rev. Lett.* (1995), 74,

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### **Acknowledgements**

We thank Mr. M. Yoshida, Prof. N. Yamamoto and Prof. Shin-ya Koshihara of Tokyo Insti‐ tute of Technology for their contributions in this work. We acknowledge the support of a Grant-in-Aid for Scientific Research, MEXT, Japan.

### **Author details**

Desheng Fu1\*, Hiroki Taniguchi2 , Mitsuru Itoh2 and Shigeo Mori3

\*Address all correspondence to: ddsfu@ipc. shizuoka. ac. jp

1 Division of Global Research Leaders, Shizuoka University, Johoku, Naka-ku, Hamamatsu, Japan

2 Materials and Structures Laboratory, Tokyo Institute of Technology, Nagatsuta, Yokoha‐ ma, Japan

3 Department of Materials Science, Osaka Prefecture University, Sakai, Osaka, Japan

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[6] Vugmeister, B. E. Polarization dynamics and formation of polar nanoregions in relax‐ or ferroelectrics. *Phys. Rev. B* (2006), 73, 174117.

shape tend to team up together to form a larger domain at *T*<*T*<sup>c</sup> at which a ferroelectric state occurs. The existence of a multi-scale inhomogeneity of ferroelectric domain structure pro‐ vides a key point to understand the huge electromechanical coupling effects in relaxors and piezoelectrics with morphotropic phase boundary (MPB). This also gives idea to design new material with domain structure to have large elastic deformation with the application of an

We thank Mr. M. Yoshida, Prof. N. Yamamoto and Prof. Shin-ya Koshihara of Tokyo Insti‐ tute of Technology for their contributions in this work. We acknowledge the support of a

1 Division of Global Research Leaders, Shizuoka University, Johoku, Naka-ku, Hamamatsu,

2 Materials and Structures Laboratory, Tokyo Institute of Technology, Nagatsuta, Yokoha‐

3 Department of Materials Science, Osaka Prefecture University, Sakai, Osaka, Japan

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\*Address all correspondence to: ddsfu@ipc. shizuoka. ac. jp

**Acknowledgements**

64 Advances in Ferroelectrics

**Author details**

Japan

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**Chapter 4**

**Self Assembled Nanoscale Relaxor Ferroelectrics**

Worldwide research on relaxor ferroelectric (RFE) has been carried out since 1950s. There are several schools in the world who have defined the evolution and origin of relaxor prop‐ erties in the ferroelectric materials in their own way. One common consensus among the sci‐ entists is the presence of polar nano regions (PNRs) i.e. self assembled domains of short range ordering typically of less than 20-50 nm in the ferroelectric relaxor materials which causes the dielectric dispersion near the phase transition temperature. Another common ap‐ proach has been also believed among the relaxor ferroelectric scientists i.e. the existence of random field at nanoscale. Random field model is considered on the experimental and theo‐ retical facts driven from the different dielectric dispersion response under zero field cooled

Overall relaxor ferroelectrics have been divided into two main categories such as (i) classical relaxors (only short range ordering), the most common example is PbMg1*/*3Nb2*/*3O3 (PMN), (ii) Semi-classical relaxor ferroelectrics (a combination of short and long range ordering). In the latter case, the relaxor properties can be arises due to compositional inhomogeneities, ar‐ tificially induced strain, growth conditions (temperature, pressure, medium, etc.), and due to different ionic radii mismatched based chemical pressure in the matrix. The local domains (PNRs) reorientation induces polar-strain coupling which makes RFE the potentially high piezoelectric coefficient materials widely used in Micro/Nanoelectromechanical system (MEMS/NEMS). The basic features of the RFE over a wide range of temperatures and fre‐ quencies are as follows (i) dispersive and diffuse phase transition, (ii) partially disordered structure, (iii) existence of polar nano-ordered regions, etc. over a wide range of tempera‐ tures and frequencies. Physical and functional properties of relaxors are very different from

> © 2013 Kumar et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

Ashok Kumar, Margarita Correa, Nora Ortega,

Additional information is available at the end of the chapter

(ZFC) and field cooled (FC) among the relaxors.

Salini Kumari and R. S. Katiyar

http://dx.doi.org/10.5772/54298

**1. Introduction**

### **Self Assembled Nanoscale Relaxor Ferroelectrics**

Ashok Kumar, Margarita Correa, Nora Ortega, Salini Kumari and R. S. Katiyar

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54298

### **1. Introduction**

Worldwide research on relaxor ferroelectric (RFE) has been carried out since 1950s. There are several schools in the world who have defined the evolution and origin of relaxor prop‐ erties in the ferroelectric materials in their own way. One common consensus among the sci‐ entists is the presence of polar nano regions (PNRs) i.e. self assembled domains of short range ordering typically of less than 20-50 nm in the ferroelectric relaxor materials which causes the dielectric dispersion near the phase transition temperature. Another common ap‐ proach has been also believed among the relaxor ferroelectric scientists i.e. the existence of random field at nanoscale. Random field model is considered on the experimental and theo‐ retical facts driven from the different dielectric dispersion response under zero field cooled (ZFC) and field cooled (FC) among the relaxors.

Overall relaxor ferroelectrics have been divided into two main categories such as (i) classical relaxors (only short range ordering), the most common example is PbMg1*/*3Nb2*/*3O3 (PMN), (ii) Semi-classical relaxor ferroelectrics (a combination of short and long range ordering). In the latter case, the relaxor properties can be arises due to compositional inhomogeneities, ar‐ tificially induced strain, growth conditions (temperature, pressure, medium, etc.), and due to different ionic radii mismatched based chemical pressure in the matrix. The local domains (PNRs) reorientation induces polar-strain coupling which makes RFE the potentially high piezoelectric coefficient materials widely used in Micro/Nanoelectromechanical system (MEMS/NEMS). The basic features of the RFE over a wide range of temperatures and fre‐ quencies are as follows (i) dispersive and diffuse phase transition, (ii) partially disordered structure, (iii) existence of polar nano-ordered regions, etc. over a wide range of tempera‐ tures and frequencies. Physical and functional properties of relaxors are very different from

© 2013 Kumar et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

normal ferroelectrics due to the presence of self assembled ordered regions in the configura‐ tional disorder matrix.

col [5] who studied the electro-optical effect in a PMN crystal. In case of normal FE, the tem‐ perature dependence of the refraction index *(n)* show a linear decrease of *n* from the paraelectric phase down to phase transition temperature *(Tc)*, below *Tc n* deviates from line‐

with temperature. Burns and Dacol observed deviation from linear *n(T)* in relaxor PMN crystal well above *Tm,* the onset of the deviation was observed at 620 K, almost 350 K above *Tm*. The temperature in which onset of deviation from linearity of *n(T)* occurs (in any relax‐

two components, one due to the linear coefficient of thermal expansion, α, and another due to electrostriction accounted by the electrostrictive coefficients Qijkl. Measures on the temper‐ ature dependence of thermal expansion in RFE crystals have shown that the contribution

The existence of PNRs well above *Tm* and the growth of these domains with decreasing tem‐ perature have been demonstrated by TEM [8 - 12]. Transmission electron studies also ac‐ count for the B-cation order in complex perovskites [11 - 12]. Ordering of the B-site cations

The general formula of perovskite having A and B-site cations with different charge states can be written as A′xA′′1−xB′yB′′1−yO3. The randomness occupancy of A′,A′′ and B′,B′′ in A, B site respectively, depends on the ionic sizes, distribution of cations ordering at sub lattices and charge of cations. If the charges of cations at B-sites are same it is unlikely to have the polar ran‐ domness at nanoscale. Long range order (LRO) is defined as a continuous and ordered distri‐ bution of the B cations on the nearest neighbor sites. The short coherence length of LRO occurs when the size of the ordered domains are in a range of 20 to 800 Å in diameter. The long coher‐ ence range of LRO occur when the size of the ordered domains are much greater than 1000 Å. Randall et al. made a classification of complex lead perovskite and their solid solutions based on B-cation order studied by TEM and respective dielectric, X-rays and optical properties [11]- [12]. This classification divides the complex lead perovskites into three subgroups; random oc‐ cupation or disordered, nanoscale or short coherent long-range order and long coherent long-

Diffuse phase transition behavior is characteristic of the disordered structures. In these structures, random lattice disorder introduces dipolar impurities and defects that influence the static and dynamic properties of these materials. The presence of the dipolar entities on a lattice site of the highly polarizable FE structure, induced dipoles in a region determined by the correlation length (*rc*). The correlation length is a measure of the extent of dipoles that respond in a correlated manner. In normal FE, *rc* is larger than the lattice parameter *(a)* and it is strongly temperature dependent. On decreasing the temperature, a faster increase of *rc* promotes growing of polar domains yielding a static cooperative long-range ordered FE

occurs if there is sufficiently large interaction energy between neighboring cations.

or) had been identified in the literature as the Burns temperature (*TB* or *Td*).

and it increases as the polarization changes

Self Assembled Nanoscale Relaxor Ferroelectrics

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71

. Thermal strain in a cubic perovskite has

arity. The deviation is proportional to the *P2*

Electrostriction is a property which depends on *P2*

**3. Theory of relaxor ferroelectricity**

range order of B-site cations.

due to electrostriction vanishes only above *TB* where P=0.

This article deals with the historical development of the relaxor ferroelectrics, microstructur‐ al origin of RFE, strain mediated conversion of normal ferroelectric to relaxor ferroelectrics, superlattice relaxors, lead free classical relaxor ferroelectrics, distinction of classical relaxors and semiclassical ferroelectric relaxors based on the polarization study, and their potential applications in microelectronic industry.

### **2. History**

Relaxor ferroelectric materials were discovered in complex perovskites by Smolenskii [1] in early fifties of twentieth centuries. Since then a vast scientific communities have been work‐ ing on relaxor materials and related phenomena but the plethora of mesoscopic and micro‐ scopic heterogeneities over a range of lengths and timescales made difficult to systematic observations [2]. The microscopic image and the dielectric response of relaxor are qualita‐ tively different for that of normal ferroelectrics. The universal signature of RFE is as follows [3 - 18]: (i) the occurrence of broad frequency-dependent peak in the real and imaginary part of the temperature dependent dielectric susceptibility (χ') or permittivity (ε') which shifts to higher temperatures with increasing frequency, G. A. Smolenskii and A. I. Agranokskaya, Soviet Physics Solid State 1, 1429 (1959), (ii) Curie Weiss law is observed at temperatures far above dielectric maxima temperature (*Tm*), (iii) slim hysteresis loops because polar domains are nanosized and randomly oriented, (iv) existence of polar nano domains far above *Tm*, (v) no structural phase transition (overall structure remains pseudo-cubic on decreasing tem‐ perature but rhombohedral-type distortion occurs in crystal at local level) across *Tm* in relax‐ ors in contrast with the normal ferroelectric in which phase transition implies a macroscopic symmetry change,(vii) history dependent functional properties and skin effects (surface be‐ havior is quite different than the interior of the system) (vi) in most cases, relaxor crystal showed no optical birefringence (either far below the freezing temperature and/or under ex‐ ternal electric field).

As we know ferroelectric and related phenomenon are synonyms of ferromagnetism, anti‐ ferromagnetism, spin glass, and random field model which have been originally studied for magnetic materials. Relaxors ferroelectrics possess a random field state, as initially proposed by Westphal, Kleemann and Glinchuk [16-18]. Near Burns temperature, small polar nano re‐ gions start originating in different directions inside the crystal at mesoscopic scale; however, these polar nano regions are not static in nature at Burns temperature. At low temperature these polar nanoregions (PNRs) become static and developed a more defined regions called random field, the nature of these fields are short range order. The evidence of the existence of nanosized polar domains at temperatures well above the dynamic transition temperature comes for experimental observations. The presence of these polar nanoregions (PNRs) was established by measuring properties that depend on square of the polarization (*P2* ) and by direct imaging with TEM studies. The first evidences came from a report by Burns and Da‐ col [5] who studied the electro-optical effect in a PMN crystal. In case of normal FE, the tem‐ perature dependence of the refraction index *(n)* show a linear decrease of *n* from the paraelectric phase down to phase transition temperature *(Tc)*, below *Tc n* deviates from line‐ arity. The deviation is proportional to the *P2* and it increases as the polarization changes with temperature. Burns and Dacol observed deviation from linear *n(T)* in relaxor PMN crystal well above *Tm,* the onset of the deviation was observed at 620 K, almost 350 K above *Tm*. The temperature in which onset of deviation from linearity of *n(T)* occurs (in any relax‐ or) had been identified in the literature as the Burns temperature (*TB* or *Td*).

Electrostriction is a property which depends on *P2* . Thermal strain in a cubic perovskite has two components, one due to the linear coefficient of thermal expansion, α, and another due to electrostriction accounted by the electrostrictive coefficients Qijkl. Measures on the temper‐ ature dependence of thermal expansion in RFE crystals have shown that the contribution due to electrostriction vanishes only above *TB* where P=0.

The existence of PNRs well above *Tm* and the growth of these domains with decreasing tem‐ perature have been demonstrated by TEM [8 - 12]. Transmission electron studies also ac‐ count for the B-cation order in complex perovskites [11 - 12]. Ordering of the B-site cations occurs if there is sufficiently large interaction energy between neighboring cations.

### **3. Theory of relaxor ferroelectricity**

normal ferroelectrics due to the presence of self assembled ordered regions in the configura‐

This article deals with the historical development of the relaxor ferroelectrics, microstructur‐ al origin of RFE, strain mediated conversion of normal ferroelectric to relaxor ferroelectrics, superlattice relaxors, lead free classical relaxor ferroelectrics, distinction of classical relaxors and semiclassical ferroelectric relaxors based on the polarization study, and their potential

Relaxor ferroelectric materials were discovered in complex perovskites by Smolenskii [1] in early fifties of twentieth centuries. Since then a vast scientific communities have been work‐ ing on relaxor materials and related phenomena but the plethora of mesoscopic and micro‐ scopic heterogeneities over a range of lengths and timescales made difficult to systematic observations [2]. The microscopic image and the dielectric response of relaxor are qualita‐ tively different for that of normal ferroelectrics. The universal signature of RFE is as follows [3 - 18]: (i) the occurrence of broad frequency-dependent peak in the real and imaginary part of the temperature dependent dielectric susceptibility (χ') or permittivity (ε') which shifts to higher temperatures with increasing frequency, G. A. Smolenskii and A. I. Agranokskaya, Soviet Physics Solid State 1, 1429 (1959), (ii) Curie Weiss law is observed at temperatures far above dielectric maxima temperature (*Tm*), (iii) slim hysteresis loops because polar domains are nanosized and randomly oriented, (iv) existence of polar nano domains far above *Tm*, (v) no structural phase transition (overall structure remains pseudo-cubic on decreasing tem‐ perature but rhombohedral-type distortion occurs in crystal at local level) across *Tm* in relax‐ ors in contrast with the normal ferroelectric in which phase transition implies a macroscopic symmetry change,(vii) history dependent functional properties and skin effects (surface be‐ havior is quite different than the interior of the system) (vi) in most cases, relaxor crystal showed no optical birefringence (either far below the freezing temperature and/or under ex‐

As we know ferroelectric and related phenomenon are synonyms of ferromagnetism, anti‐ ferromagnetism, spin glass, and random field model which have been originally studied for magnetic materials. Relaxors ferroelectrics possess a random field state, as initially proposed by Westphal, Kleemann and Glinchuk [16-18]. Near Burns temperature, small polar nano re‐ gions start originating in different directions inside the crystal at mesoscopic scale; however, these polar nano regions are not static in nature at Burns temperature. At low temperature these polar nanoregions (PNRs) become static and developed a more defined regions called random field, the nature of these fields are short range order. The evidence of the existence of nanosized polar domains at temperatures well above the dynamic transition temperature comes for experimental observations. The presence of these polar nanoregions (PNRs) was

established by measuring properties that depend on square of the polarization (*P2*

direct imaging with TEM studies. The first evidences came from a report by Burns and Da‐

) and by

tional disorder matrix.

70 Advances in Ferroelectrics

**2. History**

ternal electric field).

applications in microelectronic industry.

The general formula of perovskite having A and B-site cations with different charge states can be written as A′xA′′1−xB′yB′′1−yO3. The randomness occupancy of A′,A′′ and B′,B′′ in A, B site respectively, depends on the ionic sizes, distribution of cations ordering at sub lattices and charge of cations. If the charges of cations at B-sites are same it is unlikely to have the polar ran‐ domness at nanoscale. Long range order (LRO) is defined as a continuous and ordered distri‐ bution of the B cations on the nearest neighbor sites. The short coherence length of LRO occurs when the size of the ordered domains are in a range of 20 to 800 Å in diameter. The long coher‐ ence range of LRO occur when the size of the ordered domains are much greater than 1000 Å. Randall et al. made a classification of complex lead perovskite and their solid solutions based on B-cation order studied by TEM and respective dielectric, X-rays and optical properties [11]- [12]. This classification divides the complex lead perovskites into three subgroups; random oc‐ cupation or disordered, nanoscale or short coherent long-range order and long coherent longrange order of B-site cations.

Diffuse phase transition behavior is characteristic of the disordered structures. In these structures, random lattice disorder introduces dipolar impurities and defects that influence the static and dynamic properties of these materials. The presence of the dipolar entities on a lattice site of the highly polarizable FE structure, induced dipoles in a region determined by the correlation length (*rc*). The correlation length is a measure of the extent of dipoles that respond in a correlated manner. In normal FE, *rc* is larger than the lattice parameter *(a)* and it is strongly temperature dependent. On decreasing the temperature, a faster increase of *rc* promotes growing of polar domains yielding a static cooperative long-range ordered FE state at *T < Tc*. This is not the case for RFE where a small correlation length of dipoles leads to formation of polar nanodomains frustrating the establishment of long-range FE state. Therefore, the dipolar nanoregions form a dipolar-glass like or relaxor state at low tempera‐ ture with some correlation among nanodomains.

The temperature dependence of the dielectric constant below *TB* and in the vicinity of *Tm* is

where γ and δ are parameters describing the degree of relaxation and diffuseness of the transition respectively. The parameter γ varies between 1 and 2, where values closer to 1 in‐ dicate normal ferroelectric behavior whereas values close to 2 indicate good relaxor behav‐

Last sixty years of extensive research in the field of relaxor ferroelectrics, several models have been proposed to explain the unusual dielectric behavior of these materials. Some of these models are: statistical composition fluctuations [20, 21], superparaelectric model [22], dipolar glass model [23], random field model [24], spherical random field model [25] etc. These models can explain much of the experimental observed facts but a clear understand‐ ing of the relaxor nature or even a comprehensive theory is not available yet. Despite the absence of a comprehensive theory of relaxor ferroelectricity, the literature agree to define relaxor materials in terms of the existence of polar nano regions as motioned above, these

The influence of epitaxial clamping on ferroelectric properties of thin films to rigid sub‐ strates has been largely studied and successfully explained in a Landau-Ginzburg-Devon‐ shire (LGD) framework [26-30]. However, when the same treatment was applied to RFE, discrepancies among the predicted values and those observed in experiments were found [30]. For instance, a downward shift in *Tm* of PMN films in the presence of compressive inplane strain was found, contrary to the expected upward shift. Catalan [30] et al. developed a model to analyze the influence of epitaxial strain on RFE. This model is based on LGD theory but assuming a quadratic dependence on (*T-Tm*), rather than linear, for the first coeffi‐ cient of the free energy. Catalan's model shows that the shift in *Tm* for relaxors thin films do not depend on the sign of the epitaxial mismatch strain, but rather on the thermal expansion mismatch between substrate and film. Reference [30] also provided a list of compounds which exhibited shift in *Tm* when they were grown in film forms. We have already observed in-plane strain in our Pb(Sc0.5Nb0.25Ta0.25)O3(PSNT)/La0.5Sr0.5CoO3 (LSCO)/MgO heterostruc‐ ture, which may have caused the measured shift of *Tm* with respect to bulk values [31-33].

(*T* - *T max*)*<sup>γ</sup>*

<sup>2</sup>*<sup>δ</sup>* <sup>2</sup> ) (2)

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73

*<sup>T</sup>* - <sup>Θ</sup> (3)

*<sup>ε</sup>max*(*ω*, *<sup>T</sup>* ) )(1 <sup>+</sup>

ior. Above *TB* the inverse of dielectric permittivity is fitted by the Curie-Weiss law:

*<sup>χ</sup>* <sup>=</sup> *<sup>C</sup>*

where C and Θ are the Curie constant and Curie temperature respectively.

normally fitted by the empirical power law [19]:

ordered regions exist in a disordered environment.

**4. Strain induced relaxor phenomenon**

1 *<sup>ε</sup>*(*ω*, *<sup>T</sup>* ) =( <sup>1</sup>

**Figure 1.** Temperature evolution of dielectric constant showing the characteristic temperatures in RFE. Representative hysteresis loops for each temperature interval are showed below.

These PNRs are dynamic and they experience a slowing down of their fluctuations at *T ≤ Tm*. The dielectric relaxation does not fit the classical Debye relaxation model; instead there is a distribution of relaxation times related to the sizes of the nanodomains. The temperature de‐ pendence of dielectric constant as shown in Fig. 1 indentifies the main temperatures associ‐ ated with relaxor ferroelectric behavior. These temperatures are *TB*, *Tm* and *Tf* , we already defined *TB* and *Tm* but the definition of *Tf* follow from the fit of the frequency dispersion of *Tm* with the Vogel-Fulcher law [15]. The dynamics of polar nanoregions does not follow Ar‐ rhenius type temperature dependence; instead nice fit of the frequency dispersion for each relaxor system is obtained with the Vogel-Fulcher law:

$$f = f\_0 \exp\left(\frac{-E\_s}{k\_B(T\_w - T\_{f\_f})}\right) \tag{1}$$

where *f0* is the attempt frequency which is related to the cut-off frequency of the distribution of relaxation times, *Ea* is the activation barrier to dipole reorientation, *Tm* is the dielectric maxima temperature, and *Tf* is the freezing temperature where polarization fluctuations "frozen-in".

The temperature dependence of the dielectric constant below *TB* and in the vicinity of *Tm* is normally fitted by the empirical power law [19]:

$$\frac{1}{\varepsilon\_{\varepsilon}(\omega,\ \ T)} = \left(\frac{1}{\varepsilon\_{\max}(\omega,\ \ T)}\right) \left(1 + \frac{\langle T \,\ \cdot \ T\_{\max}\rangle^{\gamma}}{2\delta^2}\right) \tag{2}$$

where γ and δ are parameters describing the degree of relaxation and diffuseness of the transition respectively. The parameter γ varies between 1 and 2, where values closer to 1 in‐ dicate normal ferroelectric behavior whereas values close to 2 indicate good relaxor behav‐ ior. Above *TB* the inverse of dielectric permittivity is fitted by the Curie-Weiss law:

$$X = \frac{C}{T \cdot \Theta} \tag{3}$$

where C and Θ are the Curie constant and Curie temperature respectively.

Last sixty years of extensive research in the field of relaxor ferroelectrics, several models have been proposed to explain the unusual dielectric behavior of these materials. Some of these models are: statistical composition fluctuations [20, 21], superparaelectric model [22], dipolar glass model [23], random field model [24], spherical random field model [25] etc. These models can explain much of the experimental observed facts but a clear understand‐ ing of the relaxor nature or even a comprehensive theory is not available yet. Despite the absence of a comprehensive theory of relaxor ferroelectricity, the literature agree to define relaxor materials in terms of the existence of polar nano regions as motioned above, these ordered regions exist in a disordered environment.

### **4. Strain induced relaxor phenomenon**

state at *T < Tc*. This is not the case for RFE where a small correlation length of dipoles leads to formation of polar nanodomains frustrating the establishment of long-range FE state. Therefore, the dipolar nanoregions form a dipolar-glass like or relaxor state at low tempera‐

**Figure 1.** Temperature evolution of dielectric constant showing the characteristic temperatures in RFE. Representative

These PNRs are dynamic and they experience a slowing down of their fluctuations at *T ≤ Tm*. The dielectric relaxation does not fit the classical Debye relaxation model; instead there is a distribution of relaxation times related to the sizes of the nanodomains. The temperature de‐ pendence of dielectric constant as shown in Fig. 1 indentifies the main temperatures associ‐

*Tm* with the Vogel-Fulcher law [15]. The dynamics of polar nanoregions does not follow Ar‐ rhenius type temperature dependence; instead nice fit of the frequency dispersion for each

where *f0* is the attempt frequency which is related to the cut-off frequency of the distribution of relaxation times, *Ea* is the activation barrier to dipole reorientation, *Tm* is the dielectric

, we already

follow from the fit of the frequency dispersion of

*kB*(*<sup>T</sup> <sup>m</sup>* - *<sup>T</sup> <sup>f</sup>* ) ) (1)

is the freezing temperature where polarization fluctuations

ated with relaxor ferroelectric behavior. These temperatures are *TB*, *Tm* and *Tf*

*<sup>f</sup>* <sup>=</sup> *<sup>f</sup>* <sup>0</sup>*exp*( -*Ea*

ture with some correlation among nanodomains.

72 Advances in Ferroelectrics

hysteresis loops for each temperature interval are showed below.

relaxor system is obtained with the Vogel-Fulcher law:

defined *TB* and *Tm* but the definition of *Tf*

maxima temperature, and *Tf*

"frozen-in".

The influence of epitaxial clamping on ferroelectric properties of thin films to rigid sub‐ strates has been largely studied and successfully explained in a Landau-Ginzburg-Devon‐ shire (LGD) framework [26-30]. However, when the same treatment was applied to RFE, discrepancies among the predicted values and those observed in experiments were found [30]. For instance, a downward shift in *Tm* of PMN films in the presence of compressive inplane strain was found, contrary to the expected upward shift. Catalan [30] et al. developed a model to analyze the influence of epitaxial strain on RFE. This model is based on LGD theory but assuming a quadratic dependence on (*T-Tm*), rather than linear, for the first coeffi‐ cient of the free energy. Catalan's model shows that the shift in *Tm* for relaxors thin films do not depend on the sign of the epitaxial mismatch strain, but rather on the thermal expansion mismatch between substrate and film. Reference [30] also provided a list of compounds which exhibited shift in *Tm* when they were grown in film forms. We have already observed in-plane strain in our Pb(Sc0.5Nb0.25Ta0.25)O3(PSNT)/La0.5Sr0.5CoO3 (LSCO)/MgO heterostruc‐ ture, which may have caused the measured shift of *Tm* with respect to bulk values [31-33].

We have calculated shift in the peak position of the dielectric maxima of PSNT films com‐ pared to bulk values following the model developed by Catalan et.al. The model is as fol‐ low: In LGD formalism, the thermodynamical potential of a perovskite dielectric thin films is described as [27-28].

$$
\Delta G = \left(\frac{1}{2}\alpha \text{ - } Q\_{i3}X\_i\right)P\_3^2 + \frac{1}{4}\beta P\_3^2 \text{ - } \frac{1}{2}s\_{ij}\{X\_i X\_j\} \tag{4}
$$

*<sup>χ</sup>*(*<sup>T</sup>* , *<sup>f</sup>* )=*α*(*<sup>T</sup>* , *<sup>f</sup>* )= <sup>1</sup>

*'*

*<sup>ε</sup>*0*εm<sup>δ</sup>* <sup>2</sup> (*Tm '*

ture under strain (*Tm*

expansion:

ed shift in *Tm* is:

experimental observation.

perature of the dielectric maxima.

spect to temperature at *Tm*

*'*

). To calculate *Tm*

*'*

∂ *χ '* ∂ *T <sup>m</sup> '*=0= <sup>1</sup>

> ∂ *xm* <sup>∂</sup> *<sup>T</sup>* <sup>=</sup> <sup>∂</sup> ∂ *T*

<sup>Δ</sup>*Tm* =4*<sup>δ</sup>* <sup>2</sup>

*εmε*0*Q*<sup>13</sup>

**5. Experimental observation of strain induced relaxor phenomenon**

Fig. 2 shows the dielectric behavior and the microstructures of the PSNT thin films and their bulk matrix [31-33]. It has been shown that the two-dimensional (2D) clamping of films by the substrate may change profoundly the physical properties of ferroelectric heterostructure with respect to the bulk material. We observed a transfer and relaxing of epitaxial strain be‐ tween the layers due to in-plane oriented heterostructure. However the mismatch strain due to different thermal expansion properties of each layer provokes a shift of 62 K in the tem‐

.

2*εmε*<sup>0</sup>

We substituted the values of α in Equation 5 and calculated the dielectric maxima tempera‐


where it is assumed that *Q13*, *Y*, and ν do not change substantially [34] with temperature in the dielectric maxima regions; only the mismatch strain changes due to differential thermal

(*a film* 1 - *λsubstrate*(*T* - *T*0) - *abulk*

*abulk*

where *λsubstrate* is the coefficient of thermal expansion of the substrate, *afilm* and *abulk* are the lat‐ tice constant of the film and the bulk respectively. Using Equations, 9, 10, and 11 the expect‐

> *Y* 1 - *ν a film*

We have used the above equation to calculate the shift in the position of dielectric maxima of PSNT films. Due to lack of experimental mechanical data in literature for PSNT films or ceramics, we have used the standard *Q13* for PST thin films [34] and *Y* value for the PMN single crystal [35] other values we got from our experimental observation i.e. (δ, εm, *afilm*, *abulk*,). For MgO substrate, λsubstrate = 1.2x10-5 K-1. Putting all these data in Equation 12 we got a shift in the dielectric maximum temperature (*Tm*) of the same magnitude order than those of

*Y* 1 - *ν* ∂ *xm*

)

(*T* - *Tm*)<sup>2</sup> (9)

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<sup>∂</sup> *<sup>T</sup>* (10)

*abulk <sup>λ</sup>substrate* (12)

(11)

75

, we differentiated the inverse of permittivity with re‐

where α and β correspond to linear and nonlinear terms of the inverse permittivity, and *sij*, *Xi* , and *Qi3* are the elastic compliances, the stress tensor, and the electrostriction tensor in Voigt notation; P3 is out of the plane polarization. Strain gradients across the films are ne‐ glected. The inverse permittivity is the second derivative of the free energy with respect to the polarization:

$$\chi\_3 = \frac{\delta^2 \Lambda G}{\delta P^2} = \alpha \cdot 4Q\_{13} \frac{\chi}{1 \cdot \nu} \chi\_m + 3\beta P\_3^{\prime 2} \tag{5}$$

where *xm*, *Y*, and ν are the mismatch strain, Young's modulus, and Poisson's ratio of the film respectively. For conventional ferroelectrics, α is usually expanded in a power series of (T-Tc):

$$\alpha = \frac{1}{\mathbb{C}\varepsilon\_0} \{T \, - \, T\_c\} + f\left\{T \, - \, T\_c\right\}^2 \tag{6}$$

which directly leads to Curie-Weiss behavior, where C is the Curie constant. Substituting α into the above equation showing that the strain shifts the critical temperature for a ferroelec‐ tric by:

$$
\Delta T\_c = 4C\varepsilon\_0 Q\_{13} \frac{Y}{1 - \nu} \mathbf{x}\_m \tag{7}
$$

Since *Q13* is always negative, it has been observed that the shift in the critical temperature depends on the sign of in-plain strain: *Tc* decreases for the tensile strain and it increases for compressive strain. Equation 7 is not adequate to explain the shifts in the dielectric maxi‐ mum temperature for the relaxor ferroelectric materials

In case of PSNT films, we observed a quadratic temperature dependence of the permittivity:

$$\frac{1}{\frac{1}{\varepsilon\_{\varepsilon}(\omega\_{\ast},T)} - \frac{1}{\varepsilon\_{m}(\omega\_{\ast},T)}} = \frac{1}{\frac{1}{2\varepsilon\_{m}\varepsilon\_{0}}(T - T\_{m})^{2}} (T - T\_{m})^{2} \tag{8}$$

Since the inverse of the dielectric constant is the second derivative of free energy with re‐ spect to polarization (Equation 5), the measured dielectric constant can be related to the ap‐ propriate coefficients in the LGD thermodynamical potential. For P=0 and *xm*=0 (in the absence of strain), χ correlates with the first coefficient of the Landau equation, i.e.

$$\alpha\_{\lambda}(T, \; f) = \alpha(T, \; f) = \frac{1}{2\varepsilon\_{m}\varepsilon\_{0}}(T - T\_{m})^{2} \tag{9}$$

We substituted the values of α in Equation 5 and calculated the dielectric maxima tempera‐ ture under strain (*Tm '* ). To calculate *Tm '* , we differentiated the inverse of permittivity with re‐ spect to temperature at *Tm '* .

We have calculated shift in the peak position of the dielectric maxima of PSNT films com‐ pared to bulk values following the model developed by Catalan et.al. The model is as fol‐ low: In LGD formalism, the thermodynamical potential of a perovskite dielectric thin films

where α and β correspond to linear and nonlinear terms of the inverse permittivity, and *sij*,

, and *Qi3* are the elastic compliances, the stress tensor, and the electrostriction tensor in Voigt notation; P3 is out of the plane polarization. Strain gradients across the films are ne‐ glected. The inverse permittivity is the second derivative of the free energy with respect to

*Y*

where *xm*, *Y*, and ν are the mismatch strain, Young's modulus, and Poisson's ratio of the film respectively. For conventional ferroelectrics, α is usually expanded in a power series

which directly leads to Curie-Weiss behavior, where C is the Curie constant. Substituting α into the above equation showing that the strain shifts the critical temperature for a ferroelec‐

*Y*

Since *Q13* is always negative, it has been observed that the shift in the critical temperature depends on the sign of in-plain strain: *Tc* decreases for the tensile strain and it increases for compressive strain. Equation 7 is not adequate to explain the shifts in the dielectric maxi‐

In case of PSNT films, we observed a quadratic temperature dependence of the permittivity:

2*εmε*<sup>0</sup>

Since the inverse of the dielectric constant is the second derivative of free energy with re‐ spect to polarization (Equation 5), the measured dielectric constant can be related to the ap‐ propriate coefficients in the LGD thermodynamical potential. For P=0 and *xm*=0 (in the

*<sup>ε</sup>m*(*ω*, *<sup>T</sup>* ) <sup>=</sup> <sup>1</sup>

absence of strain), χ correlates with the first coefficient of the Landau equation, i.e.

<sup>1</sup> - *<sup>ν</sup> xm* + 3*βP*<sup>3</sup>

(*XiX <sup>j</sup>*

) (4)

<sup>2</sup> (5)

(*T* - *Tc*) + *f* (*T* - *Tc*)<sup>2</sup> (6)

<sup>1</sup> - *<sup>ν</sup> xm* (7)

(*T* - *Tm*)<sup>2</sup> (8)

is described as [27-28].

74 Advances in Ferroelectrics

the polarization:

of (T-Tc):

tric by:

*Xi*

<sup>∆</sup>*<sup>G</sup>* =( <sup>1</sup>

*χ*3 *'*

mum temperature for the relaxor ferroelectric materials

1 *<sup>ε</sup>*(*ω*, *<sup>T</sup>* ) - <sup>1</sup>

<sup>=</sup> <sup>∂</sup>2Δ*<sup>G</sup>*

*<sup>α</sup>* <sup>=</sup> <sup>1</sup> *Cε*<sup>0</sup>

<sup>∂</sup> *<sup>P</sup>* <sup>2</sup> =*α* - 4*Q*<sup>13</sup>

Δ*Tc* =4*Cε*0*Q*<sup>13</sup>

<sup>2</sup> *α* - *Qi*3*Xi*

)*P*3 2 + 1 <sup>4</sup> *βP*<sup>3</sup> <sup>2</sup> - <sup>1</sup> <sup>2</sup> *sij*

$$\frac{\partial \chi^{'}}{\partial \ ^{\circ}T\_{m}} = 0 = \frac{1}{\varepsilon\_{0}\varepsilon\_{m}\delta^{'}} \left(T\_{m}^{'} - T\_{m}\right) - 4Q\_{13}\frac{\chi}{1 - \nu} \frac{\partial \chi\_{m}}{\partial T} \tag{10}$$

where it is assumed that *Q13*, *Y*, and ν do not change substantially [34] with temperature in the dielectric maxima regions; only the mismatch strain changes due to differential thermal expansion:

$$\frac{\partial \ln \mathbf{x}\_w}{\partial T} = \frac{\partial}{\partial T} \left[ \frac{\left( a\_{film} \prod \mathbf{1} - \lambda\_{substrate} (T - T\_0) \right) \cdot a\_{bulk}}{a\_{bulk}} \right] \tag{11}$$

where *λsubstrate* is the coefficient of thermal expansion of the substrate, *afilm* and *abulk* are the lat‐ tice constant of the film and the bulk respectively. Using Equations, 9, 10, and 11 the expect‐ ed shift in *Tm* is:

$$
\Delta T\_m = 4\delta \,^2 \varepsilon\_m \varepsilon\_0 \mathbf{Q}\_{13} \frac{\chi}{1 \cdot \nu} \frac{a\_{film}}{a\_{bul}} \lambda\_{substrate} \tag{12}
$$

We have used the above equation to calculate the shift in the position of dielectric maxima of PSNT films. Due to lack of experimental mechanical data in literature for PSNT films or ceramics, we have used the standard *Q13* for PST thin films [34] and *Y* value for the PMN single crystal [35] other values we got from our experimental observation i.e. (δ, εm, *afilm*, *abulk*,). For MgO substrate, λsubstrate = 1.2x10-5 K-1. Putting all these data in Equation 12 we got a shift in the dielectric maximum temperature (*Tm*) of the same magnitude order than those of experimental observation.

### **5. Experimental observation of strain induced relaxor phenomenon**

Fig. 2 shows the dielectric behavior and the microstructures of the PSNT thin films and their bulk matrix [31-33]. It has been shown that the two-dimensional (2D) clamping of films by the substrate may change profoundly the physical properties of ferroelectric heterostructure with respect to the bulk material. We observed a transfer and relaxing of epitaxial strain be‐ tween the layers due to in-plane oriented heterostructure. However the mismatch strain due to different thermal expansion properties of each layer provokes a shift of 62 K in the tem‐ perature of the dielectric maxima.

dering response, induced dielectric dispersion and shifting in dielectric maxima temperature towards lower temperature side. We have observed from the temperature evolution of the Raman spectra of bulk PSNT ceramic [31-33], the competitions between ordered domains of short and long-range order due to Nb- and Ta-rich regions respective‐ ly. The average disorder arrangement of Sc3+/Nb5+/Ta5+ ions in the B-site octahedra leads to the observed diffuse phase transition. In film only short range ordering in B'/B'' ions was developed within the disordered matrix. Although bulk matrix exhibited nano-ordered re‐ gions its average size must be higher than film. Stress effects change the ionic positions somehow favoring short-range ordering. These microstructural differences trigger the dif‐ ferent observed dielectric responses. The microstructure-property relation of PSNT thin films and ceramics, one can build conclusions that perovskite with similar compositions

We have discussed the origin and functional properties of relaxor, in nutshell the basic charac‐ teristic of relaxors are the existence of ordered PNRs in a disordered matrix [37-42]. Similarly, relaxor ferromagnets (synonyms: "Mictomagnets") are also known in the literature since the 1970s. A mictomagnet is described as having magnetic clusters (superparamagnetism) which form a spin glass and have a tendency to form short-range ordering [40, 41]. The materials hold both ferroelectric and magnetic orders at nanoscale are called "birelaxors". Birelaxors are sup‐ pose to possess only short range order over the entire temperature range with broken symme‐ try such as "global spatial inversion" and "time reversal symmetry" at nanoscale. Nanoscale broken symmetries and length scale coherency make birelaxors difficult to investigate micro‐ scopically; it is advisable to use optical tools for proper probing. Birelaxors do not usually show

coupling is not allowed, local strain-mediated PNR-PNR, PNR-MNR (magnetic nanoregions),

A single-phase perovskite PbZr0.53Ti0.47O3 (PZT) and PbFe2/3W1/3O3 (PFW), (40% PZT-60 % PFW) solid solutions thin film grown by pulsed laser deposition system have shown inter‐ esting birelaxor properties. Raman spectroscopy, dielectric spectroscopy and temperaturedependent zero- field magnetic susceptibility indicate the presence of both ferroic orders

Dipolar glass and spin glass properties are confirmed from the dielectric and magnetic re‐ sponse of the PZT-PFW system, the micro Raman spectra of this system also revealed the presence of polar nanoregions. Dielectric constant and tangent loss show the dielectric dis‐ persion near the dielectric maxima temperature that confirms near-room-temperature re‐ laxor behavior. Magnetic irreversibility is defined as *(Mirr= MFC-MZFC*) and it represents the degree of spin glass behavior. PZT-PFW demonstrates special features in *Mirr* data at 100 Oe indicate that irreversibility persists above 220 K up to 4th or 5th order of the magni‐ tude, *Tirr* (inset Fig. 3 (b)). The evidence of ZFC and FC splitting at low field ~100 Oe and

) but may have large nonlinear terms such as *bP2*

*M2*

Self Assembled Nanoscale Relaxor Ferroelectrics

http://dx.doi.org/10.5772/54298

77

. Since linear

have well defined relaxor behavior than their bulk matrix.

 *Mj*

and MNR-MNR interactions may provide very strong ME effects [43-44].

**6. Birelaxors**

linear ME coupling (*aij Pi*

at nanoscale [45].

Fig.2 Microstructure of the PSNT bulk nanoceramics (a) and the thin films (b) grown on the LSCO coated MgO substrate, and a drastic shift in dielectric maxima temperature (c) in PSNT thin film with respect same compositions of their bulk counterpart. The microstructures of film **Figure 2.** Microstructure of the PSNT bulk nanoceramics (a) and the thin films (b) grown on the LSCO coated MgO substrate, and a drastic shift in dielectric maxima temperature (c) in PSNT thin film with respect same compositions of their bulk counterpart. The microstructures of film and ceramic support the existence of strain induced relaxor phe‐ nomenon, as their microstructures- property correlation indicate the presence of some ordered polar nano-regions in the thin films whereas absence of such effect in bulk. Figure 2(a) was adapted from Ref. [31] and reproduced with permission (© 2008 John Wiley and Sons).

13 and ceramic support the existence of strain induced relaxor phenomenon, as their Transmission electron microscopy (TEM) images of PSNT in bulk (Fig. 2(a)) and thin film (Fig. 2(b)) form are shown in Fig. 2. The interlayer of about 20 nm between MgO and LSCO and the fibril structures along a single direction in the film confirm the in-plane strain state of the film. TEM image of bulk shows well ordered nanoregions in the grain matrix that meet a basic ingredient to produces relaxor behavior: the existence of nano-or‐ dered regions surrounding by a disordered structure, however, it is unable to produce it. It has been observed from the neutron diffraction data that if the ordered nanoregions are in the range of ~ 5-10 nm, then it is capable of yielding frequency dispersion in the dielec‐ tric spectra [30]. But the relaxor state also depends on the coherency with what the di‐ poles respond to the probing field (and frequency). They can establish long range or short range coherent response. PSNT ceramic was incapable to produce the frequency disper‐ sion behavior, but in thin films form the in plane strain and the breaking of long range or‐ dering response, induced dielectric dispersion and shifting in dielectric maxima temperature towards lower temperature side. We have observed from the temperature evolution of the Raman spectra of bulk PSNT ceramic [31-33], the competitions between ordered domains of short and long-range order due to Nb- and Ta-rich regions respective‐ ly. The average disorder arrangement of Sc3+/Nb5+/Ta5+ ions in the B-site octahedra leads to the observed diffuse phase transition. In film only short range ordering in B'/B'' ions was developed within the disordered matrix. Although bulk matrix exhibited nano-ordered re‐ gions its average size must be higher than film. Stress effects change the ionic positions somehow favoring short-range ordering. These microstructural differences trigger the dif‐ ferent observed dielectric responses. The microstructure-property relation of PSNT thin films and ceramics, one can build conclusions that perovskite with similar compositions have well defined relaxor behavior than their bulk matrix.

### **6. Birelaxors**

13

Fig.2 Microstructure of the PSNT bulk nanoceramics (a) and the thin films (b) grown on the LSCO coated MgO substrate, and a drastic shift in dielectric maxima temperature (c) in PSNT thin film with respect same compositions of their bulk counterpart. The microstructures of film and ceramic support the existence of strain induced relaxor phenomenon, as their

Transmission electron microscopy (TEM) images of PSNT in bulk (Fig. 2(a)) and thin film (Fig. 2(b)) form are shown in Fig. 2. The interlayer of about 20 nm between MgO and LSCO and the fibril structures along a single direction in the film confirm the in-plane strain state of the film. TEM image of bulk shows well ordered nanoregions in the grain matrix that meet a basic ingredient to produces relaxor behavior: the existence of nano-or‐ dered regions surrounding by a disordered structure, however, it is unable to produce it. It has been observed from the neutron diffraction data that if the ordered nanoregions are in the range of ~ 5-10 nm, then it is capable of yielding frequency dispersion in the dielec‐ tric spectra [30]. But the relaxor state also depends on the coherency with what the di‐ poles respond to the probing field (and frequency). They can establish long range or short range coherent response. PSNT ceramic was incapable to produce the frequency disper‐ sion behavior, but in thin films form the in plane strain and the breaking of long range or‐

permission (© 2008 John Wiley and Sons).

76 Advances in Ferroelectrics

**Figure 2.** Microstructure of the PSNT bulk nanoceramics (a) and the thin films (b) grown on the LSCO coated MgO substrate, and a drastic shift in dielectric maxima temperature (c) in PSNT thin film with respect same compositions of their bulk counterpart. The microstructures of film and ceramic support the existence of strain induced relaxor phe‐ nomenon, as their microstructures- property correlation indicate the presence of some ordered polar nano-regions in the thin films whereas absence of such effect in bulk. Figure 2(a) was adapted from Ref. [31] and reproduced with

**(c**

**(a (b**

We have discussed the origin and functional properties of relaxor, in nutshell the basic charac‐ teristic of relaxors are the existence of ordered PNRs in a disordered matrix [37-42]. Similarly, relaxor ferromagnets (synonyms: "Mictomagnets") are also known in the literature since the 1970s. A mictomagnet is described as having magnetic clusters (superparamagnetism) which form a spin glass and have a tendency to form short-range ordering [40, 41]. The materials hold both ferroelectric and magnetic orders at nanoscale are called "birelaxors". Birelaxors are sup‐ pose to possess only short range order over the entire temperature range with broken symme‐ try such as "global spatial inversion" and "time reversal symmetry" at nanoscale. Nanoscale broken symmetries and length scale coherency make birelaxors difficult to investigate micro‐ scopically; it is advisable to use optical tools for proper probing. Birelaxors do not usually show linear ME coupling (*aij Pi Mj* ) but may have large nonlinear terms such as *bP2 M2* . Since linear coupling is not allowed, local strain-mediated PNR-PNR, PNR-MNR (magnetic nanoregions), and MNR-MNR interactions may provide very strong ME effects [43-44].

A single-phase perovskite PbZr0.53Ti0.47O3 (PZT) and PbFe2/3W1/3O3 (PFW), (40% PZT-60 % PFW) solid solutions thin film grown by pulsed laser deposition system have shown inter‐ esting birelaxor properties. Raman spectroscopy, dielectric spectroscopy and temperaturedependent zero- field magnetic susceptibility indicate the presence of both ferroic orders at nanoscale [45].

Dipolar glass and spin glass properties are confirmed from the dielectric and magnetic re‐ sponse of the PZT-PFW system, the micro Raman spectra of this system also revealed the presence of polar nanoregions. Dielectric constant and tangent loss show the dielectric dis‐ persion near the dielectric maxima temperature that confirms near-room-temperature re‐ laxor behavior. Magnetic irreversibility is defined as *(Mirr= MFC-MZFC*) and it represents the degree of spin glass behavior. PZT-PFW demonstrates special features in *Mirr* data at 100 Oe indicate that irreversibility persists above 220 K up to 4th or 5th order of the magni‐ tude, *Tirr* (inset Fig. 3 (b)). The evidence of ZFC and FC splitting at low field ~100 Oe and even for high 1 kOe field suggests and supports glass-like behavior with competition be‐ tween long-range ordering and short-range order. The behavior of the ZFC cusp is the same for the entire range of field (>100 Oe), with a shift in blocking transition tempera‐ ture (*TB*) (maximum value of magnetization in ZFC cusp) to lower temperature, i.e. from 48 K to 29.8 K with increasing magnetic field. The ZFC cusps disappeared in the FC proc‐ ess, suggesting that competing forces were stabilized by the field- cooled process. Magnet‐ ic irreversibilities were found over a wide range of temperature with a sharp cusp in the ZFC data, suggesting the presence of MNRs with spin-glass-like behavior. The detailed fabrication process, the complete characterization process and the functional properties are reported in reference [43]. This is not only the unique system to have birelaxor proper‐ ties, however, Levstik et al also observed the birelaxor properties in 0.8Pb(Fe1/2Nb1/2)O3– 0.2Pb(Mg1/2W1/2)O3 thin films [38].

**7. Superlattice relaxors and high energy density capacitors**

lectrics [46-51].

energy density of 1.5 J/cm2

60-70% dielectric saturation.

40 J/cm3 energy density.

and low current density (~ 10-25 mA/cm2

density storage capacity (~ 46 J/cm3

zation techniques are presented elsewhere [52].

Material scientists are looking for a system or some novel materials that possess high power density and high energy density or both. Relaxors demonstrate high dielectric constant, low loss, non linear polarization under external electric field, high bipolar den‐ sity, nano dipoles (polar nano regions (PNRs) and moderate dielectric saturation, these properties support their potential candidature for the high power as well as high ener‐ gy devices. At present high-k dielectric (dielectric constant less than 100, with linear di‐ electric) dominates in the high energy, high power density capacitor market. The high-k dielectrics show very high electric breakdown strength (> 3 MV/cm to 12 MV/cm) but their dielectric constant is relatively very low which in turns offer 1-2 J/cm3 volume en‐ ergy density. Recently, polymer ferroelectric, antiferroelectric, and relaxor ferroelectric have shown better potential and high energy density compare to the existing linear die‐

BaTiO3/Ba0.30Sr0.70TiO3 (BT/BST) superlattices (SLs) with a constant modulation period of Λ= 80 Å were grown on (001) MgO substrate by pulsed laser deposition. The modulation perio‐ dicity Λ/2 was precisely maintained by controlling the number of laser shots; the total thick‐ ness of each SL film was ~6,000 Å = 0.6 μm. An excimer laser (KrF, 248 nm) with a laser

oxygen pressure at 200 mTorr was used for SL growth. The detailed growth and characteri‐

Dielectric responses of BT/BST SLs display the similar response as for normal relaxor fer‐ roelectric which can be seen in Figure 4. It follows the non linear Vogel- Fulcher relation‐ ship (inset Figure. 4 (a)), frequency dispersion near and below the dielectric maxima temperature (*Tm*), merger of frequency far above the *Tm*, shift in *Tm* towards higher tem‐ perature side (about 50-60 K) with increase in frequencies, low dielectric loss, and about

BT/BST SLs demonstrate a "in-built" field in as grown samples at low probe frequency (<1 kHz), whereas it becomes more symmetric and centered with increase in probe frequency system (>1 kHz) that ruled out the effect of any space charge and interfacial polarization. Energy density were calculated for the polarization-electric field (P-E) loops provide ~ 12.24 J/cm3 energy density within the experimental limit, but extrapolation of this data in the energy density as function of applied field graph for different frequencies (see inset in Figure 4) suggests huge potential in the system such as it can hold and release more than

Experimental limitations restrict to proof the extrapolated data, however the current density versus applied electric field indicates exceptionally high breakdown field (5.8 to 6.0 MV/cm)

rect measurements of the energy density indicate that it has ability to store very high energy

).

, pulse repetition rate of 10 Hz, substrate temperature 830 °C, and

Self Assembled Nanoscale Relaxor Ferroelectrics

http://dx.doi.org/10.5772/54298

79

) near the breakdown voltage. Both direct and indi‐

**Figure 3.** (a) Dielectric response and Vogel-Fulcher fitting (inset), (b) Zero field cooled and field cool magnetic re‐ sponse of PZT-PFW thin films indicate the presence of PNRs and MNRs at nanoscale. Adapted from Ref. [45] and repro‐ duced with permission (© 2011 AIP).

### **7. Superlattice relaxors and high energy density capacitors**

even for high 1 kOe field suggests and supports glass-like behavior with competition be‐ tween long-range ordering and short-range order. The behavior of the ZFC cusp is the same for the entire range of field (>100 Oe), with a shift in blocking transition tempera‐ ture (*TB*) (maximum value of magnetization in ZFC cusp) to lower temperature, i.e. from 48 K to 29.8 K with increasing magnetic field. The ZFC cusps disappeared in the FC proc‐ ess, suggesting that competing forces were stabilized by the field- cooled process. Magnet‐ ic irreversibilities were found over a wide range of temperature with a sharp cusp in the ZFC data, suggesting the presence of MNRs with spin-glass-like behavior. The detailed fabrication process, the complete characterization process and the functional properties are reported in reference [43]. This is not only the unique system to have birelaxor proper‐ ties, however, Levstik et al also observed the birelaxor properties in 0.8Pb(Fe1/2Nb1/2)O3–

**150 300 450 600**

**0 100 200 300**

 **@ 100 Oe**

**T (K)**

**Experimental VF fitting**

**4 6 8 10 12**

**Ln(f)**

**Temperature (K)**

**0 100 200 300**

**Temperature (K)**

**Figure 3.** (a) Dielectric response and Vogel-Fulcher fitting (inset), (b) Zero field cooled and field cool magnetic re‐ sponse of PZT-PFW thin films indicate the presence of PNRs and MNRs at nanoscale. Adapted from Ref. [45] and repro‐

**0.00 0.30 0.60 0.90 1.20**

 **ZFC @ 100 Oe**

**Mirr**

 **FC**

**(emu/cm**

**)**

**3**

**(b)** 

**3.0**

**(103/T**

**m)**

**3.2**

0.2Pb(Mg1/2W1/2)O3 thin films [38].

78 Advances in Ferroelectrics

**300**

**5.6**

**6.1**

**Magnetization (emu/cm**

duced with permission (© 2011 AIP).

**3**

**)**

**6.5**

**7.0**

**600**

**900**

**Dielectric constant**

**(**e**)**

**1200**

**(a)** 

**1500**

Material scientists are looking for a system or some novel materials that possess high power density and high energy density or both. Relaxors demonstrate high dielectric constant, low loss, non linear polarization under external electric field, high bipolar den‐ sity, nano dipoles (polar nano regions (PNRs) and moderate dielectric saturation, these properties support their potential candidature for the high power as well as high ener‐ gy devices. At present high-k dielectric (dielectric constant less than 100, with linear di‐ electric) dominates in the high energy, high power density capacitor market. The high-k dielectrics show very high electric breakdown strength (> 3 MV/cm to 12 MV/cm) but their dielectric constant is relatively very low which in turns offer 1-2 J/cm3 volume en‐ ergy density. Recently, polymer ferroelectric, antiferroelectric, and relaxor ferroelectric have shown better potential and high energy density compare to the existing linear die‐ lectrics [46-51].

BaTiO3/Ba0.30Sr0.70TiO3 (BT/BST) superlattices (SLs) with a constant modulation period of Λ= 80 Å were grown on (001) MgO substrate by pulsed laser deposition. The modulation perio‐ dicity Λ/2 was precisely maintained by controlling the number of laser shots; the total thick‐ ness of each SL film was ~6,000 Å = 0.6 μm. An excimer laser (KrF, 248 nm) with a laser energy density of 1.5 J/cm2 , pulse repetition rate of 10 Hz, substrate temperature 830 °C, and oxygen pressure at 200 mTorr was used for SL growth. The detailed growth and characteri‐ zation techniques are presented elsewhere [52].

Dielectric responses of BT/BST SLs display the similar response as for normal relaxor fer‐ roelectric which can be seen in Figure 4. It follows the non linear Vogel- Fulcher relation‐ ship (inset Figure. 4 (a)), frequency dispersion near and below the dielectric maxima temperature (*Tm*), merger of frequency far above the *Tm*, shift in *Tm* towards higher tem‐ perature side (about 50-60 K) with increase in frequencies, low dielectric loss, and about 60-70% dielectric saturation.

BT/BST SLs demonstrate a "in-built" field in as grown samples at low probe frequency (<1 kHz), whereas it becomes more symmetric and centered with increase in probe frequency system (>1 kHz) that ruled out the effect of any space charge and interfacial polarization. Energy density were calculated for the polarization-electric field (P-E) loops provide ~ 12.24 J/cm3 energy density within the experimental limit, but extrapolation of this data in the energy density as function of applied field graph for different frequencies (see inset in Figure 4) suggests huge potential in the system such as it can hold and release more than 40 J/cm3 energy density.

Experimental limitations restrict to proof the extrapolated data, however the current density versus applied electric field indicates exceptionally high breakdown field (5.8 to 6.0 MV/cm) and low current density (~ 10-25 mA/cm2 ) near the breakdown voltage. Both direct and indi‐ rect measurements of the energy density indicate that it has ability to store very high energy density storage capacity (~ 46 J/cm3 ).

**8. Conclusions**

energy density applications.

**Acknowledgement**

FG02-08ER46526 grants.

Ashok Kumar1,2, Margarita Correa2

Rico, San Juan, Puerto Rico

\*Address all correspondence to: ashok553@gmail.com

**Author details**

Delhi, India

Extensive studies on the relaxor ferroelectrics suggest the presence of localized nano size ordered regions in the disordered matrix or static random field are responsible for the die‐ lectric dispersion near *Tm*. The nano regions and their coherence length are also critical for the relaxor behavior. A normal ferroelectric with diffuse phase transition system can be turn to relaxor ferroeletrics in their thin film forms under the suitable applications of strain, utilizing highly lattice mismatch substrate, growth conditions (thermal, oxygen par‐ tial pressure, atmosphere), etc.. It also indicates that the defects, oxygen vacancies, order‐ ing of cations at A and B site of perovskite, tensile or comprehensive strain across the interface, etc. originate the ordered nano regions in films mainly responsible for dielectric dispersion. Materials with same compositions can have normal ferroelectric in bulk form whereas become relaxor in the thin film. Birelaxors hold both ferroics orders at nano scale with only short range ordering (SRO). These nanocale SRO make it potential candidates for non linear biquadratic magneto-electric coupling suitable for magnetic field sensors and non volatile memory applications. BT/BST superlattice has shown very high dielectric constant, high breakdown field, relaxor, and high energy density functional properties. A Relaxor superlattice with high functional properties is capable for both high power and

This work was partially supported by NSF-EFRI-RPI-1038272 and DOE-DE-

, Nora Ortega2

1 National Physical Laboratory, Council of Scientific and Industrial Research (CSIR), New

2 Department of Physics and Institute for Functional Nanomaterials, University of Puerto

, Salini Kumari2

and R. S. Katiyar2

Self Assembled Nanoscale Relaxor Ferroelectrics

http://dx.doi.org/10.5772/54298

81

**Figure 4.** (a) Dielectric responses of BT/BST superlattice relaxors as function of temperature over wide range of fre‐ quencies, nonlinear Vogel-Fulcher relationship (inset), (b) Energy density capacity as function of applied electric filed, red dots show the extrapolated data. Adapted from Ref. [46] and reproduced with permission (© 2012 IOP).

The above experimental facts suggest the relaxor nature of BT/BST ferroelectric superlatti‐ ces. It also indicates their potential to store and fast release of energy density which is com‐ parable to that of high-k (<100) dielectrics, hopefully in the coming years it might be suitable for high power energy applications. These functional properties of relaxor superlattices make it plausible high energy density dielectrics capable of both high power and energy density applications.

### **8. Conclusions**

Extensive studies on the relaxor ferroelectrics suggest the presence of localized nano size ordered regions in the disordered matrix or static random field are responsible for the die‐ lectric dispersion near *Tm*. The nano regions and their coherence length are also critical for the relaxor behavior. A normal ferroelectric with diffuse phase transition system can be turn to relaxor ferroeletrics in their thin film forms under the suitable applications of strain, utilizing highly lattice mismatch substrate, growth conditions (thermal, oxygen par‐ tial pressure, atmosphere), etc.. It also indicates that the defects, oxygen vacancies, order‐ ing of cations at A and B site of perovskite, tensile or comprehensive strain across the interface, etc. originate the ordered nano regions in films mainly responsible for dielectric dispersion. Materials with same compositions can have normal ferroelectric in bulk form whereas become relaxor in the thin film. Birelaxors hold both ferroics orders at nano scale with only short range ordering (SRO). These nanocale SRO make it potential candidates for non linear biquadratic magneto-electric coupling suitable for magnetic field sensors and non volatile memory applications. BT/BST superlattice has shown very high dielectric constant, high breakdown field, relaxor, and high energy density functional properties. A Relaxor superlattice with high functional properties is capable for both high power and energy density applications.

### **Acknowledgement**

**200 300 400 500**

 **VF Fit**

**6 7 8 9 10 11**

**Ln (f)**

**Temperature (K)**

**10-3 10-2 10-1 100 101**

**Electric Field (MV/cm)**

**Figure 4.** (a) Dielectric responses of BT/BST superlattice relaxors as function of temperature over wide range of fre‐ quencies, nonlinear Vogel-Fulcher relationship (inset), (b) Energy density capacity as function of applied electric filed, red dots show the extrapolated data. Adapted from Ref. [46] and reproduced with permission (© 2012 IOP).

The above experimental facts suggest the relaxor nature of BT/BST ferroelectric superlatti‐ ces. It also indicates their potential to store and fast release of energy density which is com‐ parable to that of high-k (<100) dielectrics, hopefully in the coming years it might be suitable for high power energy applications. These functional properties of relaxor superlattices make it plausible high energy density dielectrics capable of both high power and energy

 **1 kHz 12 kHz**

**3.08 Exp. data** 

**50 kHz**

**2.92 2.96 3.00 3.04**

**1000/Tm (K-1**

**(b)** 

**)**

**1 kHz (a)**

**0**

**10-4 10-3 10-2 10-1 100 101 102 103**

**Energy Density (J/cm**

density applications.

**3**

**)**

**250**

**500**

**Dielectric Constant**

80 Advances in Ferroelectrics

**750**

**1000**

This work was partially supported by NSF-EFRI-RPI-1038272 and DOE-DE-FG02-08ER46526 grants.

### **Author details**

Ashok Kumar1,2, Margarita Correa2 , Nora Ortega2 , Salini Kumari2 and R. S. Katiyar2

\*Address all correspondence to: ashok553@gmail.com

1 National Physical Laboratory, Council of Scientific and Industrial Research (CSIR), New Delhi, India

2 Department of Physics and Institute for Functional Nanomaterials, University of Puerto Rico, San Juan, Puerto Rico

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čič, Appl. Phys. Lett. 91, 012905, (2007).

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demic, New York, 2001, Vol. 56, p. 239.

dens. Matter 21, 382204 (2009).

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[27] G.A. Rossetti Jr., L.E. Cross, and K. Kushida, Appl. Phys. Lett., 59, 2504 (1991)

[28] N.A. Perstev, A.G. Zembilgotov, and A. K. Tagantsev, Phys. Rev. Lett., 80, 1988

Self Assembled Nanoscale Relaxor Ferroelectrics

http://dx.doi.org/10.5772/54298

83

[30] G. Catalan, M. H. Corbett, R. M. Bowman, and J. M. Gregg, J. App. Phys., 91, [4] 2295

[32] Margarita Correa, Ashok Kumar, and R. S. Katiyar, Applied Physic Letters, 91,

[33] Margarita Correa, Ashok Kumar, and R. S. Katiyar, Integrated Ferroelectrics, 100, 297

[34] K. Brinkman, A. Tagantsev, P. Muralt, and N. Setter, Jpn. J. App. Phys., 45, [9B] 7288

[36] D. L. Orauttapong, J. Toulouse, J. L. Robertson, and Z. G. Ye, Physical Rev. B , 64,

[38] A. Levstik, V. Bobnar, C. Filipič, J. Holc, M. Kosec, R. Blinc, Z. Trontelj, and Z. Jagli‐

[39] V. V. Shvartsman, S. Bedanta, P. Borisov, W. Kleemann, A. Tkach, and P. M. Vilarin‐

[40] G. A. Samara, in Solid State Physics, edited by H. Ehrenreich and R. Spaepen Aca‐

[43] A. Kumar, G. L. Sharma, R. S. Katiyar, R. Pirc, R. Blinc, and J. F. Scott, J. Phys.: Con‐

[41] J. Dho, W. S. Kim, and N. H. Hur, Phys. Rev. Lett. 89, 027202 (2002).

[44] R. Pirc, R. Blinc, and J. F. Scott, Phys. Rev. B 79, 214114 (2009).

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**Chapter 5**

**Relaxor Behaviour in Ferroelectric Ceramics**

Ferroelectric materials are commonly characterized by high dielectric permittivity values [1]. Usually, for the well-known 'normal' ferroelectrics the temperature of the maximum real di‐ electric permittivity (Tm) corresponds to the ferroelectric-paraelectric (FE-PE) phase transi‐ tion temperature (TC) [2]. On the other hand, there are some kinds of ferroelectrics, so-called relaxor ferroelectrics, which have received special attention in the last years because of the observed intriguing and extraordinary dielectric properties [3-25], which remain not clearly understood nowadays. For instance, some remarkable characteristics of the dielectric re‐ sponse of relaxor materials can be summarized as follows: i- they are characterized by wide peaks in the temperature dependence of the dielectric permittivity, ii- the temperature of the corresponding maximum for the real (ε') and imaginary (ε'') component of the dielectric per‐ mittivity (Tm and Tε''max, respectively) appears at different values, showing a frequency de‐ pendent behaviour, and iii- the Curie-Weiss law is not fulfilled for temperatures around Tm. So that, the temperature of the maximum real dielectric permittivity, which depends on the

measurement frequency, cannot be associated with a FE-PE phase transition.

needs the addition of BaTiO3 (BT) and PbTiO3 (PT) to obtain pure phases [16,21].

Lead zirconatetitanate (PZT) system is a typical ferroelectric perovskite showing 'normal' FE-PE phase transition [1]. Nevertheless, the partial substitution by different elements, such as lanthanum, contributes to enhance such relaxor characteristics [11-15]. In fact, for some lanthanum concentrations, the distortion of the crystalline lattice in the PZT system due to ions displacement could promote the formation of the so-called polar nanoregions (PNRs). Another interesting relaxor ferroelectric perovskite is known as PZN-PT-BT [16-19]. The Pb(Zn1/3Nb2/3)O3 ferroelectric material (PZN) belongs to the relaxor ferroelectrics family, re‐ ceiving special attention for its technical importance [20]. However, its preparation usually

and reproduction in any medium, provided the original work is properly cited.

© 2013 Peláiz-Barranco et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

A. Peláiz-Barranco, F. Calderón-Piñar,

http://dx.doi.org/10.5772/52149

**1. Introduction**

O. García-Zaldívar and Y. González-Abreu

Additional information is available at the end of the chapter

### **Relaxor Behaviour in Ferroelectric Ceramics**

A. Peláiz-Barranco, F. Calderón-Piñar,

O. García-Zaldívar and Y. González-Abreu

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52149

### **1. Introduction**

[48] X. Hao, J. Zhai, and, X. Yao, J. Am. Ceram. Soc., 92, 1133 (2009).

[49] J. Parui, S. B. Krupannidhi, Appl. Phys. Lett., 92, 192901 (2008).

[51] G. D. Wilk, R. M. Wallace, J. M. Anthony, J. Appl. Phys. 89, 5243 (2001).

[52] N. Ortega, A Kumar, J. F. Scott, Douglas B. Chrisey, M. Tomazawa, Shalini Kumari, D. G. B. Diestra and R. S. Katiyar. J. Phys.: Condens. Matter. 24 445901 (2012).

[50] R. M. Wallace, G. Wilk, MRS Bulletin. 27, 186 (2002).

84 Advances in Ferroelectrics

Ferroelectric materials are commonly characterized by high dielectric permittivity values [1]. Usually, for the well-known 'normal' ferroelectrics the temperature of the maximum real di‐ electric permittivity (Tm) corresponds to the ferroelectric-paraelectric (FE-PE) phase transi‐ tion temperature (TC) [2]. On the other hand, there are some kinds of ferroelectrics, so-called relaxor ferroelectrics, which have received special attention in the last years because of the observed intriguing and extraordinary dielectric properties [3-25], which remain not clearly understood nowadays. For instance, some remarkable characteristics of the dielectric re‐ sponse of relaxor materials can be summarized as follows: i- they are characterized by wide peaks in the temperature dependence of the dielectric permittivity, ii- the temperature of the corresponding maximum for the real (ε') and imaginary (ε'') component of the dielectric per‐ mittivity (Tm and Tε''max, respectively) appears at different values, showing a frequency de‐ pendent behaviour, and iii- the Curie-Weiss law is not fulfilled for temperatures around Tm. So that, the temperature of the maximum real dielectric permittivity, which depends on the measurement frequency, cannot be associated with a FE-PE phase transition.

Lead zirconatetitanate (PZT) system is a typical ferroelectric perovskite showing 'normal' FE-PE phase transition [1]. Nevertheless, the partial substitution by different elements, such as lanthanum, contributes to enhance such relaxor characteristics [11-15]. In fact, for some lanthanum concentrations, the distortion of the crystalline lattice in the PZT system due to ions displacement could promote the formation of the so-called polar nanoregions (PNRs). Another interesting relaxor ferroelectric perovskite is known as PZN-PT-BT [16-19]. The Pb(Zn1/3Nb2/3)O3 ferroelectric material (PZN) belongs to the relaxor ferroelectrics family, re‐ ceiving special attention for its technical importance [20]. However, its preparation usually needs the addition of BaTiO3 (BT) and PbTiO3 (PT) to obtain pure phases [16,21].

© 2013 Peláiz-Barranco et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

On the other hand, a considerable number of compositions from the Aurivillius family exhibit a relaxor ferroelectric behaviour [22-26]. The Aurivillius compounds are layered bismuth [Bi2O2] 2+[An−1BnO3n+1] 2−, where the sites *A* and *B* can be occupied for different elements. These are formed by the regular stacking of Bi2O2 slabs and perovskite-like blocks An−1BnO3n+1. These materials have received great attention due to their large remanent polarization, lead-free na‐ ture, relatively low processing temperatures, high Curie temperatures and excellent piezo‐ electric properties, which made them good candidates for high-temperature piezoelectric applications and memory storage [27]. The origin of the relaxor behaviour for these materials have been associated to a positional disorder of cations on *A* or *B* sites of the perovskite blocks that delay the evolution of long-rage polar ordering [28].

**2. Lanthanum modified Lead Zirconate Titanate (PLZT)**

Several researches concerning PZT properties have shown that very good properties can be obtained by suitable selection of the Zr/Ti ratio and the substitution of a small amount of isovalent or heterovalent elements for the Pb or (Zr,Ti) sublattices [1]. Especial attention re‐ ceives the Zr-rich PZT-type ceramics, which exhibit interesting dielectric and pyroelectric characteristics, suitable for pyroelectric detectors, energy converters, imaging systems, etc [1,33-34]. They show orthorhombic antiferroelectric (AFE-O), low temperature rhombohe‐ dral ferroelectric (FR-LT), high temperature rhombohedral ferroelectric (FR-HT) and cubic

Relaxor Behaviour in Ferroelectric Ceramics http://dx.doi.org/10.5772/52149 87

One of the most common additives to the PZT ceramics is the lanthanum cation, which sub‐ stitute the Pb-ions in the A-site of the perovskite structure [1]. The lanthanum-modified lead zirconatetitanate Pb1-xLax(Zr1-yTiy)1-x/4O3 (PLZT) ferroelectric ceramics show excellent proper‐ ties to be considered for practical applications, especially PLZT x/65/35, x/70/30 and x/80/20 compositions due to the relaxor ferroelectric behaviour [35-36]. For Zr-rich PZT ceramics, which are close to the antiferroelectric-ferroelectric (AFE-FE) phase boundary, the FE phase is only marginally stable over the AFE [33]. In this way, as the FE state is disrupted by lan‐

**Figure 1.** Temperature dependence of the real (ε') and imaginary (ε'') parts of the dielectric permittivity at several fre‐

Figure 1 shows the temperature dependence of the real (ε') and imaginary (ε'') parts of the dielectric permittivity, at several frequencies, for PLZT 6/80/20 ferroelectric ceramic system. Typical characteristics of relaxor ferroelectric behaviour are observed. The frequency de‐ pendence of ε' is very weak but the frequency dependence of ε'' clearly reflects the relaxor nature of the system. The asymmetrical shape of the curves could suggest the existence of more than one contribution. X-ray diffraction analysis for the PLZT 6/80/20 ceramic samples

**2.1. Relaxor behaviour and coexistence of AFE and FE phases**

paraelectric (PC) phase sequence [1].

thanum modification, the AFE state is stabilized [33].

quencies for PLZT 6/80/20 ferroelectric ceramic system.

Several models have been proposed to explain the dielectric behaviour of relaxor ferroelec‐ trics. The basic ideas have been related to the dynamics and formation of the polar nanore‐ gions (PNRs). In this way, Smolenskii proposed the existence of compositional fluctuations on the nanometer scale taking into account a statistical distribution for the phase transition temperature [29]. On the other hand, Cross extended the Smolenskii's theory to a superpara‐ electric model associating the relaxor behaviour to a thermally activated ensemble of super‐ paraelectric clusters [30]. Viehland et al. have showed that cooperative interactions among these superparaelectric clusters could produce a glass-like freezing behaviour, commonly exhibited in spin-glass systems [31]. Later, Qian and Bursill [32] analyzed the possible influ‐ ence of random electric fields on the formation and dynamics of the polar clusters, which can be originated from nano-scaled chemical defects. They have also proposed that the re‐ laxor behaviour can be associated to a dipolar moment in an anisotropic double-well poten‐ tial, taking into account only two characteristic relaxation times. According to this model, the dispersive behaviour is produced by changes in the clusters size and the correlation length (defined as the distance, above which such PNRs become non-interactive regions) as a function of the temperature, which provides a distribution function for the activation ener‐ gy. However, despite their very attractive physical properties, the identification of the na‐ ture of the dielectric response in relaxors systems still remains open and requires additional theoretical and experimental information, which can be very interesting to contribute to the explanation of the origin of the observed anomalies.

The present chapter shows the studies carried out on several relaxor ferroelectric ceramic materials, which have been developed by the present authors. Lanthanum modified lead zirconatetitanate ceramics will be evaluated considering different La3+ concentrations and Zr/Ti ratios. It will be discussed the dynamical behaviour of the PNRs taking into account a relaxation model, which considers a distribution function for the relaxation times. The influ‐ ence of *A* or *B* vacancies on the relaxor behaviour will be also analyzed considering the de‐ coupling effects of these defects in the Pb-O-Ti/Zr bounding. For the PZN-PT-BT system, the relaxor behaviour will be explained considering the presence of local compositional fluctua‐ tion on a macroscopic scale. Finally, the relaxor behaviour of Sr0.50Ba0.50Bi2Nb2O9 ferroelectric ceramics, which belong to the Aurivillius family, will be discussed considering a positional disorder of cations on *A*or *B* sites of the perovskite blocks.

### **2. Lanthanum modified Lead Zirconate Titanate (PLZT)**

### **2.1. Relaxor behaviour and coexistence of AFE and FE phases**

On the other hand, a considerable number of compositions from the Aurivillius family exhibit a relaxor ferroelectric behaviour [22-26]. The Aurivillius compounds are layered bismuth

are formed by the regular stacking of Bi2O2 slabs and perovskite-like blocks An−1BnO3n+1. These materials have received great attention due to their large remanent polarization, lead-free na‐ ture, relatively low processing temperatures, high Curie temperatures and excellent piezo‐ electric properties, which made them good candidates for high-temperature piezoelectric applications and memory storage [27]. The origin of the relaxor behaviour for these materials have been associated to a positional disorder of cations on *A* or *B* sites of the perovskite blocks

Several models have been proposed to explain the dielectric behaviour of relaxor ferroelec‐ trics. The basic ideas have been related to the dynamics and formation of the polar nanore‐ gions (PNRs). In this way, Smolenskii proposed the existence of compositional fluctuations on the nanometer scale taking into account a statistical distribution for the phase transition temperature [29]. On the other hand, Cross extended the Smolenskii's theory to a superpara‐ electric model associating the relaxor behaviour to a thermally activated ensemble of super‐ paraelectric clusters [30]. Viehland et al. have showed that cooperative interactions among these superparaelectric clusters could produce a glass-like freezing behaviour, commonly exhibited in spin-glass systems [31]. Later, Qian and Bursill [32] analyzed the possible influ‐ ence of random electric fields on the formation and dynamics of the polar clusters, which can be originated from nano-scaled chemical defects. They have also proposed that the re‐ laxor behaviour can be associated to a dipolar moment in an anisotropic double-well poten‐ tial, taking into account only two characteristic relaxation times. According to this model, the dispersive behaviour is produced by changes in the clusters size and the correlation length (defined as the distance, above which such PNRs become non-interactive regions) as a function of the temperature, which provides a distribution function for the activation ener‐ gy. However, despite their very attractive physical properties, the identification of the na‐ ture of the dielectric response in relaxors systems still remains open and requires additional theoretical and experimental information, which can be very interesting to contribute to the

The present chapter shows the studies carried out on several relaxor ferroelectric ceramic materials, which have been developed by the present authors. Lanthanum modified lead zirconatetitanate ceramics will be evaluated considering different La3+ concentrations and Zr/Ti ratios. It will be discussed the dynamical behaviour of the PNRs taking into account a relaxation model, which considers a distribution function for the relaxation times. The influ‐ ence of *A* or *B* vacancies on the relaxor behaviour will be also analyzed considering the de‐ coupling effects of these defects in the Pb-O-Ti/Zr bounding. For the PZN-PT-BT system, the relaxor behaviour will be explained considering the presence of local compositional fluctua‐ tion on a macroscopic scale. Finally, the relaxor behaviour of Sr0.50Ba0.50Bi2Nb2O9 ferroelectric ceramics, which belong to the Aurivillius family, will be discussed considering a positional

2−, where the sites *A* and *B* can be occupied for different elements. These

[Bi2O2]

86 Advances in Ferroelectrics

2+[An−1BnO3n+1]

that delay the evolution of long-rage polar ordering [28].

explanation of the origin of the observed anomalies.

disorder of cations on *A*or *B* sites of the perovskite blocks.

Several researches concerning PZT properties have shown that very good properties can be obtained by suitable selection of the Zr/Ti ratio and the substitution of a small amount of isovalent or heterovalent elements for the Pb or (Zr,Ti) sublattices [1]. Especial attention re‐ ceives the Zr-rich PZT-type ceramics, which exhibit interesting dielectric and pyroelectric characteristics, suitable for pyroelectric detectors, energy converters, imaging systems, etc [1,33-34]. They show orthorhombic antiferroelectric (AFE-O), low temperature rhombohe‐ dral ferroelectric (FR-LT), high temperature rhombohedral ferroelectric (FR-HT) and cubic paraelectric (PC) phase sequence [1].

One of the most common additives to the PZT ceramics is the lanthanum cation, which sub‐ stitute the Pb-ions in the A-site of the perovskite structure [1]. The lanthanum-modified lead zirconatetitanate Pb1-xLax(Zr1-yTiy)1-x/4O3 (PLZT) ferroelectric ceramics show excellent proper‐ ties to be considered for practical applications, especially PLZT x/65/35, x/70/30 and x/80/20 compositions due to the relaxor ferroelectric behaviour [35-36]. For Zr-rich PZT ceramics, which are close to the antiferroelectric-ferroelectric (AFE-FE) phase boundary, the FE phase is only marginally stable over the AFE [33]. In this way, as the FE state is disrupted by lan‐ thanum modification, the AFE state is stabilized [33].

**Figure 1.** Temperature dependence of the real (ε') and imaginary (ε'') parts of the dielectric permittivity at several fre‐ quencies for PLZT 6/80/20 ferroelectric ceramic system.

Figure 1 shows the temperature dependence of the real (ε') and imaginary (ε'') parts of the dielectric permittivity, at several frequencies, for PLZT 6/80/20 ferroelectric ceramic system. Typical characteristics of relaxor ferroelectric behaviour are observed. The frequency de‐ pendence of ε' is very weak but the frequency dependence of ε'' clearly reflects the relaxor nature of the system. The asymmetrical shape of the curves could suggest the existence of more than one contribution. X-ray diffraction analysis for the PLZT 6/80/20 ceramic samples have previously shown a mixture of FE and AFE phases at room temperature [13]. Polarized light microscopy studies were performed on the ceramic system in a wide temperature range. Aggregates of individual regions were observed, which were associated to FE and AFE phases. As the temperature increased, a gradual change was observed, suggesting a transition to a single phase [13].

typically hundreds degrees above the temperature of the maximum real dielectric permittiv‐ ity (Tm). On cooling, the number, size (some of them) and the interaction of the PNRs in‐ crease [39]. The increase of such interactions promotes the freezing of some regions around certain temperatures below Tm, known as the freezing temperature (TF). Two fundamental polarization mechanisms have been reported, which have been associated with the dynam‐ ics of the PNRs; (i) dipole reorientation [37,40] and (ii) domain wall vibrations [32]. Both mechanisms have a characteristic time response, which depends on the temperature and size of the PNRs. If the contributions of an ensemble of these regions are considered, the

*t t t t*


with the Debye single relaxation time model, which takes into account a distribution func‐

<sup>0</sup> ( ) *<sup>s</sup> i i i P g* e e

t

2 1 ( )

s

p

*i*

=

t

*g*

ew

 e e e t

is the relaxation time of the ith PNR and P0i takes the form of equation (2), in analogy

t

2 2

s

) is a distribution function for the logarithms of the relaxation times,

w

¥ \* =- = + ¢ ¢¢ ò (4)

0

which has been assumed to be a Lorentz rather than a Gaussian distribution function; τ0 and σ are the mean relaxation time and the Standard Deviation, respectively. According to the Debye model, the frequency dependence of the complex dielectric permittivity can be ex‐

( ) ( ) *i t i P t e dt*

where εs and ε∞ are the low (static) and high (optical) dielectric permittivity, respectively, ω the measurement frequency, and P(t) the decay polarization function. Substituting equations (1), (2) and (3) into equation (4), the real and imaginary component of the dielectric permit‐ tivity for the multi-relaxation times approximation can be obtained and expressed as in equations (5) and (6). It is important to point out that equations (5) and (6) have been de‐


*i*

t

t

4(ln )

*i i*

= + + += å (1)

 t

Relaxor Behaviour in Ferroelectric Ceramics http://dx.doi.org/10.5772/52149 89

¥ - <sup>=</sup> (2)

<sup>+</sup> (3)

macroscopic polarization function, at fixed temperature, can be expressed by [41]:

1 2 3 01 02 03 <sup>0</sup> ( ) ..... *<sup>i</sup>*

 t

t

where τ<sup>i</sup>

tion of relaxation times [41]:

In these relations g(τ<sup>i</sup>

pressed as:

*Pt P e P e P e P e*

Figure 2 shows the hysteresis loops obtained for the PLZT 6/80/20 ceramics for two tempera‐ tures, as example of the obtained behaviour in a wide temperature range. Around 70o C a double-loop-like behaviour was observed suggesting an induced FE-AFE transforma‐ tion.The FE state is disrupted by the lanthanum modification and the AFE state is stabilized, which may be a consequence of the short-range nature of the interaction between the AFE sublattices. The double-loop remains for higher temperature and around 120o C the loop dis‐ appears suggesting a transition to a PE phase. Then, it is possible to consider the coexistence of two phase transitions: a FE-AFE phase transition observed around 70o C and an AFE-PE phase transition around 120o C. Note, that the maximum of ε' is observed around 120o C. Negative values for TC were found, which has confirmed that around 120o C an AFE-PE tran‐ sition take places [13]. Thus, the relaxor ferroelectric behaviour for this system could be as‐ sociated to the coexistence of FE and AFE phases in the studied material.

**Figure 2.** Hysteresis loops for PLZT 6/80/20 ferroelectric ceramic system.

#### **2.2. A relaxation model**

The nature of the relaxor behaviour is determined by the existence of PNRs, which possess different relaxation times [32,37]. The relaxation time (τ) represents the time response of such PNRs or polarization mechanisms to change with the applied electric field. However, this process does not occur instantaneously. Indeed, there exists certain inertia, which is the cause of the pronounced dielectric relaxation in relaxor ferroelectrics. The polar nanoregions appear below a certain temperature, the so-called the Burns' temperature (TB) [38], which is typically hundreds degrees above the temperature of the maximum real dielectric permittiv‐ ity (Tm). On cooling, the number, size (some of them) and the interaction of the PNRs in‐ crease [39]. The increase of such interactions promotes the freezing of some regions around certain temperatures below Tm, known as the freezing temperature (TF). Two fundamental polarization mechanisms have been reported, which have been associated with the dynam‐ ics of the PNRs; (i) dipole reorientation [37,40] and (ii) domain wall vibrations [32]. Both mechanisms have a characteristic time response, which depends on the temperature and size of the PNRs. If the contributions of an ensemble of these regions are considered, the macroscopic polarization function, at fixed temperature, can be expressed by [41]:

have previously shown a mixture of FE and AFE phases at room temperature [13]. Polarized light microscopy studies were performed on the ceramic system in a wide temperature range. Aggregates of individual regions were observed, which were associated to FE and AFE phases. As the temperature increased, a gradual change was observed, suggesting a

Figure 2 shows the hysteresis loops obtained for the PLZT 6/80/20 ceramics for two tempera‐ tures, as example of the obtained behaviour in a wide temperature range. Around 70o

double-loop-like behaviour was observed suggesting an induced FE-AFE transforma‐ tion.The FE state is disrupted by the lanthanum modification and the AFE state is stabilized, which may be a consequence of the short-range nature of the interaction between the AFE

appears suggesting a transition to a PE phase. Then, it is possible to consider the coexistence

sition take places [13]. Thus, the relaxor ferroelectric behaviour for this system could be as‐

The nature of the relaxor behaviour is determined by the existence of PNRs, which possess different relaxation times [32,37]. The relaxation time (τ) represents the time response of such PNRs or polarization mechanisms to change with the applied electric field. However, this process does not occur instantaneously. Indeed, there exists certain inertia, which is the cause of the pronounced dielectric relaxation in relaxor ferroelectrics. The polar nanoregions appear below a certain temperature, the so-called the Burns' temperature (TB) [38], which is

C. Note, that the maximum of ε' is observed around 120o

sublattices. The double-loop remains for higher temperature and around 120o

of two phase transitions: a FE-AFE phase transition observed around 70o

Negative values for TC were found, which has confirmed that around 120o

sociated to the coexistence of FE and AFE phases in the studied material.

**Figure 2.** Hysteresis loops for PLZT 6/80/20 ferroelectric ceramic system.

C a

C.

C the loop dis‐

C and an AFE-PE

C an AFE-PE tran‐

transition to a single phase [13].

88 Advances in Ferroelectrics

phase transition around 120o

**2.2. A relaxation model**

$$P(t) = P\_{01}e^{-\frac{t}{\tau\_1}} + P\_{02}e^{-\frac{t}{\tau\_2}} + P\_{03}e^{-\frac{t}{\tau\_3}} + \dots = \sum\_{i} P\_{0i}e^{-\frac{t}{\tau\_i}}\tag{1}$$

where τ<sup>i</sup> is the relaxation time of the ith PNR and P0i takes the form of equation (2), in analogy with the Debye single relaxation time model, which takes into account a distribution func‐ tion of relaxation times [41]:

$$P\_{0i} = \frac{\mathcal{E}\_s - \mathcal{E}\_\infty}{\pi\_i} g(\tau\_i) \tag{2}$$

$$\log(\tau\_i) = \frac{2\sigma}{\pi} \frac{1}{4(\ln\frac{\tau\_i}{\tau\_0})^2 + \sigma^2} \tag{3}$$

In these relations g(τ<sup>i</sup> ) is a distribution function for the logarithms of the relaxation times, which has been assumed to be a Lorentz rather than a Gaussian distribution function; τ0 and σ are the mean relaxation time and the Standard Deviation, respectively. According to the Debye model, the frequency dependence of the complex dielectric permittivity can be ex‐ pressed as:

$$\varepsilon \* \langle \alpha \rangle = \varepsilon' - \mathrm{i}\varepsilon'' = \varepsilon\_{\alpha} + \int P(t)e^{-i\alpha t}dt\tag{4}$$

where εs and ε∞ are the low (static) and high (optical) dielectric permittivity, respectively, ω the measurement frequency, and P(t) the decay polarization function. Substituting equations (1), (2) and (3) into equation (4), the real and imaginary component of the dielectric permit‐ tivity for the multi-relaxation times approximation can be obtained and expressed as in equations (5) and (6). It is important to point out that equations (5) and (6) have been de‐ rived from the discrete expression (1), taking into account the values of the relaxation times (τi ) close to each other [41].

$$
\omega'(\alpha, T) = \varepsilon\_{\alpha} + (\varepsilon\_s - \varepsilon\_{\alpha}) \frac{2\sigma}{\pi} \int\_{-\alpha}^{+\alpha} \frac{1}{(4z^2 + \sigma^2)(1 + \alpha^2 \tau\_0^2 \exp(2z))} dz \tag{5}
$$

$$
\varepsilon''(\boldsymbol{\alpha}, T) = (\varepsilon\_s - \varepsilon\_\alpha) \frac{2\sigma}{\pi} \int\_{-\alpha}^{+\alpha} \frac{\alpha \tau\_0 \exp(z)}{(4z^2 + \sigma^2)(1 + \alpha^2 \tau\_0^2 \exp(2z)} dz \tag{6}
$$

**Figure 4.** Temperature dependence of the imaginary part (ε'') of the dielectric permittivity at several frequencies for PLZT 10/80/20 ferroelectric ceramic system. The experimental values are represented by solid points and the theoreti‐

Relaxor Behaviour in Ferroelectric Ceramics http://dx.doi.org/10.5772/52149 91

The number of PNRs able to follow the applied external electric field switching decreases with the increase of the frequency, so that only such inertial-less regions contribute to the dielectric permittivity (those regions whose activation energy is close to thermal energy kBT, being kBthe Boltzmann constant). Therefore, due to the cooperative nature, the real compo‐ nent of the dielectric permittivity should decrease with the increase of the measurement fre‐

The maximum for ε' in relaxor systems is not related to a crystallographic transition. Indeed, such a maximum corresponds to rapid changes of the fraction of the frozen Polar Regions [39]. Hence, the shift up to higher temperatures of the maximum for ε' with the increase of the fre‐

By using equations (5) and (6) and the experimental data of Figures 3 and 4, the temperature dependences of εs, σ and τ<sup>0</sup> were obtained. The results were obtained by numerical methods because there was no analytical solution for the equation system. The theoretical curves for ε' and ε'' were obtained for all the studied frequency range, and shown in the same Figures 3 and 4 as solid lines, for the studied system. The temperature dependence of the static dielec‐ tric permittivity (εs) is also shown in Figure 3. A good agreement between experimental and theoretical results can be observed. It is important to point out that the deviation between the experimental and theoretical results, observed at low frequencies for the temperature de‐ pendence of ε'', can be associated with the contribution of the electric conductivity to the di‐

Figure 5 shows the temperature dependence of lnτ0 and σ for the studied composition. As can be seen, lnτ0 increases with the increase of the temperature, passes through a maximum and then decreases for higher temperatures. Similar results have been previously reported by Lin et al [41]. According to the model, the logarithmic mean relaxation time (lnτ0) should have has approximately the same tendency as that of σ(T). Both parameters increases when

quency is a direct consequence of the delay in dielectric response of the frozen regions.

electric response, which has not been considered in the proposed model.

cal results by solid lines.

quency.

It has been considered in equations (5) and (6) that *z* = ln*τ/τ*0. By using the experimental re‐ sults of ε'(ω,T) and ε''(ω,T) for two frequencies, the temperature dependence of dielectric pa‐ rameters, such as, τ0, εs and σ can be obtained as a solution of the equations system. After that, by using the theoretical results of τ0(T), εs(T) and σ(T), the theoretical dependences of ε'(ω,T) and ε''(ω,T) can be obtained for the studied frequency and temperature ranges. The parameter ε∞ has been considered negligible because of the high values of the dielectric per‐ mittivity in ferroelectric systems [1].

Figures 3 and 4 show the temperature dependence of the real (ε') and imaginary (ε'') compo‐ nents of the dielectric permittivity (symbols), at several frequencies, for the studied PLZT 10/80/20 composition. A relaxor characteristic behaviour can be observed. The maximum re‐ al dielectric permittivity decreases, while its corresponding temperature (Tm) increases, with the increase of the measurement frequency. There is not any peak for ε'' due to the low tem‐ perature range where the maximum of the real part of the dielectric permittivity appears, i.e. the maximum for ε'' should appear below the room temperature.

**Figure 3.** Temperature dependence of the real part (ε') of the dielectric permittivity at several frequencies for PLZT 10/80/20 ferroelectric ceramic system. The experimental values are represented by solid points and the theoretical re‐ sults by solid lines. It has been included the theoretical temperature dependence of the static dielectric permittivity (εS).

rived from the discrete expression (1), taking into account the values of the relaxation times

2 1 (,) ( ) (4 )(1 exp(2 )) *<sup>s</sup> <sup>T</sup> dz*

<sup>2</sup> exp( ) (,) ( ) (4 )(1 exp(2 ) *<sup>s</sup> <sup>z</sup> <sup>T</sup> dz*

s

+¥


p

s +¥


p

¥

the maximum for ε'' should appear below the room temperature.

2 2 22

0 2 2 22

wt

s

It has been considered in equations (5) and (6) that *z* = ln*τ/τ*0. By using the experimental re‐ sults of ε'(ω,T) and ε''(ω,T) for two frequencies, the temperature dependence of dielectric pa‐ rameters, such as, τ0, εs and σ can be obtained as a solution of the equations system. After that, by using the theoretical results of τ0(T), εs(T) and σ(T), the theoretical dependences of ε'(ω,T) and ε''(ω,T) can be obtained for the studied frequency and temperature ranges. The parameter ε∞ has been considered negligible because of the high values of the dielectric per‐

Figures 3 and 4 show the temperature dependence of the real (ε') and imaginary (ε'') compo‐ nents of the dielectric permittivity (symbols), at several frequencies, for the studied PLZT 10/80/20 composition. A relaxor characteristic behaviour can be observed. The maximum re‐ al dielectric permittivity decreases, while its corresponding temperature (Tm) increases, with the increase of the measurement frequency. There is not any peak for ε'' due to the low tem‐ perature range where the maximum of the real part of the dielectric permittivity appears, i.e.

**Figure 3.** Temperature dependence of the real part (ε') of the dielectric permittivity at several frequencies for PLZT 10/80/20 ferroelectric ceramic system. The experimental values are represented by solid points and the theoretical re‐ sults by solid lines. It has been included the theoretical temperature dependence of the static dielectric permittivity (εS).

*z z*

¢¢ = - + + ò (6)

 wt

s

*z z*

 wt

0

0

+ + ò (5)

(τi

) close to each other [41].

90 Advances in Ferroelectrics

ew

ew

mittivity in ferroelectric systems [1].

 e

¢ =+-

 e e

 e e

¥ ¥

**Figure 4.** Temperature dependence of the imaginary part (ε'') of the dielectric permittivity at several frequencies for PLZT 10/80/20 ferroelectric ceramic system. The experimental values are represented by solid points and the theoreti‐ cal results by solid lines.

The number of PNRs able to follow the applied external electric field switching decreases with the increase of the frequency, so that only such inertial-less regions contribute to the dielectric permittivity (those regions whose activation energy is close to thermal energy kBT, being kBthe Boltzmann constant). Therefore, due to the cooperative nature, the real compo‐ nent of the dielectric permittivity should decrease with the increase of the measurement fre‐ quency.

The maximum for ε' in relaxor systems is not related to a crystallographic transition. Indeed, such a maximum corresponds to rapid changes of the fraction of the frozen Polar Regions [39]. Hence, the shift up to higher temperatures of the maximum for ε' with the increase of the fre‐ quency is a direct consequence of the delay in dielectric response of the frozen regions.

By using equations (5) and (6) and the experimental data of Figures 3 and 4, the temperature dependences of εs, σ and τ<sup>0</sup> were obtained. The results were obtained by numerical methods because there was no analytical solution for the equation system. The theoretical curves for ε' and ε'' were obtained for all the studied frequency range, and shown in the same Figures 3 and 4 as solid lines, for the studied system. The temperature dependence of the static dielec‐ tric permittivity (εs) is also shown in Figure 3. A good agreement between experimental and theoretical results can be observed. It is important to point out that the deviation between the experimental and theoretical results, observed at low frequencies for the temperature de‐ pendence of ε'', can be associated with the contribution of the electric conductivity to the di‐ electric response, which has not been considered in the proposed model.

Figure 5 shows the temperature dependence of lnτ0 and σ for the studied composition. As can be seen, lnτ0 increases with the increase of the temperature, passes through a maximum and then decreases for higher temperatures. Similar results have been previously reported by Lin et al [41]. According to the model, the logarithmic mean relaxation time (lnτ0) should have has approximately the same tendency as that of σ(T). Both parameters increases when the temperature decreases which can be understood as a consequence of the increased iner‐ tia of the dipolar clusters as they become more correlated with each other upon cooling. This observation also could reflect a dynamic change in the well potential for the shifting ions. However, the reduction in lnτ0(T) at lower temperatures possibly could reflect a freezing phenomenon of some clusters that are saturated in correlation, which leads to the frustration of cooperative interactions and, hence, leaves only relatively unstrained or smaller clusters available to couple with the electric field. Also, one could speculate that such clusters are the regions within the domain boundaries between the frozen regions.

So that, a decrement of the correlation between the PNRs is promoted by an increase of the temperature and the Standard deviation decreases up to a relatively constant value around

**Figure 6.** Temperature dependence of the frequency (lnf versus 1/(Tm–TF) curve) for PLZT 10/80/20 ferroelectric ce‐

The freezing temperature (TF) was determined by using the Vogel–Fulcher relation [44], as

*Bm F* ( ) *U kT T*

where fD is the Debye frequency and U is the activation energy. Figure 6 show the tempera‐ ture dependence of the frequency (lnf versus 1/(Tm–TF) curve) and the fitting by using equa‐ tion (7). In order to maintain the standard representation, the temperature in Figure 6 was expressed in Kelvin. The fitting parameters U, fD and TF have been included in the figure. The results are in agreement with previously reported results in the literature [42]. The disa‐ greement between the TF value, which has been determined by using the Vogel-Fulcherrela‐ tion, and the temperature corresponding to the maximum value of lnτ0 could be associated

Relaxor ferroelectrics are formed by temperature dependent Polar Nanometric Regions (PNRs), which possess different volumes [45] and orientations for the polarization [30,37,46-47]. The PNRs appear at elevated temperatures (in the paraelectric state), at the socalled Burns' temperature (TB) [38,48-49], due to short-range interactions, which establish a


Relaxor Behaviour in Ferroelectric Ceramics http://dx.doi.org/10.5772/52149 93

*<sup>D</sup> f fe*

ramic system. Solid lines represent the fitting by using the Vogel–Fulcher relation.

to the diffusivity of the phase transition in the studied system.

local polarization thermally fluctuating between equivalents polar states.

**2.3. Relaxor behaviour and vacancies.**

expressed by equation (7):

the Burns' temperature (TB) [38,43], where the PNRs completely disappear.

**Figure 5.** Temperature dependence of the main relaxation time (logarithmic representation, lnτ0) and its standard de‐ viation (σ), for PLZT 10/80/20 ferroelectric ceramic system.

In this way, the size and the interaction between PNRs increase on cooling from the high temperature region and its contribution to the dielectric permittivity becomes negligible be‐ low the freezing temperature (TF) [39]. In this temperature range (T < TF), the applied electric field is not strong enough to break such interactions and only the smallest regions can switch with the electric field, i.e. that is to say, there is a frustration of the cooperative effect. For temperatures above TF, there are fluctuations between equivalent polarization states [30], leading to a decrease of the macroscopic polarization upon heating [42]. The applied electric field cannot reorient the ferroelectric dipoles because of the thermal fluctuation. The potential barrier between equivalent polarization states decreases with the increase of the temperature and the thermal energy promotes the spontaneous switching of the dipoles, even when an electric field is applied. The contribution to the dielectric permittivity is due to those PNRs, which can switch with the applied electric field. Thus, for temperatures slightly above TF all the PNRs could contribute to the dielectric permittivity and a maximum value of the mean relaxation time could be expected. Therefore, the temperature corre‐ sponding to the maximum of lnτ0 can be related to the freezing temperature. The standard deviation (σ), which can be interpreted as the correlation between the PNRs [32,41], decreas‐ es upon heating and it is relatively small at high temperatures. When the temperature in‐ creases the thermal energy is high enough to break down the interaction between the PNRs. So that, a decrement of the correlation between the PNRs is promoted by an increase of the temperature and the Standard deviation decreases up to a relatively constant value around the Burns' temperature (TB) [38,43], where the PNRs completely disappear.

**Figure 6.** Temperature dependence of the frequency (lnf versus 1/(Tm–TF) curve) for PLZT 10/80/20 ferroelectric ce‐ ramic system. Solid lines represent the fitting by using the Vogel–Fulcher relation.

The freezing temperature (TF) was determined by using the Vogel–Fulcher relation [44], as expressed by equation (7):

$$f = f\_D e^{-\frac{U}{k\_B \left(T\_m - T\_F\right)}}\tag{7}$$

where fD is the Debye frequency and U is the activation energy. Figure 6 show the tempera‐ ture dependence of the frequency (lnf versus 1/(Tm–TF) curve) and the fitting by using equa‐ tion (7). In order to maintain the standard representation, the temperature in Figure 6 was expressed in Kelvin. The fitting parameters U, fD and TF have been included in the figure. The results are in agreement with previously reported results in the literature [42]. The disa‐ greement between the TF value, which has been determined by using the Vogel-Fulcherrela‐ tion, and the temperature corresponding to the maximum value of lnτ0 could be associated to the diffusivity of the phase transition in the studied system.

### **2.3. Relaxor behaviour and vacancies.**

the temperature decreases which can be understood as a consequence of the increased iner‐ tia of the dipolar clusters as they become more correlated with each other upon cooling. This observation also could reflect a dynamic change in the well potential for the shifting ions. However, the reduction in lnτ0(T) at lower temperatures possibly could reflect a freezing phenomenon of some clusters that are saturated in correlation, which leads to the frustration of cooperative interactions and, hence, leaves only relatively unstrained or smaller clusters available to couple with the electric field. Also, one could speculate that such clusters are the

**Figure 5.** Temperature dependence of the main relaxation time (logarithmic representation, lnτ0) and its standard de‐

In this way, the size and the interaction between PNRs increase on cooling from the high temperature region and its contribution to the dielectric permittivity becomes negligible be‐ low the freezing temperature (TF) [39]. In this temperature range (T < TF), the applied electric field is not strong enough to break such interactions and only the smallest regions can switch with the electric field, i.e. that is to say, there is a frustration of the cooperative effect. For temperatures above TF, there are fluctuations between equivalent polarization states [30], leading to a decrease of the macroscopic polarization upon heating [42]. The applied electric field cannot reorient the ferroelectric dipoles because of the thermal fluctuation. The potential barrier between equivalent polarization states decreases with the increase of the temperature and the thermal energy promotes the spontaneous switching of the dipoles, even when an electric field is applied. The contribution to the dielectric permittivity is due to those PNRs, which can switch with the applied electric field. Thus, for temperatures slightly above TF all the PNRs could contribute to the dielectric permittivity and a maximum value of the mean relaxation time could be expected. Therefore, the temperature corre‐ sponding to the maximum of lnτ0 can be related to the freezing temperature. The standard deviation (σ), which can be interpreted as the correlation between the PNRs [32,41], decreas‐ es upon heating and it is relatively small at high temperatures. When the temperature in‐ creases the thermal energy is high enough to break down the interaction between the PNRs.

regions within the domain boundaries between the frozen regions.

viation (σ), for PLZT 10/80/20 ferroelectric ceramic system.

92 Advances in Ferroelectrics

Relaxor ferroelectrics are formed by temperature dependent Polar Nanometric Regions (PNRs), which possess different volumes [45] and orientations for the polarization [30,37,46-47]. The PNRs appear at elevated temperatures (in the paraelectric state), at the socalled Burns' temperature (TB) [38,48-49], due to short-range interactions, which establish a local polarization thermally fluctuating between equivalents polar states.

The disorder in the arrangement of different ions on the crystallographic equivalent sites is the common feature of relaxors [50]. This is associated, in general, with a complex perov‐ skite of the type A(B'B'')O3, where the *B* sites of the structure are occupied by different cati‐ ons. Nevertheless, in lead zirconate titanate based ceramics (PZT) the disorder due to the arrangement of the isovalent cations Zr4+ and Ti4+, in the *B* site of the structure, does not lead to relaxor behaviour for any Zr/Ti ratio. The lanthanum modification on this system, above certain La3+ concentrations, provides a relaxor ferroelectric state [11-12]. Electrical neutrality can be achieved, stoichiometrically, considering the vacancies formation either in the *A*-site (Pb2+) or in the *B*-site (Zr4+, Ti4+), or on both. The distortion of the crystalline lattice in the PZT system could promote the formation of the PNRs. In this section, the results concerning the influence of the *A* or *B* vacancies defects in the relaxor ferroelectric behaviour of lanthanum modified PZT ceramic samples will be presented. The studied compositions will be Pb0.85La0.10(Zr0.60Ti0.40)O3 and Pb0.90La0.10(Zr0.60Ti0.40)0.975O3, labelled as PLZT10-VA and PLZT10- VB, respectively.

The temperature dependence of the real dielectric permittivity (ε') and the dielectric losses (tan δ) are shown in the Figures 7 and 8, respectively, for several frequencies. Typical char‐ acteristics of relaxor ferroelectrics are exhibited for both samples, i.e. wide peaks for ε', the maximum real dielectric permittivity shifts to higher temperatures with the increase of the frequency, and the maximum dielectric losses temperature appears at temperatures below Tm. On the other hand, it can be observed that the dielectric response is highly affected by the type of compensation. The maximum values of the real dielectric permittivity in the PLZT10-VA sample are lower than those obtained for the PLZT-10VB sample, and appear at lower temperatures. The PLZT10-VA sample also exhibits a larger temperature shift of Tm (ΔT=15<sup>o</sup> C), from 1 kHz to 1 MHz, than that of the PLZT10-VB sample (ΔT=9<sup>o</sup> C).

**Figure 8.** Temperature dependence of the dielectric losses (tan δ) at several frequencies for PLZT10-VA and PLZT10-VB

Relaxor Behaviour in Ferroelectric Ceramics http://dx.doi.org/10.5772/52149 95

It has been previously commented that, in relaxors, the maximum real dielectric permittivity is not a consequence of crystallographic changes but it is associated with a rapid change in the volume fraction of frozen polar nanoregions [39]. During cooling, the PNRs grow in size and number [39,43] as well as the interactions among them increases. Thus, a decrement of its mobility is expected; the thermal energy (kBT) is not enough to switch the polar state of the PNRs (freezing phenomenon). While the regions grow, with the decreasing of the tem‐ perature, the thermal energy kBT decreases. The freezing process begins at high tempera‐ tures and finish at low temperatures, below Tm, in the so called freezing temperature (TF),

The dielectric and ferroelectric response in relaxor ferroelectrics strongly depend of the tem‐ perature dependence of the freezing process (dynamic of the PNRs) as well as on the inter‐ actions between external electric field and the dipole moment of the PNRs. When a weak alternating electric field is applied, only the unfrozen regions and the frozen ones with acti‐ vation energies close to kBT could switch with the applied alternating electric field. There‐ fore, only these regions contribute to the dielectric permittivity. However, not all these regions can switch at any excitation frequency; the number of PNRs able to follow the ap‐ plied external electric field switching decreases with the increase of the frequency [14]. The response also depends on the relaxation time of each region [14, 41]. Thus, the dielectric re‐ sponse in relaxors depends of the number of regions that can contribute at each temperature

The Pb2+ ions (*A*-sites of the perovskite structure) establish the long-range order in PZT based ceramics due to strong coupling of Pb-O-Ti/Zr bonding [51]. The coupling defines the height of the energy barrier between polar states via Ti/Zr-hopping [51]. The Pb2+ substitu‐ tion by La3+ ions, above certain level of La3+ ions concentration, weakens this bonding result‐ ing in small and broadly distributed energy barriers [51], which disrupts the long-range

ceramic systems.

where all the PNRs are frozen.

and the distribution of the corresponding relaxation times.

**Figure 7.** Temperature dependence of the real (ε') component of the dielectric permittivity at several frequencies, for PLZT10-VA and PLZT10-VB ceramic samples.

The disorder in the arrangement of different ions on the crystallographic equivalent sites is the common feature of relaxors [50]. This is associated, in general, with a complex perov‐ skite of the type A(B'B'')O3, where the *B* sites of the structure are occupied by different cati‐ ons. Nevertheless, in lead zirconate titanate based ceramics (PZT) the disorder due to the arrangement of the isovalent cations Zr4+ and Ti4+, in the *B* site of the structure, does not lead to relaxor behaviour for any Zr/Ti ratio. The lanthanum modification on this system, above certain La3+ concentrations, provides a relaxor ferroelectric state [11-12]. Electrical neutrality can be achieved, stoichiometrically, considering the vacancies formation either in the *A*-site (Pb2+) or in the *B*-site (Zr4+, Ti4+), or on both. The distortion of the crystalline lattice in the PZT system could promote the formation of the PNRs. In this section, the results concerning the influence of the *A* or *B* vacancies defects in the relaxor ferroelectric behaviour of lanthanum modified PZT ceramic samples will be presented. The studied compositions will be Pb0.85La0.10(Zr0.60Ti0.40)O3 and Pb0.90La0.10(Zr0.60Ti0.40)0.975O3, labelled as PLZT10-VA and PLZT10-

The temperature dependence of the real dielectric permittivity (ε') and the dielectric losses (tan δ) are shown in the Figures 7 and 8, respectively, for several frequencies. Typical char‐ acteristics of relaxor ferroelectrics are exhibited for both samples, i.e. wide peaks for ε', the maximum real dielectric permittivity shifts to higher temperatures with the increase of the frequency, and the maximum dielectric losses temperature appears at temperatures below Tm. On the other hand, it can be observed that the dielectric response is highly affected by the type of compensation. The maximum values of the real dielectric permittivity in the PLZT10-VA sample are lower than those obtained for the PLZT-10VB sample, and appear at lower temperatures. The PLZT10-VA sample also exhibits a larger temperature shift of Tm

C), from 1 kHz to 1 MHz, than that of the PLZT10-VB sample (ΔT=9<sup>o</sup>

**Figure 7.** Temperature dependence of the real (ε') component of the dielectric permittivity at several frequencies, for

C).

VB, respectively.

94 Advances in Ferroelectrics

(ΔT=15<sup>o</sup>

PLZT10-VA and PLZT10-VB ceramic samples.

**Figure 8.** Temperature dependence of the dielectric losses (tan δ) at several frequencies for PLZT10-VA and PLZT10-VB ceramic systems.

It has been previously commented that, in relaxors, the maximum real dielectric permittivity is not a consequence of crystallographic changes but it is associated with a rapid change in the volume fraction of frozen polar nanoregions [39]. During cooling, the PNRs grow in size and number [39,43] as well as the interactions among them increases. Thus, a decrement of its mobility is expected; the thermal energy (kBT) is not enough to switch the polar state of the PNRs (freezing phenomenon). While the regions grow, with the decreasing of the tem‐ perature, the thermal energy kBT decreases. The freezing process begins at high tempera‐ tures and finish at low temperatures, below Tm, in the so called freezing temperature (TF), where all the PNRs are frozen.

The dielectric and ferroelectric response in relaxor ferroelectrics strongly depend of the tem‐ perature dependence of the freezing process (dynamic of the PNRs) as well as on the inter‐ actions between external electric field and the dipole moment of the PNRs. When a weak alternating electric field is applied, only the unfrozen regions and the frozen ones with acti‐ vation energies close to kBT could switch with the applied alternating electric field. There‐ fore, only these regions contribute to the dielectric permittivity. However, not all these regions can switch at any excitation frequency; the number of PNRs able to follow the ap‐ plied external electric field switching decreases with the increase of the frequency [14]. The response also depends on the relaxation time of each region [14, 41]. Thus, the dielectric re‐ sponse in relaxors depends of the number of regions that can contribute at each temperature and the distribution of the corresponding relaxation times.

The Pb2+ ions (*A*-sites of the perovskite structure) establish the long-range order in PZT based ceramics due to strong coupling of Pb-O-Ti/Zr bonding [51]. The coupling defines the height of the energy barrier between polar states via Ti/Zr-hopping [51]. The Pb2+ substitu‐ tion by La3+ ions, above certain level of La3+ ions concentration, weakens this bonding result‐ ing in small and broadly distributed energy barriers [51], which disrupts the long-range order and promotes the PNRs formations. An additional contribution, which plays a funda‐ mental role in the relaxor behaviour, can be originated from the *A*- or *B*-sites vacancies for‐ mations. There are two important differences between the *A*- or *B*-site compensation, which can explain the different dielectric behaviour observed in the studied samples: *i*) for the same lanthanum concentration the number of lead vacancies is twice the Zr/Ti vacancies, if it is considered *A*- or *B*-site compensation, respectively, i.e. 2La3+→1VPb and 4La3+→1VZr/Ti; *ii*) the *A*-site vacancies generate large inhomogeneous electric fields, which reduce the barriers between energy minima for different polarization directions [20], and promote the decou‐ pling of the Pb-O-Ti/Zr bonding, making the corresponding Ti/Zr-hopping easier due to the lack of lead ions. For the case of the *B*-site vacancies, all the Ti/Zr ions are well coupled to the lead or the lanthanum ions.

From this point of view, it could be pointed out that for the PLZT10-VA sample the defects con‐ centration, which could promote the PNRs formation, is higher than that for the PLZT10-VB sample. On the other hand, the thermal energy, which could change the nanoregions polariza‐ tion state, is smaller for the PLZT10-VA sample as a consequence of the smaller hopping barri‐ er. Considering these analyses, it could be explained the lower Tm values of the real part of the dielectric permittivity for the PLZT10-VA. The observed results are in agreement with previ‐ ous reports for lead based systems with and without Pb2+ vacancies [52].

**Figure 9.** Temperature dependence of remnant polarization for the PLZT10-VA and PLZT10-VB samples. The lines be‐

Relaxor Behaviour in Ferroelectric Ceramics http://dx.doi.org/10.5772/52149 97

Relaxor-based ferroelectric materials PbMg1/3Nb2/3O3–PbTiO3 (PMN–PT) and PbZn1/3Nb2/3O3–PbTiO3 (PZN–PT) have been extensively studied due to their high electrome‐

It is known that the large dielectric permittivity of perovskite relaxor ferroelectrics, such as (PbMg1/3Nb2/3O3)1−*<sup>x</sup>*(PbTiO3)*x*, comes from ferroelectric like polar nanoregions [30]. Upon lowering the temperature these nanoregions grow slightly [43] but do not form long-range ferroelectric order. The temperature dependence of the dielectric permittivity shows a fre‐ quency-dependent peak as the polar nanoregion orientations undergo some sort of glassy kinetic freezing. Analyses of the phase and microstructure evolution for these materials have showed that the increase in the sintering temperature caused the weakening of the die‐ lectric relaxor behaviour [54]. It has been related to compositional fluctuation and the release

The PNRs freezing process in PbMg1/3Nb2/3O3 has been analyzed considering both polariza‐ tion and strain operating on subtly different timescales and length scales [57]. The strain fields are considered as relatively weak but longer ranging, while the dipole interactions tend to be short range and relatively strong. It has been discussed that the elastic shear strain fields is partly suppressed by cation disordering and screened by the presence of anti‐ phase boundaries with their own distinctive strain and elastic properties, which provide a

On the other hand, the barium modified PZN-PT system has shown excellent piezoelectric properties [58]. The preparation of PZN system usually needs the addition of BaTiO3 (BT) to obtain pure phases. Therefore, the ternary system PZN-PT-BT has received wider attention,

chanical coupling factor and piezoelectric coefficient [3,8,16-21,43-44,54-57].

of the internal stresses, leading to the decrease in the short-range *B*-site order.

mechanism for suppressing longer ranging correlations of strain fields.

tween dots are only a guide to the eyes.

**3. Other typical relaxor ferroelectric perovskites**

The larger temperature shift of Tm, which was analyzed for the PLZT10-VA, could be related to a higher temperature dependence of the freezing process around Tm, i.e. higher tempera‐ ture dependence in the change of the volume fraction of frozen polar nanoregions around Tm. The number of PNRs, which can follow the electric field switching, decreases with the increase of the frequency. Thus, the maximum value of ε' decreases when the frequency in‐ creases, and also shifts to higher temperatures.

From the Figure 8, it is observed that the maximum values of the dielectric losses, for both samples, are observed in the same temperature range. No remarkable differences exist as can be expected from the difference observed in the Tm values between both systems. This could suggest similar dipolar dynamic in the temperature range for both ceramics.

Figure 9 shows the temperature dependence of the remanent polarization (PR) for both sam‐ ples. The PLZT10-VB sample exhibits higher values, which is in agreement with the previ‐ ous discussion concerning the hopping barrier. The decoupling introduced by *A*-site vacancies affects the total dipolar moment of the system and, consequently, the macroscopic polarization is affected. Thus, the magnitude of the polarization decreases with the increase of the *A*-site vacancies concentration.

On the other hand, the remanent polarization for both samples shows an anomaly in the same temperature region (70–85o C), which could be associated with the 'offset' of the freez‐ ing process [14, 53]. This result suggests that the freezing temperature (TF) could be the same for both samples or, at less the dynamic of PNRs is the same in both systems for that tem‐ perature range. Furthermore, the anomaly is observed in the same region where the maxi‐ mum values of the dielectric losses have been observed, which could suggest a relation between the freezing phenomenon and the temperature evolution of dielectric losses.

**Figure 9.** Temperature dependence of remnant polarization for the PLZT10-VA and PLZT10-VB samples. The lines be‐ tween dots are only a guide to the eyes.

### **3. Other typical relaxor ferroelectric perovskites**

order and promotes the PNRs formations. An additional contribution, which plays a funda‐ mental role in the relaxor behaviour, can be originated from the *A*- or *B*-sites vacancies for‐ mations. There are two important differences between the *A*- or *B*-site compensation, which can explain the different dielectric behaviour observed in the studied samples: *i*) for the same lanthanum concentration the number of lead vacancies is twice the Zr/Ti vacancies, if it is considered *A*- or *B*-site compensation, respectively, i.e. 2La3+→1VPb and 4La3+→1VZr/Ti; *ii*) the *A*-site vacancies generate large inhomogeneous electric fields, which reduce the barriers between energy minima for different polarization directions [20], and promote the decou‐ pling of the Pb-O-Ti/Zr bonding, making the corresponding Ti/Zr-hopping easier due to the lack of lead ions. For the case of the *B*-site vacancies, all the Ti/Zr ions are well coupled to

From this point of view, it could be pointed out that for the PLZT10-VA sample the defects con‐ centration, which could promote the PNRs formation, is higher than that for the PLZT10-VB sample. On the other hand, the thermal energy, which could change the nanoregions polariza‐ tion state, is smaller for the PLZT10-VA sample as a consequence of the smaller hopping barri‐ er. Considering these analyses, it could be explained the lower Tm values of the real part of the dielectric permittivity for the PLZT10-VA. The observed results are in agreement with previ‐

The larger temperature shift of Tm, which was analyzed for the PLZT10-VA, could be related to a higher temperature dependence of the freezing process around Tm, i.e. higher tempera‐ ture dependence in the change of the volume fraction of frozen polar nanoregions around Tm. The number of PNRs, which can follow the electric field switching, decreases with the increase of the frequency. Thus, the maximum value of ε' decreases when the frequency in‐

From the Figure 8, it is observed that the maximum values of the dielectric losses, for both samples, are observed in the same temperature range. No remarkable differences exist as can be expected from the difference observed in the Tm values between both systems. This

Figure 9 shows the temperature dependence of the remanent polarization (PR) for both sam‐ ples. The PLZT10-VB sample exhibits higher values, which is in agreement with the previ‐ ous discussion concerning the hopping barrier. The decoupling introduced by *A*-site vacancies affects the total dipolar moment of the system and, consequently, the macroscopic polarization is affected. Thus, the magnitude of the polarization decreases with the increase

On the other hand, the remanent polarization for both samples shows an anomaly in the

ing process [14, 53]. This result suggests that the freezing temperature (TF) could be the same for both samples or, at less the dynamic of PNRs is the same in both systems for that tem‐ perature range. Furthermore, the anomaly is observed in the same region where the maxi‐ mum values of the dielectric losses have been observed, which could suggest a relation

between the freezing phenomenon and the temperature evolution of dielectric losses.

C), which could be associated with the 'offset' of the freez‐

could suggest similar dipolar dynamic in the temperature range for both ceramics.

ous reports for lead based systems with and without Pb2+ vacancies [52].

creases, and also shifts to higher temperatures.

of the *A*-site vacancies concentration.

same temperature region (70–85o

the lead or the lanthanum ions.

96 Advances in Ferroelectrics

Relaxor-based ferroelectric materials PbMg1/3Nb2/3O3–PbTiO3 (PMN–PT) and PbZn1/3Nb2/3O3–PbTiO3 (PZN–PT) have been extensively studied due to their high electrome‐ chanical coupling factor and piezoelectric coefficient [3,8,16-21,43-44,54-57].

It is known that the large dielectric permittivity of perovskite relaxor ferroelectrics, such as (PbMg1/3Nb2/3O3)1−*<sup>x</sup>*(PbTiO3)*x*, comes from ferroelectric like polar nanoregions [30]. Upon lowering the temperature these nanoregions grow slightly [43] but do not form long-range ferroelectric order. The temperature dependence of the dielectric permittivity shows a fre‐ quency-dependent peak as the polar nanoregion orientations undergo some sort of glassy kinetic freezing. Analyses of the phase and microstructure evolution for these materials have showed that the increase in the sintering temperature caused the weakening of the die‐ lectric relaxor behaviour [54]. It has been related to compositional fluctuation and the release of the internal stresses, leading to the decrease in the short-range *B*-site order.

The PNRs freezing process in PbMg1/3Nb2/3O3 has been analyzed considering both polariza‐ tion and strain operating on subtly different timescales and length scales [57]. The strain fields are considered as relatively weak but longer ranging, while the dipole interactions tend to be short range and relatively strong. It has been discussed that the elastic shear strain fields is partly suppressed by cation disordering and screened by the presence of anti‐ phase boundaries with their own distinctive strain and elastic properties, which provide a mechanism for suppressing longer ranging correlations of strain fields.

On the other hand, the barium modified PZN-PT system has shown excellent piezoelectric properties [58]. The preparation of PZN system usually needs the addition of BaTiO3 (BT) to obtain pure phases. Therefore, the ternary system PZN-PT-BT has received wider attention, showing high values of the dielectric permittivity and the pyroelectric coefficient [21,59]. This material shows lower diffuseness of the phase transition and weaker frequency disper‐ sion of the dielectric response than that of the PZN-BT system [16]. It has been shown that it could be interesting try to decrease the transition temperature and to grow the value of the figure of merit in order to obtain better materials for practical applications [17-18,21,60-61].

ordering induces strong charge effects; i.e., the ordered domains have a negative charge with respect to the disordered matrix. Thus, the charge imbalance inhibits the growth of do‐ mains and nanodomains are obtained in a disordered matrix with polar micro-regions.

Relaxor Behaviour in Ferroelectric Ceramics http://dx.doi.org/10.5772/52149 99

**Figure 11.** Hysteresis loops for (Pb0.8Ba0.2)[(Zn1/3Nb2/3)0.7Ti0.3]O3 ferroelectric ceramic system.

means of an electron beam inducing local stresses to align the domains [66].

cy range. This behaviour could be associated to the conductivity losses.

**4. Lead free Aurivillius relaxor ferroelectrics**

Figure 11 shows the hysteresis loops at several temperatures for the studied system. Slim P-E loops and small remanent polarization values are observed. The slim-loop nature suggests that most of the aligned dipole moments switch back to a randomly oriented state upon re‐ moval of the field. It could be interpreted in terms of correlated polar nanodomains embed‐ ded in a paraelectric matrix [44]. For relaxor ferroelectric materials, there is a micro- to macro-domain transition [66]. In the absence of any external field, the domain structure of relaxor ferroelectrics contains randomly oriented micropolar regions. When an electric field is applied, the micro-domains are oriented along the field direction and the macro-domains appear. The micro- to macro-domain transition has been confirmed by 'in-situ' switching by

The temperature dependence of the real (ε') and imaginary (ε'') parts of the dielectric permit‐ tivity, at several frequencies, for the Sr0.5Ba0.5Bi2Nb2O9 sample are showed in the Figure 12. The maximum of ε' decreases with the increase of the frequency; its corresponding temperature (Tm) shifts with the frequency, showing a high frequency dispersion. For the imaginary part (ε'') the maximum values are observed at lower temperatures than those the observed for ε' and the corresponding temperature again shows a significant frequency dispersion. These characteristics are typical of relaxor ferroelectric materials. On the other hand, an abrupt in‐ crease of ε'' is observed in the higher temperature zone, which is more clear in the low frequen‐

The present authors have developed several researches concerning (Pb0.8Ba0.2) [(Zn1/3Nb2/3)0.7Ti0.3]O3 (PZN-PT-BT) system [19,62-63]. A Positive Temperature Coefficient of Resistivity (PTCR) effect has been studied. This effect has found extensive applications as ther‐ mal fuses, thermistors, safety circuits and other overload protection devices [64]. It has been discussed that the PTCR behaviour primarily arise from the Schottky barrier formed at grain boundary regions, which act as effective electron traps of the available electrons from the oxy‐ gen vacancies in the ceramic, increasing the Schottky barrier height of the material. Excellent properties to be used in dielectric and pyroelectric applications have been also reported [63].

Figure 10 shows the temperature dependence of the real part (ε') of the dielectric permittivi‐ ty for the (Pb0.8Ba0.2)[(Zn1/3Nb2/3)0.7Ti0.3]O3 ferroelectric ceramic system at several frequencies. Typical characteristics of relaxor ferroelectrics are observed. There is a strong dispersion of the maximum of ε' and its corresponding temperature shift towards higher temperatures when the frequency increases. On the other hand, the system did not follow a Curie–Weiss– like behaviour above Tm. For relaxor ferroelectrics, some of the dipoles are frozen during the time scale of measurement. The fraction of frozen dipoles in itself is a function of the tem‐ perature, so Curie-Weiss's law is no longer valid.

**Figure 10.** Temperature dependence of the real (ε') part of the dielectric permittivity at several frequencies for (Pb0.8Ba0.2)[(Zn1/3Nb2/3)0.7Ti0.3]O3 ferroelectric ceramic system.

For Pb(B´1/3B"2/3)O3 type pervoskites, it has been reported a particular microstructure where the B-site 1:1 short-range-ordered nanodomains (rich in B' ions) are embedded in a matrix rich in B" ions, which promotes a compositional inhomogeneity [65]. The nonstoichiometric ordering induces strong charge effects; i.e., the ordered domains have a negative charge with respect to the disordered matrix. Thus, the charge imbalance inhibits the growth of do‐ mains and nanodomains are obtained in a disordered matrix with polar micro-regions.

**Figure 11.** Hysteresis loops for (Pb0.8Ba0.2)[(Zn1/3Nb2/3)0.7Ti0.3]O3 ferroelectric ceramic system.

showing high values of the dielectric permittivity and the pyroelectric coefficient [21,59]. This material shows lower diffuseness of the phase transition and weaker frequency disper‐ sion of the dielectric response than that of the PZN-BT system [16]. It has been shown that it could be interesting try to decrease the transition temperature and to grow the value of the figure of merit in order to obtain better materials for practical applications [17-18,21,60-61].

The present authors have developed several researches concerning (Pb0.8Ba0.2) [(Zn1/3Nb2/3)0.7Ti0.3]O3 (PZN-PT-BT) system [19,62-63]. A Positive Temperature Coefficient of Resistivity (PTCR) effect has been studied. This effect has found extensive applications as ther‐ mal fuses, thermistors, safety circuits and other overload protection devices [64]. It has been discussed that the PTCR behaviour primarily arise from the Schottky barrier formed at grain boundary regions, which act as effective electron traps of the available electrons from the oxy‐ gen vacancies in the ceramic, increasing the Schottky barrier height of the material. Excellent properties to be used in dielectric and pyroelectric applications have been also reported [63].

Figure 10 shows the temperature dependence of the real part (ε') of the dielectric permittivi‐ ty for the (Pb0.8Ba0.2)[(Zn1/3Nb2/3)0.7Ti0.3]O3 ferroelectric ceramic system at several frequencies. Typical characteristics of relaxor ferroelectrics are observed. There is a strong dispersion of the maximum of ε' and its corresponding temperature shift towards higher temperatures when the frequency increases. On the other hand, the system did not follow a Curie–Weiss– like behaviour above Tm. For relaxor ferroelectrics, some of the dipoles are frozen during the time scale of measurement. The fraction of frozen dipoles in itself is a function of the tem‐

**Figure 10.** Temperature dependence of the real (ε') part of the dielectric permittivity at several frequencies for

For Pb(B´1/3B"2/3)O3 type pervoskites, it has been reported a particular microstructure where the B-site 1:1 short-range-ordered nanodomains (rich in B' ions) are embedded in a matrix rich in B" ions, which promotes a compositional inhomogeneity [65]. The nonstoichiometric

perature, so Curie-Weiss's law is no longer valid.

98 Advances in Ferroelectrics

(Pb0.8Ba0.2)[(Zn1/3Nb2/3)0.7Ti0.3]O3 ferroelectric ceramic system.

Figure 11 shows the hysteresis loops at several temperatures for the studied system. Slim P-E loops and small remanent polarization values are observed. The slim-loop nature suggests that most of the aligned dipole moments switch back to a randomly oriented state upon re‐ moval of the field. It could be interpreted in terms of correlated polar nanodomains embed‐ ded in a paraelectric matrix [44]. For relaxor ferroelectric materials, there is a micro- to macro-domain transition [66]. In the absence of any external field, the domain structure of relaxor ferroelectrics contains randomly oriented micropolar regions. When an electric field is applied, the micro-domains are oriented along the field direction and the macro-domains appear. The micro- to macro-domain transition has been confirmed by 'in-situ' switching by means of an electron beam inducing local stresses to align the domains [66].

### **4. Lead free Aurivillius relaxor ferroelectrics**

The temperature dependence of the real (ε') and imaginary (ε'') parts of the dielectric permit‐ tivity, at several frequencies, for the Sr0.5Ba0.5Bi2Nb2O9 sample are showed in the Figure 12. The maximum of ε' decreases with the increase of the frequency; its corresponding temperature (Tm) shifts with the frequency, showing a high frequency dispersion. For the imaginary part (ε'') the maximum values are observed at lower temperatures than those the observed for ε' and the corresponding temperature again shows a significant frequency dispersion. These characteristics are typical of relaxor ferroelectric materials. On the other hand, an abrupt in‐ crease of ε'' is observed in the higher temperature zone, which is more clear in the low frequen‐ cy range. This behaviour could be associated to the conductivity losses.

whereε'max is the maximum value for the real part of the dielectric permittivity, and δ and C are constants. For an ideal relaxor ferroelectric δ =2, while for a normal ferroelectric δ =1 and

the obtained values for the PMN and PLZT 8/65/35 systems [67]. Thus, the studied material shows higher frequency dispersion. However, ΔT is lower than that of the BaBi2Nb2O9 sys‐ tem (BBN), which is in agreement with previous reports where ΔT decreases with the de‐

The Figure 13 shows the dependency of the log (1/ε'-1/ε'max) vs. log (T-Tm) at 1 MHz for the study ceramic sample. The solid points represent the experimental data and the line the fit‐ ting, which was carried out by using the equation 8. The value obtained for δ parameter is

1.71, which is in agreement with other reports for the studied system [67].

**Figure 13.** Dependence of the log (1/ε'-1/ε''max) vs. log (T-Tm) at 1 MHz for the Sr0.5Ba0.5Bi2Nb2O9.

MHz. The fitting was carried out by using the Vogel-Fulcher law.

**Figure 14.** Dependence of the log f with Tm for the Sr0.5Ba0.5Bi2Nb2O9 considering a frequency range from 1 kHz to 1

C between 1 kHz and 1 MHz, which is higher than that of

Relaxor Behaviour in Ferroelectric Ceramics http://dx.doi.org/10.5772/52149 101

the system follows the Curie-Weiss law.

crease of the barium concentration [67].

For the studied material, ΔT=50<sup>o</sup>

**Figure 12.** Temperature dependence of the real (ε') and imaginary (ε'') parts of the dielectric permittivity, at several frequencies, for the Sr0.5Ba0.5Bi2Nb2O9.

The origin of the relaxor behaviour for this material can be explained from a positional dis‐ order of cations on *A* or *B* sites of the perovskite blocks that delay the evolution of longrange polar ordering [28]. Previous studies in the SrBi2Nb2O9 system have showed that the incorporation of barium to this system conduces to a relaxor behaviour in the ferroelectricparaelectric phase transition [28,67] and the increases of barium content enhance the degree of the frequency dispersion of the dielectric parameters [67]. This behaviour can be ex‐ plained considering that the barium ions can substitute the strontium ions in the *A* site of the perovskite block but enter in the Bi2O2 2+ layers, conducting to an inhomogeneous distri‐ bution of barium and local charge imbalance in the layered structure. The incorporation of barium ions to bismuth sites takes care to reduce the constrain existent between the perov‐ skite blocks and the layered structure [68].

The relaxor behaviour is usually characterized by using the variation of Tm with frequency (ΔT), degree of the frequency dispersion, and the critical exponent (δ) obtained from the fol‐ lowing law [67]:

$$\frac{1}{\varepsilon'} - \frac{1}{\varepsilon'\_{\text{max}}} = \frac{\left(T - T\_m\right)^\delta}{\mathbb{C}} \tag{8}$$

whereε'max is the maximum value for the real part of the dielectric permittivity, and δ and C are constants. For an ideal relaxor ferroelectric δ =2, while for a normal ferroelectric δ =1 and the system follows the Curie-Weiss law.

For the studied material, ΔT=50<sup>o</sup> C between 1 kHz and 1 MHz, which is higher than that of the obtained values for the PMN and PLZT 8/65/35 systems [67]. Thus, the studied material shows higher frequency dispersion. However, ΔT is lower than that of the BaBi2Nb2O9 sys‐ tem (BBN), which is in agreement with previous reports where ΔT decreases with the de‐ crease of the barium concentration [67].

The Figure 13 shows the dependency of the log (1/ε'-1/ε'max) vs. log (T-Tm) at 1 MHz for the study ceramic sample. The solid points represent the experimental data and the line the fit‐ ting, which was carried out by using the equation 8. The value obtained for δ parameter is 1.71, which is in agreement with other reports for the studied system [67].

**Figure 13.** Dependence of the log (1/ε'-1/ε''max) vs. log (T-Tm) at 1 MHz for the Sr0.5Ba0.5Bi2Nb2O9.

**Figure 12.** Temperature dependence of the real (ε') and imaginary (ε'') parts of the dielectric permittivity, at several

The origin of the relaxor behaviour for this material can be explained from a positional dis‐ order of cations on *A* or *B* sites of the perovskite blocks that delay the evolution of longrange polar ordering [28]. Previous studies in the SrBi2Nb2O9 system have showed that the incorporation of barium to this system conduces to a relaxor behaviour in the ferroelectricparaelectric phase transition [28,67] and the increases of barium content enhance the degree of the frequency dispersion of the dielectric parameters [67]. This behaviour can be ex‐ plained considering that the barium ions can substitute the strontium ions in the *A* site of

bution of barium and local charge imbalance in the layered structure. The incorporation of barium ions to bismuth sites takes care to reduce the constrain existent between the perov‐

The relaxor behaviour is usually characterized by using the variation of Tm with frequency (ΔT), degree of the frequency dispersion, and the critical exponent (δ) obtained from the fol‐

( )

d

*C*

max

e e

1 1 *<sup>m</sup> T T*


2+ layers, conducting to an inhomogeneous distri‐

¢ ¢ (8)

frequencies, for the Sr0.5Ba0.5Bi2Nb2O9.

100 Advances in Ferroelectrics

the perovskite block but enter in the Bi2O2

skite blocks and the layered structure [68].

lowing law [67]:

**Figure 14.** Dependence of the log f with Tm for the Sr0.5Ba0.5Bi2Nb2O9 considering a frequency range from 1 kHz to 1 MHz. The fitting was carried out by using the Vogel-Fulcher law.

Figure 14 shows the dependence of log f with Tm for the studied material. The experimental data was fitted by using the Vogel-Fulcher law (equation 7). The fitting parameters have been included in the figure. The difference between Tm at 1 kHz and TF is about 100 K, a higher value considering the studies carried out in PMN type relaxors [69]. On the other hand, the BBN system has showed a difference higher than 300 K, which is agreement with its strong shift of Tm [28, 69].

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[2] Strukov BA., Levanyuk AP. Ferroelectric Phenomena in Crystals. Berlin: Springer-

[3] Tai CW., Baba-Kishi KZ. Relationship between dielectric properties and structural long-range order in (*x*)Pb(In1/2Nb1/2)O3:(1 − *x*)Pb(Mg1/3Nb2/3)O3 relaxor ceramics. Acta

[4] Huang S., Sun L., Feng Ch., Chen L. Relaxor behaviour of layer structured

[5] Ko JH., Kim DH., Kojima S. Correlation between the dynamics of polar nanoregions and temperature evolution of central peaks in Pb[(Zn1/3Nb2/3)0.91Ti0.09]O3 ferroelectric

[6] Kumar P., Singh S., Juneja JK., Prakash Ch., Raina KK. Dielectric behaviour of La

[7] Craciun F. Strong variation of electrostrictive coupling near an intermediate tempera‐

[8] Ko JH., Kim DH., Kojima S. Low-frequency central peaks in relaxor ferroelectric

[9] Samara GA. The relaxational properties of compositionally disordered ABO3 perov‐

[10] Shvartsman VV., Lupascu DC. Lead-free relaxor ferroelectrics. J. Am. Ceram. Soc.

[11] Dai X., Xu Z., Li JF., Viehland D. Effects of lanthanum modification on rombohedral Pb(Zr1-xTix)O3 ceramics: Part I. Transformation from normal to relaxor ferroelectric

[12] Dai X., Xu Z., Li JF., Viehland D. Effects of lanthanum modification on rombohedral Pb(Zr1-xTix)O3 ceramics: Part II. Relaxor behaviour versus enhanced antiferroelectric

[13] Peláiz-Barranco A., Mendoza ME., Calderón-Piñar F., García-Zaldívar O., López-No‐ da R., de los Santos-Guerra J., Eiras JA., Features on phase transitions in lanthanummodified lead zirconate titanate ferroelectric ceramics. Sol. Sta. Comm.

[14] García-Zaldívar O., Peláiz-Barranco A., Calderón-Piñar F., Fundora-Cruz A., Guerra JDS., Hall DA., Mendoza ME.. Modelling the dielectric response of lanthanum modi‐

### **5. Conclusions**

The relaxor behaviour has been discussed in ferroelectric perovskites-related structures. The influence of the coexistence of ferroelectric and antiferroelectric phases on the relaxor behav‐ iour of lanthanum modified lead zirconate titanate ceramics has been analyzed. A relaxation model to evaluate the dynamical behaviour of the polar nanoregions in a relaxor ferroelec‐ tric PLZT system has been presented and the results have been discussed considering the correlation between the polar nanoregions and the freezing temperature. On the other hand, the influence of *A* or *B* vacancies on the relaxor behaviour for PLZT materials have been studied considering the decoupling effects of these defects in the Pb-O-Ti/Zr bounding and in terms of the dynamic of PNRs. Other typical relaxor ferroelectrics were analyzed too. For the PZN-PT-BT system, the relaxor behaviour was discussed considering the presence of lo‐ cal compositional fluctuation on a macroscopic scale. The Sr0.50Ba0.50Bi2Nb2O9 ferroelectric ce‐ ramic, which is a typical relaxor from the Aurivillius family, was analyzed considering a positional disorder of cations on A or B sites of the perovskite blocks, which delay the evolu‐ tion of long-rage polar ordering.

### **Acknowledgement**

The authors wish to thank to TWAS for financial support (RG/PHYS/LA No. 99-050, 02-225 and 05-043) and to ICTP for financial support of Latin-American Network of Ferroelectric Materials (NET-43). Thanks to FAPESP, CNPq and FAPEMIG Brazilian agencies. Thanks to CONACyT, Mexico. Dr. A. Peláiz-Barranco wishes to thank the Royal Society. M.Sc. O. García-Zaldívar wishes to thanks to Red de Macrouniversidades / 2007 and to CLAF/ICTP fellowships NS 32/08.

### **Author details**

A. Peláiz-Barranco, F. Calderón-Piñar, O. García-Zaldívar and Y. González-Abreu

\*Address all correspondence to: pelaiz@fisica.uh.cu

Physics Faculty−Institute of Science and Technology of Materials, Havana University, San Lázaro y L, Vedado, La Habana, Cuba

### **References**

Figure 14 shows the dependence of log f with Tm for the studied material. The experimental data was fitted by using the Vogel-Fulcher law (equation 7). The fitting parameters have been included in the figure. The difference between Tm at 1 kHz and TF is about 100 K, a higher value considering the studies carried out in PMN type relaxors [69]. On the other hand, the BBN system has showed a difference higher than 300 K, which is agreement with

The relaxor behaviour has been discussed in ferroelectric perovskites-related structures. The influence of the coexistence of ferroelectric and antiferroelectric phases on the relaxor behav‐ iour of lanthanum modified lead zirconate titanate ceramics has been analyzed. A relaxation model to evaluate the dynamical behaviour of the polar nanoregions in a relaxor ferroelec‐ tric PLZT system has been presented and the results have been discussed considering the correlation between the polar nanoregions and the freezing temperature. On the other hand, the influence of *A* or *B* vacancies on the relaxor behaviour for PLZT materials have been studied considering the decoupling effects of these defects in the Pb-O-Ti/Zr bounding and in terms of the dynamic of PNRs. Other typical relaxor ferroelectrics were analyzed too. For the PZN-PT-BT system, the relaxor behaviour was discussed considering the presence of lo‐ cal compositional fluctuation on a macroscopic scale. The Sr0.50Ba0.50Bi2Nb2O9 ferroelectric ce‐ ramic, which is a typical relaxor from the Aurivillius family, was analyzed considering a positional disorder of cations on A or B sites of the perovskite blocks, which delay the evolu‐

The authors wish to thank to TWAS for financial support (RG/PHYS/LA No. 99-050, 02-225 and 05-043) and to ICTP for financial support of Latin-American Network of Ferroelectric Materials (NET-43). Thanks to FAPESP, CNPq and FAPEMIG Brazilian agencies. Thanks to CONACyT, Mexico. Dr. A. Peláiz-Barranco wishes to thank the Royal Society. M.Sc. O. García-Zaldívar wishes to thanks to Red de Macrouniversidades / 2007 and to CLAF/ICTP fellowships NS 32/08.

A. Peláiz-Barranco, F. Calderón-Piñar, O. García-Zaldívar and Y. González-Abreu

Physics Faculty−Institute of Science and Technology of Materials, Havana University, San

\*Address all correspondence to: pelaiz@fisica.uh.cu

Lázaro y L, Vedado, La Habana, Cuba

its strong shift of Tm [28, 69].

tion of long-rage polar ordering.

**Acknowledgement**

**Author details**

**5. Conclusions**

102 Advances in Ferroelectrics


fied lead zirconate titanate ferroelectric ceramics - an approach to the phase transi‐ tions in relaxor ferroelectrics. J. Phys.: Cond. Matt. 2008;20; 445230.

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[31] Viehland D., Jang SJ., Cross LE., Wutting M. Freezing of the polarization fluctuations

[32] Quian H., Bursill LA., Phenomenological theory of the dielectric response of lead magnesium niobate and lead scandium tantalate. Int. J. Mod Phys. B 1996;10;

[33] Ishchuk VM., Baumer VN., Sobolev VL., The influence of the coexistence of ferroelec‐ tric and antiferroelectric states on the lead lanthanum zirconate titanate crystal struc‐

[34] Peláiz-Barranco A., Hall DA. Influence of composition and pressure on the electric field-induced antiferroelectric to ferroelectric phase transformations in lanthanum modified lead zirconate titanate ceramics. IEEE Trans. Ultras. Ferroel. Freq. Cont.

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**Chapter 6**

**Provisional chapter**

**Electronic Band Structures and Phase Transitions of**

**Electronic Band Structures and Phase Transitions of**

Perovskite ferroelectric (FE) materials have attracted considerable attention for a wide range of applications, such as dynamic random access memories (DRAM), microwave tunable phase shifters and second harmonic generators (SHGs). [1–3] Moreover, materials that have coupled electric, magnetic, and structural order parameters that result in simultaneous ferroelectricity, ferromagnetism, and ferroelasticity are known as multiferroics. [4–6] These multiferroics materials have attracted a lot of attention in recent years because they can potentially offer a whole range of new applications, including nonvolatile ferroelectric memories, novel multiple state memories, and devices based on magnetoeletric effects. Although there are some reports on the electrical and magnetic properties of perovskite-type ferroelectric and multiferroics materials, optical properties and electronic transitions have not been well investigated up to now. On the other hand, phase transition is one of the important characteristics for the ferroelectric/multiferroics system. As we know, the phase transition is strongly related to the structural variation, which certainly can result in the electronic band modifications. Therefore, one can study the phase transition of the above material systems

Among these materials, barium strontium titanate (BST) has been considered to be one of the most promising candidates for devices due to its excellent dielectric properties of high dielectric functions, low leakage current and an adjustable Curie temperature *T*<sup>c</sup> through variation of the composition between barium titanate (BT) and strontium titanate (ST). However, the limited figure of merit at high frequency microwave region restricts the BST practical applications. In order to improve the physical properties of the BST materials, introducing small compositions of dopants has been used for several decades. Many experimental and theoretical studies have been performed on the dielectric properties of BST in the ferroelectric state by adding dopants such as Magnesium [7, 8], Aluminum [9], Manganese (Mn) [10–12], Samarium [13], and different rare earth [14], In particular, the Mn

> ©2012 Hu et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Hu et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

by the corresponding spectral response behavior at different temperatures.

**Ferroelectric and Multiferroic Oxides**

**Ferroelectric and Multiferroic Oxides**

Zhigao Hu, Junhao Chu, Yawei Li, Kai Jiang and

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Ziqiang Zhu

**1. Introduction**

Zhigao Hu, Junhao Chu,

Yawei Li, Kai Jiang and Ziqiang Zhu

http://dx.doi.org/10.5772/52189

**Provisional chapter**

### **Electronic Band Structures and Phase Transitions of Ferroelectric and Multiferroic Oxides Electronic Band Structures and Phase Transitions of Ferroelectric and Multiferroic Oxides**

Zhigao Hu, Junhao Chu, Yawei Li, Kai Jiang and Ziqiang Zhu Yawei Li, Kai Jiang and Ziqiang Zhu Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52189

Zhigao Hu, Junhao Chu,

### **1. Introduction**

Perovskite ferroelectric (FE) materials have attracted considerable attention for a wide range of applications, such as dynamic random access memories (DRAM), microwave tunable phase shifters and second harmonic generators (SHGs). [1–3] Moreover, materials that have coupled electric, magnetic, and structural order parameters that result in simultaneous ferroelectricity, ferromagnetism, and ferroelasticity are known as multiferroics. [4–6] These multiferroics materials have attracted a lot of attention in recent years because they can potentially offer a whole range of new applications, including nonvolatile ferroelectric memories, novel multiple state memories, and devices based on magnetoeletric effects. Although there are some reports on the electrical and magnetic properties of perovskite-type ferroelectric and multiferroics materials, optical properties and electronic transitions have not been well investigated up to now. On the other hand, phase transition is one of the important characteristics for the ferroelectric/multiferroics system. As we know, the phase transition is strongly related to the structural variation, which certainly can result in the electronic band modifications. Therefore, one can study the phase transition of the above material systems by the corresponding spectral response behavior at different temperatures.

Among these materials, barium strontium titanate (BST) has been considered to be one of the most promising candidates for devices due to its excellent dielectric properties of high dielectric functions, low leakage current and an adjustable Curie temperature *T*<sup>c</sup> through variation of the composition between barium titanate (BT) and strontium titanate (ST). However, the limited figure of merit at high frequency microwave region restricts the BST practical applications. In order to improve the physical properties of the BST materials, introducing small compositions of dopants has been used for several decades. Many experimental and theoretical studies have been performed on the dielectric properties of BST in the ferroelectric state by adding dopants such as Magnesium [7, 8], Aluminum [9], Manganese (Mn) [10–12], Samarium [13], and different rare earth [14], In particular, the Mn

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Hu et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Hu et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative

doping BST shows some advantages in reducing the dielectric loss, enhancing the resistivity, and increasing dielectric tunability. It can significantly improve the dielectric properties, which makes it a potential candidate for microwave elements. For example, the Mn doping can cause the variation of the oxygen vacancy, which is the crucial role in modifying the dielectric loss mechanism. On the other hand, the doping of Mn can also reduce the dielectric constant peak and broaden the dielectric phase transition temperature range, which results in a smaller temperature coefficient of capacitance in BSMT materials. [10] Therefore, it is important to further investigate the physical properties of BSMT materials in order to develop the potential applications.

In spite of the promising properties, there are no systematical reports focused on the optical properties of BFO films. In order to make BFO useful in actual electrical and optoelectronic devices, the physical properties, especially for electronic band structure and optical response

Electronic Band Structures and Phase Transitions of Ferroelectric and Multiferroic Oxides

http://dx.doi.org/10.5772/52189

111

The objectives of the chapter will tentatively answer the interesting questions: (1) Is there an effective method to directly analyze electronic structure of FE materials by optical spectroscopy? (2) What kind of temperature dependence have FE oxides from band-to-band transitions? (3) Can spectral response at high-temperature be used to judge phase transition? Correspondingly, this chapter is arranged in the following way. In Sec. 2, detailed growths of Ba0.4Sr0.6−*x*Mn*x*TiO3 (BSMT), SrBi2−*x*Nd*x*Nb2O9 (SBNN) ceramics and BFO films are described; In Sec. 3, solid state spectroscopic techniques are introduced; In Sec. 4, electronic band structures of BSMT ceramics are presented; In Sec. 5, phase transitions of SBNN ceramics are derived; In Sec. 6, temperature effects on electronic transitions of BFO films

The ceramics based on Ba0.4Sr0.6−*x*Mn*x*TiO3 (with *x* = 1, 2, 5 and 10%) specimens were prepared by the conventional solid-state reaction sintering. High purity BaCO3 (99.8%), SrCO3 (99.0%), TiO2 (99.9%), and MnCO3 were used as the starting materials. Weighted powers were mixed by ball milling with zirconia media in the ethanol as a solvent for 24 h and then dried at 110 ◦C for 12 h. After drying, the powders were calcined at 1200 ◦C for 4 h, and then remilled for 24 h to reduce the particle size for sintering. The calcined powders were added with 8 wt.% polyvinyl alcohol (PVA) as a binder. The granulated powders were pressed into discs in diameter of 10 mm and thickness of 1.0 mm. The green pellets were kept at 550 ◦C for 6 h to remove the solvent and binder, followed by sintering at 1400 ◦C for 4 h. More details of the preparation process can be found in Ref. [39]. On the other hand, the SBNN (*x*=0, 0.05, 0.1, and 0.2) ceramics were prepared by a similar method, and SrCO3, Bi2O3, Nb2O5, and Nd2O3 were used as the starting materials. Details of the fabrication

The BiFeO3 films were deposited on *c*-sapphire substrates by the PLD technique. The BiFeO3 targets with a diameter of 3 cm were prepared through a conventional solid state reaction method using reagent-grade Bi2O3 (99.9%) and Fe2O3 (99.9%) powders. Weighed powders were mixed for 24 h by ball milling with zirconia media in ethanol and then dried at 100 ◦<sup>C</sup> for 12 h. The dried powders were calcined at about 680 ◦C in air for 6 h to form the desired phase, and followed by sintering at about 830 ◦C for 2 h. Before the deposition of the BFO films, *c*-sapphire substrates need to be cleaned in pure ethanol with an ultrasonic bath to remove physisorbed organic molecules from the surfaces, followed by rinsing several times with de-ionized water. Then the substrates were dried in a pure nitrogen stream before the film deposition. A pulsed Nd:YAG (yttrium aluminum garnet) laser (532 nm wavelength, 5 ns duration) operated with an energy of 60 mJ/pulse and repetition rate of 10 Hz was used as the ablation source. The films were deposited immediately after the target was

have been discussed; In Sec. 7, the main results and remarks are summarized.

behavior, need to be further clarified.

**2.1. Fabrications of FE ceramics**

**2.2. Depositions of BFO films**

process for the ceramics can be found elsewhere. [17, 40]

**2. Experimental**

Meanwhile, many research groups have focused on the doping effect on the fabrications and dielectric properties for strontium barium niobate SrBi2Nb2O9 (SBN) materials. [15–20] The substitution of Ca ions in the Sr site for the SBN ceramic induced the *T*<sup>C</sup> increasing, which is useful for the application in high-temperature resonators. [15, 18] However, the substitution of some rare earth ions such as La3<sup>+</sup> or Pr3<sup>+</sup> for Bi3<sup>+</sup> in the Bi2O2 layers can result in a shift for the *T*<sup>C</sup> to lower temperature. [16, 20, 21] It is found that the behavior of the Nd-doped SBN ceramic tends to change from a normal ferroelectrics to a relaxor type ferroelectrics owing to the introduction of Nd ions in the Bi2O2 layers. [16, 17, 20, 22] Up to now, a detailed understanding of the lattice dynamic properties and the phase transition behavior of Nd-doped SBN ceramics are still lacking. Raman spectroscopy is a sensitive technique for investigating the structure modifications and lattice vibration modes, which can give the information on the changes of lattice vibrations and the occupying positions of doping ions. Thus, it is a powerful tool for the detection of phase transition in the doping-related ferroelectric materials. [22, 23]

The optical properties such as the dielectric functions provide an important insight on dielectric and ferroelectric behaviors of the material and play an important role in design, optimization, and evaluation of optoelectronic devices. [24–27] In addition, the doping of Mn or Nd can induce more defects in the lattice structure, which can affect its electronic band structures and optoelectronic properties. Hence, the doping composition dependence of optical properties for BST and SBN ceramics is technically important for practical optoelectronic device development. Compared to film structure, the optical properties of bulk material (single crystals and dense ceramics) are not affected by interface layer, stress from clamping by the substrate, non-stoichiometry and lattice mismatch between film and substrate. Hence, it is desirable to carry out a delicate investigation regarding the optical properties of the BSMT and SBNN ceramics. Note that spectroscopic ellipsometry (SE), Raman scattering, and transmittance spectra are potentially valuable techniques for the studies of ferroelectric materials due to their high sensitivity of local structure and symmetry. Compared with the other techniques, they can provide dielectric functions of the materials. SE and transmittance spectra can provide optical band gap and optical conductivity, whereas Raman spectra can provide Raman-active modes of the materials. [28–31]

On the other hand, bismuth ferrite (BiFeO3, BFO) is known to be the only perovskite material that exhibits multiferroic at room temperature (RT). At RT, it is a rhombohedrally distorted ferroelectric perovskite with the space group *R3c* and a Curie temperature (*TC*) of about 1100 K. [32–36] Since the physical properties of BFO films are related to their domain structure and phase states, which is sensitive to the applied stress, composition, and fabrication condition for BFO materials. PLD technique has the ability to exceed the solubility of magnetic impurity and to permit high quality film grown at low substrate temperature. Recent studies of photoconductivity, [35] photovoltaic effect, [37] and low open circuit voltage in a working solar device, [38] illustrate the potential of polar oxides as the active photovoltaic material. In spite of the promising properties, there are no systematical reports focused on the optical properties of BFO films. In order to make BFO useful in actual electrical and optoelectronic devices, the physical properties, especially for electronic band structure and optical response behavior, need to be further clarified.

The objectives of the chapter will tentatively answer the interesting questions: (1) Is there an effective method to directly analyze electronic structure of FE materials by optical spectroscopy? (2) What kind of temperature dependence have FE oxides from band-to-band transitions? (3) Can spectral response at high-temperature be used to judge phase transition? Correspondingly, this chapter is arranged in the following way. In Sec. 2, detailed growths of Ba0.4Sr0.6−*x*Mn*x*TiO3 (BSMT), SrBi2−*x*Nd*x*Nb2O9 (SBNN) ceramics and BFO films are described; In Sec. 3, solid state spectroscopic techniques are introduced; In Sec. 4, electronic band structures of BSMT ceramics are presented; In Sec. 5, phase transitions of SBNN ceramics are derived; In Sec. 6, temperature effects on electronic transitions of BFO films have been discussed; In Sec. 7, the main results and remarks are summarized.

### **2. Experimental**

2 Advances in Ferroelectrics

the potential applications.

ferroelectric materials. [22, 23]

doping BST shows some advantages in reducing the dielectric loss, enhancing the resistivity, and increasing dielectric tunability. It can significantly improve the dielectric properties, which makes it a potential candidate for microwave elements. For example, the Mn doping can cause the variation of the oxygen vacancy, which is the crucial role in modifying the dielectric loss mechanism. On the other hand, the doping of Mn can also reduce the dielectric constant peak and broaden the dielectric phase transition temperature range, which results in a smaller temperature coefficient of capacitance in BSMT materials. [10] Therefore, it is important to further investigate the physical properties of BSMT materials in order to develop

Meanwhile, many research groups have focused on the doping effect on the fabrications and dielectric properties for strontium barium niobate SrBi2Nb2O9 (SBN) materials. [15–20] The substitution of Ca ions in the Sr site for the SBN ceramic induced the *T*<sup>C</sup> increasing, which is useful for the application in high-temperature resonators. [15, 18] However, the substitution of some rare earth ions such as La3<sup>+</sup> or Pr3<sup>+</sup> for Bi3<sup>+</sup> in the Bi2O2 layers can result in a shift for the *T*<sup>C</sup> to lower temperature. [16, 20, 21] It is found that the behavior of the Nd-doped SBN ceramic tends to change from a normal ferroelectrics to a relaxor type ferroelectrics owing to the introduction of Nd ions in the Bi2O2 layers. [16, 17, 20, 22] Up to now, a detailed understanding of the lattice dynamic properties and the phase transition behavior of Nd-doped SBN ceramics are still lacking. Raman spectroscopy is a sensitive technique for investigating the structure modifications and lattice vibration modes, which can give the information on the changes of lattice vibrations and the occupying positions of doping ions. Thus, it is a powerful tool for the detection of phase transition in the doping-related

The optical properties such as the dielectric functions provide an important insight on dielectric and ferroelectric behaviors of the material and play an important role in design, optimization, and evaluation of optoelectronic devices. [24–27] In addition, the doping of Mn or Nd can induce more defects in the lattice structure, which can affect its electronic band structures and optoelectronic properties. Hence, the doping composition dependence of optical properties for BST and SBN ceramics is technically important for practical optoelectronic device development. Compared to film structure, the optical properties of bulk material (single crystals and dense ceramics) are not affected by interface layer, stress from clamping by the substrate, non-stoichiometry and lattice mismatch between film and substrate. Hence, it is desirable to carry out a delicate investigation regarding the optical properties of the BSMT and SBNN ceramics. Note that spectroscopic ellipsometry (SE), Raman scattering, and transmittance spectra are potentially valuable techniques for the studies of ferroelectric materials due to their high sensitivity of local structure and symmetry. Compared with the other techniques, they can provide dielectric functions of the materials. SE and transmittance spectra can provide optical band gap and optical conductivity, whereas

On the other hand, bismuth ferrite (BiFeO3, BFO) is known to be the only perovskite material that exhibits multiferroic at room temperature (RT). At RT, it is a rhombohedrally distorted ferroelectric perovskite with the space group *R3c* and a Curie temperature (*TC*) of about 1100 K. [32–36] Since the physical properties of BFO films are related to their domain structure and phase states, which is sensitive to the applied stress, composition, and fabrication condition for BFO materials. PLD technique has the ability to exceed the solubility of magnetic impurity and to permit high quality film grown at low substrate temperature. Recent studies of photoconductivity, [35] photovoltaic effect, [37] and low open circuit voltage in a working solar device, [38] illustrate the potential of polar oxides as the active photovoltaic material.

Raman spectra can provide Raman-active modes of the materials. [28–31]

### **2.1. Fabrications of FE ceramics**

The ceramics based on Ba0.4Sr0.6−*x*Mn*x*TiO3 (with *x* = 1, 2, 5 and 10%) specimens were prepared by the conventional solid-state reaction sintering. High purity BaCO3 (99.8%), SrCO3 (99.0%), TiO2 (99.9%), and MnCO3 were used as the starting materials. Weighted powers were mixed by ball milling with zirconia media in the ethanol as a solvent for 24 h and then dried at 110 ◦C for 12 h. After drying, the powders were calcined at 1200 ◦C for 4 h, and then remilled for 24 h to reduce the particle size for sintering. The calcined powders were added with 8 wt.% polyvinyl alcohol (PVA) as a binder. The granulated powders were pressed into discs in diameter of 10 mm and thickness of 1.0 mm. The green pellets were kept at 550 ◦C for 6 h to remove the solvent and binder, followed by sintering at 1400 ◦C for 4 h. More details of the preparation process can be found in Ref. [39]. On the other hand, the SBNN (*x*=0, 0.05, 0.1, and 0.2) ceramics were prepared by a similar method, and SrCO3, Bi2O3, Nb2O5, and Nd2O3 were used as the starting materials. Details of the fabrication process for the ceramics can be found elsewhere. [17, 40]

### **2.2. Depositions of BFO films**

The BiFeO3 films were deposited on *c*-sapphire substrates by the PLD technique. The BiFeO3 targets with a diameter of 3 cm were prepared through a conventional solid state reaction method using reagent-grade Bi2O3 (99.9%) and Fe2O3 (99.9%) powders. Weighed powders were mixed for 24 h by ball milling with zirconia media in ethanol and then dried at 100 ◦<sup>C</sup> for 12 h. The dried powders were calcined at about 680 ◦C in air for 6 h to form the desired phase, and followed by sintering at about 830 ◦C for 2 h. Before the deposition of the BFO films, *c*-sapphire substrates need to be cleaned in pure ethanol with an ultrasonic bath to remove physisorbed organic molecules from the surfaces, followed by rinsing several times with de-ionized water. Then the substrates were dried in a pure nitrogen stream before the film deposition. A pulsed Nd:YAG (yttrium aluminum garnet) laser (532 nm wavelength, 5 ns duration) operated with an energy of 60 mJ/pulse and repetition rate of 10 Hz was used as the ablation source. The films were deposited immediately after the target was

**Figure 1.** The XRD patterns of the BSMT ceramics with the Mn composition of 1, 2, 5 and 10%, respectively.

preablated in order to remove any surface contaminants. The distance between the target and the substrate was kept at 3 cm. The deposition time was set to about 30 min. Finally, the films were annealed at 600 ◦C in air atmosphere by a rapid thermal annealing process. A detailed preparation of the films can be found in Ref. [6].

**Figure 2.** Experimental ellipsometric data (a) Ψ and (b) ∆ for the BSMT ceramics from near-infrared to ultraviolet photon energy

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**Figure 3.** The (a) real and (b) imaginary parts of the NIR-UV dielectric functions for the BSMT ceramics with different Mn

ascribed to the fact that Mn occupies the A site together with Ba2<sup>+</sup> and Sr2+. In order to reduce distortion of the crystal lattice, the Mn mainly substitutes Sr2<sup>+</sup> because the differences in ionic radius between Mn ion and Ba ion is larger than that between Mn and Sr. On the other hand, when the Mn composition is larger than 5%, the excess Mn can also substitute the Ti site, which results in the increase of the lattice constant and the smaller shift of the peak position. It can be concluded that the Mn ions substitute Sr sites of the BST lattice at

The experimental ellipsometric spectra of <sup>Ψ</sup> and <sup>∆</sup> recorded at an incident angle of 67◦ for the BSMT ceramics are depicted in Figs. 2(a) and (b), respectively. The observed changes in the Ψ and ∆ data for different Mn composition may be attributed to the lattice distortion and variation in atomic coordinate. Because the sample is bulk material with a thickness of several millimeters, the dielectric functions of the BSMT ceramics can be directly calculated

composition. (Figure reproduced with permission from [39]. Copyright 2012, Springer.)

first, then occupy Ti sites when the Mn composition is beyond 5%.

region at the incident angle of 67◦. (Figure reproduced with permission from [39]. Copyright 2012, Springer.)

### **3. Optical spectroscopy**

The ellipsometric measurements were carried out in the photon energy range of 0.7-4.2 eV (300-1700 nm) with a spectral resolution of 5 nm by near-infrared-ultraviolet (NIR-UV) SE (SC630UVN by Shanghai Sanco Instrument, Co., Ltd.). The measurements were performed under the incident angle of 67◦ for all the ceramics corresponding to the experimental optimization near the Brewster angle of the BSMT. Raman scattering experiments were carried out using a Jobin-Yvon LabRAM HR 800 UV micro-Raman spectrometer, excited by a 632.8 nm He-Ne laser with a spectral resolution of 0.5 cm<sup>−</sup>1. Temperature dependent measurements from 80 to 873 K were performed using the Linkam THMSE 600 heating stage, and the set-point stability is of better than 0.5 K. The normal-incident transmittance spectra were recorded using a double beam ultraviolet-infrared spectrophotometer (PerkinElmer Lambda 950) at the photon energy from 0.5 to 6.5 eV (190-2650 nm) with a spectral resolution of 2 nm. The samples at 5.3-300 K were mounted into an optical cryostat (Janis SHI-4-1) for variable temperature experiments. [41]

### **4. Electronic band structures of BSMT ceramics**

The XRD patterns of the BSMT ceramics with different Mn composition are shown in Fig. 1 and no secondary phase appears within the detection limit of the XRD. Besides the strongest (110) peak, some weaker peaks (100), (111), (200), (210), (211), (220) can be also observed, which indicate that the ceramics are polycrystalline with single perovskite phase. The diffraction patterns are fitted by the Gaussian lineshape analysis to extract the peak positions and full width at half maximum (FWHM). The lattice constant *a* of the BSMT ceramics, which can be estimated from the (110) diffraction peak, is calculated to be about 3.954 Å. [39] The ionic radius of Mn2<sup>+</sup> (1.27 Å) is smaller than that of Sr2<sup>+</sup> (1.44 Å) and Ba2<sup>+</sup> (1.61 Å), and is larger than that of Ti4<sup>+</sup> (0.61 Å), which can be attributed to the change of the lattice constant. When the Mn composition is below 5%, the (110) diffraction peak positions shift from smaller angles to larger angles and the lattice constant slightly decreases, which can be

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**Figure 1.** The XRD patterns of the BSMT ceramics with the Mn composition of 1, 2, 5 and 10%, respectively.

detailed preparation of the films can be found in Ref. [6].

**4. Electronic band structures of BSMT ceramics**

**3. Optical spectroscopy**

variable temperature experiments. [41]

preablated in order to remove any surface contaminants. The distance between the target and the substrate was kept at 3 cm. The deposition time was set to about 30 min. Finally, the films were annealed at 600 ◦C in air atmosphere by a rapid thermal annealing process. A

The ellipsometric measurements were carried out in the photon energy range of 0.7-4.2 eV (300-1700 nm) with a spectral resolution of 5 nm by near-infrared-ultraviolet (NIR-UV) SE (SC630UVN by Shanghai Sanco Instrument, Co., Ltd.). The measurements were performed under the incident angle of 67◦ for all the ceramics corresponding to the experimental optimization near the Brewster angle of the BSMT. Raman scattering experiments were carried out using a Jobin-Yvon LabRAM HR 800 UV micro-Raman spectrometer, excited by a 632.8 nm He-Ne laser with a spectral resolution of 0.5 cm<sup>−</sup>1. Temperature dependent measurements from 80 to 873 K were performed using the Linkam THMSE 600 heating stage, and the set-point stability is of better than 0.5 K. The normal-incident transmittance spectra were recorded using a double beam ultraviolet-infrared spectrophotometer (PerkinElmer Lambda 950) at the photon energy from 0.5 to 6.5 eV (190-2650 nm) with a spectral resolution of 2 nm. The samples at 5.3-300 K were mounted into an optical cryostat (Janis SHI-4-1) for

The XRD patterns of the BSMT ceramics with different Mn composition are shown in Fig. 1 and no secondary phase appears within the detection limit of the XRD. Besides the strongest (110) peak, some weaker peaks (100), (111), (200), (210), (211), (220) can be also observed, which indicate that the ceramics are polycrystalline with single perovskite phase. The diffraction patterns are fitted by the Gaussian lineshape analysis to extract the peak positions and full width at half maximum (FWHM). The lattice constant *a* of the BSMT ceramics, which can be estimated from the (110) diffraction peak, is calculated to be about 3.954 Å. [39] The ionic radius of Mn2<sup>+</sup> (1.27 Å) is smaller than that of Sr2<sup>+</sup> (1.44 Å) and Ba2<sup>+</sup> (1.61 Å), and is larger than that of Ti4<sup>+</sup> (0.61 Å), which can be attributed to the change of the lattice constant. When the Mn composition is below 5%, the (110) diffraction peak positions shift from smaller angles to larger angles and the lattice constant slightly decreases, which can be

**Figure 2.** Experimental ellipsometric data (a) Ψ and (b) ∆ for the BSMT ceramics from near-infrared to ultraviolet photon energy region at the incident angle of 67◦. (Figure reproduced with permission from [39]. Copyright 2012, Springer.)

**Figure 3.** The (a) real and (b) imaginary parts of the NIR-UV dielectric functions for the BSMT ceramics with different Mn composition. (Figure reproduced with permission from [39]. Copyright 2012, Springer.)

ascribed to the fact that Mn occupies the A site together with Ba2<sup>+</sup> and Sr2+. In order to reduce distortion of the crystal lattice, the Mn mainly substitutes Sr2<sup>+</sup> because the differences in ionic radius between Mn ion and Ba ion is larger than that between Mn and Sr. On the other hand, when the Mn composition is larger than 5%, the excess Mn can also substitute the Ti site, which results in the increase of the lattice constant and the smaller shift of the peak position. It can be concluded that the Mn ions substitute Sr sites of the BST lattice at first, then occupy Ti sites when the Mn composition is beyond 5%.

The experimental ellipsometric spectra of <sup>Ψ</sup> and <sup>∆</sup> recorded at an incident angle of 67◦ for the BSMT ceramics are depicted in Figs. 2(a) and (b), respectively. The observed changes in the Ψ and ∆ data for different Mn composition may be attributed to the lattice distortion and variation in atomic coordinate. Because the sample is bulk material with a thickness of several millimeters, the dielectric functions of the BSMT ceramics can be directly calculated

**Figure 4.** Absorption coefficient *vs*. incident photon energy near the optical band gap of the BSMT ceramics. The insert is the optical band gap *Eg* with the different Mn composition. (Figure reproduced with permission from [39]. Copyright 2012, Springer.)

**Figure 5.** Raman scattering spectra of the SBNN ceramics with different Nd composition (*x*) recorded at RT. The dashed lines clearly indicate some Raman-active phonon modes. The inset shows the peak frequency variation of the *A*1*<sup>g</sup>* [Nb] phonon mode at about <sup>207</sup> cm−<sup>1</sup> and the *<sup>A</sup>*1*<sup>g</sup>* [O] phonon mode at about <sup>836</sup> cm−<sup>1</sup> as a function of Nd composition. (Figure reproduced with

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generated in the system, the lowest lying states in the CB are filled and the vertical distance needed for the optical transition increases. [44] Hence, it can be concluded that, when the Mn composition is 10%, the sharply increase in the band gap may be attributed to the BM shift caused by electrons generated by oxygen vacancies. Besides, the grain size for the ceramic decreases induced by the heavy doping with the Mn composition of 10%, which may also result in a sharply increase of the optical band gap. It can be concluded that the difference of the optical band gap could be due to the dopant composition, the oxygen vacancies, and

The general formula of bismuth layer structure ferroelectrics (BLSFs) is given as (Bi2O2)2<sup>+</sup> (*Am*<sup>−</sup>1*BmO*3*m*+1)2−, where *<sup>A</sup>* and *<sup>B</sup>* are the two types of cations that enter the perovskite unit, and *m* is the the number of perovskite unit cell between bismuth oxide layers. [20] SBN, which is known to be *m*=2 member of BLSFs family, has been regarded as a promising ferroelectric material due to low dielectric constants and excellent fatigue resistance. [40] Fig. 5(a) shows the Raman spectra of the SBNN ceramics with different Nd compositions at RT in the spectral range of 50 − 950 cm<sup>−</sup>1. The Raman selection rules allow 18 phonon modes (4*A*1*<sup>g</sup>* + 2*B*1*<sup>g</sup>* + 6*B*2*<sup>g</sup>* + 6*B*3*g*) for SBN ceramics at RT. [45] However, less than 10 phonon modes are observed because of the possible overlap of the same symmetry vibration or the weak feature of some vibration bands. [46, 47] According to the assignment of SrBi2Nb2O9 single crystal, [45] the Raman phonon modes at about 61, 207 and 835 cm−<sup>1</sup> can be assigned to the *<sup>A</sup>*1*<sup>g</sup>* phonon mode, the vibrations at about 179 and 579 cm−<sup>1</sup> can be assigned to the *Eg* phonon mode. However, the assignment of other phonon modes are still not clear now. The internal vibrations of NbO6 octahedra occur in the high-frequency mode region above 200 cm−<sup>1</sup> because the intragroup binding energy within the NbO6 octahedra is much larger than the intergroup or crystal binding energy. [15] The composition dependence of the frequencies for two typical phonon modes is illustrated in Fig. 5(b). Note that the *A*1*g*[Nb] phonon mode at 207 cm<sup>−</sup>1, which arises from the distortion of NbO6 octahedra, generally decreases with the Nd composition, whereas the *<sup>A</sup>*1*g*[O] phonon mode at 835 cm−<sup>1</sup> mode corresponding to the symmetric Nb−O stretching vibration, increases with the introduction of Nd ions.

permission from [46]. Copyright 2012, Wiley.)

the crystallinity of the BSMT ceramics.

**5. Phase transitions of SBNN ceramics**

according to the ellipsometric spectra Figs. 3(a) and (b) show the real ( ˙ *ε*1) and imaginary (*ε*2) parts of the dielectric functions in the photon energy range of 0.7-4.2 eV, respectively. The evolution of *ε* with the photon energy is a typical optical response behavior of ferroelectric and/or semiconductors. The optical band gap (*Eg*) of the BSMT ceramics is one of the important optical behaviors, which is calculated by considering a direct transition from the VB to the CB when the photon energy falls on the materials. In the BSMT system, the VB is mainly composed of the O 2*p* orbital and the CB is mainly composed of the Ti 3*d* orbital. It should be noted that because of the splitting of the Ti 3*d* conduction bands into *t*2*<sup>g</sup>* and *eg* subbands, the lowest CB arises from the threefold degenerate Ti 3*d t*2*<sup>g</sup>* orbital, which has lower energy than the twofold degenerate Ti 3*d eg* orbital. [42] The absorption coefficient related *Eg* of the BSMT ceramics can be determined according to the Tauc's law: (*αE*)<sup>2</sup> = *A*(*E* − *Eg*), where *A* is a constant, *α* and *E* are the absorption coefficient and incident photon energy, respectively. For the allowed direct transition, the straight line between (*αE*)<sup>2</sup> and *E* will provide the value of the band gap, which is extrapolated by the linear portion of the plot to (*αE*)2=0, as seen in Fig. 4. The *Eg* is estimated to 3.65, 3.57, 3.40 and 3.60 eV corresponding to *x*=1, 2, 5 and 10% for the BSMT ceramics, respectively, as shown in the inset of Fig. 4. The results suggest that the band gap of the BSMT ceramics decreases and then increases with increasing Mn composition.

As we know, the optical band gap can be affected by some factors such as grain size, oxygen vacancy, stress and amorphous nature of the materials. [2] The decreasing trend of the band gap with the Mn composition below 5% can be attributed to the increase of the grain size and the smaller lattice constant, which are caused by the Mn introduction in the A sites. When the Mn composition is 10%, there is a sharp increase in the optical band gap because the excess Mn will substitute the Ti site at the Mn composition of 10%, causing the increase of the oxygen vacancies. In addition, the creation of an oxygen vacancy which is associated with the generation of free charge carriers can be described as the following: MnO(−TiO2) <sup>→</sup> Mn′′ Ti <sup>+</sup> <sup>V</sup>·· <sup>O</sup> <sup>+</sup> OO; OO <sup>→</sup> <sup>V</sup>·· <sup>O</sup> <sup>+</sup> 2e<sup>−</sup> <sup>+</sup> 1/2O2, where V·· <sup>O</sup> represents the doubly charged oxygen vacancy, OO is an oxygen ion at its normal site, *<sup>e</sup>*<sup>−</sup> is the free electronic charge generated through the vacancy formation. [9, 43] The heavy doping blocks the lowest states in the CB and the effective band-gap increases, which is known as the Burstein-Moss (BM) effect. [44] When a large number of vacancy-related charge carriers are

**Figure 5.** Raman scattering spectra of the SBNN ceramics with different Nd composition (*x*) recorded at RT. The dashed lines clearly indicate some Raman-active phonon modes. The inset shows the peak frequency variation of the *A*1*<sup>g</sup>* [Nb] phonon mode at about <sup>207</sup> cm−<sup>1</sup> and the *<sup>A</sup>*1*<sup>g</sup>* [O] phonon mode at about <sup>836</sup> cm−<sup>1</sup> as a function of Nd composition. (Figure reproduced with permission from [46]. Copyright 2012, Wiley.)

generated in the system, the lowest lying states in the CB are filled and the vertical distance needed for the optical transition increases. [44] Hence, it can be concluded that, when the Mn composition is 10%, the sharply increase in the band gap may be attributed to the BM shift caused by electrons generated by oxygen vacancies. Besides, the grain size for the ceramic decreases induced by the heavy doping with the Mn composition of 10%, which may also result in a sharply increase of the optical band gap. It can be concluded that the difference of the optical band gap could be due to the dopant composition, the oxygen vacancies, and the crystallinity of the BSMT ceramics.

### **5. Phase transitions of SBNN ceramics**

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Springer.)

**Figure 4.** Absorption coefficient *vs*. incident photon energy near the optical band gap of the BSMT ceramics. The insert is the optical band gap *Eg* with the different Mn composition. (Figure reproduced with permission from [39]. Copyright 2012,

parts of the dielectric functions in the photon energy range of 0.7-4.2 eV, respectively. The evolution of *ε* with the photon energy is a typical optical response behavior of ferroelectric and/or semiconductors. The optical band gap (*Eg*) of the BSMT ceramics is one of the important optical behaviors, which is calculated by considering a direct transition from the VB to the CB when the photon energy falls on the materials. In the BSMT system, the VB is mainly composed of the O 2*p* orbital and the CB is mainly composed of the Ti 3*d* orbital. It should be noted that because of the splitting of the Ti 3*d* conduction bands into *t*2*<sup>g</sup>* and *eg* subbands, the lowest CB arises from the threefold degenerate Ti 3*d t*2*<sup>g</sup>* orbital, which has lower energy than the twofold degenerate Ti 3*d eg* orbital. [42] The absorption coefficient related *Eg* of the BSMT ceramics can be determined according to the Tauc's law: (*αE*)<sup>2</sup> = *A*(*E* − *Eg*), where *A* is a constant, *α* and *E* are the absorption coefficient and incident photon energy, respectively. For the allowed direct transition, the straight line between (*αE*)<sup>2</sup> and *E* will provide the value of the band gap, which is extrapolated by the linear portion of the plot to (*αE*)2=0, as seen in Fig. 4. The *Eg* is estimated to 3.65, 3.57, 3.40 and 3.60 eV corresponding to *x*=1, 2, 5 and 10% for the BSMT ceramics, respectively, as shown in the inset of Fig. 4. The results suggest that the band gap of the BSMT ceramics decreases and

As we know, the optical band gap can be affected by some factors such as grain size, oxygen vacancy, stress and amorphous nature of the materials. [2] The decreasing trend of the band gap with the Mn composition below 5% can be attributed to the increase of the grain size and the smaller lattice constant, which are caused by the Mn introduction in the A sites. When the Mn composition is 10%, there is a sharp increase in the optical band gap because the excess Mn will substitute the Ti site at the Mn composition of 10%, causing the increase of the oxygen vacancies. In addition, the creation of an oxygen vacancy which is associated with the generation of free charge carriers can be described as the following:

doubly charged oxygen vacancy, OO is an oxygen ion at its normal site, *<sup>e</sup>*<sup>−</sup> is the free electronic charge generated through the vacancy formation. [9, 43] The heavy doping blocks the lowest states in the CB and the effective band-gap increases, which is known as the Burstein-Moss (BM) effect. [44] When a large number of vacancy-related charge carriers are

<sup>O</sup> <sup>+</sup> OO; OO <sup>→</sup> <sup>V</sup>··

˙ *ε*1) and imaginary (*ε*2)

<sup>O</sup> <sup>+</sup> 2e<sup>−</sup> <sup>+</sup> 1/2O2, where V··

<sup>O</sup> represents the

according to the ellipsometric spectra Figs. 3(a) and (b) show the real (

then increases with increasing Mn composition.

Ti <sup>+</sup> <sup>V</sup>··

MnO(−TiO2) <sup>→</sup> Mn′′

The general formula of bismuth layer structure ferroelectrics (BLSFs) is given as (Bi2O2)2<sup>+</sup> (*Am*<sup>−</sup>1*BmO*3*m*+1)2−, where *<sup>A</sup>* and *<sup>B</sup>* are the two types of cations that enter the perovskite unit, and *m* is the the number of perovskite unit cell between bismuth oxide layers. [20] SBN, which is known to be *m*=2 member of BLSFs family, has been regarded as a promising ferroelectric material due to low dielectric constants and excellent fatigue resistance. [40] Fig. 5(a) shows the Raman spectra of the SBNN ceramics with different Nd compositions at RT in the spectral range of 50 − 950 cm<sup>−</sup>1. The Raman selection rules allow 18 phonon modes (4*A*1*<sup>g</sup>* + 2*B*1*<sup>g</sup>* + 6*B*2*<sup>g</sup>* + 6*B*3*g*) for SBN ceramics at RT. [45] However, less than 10 phonon modes are observed because of the possible overlap of the same symmetry vibration or the weak feature of some vibration bands. [46, 47] According to the assignment of SrBi2Nb2O9 single crystal, [45] the Raman phonon modes at about 61, 207 and 835 cm−<sup>1</sup> can be assigned to the *<sup>A</sup>*1*<sup>g</sup>* phonon mode, the vibrations at about 179 and 579 cm−<sup>1</sup> can be assigned to the *Eg* phonon mode. However, the assignment of other phonon modes are still not clear now. The internal vibrations of NbO6 octahedra occur in the high-frequency mode region above 200 cm−<sup>1</sup> because the intragroup binding energy within the NbO6 octahedra is much larger than the intergroup or crystal binding energy. [15] The composition dependence of the frequencies for two typical phonon modes is illustrated in Fig. 5(b). Note that the *A*1*g*[Nb] phonon mode at 207 cm<sup>−</sup>1, which arises from the distortion of NbO6 octahedra, generally decreases with the Nd composition, whereas the *<sup>A</sup>*1*g*[O] phonon mode at 835 cm−<sup>1</sup> mode corresponding to the symmetric Nb−O stretching vibration, increases with the introduction of Nd ions.

**Figure 6.** Temperature dependence of the SBNN ceramics with the composition of (a)(b) *x*=0 and (c)(d) *x*=0.2. The solid arrows indicate the temperature increasing from 80 K to 873 K, and the dash arrows show the shift of the frequency for the phonon modes with the temperature. The symbol asterisk (∗) and pound sign (#) indicate the two weak *Eg* phonon modes in the range of 281-310 cm−<sup>1</sup> and 435-456 cm−1, respectively. (Figure reproduced with permission from [46]. Copyright 2012, Wiley.)

**Figure 7.** (a)-(d) Raman shift and (e) intensities of the *A*1*<sup>g</sup>* [Nb] phonon mode as a function of the temperature for the SBNN ceramics. The arrows indicate that the anomalous vibration occurs around the ferroelectric to paraelectric phase transition

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**Figure 8.** The XRD patterns of the BFO films deposited on STO (111) substrate. (Figure reproduced with permission from [5].

of the anomalous points is different for the four ceramics: 710 K for *x*=0, 665 K for *x*=0.05, 625 K for *x*=0.1 and 550 K for *x*=0.2. The different anomalous points can be ascribed to the Nd3<sup>+</sup> incorporation in the Bi2O2 layers. In addition, the phase transition from the orthorhombic to the tetragonal phase may occur at the temperature when the frequency of the *A*1*g*[Nb] phonon mode shows the anomalous behavior. Therefore, the sharp change in the temperature dependence of both wavenumber and intensity of the *A*1*g*[Nb] phonon

BFO film with the nominal thickness of about 330 nm was prepared on STO (111) substrate by pulsed laser deposition. [5] Fig. 8 shows the XRD pattern of the BFO film and there is no impurity phase. As can be seen, the film is well crystallized with the rhombohedral phase and presents a (111) single crystalline orientation. According to the known Scherrer's equation, the grain size from the (111) diffraction peak was evaluated to about 32 nm. A three-phase layered structure (air/film/substrate) was constructed to

mode was successfully applied to probe the phase transition of the SBNN ceramics.

**6. Temperature effects on electronic transitions of BFO films**

temperatures. (Figure reproduced with permission from [46]. Copyright 2012, Wiley.)

Copyright 2010, American Institute of Physics.)

In order to further understand the effect of Nd3<sup>+</sup> ion substitution on the phonon modes, Fig. 6 presents the temperature dependence of the Raman spectra for the SBNN ceramics with two Nd compositions of *x*=0 and 0.2 in the temperature range from 80 to 873 K. It suggests that the intensities for all the phonon modes increase with the temperature except for the phonon mode at about 579 cm<sup>−</sup>1, whose peak has been overwhelmed at high temperature. The broadening band can be assigned to a rigid sublattice mode, in which all the positive and negative ion displacements are equal and opposite. [48] It is found that a strong broadening peak can be observed at 80 K due to the combined effects of two modes splitting from the *Eg* character mode. However, with increasing the temperature, the frequency and intensity of the mode present a decreasing trend. Because the mode is assigned to the asymmetric Nb−O vibration, it can be concluded that the NbO6 octahedra is sensitive to the temperature. On the other band, the phonon modes at about 281 and 310 cm−<sup>1</sup> (labeled by ∗), which are associated with the O−Nb−O bending, become more difficult to be distinguished as the temperature increases and disappear at high temperature. Similar phenomena can be observed for the phonon modes in the range of 435-456 cm−<sup>1</sup> (labeled by #). The band at about 456 cm<sup>−</sup>1, which is described to a Ti−O torsional mode, has been assigned as the *Eg* character and splits into two phonon modes centered at 435 and 456 cm−<sup>1</sup> at lower temperature. As pointed out by Graves *et al*., [45] it can be ascribed to the fact that the several *Eg* phonon modes split into the *B*2*<sup>g</sup>* and *B*3*<sup>g</sup>* phonon modes during the tetragonal to orthorhombic transition. Moreover, the splitting of the phonon modes reveals the structural changes in the SBNN ceramics with the temperature.

Considering that the phase transition temperature is related to the distortion extent of the NbO6 octahedra for the SBNN ceramics, the temperature dependence of the Raman shift for the *A*1*g*[Nb] phonon mode is plotted in Fig. 7(a)-(d). For all the SBNN ceramics, the decrease of the *A*1*g*[Nb] phonon mode can be observed as the temperature is increased. Note that an obviously anomalous vibration occurs around the phase transition temperatures: the Raman shift sharply increases with increasing the temperature. In addition, the temperature

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the temperature.

**Figure 6.** Temperature dependence of the SBNN ceramics with the composition of (a)(b) *x*=0 and (c)(d) *x*=0.2. The solid arrows indicate the temperature increasing from 80 K to 873 K, and the dash arrows show the shift of the frequency for the phonon modes with the temperature. The symbol asterisk (∗) and pound sign (#) indicate the two weak *Eg* phonon modes in the range of 281-310 cm−<sup>1</sup> and 435-456 cm−1, respectively. (Figure reproduced with permission from [46]. Copyright 2012, Wiley.) In order to further understand the effect of Nd3<sup>+</sup> ion substitution on the phonon modes, Fig. 6 presents the temperature dependence of the Raman spectra for the SBNN ceramics with two Nd compositions of *x*=0 and 0.2 in the temperature range from 80 to 873 K. It suggests that the intensities for all the phonon modes increase with the temperature except for the phonon mode at about 579 cm<sup>−</sup>1, whose peak has been overwhelmed at high temperature. The broadening band can be assigned to a rigid sublattice mode, in which all the positive and negative ion displacements are equal and opposite. [48] It is found that a strong broadening peak can be observed at 80 K due to the combined effects of two modes splitting from the *Eg* character mode. However, with increasing the temperature, the frequency and intensity of the mode present a decreasing trend. Because the mode is assigned to the asymmetric Nb−O vibration, it can be concluded that the NbO6 octahedra is sensitive to the temperature. On the other band, the phonon modes at about 281 and 310 cm−<sup>1</sup> (labeled by ∗), which are associated with the O−Nb−O bending, become more difficult to be distinguished as the temperature increases and disappear at high temperature. Similar phenomena can be observed for the phonon modes in the range of 435-456 cm−<sup>1</sup> (labeled by #). The band at about 456 cm<sup>−</sup>1, which is described to a Ti−O torsional mode, has been assigned as the *Eg* character and splits into two phonon modes centered at 435 and 456 cm−<sup>1</sup> at lower temperature. As pointed out by Graves *et al*., [45] it can be ascribed to the fact that the several *Eg* phonon modes split into the *B*2*<sup>g</sup>* and *B*3*<sup>g</sup>* phonon modes during the tetragonal to orthorhombic transition. Moreover, the splitting of the phonon modes reveals the structural changes in the SBNN ceramics with

Considering that the phase transition temperature is related to the distortion extent of the NbO6 octahedra for the SBNN ceramics, the temperature dependence of the Raman shift for the *A*1*g*[Nb] phonon mode is plotted in Fig. 7(a)-(d). For all the SBNN ceramics, the decrease of the *A*1*g*[Nb] phonon mode can be observed as the temperature is increased. Note that an obviously anomalous vibration occurs around the phase transition temperatures: the Raman shift sharply increases with increasing the temperature. In addition, the temperature

**Figure 7.** (a)-(d) Raman shift and (e) intensities of the *A*1*<sup>g</sup>* [Nb] phonon mode as a function of the temperature for the SBNN ceramics. The arrows indicate that the anomalous vibration occurs around the ferroelectric to paraelectric phase transition temperatures. (Figure reproduced with permission from [46]. Copyright 2012, Wiley.)

**Figure 8.** The XRD patterns of the BFO films deposited on STO (111) substrate. (Figure reproduced with permission from [5]. Copyright 2010, American Institute of Physics.)

of the anomalous points is different for the four ceramics: 710 K for *x*=0, 665 K for *x*=0.05, 625 K for *x*=0.1 and 550 K for *x*=0.2. The different anomalous points can be ascribed to the Nd3<sup>+</sup> incorporation in the Bi2O2 layers. In addition, the phase transition from the orthorhombic to the tetragonal phase may occur at the temperature when the frequency of the *A*1*g*[Nb] phonon mode shows the anomalous behavior. Therefore, the sharp change in the temperature dependence of both wavenumber and intensity of the *A*1*g*[Nb] phonon mode was successfully applied to probe the phase transition of the SBNN ceramics.

### **6. Temperature effects on electronic transitions of BFO films**

BFO film with the nominal thickness of about 330 nm was prepared on STO (111) substrate by pulsed laser deposition. [5] Fig. 8 shows the XRD pattern of the BFO film and there is no impurity phase. As can be seen, the film is well crystallized with the rhombohedral phase and presents a (111) single crystalline orientation. According to the known Scherrer's equation, the grain size from the (111) diffraction peak was evaluated to about 32 nm. A three-phase layered structure (air/film/substrate) was constructed to

**Figure 9.** Experimental (dotted curves) and fitting (solid curves) transmittance spectra at temperatures of 300 and 5.3 K, respectively. The inset shows the enlarged band gap region of the BFO film at temperatures of 300, 200, and 5.3 K, respectively. (Figure reproduced with permission from [5]. Copyright 2010, American Institute of Physics.)

**Figure 10.** The dielectric functions of the BFO films in the photon energy range of 0.5-6.5 eV at 300, 200, and 5.3 K, respectively. The inset shows the temperature dependence of the *Eg* (dotted curve) and Bose-Einstein model fitting result (solid

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the energy space, which can affect the electronic excited ability of the Bi, Fe, and O states. On the other hand, the evaluated optical constants of the BFO film is presented in Fig. 10. The real and imaginary parts of dielectric functions increase with the temperature at the lower photon energy while decrease with further increasing the photon energy. With decreasing the temperature, both the *ε*<sup>1</sup> and *ε*<sup>2</sup> shift toward the higher energy. The phenomena are related to the modification of the electronic structure such as the fundamental band gap absorption under the lower temperature. From the inset of Fig. 10, it can be found that the *Eg* value increases from 2.65 ± 0.01 to 2.69 ± 0.01 eV, corresponding to decreasing the temperature from 300 to 5.3 K, which indicates that the total shift value of the *Eg* is about 40 meV. The observed decrease in the *Eg* with the temperature can be described using the Bose-Einstein model. It is widely recognized that the electron-phonon interaction and the lattice thermal expansion are responsible for the shrinkage in the optical band gap with the temperature.

In summary, electronic band structures and phase transitions of perovskite (ABO3)-type Manganese (Mn) doped Ba0.4Sr0.6TiO3 ceramics, SrBi2−*x*Nd*x*Nb2O9, and BiFeO3 materials have been investigated from infrared to ultraviolet transmittance, SE and temperature-dependent Raman scattering techniques. The interband electronic transitions and dielectric functions of these materials could be readily obtained in the wider photon energy region. Meanwhile, the phase transition temperature can be easily determined by phonon scattering measurements, indicating that the present solid state spectroscopy is useful to further clarify the physical phenomena for perovskite-type

It should be emphasized that optical properties and their related phenomena, such as phase transition and domain status have not been well investigated. The following factors must be addressed: (1) Growth of high-quality perovskite FE materials (crystal, ceramic and

curve). (Figure reproduced with permission from [5]. Copyright 2010, American Institute of Physics.)

**7. Conclusion and remarks**

ferroelectric/multiferroics oxides.

simulate the transmittance spectra of the BFO film. It should be emphasized that the normal-incident transmittance spectra cannot be sensitive to the thinner surface rough layer, which could be several nanometers and much less than the film thickness. Therefore, the surface rough layer can be reasonably neglected owing to a slight contribution in the evaluation of the optical properties. The optical constants of the BFO film can be expressed using four Tauc-Lorentz (TL) oscillators. [50] As an example, the experimental and fitting transmittance spectra of the BFO film at 300 and 5.3 K are shown in Fig. 9 with the dotted and solid curves, respectively. Note that the symmetrical interference period indicates that the film is of good uniformity and crystallization. From Fig. 9, it can be observed that the absorption edge remarkably shift toward the lower energy with increasing the temperature, suggesting that the OBG of the film has a negative temperature coefficient. Note that the shift at high temperature region (100-300 K) is larger than that at low temperature region (5.3-100 K). This is because the quantities of the conduction band downward and the valence band upward are different under the distinct temperature regions. Especially, two broadening shoulder structures appear and the intensities become much stronger with decreasing the temperature. The similar phenomena have been observed at 2.5 eV when the temperature decreases to about 4 K, which represents the onset of the optical absorption. [51] Furthermore, the shoulders are simply low-lying features of the electronic structure or evidence for excitonic character.

Based on the theoretical calculations and experimental observations, the four energy bands can be uniquely assigned to the following electronic transitions: (1) on-site Fe3<sup>+</sup> *d* to *d* crystal field transition; (2) majority channel Fe 3*d* to O 2*p* charge transfer excitation; (3) minority channel dipole-allowed O *p* to Fe *d* charge transfer excitation; and (4) strong hybridized majority channel O *p* and Fe *d* to Bi *p* state excitation, respectively. [51–55] Within the experimental error bars, the energy positions shift toward the higher energy at the temperature of 5.3 K except for the second excitation, which can be attributed to the energy band variations. Nevertheless, the origin of the abnormal shift for the second excitation is unclear in the present work. Under the influence of the tetrahedral crystal field, the Fe 3*d* orbital states split into *t*2*<sup>g</sup>* and *eg* state and the *t*2*<sup>g</sup>* state strongly hybrided with the O *p* orbital. [52] With decreasing the temperature, the *t*2*<sup>g</sup>* and *eg* states can be located at different level in

**Figure 10.** The dielectric functions of the BFO films in the photon energy range of 0.5-6.5 eV at 300, 200, and 5.3 K, respectively. The inset shows the temperature dependence of the *Eg* (dotted curve) and Bose-Einstein model fitting result (solid curve). (Figure reproduced with permission from [5]. Copyright 2010, American Institute of Physics.)

the energy space, which can affect the electronic excited ability of the Bi, Fe, and O states. On the other hand, the evaluated optical constants of the BFO film is presented in Fig. 10. The real and imaginary parts of dielectric functions increase with the temperature at the lower photon energy while decrease with further increasing the photon energy. With decreasing the temperature, both the *ε*<sup>1</sup> and *ε*<sup>2</sup> shift toward the higher energy. The phenomena are related to the modification of the electronic structure such as the fundamental band gap absorption under the lower temperature. From the inset of Fig. 10, it can be found that the *Eg* value increases from 2.65 ± 0.01 to 2.69 ± 0.01 eV, corresponding to decreasing the temperature from 300 to 5.3 K, which indicates that the total shift value of the *Eg* is about 40 meV. The observed decrease in the *Eg* with the temperature can be described using the Bose-Einstein model. It is widely recognized that the electron-phonon interaction and the lattice thermal expansion are responsible for the shrinkage in the optical band gap with the temperature.

### **7. Conclusion and remarks**

10 Advances in Ferroelectrics

evidence for excitonic character.

**Figure 9.** Experimental (dotted curves) and fitting (solid curves) transmittance spectra at temperatures of 300 and 5.3 K, respectively. The inset shows the enlarged band gap region of the BFO film at temperatures of 300, 200, and 5.3 K, respectively.

simulate the transmittance spectra of the BFO film. It should be emphasized that the normal-incident transmittance spectra cannot be sensitive to the thinner surface rough layer, which could be several nanometers and much less than the film thickness. Therefore, the surface rough layer can be reasonably neglected owing to a slight contribution in the evaluation of the optical properties. The optical constants of the BFO film can be expressed using four Tauc-Lorentz (TL) oscillators. [50] As an example, the experimental and fitting transmittance spectra of the BFO film at 300 and 5.3 K are shown in Fig. 9 with the dotted and solid curves, respectively. Note that the symmetrical interference period indicates that the film is of good uniformity and crystallization. From Fig. 9, it can be observed that the absorption edge remarkably shift toward the lower energy with increasing the temperature, suggesting that the OBG of the film has a negative temperature coefficient. Note that the shift at high temperature region (100-300 K) is larger than that at low temperature region (5.3-100 K). This is because the quantities of the conduction band downward and the valence band upward are different under the distinct temperature regions. Especially, two broadening shoulder structures appear and the intensities become much stronger with decreasing the temperature. The similar phenomena have been observed at 2.5 eV when the temperature decreases to about 4 K, which represents the onset of the optical absorption. [51] Furthermore, the shoulders are simply low-lying features of the electronic structure or

Based on the theoretical calculations and experimental observations, the four energy bands can be uniquely assigned to the following electronic transitions: (1) on-site Fe3<sup>+</sup> *d* to *d* crystal field transition; (2) majority channel Fe 3*d* to O 2*p* charge transfer excitation; (3) minority channel dipole-allowed O *p* to Fe *d* charge transfer excitation; and (4) strong hybridized majority channel O *p* and Fe *d* to Bi *p* state excitation, respectively. [51–55] Within the experimental error bars, the energy positions shift toward the higher energy at the temperature of 5.3 K except for the second excitation, which can be attributed to the energy band variations. Nevertheless, the origin of the abnormal shift for the second excitation is unclear in the present work. Under the influence of the tetrahedral crystal field, the Fe 3*d* orbital states split into *t*2*<sup>g</sup>* and *eg* state and the *t*2*<sup>g</sup>* state strongly hybrided with the O *p* orbital. [52] With decreasing the temperature, the *t*2*<sup>g</sup>* and *eg* states can be located at different level in

(Figure reproduced with permission from [5]. Copyright 2010, American Institute of Physics.)

In summary, electronic band structures and phase transitions of perovskite (ABO3)-type Manganese (Mn) doped Ba0.4Sr0.6TiO3 ceramics, SrBi2−*x*Nd*x*Nb2O9, and BiFeO3 materials have been investigated from infrared to ultraviolet transmittance, SE and temperature-dependent Raman scattering techniques. The interband electronic transitions and dielectric functions of these materials could be readily obtained in the wider photon energy region. Meanwhile, the phase transition temperature can be easily determined by phonon scattering measurements, indicating that the present solid state spectroscopy is useful to further clarify the physical phenomena for perovskite-type ferroelectric/multiferroics oxides.

It should be emphasized that optical properties and their related phenomena, such as phase transition and domain status have not been well investigated. The following factors must be addressed: (1) Growth of high-quality perovskite FE materials (crystal, ceramic and film); (2) Theoretical model and explanations; (3) Improved experimental methods. As for solid state spectroscopic technique, however, it can uniquely discover the electronic band structure of FE system. One can think that the electronic transition will be changed during the phase transition process owing to the crystal structure variation. Thus, we can check the status by recording the spectral response. Evidently, some optical setup at elevated temperatures are necessary because most of FE materials have the high *TC*. In our research group, Transmittane/Reflectance, SE, and Raman systems from LHe temperature to about 800 K have been developed. Our next goal is to characterize the physical information of some typical FE materials in a wider spectral and temperature ranges, which play an important role in clarifying the structure transitions and the intrinsic origin.

[4] Catalan, G. & Scott, J. F. (2009). Physics and applications of bismuth ferrite. *Adv. Mater.*,

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[6] Jiang, K.; Zhu, J. J.; Wu, J. D.; Sun, J.; Hu, Z. G. & Chu, J. H. (2011). Influences of oxygen pressure on optical properties and interband electronic transitions in multiferroic bismuth ferrite nanocrystalline films grown by pulsed laser deposition. *ACS Appl.*

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### **Acknowledgments**

The authors would like to thank Dr. Wenwu Li for valuable discussions. This work was financially supported by the Major State Basic Research Development Program of China (Grant Nos. 2011CB922200 and 2013CB922300), the Natural Science Foundation of China (Grant Nos. 11074076, 60906046 and 61106122), the Program of New Century Excellent Talents, MOE (Grant No. NCET-08-0192) and PCSIRT, the Projects of Science and Technology Commission of Shanghai Municipality (Grant Nos. 11520701300, 10DJ1400201 and 10SG28), and the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning.

### **Author details**

Zhigao Hu1,<sup>⋆</sup>, Junhao Chu1,2,⋆, Yawei Li1, Kai Jiang1 and Ziqiang Zhu1

1 Key Laboratory of Polar Materials and Devices, Ministry of Education, Department of Electronic Engineering, East China Normal University, Shanghai, People's Republic of China 2 National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai, People's Republic of China

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12 Advances in Ferroelectrics

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The authors would like to thank Dr. Wenwu Li for valuable discussions. This work was financially supported by the Major State Basic Research Development Program of China (Grant Nos. 2011CB922200 and 2013CB922300), the Natural Science Foundation of China (Grant Nos. 11074076, 60906046 and 61106122), the Program of New Century Excellent Talents, MOE (Grant No. NCET-08-0192) and PCSIRT, the Projects of Science and Technology Commission of Shanghai Municipality (Grant Nos. 11520701300, 10DJ1400201 and 10SG28), and the Program for Professor of Special Appointment (Eastern Scholar) at

1 Key Laboratory of Polar Materials and Devices, Ministry of Education, Department of Electronic Engineering, East China Normal University, Shanghai, People's Republic of China 2 National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese

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grown by pulsed laser deposition. *Appl. Phys. Lett.*, 100, 07, 071905(1-4)

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layer-structured ceramics determined by infrared reflectance spectra. *Mater. Res. Bull.*, 45, 11, 1654-1658

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**Chapter 7**

**Phase Diagram of The Ternary BaO-Bi2O3-B2O3 System:**

**New Compounds and Glass Ceramics Characterisation**

Restriction and an interdiction on use of toxic materials in electronics products since 2006 have promoted an intensification of development new ecologically friendly materials (glasses, glass ceramics, ceramics) with attractive properties. It has stimulated new lead (cadmium) free systems with good glass forming abilities investigations and new stoiciometric and eutectic points revealing and characterization. Alkaline-earth bismuth borate ternary systems were a good candidate for this purpose, because the binary Bi2O3-B2O3 system have propensity for glass formation and set of binary compounds and eutectics [1 - 3]. Furthermore, bismuth borate single crystals and glass ceramics have nonlinear optical (NLO) properties and other attrac‐ tive properties [4 - 7]. Both these factors are reasons for further study of binary and ternary

The phase diagram of the Bi2O3–B2O3 system was first determined by Levin &Daniel in 1962 [2] and five crystalline compounds, Bi24B12O39, Bi4B2O9, Bi3B5O12, BiB3O6 and Bi2B8O15, were identified. Later Pottier revealed a sixth compound, BiBO3 (bismuth orthoborate) [8], which was missing in the original phase diagram [2]. There are no doubts about the existence of BiBO3 now: Becker with co-workers have confirmed existence of bismuth orthoborate [5, 9, 10] and its transparent colourless single crystals of BiBO3 have recently been grown from the melt and characterized by Becker & Froehlich [10]. Monophase samples of both crystalline BiBO3 modifications were obtained by crystallisation below 550°C of bismuth borate glasses with 50-57 mol% B2O3 [10]. However, these authors did not correct the phase diagram, and did not determine the melting point of BiBO3 or the eutectic composition between BiBO3 and Bi3B5O12. The compound BiBO3 and this eutectic point are clearly given on the Zargarova & Kasumova's version of the B2O3–Bi2O3 phase diagram, without indication of their melting

> © 2013 Hovhannisyan; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

Martun Hovhannisyan

http://dx.doi.org/10.5772/52405

**1. Introduction**

Additional information is available at the end of the chapter

bismuth borate systems, and the glasses which they form.

points and the eutectic composition [11].

**Chapter 7**

### **Phase Diagram of The Ternary BaO-Bi2O3-B2O3 System: New Compounds and Glass Ceramics Characterisation**

Martun Hovhannisyan

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52405

### **1. Introduction**

Restriction and an interdiction on use of toxic materials in electronics products since 2006 have promoted an intensification of development new ecologically friendly materials (glasses, glass ceramics, ceramics) with attractive properties. It has stimulated new lead (cadmium) free systems with good glass forming abilities investigations and new stoiciometric and eutectic points revealing and characterization. Alkaline-earth bismuth borate ternary systems were a good candidate for this purpose, because the binary Bi2O3-B2O3 system have propensity for glass formation and set of binary compounds and eutectics [1 - 3]. Furthermore, bismuth borate single crystals and glass ceramics have nonlinear optical (NLO) properties and other attrac‐ tive properties [4 - 7]. Both these factors are reasons for further study of binary and ternary bismuth borate systems, and the glasses which they form.

The phase diagram of the Bi2O3–B2O3 system was first determined by Levin &Daniel in 1962 [2] and five crystalline compounds, Bi24B12O39, Bi4B2O9, Bi3B5O12, BiB3O6 and Bi2B8O15, were identified. Later Pottier revealed a sixth compound, BiBO3 (bismuth orthoborate) [8], which was missing in the original phase diagram [2]. There are no doubts about the existence of BiBO3 now: Becker with co-workers have confirmed existence of bismuth orthoborate [5, 9, 10] and its transparent colourless single crystals of BiBO3 have recently been grown from the melt and characterized by Becker & Froehlich [10]. Monophase samples of both crystalline BiBO3 modifications were obtained by crystallisation below 550°C of bismuth borate glasses with 50-57 mol% B2O3 [10]. However, these authors did not correct the phase diagram, and did not determine the melting point of BiBO3 or the eutectic composition between BiBO3 and Bi3B5O12. The compound BiBO3 and this eutectic point are clearly given on the Zargarova & Kasumova's version of the B2O3–Bi2O3 phase diagram, without indication of their melting points and the eutectic composition [11].

© 2013 Hovhannisyan; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Kargin with co-authors [12] by DTA and X-ray analysis have studied conditions of metasta‐ ble phases formation at system Bi2O3-B2O3 melts crystallization. They have confirmed exis‐ tence of metastable BiBO3 compound and for the first time have specified on congruent character of its melting. Authors also establish formation of a metastable phase of 5Bi2O3•3B2O3 composition. Both compounds together with initial Bi2O3 and B2O3 are present on the metastable state diagram of the Bi2O3-B2O3 system constructed by them.

Hubner confirmed an existence of the congruent melted Ba2B5O17 compound with m.p. 890

Phase Diagram of The Ternary BaO-Bi2O3-B2O3 System: New Compounds and Glass Ceramics Characterisation

searches have used the melting diagram of the BaO-B2O3 system created by Levin & McMur‐ die up to now, without the indication in it the specific areas of existence of new compounds

Both these factors were the reason of the BaO-B2O3 system phase diagram correction made by Hovhannisyan R.M. [22]. Author has revealed fields of Ba2B5O17 and BaB4O7 compounds crystallisation and new eutectic points which are absent on the diagram constructed by Lev‐ in & McMurdie [18]. Six binary eutectic compositions containing 31.5, 37.5, 63.5, 68.5, 76.0, 83.4 mol % B2O3 with melting points 1025, 915, 905, 895, 869 and 878°C accordingly were on

Area of two immiscible liquids established by Levin & McMurdie [35] in the BaO-B2O3 in an interval of 1.5 to 30wt. % BaO content, it has been confirmed in the subsequent by other au‐ thors. However, the temperature of the liquation couple, which are 1150, 1180, 1256 °C ac‐

There are no full version of the phase diagram of the BaO-Bi2O3 system till now [1]. It is very complex system, which is very critical to atmosphere and pressure at experiment carrying out [25 - 28]. Two low melted eutectic areas (740-790°C) clear observed on phase diagram studied in air or oxigen in high bismuth content region around 5-7 mol%BaO and 25-30 mol

All research groups payd special attention to BaO-Bi2O3-B2O3 system studies and new ter‐ nary compounds revealing and characterisation. Barbier *et al.* have studied seven compo‐ sitions in the ternary BaO-B2O3-Bi2O3 system by solid state synthesis at temperatures below 650°C and BaBiBO4, or BaBi(BO3)O, a novel borate compound, has been made and chrac‐ terisised [29]. Above 650°C it decays with bismuth borate glass formation. A powder sam‐ ple of BaBiBO4 had a second harmonic signal with a NLO efficiency equal to five times

Practically in parallel, Egorysheva with co-workers have been investigated phase equilibri‐ um in the Bi2O3-BaB2O4-B2O3 system by X-ray analysis and DTA [30, 31]. Studies were spent by the samples solid state synthesis in closed Pt crucibles in muffle furnaces at the tempera‐

termediate cakes regrinding) were 6-16 days. They confirmed presence of BaBiBO4 and have revealed three new compounds: BaBiB11O19, BaBi2B4O10, Ba3BiB3O9. BaBiB11O19, BaBi2B4O10 have congruent melting at 830 and 730 °C respectively and BaBiBO4 melt incongruently at

Recently single crystals of BaBi2B4O10 composition were grown by cooling of a melt with the stoichiometric composition with cooling rate 0.5 K/h [32]. They have once again confirmed existence of BaBi2B4O10 stoichometric compound earlier obtained by solid state synthesis.

C. Ba3BiB3O9 undergoes a phase transition at 850°C and exist up to 885°C, were decom‐

C, that corresponds to sub-solidus area. The synthesis duration (with in‐

However, all scientists and re‐

http://dx.doi.org/10.5772/52405

129

°C, and revealed two new compounds Ba4B2O7, Ba2B2O5 [21].

cording to [23] and 1539°C according to [24] is discussed till now.

and eutectic points among them [18].

the diagram after correction.

%BaO [26 - 28].

that of KDP.

7800

ture range 500-750 0

pose in the solid state [31].

Presence of five compounds on the known Bi2O3-B2O3 phase diagram has naturally led to formation of five eutectics compositions containing (mol % B2O3): 19.14 (622°C), 44.4 (646 °C), 73.5 (698°C), 76.6 (695°C) and 81.04 (709°C). There is an area of phase separation traditional for borate systems, observed for compositions containing 81-100 mol % B2O3 [2]. Though according to [12], the area of stable phase separation is stretched to 58-95 mol % B2O3.

Interest to ternary alkali free bismuth borate systems MxOy-Bi2O3-B2O3 (M=Zn,Sr,Ca,Ba) studies has amplified recently. Various research groups (Russian, Canadian, Armenian) worked in this area during 1990-2009 and revealed a number of ternary compounds, deter‐ mined their structure, optical and nonlinear optical properties. Thus, three ternary zinc bis‐ muth borate compounds have been revealed in the ZnO-Bi2O3-B2O3 system. At first Zargarova& Kasumova have revealed ZnBi4B2O10 and ZnBiBO4 compounds [11]. Later Barb‐ ier with co-authors by solid-state reaction have synthesized third melilite type ZnBi2B2O7 compound with large SHG (four time higher as KDP) [13].

Barbier & Cranswick at first two novel noncentrosymmetric MBi2B2O7 or MBi2O(BO3)2 (M=Ca, Sr) compounds have synthesized by solid-state reactions in air at temperatures in the 600–700°C range [14]. Their crystal structures have been determined and refined using powder neutron diffraction data. CaBi2B2O7 compound has SHG response two time higher as KDP [14]. However, authors didn't pay attention for both compounds melting behavior.

Egorisheva with co-authors have studied phase relation in the CaO-Bi2O3-B2O3 system and constructs the 600 °C (subsolidus) section of its phase diagram [15]. A new ternary com‐ pound of composition CaBi2B4O10 was identified and the existence of CaBi2B2O7 ternary com‐ pound was comfirmed. Both compounds had incongruent melting at 700 and 783 °C respectively and liquidus temperature about 900-930 °C.

Kargin with co-workers have studied phase relation in the SrO-Bi2O3-B2O3 system in subsoli‐ dus at 600 °C [16]. Two new ternary compound of Sr7Bi8B18O46 and SrBiBO4 compositions were identified. Both compounds had incongruent melting at 760 and 820 °C without indication liquidus temperature. However, later Barbier et el. have discribe new novel centrosymmet‐ ric borate SrBi2OB4O9 (SrBi2B4O10) forming in the SrO–Bi2O3–B2O3 system [17], thereby hav‐ ing substituted under doubt existence of previously reported Sr7Bi8B18O46 compound [16].

The uniqueness of the BaO-Bi2O3-B2O3 system is shown by the available sets of compounds and eutectics both in the binary Bi2O3–B2O3 and BaO–B2O3 systems. Seven compounds are known in the BaO-B2O3 system. Four congruent melting binary compounds Ba3B2O6, BaB2O4, BaB4O7, BaB8O13 with melting points(m.p.) 1383, 1105, 910, 889°C accordingly were found by Levin & McMurdie [18, 19]. Further, Green and Wahler have found out new congruent melt‐ ed at 890°C Ba2B5O17 compound at the ternary BaO-B2O3-Al2O3 system investigation [20]. Hubner confirmed an existence of the congruent melted Ba2B5O17 compound with m.p. 890 °C, and revealed two new compounds Ba4B2O7, Ba2B2O5 [21]. However, all scientists and re‐ searches have used the melting diagram of the BaO-B2O3 system created by Levin & McMur‐ die up to now, without the indication in it the specific areas of existence of new compounds and eutectic points among them [18].

Kargin with co-authors [12] by DTA and X-ray analysis have studied conditions of metasta‐ ble phases formation at system Bi2O3-B2O3 melts crystallization. They have confirmed exis‐ tence of metastable BiBO3 compound and for the first time have specified on congruent character of its melting. Authors also establish formation of a metastable phase of 5Bi2O3•3B2O3 composition. Both compounds together with initial Bi2O3 and B2O3 are present

Presence of five compounds on the known Bi2O3-B2O3 phase diagram has naturally led to formation of five eutectics compositions containing (mol % B2O3): 19.14 (622°C), 44.4 (646 °C), 73.5 (698°C), 76.6 (695°C) and 81.04 (709°C). There is an area of phase separation traditional for borate systems, observed for compositions containing 81-100 mol % B2O3 [2]. Though according to [12], the area of stable phase separation is stretched to 58-95 mol % B2O3.

Interest to ternary alkali free bismuth borate systems MxOy-Bi2O3-B2O3 (M=Zn,Sr,Ca,Ba) studies has amplified recently. Various research groups (Russian, Canadian, Armenian) worked in this area during 1990-2009 and revealed a number of ternary compounds, deter‐ mined their structure, optical and nonlinear optical properties. Thus, three ternary zinc bis‐ muth borate compounds have been revealed in the ZnO-Bi2O3-B2O3 system. At first Zargarova& Kasumova have revealed ZnBi4B2O10 and ZnBiBO4 compounds [11]. Later Barb‐ ier with co-authors by solid-state reaction have synthesized third melilite type ZnBi2B2O7

Barbier & Cranswick at first two novel noncentrosymmetric MBi2B2O7 or MBi2O(BO3)2 (M=Ca, Sr) compounds have synthesized by solid-state reactions in air at temperatures in the 600–700°C range [14]. Their crystal structures have been determined and refined using powder neutron diffraction data. CaBi2B2O7 compound has SHG response two time higher as KDP [14]. However, authors didn't pay attention for both compounds melting behavior. Egorisheva with co-authors have studied phase relation in the CaO-Bi2O3-B2O3 system and constructs the 600 °C (subsolidus) section of its phase diagram [15]. A new ternary com‐ pound of composition CaBi2B4O10 was identified and the existence of CaBi2B2O7 ternary com‐ pound was comfirmed. Both compounds had incongruent melting at 700 and 783 °C

Kargin with co-workers have studied phase relation in the SrO-Bi2O3-B2O3 system in subsoli‐ dus at 600 °C [16]. Two new ternary compound of Sr7Bi8B18O46 and SrBiBO4 compositions were identified. Both compounds had incongruent melting at 760 and 820 °C without indication liquidus temperature. However, later Barbier et el. have discribe new novel centrosymmet‐ ric borate SrBi2OB4O9 (SrBi2B4O10) forming in the SrO–Bi2O3–B2O3 system [17], thereby hav‐ ing substituted under doubt existence of previously reported Sr7Bi8B18O46 compound [16].

The uniqueness of the BaO-Bi2O3-B2O3 system is shown by the available sets of compounds and eutectics both in the binary Bi2O3–B2O3 and BaO–B2O3 systems. Seven compounds are known in the BaO-B2O3 system. Four congruent melting binary compounds Ba3B2O6, BaB2O4, BaB4O7, BaB8O13 with melting points(m.p.) 1383, 1105, 910, 889°C accordingly were found by Levin & McMurdie [18, 19]. Further, Green and Wahler have found out new congruent melt‐ ed at 890°C Ba2B5O17 compound at the ternary BaO-B2O3-Al2O3 system investigation [20].

on the metastable state diagram of the Bi2O3-B2O3 system constructed by them.

compound with large SHG (four time higher as KDP) [13].

128 Advances in Ferroelectrics

respectively and liquidus temperature about 900-930 °C.

Both these factors were the reason of the BaO-B2O3 system phase diagram correction made by Hovhannisyan R.M. [22]. Author has revealed fields of Ba2B5O17 and BaB4O7 compounds crystallisation and new eutectic points which are absent on the diagram constructed by Lev‐ in & McMurdie [18]. Six binary eutectic compositions containing 31.5, 37.5, 63.5, 68.5, 76.0, 83.4 mol % B2O3 with melting points 1025, 915, 905, 895, 869 and 878°C accordingly were on the diagram after correction.

Area of two immiscible liquids established by Levin & McMurdie [35] in the BaO-B2O3 in an interval of 1.5 to 30wt. % BaO content, it has been confirmed in the subsequent by other au‐ thors. However, the temperature of the liquation couple, which are 1150, 1180, 1256 °C ac‐ cording to [23] and 1539°C according to [24] is discussed till now.

There are no full version of the phase diagram of the BaO-Bi2O3 system till now [1]. It is very complex system, which is very critical to atmosphere and pressure at experiment carrying out [25 - 28]. Two low melted eutectic areas (740-790°C) clear observed on phase diagram studied in air or oxigen in high bismuth content region around 5-7 mol%BaO and 25-30 mol %BaO [26 - 28].

All research groups payd special attention to BaO-Bi2O3-B2O3 system studies and new ter‐ nary compounds revealing and characterisation. Barbier *et al.* have studied seven compo‐ sitions in the ternary BaO-B2O3-Bi2O3 system by solid state synthesis at temperatures below 650°C and BaBiBO4, or BaBi(BO3)O, a novel borate compound, has been made and chrac‐ terisised [29]. Above 650°C it decays with bismuth borate glass formation. A powder sam‐ ple of BaBiBO4 had a second harmonic signal with a NLO efficiency equal to five times that of KDP.

Practically in parallel, Egorysheva with co-workers have been investigated phase equilibri‐ um in the Bi2O3-BaB2O4-B2O3 system by X-ray analysis and DTA [30, 31]. Studies were spent by the samples solid state synthesis in closed Pt crucibles in muffle furnaces at the tempera‐ ture range 500-750 0 C, that corresponds to sub-solidus area. The synthesis duration (with in‐ termediate cakes regrinding) were 6-16 days. They confirmed presence of BaBiBO4 and have revealed three new compounds: BaBiB11O19, BaBi2B4O10, Ba3BiB3O9. BaBiB11O19, BaBi2B4O10 have congruent melting at 830 and 730 °C respectively and BaBiBO4 melt incongruently at 7800 C. Ba3BiB3O9 undergoes a phase transition at 850°C and exist up to 885°C, were decom‐ pose in the solid state [31].

Recently single crystals of BaBi2B4O10 composition were grown by cooling of a melt with the stoichiometric composition with cooling rate 0.5 K/h [32]. They have once again confirmed existence of BaBi2B4O10 stoichometric compound earlier obtained by solid state synthesis.

In 1972 Elwell with co-workers investigated the BaO–B2O3–Bi2O3 system by hot stage micro‐ scopy and a new ternary eutectic composition, 23.4BaO•62.4Bi2O3•14.2B2O3 (wt%), with a low liquidus temperature of 600°C, was revealed for ferrite spinel growth [33].

frayed in an agate mortar, pressed as tablets, located on platinum plates and passed the thermal treatment in "Naber"firm electric muffles. After regrinding powders were tested by DTA and X-ray methods. The synthesized samples of binary barium borate system composi‐ tions containing 60 mol% and more of BaO and also compositions containing over 90mol % B2O3 had very low chemical resistance and were hydrolyzed on air at room temperature. In

Phase Diagram of The Ternary BaO-Bi2O3-B2O3 System: New Compounds and Glass Ceramics Characterisation

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131

DTA and X-ray diffraction data of glass and crystallized glass samples have been used for phase diagram construction in the ternary BaO- Bi2O3 -B2O3 system. The DTA analysis (pure Al2O3 crucible, powder samples weight ~600 mg, heating rates 10 K/min) on Q-1500 type de‐ rivatograph were carried out. Glass transition -Tg, crystallization peaks -Tcr, melting -Tm and liquidus -TL temperatures have been determined from DTA curves. Reproducibility of tem‐ peratures effects on DTA curves from melting to melting was ±10K. The accuracy of temper‐

Thermal expansion coefficient (TEC) and glass transition temperature (Tg) measurements were made on a DKV-4A type vertical quartz dilatometer with a heating rate of 3K/min. Glass samples in the size of 4×4×50 millimeters have been prepared for TEC measurement. The dilatometer was graduated by the quartz glass and sapphire standards. The TEC meas‐

X-ray patterns were obtained on a DRON-3 type diffractometer (powder method, CuKα–ra‐ diation, Ni-filter). Samples for glass crystallization were prepared with glass powder press‐ ed in the form of tablets. Crystallization process was done in the electrical muffles of "Naber" firm by a single-stage heat treatment. This was done within 6-12 hours around a temperature at which the maximum exothermal effects on glasses by DTA were observed.

Crystalline phases of binary and ternary compounds formed both at glasses crystalliza‐ tion and at solid-phase synthesis have been identified by using JCPDS-ICDD PDF-2 re‐

Computerized methodic of ferroelectric hysteresis test and measurement of ferroelectric properties such as coercive field and remanent polarization at wide temperature (up to 250°C and frequency (10-5000Hz) ranges was used. Methodic based on the well known Saw‐ yer – Tower's [44] modified scheme, which is allowing to compensate phase shifts con‐ cerned with dielectric losses and conductivity. The desired frequency signal from waveform generator is amplifying by high voltage amplifier and applying to sample. The signals, from the measuring circuit output, proportional to applied field and spontaneous polarization are passing throw high impedance conditioning amplifiers, converting by ADC and operating and analyzing in PC. The technique allows to perform tests of synthesized glass ceramics obtained by means of controlling crystallization of thin (above 30 micrometer thick) mono‐ lithic tape (film) specimens by applying up to 300kV/cm field to our thin samples (~50 mi‐

crometer thick) and obtain hysteresis loops for wide diversity of hard FE materials.

this connection the synthesized samples were kept in a dryer at 200°C.

ature measurement is ±5 K.

lease 2008 database [43].

urement accuracy is ±(3÷4)∙10-7K-1, Tg ±5 °C.

Using different melts cooling rates Hovhannisyan, M. with co-authors at first have deter‐ mined large glass-forming field in the BaO-Bi2O3-B2O3 system, which includes all eutectics in the binary Bi2O3-B2O3, BaO-B2O3 and BaO-Bi2O3 systems and covers majority of the concen‐ tration triangles, reaching up to 90 mol% Bi2O3. [34, 35].

The methodology based on glass samples investigation was more effective at BaO-Bi2O3- B2O3 system phase diagram construction, than a traditional technique based on solid state sintered samples studies. Because DTA curves of glasses, to the contrary DTA curves of sol‐ id state sintered samples, indicates their all characteristics temperatures, includes exother‐ mal effects of glass crystallizations and endothermic effects of formed crystalline phases melting. It has allowed us to reveal two new BaBi2B2O7 and BaBi10B6O25 congruent melted at 725 and 690°C respectively compounds in the BaO-Bi2O3-B2O3 system [34 - 36].

However, our further studies of glasses and glass ceramics in this system have shown neces‐ sity of glass forming diagram correction and phase diagram construction in the ternary BaO-Bi2O3-B2O3 system and present these data to scientific communitty. Another aim of this work is both known and novel stoichiometric ternary barium bismuth borates compounds charac‐ terisation in glassy, glass ceramic and ceramic states for further practical application.

### **2. Experimental**

About three hundred samples of various binary and ternary compositions have been synthe‐ sized and tested in BaO- Bi2O3-B2O3 system. Compositions were prepared from "chemically pure" grade BaCO3, H3BO3 and Bi2O3 at 2.5-5.0 mol % intervals. The most part of samples has been obtained as glasses by various cooling rates depending on melts glass forming abil‐ ities: as bulk glass plates with thickness 6,5 ÷7mm by casting on metallic plate (up to 10 K/s), as monolithic glass plates with thickness up to 3mm by casting between two steel plates(~10<sup>2</sup> <sup>K</sup>/s), and glass tapes samples with thickness 30–400 μm through super cooling method ( 10<sup>3</sup> ÷104 K/s). Glass formation was determined visually or by x-ray analysis. The glass melt‐ ing was performed at 800–1200°C for 15–20 min with a 20–50 g batch in a 20–50 ml uncov‐ ered quartz glass or corundum crucible, using an air atmosphere and a "Superterm 17/08" electric furnace. Compositions in the BaO-B2O3 system were melted in a 25 or 50 ml uncov‐ ered Pt crucibles at 1400–1500°C for 30 min with a 20–50 g batch. The chemical composition of some glasses was determined by traditional chemical analysis, and the results indicate a good compatibility between the calculated and analytical amounts of B2O3, BaO and Bi2O3. SiO<sup>2</sup> contamination from quartz glass crucibles did not exceed 2 wt%, and alumina contami‐ nation did not exceed 0 5–1 wt%, according to the chemical analysis data.

Samples of compositions laying outside of a glass formation field or having high melting temperature, have been obtained by solid-phase synthesis. Mixes (15-20 g) were carefully frayed in an agate mortar, pressed as tablets, located on platinum plates and passed the thermal treatment in "Naber"firm electric muffles. After regrinding powders were tested by DTA and X-ray methods. The synthesized samples of binary barium borate system composi‐ tions containing 60 mol% and more of BaO and also compositions containing over 90mol % B2O3 had very low chemical resistance and were hydrolyzed on air at room temperature. In this connection the synthesized samples were kept in a dryer at 200°C.

In 1972 Elwell with co-workers investigated the BaO–B2O3–Bi2O3 system by hot stage micro‐ scopy and a new ternary eutectic composition, 23.4BaO•62.4Bi2O3•14.2B2O3 (wt%), with a

Using different melts cooling rates Hovhannisyan, M. with co-authors at first have deter‐ mined large glass-forming field in the BaO-Bi2O3-B2O3 system, which includes all eutectics in the binary Bi2O3-B2O3, BaO-B2O3 and BaO-Bi2O3 systems and covers majority of the concen‐

The methodology based on glass samples investigation was more effective at BaO-Bi2O3- B2O3 system phase diagram construction, than a traditional technique based on solid state sintered samples studies. Because DTA curves of glasses, to the contrary DTA curves of sol‐ id state sintered samples, indicates their all characteristics temperatures, includes exother‐ mal effects of glass crystallizations and endothermic effects of formed crystalline phases melting. It has allowed us to reveal two new BaBi2B2O7 and BaBi10B6O25 congruent melted at

However, our further studies of glasses and glass ceramics in this system have shown neces‐ sity of glass forming diagram correction and phase diagram construction in the ternary BaO-Bi2O3-B2O3 system and present these data to scientific communitty. Another aim of this work is both known and novel stoichiometric ternary barium bismuth borates compounds charac‐

About three hundred samples of various binary and ternary compositions have been synthe‐ sized and tested in BaO- Bi2O3-B2O3 system. Compositions were prepared from "chemically pure" grade BaCO3, H3BO3 and Bi2O3 at 2.5-5.0 mol % intervals. The most part of samples has been obtained as glasses by various cooling rates depending on melts glass forming abil‐ ities: as bulk glass plates with thickness 6,5 ÷7mm by casting on metallic plate (up to 10 K/s), as monolithic glass plates with thickness up to 3mm by casting between two steel plates(~10<sup>2</sup> <sup>K</sup>/s), and glass tapes samples with thickness 30–400 μm through super cooling method

÷104 K/s). Glass formation was determined visually or by x-ray analysis. The glass melt‐ ing was performed at 800–1200°C for 15–20 min with a 20–50 g batch in a 20–50 ml uncov‐ ered quartz glass or corundum crucible, using an air atmosphere and a "Superterm 17/08" electric furnace. Compositions in the BaO-B2O3 system were melted in a 25 or 50 ml uncov‐ ered Pt crucibles at 1400–1500°C for 30 min with a 20–50 g batch. The chemical composition of some glasses was determined by traditional chemical analysis, and the results indicate a good compatibility between the calculated and analytical amounts of B2O3, BaO and Bi2O3. SiO<sup>2</sup> contamination from quartz glass crucibles did not exceed 2 wt%, and alumina contami‐

Samples of compositions laying outside of a glass formation field or having high melting temperature, have been obtained by solid-phase synthesis. Mixes (15-20 g) were carefully

nation did not exceed 0 5–1 wt%, according to the chemical analysis data.

terisation in glassy, glass ceramic and ceramic states for further practical application.

low liquidus temperature of 600°C, was revealed for ferrite spinel growth [33].

725 and 690°C respectively compounds in the BaO-Bi2O3-B2O3 system [34 - 36].

tration triangles, reaching up to 90 mol% Bi2O3. [34, 35].

**2. Experimental**

130 Advances in Ferroelectrics

( 10<sup>3</sup>

DTA and X-ray diffraction data of glass and crystallized glass samples have been used for phase diagram construction in the ternary BaO- Bi2O3 -B2O3 system. The DTA analysis (pure Al2O3 crucible, powder samples weight ~600 mg, heating rates 10 K/min) on Q-1500 type de‐ rivatograph were carried out. Glass transition -Tg, crystallization peaks -Tcr, melting -Tm and liquidus -TL temperatures have been determined from DTA curves. Reproducibility of tem‐ peratures effects on DTA curves from melting to melting was ±10K. The accuracy of temper‐ ature measurement is ±5 K.

Thermal expansion coefficient (TEC) and glass transition temperature (Tg) measurements were made on a DKV-4A type vertical quartz dilatometer with a heating rate of 3K/min. Glass samples in the size of 4×4×50 millimeters have been prepared for TEC measurement. The dilatometer was graduated by the quartz glass and sapphire standards. The TEC meas‐ urement accuracy is ±(3÷4)∙10-7K-1, Tg ±5 °C.

X-ray patterns were obtained on a DRON-3 type diffractometer (powder method, CuKα–ra‐ diation, Ni-filter). Samples for glass crystallization were prepared with glass powder press‐ ed in the form of tablets. Crystallization process was done in the electrical muffles of "Naber" firm by a single-stage heat treatment. This was done within 6-12 hours around a temperature at which the maximum exothermal effects on glasses by DTA were observed.

Crystalline phases of binary and ternary compounds formed both at glasses crystalliza‐ tion and at solid-phase synthesis have been identified by using JCPDS-ICDD PDF-2 re‐ lease 2008 database [43].

Computerized methodic of ferroelectric hysteresis test and measurement of ferroelectric properties such as coercive field and remanent polarization at wide temperature (up to 250°C and frequency (10-5000Hz) ranges was used. Methodic based on the well known Saw‐ yer – Tower's [44] modified scheme, which is allowing to compensate phase shifts con‐ cerned with dielectric losses and conductivity. The desired frequency signal from waveform generator is amplifying by high voltage amplifier and applying to sample. The signals, from the measuring circuit output, proportional to applied field and spontaneous polarization are passing throw high impedance conditioning amplifiers, converting by ADC and operating and analyzing in PC. The technique allows to perform tests of synthesized glass ceramics obtained by means of controlling crystallization of thin (above 30 micrometer thick) mono‐ lithic tape (film) specimens by applying up to 300kV/cm field to our thin samples (~50 mi‐ crometer thick) and obtain hysteresis loops for wide diversity of hard FE materials.

### **3. Results**

### **3.1. Glass forming and phase diagrams of the BaO-Bi2O3-B2O3 system**

The traditional method of phase diagram construction based on solid-phase sintered sam‐ ples investigation takes long time and is not effective. The glass samples investigation tech‐ nique is progressive, because the DTA curves have registered all processes taking place in glass samples, including the processes of glass crystallizations, quantity of crystal phases and temperature intervals of their formation and melting. However, inadequate amount of glass samples restrict their use during phase diagram construction. The super-cooling meth‐ od promotes the mentioned problem solving and open new possibilities for phase dia‐ grams constructions.

Hovhannisyan R.M. with co-workes successfully developed this direction last time and have constructed phase diagrams in binary and ternary alkiline-eath bismuth borate, barium bor‐ on titanate, barium aluminum boron titanate, barium gallium borate, yttrium aluminum bo‐ rate, yttrium gallium borate, lantanum gallium borate, zinc tellurium molibdate and other systems [34 - 42].

**Figure 1.** Glass forming regions in the BaO-Bi2O3-B2O3 system according to the data of the authors: 1- [45], 2-[ 46],

Phase Diagram of The Ternary BaO-Bi2O3-B2O3 System: New Compounds and Glass Ceramics Characterisation

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133

Figure 2 shows corrected glass formation diagram in the BaO-Bi2O3-B2O3 system based on phase diagrams of the BaO–B2O3, BaO–Bi2O3, and B2O3–Bi2O3 binary systems and controlla‐ ble melt cooling rates. Using the term "diagram," but not the glass formation region, we take into account the interrelation between the phase diagram and the glass forming ability

**Figure 2.** Glass forming diagram in the BaO-Bi2O3-B2O3 system depending of melts cooling rates: 1-up to 10 K/s;

Area of glass compositions with low crystallization ability and stable glass formation in the studied ternary BaO-Bi2O3 -B2O3 system have been determined at melts cooling rate ~ 10 K/s

2~102K/s; 3-(103-104) K/s; 4- stable phase separation region.

3-[ 47], 4-[ 48], 5-[ 49].

of the system.

### *3.1.1. Glass forming diagram of the BaO-Bi2O3-B2O3 system*

Figure 1 shows the experimental data on glass formation in the BaO-Bi2O3-B2O3 system ob‐ tained by different authors from 1958 to 2007 [45 - 49]. For defining the glass forming ability of the pointed system, the authors of the mentioned works used different amounts of melt, glass melting crucibles, temperature–time melting regimes, and technological methods of melt cooling. Imaoka & Yamazaki studied glass formation by melts cooling on air Glasses were melted at temperatures below 1200 °C in gold-palladium or platinum-rhodium cruci‐ bles (Fig.1.1) [45]. Janakirama-Rao glass formation studied by melts cooling on air. Glasses melted in platinum crucibles at 600- 1400 °C with 0.5-1.0 h melts exposition and its cooling in air (Fig. 1.2) [46]. Izumitani [47] experiments spent in 10g crucibles at 1100-1350 °C with melts cooling on air (Fig. 1.3). Milyukov with co-authors glass formation studied by melts casting in steel mold. Glasses melted in platinum crucibles at 600-1400 °C with melts stirring by Pt stirrer for 1h (Fig. 1.4) [48]. Kawanaka & Matusita glass formation studied by silica rod stirred melts pouring into preheated to 250-300°C carbon mold (Fig. 1.5) [49].

Authors used different weights of glass forming melts, melting crucibles, temperature-time of melting regimes and technological methods of melts cooling. Obtained data are difficultly comparable and remote from two basic criteria promoting glass formation: liquidus temper‐ ature and speeds of melts cooling.

Phase Diagram of The Ternary BaO-Bi2O3-B2O3 System: New Compounds and Glass Ceramics Characterisation http://dx.doi.org/10.5772/52405 133

**3. Results**

132 Advances in Ferroelectrics

grams constructions.

systems [34 - 42].

**3.1. Glass forming and phase diagrams of the BaO-Bi2O3-B2O3 system**

*3.1.1. Glass forming diagram of the BaO-Bi2O3-B2O3 system*

ature and speeds of melts cooling.

The traditional method of phase diagram construction based on solid-phase sintered sam‐ ples investigation takes long time and is not effective. The glass samples investigation tech‐ nique is progressive, because the DTA curves have registered all processes taking place in glass samples, including the processes of glass crystallizations, quantity of crystal phases and temperature intervals of their formation and melting. However, inadequate amount of glass samples restrict their use during phase diagram construction. The super-cooling meth‐ od promotes the mentioned problem solving and open new possibilities for phase dia‐

Hovhannisyan R.M. with co-workes successfully developed this direction last time and have constructed phase diagrams in binary and ternary alkiline-eath bismuth borate, barium bor‐ on titanate, barium aluminum boron titanate, barium gallium borate, yttrium aluminum bo‐ rate, yttrium gallium borate, lantanum gallium borate, zinc tellurium molibdate and other

Figure 1 shows the experimental data on glass formation in the BaO-Bi2O3-B2O3 system ob‐ tained by different authors from 1958 to 2007 [45 - 49]. For defining the glass forming ability of the pointed system, the authors of the mentioned works used different amounts of melt, glass melting crucibles, temperature–time melting regimes, and technological methods of melt cooling. Imaoka & Yamazaki studied glass formation by melts cooling on air Glasses were melted at temperatures below 1200 °C in gold-palladium or platinum-rhodium cruci‐ bles (Fig.1.1) [45]. Janakirama-Rao glass formation studied by melts cooling on air. Glasses melted in platinum crucibles at 600- 1400 °C with 0.5-1.0 h melts exposition and its cooling in air (Fig. 1.2) [46]. Izumitani [47] experiments spent in 10g crucibles at 1100-1350 °C with melts cooling on air (Fig. 1.3). Milyukov with co-authors glass formation studied by melts casting in steel mold. Glasses melted in platinum crucibles at 600-1400 °C with melts stirring by Pt stirrer for 1h (Fig. 1.4) [48]. Kawanaka & Matusita glass formation studied by silica rod

stirred melts pouring into preheated to 250-300°C carbon mold (Fig. 1.5) [49].

Authors used different weights of glass forming melts, melting crucibles, temperature-time of melting regimes and technological methods of melts cooling. Obtained data are difficultly comparable and remote from two basic criteria promoting glass formation: liquidus temper‐

**Figure 1.** Glass forming regions in the BaO-Bi2O3-B2O3 system according to the data of the authors: 1- [45], 2-[ 46], 3-[ 47], 4-[ 48], 5-[ 49].

Figure 2 shows corrected glass formation diagram in the BaO-Bi2O3-B2O3 system based on phase diagrams of the BaO–B2O3, BaO–Bi2O3, and B2O3–Bi2O3 binary systems and controlla‐ ble melt cooling rates. Using the term "diagram," but not the glass formation region, we take into account the interrelation between the phase diagram and the glass forming ability of the system.

**Figure 2.** Glass forming diagram in the BaO-Bi2O3-B2O3 system depending of melts cooling rates: 1-up to 10 K/s; 2~102K/s; 3-(103-104) K/s; 4- stable phase separation region.

Area of glass compositions with low crystallization ability and stable glass formation in the studied ternary BaO-Bi2O3 -B2O3 system have been determined at melts cooling rate ~ 10 K/s (Fig.2-1). It included binary Bi4B2O9, BiBO3, Bi3B5O12, BiB3O6, Bi2B8O15, Ba2B10O17, BaB8O13 com‐ pounds in the BaO-Bi2O3 and BaO-B2O3 systems and five ternary BaBiB11O19(D), Ba‐ Bi2B4O10(C), BaBi2B2O7 (E), BaBi10B6O25 (F), and BaBiBO4 (B) compounds. However, we didn't comfirm presence of Ba3BiB3O9 (A) compound in area of stable glasses at melts cooling rate ~ 10 K/s, which was reported earlier [34, 35].

Increasing of melts cooling speed up to ~102 K/s has led to glass formation area expansion (Fig.2-2). This cooling rate is enough for monolithic glass plates with thickness up to 3mm fabrication by melts casting between two steel plates (Fig.2-2). The glass plates of composi‐ tions correspondings to Ba3BiB3O9 (A) and supposed BaBi8B2O16 (G) compounds have been obtained by this way.

Super cooling technique constructed by our group allowed to expand the borders of glass formation in studied system under high melts cooling rates equal to (103 -104 ) K/s (Fig.2-3). Determined glassforming area include compositions content: 80 – 95 mol% Bi2O3 in the bina‐ ry B2O3–Bi2O3 system; 43-70 mol% BaO in the binary BaO–B2O3 system, including BaB2O4 composition. Area of glass formation from both these areas moves to 55-95 mol% content compositions in the binary BaO–Bi2O3 system (Fig.2-3).

**Figure 3.** The BaO-B2O3-Bi2O3 system triangulation

Bi2B8O15, D-Bi2B8O15-B2O3.

gruently at 685±5°С.

into elementary triangles, i.e. to make triangulation.

*3.1.2.1. Phase diagram of the binary Bi2O3–B2O3 system*

Experimental data concerning phase diagrams of binary systems Bi2O3-B2O3, BaO-B2O3, BaO-Bi2O3 and pseudo-binary sections in the BaO-B2O3-Bi2O3 system have allowed us to estimate fields of primary crystallization of co-existing phases and divided all concentration triangle

Phase Diagram of The Ternary BaO-Bi2O3-B2O3 System: New Compounds and Glass Ceramics Characterisation

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135

The triangulation scheme in the BaO-B2O3-Bi2O3 system is presented on Fig. 3. By means of a triangulation all concentration triangle is divided into following elementary trian‐ gles: BiBO3-F-C, BiBO3-F-Bi4B2O9, BiBO3-C-Bi3B5O12, F-E-C, C-BaB2O4-E, C-Bi3B5O12-D, Bi4B2O9-F-Bi2O3, E-Bi2O3-F, C-BaB2O4-Ba2B10O17, Ba2B10O17-D-C, Ba2B10O17-D-B2O3, Bi3B5O12-D-

First of all we have attempt to finished phase diagram construction in area of compositions around of BiBO3 compound. Compositions containing 45–65 mol% B2O3 in the Bi2O3–B2O3 system were tested to determine the melting point of BiBO3 and to determine the eutectic composition between BiBO3 and Bi3B5O12. The compositions used to correct the B2O3–Bi2O3 phase diagram were prepared by solid state synthesis at 520°C, with steps of 0.5–1.0 mol% В2О3 over the interval 45–55 mol% В2О3. As a result, the eutectic composition, 48.5Bi2O3•51.5B2O3 (mol%), between BiBO3 and Bi3B5O12, was determined, and its melting point was measured by DTA as 665±5°С (Fig. 4). It was also found that BiBO3 melts con‐

Traditional for borate systems a stable phase separation region was also observed for high B2O3 content compositions contents more than 84 -87 mol%B2O3 (Fig.2-4).

### *3.1.2. Phase diagram of the BaO-Bi2O3-B2O3 system*

Our investigation of the ternary BaO-B2O3-Bi2O3 system have purposefully been directed on construction of the phase diagram through first of all glass forming diagram construction and revealing both new compounds and eutectic compositions. Constructed by us glass forming diagram (Fig.2) practically occupies the most part of the BaO-B2O3-Bi2O3 concentra‐ tion triangles. It has allowed to use synthesized glasses as initial compositions at phase dia‐ gram construction. It was basic difference of our methodology from technologies used by other authors. Phase equilibriums reached at isothermal sections construction do not allow to have a full picture of processes in cases of the solid state synthesized samples investiga‐ tions. Whereas at glass samples studies we determine not only characteristic points of glasses(Tg and Ts) by DTA, but also quantity of crystal phases, temperatures of their crystal‐ lization and then temperatures of their melting. It has allowed us to reveal new stoichiomet‐ ric compositions which have been lost by other research groups at isothermal sections construction by traditional methods. In some cases we also in parallel used samples ob‐ tained by solid state synthesis for comparison with their glassy analogues or in those cases, when their obtaining in the glassy form was impossible.

Phase Diagram of The Ternary BaO-Bi2O3-B2O3 System: New Compounds and Glass Ceramics Characterisation http://dx.doi.org/10.5772/52405 135

**Figure 3.** The BaO-B2O3-Bi2O3 system triangulation

(Fig.2-1). It included binary Bi4B2O9, BiBO3, Bi3B5O12, BiB3O6, Bi2B8O15, Ba2B10O17, BaB8O13 com‐ pounds in the BaO-Bi2O3 and BaO-B2O3 systems and five ternary BaBiB11O19(D), Ba‐ Bi2B4O10(C), BaBi2B2O7 (E), BaBi10B6O25 (F), and BaBiBO4 (B) compounds. However, we didn't comfirm presence of Ba3BiB3O9 (A) compound in area of stable glasses at melts cooling rate ~

(Fig.2-2). This cooling rate is enough for monolithic glass plates with thickness up to 3mm fabrication by melts casting between two steel plates (Fig.2-2). The glass plates of composi‐ tions correspondings to Ba3BiB3O9 (A) and supposed BaBi8B2O16 (G) compounds have been

Super cooling technique constructed by our group allowed to expand the borders of glass

Determined glassforming area include compositions content: 80 – 95 mol% Bi2O3 in the bina‐ ry B2O3–Bi2O3 system; 43-70 mol% BaO in the binary BaO–B2O3 system, including BaB2O4 composition. Area of glass formation from both these areas moves to 55-95 mol% content

Traditional for borate systems a stable phase separation region was also observed for high

Our investigation of the ternary BaO-B2O3-Bi2O3 system have purposefully been directed on construction of the phase diagram through first of all glass forming diagram construction and revealing both new compounds and eutectic compositions. Constructed by us glass forming diagram (Fig.2) practically occupies the most part of the BaO-B2O3-Bi2O3 concentra‐ tion triangles. It has allowed to use synthesized glasses as initial compositions at phase dia‐ gram construction. It was basic difference of our methodology from technologies used by other authors. Phase equilibriums reached at isothermal sections construction do not allow to have a full picture of processes in cases of the solid state synthesized samples investiga‐ tions. Whereas at glass samples studies we determine not only characteristic points of glasses(Tg and Ts) by DTA, but also quantity of crystal phases, temperatures of their crystal‐ lization and then temperatures of their melting. It has allowed us to reveal new stoichiomet‐ ric compositions which have been lost by other research groups at isothermal sections construction by traditional methods. In some cases we also in parallel used samples ob‐ tained by solid state synthesis for comparison with their glassy analogues or in those cases,

formation in studied system under high melts cooling rates equal to (103

B2O3 content compositions contents more than 84 -87 mol%B2O3 (Fig.2-4).

K/s has led to glass formation area expansion


) K/s (Fig.2-3).

10 K/s, which was reported earlier [34, 35].

obtained by this way.

134 Advances in Ferroelectrics

Increasing of melts cooling speed up to ~102

compositions in the binary BaO–Bi2O3 system (Fig.2-3).

*3.1.2. Phase diagram of the BaO-Bi2O3-B2O3 system*

when their obtaining in the glassy form was impossible.

Experimental data concerning phase diagrams of binary systems Bi2O3-B2O3, BaO-B2O3, BaO-Bi2O3 and pseudo-binary sections in the BaO-B2O3-Bi2O3 system have allowed us to estimate fields of primary crystallization of co-existing phases and divided all concentration triangle into elementary triangles, i.e. to make triangulation.

The triangulation scheme in the BaO-B2O3-Bi2O3 system is presented on Fig. 3. By means of a triangulation all concentration triangle is divided into following elementary trian‐ gles: BiBO3-F-C, BiBO3-F-Bi4B2O9, BiBO3-C-Bi3B5O12, F-E-C, C-BaB2O4-E, C-Bi3B5O12-D, Bi4B2O9-F-Bi2O3, E-Bi2O3-F, C-BaB2O4-Ba2B10O17, Ba2B10O17-D-C, Ba2B10O17-D-B2O3, Bi3B5O12-D-Bi2B8O15, D-Bi2B8O15-B2O3.

### *3.1.2.1. Phase diagram of the binary Bi2O3–B2O3 system*

First of all we have attempt to finished phase diagram construction in area of compositions around of BiBO3 compound. Compositions containing 45–65 mol% B2O3 in the Bi2O3–B2O3 system were tested to determine the melting point of BiBO3 and to determine the eutectic composition between BiBO3 and Bi3B5O12. The compositions used to correct the B2O3–Bi2O3 phase diagram were prepared by solid state synthesis at 520°C, with steps of 0.5–1.0 mol% В2О3 over the interval 45–55 mol% В2О3. As a result, the eutectic composition, 48.5Bi2O3•51.5B2O3 (mol%), between BiBO3 and Bi3B5O12, was determined, and its melting point was measured by DTA as 665±5°С (Fig. 4). It was also found that BiBO3 melts con‐ gruently at 685±5°С.

**Figure 4.** Corrected phase diagram of the Bi2O3-B2O3 system in the interval 30–65 mol% B2O3.

### *3.1.2.2. Phase diagram of the pseudo-binary BiBO3–BaB2O4 system*

BaBi2B4O10 is a congruently melting compound, with a melting point of 730°C, and it occu‐ pies the central area of the BiBO3–BaB2O4 pseudo-binary system (Fig. 5). This system forms two simple pseudo-binary eutectics, E1 at 15 mol% BaB2O4, with a melting point of 620°C, and E2 at 60 mol% BaB2O4, with a melting point of 718°C.

**Figure 6.** Phase diagram of the pseudo-binary system Bi4B2O9-BaBi2B2O7.

BaBi10B6O25 were determined and are given in Tables 1 and 2.

19 2.254 22 113 38 1.373 11 821

crystallization (640°C, 20 h).

**Table 1.** X-ray characteristics of the new ternary compound BaBi2B2O7, synthesized at the same glass composition

Two new crystalline ternary compounds, BaBi2B2O7 and BaBi10B6O25, were revealed by crystal‐ lisation at the same glass composition. Both compounds, BaBi2B2O7 and BaBi10B6O25, melt congruently at 725±5°С and 690±5°С, respectively. The X-ray characteristics of BaBi2B2O7 and

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**No. dexp I/I0 hkl No. dexp I/I0 hkl No. dexp I/I0 hkl** 6.23 9 101 20 2.15 25 140 39 1.349 7 125 5.02 9 111 21 2.12 5 232 40 1.33 3 543 4.80 5 201 22 2.06 24 123 41 1.28 8 035 4.29 5 020 23 2.01 6 523 42 1.24 6 560 4.11 14 120 24 1.977 25 241 43 1.217 5 263 3.88 6 300 25 1.84 15 142 44 1.21 9 843 3.67 4 021 26 1.826 7 133 45 1.206 13 271 3.59 26 301 27 1.786 6 004 46 1.19 14 145 3.56 50 121 28 1.729 52 114 47 1.173 12 245 3.52 23 220 29 1.679 23 250 48 1.17 4 1010 3.19 100 112 30 1.636 34 251 49 1.14 6 126 3.12 8 221 31 1.63 5 532 50 1.11 6 662 3.05 9 202 32 1.57 4 052 51 1.10 6 326 2.91 43 030 33 1.556 10 243 52 1.09 4 180 2.696 90 122 34 1.522 8 034 53 1.042 6 146 2.51 12 222 35 1.488 23 632 54 1.021 9 065 2.376 21 003 36 1.458 6 060 55 1.018 9 943 2.31 5 421 37 1.428 10 811 56 1.01 5 274

**Figure 5.** Phase diagram of the pseudo-binary system BiBO3-BaB2O4

### *3.1.2.3. Phase diagram of the pseudo-binary Bi4B2O9–BaBi2B2O7 system*

The introduction of 12 mol% BaBi2B2O7 in the pseudo-binary system Bi4B2O9-BaBi2B2O7 re‐ duced the melting point of initial Bi4B2O9, and resulted in the formation of a simple pseudobinary eutectic, E3, with melting point 605°C (Fig. 6). A maximum of the liquidus with melting point of 690°C is seen at 33.33 mol% BaBi2B2O7, which indicates the formation of the new congruently melting ternary compound BaBi10B6O25 (11.11BaO•55.55Bi2O3•33.33B2O3). Fur‐ ther increase of the BaBi2B2O7 content (49 mol%) leads to a second pseudo-binary eutectic, E4, with melting point 660°C. Increasing of the liquidus temperature is observed in the post eutectic region of composition, with a maximum at 725°C. It corresponds to the formation of the new congruently melting ternary compound BaBi2B2O7 (Fig. 6).

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**Figure 6.** Phase diagram of the pseudo-binary system Bi4B2O9-BaBi2B2O7.

**Figure 4.** Corrected phase diagram of the Bi2O3-B2O3 system in the interval 30–65 mol% B2O3.

BaBi2B4O10 is a congruently melting compound, with a melting point of 730°C, and it occu‐ pies the central area of the BiBO3–BaB2O4 pseudo-binary system (Fig. 5). This system forms two simple pseudo-binary eutectics, E1 at 15 mol% BaB2O4, with a melting point of 620°C, and

The introduction of 12 mol% BaBi2B2O7 in the pseudo-binary system Bi4B2O9-BaBi2B2O7 re‐ duced the melting point of initial Bi4B2O9, and resulted in the formation of a simple pseudobinary eutectic, E3, with melting point 605°C (Fig. 6). A maximum of the liquidus with melting point of 690°C is seen at 33.33 mol% BaBi2B2O7, which indicates the formation of the new congruently melting ternary compound BaBi10B6O25 (11.11BaO•55.55Bi2O3•33.33B2O3). Fur‐ ther increase of the BaBi2B2O7 content (49 mol%) leads to a second pseudo-binary eutectic, E4, with melting point 660°C. Increasing of the liquidus temperature is observed in the post eutectic region of composition, with a maximum at 725°C. It corresponds to the formation of the new

*3.1.2.2. Phase diagram of the pseudo-binary BiBO3–BaB2O4 system*

E2 at 60 mol% BaB2O4, with a melting point of 718°C.

136 Advances in Ferroelectrics

**Figure 5.** Phase diagram of the pseudo-binary system BiBO3-BaB2O4

*3.1.2.3. Phase diagram of the pseudo-binary Bi4B2O9–BaBi2B2O7 system*

congruently melting ternary compound BaBi2B2O7 (Fig. 6).

Two new crystalline ternary compounds, BaBi2B2O7 and BaBi10B6O25, were revealed by crystal‐ lisation at the same glass composition. Both compounds, BaBi2B2O7 and BaBi10B6O25, melt congruently at 725±5°С and 690±5°С, respectively. The X-ray characteristics of BaBi2B2O7 and BaBi10B6O25 were determined and are given in Tables 1 and 2.


**Table 1.** X-ray characteristics of the new ternary compound BaBi2B2O7, synthesized at the same glass composition crystallization (640°C, 20 h).

Single crystals of BaBi10B6O25 were obtained by cooling of a melt with the stoichiometric composition. Glass powder of composition 11.11BaO•55.55Bi2O3•33.33B2O3 (mol%) was heated in a quartz glass ampoule up to 750°C at a rate 10 K/min. After 2 h exposition at high temperature, the melt was cooled at a rate 0.5 K/h. Single crystals with sizes up to 1.66×0.38×0.19 mm3 were grown.

mation of the new ternary compound BaBi10B6O25. There is a simple pseudo-binary eutectic E5 between these two compounds at 54 mol%BaBi10B6O25, with a melting point of 595°C.

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**Figure 7.** Phase diagram of the pseudo-binary system BiBO3- BaBi10B6O25.

pounds at 28 mol % BaBi2B4O10, with a melting point of 660°C.

**Figure 8.** Phase diagrams of the pseudo-binary system BaBi10B6O25 -BaBi2B4O10.

*3.1.2.6. Phase diagram of the pseudo-binary BaBi2B4O10 – 50BaO•50Bi2O3 section*

This pseudo-binary section consists of two ternary compounds BaBi2B4O10, BaBi2B2O7 and two eutectics Е7, Е8 dividing fields primary crystallisations these compounds. Initial composition is BaBi2B4O10 (Fig.9). The introduction of 20 mol% 50%BaO•50%Bi2O3 in the pseudo-binary

*3.1.2.5. Phase diagram of the pseudo-binary BaBi10B6O25 - BaBi2B4O10 system*

The BaBi10B6O25–BaBi2B4O10 system confirms the presence of the new congruently melting ternary compound BaBi10B6O25, with a melting point of 690°C (Fig.8). BaBi2B4O10 melts con‐ gruently at 730°C. There is a simple pseudo-binary eutectic E6 between these two com‐


**Table 2.** X-ray characteristics of the new ternary BaBi10B6O25 single crystals.

The x-ray powder diffraction patterns of BaBi2B2O7 and BaBi10B6O25 could be indexed on an orthorhombic cell with lattice parameters as follows:


*3.1.2.4. Phase diagram of the pseudo-binary BiBO3–BaBi10B6O25 system*

BiBO3– BaBi10B6O25 is a very important system (Fig.7). Initial BiBO3 has a melting point of 685°C. The second maximum in the liquidus curve (Fig.7) of 690°C is connected with the for‐

mation of the new ternary compound BaBi10B6O25. There is a simple pseudo-binary eutectic E5 between these two compounds at 54 mol%BaBi10B6O25, with a melting point of 595°C.

**Figure 7.** Phase diagram of the pseudo-binary system BiBO3- BaBi10B6O25.

Single crystals of BaBi10B6O25 were obtained by cooling of a melt with the stoichiometric composition. Glass powder of composition 11.11BaO•55.55Bi2O3•33.33B2O3 (mol%) was heated in a quartz glass ampoule up to 750°C at a rate 10 K/min. After 2 h exposition at high temperature, the melt was cooled at a rate 0.5 K/h. Single crystals with sizes up to

**No. dexp I/I0 hkl No. dexp I/I0 hkl No. dexp I/I0 hkl** 9.21 3.0 012 23 2.91 31 0110 45 2.01 5.2 321 6.26 3.0 101 24 2.8 6 221 46 1.98 23.7 323 6.02 3.0 005 25 2.7 75.9 206 47 1.92 3.0 1214 5.01 7.3 006 26 2.64 3.4 045 48 1.88 3.0 0314 4.89 10.8 104 27 2.57 3.0 2.07 49 1.86 3.0 162 4.63 6.5 024 28 2.53 4.3 046 50 1.84 15.1 254 4.19 4.3 025 29 2.52 5.2 144 51 1.83 3.0 1412 4.18 4.3 122 30 2.49 6.9 230 52 1.82 3.0 164 4.11 10 115 31 2.47 9.5 1210 53 1.81 3.0 255 3.92 6.0 030 32 2.45 4.7 232 54 1.79 3.4 165 3.80 3.4 032 33 2.38 16 0310 55 1.77 3.0 328 3.65 10.8 033 34 2.35 4.3 050 56 1.75 3.4 1413 3.56 39.7 107 35 2.34 5.2 051 57 1.73 55.2 341 3.51 9.5 125 36 2.33 3.9 1012 58 1.71 4.3 343 3.41 3.4 117 37 2.31 3.4 1212 59 1.69 3.0 069 3.38 7.3 130 38 2.25 17.2 228 60 1.68 16.4 070 3.33 8.6 131 39 2.21 7.8 150 61 1.65 3.0 259 3.27 9.1 132 40 2.19 3.0 055 62 1.64 31.0 074 3.18 100 133 41 2.15 21.2 237 63 1.61 2.9 400 3.07 14.7 203 42 2.09 3.4 312 64 1.6 6.0 402 3.04 9.2 212 43 2.06 21.6 304 65 1.55 6.5 421 2.94 14.2 040 44 2.03 3.0 314 66 1.52 9.5 177

The x-ray powder diffraction patterns of BaBi2B2O7 and BaBi10B6O25 could be indexed on an

BiBO3– BaBi10B6O25 is a very important system (Fig.7). Initial BiBO3 has a melting point of 685°C. The second maximum in the liquidus curve (Fig.7) of 690°C is connected with the for‐

, *Z*=4;

, *Z*=8.

**•** for BaBi2B2O7 *a*=11.818 Å, *b*=8.753 Å, *c*=7.146 Å, cell volume *V*=739.203 Å3

*3.1.2.4. Phase diagram of the pseudo-binary BiBO3–BaBi10B6O25 system*

**•** for BaBi10B6O25 *a*=6.434 Å, *b*=11.763 Å, *c*=29.998 Å, cell volume *V*=2270.34 Å3

1.66×0.38×0.19 mm3

138 Advances in Ferroelectrics

were grown.

**Table 2.** X-ray characteristics of the new ternary BaBi10B6O25 single crystals.

orthorhombic cell with lattice parameters as follows:

### *3.1.2.5. Phase diagram of the pseudo-binary BaBi10B6O25 - BaBi2B4O10 system*

The BaBi10B6O25–BaBi2B4O10 system confirms the presence of the new congruently melting ternary compound BaBi10B6O25, with a melting point of 690°C (Fig.8). BaBi2B4O10 melts con‐ gruently at 730°C. There is a simple pseudo-binary eutectic E6 between these two com‐ pounds at 28 mol % BaBi2B4O10, with a melting point of 660°C.

**Figure 8.** Phase diagrams of the pseudo-binary system BaBi10B6O25 -BaBi2B4O10.

### *3.1.2.6. Phase diagram of the pseudo-binary BaBi2B4O10 – 50BaO•50Bi2O3 section*

This pseudo-binary section consists of two ternary compounds BaBi2B4O10, BaBi2B2O7 and two eutectics Е7, Е8 dividing fields primary crystallisations these compounds. Initial composition is BaBi2B4O10 (Fig.9). The introduction of 20 mol% 50%BaO•50%Bi2O3 in the pseudo-binary system BaBi2B4O10 – 50BaO•50Bi2O3 reduced the melting point of initial BaBi2B4O10, and result‐ ed in the formation of a simple pseudo-binary eutectic, E7, with melting point 680°C (Fig.9). A maximum of the liquidus with melting point of 725°C is seen at 33.33 mol% of 50%BaO•50%Bi2O3, which indicates the formation of the new congruently melting ternary compound BaBi2B2O7. Further increase of the 50%BaO•50%Bi2O3 content (52.5 mol%) leads to a second pseudobinary eutectic, E8, with melting point 700°C. Increasing of the liquidus temperature is ob‐ served in the post eutectic region of composition (more than 52.5mol% of 50BaO•50Bi2O3). Un identified phase is in the post eutectic (E8) region of composition (Fig.9).

**Figure 10.** Phase diagram of the BaB2O4 – Bi2O3 section.

melting point 680°C (Fig.11).

*3.1.2.8. Phase diagram of the pseudo-binary Bi3B5O12 – BaBi2B4O10 system*

**Figure 11.** Phase diagram of the pseudo-binary system Bi3B5O12 – BaBi2B4O10

Bi2B4O10, with a melting point of 695°C.

*3.1.2.9. Phase diagram of the pseudo-binary BaBi2B4O10 – BaBiB11O19 system*

Initial BaBi2B4O10 has a melting point of 730°C. The second maximum in the liquidus curve (Fig.12) of 807°C is connected with the formation of the ternary compound BaBiB11O19. There is a simple pseudo-binary eutectic, E12, between these two compounds at 65.5 mol% Ba‐

It is very simple system with pseudo-binary eutectic Е<sup>11</sup> between two congruent melted Bi3B5O12 and BaBi2B4O10 compounds. Eutectic Е11 content 38 mol% of BaBi2B4O10 and has

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**Figure 9.** Phase diagram of the pseudo-binary section BaBi2B4O10 – 50BaO•50Bi2O3.

### *3.1.2.7. Phase diagram of the pseudo-binary BaB2O4–Bi2O3 section*

BaB2O4 - Bi2O3 section consist of two pseudo-binary BaB2O4- BaBi2B2O7, BaBi2B2O7- Bi2O3 sys‐ tems (Fig.10). There are two eutectics: Е9 between BaB2O4 and BaBi2B2O7, Е10 between Ba‐ Bi2B2O7 and BaBi8B2O16, and peritectic point P1 between BaBi8B2O16 and Bi2O3 (Fig.10). The introduction of 26 mol% Bi2O3 in the pseudo-binary system BaB2O4 – Bi2O3 sharp reduced the melting point of initial BaB2O4 on 4450 C, and resulted in the formation of a simple eutec‐ tic, E9, with melting point 685°C (Fig.10). A maximum of the liquidus with melting point of 725°C is seen at 33 33 mol% of Bi2O3, which indicates the formation of the new congruently melting ternary compound BaBi2B2O7.

Further increase of the Bi2O3 content (42 mol%) leads to a second eutectic, E10, formation with melting point 690°C. Increasing of the liquidus temperature is observed in the post eu‐ tectic region of composition (more than 42mol% of Bi2O3) and formation of new incongruent melted at 725 °C BaBi8B2O16 ternary compound (Fig.10). It is very difficult determined of Ba‐ Bi8B2O16 X-ray characteristics, because they very closed to Bi2O3 characteristics.

Constructed by us this section's diagram essentially differs from that constructed by Russi‐ an researches Egorisheva & Kargin [30] because they could not find out two new com‐ pounds revealed by us: congruent melted at 725°C BaBi2B2O7 and incongruent melted at 725°C BaBi8B2O16(Fig.10). At repeated, even more detailed studies they also could not find out these compounds [31].

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**Figure 10.** Phase diagram of the BaB2O4 – Bi2O3 section.

system BaBi2B4O10 – 50BaO•50Bi2O3 reduced the melting point of initial BaBi2B4O10, and result‐ ed in the formation of a simple pseudo-binary eutectic, E7, with melting point 680°C (Fig.9). A maximum of the liquidus with melting point of 725°C is seen at 33.33 mol% of 50%BaO•50%Bi2O3, which indicates the formation of the new congruently melting ternary compound BaBi2B2O7. Further increase of the 50%BaO•50%Bi2O3 content (52.5 mol%) leads to a second pseudobinary eutectic, E8, with melting point 700°C. Increasing of the liquidus temperature is ob‐ served in the post eutectic region of composition (more than 52.5mol% of 50BaO•50Bi2O3). Un

BaB2O4 - Bi2O3 section consist of two pseudo-binary BaB2O4- BaBi2B2O7, BaBi2B2O7- Bi2O3 sys‐ tems (Fig.10). There are two eutectics: Е9 between BaB2O4 and BaBi2B2O7, Е10 between Ba‐ Bi2B2O7 and BaBi8B2O16, and peritectic point P1 between BaBi8B2O16 and Bi2O3 (Fig.10). The introduction of 26 mol% Bi2O3 in the pseudo-binary system BaB2O4 – Bi2O3 sharp reduced

tic, E9, with melting point 685°C (Fig.10). A maximum of the liquidus with melting point of 725°C is seen at 33 33 mol% of Bi2O3, which indicates the formation of the new congruently

Further increase of the Bi2O3 content (42 mol%) leads to a second eutectic, E10, formation with melting point 690°C. Increasing of the liquidus temperature is observed in the post eu‐ tectic region of composition (more than 42mol% of Bi2O3) and formation of new incongruent melted at 725 °C BaBi8B2O16 ternary compound (Fig.10). It is very difficult determined of Ba‐

Constructed by us this section's diagram essentially differs from that constructed by Russi‐ an researches Egorisheva & Kargin [30] because they could not find out two new com‐ pounds revealed by us: congruent melted at 725°C BaBi2B2O7 and incongruent melted at 725°C BaBi8B2O16(Fig.10). At repeated, even more detailed studies they also could not find

Bi8B2O16 X-ray characteristics, because they very closed to Bi2O3 characteristics.

C, and resulted in the formation of a simple eutec‐

identified phase is in the post eutectic (E8) region of composition (Fig.9).

**Figure 9.** Phase diagram of the pseudo-binary section BaBi2B4O10 – 50BaO•50Bi2O3.

*3.1.2.7. Phase diagram of the pseudo-binary BaB2O4–Bi2O3 section*

the melting point of initial BaB2O4 on 4450

melting ternary compound BaBi2B2O7.

140 Advances in Ferroelectrics

out these compounds [31].

### *3.1.2.8. Phase diagram of the pseudo-binary Bi3B5O12 – BaBi2B4O10 system*

It is very simple system with pseudo-binary eutectic Е<sup>11</sup> between two congruent melted Bi3B5O12 and BaBi2B4O10 compounds. Eutectic Е11 content 38 mol% of BaBi2B4O10 and has melting point 680°C (Fig.11).

**Figure 11.** Phase diagram of the pseudo-binary system Bi3B5O12 – BaBi2B4O10

### *3.1.2.9. Phase diagram of the pseudo-binary BaBi2B4O10 – BaBiB11O19 system*

Initial BaBi2B4O10 has a melting point of 730°C. The second maximum in the liquidus curve (Fig.12) of 807°C is connected with the formation of the ternary compound BaBiB11O19. There is a simple pseudo-binary eutectic, E12, between these two compounds at 65.5 mol% Ba‐ Bi2B4O10, with a melting point of 695°C.

**Figure 12.** Phase diagram of the pseudo-binary system BaBi2B4O10 – BaBiB11O19.

### *3.1.2.10. Phase diagram of the pseudo-binary Bi3B5O12 – BaBiB11O19 system*

Pseudo-binary system Bi3B5O12 -BaBiB11O19 has simple eutectic Е<sup>13</sup> formed between two con‐ gruent melted BaBiB11O19 and Bi3B5O12 compounds. According to DTA eutectic Е13 has melt‐ ing point 705°C and content 28 mol% BaBiB11O19 (Fig.13).

**Figure 14.** Phase diagram of the pseudo-binary system BaBiB11O19 – Ba2B10O17.

**Figure 15.** Phase diagram of the pseudo-binary system BaBi2B4O10 – Ba2B10O17.

BaB4O7 and has melting point 715°C (Fig. 16).

*3.1.2.13. Phase diagram of the pseudo-binary BaBi2B4O10 – BaB4O7 system*

melting point 710°C (Fig. 15).

*3.1.2.12. Phase diagram of the pseudo-binary BaBi2B4O10 – Ba2B10O17 system*

It is very simple system with eutectic Е15 formed between two congruent melted Ba2B10O17 and BaBi2B4O10 compounds. Pseudo-binary eutectic Е<sup>15</sup> content 24 mol% of Ba2B10O17 and has

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The same picture is observe for BaBi2B4O10 – BaB4O7 system: simple eutectic Е16 is formed be‐ tween two congruent melted compounds. Pseudo-binary eutectic Е<sup>16</sup> content 24 mol% of

**Figure 13.** Phase diagram of the pseudo-binary system Bi3B5O12 - BaBiB11O19.

### *3.1.2.11. Phase diagram of the pseudo-binary BaBiB11O19 – Ba2B10O17 system*

Pseudo-binary system BaBiB11O19 – Ba2B10O17 has simple eutectic Е<sup>14</sup> formed between two congruent melted compounds Ba2B10O17 and BaBiB11O19. According to DTA eutectic Е<sup>14</sup> has melting point 780°C and content 26 mol% Ba2B10O17(Fig.14).

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**Figure 14.** Phase diagram of the pseudo-binary system BaBiB11O19 – Ba2B10O17.

**Figure 12.** Phase diagram of the pseudo-binary system BaBi2B4O10 – BaBiB11O19.

142 Advances in Ferroelectrics

ing point 705°C and content 28 mol% BaBiB11O19 (Fig.13).

**Figure 13.** Phase diagram of the pseudo-binary system Bi3B5O12 - BaBiB11O19.

melting point 780°C and content 26 mol% Ba2B10O17(Fig.14).

*3.1.2.11. Phase diagram of the pseudo-binary BaBiB11O19 – Ba2B10O17 system*

Pseudo-binary system BaBiB11O19 – Ba2B10O17 has simple eutectic Е<sup>14</sup> formed between two congruent melted compounds Ba2B10O17 and BaBiB11O19. According to DTA eutectic Е<sup>14</sup> has

*3.1.2.10. Phase diagram of the pseudo-binary Bi3B5O12 – BaBiB11O19 system*

Pseudo-binary system Bi3B5O12 -BaBiB11O19 has simple eutectic Е<sup>13</sup> formed between two con‐ gruent melted BaBiB11O19 and Bi3B5O12 compounds. According to DTA eutectic Е13 has melt‐

### *3.1.2.12. Phase diagram of the pseudo-binary BaBi2B4O10 – Ba2B10O17 system*

It is very simple system with eutectic Е15 formed between two congruent melted Ba2B10O17 and BaBi2B4O10 compounds. Pseudo-binary eutectic Е<sup>15</sup> content 24 mol% of Ba2B10O17 and has melting point 710°C (Fig. 15).

**Figure 15.** Phase diagram of the pseudo-binary system BaBi2B4O10 – Ba2B10O17.

#### *3.1.2.13. Phase diagram of the pseudo-binary BaBi2B4O10 – BaB4O7 system*

The same picture is observe for BaBi2B4O10 – BaB4O7 system: simple eutectic Е16 is formed be‐ tween two congruent melted compounds. Pseudo-binary eutectic Е<sup>16</sup> content 24 mol% of BaB4O7 and has melting point 715°C (Fig. 16).

Fields of binary bismuth and barium borates as well as all ternary barium bismuth borates compounds crystallizations have been determined and outlined and sixteen ternary eutectic points E1-E16 have been revealed as result of phase diagram construction (Fig. 18, table 3). The phase diagram evidently represents interaction of binary and ternary compounds tak‐ ing place in the pseudo-ternary systems. The ternary eutectic E1 with m.p 590°C has been determined among BiBO3, F and Bi4B2O9 compounds; ternary eutectic E2 with m.p. 585°C has been formed among BiBO3, F and C compounds; ternary eutectic E3 with m.p. 640°C has been formed among F, E and C compounds; ternary eutectic E4 with m.p. 622°C has been formed among C, BaB2O4 and E compounds; ternary eutectic E5 with m.p. 610°C has been formed among BiBO3, C and Bi3B5O12 compounds; ternary eutectic E6 with m.p. 675°C has been formed among C, Bi3B5O12 and D compounds; ternary eutectic E7 with m.p. 680°C has been formed among Bi3B5O12, D and BiB3O6 compounds; ternary eutectic E8 with m.p. 675°C has been formed among BiB3O6, D and Bi2B8O15 compounds; ternary eutectic E9 with m.p. 680°C has been formed among Bi2B8O15, D and B2O3 compounds; ternary eutectic E10 with m.p. 730°C has been formed among BaB8O13, D and B2O3 compounds; ternary eutectic E11 with m.p. 750 °C has been formed among Ba2B10O17-D- BaB8O13 compounds; ternary eutectic E12 with m.p. 680°C has been formed among C, Ba2B10O17 and D compounds; ternary eutectic E13 with m.p. 690°C has been formed among Ba2B10O17, C and BaB4O7 compounds; ternary eutectic E14 with m.p. 700°C has been formed among BaB4O7, C and BaB2O4 compounds; ter‐ nary eutectic E15 with m.p. 645 °C has been formed among G, F and C compounds; ternary eutectic E16 with m.p. 615°C has been formed among Bi24B2O39, F and Bi4B2O9 compounds

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(Fig18, Table 3).

**Figure 18.** Phase diagram of the BaO-Bi2O3-B2O3 system

**Figure 16.** Phase diagram of the pseudo-binary system BaBi2B4O10 – BaB4O7.

### *3.1.2.14. Phase diagram of the pseudo-binary BaBiB11O19 – BaB8O13 system*

It is simple system with pseudo-binary eutectic Е17 between two congruent melted Ba‐ BiB11O19 and BaB8O13 compounds. Eutectic Е17 content 38 mol% of BaB8O13 and has melting point 770°C (Fig. 17).

**Figure 17.** Phase diagram of the pseudo-binary system BaBiB11O19 – BaB8O13.

### *3.1.2.15. Phase diagram of the BaO-Bi2O3-B2O3 ternary system*

As result of huge work under project the phase diagram in the ternary BaO-B2O3-Bi2O3 sys‐ tem has been constructed for the first time and presented on Fig.18. Three new compounds BaBi2B2O7, BaBi10B6O25 and BaBi8B2O16 have been revealed and characterized.

Fields of binary bismuth and barium borates as well as all ternary barium bismuth borates compounds crystallizations have been determined and outlined and sixteen ternary eutectic points E1-E16 have been revealed as result of phase diagram construction (Fig. 18, table 3). The phase diagram evidently represents interaction of binary and ternary compounds tak‐ ing place in the pseudo-ternary systems. The ternary eutectic E1 with m.p 590°C has been determined among BiBO3, F and Bi4B2O9 compounds; ternary eutectic E2 with m.p. 585°C has been formed among BiBO3, F and C compounds; ternary eutectic E3 with m.p. 640°C has been formed among F, E and C compounds; ternary eutectic E4 with m.p. 622°C has been formed among C, BaB2O4 and E compounds; ternary eutectic E5 with m.p. 610°C has been formed among BiBO3, C and Bi3B5O12 compounds; ternary eutectic E6 with m.p. 675°C has been formed among C, Bi3B5O12 and D compounds; ternary eutectic E7 with m.p. 680°C has been formed among Bi3B5O12, D and BiB3O6 compounds; ternary eutectic E8 with m.p. 675°C has been formed among BiB3O6, D and Bi2B8O15 compounds; ternary eutectic E9 with m.p. 680°C has been formed among Bi2B8O15, D and B2O3 compounds; ternary eutectic E10 with m.p. 730°C has been formed among BaB8O13, D and B2O3 compounds; ternary eutectic E11 with m.p. 750 °C has been formed among Ba2B10O17-D- BaB8O13 compounds; ternary eutectic E12 with m.p. 680°C has been formed among C, Ba2B10O17 and D compounds; ternary eutectic E13 with m.p. 690°C has been formed among Ba2B10O17, C and BaB4O7 compounds; ternary eutectic E14 with m.p. 700°C has been formed among BaB4O7, C and BaB2O4 compounds; ter‐ nary eutectic E15 with m.p. 645 °C has been formed among G, F and C compounds; ternary eutectic E16 with m.p. 615°C has been formed among Bi24B2O39, F and Bi4B2O9 compounds (Fig18, Table 3).

**Figure 18.** Phase diagram of the BaO-Bi2O3-B2O3 system

**Figure 16.** Phase diagram of the pseudo-binary system BaBi2B4O10 – BaB4O7.

**Figure 17.** Phase diagram of the pseudo-binary system BaBiB11O19 – BaB8O13.

*3.1.2.15. Phase diagram of the BaO-Bi2O3-B2O3 ternary system*

point 770°C (Fig. 17).

144 Advances in Ferroelectrics

*3.1.2.14. Phase diagram of the pseudo-binary BaBiB11O19 – BaB8O13 system*

It is simple system with pseudo-binary eutectic Е17 between two congruent melted Ba‐ BiB11O19 and BaB8O13 compounds. Eutectic Е17 content 38 mol% of BaB8O13 and has melting

As result of huge work under project the phase diagram in the ternary BaO-B2O3-Bi2O3 sys‐ tem has been constructed for the first time and presented on Fig.18. Three new compounds

BaBi2B2O7, BaBi10B6O25 and BaBi8B2O16 have been revealed and characterized.


**Figure 19.** DTA curves (heating rate 10K/min) of glasses corresponding to ternary compounds in the BaO-Bi2O3-B2O3

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1 BaBiB11O19 (glass) 72 498 535 485 640; 770 807 807

2 BaBi2B4O10 (glass) 96 445 475 450 545; 640 730 730

3 BaBi2B2O7 (glass) 108 415 455 420 570; 645 725 725

4 BaBi10B6O25 (glass) 99 350 380 340 460; 620 690 690

5 BaBiBO4 (glass) 120 400 450 390 450 680 760

6 Ba3BiB3O9 (glass) 127 460 490 460 540; 670 827 865

Two exothermic effects were observed on DTA curve of 14.28BaO•7.14Bi2O3•78.57B2O3 (mol

%) glass composition: first weak effect at 640°C and second strong effect at 770°C(Fig.19,

**Table 4.** Chemical compositions, DTA ( glass transition -Tg, crystallization peak -Tcr, melting -Tm, liquidus- TL) and dilatometric characteristics (glass transition temperature -Tg, softening point - TS, thermal expansion coefficient -TEC)

**Dilatometric characteristics DTA characteristics** TEC(α20-300)∙107K-1 Tg,ºC Ts ,ºC Tg, ºC Tcr, ºC Tm, °С TL, °С

system: 1-BaBiB11O19, 2-BaBi2B4O10, 3-BaBi2B2O7, 4-BaBi10B6O25,5-BaBiBO4, 6-Ba3BiB3O9.

**## Glass compositions, correspoding to stoichometric compounds**

BaBiB11O19 (615°C 24h) 49.8

BaBi2B4O10 (640°C 24h) 77.9

BaBi2B2O7 (640°C 24h) 96

BaBi10B6O25(590°C 24h) 97

BaBiBO4 (570°C 24h) 110.8

Ba3BiB3O9 (690°C 24h) 109.8

of BaO-Bi2O3-B2O3system glasses and crystallaised glasses.

**Table 3.** The temperature and compositions for ternary eutectic points in the BaO-Bi2O3-B2O3 system

### **3.2. DTA and X-ray characterisation of ternary stoichiometric glasses and glass ceramics from the BaO-Bi2O3-B2O3 system**

The glasses corresponding to known sixth stoichiometric compounds in the BaO-Bi2O3- B2O3 system examined in the present study and following glass compositions (mol%) have been melted: 14.28BaO•7.14Bi2O3•78.57B2O3 (BaBiB11O19), 25BaO•25Bi2O3•50B2O3(Ba‐ Bi2B4O10), 33.33BaO•33.33Bi2O3•33.33B2O3(BaBi2B2O7), 11.11BaO•55.55Bi2O3•33.33B2O3 (Ba‐ Bi10B6O25), 50BaO• 25Bi2O3• 25B2O3 (BaBiBO4) and 60BaO•10Bi2O3•30B2O3 (Ba3BiB3O9). These glasses DTA curves are shown in Fig. 19, giving the peaks due to the glass transition, crystallization, melting, and liquidus temperatures. The glass characteristics points Tg (glass transition), Ts(glass softening), Tc (peak of exothermal effects connected with crystalline phases crystallizations) and Tm (minimum of endothermic effects associated with these phas‐ es melting) observed on DTA curves (Fig. 19, curves 1-6) of all tested powder samples summarized on table 4.

Phase Diagram of The Ternary BaO-Bi2O3-B2O3 System: New Compounds and Glass Ceramics Characterisation http://dx.doi.org/10.5772/52405 147

**Point Composition, mol% Tm, °C BaO B2O3 Bi2O3** E1 4.5 40.5 55 590 E2 7.3 45.1 47.6 585 E3 15 38 47 640 E4 32.4 41.3 26.3 622 E5 5.4 52 42.6 610 E6 9 63 28 675 E7 3 71 26 680 E8 3.5 78 18.5 675 E9 3.2 81.8 15 680 E10 15 81.2 3.8 730 E11 18.9 77.5 3.6 750 E12 22.4 63.2 14.4 680 E13 26.6 54.7 18.7 690 E14 28.8 52 19.2 700 E15 20 32 48 645 E16 4.5 30 65.5 615

**Table 3.** The temperature and compositions for ternary eutectic points in the BaO-Bi2O3-B2O3 system

**from the BaO-Bi2O3-B2O3 system**

146 Advances in Ferroelectrics

summarized on table 4.

**3.2. DTA and X-ray characterisation of ternary stoichiometric glasses and glass ceramics**

The glasses corresponding to known sixth stoichiometric compounds in the BaO-Bi2O3- B2O3 system examined in the present study and following glass compositions (mol%) have been melted: 14.28BaO•7.14Bi2O3•78.57B2O3 (BaBiB11O19), 25BaO•25Bi2O3•50B2O3(Ba‐ Bi2B4O10), 33.33BaO•33.33Bi2O3•33.33B2O3(BaBi2B2O7), 11.11BaO•55.55Bi2O3•33.33B2O3 (Ba‐ Bi10B6O25), 50BaO• 25Bi2O3• 25B2O3 (BaBiBO4) and 60BaO•10Bi2O3•30B2O3 (Ba3BiB3O9). These glasses DTA curves are shown in Fig. 19, giving the peaks due to the glass transition, crystallization, melting, and liquidus temperatures. The glass characteristics points Tg (glass transition), Ts(glass softening), Tc (peak of exothermal effects connected with crystalline phases crystallizations) and Tm (minimum of endothermic effects associated with these phas‐ es melting) observed on DTA curves (Fig. 19, curves 1-6) of all tested powder samples

**Figure 19.** DTA curves (heating rate 10K/min) of glasses corresponding to ternary compounds in the BaO-Bi2O3-B2O3 system: 1-BaBiB11O19, 2-BaBi2B4O10, 3-BaBi2B2O7, 4-BaBi10B6O25,5-BaBiBO4, 6-Ba3BiB3O9.


**Table 4.** Chemical compositions, DTA ( glass transition -Tg, crystallization peak -Tcr, melting -Tm, liquidus- TL) and dilatometric characteristics (glass transition temperature -Tg, softening point - TS, thermal expansion coefficient -TEC) of BaO-Bi2O3-B2O3system glasses and crystallaised glasses.

Two exothermic effects were observed on DTA curve of 14.28BaO•7.14Bi2O3•78.57B2O3 (mol %) glass composition: first weak effect at 640°C and second strong effect at 770°C(Fig.19, curve1). The melting temperature (Tm) is equal to 807°C and corresponding to Egorisheva and Kargin's data [30]. X-ray patterns of this glass crystallization products show one Ba‐ BiB11O19 crystalline phase presence [30], which formed at powder samples crystallization in an temperature interval 640-770°C (Fig.20, curve1). It is possible to assume, that weak exo‐ thermic effect at 640°C apparently is connected with pre-crystallisation fluctuations taking place in glass matrix [50]. Diffuse character of second exothermic effect at 770°C testifies about dominating surface crystallisation of the given glass particles.

exothermic peaks temperature (640°C 24h) was indexed on an orthorhombic cell with fol‐ lowing lattice parameters: a=11.818Å, b=8.753 Å, c=7.146Å, cell volume V=739.203 Å, Z=4 (Fig. 21, curve2). XRD-patterns of products of same glass crystallization at 570°C 24h keeps all diffraction lines of its analogue obtained at 640°C 24h (Fig. 21, curve1). Difference is ob‐ served only in sharp increasing of intensity (I/Io) of [030] diffraction line from 4 to 43 at high temperature crystallization. That leads to reorientation of crystal structure, decreasing [030] diffraction line and accompanied with occurrence of the second exothermal effect on DTA

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**Figure 21.** XRD-patterns of the crystallized glasses corresponding to ternary BaBi2B2O7 composition:

X-ray powder diffraction patterns of BaBi10B6O25 single crystals was indexed on an ortho‐ rhombic cell with following lattice parameters: a=6.434Å, b=11.763 Å, c=29.998Å, cell volume V=2270.34 Å, Z=8 [34]. XRD-patterns of products of same compositions (11.11BaO• 55.55Bi2O3• 33.33B2O3 mol%) glass crystallization at both exothermal effects (420°C 24h and 620°C 24h) have all diffraction lines of the BaBi10B6O25 single crystals (Fig. 22, curves1-3). Naturally, BaBi10B6O25 single crystal has well generated planes and clear observed diffrac‐ tion lines on XRD-patterns in contrast to crystalline phases formed at same composition glasses crystallization. However, the most intensive diffraction line (I/Io=100) of single crys‐ tals is [133], whereas products of glass crystallizations have [203] strongest diffraction line and [133] diffraction line became 5-10 times less (Fig. 22, curves1-3). Now it is difficult to us only on the basis of XRD-patterns analysis of glass crystallizations products to assume the nature of the second exothermal effect at 620°C on DTA curve of BaBi10B6O25 of glass compo‐ sition (Fig.19, curve 4). Their XRD-patterns are identical each other and to single crystals,

1-570°C 24h, cooling in the muffle; 2-640°C 24h, casting in the cold water

curve (Fig. 19, curve 3).

One sharp exothermic effect at 640°C and sharp endothermic effect at 730°C were observed on DTA curve of BaBi2B4O10 glass composition (Fig. 19, curve2). The melting temperature (Tm) is equal to 730°C and corresponding to Egorisheva's data [30]. X-ray diffraction patterns of this glass crystallization products show one BaBi2B4O10 crystalline phase crystalliza‐ tion[30], which formed at glass powder samples crystallization at temperature 640°C (Fig. 20, curve2) and its melting. Tm is equal to 730°C and corresponding to Egorisheva's data [30]. Hardly visible pre-crystallisation fluctuation exothermal effect is observed also at 545°C (Fig. 19, curve2).

**Figure 20.** XRD-patterns of the crystallized glasses corresponding to ternary BaBiB11O19 (1- 760 °C 24h,cooling in the muffle) and BaBi2B4O10 (2- 640°C 24h,cooling in the muffle) compounds

On the DTA curves of stoichiometric BaBi2B2O7 and BaBi10B6O25 glass compositions observed two exothermal effects at 570 and 645°C for BaBi2B2O7 and at 460 and 620°C for BaBi10B6O25 (Fig. 19, curves 3,4). But both compositions have one endothermic effect of melting at 725 and 690°C respectively for BaBi2B2O7 and BaBi10B6O25 testifying to one formed crystalline phase melting (Fig. 19, curves 3,4). X-ray data of these samples confirmed monophase crys‐ tallizations in each samples (Fig. 21, curves 1,2; Fig.22, curves 1,2).

According to [34] the X-ray powder diffraction patterns of formed BaBi2B2O7 crystalline phase at stoichiometric glass composition (33.33BaO•33.33Bi2O3• 33.33B2O3 mol%) at second exothermic peaks temperature (640°C 24h) was indexed on an orthorhombic cell with fol‐ lowing lattice parameters: a=11.818Å, b=8.753 Å, c=7.146Å, cell volume V=739.203 Å, Z=4 (Fig. 21, curve2). XRD-patterns of products of same glass crystallization at 570°C 24h keeps all diffraction lines of its analogue obtained at 640°C 24h (Fig. 21, curve1). Difference is ob‐ served only in sharp increasing of intensity (I/Io) of [030] diffraction line from 4 to 43 at high temperature crystallization. That leads to reorientation of crystal structure, decreasing [030] diffraction line and accompanied with occurrence of the second exothermal effect on DTA curve (Fig. 19, curve 3).

curve1). The melting temperature (Tm) is equal to 807°C and corresponding to Egorisheva and Kargin's data [30]. X-ray patterns of this glass crystallization products show one Ba‐ BiB11O19 crystalline phase presence [30], which formed at powder samples crystallization in an temperature interval 640-770°C (Fig.20, curve1). It is possible to assume, that weak exo‐ thermic effect at 640°C apparently is connected with pre-crystallisation fluctuations taking place in glass matrix [50]. Diffuse character of second exothermic effect at 770°C testifies

One sharp exothermic effect at 640°C and sharp endothermic effect at 730°C were observed on DTA curve of BaBi2B4O10 glass composition (Fig. 19, curve2). The melting temperature (Tm) is equal to 730°C and corresponding to Egorisheva's data [30]. X-ray diffraction patterns of this glass crystallization products show one BaBi2B4O10 crystalline phase crystalliza‐ tion[30], which formed at glass powder samples crystallization at temperature 640°C (Fig. 20, curve2) and its melting. Tm is equal to 730°C and corresponding to Egorisheva's data [30]. Hardly visible pre-crystallisation fluctuation exothermal effect is observed also at 545°C

**Figure 20.** XRD-patterns of the crystallized glasses corresponding to ternary BaBiB11O19 (1- 760 °C 24h,cooling in the

On the DTA curves of stoichiometric BaBi2B2O7 and BaBi10B6O25 glass compositions observed two exothermal effects at 570 and 645°C for BaBi2B2O7 and at 460 and 620°C for BaBi10B6O25 (Fig. 19, curves 3,4). But both compositions have one endothermic effect of melting at 725 and 690°C respectively for BaBi2B2O7 and BaBi10B6O25 testifying to one formed crystalline phase melting (Fig. 19, curves 3,4). X-ray data of these samples confirmed monophase crys‐

According to [34] the X-ray powder diffraction patterns of formed BaBi2B2O7 crystalline phase at stoichiometric glass composition (33.33BaO•33.33Bi2O3• 33.33B2O3 mol%) at second

muffle) and BaBi2B4O10 (2- 640°C 24h,cooling in the muffle) compounds

tallizations in each samples (Fig. 21, curves 1,2; Fig.22, curves 1,2).

about dominating surface crystallisation of the given glass particles.

(Fig. 19, curve2).

148 Advances in Ferroelectrics

**Figure 21.** XRD-patterns of the crystallized glasses corresponding to ternary BaBi2B2O7 composition:

### 1-570°C 24h, cooling in the muffle; 2-640°C 24h, casting in the cold water

X-ray powder diffraction patterns of BaBi10B6O25 single crystals was indexed on an ortho‐ rhombic cell with following lattice parameters: a=6.434Å, b=11.763 Å, c=29.998Å, cell volume V=2270.34 Å, Z=8 [34]. XRD-patterns of products of same compositions (11.11BaO• 55.55Bi2O3• 33.33B2O3 mol%) glass crystallization at both exothermal effects (420°C 24h and 620°C 24h) have all diffraction lines of the BaBi10B6O25 single crystals (Fig. 22, curves1-3). Naturally, BaBi10B6O25 single crystal has well generated planes and clear observed diffrac‐ tion lines on XRD-patterns in contrast to crystalline phases formed at same composition glasses crystallization. However, the most intensive diffraction line (I/Io=100) of single crys‐ tals is [133], whereas products of glass crystallizations have [203] strongest diffraction line and [133] diffraction line became 5-10 times less (Fig. 22, curves1-3). Now it is difficult to us only on the basis of XRD-patterns analysis of glass crystallizations products to assume the nature of the second exothermal effect at 620°C on DTA curve of BaBi10B6O25 of glass compo‐ sition (Fig.19, curve 4). Their XRD-patterns are identical each other and to single crystals, but contain slightly quantity of not indexed reflexes, which are absent in X-ray powder dif‐ fraction patterns of BaBi10B6O25 single crystals (Fig. 22, curves1-3).

water products and identified according to X-ray database [43, fail # 01-074-7523]. The disso‐ lution of this BaBiO3 phase in a melt leads to the appearance on a DTA curve the second endothermic effect in an interval 745-775°C (Fig. 19, curve 5). Above 775°C we have glass-

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forming BaBiBO4 composition melt without presence of any crystalline phase.

**Figure 23.** XRD-patterns of the crystallized glasses corresponding to ternary BaBiBO4 composition:

Two exothermal effects of glass crystallization at 540 and 670°C and one endothermic effect of crystalline phase melting at 827°C are seen on the DTA curve of the 60BaO•10Bi2O3•30B2O3 mol% (Ba3BiB3O9) glass composition (Fig. 19, curve 6). X-ray diffrac‐ tion patterns of this glass crystallization products at 540 and 670°C show one Ba3BiB3O9 crys‐ talline phase formation (Fig. 24, curves 1,2) at glass powder samples crystallization at 540 and 670°C and fully correspond to Egorisheva with co-authors data, which synthesized for the first time and have describe Ba3BiB3O9 compound [31]. However, we didn't indicate pol‐ ymorphic transition of Ba3BiB3O9 at 850°C as reported in [31]. Presence of second endother‐ mic effect within the interval of 840-890°C with minimum at 865°C is associated with Ba3BiB3O9 incongruent melting (Fig. 19, curve 6). We have revealed that the crystalline Ba3BiB3O9 compound is melted incongruently at 827°C with the glass forming melt and crys‐ talline phase formation (Fig. 24, curve 3). The Ba2B2O5 crystalline phase was observed in amorphous matrix on XRD-patterns of thermal treated at 830 °C and fast freeze in cold wa‐ ter products and identified according to X-ray database [43, fail # 024-0087]. For clear Ba2B2O5 observation on XRD-patterns the preliminary crystallized at 670°C 24h sample have been exposed at 830°C 3 h (Fig. 23, curve3). Dissolution of this Ba2B2O5 phase in a melt leads to the appearance on a DTA curve the second endothermic effect in an interval 840-890°C (Fig. 19, curve 6). Above 890°C we have glass-forming Ba3BiB3O9 composition melt without

K/s.

1-450°C 24h, cooling in the muffle; 2-720°C 3h, casting in the cold water

presence of any crystalline phase at cooling rate 102

**Figure 22.** XRD-patterns of the crystallized glasses corresponding to ternary BaBi10B6O25 composition:(1-460°C 24h,cooling in the muffle; 2-620°C 24h, casting in the cold water) and BaBi10B6O25 single crystals (3).

The DTA curve of 50BaO• 25Bi2O3• 25B2O3 mol% (BaBiBO4) glass composition contain exothermal effect of glass crystallization at 450°C and endothermic effect of this crystal‐ line phase melting at 680°C (Fig. 19, curv.5). X-ray diffraction patterns of this glass crystal‐ lization products show one BaBiBO4 crystalline phase formation at glass powder samples crystallization at temperature interval 450-640°C(Fig. 23, curve 1), which completely corre‐ spond to Barbier with co-authors data [29]. A second endothermic effect within the inter‐ val of 745-775°C with minimum at 760°C is associated with BaBiBO4 incongruent melting (Fig. 19, curve 5).

We have revealed also, that the crystalline BaBiBO4 compound is melted incongruently at 680°C with the melt and crystalline BaBiO3 formation (Fig. 23, curve 2). The BaBiO3 crystal‐ line phase was observed on XRD-patterns of thermal treated at 720 °C and fast freeze in cold water products and identified according to X-ray database [43, fail # 01-074-7523]. The disso‐ lution of this BaBiO3 phase in a melt leads to the appearance on a DTA curve the second endothermic effect in an interval 745-775°C (Fig. 19, curve 5). Above 775°C we have glassforming BaBiBO4 composition melt without presence of any crystalline phase.

but contain slightly quantity of not indexed reflexes, which are absent in X-ray powder dif‐

**Figure 22.** XRD-patterns of the crystallized glasses corresponding to ternary BaBi10B6O25 composition:(1-460°C

The DTA curve of 50BaO• 25Bi2O3• 25B2O3 mol% (BaBiBO4) glass composition contain exothermal effect of glass crystallization at 450°C and endothermic effect of this crystal‐ line phase melting at 680°C (Fig. 19, curv.5). X-ray diffraction patterns of this glass crystal‐ lization products show one BaBiBO4 crystalline phase formation at glass powder samples crystallization at temperature interval 450-640°C(Fig. 23, curve 1), which completely corre‐ spond to Barbier with co-authors data [29]. A second endothermic effect within the inter‐ val of 745-775°C with minimum at 760°C is associated with BaBiBO4 incongruent melting

We have revealed also, that the crystalline BaBiBO4 compound is melted incongruently at 680°C with the melt and crystalline BaBiO3 formation (Fig. 23, curve 2). The BaBiO3 crystal‐ line phase was observed on XRD-patterns of thermal treated at 720 °C and fast freeze in cold

24h,cooling in the muffle; 2-620°C 24h, casting in the cold water) and BaBi10B6O25 single crystals (3).

(Fig. 19, curve 5).

150 Advances in Ferroelectrics

fraction patterns of BaBi10B6O25 single crystals (Fig. 22, curves1-3).

**Figure 23.** XRD-patterns of the crystallized glasses corresponding to ternary BaBiBO4 composition:

1-450°C 24h, cooling in the muffle; 2-720°C 3h, casting in the cold water

Two exothermal effects of glass crystallization at 540 and 670°C and one endothermic effect of crystalline phase melting at 827°C are seen on the DTA curve of the 60BaO•10Bi2O3•30B2O3 mol% (Ba3BiB3O9) glass composition (Fig. 19, curve 6). X-ray diffrac‐ tion patterns of this glass crystallization products at 540 and 670°C show one Ba3BiB3O9 crys‐ talline phase formation (Fig. 24, curves 1,2) at glass powder samples crystallization at 540 and 670°C and fully correspond to Egorisheva with co-authors data, which synthesized for the first time and have describe Ba3BiB3O9 compound [31]. However, we didn't indicate pol‐ ymorphic transition of Ba3BiB3O9 at 850°C as reported in [31]. Presence of second endother‐ mic effect within the interval of 840-890°C with minimum at 865°C is associated with Ba3BiB3O9 incongruent melting (Fig. 19, curve 6). We have revealed that the crystalline Ba3BiB3O9 compound is melted incongruently at 827°C with the glass forming melt and crys‐ talline phase formation (Fig. 24, curve 3). The Ba2B2O5 crystalline phase was observed in amorphous matrix on XRD-patterns of thermal treated at 830 °C and fast freeze in cold wa‐ ter products and identified according to X-ray database [43, fail # 024-0087]. For clear Ba2B2O5 observation on XRD-patterns the preliminary crystallized at 670°C 24h sample have been exposed at 830°C 3 h (Fig. 23, curve3). Dissolution of this Ba2B2O5 phase in a melt leads to the appearance on a DTA curve the second endothermic effect in an interval 840-890°C (Fig. 19, curve 6). Above 890°C we have glass-forming Ba3BiB3O9 composition melt without presence of any crystalline phase at cooling rate 102 K/s.

The high boron content glass composition corresponds to BaBiB11O19 (14.28BaO•7.14Bi2O3•78.57B2O3 mol %) have TEC=72•10-7К-1 and Tg=498°C calculated from dilatometric curve (Table 4). Reduction the B2O3 amount together with increasing of BaO and Bi2O3 amounts in glass compositions leads to increase TEC and reduction Тg values: for glass composition 25BaO•25Bi2O3•50B2O3 mol % (BaBi2B4O10) TEC=96•10-7К-1 and Тg= 445°C; 33.3BaO•33.3 Bi2O3•33.3B2O3 mol %(BaBi2B2O7) TEC=108•10-7К-1 and Тg=415°C; 11.1BaO• 55.5Bi2O3•33.3B2O3 mol %(BaBi10B6O25) TEC=97•10-7К-1 and Тg=350°C; 16.67BaO•66.67Bi2O3•16.67B2O3 mol%(BaBi8B2O16) TEC=110•10-7К-1 and Тg=415°C. However, for 50BaO•25 Bi2O3•25B2O3 mol % (BaBiBO4) and 60BaO•10Bi2O3•30B2O3 mol % (Ba3BiB3O9) glass compositions simultaneous increase both TEC and Т<sup>g</sup> values were observed: TEC=120•10-7К-1 and Тg=400°C; TEC=127•10-7 К-1 and Тg=460°C respectively for BaBiBO4 and

Phase Diagram of The Ternary BaO-Bi2O3-B2O3 System: New Compounds and Glass Ceramics Characterisation

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TEC values of crystallized glasses corresponding to the ternary barium bismuth borates giv‐ en in Table 4. Crystallized barium bismuth borate glass samples have TEC values lower, than initial glasses and equals to: 49•10-7К-1 for BaBiB11O19 sample (750°C 24h), 78•10-7К-1 for BaBi2B4O10 sample (630°C 24 h), 96•10-7 К-1 for BaBi2B2O7 sample (640°C 24h), 97•10-7К-1 for BaBi10B6O<sup>25</sup> sample (610°C 24h), 110•10-7 К-1 for BaBiBO4 sample (450°C24h) and 109•10-7 К-1 for Ba3BiB3O9 sample (690°C 24h). The same tendency, as well as for their glassy analogues, is observed for crystallized glass samples: increase of barium and bismuth oxides amounts

**4. Ferroelectric properties of new ternary BaBi2B2O7 and BaBi10B6O25**

The ferroelectric (polarization - electric field) hysteresis, is a defining property of ferroelec‐ tric materials. Thus, the most widely studied characteristics of ferroelectric hysteresis were those of interest for this particular application: the value of the switchable polarization (the difference between the positive and negative remanent polarization, *P <sup>R</sup>* − *(*−*P <sup>R</sup> )*, depend‐ ence of the coercive field *Ec* on sample thickness, decrease of remanent or switchable polari‐ zation with number of switching cycles, polarization imprint, endurance, retention [51].

Electric field induced polarization (P) and remanent polarization(Pr) were measured at room temperature for BaBi2B2O7 and BaBi10B6O25 glass tape samples crystallized using various re‐

**a.** BaBi2B2O<sup>7</sup> glass tape sample of 0.07 mm in thickness crystallized at 450°C 24h, 2Pr = 0.15

**b.** BaBi10B6O25 glass tape sample of 0.06 mm in thickness crystallized at 380 °C 12h, 2P<sup>r</sup> =

**c.** BaBi10B6O25 glass tape sample of 0.06 mm in thickness crystallized at 410°C 12h, 2P<sup>r</sup> =

Ba3BiB3O9 (Fig. 25, table4).

gimes (Fig. 26).

μC/cm2 ;

0.32 μC/cm<sup>2</sup>

0.62 μC/cm<sup>2</sup>

;

;

in ternary compounds leads to their TEC values increase.

**stoichometric compositions glass ceramics.**

**Figure 24.** XRD-patterns of the crystallized glasses corresponding to ternary 6BaBi3BO4 composition:1.540°C 24h, cooling in the muffle; 2-670°C 24h, casting in the cold water; 3-670°C 24h+830°C 3 h, casting in the cold water

### **3.3. TEC study of the stoichiometric compositions glasses in the BaO-Bi2O3-B2O3 system**

The isolines diagram of BaO-Bi2O3-B2O3 system glasses TEC values is given on Fig. 25. It is clear observed common regularity, that the increase of barium and bismuth oxides amounts in glasses of binary BaO-Bi2O3 and Bi2O3-B2O3 systems leads to increase TEC of glasses. The same tendency is observed for glasses of ternary system: joint presence of BaO and Bi2O3 and increase their amounts leads to increase glasses TEC values from 70 to 127•10-7К-1 (Fig. 25).

**Figure 25.** BaO-Bi2O3-B2O3 system's glasses TEC (α20-300•10-7К-1) values isolines

The high boron content glass composition corresponds to BaBiB11O19 (14.28BaO•7.14Bi2O3•78.57B2O3 mol %) have TEC=72•10-7К-1 and Tg=498°C calculated from dilatometric curve (Table 4). Reduction the B2O3 amount together with increasing of BaO and Bi2O3 amounts in glass compositions leads to increase TEC and reduction Тg values: for glass composition 25BaO•25Bi2O3•50B2O3 mol % (BaBi2B4O10) TEC=96•10-7К-1 and Тg= 445°C; 33.3BaO•33.3 Bi2O3•33.3B2O3 mol %(BaBi2B2O7) TEC=108•10-7К-1 and Тg=415°C; 11.1BaO• 55.5Bi2O3•33.3B2O3 mol %(BaBi10B6O25) TEC=97•10-7К-1 and Тg=350°C; 16.67BaO•66.67Bi2O3•16.67B2O3 mol%(BaBi8B2O16) TEC=110•10-7К-1 and Тg=415°C. However, for 50BaO•25 Bi2O3•25B2O3 mol % (BaBiBO4) and 60BaO•10Bi2O3•30B2O3 mol % (Ba3BiB3O9) glass compositions simultaneous increase both TEC and Т<sup>g</sup> values were observed: TEC=120•10-7К-1 and Тg=400°C; TEC=127•10-7 К-1 and Тg=460°C respectively for BaBiBO4 and Ba3BiB3O9 (Fig. 25, table4).

TEC values of crystallized glasses corresponding to the ternary barium bismuth borates giv‐ en in Table 4. Crystallized barium bismuth borate glass samples have TEC values lower, than initial glasses and equals to: 49•10-7К-1 for BaBiB11O19 sample (750°C 24h), 78•10-7К-1 for BaBi2B4O10 sample (630°C 24 h), 96•10-7 К-1 for BaBi2B2O7 sample (640°C 24h), 97•10-7К-1 for BaBi10B6O<sup>25</sup> sample (610°C 24h), 110•10-7 К-1 for BaBiBO4 sample (450°C24h) and 109•10-7 К-1 for Ba3BiB3O9 sample (690°C 24h). The same tendency, as well as for their glassy analogues, is observed for crystallized glass samples: increase of barium and bismuth oxides amounts in ternary compounds leads to their TEC values increase.

**Figure 24.** XRD-patterns of the crystallized glasses corresponding to ternary 6BaBi3BO4 composition:1.540°C 24h, cooling in the muffle; 2-670°C 24h, casting in the cold water; 3-670°C 24h+830°C 3 h, casting in the cold water

152 Advances in Ferroelectrics

**3.3. TEC study of the stoichiometric compositions glasses in the BaO-Bi2O3-B2O3 system**

**Figure 25.** BaO-Bi2O3-B2O3 system's glasses TEC (α20-300•10-7К-1) values isolines

The isolines diagram of BaO-Bi2O3-B2O3 system glasses TEC values is given on Fig. 25. It is clear observed common regularity, that the increase of barium and bismuth oxides amounts in glasses of binary BaO-Bi2O3 and Bi2O3-B2O3 systems leads to increase TEC of glasses. The same tendency is observed for glasses of ternary system: joint presence of BaO and Bi2O3 and increase their amounts leads to increase glasses TEC values from 70 to 127•10-7К-1 (Fig. 25).

### **4. Ferroelectric properties of new ternary BaBi2B2O7 and BaBi10B6O25 stoichometric compositions glass ceramics.**

The ferroelectric (polarization - electric field) hysteresis, is a defining property of ferroelec‐ tric materials. Thus, the most widely studied characteristics of ferroelectric hysteresis were those of interest for this particular application: the value of the switchable polarization (the difference between the positive and negative remanent polarization, *P <sup>R</sup>* − *(*−*P <sup>R</sup> )*, depend‐ ence of the coercive field *Ec* on sample thickness, decrease of remanent or switchable polari‐ zation with number of switching cycles, polarization imprint, endurance, retention [51].

Electric field induced polarization (P) and remanent polarization(Pr) were measured at room temperature for BaBi2B2O7 and BaBi10B6O25 glass tape samples crystallized using various re‐ gimes (Fig. 26).


**d.** BaBi10B6O25 glass tape sample of 0.05 mm in thickness crystallized at 410°C 24h, 2Pr = 0.9 μC/cm2

2). Binary BaB4O7, Ba2B10O17, and BaB8O13 barium borates have melting temperatures (Tm) of 910, 905, and 890°C and can be found between the eutectics e7, e8, e9, and e10 with Tm = 878, 869, 895, and 899°C, respectively. The transition to a crystallization field of barium metabo‐ rate is accompanied by a sharp increase of liquidus temperature (TL) and a decrease of the glass forming ability of the melts. The final compound, forming a stable glass, contains ~43 mol %BaO and has TL = 950°C. Compounds BaB2O4 and Ba2B2O5, having higher Tm, which are 1095 and 1050°C, respectively [18, 21, 22], are found in the region of the compounds ob‐

Phase Diagram of The Ternary BaO-Bi2O3-B2O3 System: New Compounds and Glass Ceramics Characterisation

B2O3 binary system is limited by the eutectic e12 with Tm = 1025°C (Fig. 2) because of the sharp increase of the liquidus temperature during the transition to a field of crystallization of the Ba3B2O6 compound (Tm = 1383°C). BaB2O4 (Tm = 1095°C) is the dominating com‐ pound in the system and does not form stable glasses. Its considerable crystallization field narrows the region of stable glasses in the ternary system, which is only restricted by com‐

Binary bismuth borates Bi4B2O9, BiBO3, Bi3B5O12, BiB3O6, and Bi2B8O15 have Tm 675, 685, 722, 708, and 715°C can be found between the eutectics e1, e2, e3, e4, e5, e6 with a Tm of 622, 646, 665, 698, 695, and 709°C, respectively [2, 34]. The region of stable glasses in the Bi2O3–B2O3 system is limited by a compound containing ~70%mol Bi2O3 and having TL = 670°C. Com‐ pounds that are found in the range of 70–80mol% Bi2O3 (before the e1 point) are obtained in the form of glasses during cooling at a rate of ~102 K/s. During the transition to the crystalli‐ zation field of Bi24B2O3 and Bi2O3, TL increases to 825°C (Tm of Bi2O3). Glasses in this part of

have a low liquidus temperature; however, the structure factor essentially influences their

Six ternary compounds are known in the BaO–Bi2O3–B2O3 system: Ba3BiB3O9, BaBiBO4, Ba‐ Bi2B4O10, and BaBiB11O19 synthesized by Barbie and Egorysheva in 2005–2006, and BaBi2B2O7 and BaBi10B6O25 revealed by our research group in 2008–2009. BaBi2B4O10, BaBiB11O19, Ba‐ Bi2B2O7, and BaBi10B6O25 melt congruently at 730, 807, 725, and 695°C respectively, and BaBi‐ BO4 melts incongruently at 680°C and has TL = 760 °C. All these five ternary compounds along with the eutectics formed between each other and with binary barium and bismuth borates form a "plateau" with low TL, which is responsible for the formation of the region of

Compounds joining low temperature eutectic e1 (622°C) [2] in the Bi2O3–B2O3 binary system and the eutectics e13 (~790°C) and e14 (~750°C) in the BaO–Bi2O3 binary system 26 - 28] form

Bi2O3 binary system stops at 45 mol% BaO content (TL ~930°C) [26 - 28]. Along with the fac‐ tor of the liquidus temperature [52], a considerable contribution to the glass formation of the pointed compositions is made by the structural factor of the melt. The combination of the structural factors of the melt and the liquidus temperature is also considerable during the transition to the vitreous state of the compositions, which are found in the crystallization fields of BaB2O4 and Ba2B2O5, where they show the tendency towards glass formation only at

–104

–104

–104

) K/s. Glass formation in the BaO–

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155

) K/s. These compounds

) K/s. Glass formation in the BaO–

tained in the form of glasses with a cooling rate of (103

the system are obtained during melt cooling at a rate of (103

stable glasses in the ternary system BaO–Bi2O3–B2O3.

glasses only at higher cooling rates of their melts (103

high rates of melt cooling.

glass forming ability, not allowing glass formation at low melt cooling rates.

pounds with TL ~ 950°C (Fig. 2).

**Figure 26.** Dependence of polarization (P) on electric field (E) for crystallized stoichiometric glass compositions:

Linear *P*–*E* curves are observed up to fields of 40-120 kV/cm for all measured samples with thickness 0.05-0.07mm. The polarization becomes nonlinear with increasing of applied elec‐ tric field, and at 140-380 kV/cm the remanent polarization 2Pr values were found 0.15 μC/cm2 for the BaBi2B2O7 (Fig. 26, A). The remanent polarization 2Pr value for BaBi10B6O25 crestallized glass tape samples encreasing with termal treatment temperature from 0,32 to 0, 64 μC/cm2 (Fig. 26, B & C) and time (Fig. 26, D). The highest remanent polarization value (2Pr=0.9 μC/cm2 ) has BaBi10B6O25 glass tape sample crystallized at 410°C 24h (Fig.26, D). Ac‐ cording to obtained results it is possible to conclude that samples are ferroelectrics.

### **5. Discussion**

The pricipial diference of our methodology from traditional is a glass samples using as ini‐ tial testing substance for phase diagram of very complex ternary BaO-Bi2O3-B2O3 system construction. It is a very effective method, due possibility to indicate temperature intervals of all processes taking place in glass samples: glass transition, crystallization, quantity of formed crystalline phases and their melting. Whereas, samples prepared by traditional solid phase synthesis are less informative and often lose a lot of information. Super cooling tech‐ nique created by our group allowed us both to expand borders of glass formation and to have enough quantity samples for DTA and X-ray investigations and BaO-Bi2O3-B2O3 sys‐ tem phase diagram construction (Fig.2).

The region of stable glasses includes the binary compounds BaB4O7, Ba2B10O17, BaB8O13, Bi4B2O9, BiBO3, Bi3B5O12, BiB3O6, and Bi2B8O15 in the BaO–B2O3 and Bi2O3–B2O3 systems (Fig. 2). Binary BaB4O7, Ba2B10O17, and BaB8O13 barium borates have melting temperatures (Tm) of 910, 905, and 890°C and can be found between the eutectics e7, e8, e9, and e10 with Tm = 878, 869, 895, and 899°C, respectively. The transition to a crystallization field of barium metabo‐ rate is accompanied by a sharp increase of liquidus temperature (TL) and a decrease of the glass forming ability of the melts. The final compound, forming a stable glass, contains ~43 mol %BaO and has TL = 950°C. Compounds BaB2O4 and Ba2B2O5, having higher Tm, which are 1095 and 1050°C, respectively [18, 21, 22], are found in the region of the compounds ob‐ tained in the form of glasses with a cooling rate of (103 –104 ) K/s. Glass formation in the BaO– B2O3 binary system is limited by the eutectic e12 with Tm = 1025°C (Fig. 2) because of the sharp increase of the liquidus temperature during the transition to a field of crystallization of the Ba3B2O6 compound (Tm = 1383°C). BaB2O4 (Tm = 1095°C) is the dominating com‐ pound in the system and does not form stable glasses. Its considerable crystallization field narrows the region of stable glasses in the ternary system, which is only restricted by com‐ pounds with TL ~ 950°C (Fig. 2).

**d.** BaBi10B6O25 glass tape sample of 0.05 mm in thickness crystallized at 410°C 24h, 2Pr = 0.9

**Figure 26.** Dependence of polarization (P) on electric field (E) for crystallized stoichiometric glass compositions:

cording to obtained results it is possible to conclude that samples are ferroelectrics.

Linear *P*–*E* curves are observed up to fields of 40-120 kV/cm for all measured samples with thickness 0.05-0.07mm. The polarization becomes nonlinear with increasing of applied elec‐ tric field, and at 140-380 kV/cm the remanent polarization 2Pr values were found 0.15

The pricipial diference of our methodology from traditional is a glass samples using as ini‐ tial testing substance for phase diagram of very complex ternary BaO-Bi2O3-B2O3 system construction. It is a very effective method, due possibility to indicate temperature intervals of all processes taking place in glass samples: glass transition, crystallization, quantity of formed crystalline phases and their melting. Whereas, samples prepared by traditional solid phase synthesis are less informative and often lose a lot of information. Super cooling tech‐ nique created by our group allowed us both to expand borders of glass formation and to have enough quantity samples for DTA and X-ray investigations and BaO-Bi2O3-B2O3 sys‐

The region of stable glasses includes the binary compounds BaB4O7, Ba2B10O17, BaB8O13, Bi4B2O9, BiBO3, Bi3B5O12, BiB3O6, and Bi2B8O15 in the BaO–B2O3 and Bi2O3–B2O3 systems (Fig.

 for the BaBi2B2O7 (Fig. 26, A). The remanent polarization 2Pr value for BaBi10B6O25 crestallized glass tape samples encreasing with termal treatment temperature from 0,32 to 0, 64 μC/cm2 (Fig. 26, B & C) and time (Fig. 26, D). The highest remanent polarization value (2Pr=0.9 μC/cm2 ) has BaBi10B6O25 glass tape sample crystallized at 410°C 24h (Fig.26, D). Ac‐

μC/cm2

154 Advances in Ferroelectrics

μC/cm2

**5. Discussion**

tem phase diagram construction (Fig.2).

Binary bismuth borates Bi4B2O9, BiBO3, Bi3B5O12, BiB3O6, and Bi2B8O15 have Tm 675, 685, 722, 708, and 715°C can be found between the eutectics e1, e2, e3, e4, e5, e6 with a Tm of 622, 646, 665, 698, 695, and 709°C, respectively [2, 34]. The region of stable glasses in the Bi2O3–B2O3 system is limited by a compound containing ~70%mol Bi2O3 and having TL = 670°C. Com‐ pounds that are found in the range of 70–80mol% Bi2O3 (before the e1 point) are obtained in the form of glasses during cooling at a rate of ~102 K/s. During the transition to the crystalli‐ zation field of Bi24B2O3 and Bi2O3, TL increases to 825°C (Tm of Bi2O3). Glasses in this part of the system are obtained during melt cooling at a rate of (103 –104 ) K/s. These compounds have a low liquidus temperature; however, the structure factor essentially influences their glass forming ability, not allowing glass formation at low melt cooling rates.

Six ternary compounds are known in the BaO–Bi2O3–B2O3 system: Ba3BiB3O9, BaBiBO4, Ba‐ Bi2B4O10, and BaBiB11O19 synthesized by Barbie and Egorysheva in 2005–2006, and BaBi2B2O7 and BaBi10B6O25 revealed by our research group in 2008–2009. BaBi2B4O10, BaBiB11O19, Ba‐ Bi2B2O7, and BaBi10B6O25 melt congruently at 730, 807, 725, and 695°C respectively, and BaBi‐ BO4 melts incongruently at 680°C and has TL = 760 °C. All these five ternary compounds along with the eutectics formed between each other and with binary barium and bismuth borates form a "plateau" with low TL, which is responsible for the formation of the region of stable glasses in the ternary system BaO–Bi2O3–B2O3.

Compounds joining low temperature eutectic e1 (622°C) [2] in the Bi2O3–B2O3 binary system and the eutectics e13 (~790°C) and e14 (~750°C) in the BaO–Bi2O3 binary system 26 - 28] form glasses only at higher cooling rates of their melts (103 –104 ) K/s. Glass formation in the BaO– Bi2O3 binary system stops at 45 mol% BaO content (TL ~930°C) [26 - 28]. Along with the fac‐ tor of the liquidus temperature [52], a considerable contribution to the glass formation of the pointed compositions is made by the structural factor of the melt. The combination of the structural factors of the melt and the liquidus temperature is also considerable during the transition to the vitreous state of the compositions, which are found in the crystallization fields of BaB2O4 and Ba2B2O5, where they show the tendency towards glass formation only at high rates of melt cooling.

There are very stable congruent melted binary BaB2O4 and ternary BaBi2B4O10, BaBiB11O19, and BaBi10B6O25 compounds in the studied ternary BaO-Bi2O3-B2O3 system. They have domi‐ nating positions in ternary diagram and occupied the biggest part of it (Fig.18). Mutual in‐ fluence of these compounds and other binary and ternary compounds (BaB4O7, Ba2B10O17, BaB8O13, Bi4B2O9, BiBO3, Bi3B5O12, BiB3O6, Bi2B8O15, BaBi2B2O7, and BaBi8B2O16) lead to forma‐ tion of sixteen revealed at present time ternary eutectics (Fig.18& Table 3), which have es‐ sential influence on liquidus temperature decrease and to assist in glass formation. Ternary BaBi2B4O10 compound forms eight eutectics with binary and ternary compounds, its neigh‐ bors: E4(622°C), E3(640°C), E2(585°C), E5(610°C), E6(675°C), E12(680°C), E13 (690°C), and E14 (700°C) (Fig.18& Table 3). BaBiB11O19 compound forms seven eutectics with its neighbors: E6 (675°C), E7 (680°C), E8 (675°C), E9 (680°C), E10 (730°C), E11 (750°C),and E12 (680°C) (Fig.18& Table 3). BaBi10B6O25 compound forms five eutectics with its neighbors: E1 (590°C), E2 (585°C), E3 (640°C), E15 (645°C), and E16 (615°C) (Fig.18& Table 3). Determined ternary eutec‐ tics together with binary eutectics e1, e2, e3, e4, e5, and e6 of Bi2O3-B2O3 system have allowed to outline the fields of binary Bi4B2O9, BiBO3, Bi3B5O12, BiB3O6, and Bi2B8O15 bismuth borates crystallisation, as well as together with binary eutectics e7, e8, e9, and e10 of BaO-B2O3 system have allowed to outline the fields of binary BaB4O7, Ba2B10O17, BaB8O13 barium borates and partly BaB2O4 crystallization on the BaO-Bi2O3-B2O3 system phase diagram (Fig.18).

Common regularities of bulk glass samples TEC changes in studied BaO-Bi2O3-B2O3 system have been determined: the increase of barium and bismuth oxides amounts in glasses of binary BaO-Bi2O3 and Bi2O3-B2O3 systems leads to increase TEC of glasses. The same tendency is observed for glasses of ternary BaO-Bi2O3-B2O3system: joint presence of BaO and Bi2O3 and increase their amounts leads to increase glasses TEC values from 70 to 127•10-7К-1 (Fig. 25). Crystallized barium bismuth borate glass samples have TEC values lower, than initial glasses and equals to: 49•10-7К-1 for BaBiB11O19 sample (750°C 24h), 78•10-7К-1 for BaBi2B4O10 sample (630°C 24 h), 96•10-7 К-1 for BaBi2B2O7 sample (640°C 24h), 97•10-7К-1 for BaBi10B6O25 sample (610°C 24h), 110•10-7 К-1 for BaBiBO4 sample (450°C24h) and 109•10-7 К-1 for Ba3BiB3O9 sample (690°C 24h). The same tendency, as well as for their glassy analogues, is observed for crystallized glass samples: increase of barium and bismuth oxides amounts in

Phase Diagram of The Ternary BaO-Bi2O3-B2O3 System: New Compounds and Glass Ceramics Characterisation

Electric field induced polarization (P) and remanent polarization (Pr) were measured at room temperature for BaBi2B2O7 and BaBi10B6O25 glass tape samples crystallized at various

Linear *P*–*E* curves are observed up to fields of 40-120 kV/cm for all measured samples with thickness 0.05-0.07mm. The polarization becomes nonlinear with an increase of applied elec‐ tric field, and at 140-400 kV/cm the remanent polarization 2Pr values were found 0.15

crystallized glass tape samples. According to obtained results it is possible to conclude that

Effective way of new system investigation and new compounds and characteristic points re‐ vealing via simultaneous glass forming and phase diagrams construction have been shown. Phase diagram of the ternary BaO-Bi2O3-B2O3 system have been constructed for the first time us result of fourteen pseudo-binary systems and sections phase diagrams investigations.

The phase diagram of the well known binary Bi2O3–B2O3 system has been corrected in the interval between the Bi4B2O9 and Bi3B5O12 compounds. The eutectic composition, 48.5Bi2O3•51.5B2O3 (mol%), between BiBO3 and Bi3B5O12, with m.p. 665±5°С, has been deter‐ mined. It is shown that the compound BiBO3 is congruently melting with a m.p. of 685±5°С. Two new ternary BaBi2B2O7 and BaBi10B6O25 compounds have been revealed at the same glass compositions crystallisation. The new ternary compound BaBi2B2O7 synthesized at the 33.33BaO•33.33Bi2O3•33.33B2O3 (mol%) glass composition crystallization at 640°C, 20 h. The x-ray powder diffraction patterns of BaBi2B2O7 could be indexed on an orthorhombic cell with lattice parameters as follows: BaBi2B2O7 a=11.818 Å, b=8.753 Å, c=7.146 Å, cell volume

Single crystals of BaBi10B6O25 were obtained by cooling of a melt with the stoichiometric composition. Glass powder of composition 11.11BaO•55.55Bi2O3•33.33B2O3 (mol%) was

for the BaBi10B6O25 (Fig.26, B-D),

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157

ternary compounds leads to their TEC values increase.

regimes. All tested samples shown loop of hysteresis.

all tested samples are ferroelectrics.

for the BaBi2B2O7 (Fig.26, A), and 0.32- 0.9 μC/cm2

μC/cm2

**5. Conclusion**

V=739.203 Å3

, Z=4.

The clear correlation between glass forming and phase diagrams has been observed in stud‐ ied system. The glass melting temperature and level of glass formation depending on the cooling rate of the studied melts are in good conformity with boundary curves and eutectic points (Fig.2& 18).

The phase diagram of the well known binary Bi2O3–B2O3 system has been corrected in the interval between the Bi4B2O9 and Bi3B5O12 compounds. The eutectic composition, 48.5Bi2O3•51.5B2O3 (mol%), between BiBO3 and Bi3B5O12, with m.p. 665±5°С, has been deter‐ mined. It is shown that the compound BiBO3 is congruently melting with a m.p. of 685±5°С.

The next unexpected results were obtained at phase diagram construction: two new ternary BaBi2B2O7 and BaBi10B6O25 compounds have been revealed at the same glass compositions crystallisation. X-ray characteristics of the new ternary compound BaBi2B2O7, synthesized at the 33.33BaO•33.33Bi2O3•33.33B2O3 (mol%) glass composition crystallization at 640°C, 20 h. The x-ray powder diffraction patterns of BaBi2B2O7 could be indexed on an orthorhombic cell with lattice parameters as follows: BaBi2B2O7 *a*=11.818 Å, *b*=8.753 Å, *c*=7.146 Å, cell vol‐ ume *V*=739.203 Å3 , *Z*=4.

Single crystals of BaBi10B6O25 were obtained by cooling of a melt with the stoichiometric composition. Glass powder of composition 11.11BaO•55.55Bi2O3•33.33B2O3 (mol%) was heated in a quartz glass ampoule up to 750°C at a rate 10 K/min. After 2 h at high tempera‐ ture, the melt was cooled at a rate 0.5 K/h. Single crystals with sizes up to 1.66×0.38×0.19 mm3 were grown. The x-ray powder diffraction patterns of BaBi10B6O25 could be indexed on an orthorhombic cell with lattice parameters as follows: *a*=6.434 Å, *b*=11.763 Å, *c*=29.998 Å, cell volume *V*=2270.34 Å3 , *Z*=8.

Common regularities of bulk glass samples TEC changes in studied BaO-Bi2O3-B2O3 system have been determined: the increase of barium and bismuth oxides amounts in glasses of binary BaO-Bi2O3 and Bi2O3-B2O3 systems leads to increase TEC of glasses. The same tendency is observed for glasses of ternary BaO-Bi2O3-B2O3system: joint presence of BaO and Bi2O3 and increase their amounts leads to increase glasses TEC values from 70 to 127•10-7К-1 (Fig. 25).

Crystallized barium bismuth borate glass samples have TEC values lower, than initial glasses and equals to: 49•10-7К-1 for BaBiB11O19 sample (750°C 24h), 78•10-7К-1 for BaBi2B4O10 sample (630°C 24 h), 96•10-7 К-1 for BaBi2B2O7 sample (640°C 24h), 97•10-7К-1 for BaBi10B6O25 sample (610°C 24h), 110•10-7 К-1 for BaBiBO4 sample (450°C24h) and 109•10-7 К-1 for Ba3BiB3O9 sample (690°C 24h). The same tendency, as well as for their glassy analogues, is observed for crystallized glass samples: increase of barium and bismuth oxides amounts in ternary compounds leads to their TEC values increase.

Electric field induced polarization (P) and remanent polarization (Pr) were measured at room temperature for BaBi2B2O7 and BaBi10B6O25 glass tape samples crystallized at various regimes. All tested samples shown loop of hysteresis.

Linear *P*–*E* curves are observed up to fields of 40-120 kV/cm for all measured samples with thickness 0.05-0.07mm. The polarization becomes nonlinear with an increase of applied elec‐ tric field, and at 140-400 kV/cm the remanent polarization 2Pr values were found 0.15 μC/cm2 for the BaBi2B2O7 (Fig.26, A), and 0.32- 0.9 μC/cm2 for the BaBi10B6O25 (Fig.26, B-D), crystallized glass tape samples. According to obtained results it is possible to conclude that all tested samples are ferroelectrics.

### **5. Conclusion**

There are very stable congruent melted binary BaB2O4 and ternary BaBi2B4O10, BaBiB11O19, and BaBi10B6O25 compounds in the studied ternary BaO-Bi2O3-B2O3 system. They have domi‐ nating positions in ternary diagram and occupied the biggest part of it (Fig.18). Mutual in‐ fluence of these compounds and other binary and ternary compounds (BaB4O7, Ba2B10O17, BaB8O13, Bi4B2O9, BiBO3, Bi3B5O12, BiB3O6, Bi2B8O15, BaBi2B2O7, and BaBi8B2O16) lead to forma‐ tion of sixteen revealed at present time ternary eutectics (Fig.18& Table 3), which have es‐ sential influence on liquidus temperature decrease and to assist in glass formation. Ternary BaBi2B4O10 compound forms eight eutectics with binary and ternary compounds, its neigh‐ bors: E4(622°C), E3(640°C), E2(585°C), E5(610°C), E6(675°C), E12(680°C), E13 (690°C), and E14 (700°C) (Fig.18& Table 3). BaBiB11O19 compound forms seven eutectics with its neighbors: E6 (675°C), E7 (680°C), E8 (675°C), E9 (680°C), E10 (730°C), E11 (750°C),and E12 (680°C) (Fig.18& Table 3). BaBi10B6O25 compound forms five eutectics with its neighbors: E1 (590°C), E2 (585°C), E3 (640°C), E15 (645°C), and E16 (615°C) (Fig.18& Table 3). Determined ternary eutec‐ tics together with binary eutectics e1, e2, e3, e4, e5, and e6 of Bi2O3-B2O3 system have allowed to outline the fields of binary Bi4B2O9, BiBO3, Bi3B5O12, BiB3O6, and Bi2B8O15 bismuth borates crystallisation, as well as together with binary eutectics e7, e8, e9, and e10 of BaO-B2O3 system have allowed to outline the fields of binary BaB4O7, Ba2B10O17, BaB8O13 barium borates and

partly BaB2O4 crystallization on the BaO-Bi2O3-B2O3 system phase diagram (Fig.18).

points (Fig.2& 18).

156 Advances in Ferroelectrics

ume *V*=739.203 Å3

cell volume *V*=2270.34 Å3

mm3

, *Z*=4.

, *Z*=8.

The clear correlation between glass forming and phase diagrams has been observed in stud‐ ied system. The glass melting temperature and level of glass formation depending on the cooling rate of the studied melts are in good conformity with boundary curves and eutectic

The phase diagram of the well known binary Bi2O3–B2O3 system has been corrected in the interval between the Bi4B2O9 and Bi3B5O12 compounds. The eutectic composition, 48.5Bi2O3•51.5B2O3 (mol%), between BiBO3 and Bi3B5O12, with m.p. 665±5°С, has been deter‐ mined. It is shown that the compound BiBO3 is congruently melting with a m.p. of 685±5°С.

The next unexpected results were obtained at phase diagram construction: two new ternary BaBi2B2O7 and BaBi10B6O25 compounds have been revealed at the same glass compositions crystallisation. X-ray characteristics of the new ternary compound BaBi2B2O7, synthesized at the 33.33BaO•33.33Bi2O3•33.33B2O3 (mol%) glass composition crystallization at 640°C, 20 h. The x-ray powder diffraction patterns of BaBi2B2O7 could be indexed on an orthorhombic cell with lattice parameters as follows: BaBi2B2O7 *a*=11.818 Å, *b*=8.753 Å, *c*=7.146 Å, cell vol‐

Single crystals of BaBi10B6O25 were obtained by cooling of a melt with the stoichiometric composition. Glass powder of composition 11.11BaO•55.55Bi2O3•33.33B2O3 (mol%) was heated in a quartz glass ampoule up to 750°C at a rate 10 K/min. After 2 h at high tempera‐ ture, the melt was cooled at a rate 0.5 K/h. Single crystals with sizes up to 1.66×0.38×0.19

 were grown. The x-ray powder diffraction patterns of BaBi10B6O25 could be indexed on an orthorhombic cell with lattice parameters as follows: *a*=6.434 Å, *b*=11.763 Å, *c*=29.998 Å, Effective way of new system investigation and new compounds and characteristic points re‐ vealing via simultaneous glass forming and phase diagrams construction have been shown. Phase diagram of the ternary BaO-Bi2O3-B2O3 system have been constructed for the first time us result of fourteen pseudo-binary systems and sections phase diagrams investigations.

The phase diagram of the well known binary Bi2O3–B2O3 system has been corrected in the interval between the Bi4B2O9 and Bi3B5O12 compounds. The eutectic composition, 48.5Bi2O3•51.5B2O3 (mol%), between BiBO3 and Bi3B5O12, with m.p. 665±5°С, has been deter‐ mined. It is shown that the compound BiBO3 is congruently melting with a m.p. of 685±5°С.

Two new ternary BaBi2B2O7 and BaBi10B6O25 compounds have been revealed at the same glass compositions crystallisation. The new ternary compound BaBi2B2O7 synthesized at the 33.33BaO•33.33Bi2O3•33.33B2O3 (mol%) glass composition crystallization at 640°C, 20 h. The x-ray powder diffraction patterns of BaBi2B2O7 could be indexed on an orthorhombic cell with lattice parameters as follows: BaBi2B2O7 a=11.818 Å, b=8.753 Å, c=7.146 Å, cell volume V=739.203 Å3 , Z=4.

Single crystals of BaBi10B6O25 were obtained by cooling of a melt with the stoichiometric composition. Glass powder of composition 11.11BaO•55.55Bi2O3•33.33B2O3 (mol%) was heated in a quartz glass ampoule up to 750°C at a rate 10 K/min. After 2 h exposition at high temperature, the melt was cooled at a rate 0.5 K/h. Single crystals with sizes up to 1.66×0.38×0.19 mm3 were grown. The x-ray powder diffraction patterns of BaBi10B6O25 could be indexed on an orthorhombic cell with lattice parameters as follows: a=6.434 Å, b=11.763 Å, c=29.998 Å, cell volume V=2270.34 Å3 , Z=8.

Address all correspondence to: martun\_h@yahoo.com

«ENI» Institute of Electronic Materials LTD, Armenia

book., In Russian, v.III, part 2, L.,, Nauka Press.

BiB3O6 (BIBO). *Phys. Chem. Glasses*, 44, 212-214.

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CaBiGaB2O7. *Chem. Mater.*, 17, 3130-3136.

MBi2B2O7(M=Ca,Sr). *Solid State Chem*, 179, 3958-3964.

bismuth ortoborate BiBO3. *Z. Naturforsch. B*, 59, 256-258.

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[7] Oprea, I. I. (2005). Optical Properties of Borate Glass- Ceramic. *Dissertation zur Erlan‐ gung des Grades Doktor der Naturwissenschaften, Osnabruck University, Germany*.

[8] Pottier , M. J. (1974). Mise en evidence d'un compose BiBO3 et de son polymor‐

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[12] Kargin, Yu. F., Zhereb, V. P., & Egorysheva, A. V. (2002). Phase diagram of metasta‐ ble stats of the Bi2O3-B2O3 system. *Zh. Neorg. Khim*, 47(8), 1362-1364, In Russian. [13] Barbier, J., Penin, N., & Cranswick, L. M. (2005). Melilite-Type Borates Bi2ZnB2O7 and

[14] Barbier, J., & Cranswick, L. M. (2006). The Non-Centrosymmetric Borate Oxides,

Bi2O3-B2O3 projection. *Zh. Neorg. Mater*, In Russian, 26(8), 1678-1681.

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growth and some new attractive properties. *J. Cryst. Growth*, 237, 740-744.

**References**

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355-360.

*als*, 27, 403-408.

Ternary BaBi2B4O10, BaBiB11O19, and BaBi10B6O25 compounds have dominating positions in ternary diagram and occupied the biggest part of it. Mutual influence of these and other bi‐ nary and ternary compounds (BaB4O7, Ba2B10O17, BaB8O13, Bi4B2O9, BiBO3, Bi3B5O12, BiB3O6, Bi2B8O15, BaBi2B2O7, and BaBi8B2O16) lead to formation of sixteen ternary eutectics, which have essential influence on liquidus temperature decrease and to assist in glass formation.

The clear correlation between glass forming and phase diagrams has been observed: glass melting temperature and level of glass formation depending on the cooling rate of the stud‐ ied melts are in good conformity with boundary curves and eutectic points.

Common regularities of bulk glass samples TEC changes in studied BaO-Bi2O3-B2O3 system have been determined: the increase of barium and bismuth oxides amounts in glasses of bi‐ nary BaO-Bi2O3 and Bi2O3-B2O3 systems and their joint amounts increasing in ternary com‐ positions leads to increase glasses TEC values from 70 to 127•10-7К-1.

Crystallized barium bismuth borate glass samples have TEC values lower, than initial glasses. Increase of barium and bismuth oxides amounts in ternary compounds leads to their TEC values increasing from 49 to 109•10-7К-1.

Electric field induced polarization (P) and remanent polarization (Pr) were measured at room temperature for BaBi2B2O7 and BaBi10B6O25 glass tape samples crystallized at various regimes. The remanent polarization 2Pr values were found 0.15 μC/cm2 for the BaBi2B2O7, and 0.32- 0.9 μC/cm2 for the BaBi10B6O25 crystallized glass tape samples. According to ob‐ tained results it is possible to conclude that all tested samples are ferroelectrics.

### **Acknowledgements**

This work was supported by the International Science and Technology Center (Projects # A-1591). Author is very gratefull to all project team and to Dr. Rafael Hovhannisyan , Dr. Nikolay Knyazyan and Prof. Heli Jantunen for effective cooperation and fruitful discussions.

### **Author details**

Martun Hovhannisyan\*

Address all correspondence to: eni\_arm@yahoo.com

Address all correspondence to: raf.hovhannisyan@yahoo.com

Address all correspondence to: martun\_h@yahoo.com

«ENI» Institute of Electronic Materials LTD, Armenia

### **References**

heated in a quartz glass ampoule up to 750°C at a rate 10 K/min. After 2 h exposition at high temperature, the melt was cooled at a rate 0.5 K/h. Single crystals with sizes up to

be indexed on an orthorhombic cell with lattice parameters as follows: a=6.434 Å, b=11.763

Ternary BaBi2B4O10, BaBiB11O19, and BaBi10B6O25 compounds have dominating positions in ternary diagram and occupied the biggest part of it. Mutual influence of these and other bi‐ nary and ternary compounds (BaB4O7, Ba2B10O17, BaB8O13, Bi4B2O9, BiBO3, Bi3B5O12, BiB3O6, Bi2B8O15, BaBi2B2O7, and BaBi8B2O16) lead to formation of sixteen ternary eutectics, which have essential influence on liquidus temperature decrease and to assist in glass formation. The clear correlation between glass forming and phase diagrams has been observed: glass melting temperature and level of glass formation depending on the cooling rate of the stud‐

Common regularities of bulk glass samples TEC changes in studied BaO-Bi2O3-B2O3 system have been determined: the increase of barium and bismuth oxides amounts in glasses of bi‐ nary BaO-Bi2O3 and Bi2O3-B2O3 systems and their joint amounts increasing in ternary com‐

Crystallized barium bismuth borate glass samples have TEC values lower, than initial glasses. Increase of barium and bismuth oxides amounts in ternary compounds leads to

Electric field induced polarization (P) and remanent polarization (Pr) were measured at room temperature for BaBi2B2O7 and BaBi10B6O25 glass tape samples crystallized at various

This work was supported by the International Science and Technology Center (Projects # A-1591). Author is very gratefull to all project team and to Dr. Rafael Hovhannisyan , Dr. Nikolay Knyazyan and Prof. Heli Jantunen for effective cooperation and fruitful discussions.

for the BaBi10B6O25 crystallized glass tape samples. According to ob‐

for the BaBi2B2O7,

, Z=8.

ied melts are in good conformity with boundary curves and eutectic points.

positions leads to increase glasses TEC values from 70 to 127•10-7К-1.

regimes. The remanent polarization 2Pr values were found 0.15 μC/cm2

tained results it is possible to conclude that all tested samples are ferroelectrics.

their TEC values increasing from 49 to 109•10-7К-1.

Address all correspondence to: eni\_arm@yahoo.com

Address all correspondence to: raf.hovhannisyan@yahoo.com

were grown. The x-ray powder diffraction patterns of BaBi10B6O25 could

1.66×0.38×0.19 mm3

158 Advances in Ferroelectrics

and 0.32- 0.9 μC/cm2

**Acknowledgements**

**Author details**

Martun Hovhannisyan\*

Å, c=29.998 Å, cell volume V=2270.34 Å3


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**Chapter 8**

**Optical Properties of Ferroelectrics**

**and Measurement Procedures**

Additional information is available at the end of the chapter

positing successive layers of BST with different Ba/Sr ratios [8].

It is well known that the optical properties of ferroelectric materials find wide ranging appli‐ cations in laser devices. Particularly in the recent years, there has been tremendous interest in the investigation of the nonlinear optical properties of ferroelectric thin films [1-5] for pla‐ nar waveguide and integrated –optic devices. A new class of thin film waveguides has been developed using BaTiO3 thin films deposited on MgO substrates [6]. Barium strontium tita‐ nate Ba1-xSrxTiO3 (BST) is one of the most interesting thin film ferroelectric materials due to its high dielectric constant, composition dependent Curie temperature and high optical non‐ linearity. The composition dependent Tc enables a maximum infrared response to be ob‐ tained at room temperature. The BST thin films in the paraelectric phase, have characteristics such as good chemical and thermal stability and good insulating properties, due to this nature they are often considered the most suitable capacitor dielectrics for suc‐ cessful fabrication of high density Giga bit (Gbit) scale dynamic random access memories (DRAMs). Compositionally graded ferroelectric films have exhibited properties not previ‐ ously observed in conventional ferroelectric materials. The most notable property of the graded ferroelectric devices or graded Functionally Devices (GFDs) is the large DC polariza‐ tion offset they develop when driven by an alternating electric field. Such GFDs can find ap‐ plications as tunable multilayer capacitors, waveguide phase shifters and filters [7]. Recently, BST thin films were used in the formation of graded ferroelectric devices by de‐

In our work, the Barium strontium titanate (Ba0.05Sr0.95TiO3) ferroelectric thin films were pre‐ pared on single crystal [001] MgO substrates using the pulsed laser deposition method. The refractive index of BST (Ba0.05Sr0.95TiO3) thin films is determined in the wavelength range be‐ tween 1450-1580 nm at the room temperature. The dispersion curve is found to decrease

and reproduction in any medium, provided the original work is properly cited.

© 2013 Bain and Chand; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

A.K. Bain and Prem Chand

http://dx.doi.org/10.5772/52098

**1. Introduction**


**Chapter 8**

### **Optical Properties of Ferroelectrics and Measurement Procedures**

### A.K. Bain and Prem Chand

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52098

### **1. Introduction**

[43] International Center for Diffraction Data [ICDD]. (2008). Powder Diffraction Fails,

[44] Sawyer, C. B., & Tower, C. H. (1930). Rochelle Salt as a Dielectric. *Phys. Rev*, 35(3),

[45] Imaoka, M., & Yamazaki, T. (1972). Glass-formation ranges of ternary systems. *III. Borates containing b-group elements, Rep. Inst. Ind. Sci. Univ. Tokyo*, 22(3), 173-212.

[46] Janakirama-Rao, Bh. V. (1965). Unusual properties and structure of glasses in the sys‐ tem BаO-B2O3-SrO, BаO-B2O3-Bi2O3, BаO-B2O3-ZnO, BаO-B2O3-PbO, Compt. *Rend. VII*

[47] Izumitani, T. (1958). Fundamental studies on new optical glasses,. *Rep. Governm. Ind.*

[48] Kawanaka, Y, & Matusita, K. (2007). Various properties of bismuth oxide glasses for PbO free new solder glasses. *Proc.XXIth Intern.Congr.on Glass (in CD-ROM), Stras‐*

[49] Milyukov, E. M., Vilchinskaya, N. N., & Makarova, T. M. (1982). Optical constants and some other characteristics of glasses in the systems BaO-Bi2O3 and La2O3-Bi2O3-

[50] Sarkisov, P. D. (1997). The Directed Glass Crystallization- Basis for Development of Multipurpose Glass Ceramics,. *Mendeleev Russian Chemical Technological University*

[51] Damjanovic, D. (2005). Hysteresis in Piezoelectric and Ferroelectric Materials,. In: The Science of Hysteresis, Mayergoyz, I & Bertotti, G Elsevier, 0-12480-874-3 , 3,

[52] Rawson, H. (1967). Inorganic glass-forming systems. *Academic Press, London & New*

PDF-2 release database,. *Pennsylvania, USA*, 1084-3116.

*Congr. Intern. Du Verre, Bruxelles*, 1(104-104).

B2O3. Fizika i Khimiya Stekla in Russian)., 8(3), 347-349.

*Res. Inst. Osaka* [311], 127.

*Press, Moscow(In Russian*.

269-273.

162 Advances in Ferroelectrics

*bourg* [I23].

337-465.

*York*, 312.

It is well known that the optical properties of ferroelectric materials find wide ranging appli‐ cations in laser devices. Particularly in the recent years, there has been tremendous interest in the investigation of the nonlinear optical properties of ferroelectric thin films [1-5] for pla‐ nar waveguide and integrated –optic devices. A new class of thin film waveguides has been developed using BaTiO3 thin films deposited on MgO substrates [6]. Barium strontium tita‐ nate Ba1-xSrxTiO3 (BST) is one of the most interesting thin film ferroelectric materials due to its high dielectric constant, composition dependent Curie temperature and high optical non‐ linearity. The composition dependent Tc enables a maximum infrared response to be ob‐ tained at room temperature. The BST thin films in the paraelectric phase, have characteristics such as good chemical and thermal stability and good insulating properties, due to this nature they are often considered the most suitable capacitor dielectrics for suc‐ cessful fabrication of high density Giga bit (Gbit) scale dynamic random access memories (DRAMs). Compositionally graded ferroelectric films have exhibited properties not previ‐ ously observed in conventional ferroelectric materials. The most notable property of the graded ferroelectric devices or graded Functionally Devices (GFDs) is the large DC polariza‐ tion offset they develop when driven by an alternating electric field. Such GFDs can find ap‐ plications as tunable multilayer capacitors, waveguide phase shifters and filters [7]. Recently, BST thin films were used in the formation of graded ferroelectric devices by de‐ positing successive layers of BST with different Ba/Sr ratios [8].

In our work, the Barium strontium titanate (Ba0.05Sr0.95TiO3) ferroelectric thin films were pre‐ pared on single crystal [001] MgO substrates using the pulsed laser deposition method. The refractive index of BST (Ba0.05Sr0.95TiO3) thin films is determined in the wavelength range be‐ tween 1450-1580 nm at the room temperature. The dispersion curve is found to decrease

© 2013 Bain and Chand; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

gradually with increasing wavelength. The average value of the refractive index is found to be 1.985 in the wavelength range between 1450-1580 nm which is important for optoelec‐ tronic device applications [9].

temperature of 780 o

C and then annealed at 650 o

analysis revealed that the films are oriented with [001] parallel to the substrate [001] axis and thus normal to the plane of the films [9]. The films were grown to a thickness of 430 nm.

**Figure 1.** Schematic diagram of the experimental setup for the measurement of refractive index of the Ba0.05Sr0.95TiO3 thin films at room temperature. He-Ne Laser for alignment (1), Lenses (2), Polarizer (3), sample holder (4), BST sample

Figure 1 shows the schematic diagram of the experimental setup for the measurement of the refractive index of Ba0.05Sr0.95TiO3 thin films on MgO substrates through a reflection method. The He-Ne laser beam is used as a source of light to setup the alignment of the reflected beam of light from the samples to the detector. The incident beam is allowed to pass through a polarizer onto the sample. The reflected light is then passed through the same po‐

on the detector that is at 90o to the reflected/incident beam of light. The reflectivity measure‐ ment of the black metal, mirror, MgO substrate and Ba0.05Sr0.95TiO3 thin films were carried using the Agilent 8164A Light Wave Measurement system in the wavelength region of

The refractive index of substrate MgO is taken to be 1.7 [9]. The reflectivity of Ba0.05Sr0.95TiO3 film is then normalized with respect to the mirror. The value of refractive index is derived from model described in ref. [20]. The fitting is done with the calculated data of the reflectiv‐ ity of Ba0.05Sr0.95TiO3 in the wavelength range between 1450-1580 nm. Figure 2 shows the re‐ fractive index of Ba0.05Sr0.95TiO3 thin films as a function of wavelength at room temperature.

relative to the incident light and finally allowed to fall

(BST) and other materials of ferro‐

(5), Agilent light wave measurement system (6) and Tunable Laser (7).

The refractive index of Ba0.05Sr0.95TiO3 with BaxSr1-xTiO3

electric thin films at different wavelengths are presented in table 1.

larizing beam splitter oriented at 45o

1450-1580 nm at room temperature.

C for 55 min. The x-ray diffraction (XRD)

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165

Optical Properties of Ferroelectrics and Measurement Procedures

1: Present address: Director, PM College of Engineering, Kami Road, Sonepat-131001.

2: Present Address: 99 Cyril Road, Small Heath, Birmingham B10 0ST.

Lithium heptagermanate Li2Ge7O15 (LGO) is regarded as a weak ferroelectric and its curie point Tc is 283.5K [10,11]. Due to its intermediate behaviour between order-disorder and dis‐ placive types in a conventional grouping of ferroelectric materials LGO remains a subject of interest from both the theoretical and the application point of view. The paraelectric phase above Tc is orthorhombic D14 2h ~ pbcn and below Tc the ferroelectric phase is C5 2v ~ pbc21 with four formula units in a unit cell in both the phases. Below Tc LGO shows dielectric hys‐ teresis loop and the permittivity shows a sharp peak at Tc [10-12]. Below Tc the spontaneous polarization appears along the c-axis. Many interesting physical properties of LGO such as birefringence [13], elastic behaviour [14], thermal expansion [11], dielectric susceptibility [12,15, 16] and photoluminescence [17] exhibit strong anomalies around Tc. The optical properties, however vary only to such a small degree that the transition could not be detect‐ ed with the aid of a standard polarization microscope [13]. Employing a high resolution po‐ larization device, Kaminsky and HaussÜhl [13] studied the birefringence in LGO near Tc and observed anomalies at the phase transition.

The study of piezo-optic dispersion of LGO (un-irradiated and x-irradiated) in the visible re‐ gion of the spectrum of light at room temperature (RT=298 K) shows an optical zone/ window in between 5400Å and 6200Å with an enhanced piezo-optical behavior [18]. The temperature dependence of the photoelastic coefficients of the ferroelectric crystals Li2Ge7O15 (both un-irradiated and x-irradiated) in a cooling and a heating cycle between room temperature and 273K shows an interesting observation including the lowering of the Tc under uniaxial stress contrary to the increase of Tc under hydrostatic pressure and obser‐ vation of thermal photoelastic hysteresis similar to dielectric behavior [19]. The study of flu‐ orescence spectra of the crystals Li2Ge7O15:Cr3+ in the temperature interval 77-320 K shows the sharply decrease of intensities of the R1 and R2 lines (corresponding to the Cr3+ ions of types I and II) during cooling process near the temperature Tc = 283.5 K[16].

The present chapter includes optical properties of the ferroelectric BST thin films and the Lithium heptagermanate (Li2Ge7O15) single crystals, fabrication methods, measure‐ ment procedures of the refractive index of BST thin films on MgO substrates, the fluo‐ rescence spectra and the photoelastic coefficients of LGO single crystals (un-irradiated and x-irradiated) at different wave lengths and temperatures around the phase transi‐ tion temperature Tc. The potential of these materials for practical applications in the op‐ to-electronic devices will also be discussed.

### **1.1. Optical property of Barium strontium titanate thin films**

The Barium strontium titanate (Ba0.05Sr0.95TiO3) ferroelectric thin films were prepared on sin‐ gle crystal MgO substrates using the pulsed laser deposition (PLD) method at a substrate temperature of 780 o C and then annealed at 650 o C for 55 min. The x-ray diffraction (XRD) analysis revealed that the films are oriented with [001] parallel to the substrate [001] axis and thus normal to the plane of the films [9]. The films were grown to a thickness of 430 nm.

gradually with increasing wavelength. The average value of the refractive index is found to be 1.985 in the wavelength range between 1450-1580 nm which is important for optoelec‐

Lithium heptagermanate Li2Ge7O15 (LGO) is regarded as a weak ferroelectric and its curie point Tc is 283.5K [10,11]. Due to its intermediate behaviour between order-disorder and dis‐ placive types in a conventional grouping of ferroelectric materials LGO remains a subject of interest from both the theoretical and the application point of view. The paraelectric phase above Tc is orthorhombic D14 2h ~ pbcn and below Tc the ferroelectric phase is C5 2v ~ pbc21 with four formula units in a unit cell in both the phases. Below Tc LGO shows dielectric hys‐ teresis loop and the permittivity shows a sharp peak at Tc [10-12]. Below Tc the spontaneous polarization appears along the c-axis. Many interesting physical properties of LGO such as birefringence [13], elastic behaviour [14], thermal expansion [11], dielectric susceptibility [12,15, 16] and photoluminescence [17] exhibit strong anomalies around Tc. The optical properties, however vary only to such a small degree that the transition could not be detect‐ ed with the aid of a standard polarization microscope [13]. Employing a high resolution po‐ larization device, Kaminsky and HaussÜhl [13] studied the birefringence in LGO near Tc

The study of piezo-optic dispersion of LGO (un-irradiated and x-irradiated) in the visible re‐ gion of the spectrum of light at room temperature (RT=298 K) shows an optical zone/ window in between 5400Å and 6200Å with an enhanced piezo-optical behavior [18]. The temperature dependence of the photoelastic coefficients of the ferroelectric crystals Li2Ge7O15 (both un-irradiated and x-irradiated) in a cooling and a heating cycle between room temperature and 273K shows an interesting observation including the lowering of the Tc under uniaxial stress contrary to the increase of Tc under hydrostatic pressure and obser‐ vation of thermal photoelastic hysteresis similar to dielectric behavior [19]. The study of flu‐ orescence spectra of the crystals Li2Ge7O15:Cr3+ in the temperature interval 77-320 K shows the sharply decrease of intensities of the R1 and R2 lines (corresponding to the Cr3+ ions of

The present chapter includes optical properties of the ferroelectric BST thin films and the Lithium heptagermanate (Li2Ge7O15) single crystals, fabrication methods, measure‐ ment procedures of the refractive index of BST thin films on MgO substrates, the fluo‐ rescence spectra and the photoelastic coefficients of LGO single crystals (un-irradiated and x-irradiated) at different wave lengths and temperatures around the phase transi‐ tion temperature Tc. The potential of these materials for practical applications in the op‐

The Barium strontium titanate (Ba0.05Sr0.95TiO3) ferroelectric thin films were prepared on sin‐ gle crystal MgO substrates using the pulsed laser deposition (PLD) method at a substrate

types I and II) during cooling process near the temperature Tc = 283.5 K[16].

1: Present address: Director, PM College of Engineering, Kami Road, Sonepat-131001.

2: Present Address: 99 Cyril Road, Small Heath, Birmingham B10 0ST.

tronic device applications [9].

164 Advances in Ferroelectrics

and observed anomalies at the phase transition.

to-electronic devices will also be discussed.

**1.1. Optical property of Barium strontium titanate thin films**

**Figure 1.** Schematic diagram of the experimental setup for the measurement of refractive index of the Ba0.05Sr0.95TiO3 thin films at room temperature. He-Ne Laser for alignment (1), Lenses (2), Polarizer (3), sample holder (4), BST sample (5), Agilent light wave measurement system (6) and Tunable Laser (7).

Figure 1 shows the schematic diagram of the experimental setup for the measurement of the refractive index of Ba0.05Sr0.95TiO3 thin films on MgO substrates through a reflection method. The He-Ne laser beam is used as a source of light to setup the alignment of the reflected beam of light from the samples to the detector. The incident beam is allowed to pass through a polarizer onto the sample. The reflected light is then passed through the same po‐ larizing beam splitter oriented at 45o relative to the incident light and finally allowed to fall on the detector that is at 90o to the reflected/incident beam of light. The reflectivity measure‐ ment of the black metal, mirror, MgO substrate and Ba0.05Sr0.95TiO3 thin films were carried using the Agilent 8164A Light Wave Measurement system in the wavelength region of 1450-1580 nm at room temperature.

The refractive index of substrate MgO is taken to be 1.7 [9]. The reflectivity of Ba0.05Sr0.95TiO3 film is then normalized with respect to the mirror. The value of refractive index is derived from model described in ref. [20]. The fitting is done with the calculated data of the reflectiv‐ ity of Ba0.05Sr0.95TiO3 in the wavelength range between 1450-1580 nm. Figure 2 shows the re‐ fractive index of Ba0.05Sr0.95TiO3 thin films as a function of wavelength at room temperature. The refractive index of Ba0.05Sr0.95TiO3 with BaxSr1-xTiO3 (BST) and other materials of ferro‐ electric thin films at different wavelengths are presented in table 1.

structure associated with the larger lattice parameter and variations in atomic co-ordination

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167

Single crystals of Li2Ge7O15 are grown in an ambient atmosphere by Czochralski method from stoichiometric melt, employing a resistance heated furnace. Stoichiometric mixture of powdered Li2CO3 and GeO2 in the ratio of 1.03 and 7.0 respectively was heated at 1100 K for 24 hours to complete the solid state reaction for the raw material for the crystal growth. The crystals were grown by rotating the seed at the rate of 50 rpm with a pulling rate of 1.2 mm/ hour. The cooling rate of temperature in the process of growth was 0.8-1.2 K/hour. The crys‐ tals grown were colourless, fully transparent and of optical quality. The crystal axes were

**Figure 3.** Schematic diagram of the experimental setup for the measurement of fluorescence spectra of the crystals Li2Ge7O15:Cr3+. He-Ne Laser for radiation (1), Glen prism (2), crystal sample (3), condenser (4), polarizer (5), spectro‐

The desired impurities such as Cr+3, Mn+2, Bi+2, Cu2+ and Eu+2 etc are also introduced in de‐ sired concentration by mixing the appropriate amount of the desired anion salt in the growth mixture. The crystal structure of LGO above Tc is orthorhombic (psedohexagonal) with the space group D14 2h (Pbcn). The cell parameters are a: 7.406 Å, b: 16.696 Å, c: 9.610 Å, Z = 4 and b~√3c. Below Tc a small value of spontaneous polarization occurs along c-axis and the ferroelectric phase belongs to C5 2v (Pbc21) space group. The crystal structure contains strongly packed layers of GeO4 tetrahedra linked by GeO6-octahedra to form a three dimen‐ sionally bridged frame work in which Li atoms occupy the positions in the vacant channels extending three dimensionally [14, 28, 29]. The size of the unit cell (Z = 4) does not change at

[27] that is local relaxations.

**1.2. Growth and structure of Li2Ge7O15 Crystals**

determined by x-ray and optical methods.

graph (6) Multichannel analyzer(7) and computer (8).

**Figure 2.** The variation of refractive index of Ba0.05Sr0.95TiO3 thin films as a function of wavelength.


**Table 1.** The refractive index of BST and other ferroelectric thin films.

As shown in Figure 2, the dispersion curve decreases gradually with increasing wavelength. The average value of the refractive index is found to be ~ 1.985 in the wavelength range of 1450-1580 nm which is important for optoelectronic device (optical waveguide) applications. The variation of refractive index is attributed predominantly to the changes of electronic structure associated with the larger lattice parameter and variations in atomic co-ordination [27] that is local relaxations.

### **1.2. Growth and structure of Li2Ge7O15 Crystals**

**Figure 2.** The variation of refractive index of Ba0.05Sr0.95TiO3 thin films as a function of wavelength.

166 Advances in Ferroelectrics

**Table 1.** The refractive index of BST and other ferroelectric thin films.

**Sample Name Refractive index Wavelength Remarks** BaxSr1-xTiO3(x=0%) 2.37 600 nm (RT) Ref. [21] BaxSr1-xTiO3(x=30%) 2.42 600 nm (RT) Ref. [21] BaxSr1-xTiO3(x=50%) 2.37 600 nm (RT) Ref. [21] BaxSr1-xTiO3(x=70%) 2.34 600 nm (RT) Ref. [21] BaxSr1-xTiO3(x=50%) 2.45 470 nm (RT) Ref. [22] BaxSr1-xTiO3(x=80%) 2.27 430 nm (RT) Ref. [23] PbZr1-xTixO3 (PZT) 2.87 400 nm (300 0C) Ref. [24] PbZr1-xTixO3 (PZT) 2.82 400 nm (50 0C) Ref. [24] PbZr1-xTixO3 (PZT) 2.66 500 nm (300 0C) Ref. [24] PbZr1-xTixO3 (PZT) 2.67 500 nm (50 0C) Ref. [24] PbTiO3 3.10 470 nm (RT) Ref. [25] PbTiO3 2.80 490 nm (RT) Ref. [25] PbTiO3 2.75 550 nm (RT) Ref. [25] SrxBa1-xNb2O6 (x=61%) 2.35 400 nm (RT) Ref. [26]

As shown in Figure 2, the dispersion curve decreases gradually with increasing wavelength. The average value of the refractive index is found to be ~ 1.985 in the wavelength range of 1450-1580 nm which is important for optoelectronic device (optical waveguide) applications. The variation of refractive index is attributed predominantly to the changes of electronic

Single crystals of Li2Ge7O15 are grown in an ambient atmosphere by Czochralski method from stoichiometric melt, employing a resistance heated furnace. Stoichiometric mixture of powdered Li2CO3 and GeO2 in the ratio of 1.03 and 7.0 respectively was heated at 1100 K for 24 hours to complete the solid state reaction for the raw material for the crystal growth. The crystals were grown by rotating the seed at the rate of 50 rpm with a pulling rate of 1.2 mm/ hour. The cooling rate of temperature in the process of growth was 0.8-1.2 K/hour. The crys‐ tals grown were colourless, fully transparent and of optical quality. The crystal axes were determined by x-ray and optical methods.

**Figure 3.** Schematic diagram of the experimental setup for the measurement of fluorescence spectra of the crystals Li2Ge7O15:Cr3+. He-Ne Laser for radiation (1), Glen prism (2), crystal sample (3), condenser (4), polarizer (5), spectro‐ graph (6) Multichannel analyzer(7) and computer (8).

The desired impurities such as Cr+3, Mn+2, Bi+2, Cu2+ and Eu+2 etc are also introduced in de‐ sired concentration by mixing the appropriate amount of the desired anion salt in the growth mixture. The crystal structure of LGO above Tc is orthorhombic (psedohexagonal) with the space group D14 2h (Pbcn). The cell parameters are a: 7.406 Å, b: 16.696 Å, c: 9.610 Å, Z = 4 and b~√3c. Below Tc a small value of spontaneous polarization occurs along c-axis and the ferroelectric phase belongs to C5 2v (Pbc21) space group. The crystal structure contains strongly packed layers of GeO4 tetrahedra linked by GeO6-octahedra to form a three dimen‐ sionally bridged frame work in which Li atoms occupy the positions in the vacant channels extending three dimensionally [14, 28, 29]. The size of the unit cell (Z = 4) does not change at the phase transition and ferroelectric phase transition is associated with a relaxational mode as well as the soft phonon [30]. Activation of the pure crystals with impurity ions will de‐ mand charge compensating mechanism through additional defects in the pure lattice.

sample. The recording of fluorescence spectra were carried out by the optical multichannel analyzer in combination with the polychromator. The radiation beam was initially polarized with glen prism. The plane Polaroid was used as an analyzer that was placed before the in‐

The fluorescence spectra consist of narrow intensity lines referred to R1 and R2 with frequen‐ cies γ1~14348 cm-1 and γ2 ~14572 cm-1. These lines split further into two components each R1 /

wavelength region/zone is observed in the spectra. It may be related with the effect of elec‐ tron-photon interaction. It is known that Cr3+ doping ions in the structure of Li2Ge7O15:Cr3+ crystals substitute the Ge4+ host ions within oxygen octahedral (GeO6) complexes [31-36]. The optical spectra of Cr3+ ions shows the existence of two types of Cr3+ centre (type I and II with different values of effective g-factor) as observed in EPR (Electron Paramagnetic Reso‐ nance) spectra of Cr3+ ions in ferroelectric phase of the crystals Li2Ge7O15:Cr3+ [32, 33]. Two

E (at T=77 K, its positions are R1=14348 cm-1, R1 /

spectra of the crystals Li2Ge7O15:Cr3+ as shown in Fig. 4. Actually the two different types of

**Figure 6.** Part of EPR spectrum of Cr3+ doped LGO crystal in an arbitrary orientation at RT. The four EPR signals are

The intensity of fluorescence of the R1 and R2 lines of the crystals Li2Ge7O15:Cr3+ were studied near the phase transition temperature Tc at the direction E┴[001]. It is observed that the in‐ tensity of R1 and R2 lines are decreased sharply near the phase transition temperature Tc but at high temperature region (T>Tc) the intensity again increases as shown in figure 5. Such

level are duplicate [37, 38] and conform to the EPR observations.

=14593 cm-1) are observed at low temperature region (T<190 K) in the optical

) with pretty different positions below Ē and above 2Ᾱ levels of the

Optical Properties of Ferroelectrics and Measurement Procedures

=14402 cm-1, R2=14572

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169

respectively at lowering the temperature towards 77 K. Besides this, a wide long

put aperture of the polychromator.

and R2 /

pair of R lines 4A2 - 2

Cr3+ centers (R and R/

attributed to four distinct Cr3+ sites per unit cell of LGO.

cm-1 and R2 /

excited E2

**Figure 4.** Fluorescence Spectra of the crystals Li2Ge7O15:Cr3+ at T=77 K.

**Figure 5.** Temperature dependence of intensity of R1 and R2 Lines of fluorescence in Li2Ge7O15:Cr3+ crystals near the phase transition temperature Tc.

### **1.3. Study of fluorescence spectra of Li2Ge7O15:Cr3+ crystals**

The fluorescence spectra of the crystals Li2Ge7O15:Cr3+ were studied in the temperature inter‐ val 77-320 K including the phase transition temperature Tc = 283.5 K. The experimental set to record the fluorescence spectra of the crystals Li2Ge7O15:Cr3+ is shown in fig.3. A laser with the pair of mode (λ1=510.6 nm, λ2=578.2 nm) was used as a source of excitation of the crystal sample. The recording of fluorescence spectra were carried out by the optical multichannel analyzer in combination with the polychromator. The radiation beam was initially polarized with glen prism. The plane Polaroid was used as an analyzer that was placed before the in‐ put aperture of the polychromator.

the phase transition and ferroelectric phase transition is associated with a relaxational mode as well as the soft phonon [30]. Activation of the pure crystals with impurity ions will de‐

**Figure 5.** Temperature dependence of intensity of R1 and R2 Lines of fluorescence in Li2Ge7O15:Cr3+ crystals near the

The fluorescence spectra of the crystals Li2Ge7O15:Cr3+ were studied in the temperature inter‐ val 77-320 K including the phase transition temperature Tc = 283.5 K. The experimental set to record the fluorescence spectra of the crystals Li2Ge7O15:Cr3+ is shown in fig.3. A laser with the pair of mode (λ1=510.6 nm, λ2=578.2 nm) was used as a source of excitation of the crystal

mand charge compensating mechanism through additional defects in the pure lattice.

**Figure 4.** Fluorescence Spectra of the crystals Li2Ge7O15:Cr3+ at T=77 K.

**1.3. Study of fluorescence spectra of Li2Ge7O15:Cr3+ crystals**

phase transition temperature Tc.

168 Advances in Ferroelectrics

The fluorescence spectra consist of narrow intensity lines referred to R1 and R2 with frequen‐ cies γ1~14348 cm-1 and γ2 ~14572 cm-1. These lines split further into two components each R1 / and R2 / respectively at lowering the temperature towards 77 K. Besides this, a wide long wavelength region/zone is observed in the spectra. It may be related with the effect of elec‐ tron-photon interaction. It is known that Cr3+ doping ions in the structure of Li2Ge7O15:Cr3+ crystals substitute the Ge4+ host ions within oxygen octahedral (GeO6) complexes [31-36]. The optical spectra of Cr3+ ions shows the existence of two types of Cr3+ centre (type I and II with different values of effective g-factor) as observed in EPR (Electron Paramagnetic Reso‐ nance) spectra of Cr3+ ions in ferroelectric phase of the crystals Li2Ge7O15:Cr3+ [32, 33]. Two pair of R lines 4A2 - 2 E (at T=77 K, its positions are R1=14348 cm-1, R1 / =14402 cm-1, R2=14572 cm-1 and R2 / =14593 cm-1) are observed at low temperature region (T<190 K) in the optical spectra of the crystals Li2Ge7O15:Cr3+ as shown in Fig. 4. Actually the two different types of Cr3+ centers (R and R/ ) with pretty different positions below Ē and above 2Ᾱ levels of the excited E2 level are duplicate [37, 38] and conform to the EPR observations.

**Figure 6.** Part of EPR spectrum of Cr3+ doped LGO crystal in an arbitrary orientation at RT. The four EPR signals are attributed to four distinct Cr3+ sites per unit cell of LGO.

The intensity of fluorescence of the R1 and R2 lines of the crystals Li2Ge7O15:Cr3+ were studied near the phase transition temperature Tc at the direction E┴[001]. It is observed that the in‐ tensity of R1 and R2 lines are decreased sharply near the phase transition temperature Tc but at high temperature region (T>Tc) the intensity again increases as shown in figure 5. Such nature of suppression of R1 and R2 lines was not observed previously and it may be related with the mechanism of interaction of excitation spectra of light in the crystals Li2Ge7O15:Cr3+ near the phase transition temperature Tc[17].

The Crystals doped with chromium impurity give EPR (Electron Paramagnetic Resonance) signals characteristics of the trivalent chromium ions [Fig.6]. It is known that impurity Cr3+ ions substitute the Ge4+ host ions within oxygen octahedral in the basic structure of (LGO) crystal [31-36]. Incorporation of tri-positive chromium ions into GeO6-octahedra changes the local symmetry of the lattice site from monoclinic C2 group to triclinic C1 group. The local symmetry lowering is attributed to the effect of the additional Li+ defect required for com‐ pensating the charge misfit of Cr3+ ion at the Ge4+ site. Taking into account a weak coupling of lithium ions with the germanium – oxygen lattice framework, the interstitial Li+ is consid‐ ered to be the most probable charge compensating defect, located within the structural cavi‐ ty near the octahedral CrO6 complex (Fig.7). Subsequent measurements of optical spectra have confirmed the model of Cr3+– Li+ pair centers in the LGO crystal structure [32, 33].

**Figure 8.** Temperature dependence of Imaginary part ε''of permittivity ε=ε'-je'' of LGO:Cr3+, measured along crystal a axis at the following frequencies : 0.5 kHz (1); 1 kHz (2); 5 kHz (3); 10 kHz (4). Distinct peak is observed in ε'' in the temperature range 350-450 K and the peak is observed to shift to higher temperature for higher frequencies. In con‐ trast no such peak is observed for the real part ε' of the permittivity. The typical behavior is observed only along crystal

If a rectangular parallelepiped with edges parallel to x[100], y[010] and z[001] axes is stressed along z-axis and observation is made along y-axis, as shown in Fig.9, then the path retardation δzy introduced per unit length due the stress introduced birefringence is given by

where Δnz and Δn<sup>x</sup> are the changes in the corresponding refractive indices, (Δnz – Δnx) is the corresponding stress induced birefringence, Pzz is the stress along z-axis and Czy is a con‐ stant called the Brewster constant or the relative photoelastic coefficient. In general the Brewster constant is related to the stress optical and strain optical tensors of forth rank [39] and is a measure of the stress induced (piezo-optic) birefringence. It is conveniently ex‐

To study the piezo-optical birefringence the experimental set up consists of a source of light (S), a lens (L) to render the rays parallel, a polarizer (P), an analyzer Polaroid (A), a Babinet compensator (B) and a detector (D), as shown in Fig.10. The P and A combination are adjust‐ ed for optimal rejection of light. The sample with stressing arrangement and a Babinet com‐ pensator are placed between P and A. A monochromator and a gas flow temperature controlling device are used to obtain the piezo-optic coefficients (Cλ) at different wave‐ lengths and temperature. The subscript λ in the symbol C<sup>λ</sup> denotes that the piezo-optic coef‐

*δ*zy = (*Δ*nz – *Δ*nx) =CzyPzz (1)

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/dyne per cm thickness along the direction of observation is

a –axis which incidentally coincides with the proposed Cr3+- Li+ centers.

**1.5. Measurement procedure of photoelastic constants**

**1.4. Principle of Photoelasticity**

pressed in the unit of 10-13 cm2

called a Brewster [39].

**Figure 7.** Physical model of Cr3+ centers in Li2Ge7O15:Cr3+ crystals and its dipole moment d.

The available data make it possible to assume that electric dipole moments of Cr3+– Li+ pairs are directed along the crystal axis "a" of the crystal. Interstitial Li+ ions locally break the symmetry axis C2 of the sites within the oxygen octahedral complexes [34]. As a result, there are two equivalent configurations of the pair centers which are conjugated by broken C2 axis and have dipole moments with opposite orientations. It may be assumed that pair centers can reorient due to thermal activation. Reorientation of the pair centers should be accompa‐ nied by: i) shortening of the configuration life time and ii) switching of defect dipole mo‐ ments [35]. This is reflected in the typical temperature dependence of the imaginary part of dielectric permittivity of chromium doped LGO single crystals [35] along the a-axis of the crystal shown in fig.8.

**Figure 8.** Temperature dependence of Imaginary part ε''of permittivity ε=ε'-je'' of LGO:Cr3+, measured along crystal a axis at the following frequencies : 0.5 kHz (1); 1 kHz (2); 5 kHz (3); 10 kHz (4). Distinct peak is observed in ε'' in the temperature range 350-450 K and the peak is observed to shift to higher temperature for higher frequencies. In con‐ trast no such peak is observed for the real part ε' of the permittivity. The typical behavior is observed only along crystal a –axis which incidentally coincides with the proposed Cr3+- Li+ centers.

### **1.4. Principle of Photoelasticity**

nature of suppression of R1 and R2 lines was not observed previously and it may be related with the mechanism of interaction of excitation spectra of light in the crystals Li2Ge7O15:Cr3+

The Crystals doped with chromium impurity give EPR (Electron Paramagnetic Resonance) signals characteristics of the trivalent chromium ions [Fig.6]. It is known that impurity Cr3+ ions substitute the Ge4+ host ions within oxygen octahedral in the basic structure of (LGO) crystal [31-36]. Incorporation of tri-positive chromium ions into GeO6-octahedra changes the local symmetry of the lattice site from monoclinic C2 group to triclinic C1 group. The local

pensating the charge misfit of Cr3+ ion at the Ge4+ site. Taking into account a weak coupling

ered to be the most probable charge compensating defect, located within the structural cavi‐ ty near the octahedral CrO6 complex (Fig.7). Subsequent measurements of optical spectra

pair centers in the LGO crystal structure [32, 33].

of lithium ions with the germanium – oxygen lattice framework, the interstitial Li+

defect required for com‐

is consid‐

pairs

ions locally break the

near the phase transition temperature Tc[17].

170 Advances in Ferroelectrics

have confirmed the model of Cr3+– Li+

crystal shown in fig.8.

symmetry lowering is attributed to the effect of the additional Li+

**Figure 7.** Physical model of Cr3+ centers in Li2Ge7O15:Cr3+ crystals and its dipole moment d.

are directed along the crystal axis "a" of the crystal. Interstitial Li+

The available data make it possible to assume that electric dipole moments of Cr3+– Li+

symmetry axis C2 of the sites within the oxygen octahedral complexes [34]. As a result, there are two equivalent configurations of the pair centers which are conjugated by broken C2 axis and have dipole moments with opposite orientations. It may be assumed that pair centers can reorient due to thermal activation. Reorientation of the pair centers should be accompa‐ nied by: i) shortening of the configuration life time and ii) switching of defect dipole mo‐ ments [35]. This is reflected in the typical temperature dependence of the imaginary part of dielectric permittivity of chromium doped LGO single crystals [35] along the a-axis of the If a rectangular parallelepiped with edges parallel to x[100], y[010] and z[001] axes is stressed along z-axis and observation is made along y-axis, as shown in Fig.9, then the path retardation δzy introduced per unit length due the stress introduced birefringence is given by

$$
\delta\_{\rm xy} = \left(\Delta \mathbf{n}\_{\rm z} - \Delta \mathbf{n}\_{\rm x}\right) = \mathbf{C}\_{\rm xy} \mathbf{P}\_{\rm xz} \tag{1}
$$

where Δnz and Δn<sup>x</sup> are the changes in the corresponding refractive indices, (Δnz – Δnx) is the corresponding stress induced birefringence, Pzz is the stress along z-axis and Czy is a con‐ stant called the Brewster constant or the relative photoelastic coefficient. In general the Brewster constant is related to the stress optical and strain optical tensors of forth rank [39] and is a measure of the stress induced (piezo-optic) birefringence. It is conveniently ex‐ pressed in the unit of 10-13 cm2 /dyne per cm thickness along the direction of observation is called a Brewster [39].

### **1.5. Measurement procedure of photoelastic constants**

To study the piezo-optical birefringence the experimental set up consists of a source of light (S), a lens (L) to render the rays parallel, a polarizer (P), an analyzer Polaroid (A), a Babinet compensator (B) and a detector (D), as shown in Fig.10. The P and A combination are adjust‐ ed for optimal rejection of light. The sample with stressing arrangement and a Babinet com‐ pensator are placed between P and A. A monochromator and a gas flow temperature controlling device are used to obtain the piezo-optic coefficients (Cλ) at different wave‐ lengths and temperature. The subscript λ in the symbol C<sup>λ</sup> denotes that the piezo-optic coef‐ ficient depends on the wavelength of light used to measure it. The experiments are carried out for different wavelengths using white light and a monochromator and the monochro‐ matic sodium yellow light. An appropriate stress along a desired direction of the sample is applied with the help of a stressing apparatus comprising a mechanical lever and load.

**1.6. Piezo-optic Dispersion of Li2Ge7O15 Crystals**

of piezo-optic coefficients is observed.

The experimental procedure for the piezo-optic measurements is described in section 1.5. The polished optical quality samples worked out to dimensions i) 5.9 mm, 9.4 mm and 5.0 mm; ii) 3.17 mm, 5.88 mm and 6.7 mm, along the crystallographic a, b and c axes respective‐

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The values of C*λ* thus obtained at different wavelengths are given in Table 2 and the results are plotted in Fig. 11. Here Cpq is the piezo-optic coefficient with the stress direction being p and observation direction being q. The results show an interesting piezo-optic behavior. A survey of literature indicates that the piezo-optic behavior of materials studied till now shows a reduction of C*<sup>λ</sup>* with increasing wavelength in the visible region [39]. In the present case, C*λ* decreases with wavelength up to a certain wavelength as in other normal materials and then suddenly shows a peak and later on the usual behavior of reduction in the values

**Wavelengths**

To the best knowledge of the authors this behavior is unique to the LGO crystals. For the sake of convenience we denote C*λ* measured at *λ* = 5890 Å as C5890 and so on. The results show that sometimes the value of C5890 is even higher than that at C4400, the value of piezooptic coefficient obtained at the lowest wavelength studied here. This is the case with Cxy, Czx and Cxz. For other orientations the value is lower than that at 4400 Å. Further, C*λ* is found to have increased to more than 50% in the case of stress along [001] and observation along [100]. Also, it is interesting to note that the value of C6140, is less than that of C5390, in tune with usual observation of piezo-optic dispersion. Thus one can see an "optical window" in between 5400 Å and 6200 Å. The height of this optical window is different for various orien‐ tations, though the width seems approximately the same. The maximum height of about 1.5 Brewster was found for Czx followed by Cxz with about 0.9 Brewster. It should be noted here that z-axis is the ferroelectric axis for LGO. It is also interesting to note that the change in height is more in the former while the actual value of C*λ*, is less compared to that of the lat‐ ter. The percentage dispersion also is different for various orientations. It is very high, as

Obs. Cpq 4358**Å** 4880**Å** 5390**Å** 5890**Å** 6140**Å** Cxy 4.024 3.819 3.722 4.328 3.677 Cxz 5.243 4.895 4.770 5.552 4.451 Cyx 4.084 3.525 3.092 3.562 2.913 Cyz 4.353 4.118 3.946 4.261 3.866 Czy 4.179 2.814 3.177 3.713 3.172 Czx 3.312 2.991 2.650 4.190 2.618

**Table 2.** Stress optical coefficients cpq (in Brewster) for Li2Ge7015 at different wave lengths.

high as 25% for Czy, while it is just 10% for Cxy.

ly. The stress was applied with an effective load of ~23 kg in each case [40].

**Figure 9.** A solid under a linear stress of stress-optical measurements (Pzz is the applied stress and LL is the direction of light propagation and observation).

**Figure 10.** A schematic diagram of the experimental setup for the measurement of photoelastic constants of the crys‐ tals at room temperature. Source of light (S), Lense (L), Polarizer (P), Crystals (C) under stress, Babinet Compensator (B), Analyzer (A) and Detector (D).

To start with, the Babinet compensator is calibrated and the fringe width is determined for different wavelengths of light in the visible region. The crystal specimen is placed on the stressing system so that the stress could be applied along vertical axis and observation made along horizontal axis. A load on the crystal shifts the fringe in the Babinet compensator and this shift is a measure of the piezo-optic behavior. The piezo-optic coefficients (Cλ) are now calculated using the calibration of the Babinet compensator. The experiment is repeated for other orientations of the crystals and the results are obtained.

### **1.6. Piezo-optic Dispersion of Li2Ge7O15 Crystals**

ficient depends on the wavelength of light used to measure it. The experiments are carried out for different wavelengths using white light and a monochromator and the monochro‐ matic sodium yellow light. An appropriate stress along a desired direction of the sample is applied with the help of a stressing apparatus comprising a mechanical lever and load.

**Figure 9.** A solid under a linear stress of stress-optical measurements (Pzz is the applied stress and LL is the direction of

**Figure 10.** A schematic diagram of the experimental setup for the measurement of photoelastic constants of the crys‐ tals at room temperature. Source of light (S), Lense (L), Polarizer (P), Crystals (C) under stress, Babinet Compensator (B),

To start with, the Babinet compensator is calibrated and the fringe width is determined for different wavelengths of light in the visible region. The crystal specimen is placed on the stressing system so that the stress could be applied along vertical axis and observation made along horizontal axis. A load on the crystal shifts the fringe in the Babinet compensator and this shift is a measure of the piezo-optic behavior. The piezo-optic coefficients (Cλ) are now calculated using the calibration of the Babinet compensator. The experiment is repeated for

other orientations of the crystals and the results are obtained.

light propagation and observation).

172 Advances in Ferroelectrics

Analyzer (A) and Detector (D).

The experimental procedure for the piezo-optic measurements is described in section 1.5. The polished optical quality samples worked out to dimensions i) 5.9 mm, 9.4 mm and 5.0 mm; ii) 3.17 mm, 5.88 mm and 6.7 mm, along the crystallographic a, b and c axes respective‐ ly. The stress was applied with an effective load of ~23 kg in each case [40].

The values of C*λ* thus obtained at different wavelengths are given in Table 2 and the results are plotted in Fig. 11. Here Cpq is the piezo-optic coefficient with the stress direction being p and observation direction being q. The results show an interesting piezo-optic behavior. A survey of literature indicates that the piezo-optic behavior of materials studied till now shows a reduction of C*<sup>λ</sup>* with increasing wavelength in the visible region [39]. In the present case, C*λ* decreases with wavelength up to a certain wavelength as in other normal materials and then suddenly shows a peak and later on the usual behavior of reduction in the values of piezo-optic coefficients is observed.


**Table 2.** Stress optical coefficients cpq (in Brewster) for Li2Ge7015 at different wave lengths.

To the best knowledge of the authors this behavior is unique to the LGO crystals. For the sake of convenience we denote C*λ* measured at *λ* = 5890 Å as C5890 and so on. The results show that sometimes the value of C5890 is even higher than that at C4400, the value of piezooptic coefficient obtained at the lowest wavelength studied here. This is the case with Cxy, Czx and Cxz. For other orientations the value is lower than that at 4400 Å. Further, C*λ* is found to have increased to more than 50% in the case of stress along [001] and observation along [100]. Also, it is interesting to note that the value of C6140, is less than that of C5390, in tune with usual observation of piezo-optic dispersion. Thus one can see an "optical window" in between 5400 Å and 6200 Å. The height of this optical window is different for various orien‐ tations, though the width seems approximately the same. The maximum height of about 1.5 Brewster was found for Czx followed by Cxz with about 0.9 Brewster. It should be noted here that z-axis is the ferroelectric axis for LGO. It is also interesting to note that the change in height is more in the former while the actual value of C*λ*, is less compared to that of the lat‐ ter. The percentage dispersion also is different for various orientations. It is very high, as high as 25% for Czy, while it is just 10% for Cxy.

direction is on Gd2(Mo04)3 — where an anomalous peak was recorded in spontaneous bire‐

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**Figure 12.** The variation of Czx(λ) at the temperatures ranging from 298 K to 283 K on cooling process of the sample

It is well known that the photoelasticity in crystals arises due to change in number of oscilla‐ tors, effective electric field due to strain and the polarisability of the ions. In the present case, as the wavelength approaches around 5400 Å, the ionic polarisability seems to be changing enormously. There is no optical dispersion data available on LGO. We have conducted an experiment on transmission spectra of LGO along x, y and z-axes, which shows a strong ab‐ sorption around 5400 Å. The observed anomaly in the piezo-optic dispersion may be attrib‐ uted to the absorption edge falling in this region. This explanation needs further investigation in this direction. It is also known that the strain optical dispersion arises due to the shift in absorption frequencies and a change in the oscillator strength caused by the

The ferroelectric single crystals Li2Ge7O15 was irradiated by x-ray for one hour and the ex‐ perimental processes described in section 1.5 were repeated for the crystal (irradiated) LGO in order to understand the radiation effect on piezo-optical birefringence dispersion [18]. The values of Cλ of the crystal (irradiated) LGO thus obtained at different wavelengths are

**1.7. Irradiation Effect on Piezo-optic Dispersion of Li2Ge7O15 Crystals**

given in Table 3 and the results are plotted in Fig. 13.

fringence at 334.7 nm [41], an observation made for the first time.

Li2Ge7015.

physical strain in the crystal.

**Figure 11.** Stress optical dispersion of Li2Ge7015 crystals with wavelength at room temperature (298 K).

Figure 12 shows the variation of Czx(*λ*) at the temperatures ranging from 298K to 283K on cooling process of the sample LGO. It is clear from the figure that the distinct peak of Czx(*λ*) appears only at the sodium yellow wavelength of 5890 Å for the whole range of tempera‐ tures (298 K–283 K) investigated. It is also interesting to note that a temperature anomaly is also observed around 283 K. LGO undergoes a second order phase transition at 283.5 K from the high temperature paraelectric phase to the low temperature ferroelectric phase. So this anomaly is related to this phase transition of the LGO crystal.

The observed peculiarity of piezo-optic behavior could be due to many factors, viz., i) anom‐ alous behavior of refractive index or birefringence ii) anomalous ferroelastic transformation at some stage of loading iii) shift of absorption edge due to loading. The following have been done to identify the reasons for this peculiar behaviour.

Birefringence dispersion has been investigated in the visible region and no anomalies in its behavior has been observed. This rules out the first of the reasons mentioned. The reason due to ferroelastic behavior also is ruled out since the effect would be uniform over all the wavelengths investigated. It was not possible to investigate the effect of load on the absorp‐ tion edge. Hence an indirect experiment has been performed. If there is a shift in the absorp‐ tion edge due to loading the sample, the peak observed now at sodium yellow light would shift with load. No clear shift of the peak could be observed within the experimental limits. Another interesting experiment was done to identify the source of the anomaly. It is well known that Tc of LGO changes under uniaxial stress. The measurements were made near Tc under different stress (loads). Although Tc was found to shift a little with load the dispersion peak did not show any discernible shift. No particular reason could be established as to why a dispersion peak appears around sodium yellow region. Another interesting work in this direction is on Gd2(Mo04)3 — where an anomalous peak was recorded in spontaneous bire‐ fringence at 334.7 nm [41], an observation made for the first time.

**Figure 12.** The variation of Czx(λ) at the temperatures ranging from 298 K to 283 K on cooling process of the sample Li2Ge7015.

It is well known that the photoelasticity in crystals arises due to change in number of oscilla‐ tors, effective electric field due to strain and the polarisability of the ions. In the present case, as the wavelength approaches around 5400 Å, the ionic polarisability seems to be changing enormously. There is no optical dispersion data available on LGO. We have conducted an experiment on transmission spectra of LGO along x, y and z-axes, which shows a strong ab‐ sorption around 5400 Å. The observed anomaly in the piezo-optic dispersion may be attrib‐ uted to the absorption edge falling in this region. This explanation needs further investigation in this direction. It is also known that the strain optical dispersion arises due to the shift in absorption frequencies and a change in the oscillator strength caused by the physical strain in the crystal.

### **1.7. Irradiation Effect on Piezo-optic Dispersion of Li2Ge7O15 Crystals**

**Figure 11.** Stress optical dispersion of Li2Ge7015 crystals with wavelength at room temperature (298 K).

anomaly is related to this phase transition of the LGO crystal.

174 Advances in Ferroelectrics

been done to identify the reasons for this peculiar behaviour.

Figure 12 shows the variation of Czx(*λ*) at the temperatures ranging from 298K to 283K on cooling process of the sample LGO. It is clear from the figure that the distinct peak of Czx(*λ*) appears only at the sodium yellow wavelength of 5890 Å for the whole range of tempera‐ tures (298 K–283 K) investigated. It is also interesting to note that a temperature anomaly is also observed around 283 K. LGO undergoes a second order phase transition at 283.5 K from the high temperature paraelectric phase to the low temperature ferroelectric phase. So this

The observed peculiarity of piezo-optic behavior could be due to many factors, viz., i) anom‐ alous behavior of refractive index or birefringence ii) anomalous ferroelastic transformation at some stage of loading iii) shift of absorption edge due to loading. The following have

Birefringence dispersion has been investigated in the visible region and no anomalies in its behavior has been observed. This rules out the first of the reasons mentioned. The reason due to ferroelastic behavior also is ruled out since the effect would be uniform over all the wavelengths investigated. It was not possible to investigate the effect of load on the absorp‐ tion edge. Hence an indirect experiment has been performed. If there is a shift in the absorp‐ tion edge due to loading the sample, the peak observed now at sodium yellow light would shift with load. No clear shift of the peak could be observed within the experimental limits. Another interesting experiment was done to identify the source of the anomaly. It is well known that Tc of LGO changes under uniaxial stress. The measurements were made near Tc under different stress (loads). Although Tc was found to shift a little with load the dispersion peak did not show any discernible shift. No particular reason could be established as to why a dispersion peak appears around sodium yellow region. Another interesting work in this

The ferroelectric single crystals Li2Ge7O15 was irradiated by x-ray for one hour and the ex‐ perimental processes described in section 1.5 were repeated for the crystal (irradiated) LGO in order to understand the radiation effect on piezo-optical birefringence dispersion [18]. The values of Cλ of the crystal (irradiated) LGO thus obtained at different wavelengths are given in Table 3 and the results are plotted in Fig. 13.


**Table 3.** Stress Optical Coefficients Cpq (in Brewsters) for Li2Ge7015 (irradiated) at different wavelengths.

Some interesting results are obtained in the case of irradiated crystal LGO. The peak value of C/ zx has decreased about 18% and that of C/ zy has increased about 25% at the wave length λ= 5890 Å. Also, it is interesting to note that the value of C6140, is less than that of C5390 for the un-irradiated and irradiated sample of LGO crystal, in tune with usual observation of piezooptic dispersion.

**Figure 13.** Stress optical dispersion of Li2Ge7015 crystals (un-irradiated and irradiated) with wavelength at room tem‐

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**Figure 14.** Temperature dependence of the piezo-optic coefficients Cxy, Cxz, Cyz and Cyx of the crystals LGO in a cooling

The temperature dependence of piezo-optic coefficients Cpq of the crystals Li2Ge7O15 between 298 K and 273 K were determined and are shown in Fig. 14 and Fig. 15. The values of Cpq at

perature (298 K).

(0) and heating (Δ) cycle.

Irradiation of crystals can change physical properties of the crystals. Irradiation brings about many effects in the crystal such as creating defects, internal stress and electric fields etc. These irradiation effects in turn are supposed to affect the physical properties of the irradiat‐ ed crystal as compared to un-irradiated crystal. While there was no appreciable change in the lattice parameters, a significant drop in the value of dielectric constant and tan δ was ob‐ served upon x-irradiation of ferroelectric glycine phosphate. An appreciable shift in the phase transition temperature towards the lower temperature was observed. These changes are attributed to the defects produced in it by irradiation [42]. The studies of triglycine sul‐ phate (TGS) showed that very small doses of x-irradiation can give large changes of the fer‐ roelectric properties. The direct evidence of domain clamping by defects was obtained from optical studies. With increasing dosage the dielectric constant peak and polarization curve broaden and move to lower temperatures. In our present studies, the x-irradiation is be‐ lieved to produce internal stress and electric fields inside the crystals LGO due to defects that can change the values of piezo-optic constants [43].

### **1.8. Piezo-optic Birefringence in Li2Ge7O15 Crystals**

The temperature dependence of the photoelastic coefficients of the ferroelectric crystals Li2Ge7O15 in a cooling and heating cycle between 298 K and 273 K was carried out with the experimental procedure described in section 1.5 [19]. A special arrangement was made to vary the temperature of the sample. The temperature was recorded with a digital tempera‐ ture indicator and a thermocouple sensor in contact with the sample.

**Wavelengths**

xy 4.08 3.87 3.72 4.33 3.73

xz 5.35 5.00 4.88 5.59 4.55

yx 4.02 3.47 3.01 3.50 2.83

yz 4.39 4.19 4.01 4.26 3.90

zx 4.63 4.46 4.41 4.66 4.29

zy 3.71 3.26 2.97 3.43 2.72

Some interesting results are obtained in the case of irradiated crystal LGO. The peak value

λ= 5890 Å. Also, it is interesting to note that the value of C6140, is less than that of C5390 for the un-irradiated and irradiated sample of LGO crystal, in tune with usual observation of piezo-

Irradiation of crystals can change physical properties of the crystals. Irradiation brings about many effects in the crystal such as creating defects, internal stress and electric fields etc. These irradiation effects in turn are supposed to affect the physical properties of the irradiat‐ ed crystal as compared to un-irradiated crystal. While there was no appreciable change in the lattice parameters, a significant drop in the value of dielectric constant and tan δ was ob‐ served upon x-irradiation of ferroelectric glycine phosphate. An appreciable shift in the phase transition temperature towards the lower temperature was observed. These changes are attributed to the defects produced in it by irradiation [42]. The studies of triglycine sul‐ phate (TGS) showed that very small doses of x-irradiation can give large changes of the fer‐ roelectric properties. The direct evidence of domain clamping by defects was obtained from optical studies. With increasing dosage the dielectric constant peak and polarization curve broaden and move to lower temperatures. In our present studies, the x-irradiation is be‐ lieved to produce internal stress and electric fields inside the crystals LGO due to defects

The temperature dependence of the photoelastic coefficients of the ferroelectric crystals Li2Ge7O15 in a cooling and heating cycle between 298 K and 273 K was carried out with the experimental procedure described in section 1.5 [19]. A special arrangement was made to vary the temperature of the sample. The temperature was recorded with a digital tempera‐

zy has increased about 25% at the wave length

**Table 3.** Stress Optical Coefficients Cpq (in Brewsters) for Li2Ge7015 (irradiated) at different wavelengths.

zx has decreased about 18% and that of C/

that can change the values of piezo-optic constants [43].

ture indicator and a thermocouple sensor in contact with the sample.

**1.8. Piezo-optic Birefringence in Li2Ge7O15 Crystals**

pq 4358**Å** 4880**Å** 5390**Å** 5890**Å** 6140**Å**

Obs. C/

176 Advances in Ferroelectrics

1 C/

2 C/

3 C/

4 C/

5 C/

6 C/

optic dispersion.

of C/

**Figure 13.** Stress optical dispersion of Li2Ge7015 crystals (un-irradiated and irradiated) with wavelength at room tem‐ perature (298 K).

**Figure 14.** Temperature dependence of the piezo-optic coefficients Cxy, Cxz, Cyz and Cyx of the crystals LGO in a cooling (0) and heating (Δ) cycle.

The temperature dependence of piezo-optic coefficients Cpq of the crystals Li2Ge7O15 between 298 K and 273 K were determined and are shown in Fig. 14 and Fig. 15. The values of Cpq at 291 K and 278 K were reported in paper [44] and it was observed that there were large changes in the values of Czy and Cyz at 278 K and 291 K as compared to other components and Czy did not show a peak in its temperature dependence between 291K and 278 K.

perature dependence of the dielectric permittivity along the c-axis of LGO shows a sharp peak at Tc (283.5 K) and the Curie-Weiss law holds only for a narrow range of temperature (Tc ± 4 K) [11,15, 16]. The peak for piezo-optic coefficient is attributed to the paraelectric to ferroelectric phase transition of LGO at Tc. To check the curie-Weiss law like dependence

Where CT pq and C0 pq denote the value of the corresponding piezo-optic coefficients at tem‐ perature T and 273 K respectively and Kpq is a constant. Fig. 17 shows the (CT pq – C0 pq)

(T−Tc) curve for Czx and Czy. It is clear from these curves that like dielectric constant the rela‐ tion fits well only within a narrow range of temperature near Tc(Tc± 4 K). The solid lines de‐ note the theoretical curves with the following values Kzx = 1*.*05; Kzy = 0*.*92 for T *>* T*c* , Kzx =

pq <sup>=</sup> Kpq / (T – Tc) (2)

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**Temperature Range Ratio**

296K-289K -1.69 289K-283K 282K-279K -2.0 279K-276K 276K-273K

296K-289K -0.75 289K-283K 282K-279K -1.9 279K-276K 276K-273K

−1 vs

179

near Tc the following relation is used.

−0*.*40; Kzy = −0*.*34 for T *<* T*c* and Tc = 279 K.

CT pq – <sup>C</sup><sup>0</sup>

**Cpq Value of Derivative**

Czx 0.013

Czy 0.020

**(Brewster/K)**


Cxz -0.003 293K-273K


Cyz -0.026 293K-273K Cxy 0.007 293K-273K Cyz -0.023 293K-273K

Furthermore the magnitudes of the ratio of the temperature derivatives below and above Tm and Tc are given in Table 4 and we can see that the ratio near Tc comes out to be about 2. Therefore it satisfies the law of two for the ratio of such derivatives of quantities which are coupled with the spontaneous polarization in second order ferroelectric phase transition such as in the case of triglycine sulphate [45] and LGO. Therefore the peak around Tc is [ 13, 15, 16] attributed to the paraelectric to ferroelectric phase transition of LGO. The smallness

**Table 4.** The temperature derivative [dlnCpq /dT] of the piezo-optic coefficients of Li2Ge7O15.

**Figure 15.** Anomalous temperature dependence of the piezo-optic coefficients Czx and Czy of the crystals LGO in a cooling (0) and heating (Δ) cycle.

Here in contrast we observed a peak in the temperature dependence of both Czy and Czx at 279 K. The temperature dependence of Cpq are quite interesting, for example the piezo-optic coefficients Cyz, Cyx and Cxz have negative temperature derivatives but Cxy has a positive temperature derivative. In complete contrast both Czy and Czx have both positive and nega‐ tive temperature derivatives at different temperature intervals between 298 K and 273 K (Ta‐ ble: 4). Besides a clear thermal hysteresis is observed in Czy and Czx in a complete cooling and heating cycle (Fig. 15) whereas no discernible hysteresis is observed in rest of the piezooptic coefficients (Fig. 14). The two distinct anomalies in the temperature dependence of Czy and Czx are characterized by a valley at Tm (∼289 K) and a peak at Tc (∼279 K). Anomalous temperature dependence of Czx at different wave lengths is also shown in Fig. 16. The tem‐ perature dependence of the dielectric permittivity along the c-axis of LGO shows a sharp peak at Tc (283.5 K) and the Curie-Weiss law holds only for a narrow range of temperature (Tc ± 4 K) [11,15, 16]. The peak for piezo-optic coefficient is attributed to the paraelectric to ferroelectric phase transition of LGO at Tc. To check the curie-Weiss law like dependence near Tc the following relation is used.

291 K and 278 K were reported in paper [44] and it was observed that there were large changes in the values of Czy and Cyz at 278 K and 291 K as compared to other components

**Figure 15.** Anomalous temperature dependence of the piezo-optic coefficients Czx and Czy of the crystals LGO in a

Here in contrast we observed a peak in the temperature dependence of both Czy and Czx at 279 K. The temperature dependence of Cpq are quite interesting, for example the piezo-optic coefficients Cyz, Cyx and Cxz have negative temperature derivatives but Cxy has a positive temperature derivative. In complete contrast both Czy and Czx have both positive and nega‐ tive temperature derivatives at different temperature intervals between 298 K and 273 K (Ta‐ ble: 4). Besides a clear thermal hysteresis is observed in Czy and Czx in a complete cooling and heating cycle (Fig. 15) whereas no discernible hysteresis is observed in rest of the piezooptic coefficients (Fig. 14). The two distinct anomalies in the temperature dependence of Czy and Czx are characterized by a valley at Tm (∼289 K) and a peak at Tc (∼279 K). Anomalous temperature dependence of Czx at different wave lengths is also shown in Fig. 16. The tem‐

cooling (0) and heating (Δ) cycle.

178 Advances in Ferroelectrics

and Czy did not show a peak in its temperature dependence between 291K and 278 K.

$$\mathbf{C}^{\rm T}\_{\rm pq} - \mathbf{C}^{0}\_{\rm pq} = \mathbf{K}\_{\rm pq} / \{\mathbf{T} - \mathbf{T}\_{\rm c}\} \tag{2}$$

Where CT pq and C0 pq denote the value of the corresponding piezo-optic coefficients at tem‐ perature T and 273 K respectively and Kpq is a constant. Fig. 17 shows the (CT pq – C0 pq) −1 vs (T−Tc) curve for Czx and Czy. It is clear from these curves that like dielectric constant the rela‐ tion fits well only within a narrow range of temperature near Tc(Tc± 4 K). The solid lines de‐ note the theoretical curves with the following values Kzx = 1*.*05; Kzy = 0*.*92 for T *>* T*c* , Kzx = −0*.*40; Kzy = −0*.*34 for T *<* T*c* and Tc = 279 K.


**Table 4.** The temperature derivative [dlnCpq /dT] of the piezo-optic coefficients of Li2Ge7O15.

Furthermore the magnitudes of the ratio of the temperature derivatives below and above Tm and Tc are given in Table 4 and we can see that the ratio near Tc comes out to be about 2. Therefore it satisfies the law of two for the ratio of such derivatives of quantities which are coupled with the spontaneous polarization in second order ferroelectric phase transition such as in the case of triglycine sulphate [45] and LGO. Therefore the peak around Tc is [ 13, 15, 16] attributed to the paraelectric to ferroelectric phase transition of LGO. The smallness of Kpq and the applicability of relation (2) above only in a narrow range of temperature sug‐ gest that LGO may be an improper ferroelectric. The law of two does not hold for the ratio at Tm (Table 4). Therefore this anomaly is not related to the spontaneous polarization.

Czx and found a linear relationship (Fig. 18). However, a negative stress coefficient dTc /dp ∼ −22 K/kbar is obtained in this case which agrees only in magnitude with the hydrostatic pressure coefficient. The linear curve (Fig. 18) extrapolates to a Tc = 281.5 K in the unstressed state instead of 283.5K as determined by dielectric measurements [11, 15, 16]. This may be

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due to a non linear dependence of shift of Tc under stress near 283.5 K.

pq – Co pq )-1 vs (T-Tc) curve for Czx and Czy.

**Figure 18.** The stress dependence of the shift of Tc for Czx.

Now we turn to the anomaly around Tm. Morioka et al. [47] proposed that there is an inter‐ action between the soft phonon mode and a relaxational mode in the paraelectric phase in the temperature interval 300 K to Tc. The critical slowing down of the relaxational mode

**Figure 17.** Plots of (CT

From the behaviour that only Czx and Czy show anomalous it is obvious that birefringence (Δnz–Δny) and (Δnz–Δnx) show steep increase around Tc and below Tc show a (T–Tc )1*/*<sup>2</sup> be‐ haviour correlated to the spontaneous polarization which is parallel to the z-axis (crystallo‐ graphic c-axis). From the behaviour of Cxy and Cyx which do not show any temperature anomaly we may say that only nz is responsible for the anomaly in accordance with the be‐ haviour of the dielectric properties where only *ε* <sup>33</sup> is strongly affected by the phase transi‐ tion. These observations are in accordance with the results of Faraday effect and birefringence in LGO [13].

**Figure 16.** Anomalous temperature dependence of piezo-optic coefficient Czx of the crystals LGO at different wave lengths in a cooling (0) and heating (Δ) cycle.

As mentioned by Lines and Glass [43], under an external pressure Tc of a ferroelectric phase transition may be shifted. This shift may be to the higher or the lower side of normal Tc. Wa‐ da et al. [46] studied the pressure effect on the ferroelectric phase transition in LGO through the dielectric and Raman scattering measurements and found a positive pressure coefficient dTc /dp = 14.6 K/kbar. Preu and HaussÜhl [12] studied the dependences of dielectric con‐ stants on hydrostatic and uniaxial pressure as well as temperature. They observed a shift of Tc at a rate of 14.02 K/kbar for the hydrostatic pressure and ∼7 K/kbar for the uniaxial pres‐ sure. In the present case the position of the peak of Czy is found to depend on the stress ap‐ plied. If the peak position is believed to represent the Tc, it appears to shift to the lower side under the uniaxial stress. To see whether Tc shifts linearly with uniaxial stress similar to the earlier observations [12, 46], we used different stresses within the elastic limits of LGO for Czx and found a linear relationship (Fig. 18). However, a negative stress coefficient dTc /dp ∼ −22 K/kbar is obtained in this case which agrees only in magnitude with the hydrostatic pressure coefficient. The linear curve (Fig. 18) extrapolates to a Tc = 281.5 K in the unstressed state instead of 283.5K as determined by dielectric measurements [11, 15, 16]. This may be due to a non linear dependence of shift of Tc under stress near 283.5 K.

**Figure 17.** Plots of (CT pq – Co pq )-1 vs (T-Tc) curve for Czx and Czy.

of Kpq and the applicability of relation (2) above only in a narrow range of temperature sug‐ gest that LGO may be an improper ferroelectric. The law of two does not hold for the ratio at

From the behaviour that only Czx and Czy show anomalous it is obvious that birefringence (Δnz–Δny) and (Δnz–Δnx) show steep increase around Tc and below Tc show a (T–Tc )1*/*<sup>2</sup> be‐ haviour correlated to the spontaneous polarization which is parallel to the z-axis (crystallo‐ graphic c-axis). From the behaviour of Cxy and Cyx which do not show any temperature anomaly we may say that only nz is responsible for the anomaly in accordance with the be‐ haviour of the dielectric properties where only *ε* <sup>33</sup> is strongly affected by the phase transi‐ tion. These observations are in accordance with the results of Faraday effect and

**Figure 16.** Anomalous temperature dependence of piezo-optic coefficient Czx of the crystals LGO at different wave

As mentioned by Lines and Glass [43], under an external pressure Tc of a ferroelectric phase transition may be shifted. This shift may be to the higher or the lower side of normal Tc. Wa‐ da et al. [46] studied the pressure effect on the ferroelectric phase transition in LGO through the dielectric and Raman scattering measurements and found a positive pressure coefficient dTc /dp = 14.6 K/kbar. Preu and HaussÜhl [12] studied the dependences of dielectric con‐ stants on hydrostatic and uniaxial pressure as well as temperature. They observed a shift of Tc at a rate of 14.02 K/kbar for the hydrostatic pressure and ∼7 K/kbar for the uniaxial pres‐ sure. In the present case the position of the peak of Czy is found to depend on the stress ap‐ plied. If the peak position is believed to represent the Tc, it appears to shift to the lower side under the uniaxial stress. To see whether Tc shifts linearly with uniaxial stress similar to the earlier observations [12, 46], we used different stresses within the elastic limits of LGO for

Tm (Table 4). Therefore this anomaly is not related to the spontaneous polarization.

birefringence in LGO [13].

180 Advances in Ferroelectrics

lengths in a cooling (0) and heating (Δ) cycle.

**Figure 18.** The stress dependence of the shift of Tc for Czx.

Now we turn to the anomaly around Tm. Morioka et al. [47] proposed that there is an inter‐ action between the soft phonon mode and a relaxational mode in the paraelectric phase in the temperature interval 300 K to Tc. The critical slowing down of the relaxational mode near Tc is expected to cause the increase of the fluctuation of the spatially homogeneous po‐ larization and thereby the increase of the fluctuation of the hyperpolarizability with kc = 0. Wada et al. [48] measured the soft phonon mode with the help of their newly designed FR-IR spectrometer and proposed that as Tc is approached from above soft phonon mode be‐ comes over damped and transforms to a relaxational mode.

ble to attempt experiment under the simultaneous application of a suitable electric field and

 Cxz 3.74 0.28 1.25 Ref. [50] for RS Cyz 4.29 0.28 1.25 a- polar axis Cyx 3.56 1.04 4.30 Ref. [51] for KDP Czx 0.85 1.54 3.50 Ref. [52] for ADP

**Table 6.** Piezo-optic coefficients cpq (in Brewsters) for some ferroelectric crystals in their paraelectric (PE) phases.

The Stress optical coefficients Cpq of the crystals Li2Ge7O15 at paraelectric phase (RT = 298 K) and at Tc = 279 K are presented in Table 5. It is important to compare the values of Cpq for Li2Ge7O15 with other ferroelectric crystals given in Table 6 particularly with Rochelle-salt (RS) which belongs to the orthorhombic class like LGO [44]. The values of Cpq are signifi‐ cantly higher for LGO as compared to these ferroelectric systems. So, the large photoelastic coefficients and the other properties like good mechanical strength, a transition temperature close to room temperature and stability in ambient environment favour LGO as a potential

The EPR (Electron Paramagnetic Resonance) spectroscopy of the transition metal ion doped crystals of LGO (Mn2+, Cr3+) has also been studied both in Paraelectric (PE) and ferroelectric (FE) phases in the temperature interval from 298 K to 279 K during cooling and heating cy‐ cles [17, 36, 53]. It is observed that on approaching Tc in a cooling cycle, the EPR lines are slightly shifted to the high field direction and undergo substantial broadening. At the tem‐ perature Tc ( ≈ 283.4 K), the EPR lines are splitted into two components which are shifted to the higher and lower field directions progressively as a result of cooling the sample below Tc

During heating cycle (i.e. approaching Tc from below), the phenomena occurred were just opposite to the above processes observed in the cooling cycle. However, the EPR line width (peak to peak ∆Hpp) for H║c, H┴a was found to decrease to about one third of its value at Tc in a heating cycle as compared to its value in the cooling cycle. The shape of the EPR reso‐ nance lines far from Tc has a dominant Lorentzian character (a Lorentzian line shape) but very near to Tc, the line shape has been described mainly by Gaussian form of distribution (a Gaussian line shape). All the peculiarities observed are attributed to the PE ↔ FE phase transition of the LGO crystals. The line width reduction near Tc is attributed to the internal space charge (electret state) effects which produce an internal electric field inside the crystals

**KDP ADP Remarks**

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stress along z-direction.

**Obs Cpq Rochelle Salt**

candidate for photoelastic applications.

as shown in Fig.19.

**(RS)**

5 Czy 2.61 1.54 3.50 6 Cxy 3.04 1.04 4.30

On the other hand there may exist a relaxational mode with an independent degree of free‐ dom as well as the soft phonon mode and the character of the softening transfers from the phonon to the relaxational mode. This is an important problem in determining the dynamics of the peculiar ferroelectric phase transition of LGO, where both the dielectric critical slow‐ ing down characteristic of the order-disorder phase transition and the soft phonon mode characteristic of the displacive phase transition are observed [11, 14]. In the light of the above discussion we may say that the change up to Tm is caused by the softening of mode and the softening character transforms to the relaxations mode near Tm causing a change in the trend below Tm and near Tc the relaxational mode becomes dominant. The valley around Tm is perhaps caused by the interplay between the competitive relaxational mode and the soft phonon mode. It has been observed that softening of the velocity and rise of the damp‐ ing of acoustic phonon occur in the paraelectric phase of LGO even quite far from Tc, i.e.(T-Tc) ∼ 30 K and the effect is attributed to the fluctuation induced contributions [49].


**Table 5.** Stress optical coefficients cpq (in Brewster's) of Li2Ge7015 at RT=298 K and at Tc = 279 K.

Another interesting aspect is the observation of a significant thermal photoelastic hysteresis (Fig. 15). Although the peak position does not shift in the heating cycle the values of the photoelastic constants get reduced significantly in the heating cycle as compared to the cor‐ responding values in a cooling cycle. A similar kind of hysteresis was observed in the dielec‐ tric behaviour of LGO and the appearance of the dielectric hysteresis is attributed to the internal space charge (electrets state) effects which produce an internal electric field in LGO on heating from the ferroelectric phase [15-17]. It was possible to compensate the internal electric field effects in dielectric measurements by an external electric field [15-17]. It is sus‐ pected that the photoelastic hysteresis also occurs due to similar effects. Although it was not possible to try to compensate the electric field effects in the present investigation, it is possi‐


ble to attempt experiment under the simultaneous application of a suitable electric field and stress along z-direction.

near Tc is expected to cause the increase of the fluctuation of the spatially homogeneous po‐ larization and thereby the increase of the fluctuation of the hyperpolarizability with kc = 0. Wada et al. [48] measured the soft phonon mode with the help of their newly designed FR-IR spectrometer and proposed that as Tc is approached from above soft phonon mode be‐

On the other hand there may exist a relaxational mode with an independent degree of free‐ dom as well as the soft phonon mode and the character of the softening transfers from the phonon to the relaxational mode. This is an important problem in determining the dynamics of the peculiar ferroelectric phase transition of LGO, where both the dielectric critical slow‐ ing down characteristic of the order-disorder phase transition and the soft phonon mode characteristic of the displacive phase transition are observed [11, 14]. In the light of the above discussion we may say that the change up to Tm is caused by the softening of mode and the softening character transforms to the relaxations mode near Tm causing a change in the trend below Tm and near Tc the relaxational mode becomes dominant. The valley around Tm is perhaps caused by the interplay between the competitive relaxational mode and the soft phonon mode. It has been observed that softening of the velocity and rise of the damp‐ ing of acoustic phonon occur in the paraelectric phase of LGO even quite far from Tc, i.e.(T-

Tc) ∼ 30 K and the effect is attributed to the fluctuation induced contributions [49].

**phase (RT)**

Another interesting aspect is the observation of a significant thermal photoelastic hysteresis (Fig. 15). Although the peak position does not shift in the heating cycle the values of the photoelastic constants get reduced significantly in the heating cycle as compared to the cor‐ responding values in a cooling cycle. A similar kind of hysteresis was observed in the dielec‐ tric behaviour of LGO and the appearance of the dielectric hysteresis is attributed to the internal space charge (electrets state) effects which produce an internal electric field in LGO on heating from the ferroelectric phase [15-17]. It was possible to compensate the internal electric field effects in dielectric measurements by an external electric field [15-17]. It is sus‐ pected that the photoelastic hysteresis also occurs due to similar effects. Although it was not possible to try to compensate the electric field effects in the present investigation, it is possi‐

 Cxy 4.38 3.85 Cxz 5.55 5.85 Cyx 3.60 4.46 Cyz 4.26 5.50 Czy 3.71 4.83 Czx 4.19 5.45

**Table 5.** Stress optical coefficients cpq (in Brewster's) of Li2Ge7015 at RT=298 K and at Tc = 279 K.

**At Tc = 279 K**

comes over damped and transforms to a relaxational mode.

182 Advances in Ferroelectrics

**Obs. Cpq Paraelectric (PE)**

**Table 6.** Piezo-optic coefficients cpq (in Brewsters) for some ferroelectric crystals in their paraelectric (PE) phases.

The Stress optical coefficients Cpq of the crystals Li2Ge7O15 at paraelectric phase (RT = 298 K) and at Tc = 279 K are presented in Table 5. It is important to compare the values of Cpq for Li2Ge7O15 with other ferroelectric crystals given in Table 6 particularly with Rochelle-salt (RS) which belongs to the orthorhombic class like LGO [44]. The values of Cpq are signifi‐ cantly higher for LGO as compared to these ferroelectric systems. So, the large photoelastic coefficients and the other properties like good mechanical strength, a transition temperature close to room temperature and stability in ambient environment favour LGO as a potential candidate for photoelastic applications.

The EPR (Electron Paramagnetic Resonance) spectroscopy of the transition metal ion doped crystals of LGO (Mn2+, Cr3+) has also been studied both in Paraelectric (PE) and ferroelectric (FE) phases in the temperature interval from 298 K to 279 K during cooling and heating cy‐ cles [17, 36, 53]. It is observed that on approaching Tc in a cooling cycle, the EPR lines are slightly shifted to the high field direction and undergo substantial broadening. At the tem‐ perature Tc ( ≈ 283.4 K), the EPR lines are splitted into two components which are shifted to the higher and lower field directions progressively as a result of cooling the sample below Tc as shown in Fig.19.

During heating cycle (i.e. approaching Tc from below), the phenomena occurred were just opposite to the above processes observed in the cooling cycle. However, the EPR line width (peak to peak ∆Hpp) for H║c, H┴a was found to decrease to about one third of its value at Tc in a heating cycle as compared to its value in the cooling cycle. The shape of the EPR reso‐ nance lines far from Tc has a dominant Lorentzian character (a Lorentzian line shape) but very near to Tc, the line shape has been described mainly by Gaussian form of distribution (a Gaussian line shape). All the peculiarities observed are attributed to the PE ↔ FE phase transition of the LGO crystals. The line width reduction near Tc is attributed to the internal space charge (electret state) effects which produce an internal electric field inside the crystals on heating process from the ferroelectric phase. This observation is similar to the photoelas‐ tic hysteresis behavior of the crystals LGO near Tc.

**1.9. Irradiation Effect on piezo-optic Birefringence in Li2Ge7O15 Crystals**

photoelastic behaviour.

plotted in Fig.22.

lengths in a cooling (0) and heating (∆) cycle.

physical properties of the crystals.

The photoelastic coefficients Cpq of the ferroelectric crystals Li2Ge7O15 (x-irradiated) in a cool‐ ing and heating cycle between 298 K and 273 K was carried out with the experimental proce‐ dure described in section 1.5 and are shown in Fig. 20 [54]. The results show an interesting

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Peaks are observed in the temperature dependence of the photoelastic coefficients Czy and Czx at temperature ~ 279 K in a complete cooling and heating cycle whereas no discernible hystere‐ sis is observed in rest of the photoelastic coefficients. Anomalous temperature dependence of

It is observed that the peak value of Czy has increased about 25% and that of Czx has de‐ creased about 18% at the wave length λ=5890 Å during cooling process of the crystal (Fig.15 and Fig.20). The peak value of Czx of the crystal (un-irradiated and x-irradiated) LGO thus obtained at different wave lengths (Fig.16 and Fig.21) are given in Table 7 and the results are

**Figure 21.** Temperature dependence of photoelastic coefficient Czx of the crystal (x-irradiated) LGO at different wave

It has been observed that the changes in the value of photoelastic coefficients Czy and Czx of the crystal (x-irradiated) LGO in a cooling and heating cycle occur only if the crystal is stressed along the polar axis (z-axis). It is known that the irradiation of crystals can change

Czx of the crystal (x-irradiated) LGO at different wave lengths are shown in Fig.21.

**Figure 19.** Temperature dependence of EPR lines of Li2Ge7O15:Cr+3 crystals for |M|= ½ ↔3/2, H║a, H┴c near Tc during cooling process.

**Figure 20.** Temperature dependence of photoelastic coefficients Cxy, Cxz, Cyz, Cyx, Czx and Czy of the crystal (x-irradiated) LGO in a cooling (0) and heating (∆) cycle.

### **1.9. Irradiation Effect on piezo-optic Birefringence in Li2Ge7O15 Crystals**

on heating process from the ferroelectric phase. This observation is similar to the photoelas‐

**Figure 19.** Temperature dependence of EPR lines of Li2Ge7O15:Cr+3 crystals for |M|= ½ ↔3/2, H║a, H┴c near Tc during

**Figure 20.** Temperature dependence of photoelastic coefficients Cxy, Cxz, Cyz, Cyx, Czx and Czy of the crystal (x-irradiated)

tic hysteresis behavior of the crystals LGO near Tc.

cooling process.

184 Advances in Ferroelectrics

LGO in a cooling (0) and heating (∆) cycle.

The photoelastic coefficients Cpq of the ferroelectric crystals Li2Ge7O15 (x-irradiated) in a cool‐ ing and heating cycle between 298 K and 273 K was carried out with the experimental proce‐ dure described in section 1.5 and are shown in Fig. 20 [54]. The results show an interesting photoelastic behaviour.

Peaks are observed in the temperature dependence of the photoelastic coefficients Czy and Czx at temperature ~ 279 K in a complete cooling and heating cycle whereas no discernible hystere‐ sis is observed in rest of the photoelastic coefficients. Anomalous temperature dependence of Czx of the crystal (x-irradiated) LGO at different wave lengths are shown in Fig.21.

It is observed that the peak value of Czy has increased about 25% and that of Czx has de‐ creased about 18% at the wave length λ=5890 Å during cooling process of the crystal (Fig.15 and Fig.20). The peak value of Czx of the crystal (un-irradiated and x-irradiated) LGO thus obtained at different wave lengths (Fig.16 and Fig.21) are given in Table 7 and the results are plotted in Fig.22.

**Figure 21.** Temperature dependence of photoelastic coefficient Czx of the crystal (x-irradiated) LGO at different wave lengths in a cooling (0) and heating (∆) cycle.

It has been observed that the changes in the value of photoelastic coefficients Czy and Czx of the crystal (x-irradiated) LGO in a cooling and heating cycle occur only if the crystal is stressed along the polar axis (z-axis). It is known that the irradiation of crystals can change physical properties of the crystals.


tive index is found to be 1.985 in the 1450-1580 nm wavelength range which is considered to

The study of fluorescence spectra of the crystals Li2Ge7O15:Cr3+ in the temperature interval 77-320 K shows the sharply decrease of intensities of the R1 and R2 lines (corresponding to the Cr3+ ions of types I and II) during cooling process near the temperature Tc = 283.5 K as described in section 1.3. Such nature of suppression of R1 and R2 lines was not observed pre‐ viously and it may be related with the mechanism of interaction of excitation spectra of light in the crystals Li2Ge7O15:Cr3+ at the temperature Tc. The doping of chromium in LGO is be‐

two conjugate directions. The EPR, optical, dielectric and fluorescence studies conform each

The high optical quality, good mechanical strength and stability in ambient environment and large photoelastic coefficients in comparison with other ferroelectric crystals like Ro‐ chelle-salt, KDP and ADP favour the crystals LGO as a potential candidate for photoelastic applications. The piezo-optic dispersion of the crystals (un-irradiated and x-irradiated) LGO in the visible region of the spectrum of light at room temperature (298 K) have been descri‐ bed in sections 1.6 and 1.7. It shows an "optical zone or optical window" in between the wavelengths 5400 Å and 6200 Å with an enhanced piezo-optical behavior. This peculiar op‐ tical window can have a technical importance for example this window region can act as an optical switch for acousto-optical devices. From the studies undertaken it may be concluded that LGO is an attractive acousto-optic material which deserves further probe. It may be possible to understand the observed behavior if extensive piezo-optic and refractive index

The temperature dependence of the photoelastic coefficients of the crystals (un-irradiated and x-irradiated) LGO in a cooling and heating cycle between room temperature (298 K) and 273 K have shown an interesting observations: lowering of the Tc under uniaxial stress con‐ trary to the increase of Tc under hydrostatic pressure and observation of thermal photoelas‐ tic hysteresis similar to dielectric hysteresis behavior as described in sections 1.8 and 1.9. In our studies, the x-irradiation is believed to produce internal stress and electric fields inside the crystals LGO due to defects that can change the values of photoelastic coefficients, as de‐

Department of Physics Indian Institute of Technology Kanpur, Kanpur-208016, INDIA

defect pairs in the host LGO lattice at Ge4+ sites creating dipoles in

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187

be important for optoelectronic device applications.

other and pose more scope for further studies.

data become available over an extended range of wavelengths.

lieved to create Cr3+- Li+

scribed in sections 1.7 and 1.9.

and Prem Chand

**Author details**

A.K. Bain\*

**Table 7.** The peak value of Czx (in Brewster) for the Crystal (un-irradiated and x-irradiated) LGO at different wave lengths in the cooling and heating cycles.

**Figure 22.** The peak value of Czx for the un-irradiated (black colour) and x-irradiated (ash colour) crystal LGO at differ‐ ent wave lengths in a cooling (0) and heating (∆) cycle.

Irradiation brings about many effects in the crystal such as creating defects, internal stress and electric fields etc [43]. In our present studies, the x-irradiation is believed to produce in‐ ternal stress and electric fields inside the crystals Li2Ge7O15 due to defects that can change the values of photoelastic coefficients.

### **2. Summary**

It is known that the Barium strontium titanate Ba1-xSrxTiO3(BST) is one of the most interest‐ ing thin film ferroelectric materials due to its high dielectric constant, composition depend‐ ent curie temperature and high optical nonlinearity. The wavelength dependence of refractive index of BST (Ba0.05Sr0.95TiO3) thin films has shown a nonlinear dependence in the 1450-1580 nm wavelength range at room temperature as described in section 1.1. The disper‐ sion curve decreases gradually with increasing wavelength. The average value of the refrac‐ tive index is found to be 1.985 in the 1450-1580 nm wavelength range which is considered to be important for optoelectronic device applications.

The study of fluorescence spectra of the crystals Li2Ge7O15:Cr3+ in the temperature interval 77-320 K shows the sharply decrease of intensities of the R1 and R2 lines (corresponding to the Cr3+ ions of types I and II) during cooling process near the temperature Tc = 283.5 K as described in section 1.3. Such nature of suppression of R1 and R2 lines was not observed pre‐ viously and it may be related with the mechanism of interaction of excitation spectra of light in the crystals Li2Ge7O15:Cr3+ at the temperature Tc. The doping of chromium in LGO is be‐ lieved to create Cr3+- Li+ defect pairs in the host LGO lattice at Ge4+ sites creating dipoles in two conjugate directions. The EPR, optical, dielectric and fluorescence studies conform each other and pose more scope for further studies.

The high optical quality, good mechanical strength and stability in ambient environment and large photoelastic coefficients in comparison with other ferroelectric crystals like Ro‐ chelle-salt, KDP and ADP favour the crystals LGO as a potential candidate for photoelastic applications. The piezo-optic dispersion of the crystals (un-irradiated and x-irradiated) LGO in the visible region of the spectrum of light at room temperature (298 K) have been descri‐ bed in sections 1.6 and 1.7. It shows an "optical zone or optical window" in between the wavelengths 5400 Å and 6200 Å with an enhanced piezo-optical behavior. This peculiar op‐ tical window can have a technical importance for example this window region can act as an optical switch for acousto-optical devices. From the studies undertaken it may be concluded that LGO is an attractive acousto-optic material which deserves further probe. It may be possible to understand the observed behavior if extensive piezo-optic and refractive index data become available over an extended range of wavelengths.

The temperature dependence of the photoelastic coefficients of the crystals (un-irradiated and x-irradiated) LGO in a cooling and heating cycle between room temperature (298 K) and 273 K have shown an interesting observations: lowering of the Tc under uniaxial stress con‐ trary to the increase of Tc under hydrostatic pressure and observation of thermal photoelas‐ tic hysteresis similar to dielectric hysteresis behavior as described in sections 1.8 and 1.9. In our studies, the x-irradiation is believed to produce internal stress and electric fields inside the crystals LGO due to defects that can change the values of photoelastic coefficients, as de‐ scribed in sections 1.7 and 1.9.

### **Author details**

**Wave lengths Czx (un-irradiated) Czx (x-irradiated)**

lengths in the cooling and heating cycles.

186 Advances in Ferroelectrics

ent wave lengths in a cooling (0) and heating (∆) cycle.

the values of photoelastic coefficients.

**2. Summary**

(Å) Cooling Heating Cooling Heating 4.8 4.0 4.05 3.3 4.7 3.9 3.95 3.2 5.6 4.8 4.6 3.7 4.5 3.6 4.3 3.4

**Table 7.** The peak value of Czx (in Brewster) for the Crystal (un-irradiated and x-irradiated) LGO at different wave

**Figure 22.** The peak value of Czx for the un-irradiated (black colour) and x-irradiated (ash colour) crystal LGO at differ‐

Irradiation brings about many effects in the crystal such as creating defects, internal stress and electric fields etc [43]. In our present studies, the x-irradiation is believed to produce in‐ ternal stress and electric fields inside the crystals Li2Ge7O15 due to defects that can change

It is known that the Barium strontium titanate Ba1-xSrxTiO3(BST) is one of the most interest‐ ing thin film ferroelectric materials due to its high dielectric constant, composition depend‐ ent curie temperature and high optical nonlinearity. The wavelength dependence of refractive index of BST (Ba0.05Sr0.95TiO3) thin films has shown a nonlinear dependence in the 1450-1580 nm wavelength range at room temperature as described in section 1.1. The disper‐ sion curve decreases gradually with increasing wavelength. The average value of the refrac‐

#### A.K. Bain\* and Prem Chand

Department of Physics Indian Institute of Technology Kanpur, Kanpur-208016, INDIA

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Optical Properties of Ferroelectrics and Measurement Procedures

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**Chapter 9**

**Ferroelectric Domain Imaging Multiferroic Films**

Recently, multiferroic materials with the magnetoelectric coupling of ferroelectric (or anti‐ ferroelectric) properties and ferromagnetic (or antiferromagnetic) properties have attracted a lot of attention.[1-4] Among them, BiFeO3(BFO) and YMnO3has been intensively studied. For such ABO3perovskite structured ferroelectric materials, they usually show antiferromag‐ netic order because the same B site magnetic element except BiMnO3 is ferromagnet. While

ferromagnetic coupling. They are also expected to be multiferroic materials. Several bis‐

La2NiMnO6, BiFeO3-BiCrO3. But as we know, few researches are focused on Bi2FeMnO6. We believe it is particular interesting to investigate Bi2FeMnO6 (BFM). The origin of the ferro‐ magnetism in these compounds has been discussed in many reports. According to Goode‐ nough-Kanamori's (GK)rules, many ferromagnets have been designed through the coupling of two B site ions with and without eg electrons.In BFM, the 180 degree -Fe3+-O-Mn3+- bonds is quasistatic, partly because the strong Jahn-Teller uniaxial strain in an octahedral site.Be‐ cause the complication of the double perovskite system, there are still some questions about the violation of GK rules in some cases and the origin of the ferromagnetism or antiferro‐ magnetism.The other problem is the bad ferroelectric properties. In order to characterization of their ferroelectric/piezoelectric properties, scanning probe microscopy (SPM) techniques

Multiferroic materials can be classified into two categories: one is single-phase materials; the other is multilayer or composite hetero-structures that contain more than one ferroic phases [5]. The most desirable multiferroic material is the intrinsic single-phase material, which is

and reproduction in any medium, provided the original work is properly cited.

© 2013 Zhao et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

O6 double perovskite oxides, the combination between B and B' give rise to a

O6) have aroused great interest like Bi2NiMnO6,

**Using Piezoresponse Force Microscopy**

Hongyang Zhao, Hideo Kimura, Qiwen Yao, Lei Guo, Zhenxiang Cheng and Xiaolin Wang

Additional information is available at the end of the chapter

muth-based double perovskite oxides (BiBB'

http://dx.doi.org/10.5772/52519

**1. Introduction**

for the A2BB'

were used.

### **Ferroelectric Domain Imaging Multiferroic Films Using Piezoresponse Force Microscopy**

Hongyang Zhao, Hideo Kimura, Qiwen Yao, Lei Guo, Zhenxiang Cheng and Xiaolin Wang

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52519

### **1. Introduction**

Recently, multiferroic materials with the magnetoelectric coupling of ferroelectric (or anti‐ ferroelectric) properties and ferromagnetic (or antiferromagnetic) properties have attracted a lot of attention.[1-4] Among them, BiFeO3(BFO) and YMnO3has been intensively studied. For such ABO3perovskite structured ferroelectric materials, they usually show antiferromag‐ netic order because the same B site magnetic element except BiMnO3 is ferromagnet. While for the A2BB' O6 double perovskite oxides, the combination between B and B' give rise to a ferromagnetic coupling. They are also expected to be multiferroic materials. Several bis‐ muth-based double perovskite oxides (BiBB' O6) have aroused great interest like Bi2NiMnO6, La2NiMnO6, BiFeO3-BiCrO3. But as we know, few researches are focused on Bi2FeMnO6. We believe it is particular interesting to investigate Bi2FeMnO6 (BFM). The origin of the ferro‐ magnetism in these compounds has been discussed in many reports. According to Goode‐ nough-Kanamori's (GK)rules, many ferromagnets have been designed through the coupling of two B site ions with and without eg electrons.In BFM, the 180 degree -Fe3+-O-Mn3+- bonds is quasistatic, partly because the strong Jahn-Teller uniaxial strain in an octahedral site.Be‐ cause the complication of the double perovskite system, there are still some questions about the violation of GK rules in some cases and the origin of the ferromagnetism or antiferro‐ magnetism.The other problem is the bad ferroelectric properties. In order to characterization of their ferroelectric/piezoelectric properties, scanning probe microscopy (SPM) techniques were used.

Multiferroic materials can be classified into two categories: one is single-phase materials; the other is multilayer or composite hetero-structures that contain more than one ferroic phases [5]. The most desirable multiferroic material is the intrinsic single-phase material, which is

© 2013 Zhao et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

still rarely produced although significant advancements had been made recently. Therefore, it is essential to broaden the searching field for new candidates of multiferroics. This work focuses on both typesmultiferroic materials: Single-phase Bi2FeMnO6 and multilayered YM‐ nO3/SnTiO3. Scanning probe microscopy (SPM) techniques were used for ferroelectric do‐ main imaging of these multiferroic materials.

tact with the dielectric surface on a conductive substrate can be considered as a capacitor. SPM is a well-established field offering multiple opportunities for new discoveries and

Ferroelectric Domain Imaging Multiferroic Films Using Piezoresponse Force Microscopy

http://dx.doi.org/10.5772/52519

195

Piezoelectric materials provide an additional response to the applied ac electric field due to the converse piezoelectric effect: ∆l=d33V, whereΔl is the displacement, d33 is the effective longitudinal piezoelectric coefficient. [18, 19] If follows that both electrostatic and piezoelec‐ tric signals are linear with the applied voltage and thus contribute to the measured PFM re‐ sponse. These measurements are referred to as out-of-plane (OP) measurement. Measurements of local hysteresis loops are of great importance in inhomogeneous or poly‐ crystalline ferroelectrics because they are able to quantify polarization switching on a scale significantly smaller than the grain size or inhomogeneity variation. Macroscopically, the switching occurs via the nucleation and growth of a large number of reverse domains in the situation where the applied electric field is uniform. Therefore, the d33 hysteresis reflects the switching averaged over the entire sample under the electrode. [20, 21] In the PFM experi‐ mental conditions, the electric field is strongly localized and inhomogeneous; therefore, the polarization switching starts with the nucleation of a single domain just under the tip.

Multiferroic materials have been attracting considerable attention especially in the last ten years. [22-25] One of the most appealing aspects of multiferroics is their magnetoelectric coupling. Among them, the most studied is the provskite BiFeO3(BFO) with room tempera‐ ture multiferroic properties and YMnO3. Based on BFO and YMnO3, we designed new mul‐ tiferroics and fabricated the films using pulsed laser deposition (PLD) method. The material system includes double perovskitemultiferroic Bi2FeMnO6 (BFM) and multilayered YMnO3/ SnTiO3+x (YST). Compared to BFO, the magnetic properties of BFM and YST were greatly improved. However, until now there is no report about their ferroelectric properties because the difficulty of obtaining well-shaped polarization hysteresis loops. It is important to study ferroelectric properties because the possible coupling between ferroelectric and antiferro‐ magnetic domains and its lead-free nature. As is well known, the ferroelectric property is mainly determined by the domain structures and domain wall motions. The most exciting recent developments in the field of multiferroics and the most promise for future discoveries are in interfacial phenomena. [26] The interfaces include those that emerge spontaneously and those that are artificially engineered. Therefore, the domain structure and polarization switching were studied in these three new multiferroic films using piezoresponse force mi‐

The emerging technique of PFM is proved to be a powerful tool to study piezoelectric and ferroelectric materials in such cases and extensive contributions have been published In PFM, the tip contacts with the sample surface and the deformation (expansion or contrac‐ tion of the sample) is detected as a tip deflection. The local piezoresponse hysteresis loop and information on local ferroelectric behavior can be obtained because the strong coupling

breakthroughs. [15-17]

croscopy (PFM).

**3. Material designation and characterization**

As we know, most piezoelectric/ferroelectric materials with good performances are based on the perovskite-type oxides of ABO3, among them, the most extensively studied ones should be PbTiO3 and Pb(ZrTi)O3 based materials due to their high dielectric constants and good piezoelectric/ferroelectric properties. However, these materials containing lead which leads to environmental problems. Thus, it is desirable to develop new lead-free piezoelectric mate‐ rials to replace PZT based piezoelectrics for environmental protection. Through first princi‐ ple calculations, SnTiO3 containing Sn2+ ions was estimated to have excellent ferroelectric properties [6, 7]. The calculated results indicated that the spontaneous polarization and pie‐ zoelectric coefficients of SnTiO3 is comparable with those of PbTiO3. Moreover, the most sta‐ ble structure is tetragonal perovskite with a=b=3.80Å and c=4.09Å [6, 7]. The metastable SnTiO3 phase is very difficult to obtain, a good way to stabilize the SnTiO3 phase is to mix it with other compounds. Several studies have been focused on the synthesis of Sn-doped ce‐ ramics in BaTiO3 system, such as (Ba0.6Sr0.4)(Ti1-xSnx)O3 [8], (Ba1-x-yCaxSny)TiO3 [9]. However, it is difficult to obtain Sn2+ by traditional bulk synthesis techniques. On the other hand, it is easier to fabricate SnTiO3 using pulsed laser deposition method (PLD). Our interest was to find out whether the SnTiO3 phase can be stabilized through the layer-by-layer deposition using PLD, and whether it is possible to obtain such metastable materials in non-equilibri‐ um conditions.

The hexagonal manganite YMnO3, which shows an antiferromagnetic transition at TN = 75 K, and a ferroelectric transition TC = 913 K, is one of the rare existing multiferroics [10-12]. It was chosen as the basic composition to sandwich SnTiO3+x phase and to form YST multilayer film. In this work, we reported the effect of substrate orientation and layer numbers of YM‐ nO3 and SnTiO3+x on the piezoelectric/ferroelectric and magnetic properties in the designed layered YST system. It was hoped that our method would serve as a model system to intro‐ duce the use of PLD techniques in the growth of multilayer multiferroic materials and meta‐ stable materials. Such techniques could be promising for device design and the searching in fabricating new materials.

### **2. SPM system**

SPM has emerged as a powerful tool for high-resolution characterization of ferroelectrics for the first time providing an opportunity for non-destructive visualization of ferroelec‐ tric domain structures at the nanoscale. [13]These nanostructures can be used for studying the intrinsic size effects in ferroelectrics as well as for addressing such technologically im‐ portant issues as processing damage, interfacial strain, grain size, aspect ratio effect, edge effect, domain pinning and imprint. [14] The system composes of a conducting tip in con‐ tact with the dielectric surface on a conductive substrate can be considered as a capacitor. SPM is a well-established field offering multiple opportunities for new discoveries and breakthroughs. [15-17]

Piezoelectric materials provide an additional response to the applied ac electric field due to the converse piezoelectric effect: ∆l=d33V, whereΔl is the displacement, d33 is the effective longitudinal piezoelectric coefficient. [18, 19] If follows that both electrostatic and piezoelec‐ tric signals are linear with the applied voltage and thus contribute to the measured PFM re‐ sponse. These measurements are referred to as out-of-plane (OP) measurement. Measurements of local hysteresis loops are of great importance in inhomogeneous or poly‐ crystalline ferroelectrics because they are able to quantify polarization switching on a scale significantly smaller than the grain size or inhomogeneity variation. Macroscopically, the switching occurs via the nucleation and growth of a large number of reverse domains in the situation where the applied electric field is uniform. Therefore, the d33 hysteresis reflects the switching averaged over the entire sample under the electrode. [20, 21] In the PFM experi‐ mental conditions, the electric field is strongly localized and inhomogeneous; therefore, the polarization switching starts with the nucleation of a single domain just under the tip.

### **3. Material designation and characterization**

still rarely produced although significant advancements had been made recently. Therefore, it is essential to broaden the searching field for new candidates of multiferroics. This work focuses on both typesmultiferroic materials: Single-phase Bi2FeMnO6 and multilayered YM‐ nO3/SnTiO3. Scanning probe microscopy (SPM) techniques were used for ferroelectric do‐

As we know, most piezoelectric/ferroelectric materials with good performances are based on the perovskite-type oxides of ABO3, among them, the most extensively studied ones should be PbTiO3 and Pb(ZrTi)O3 based materials due to their high dielectric constants and good piezoelectric/ferroelectric properties. However, these materials containing lead which leads to environmental problems. Thus, it is desirable to develop new lead-free piezoelectric mate‐ rials to replace PZT based piezoelectrics for environmental protection. Through first princi‐ ple calculations, SnTiO3 containing Sn2+ ions was estimated to have excellent ferroelectric properties [6, 7]. The calculated results indicated that the spontaneous polarization and pie‐ zoelectric coefficients of SnTiO3 is comparable with those of PbTiO3. Moreover, the most sta‐ ble structure is tetragonal perovskite with a=b=3.80Å and c=4.09Å [6, 7]. The metastable SnTiO3 phase is very difficult to obtain, a good way to stabilize the SnTiO3 phase is to mix it with other compounds. Several studies have been focused on the synthesis of Sn-doped ce‐ ramics in BaTiO3 system, such as (Ba0.6Sr0.4)(Ti1-xSnx)O3 [8], (Ba1-x-yCaxSny)TiO3 [9]. However, it is difficult to obtain Sn2+ by traditional bulk synthesis techniques. On the other hand, it is easier to fabricate SnTiO3 using pulsed laser deposition method (PLD). Our interest was to find out whether the SnTiO3 phase can be stabilized through the layer-by-layer deposition using PLD, and whether it is possible to obtain such metastable materials in non-equilibri‐

The hexagonal manganite YMnO3, which shows an antiferromagnetic transition at TN = 75 K, and a ferroelectric transition TC = 913 K, is one of the rare existing multiferroics [10-12]. It was chosen as the basic composition to sandwich SnTiO3+x phase and to form YST multilayer film. In this work, we reported the effect of substrate orientation and layer numbers of YM‐ nO3 and SnTiO3+x on the piezoelectric/ferroelectric and magnetic properties in the designed layered YST system. It was hoped that our method would serve as a model system to intro‐ duce the use of PLD techniques in the growth of multilayer multiferroic materials and meta‐ stable materials. Such techniques could be promising for device design and the searching in

SPM has emerged as a powerful tool for high-resolution characterization of ferroelectrics for the first time providing an opportunity for non-destructive visualization of ferroelec‐ tric domain structures at the nanoscale. [13]These nanostructures can be used for studying the intrinsic size effects in ferroelectrics as well as for addressing such technologically im‐ portant issues as processing damage, interfacial strain, grain size, aspect ratio effect, edge effect, domain pinning and imprint. [14] The system composes of a conducting tip in con‐

main imaging of these multiferroic materials.

um conditions.

194 Advances in Ferroelectrics

fabricating new materials.

**2. SPM system**

Multiferroic materials have been attracting considerable attention especially in the last ten years. [22-25] One of the most appealing aspects of multiferroics is their magnetoelectric coupling. Among them, the most studied is the provskite BiFeO3(BFO) with room tempera‐ ture multiferroic properties and YMnO3. Based on BFO and YMnO3, we designed new mul‐ tiferroics and fabricated the films using pulsed laser deposition (PLD) method. The material system includes double perovskitemultiferroic Bi2FeMnO6 (BFM) and multilayered YMnO3/ SnTiO3+x (YST). Compared to BFO, the magnetic properties of BFM and YST were greatly improved. However, until now there is no report about their ferroelectric properties because the difficulty of obtaining well-shaped polarization hysteresis loops. It is important to study ferroelectric properties because the possible coupling between ferroelectric and antiferro‐ magnetic domains and its lead-free nature. As is well known, the ferroelectric property is mainly determined by the domain structures and domain wall motions. The most exciting recent developments in the field of multiferroics and the most promise for future discoveries are in interfacial phenomena. [26] The interfaces include those that emerge spontaneously and those that are artificially engineered. Therefore, the domain structure and polarization switching were studied in these three new multiferroic films using piezoresponse force mi‐ croscopy (PFM).

The emerging technique of PFM is proved to be a powerful tool to study piezoelectric and ferroelectric materials in such cases and extensive contributions have been published In PFM, the tip contacts with the sample surface and the deformation (expansion or contrac‐ tion of the sample) is detected as a tip deflection. The local piezoresponse hysteresis loop and information on local ferroelectric behavior can be obtained because the strong coupling between polarization and electromechanical response in ferroelectric materials. In the present study, we attempts to use PFM to study the ferroelectric/piezoelectric properties in films of BFM and YST. PFM response was measured with a conducting tip (Rh-coated Si cantilever, k~1.6 N m-1) by an SII Nanotechnology E-sweep AFM. PFM responses were measured as a function of applied DC bias (Vdc) with a small ac voltage applied to the bot‐ tom electrode (substrate) in the contact mode, and the resulting piezoelectric deformations transmitted to the cantilever were detected from the global deflection signal using a lock-in amplifier.

### **3.1. Characterization of BFM film**

Magnetism and ferroelectricity exclude each other in single phase multiferroics. It is difficult for designing multiferroics with good magnetic and ferroelectric properties. Our interest is to design new candidate multiferroics based on BiFeO3. BiFeO3 is a well-known multiferroic material with antiferromagntic Neel temperature of 643K, which can be synthesized in a moderate condition. [22, 23, 25] In contrast, BiMnO3 is ferromagnetic with Tc = 110K and it needs high-pressure synthesis. [27, 28] The possible magnetoelectric coupling has motivated a lot of studies on the ferroelectric and antiferromagnetic domains. The ferroelectric domain structures of BFO in the form of ceramics, single crystals and thin films have been intensive‐ ly studied using PFM, TEM and other techniques.

Single phase BFM ceramics could be synthesized by conventional solid state method as the target. For BFM ceramics, the starting materials of Bi2O3, Fe3O4, MnCO3 were weighed ac‐ cording to the molecular mole ratio with 10 mol% extra Bi2O3. They were mixed, pressed in‐ to pellets and sintered at 800 °C for 3 h. Then the ceramics were crushed, ground, pressed into pellets and sintered again at 880 °C for 1 h. BFM films were deposited on SrTiO3(STO) substrate by pulsed laser deposition (PLD) method at 650°C with 500 ~ 600mTorr dynamic oxygen. [29, 30]

The structure of BFM was calculated [31] and it is connected with the magnetic configura‐ tions as shown in Figure 1. It has three possible space groups of Pm3m, R3 and C2 and the magnetic configurations were presumedfor each structure symmetry to be G-type antiferro‐ magnetic (G-AFM) and ferromagnetic (FM) structures. The most stable structure of BFM is monoclinic with C2 space group. Mn tends to show 3+ valence which will induce a large dis‐ tortion because it is Jahn-Teller ion. The valence of Mn and Fe has been studied in the for‐ mer work. [30, 32]

Figure 2 shows the results of the PFM images of BFM film. Several features could be ob‐ served: firstly, the obvious contrast could be seen and the grains in PFM and topography are correspondence; secondly, the existence of contrasts on both OP and IP indicates multiple orientations of domains; thirdly, the IP contrast is not so clear as OP contrast, that is to say, the suppression of the in-plane response for heterostructures suggesting a constrained ferro‐ electric domain-orientation along the OP direction. In the former paper, the ferroelectric do‐ main switching and typical butterfly loops were observed. [32]

**Figure 1.** Calculated six structures of BFM.

obtain the domain morphology and do the calculation.

The domains in BFO films have been analyzed in detail in Ref [19]: the bigger spontaneous ferroelectric domains were observed in BFO than in other ferroelectrics without multiferroic properties; the domains were irregular but the the model was predicted and consistent with the experimental results. According to the present results of BFM, further study is needed to

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197

**Figure 1.** Calculated six structures of BFM.

between polarization and electromechanical response in ferroelectric materials. In the present study, we attempts to use PFM to study the ferroelectric/piezoelectric properties in films of BFM and YST. PFM response was measured with a conducting tip (Rh-coated Si cantilever, k~1.6 N m-1) by an SII Nanotechnology E-sweep AFM. PFM responses were measured as a function of applied DC bias (Vdc) with a small ac voltage applied to the bot‐ tom electrode (substrate) in the contact mode, and the resulting piezoelectric deformations transmitted to the cantilever were detected from the global deflection signal using a lock-in

Magnetism and ferroelectricity exclude each other in single phase multiferroics. It is difficult for designing multiferroics with good magnetic and ferroelectric properties. Our interest is to design new candidate multiferroics based on BiFeO3. BiFeO3 is a well-known multiferroic material with antiferromagntic Neel temperature of 643K, which can be synthesized in a moderate condition. [22, 23, 25] In contrast, BiMnO3 is ferromagnetic with Tc = 110K and it needs high-pressure synthesis. [27, 28] The possible magnetoelectric coupling has motivated a lot of studies on the ferroelectric and antiferromagnetic domains. The ferroelectric domain structures of BFO in the form of ceramics, single crystals and thin films have been intensive‐

Single phase BFM ceramics could be synthesized by conventional solid state method as the target. For BFM ceramics, the starting materials of Bi2O3, Fe3O4, MnCO3 were weighed ac‐ cording to the molecular mole ratio with 10 mol% extra Bi2O3. They were mixed, pressed in‐ to pellets and sintered at 800 °C for 3 h. Then the ceramics were crushed, ground, pressed into pellets and sintered again at 880 °C for 1 h. BFM films were deposited on SrTiO3(STO) substrate by pulsed laser deposition (PLD) method at 650°C with 500 ~ 600mTorr dynamic

The structure of BFM was calculated [31] and it is connected with the magnetic configura‐ tions as shown in Figure 1. It has three possible space groups of Pm3m, R3 and C2 and the magnetic configurations were presumedfor each structure symmetry to be G-type antiferro‐ magnetic (G-AFM) and ferromagnetic (FM) structures. The most stable structure of BFM is monoclinic with C2 space group. Mn tends to show 3+ valence which will induce a large dis‐ tortion because it is Jahn-Teller ion. The valence of Mn and Fe has been studied in the for‐

Figure 2 shows the results of the PFM images of BFM film. Several features could be ob‐ served: firstly, the obvious contrast could be seen and the grains in PFM and topography are correspondence; secondly, the existence of contrasts on both OP and IP indicates multiple orientations of domains; thirdly, the IP contrast is not so clear as OP contrast, that is to say, the suppression of the in-plane response for heterostructures suggesting a constrained ferro‐ electric domain-orientation along the OP direction. In the former paper, the ferroelectric do‐

main switching and typical butterfly loops were observed. [32]

amplifier.

196 Advances in Ferroelectrics

oxygen. [29, 30]

mer work. [30, 32]

**3.1. Characterization of BFM film**

ly studied using PFM, TEM and other techniques.

The domains in BFO films have been analyzed in detail in Ref [19]: the bigger spontaneous ferroelectric domains were observed in BFO than in other ferroelectrics without multiferroic properties; the domains were irregular but the the model was predicted and consistent with the experimental results. According to the present results of BFM, further study is needed to obtain the domain morphology and do the calculation.

**3.2. Characterization of YST film**

nutes at 700°C and then cooled to room temperature. [33]

nO3 and SnTiO3+x is the same of 40 and 20 minutes, respectively.

far, it could be a peak from the (111) TiO2 phase.

and SnTiO3.

Polycrystalline YMnO3 and TiO2 ceramics were synthesized by conventional solid state method as the targets. On addition the SnO and SnO2 commercial targets were also used at the same time in our process. The deposition of these films includes the following major steps. YST films were deposited on (100) and (110) Nb: SrTiO3 (STO) substrate using a pulsed laser deposition (PLD) system at 700°C with 10-1 ~10-5Torr dynamic oxygen. The tar‐ gets were alternately switched constantly and the films were obtained in a layer-by-layer growth mode. After deposition, the films were annealed in the same condition for 15 mi‐

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In YST multilayered films, one layer is defined to be comprised of two sub-layers: (1) YM‐ nO3 and (2) SnTiO3+x. The films deposited on (100), (111) and (110) STO are expressed as YST100, YST111 and YST110, respectively. The film YST110-4 denotes the films deposited on (110) STO with four layers, as shown clearly in Figure 3. The total deposition time for YM‐

XRD patterns for the five films of YST100-2, YST100-4 were shown in Figure 4. The peaks were identified using the XRD results of SnTiO3 (◎), YMnO3 (Y) (shown in Figure 9, FeTiO3 (▽) [34, 35]. SnTiO3 is metastable and it showed two combined phases, one is FeTiO3, and the other is the good ferroelectric phase which has tetragonal structure. XRD patterns of SnTiO3 (◎) were shown in Figure 5 in supplementary materials, which is obtained using the calculated data[6, 7]. The symbol of "?" represents the phase which we cannot identify so

**Figure 3.** Schematic explanation of films with two, four and eight layers, each layer contains two sublayers of YMnO3

**Figure 2.** PFM images of BFM film

### **3.2. Characterization of YST film**

**Figure 2.** PFM images of BFM film

198 Advances in Ferroelectrics

Polycrystalline YMnO3 and TiO2 ceramics were synthesized by conventional solid state method as the targets. On addition the SnO and SnO2 commercial targets were also used at the same time in our process. The deposition of these films includes the following major steps. YST films were deposited on (100) and (110) Nb: SrTiO3 (STO) substrate using a pulsed laser deposition (PLD) system at 700°C with 10-1 ~10-5Torr dynamic oxygen. The tar‐ gets were alternately switched constantly and the films were obtained in a layer-by-layer growth mode. After deposition, the films were annealed in the same condition for 15 mi‐ nutes at 700°C and then cooled to room temperature. [33]

In YST multilayered films, one layer is defined to be comprised of two sub-layers: (1) YM‐ nO3 and (2) SnTiO3+x. The films deposited on (100), (111) and (110) STO are expressed as YST100, YST111 and YST110, respectively. The film YST110-4 denotes the films deposited on (110) STO with four layers, as shown clearly in Figure 3. The total deposition time for YM‐ nO3 and SnTiO3+x is the same of 40 and 20 minutes, respectively.

XRD patterns for the five films of YST100-2, YST100-4 were shown in Figure 4. The peaks were identified using the XRD results of SnTiO3 (◎), YMnO3 (Y) (shown in Figure 9, FeTiO3 (▽) [34, 35]. SnTiO3 is metastable and it showed two combined phases, one is FeTiO3, and the other is the good ferroelectric phase which has tetragonal structure. XRD patterns of SnTiO3 (◎) were shown in Figure 5 in supplementary materials, which is obtained using the calculated data[6, 7]. The symbol of "?" represents the phase which we cannot identify so far, it could be a peak from the (111) TiO2 phase.

**Figure 3.** Schematic explanation of films with two, four and eight layers, each layer contains two sublayers of YMnO3 and SnTiO3.

Many reports about the YMnO3 films can be found but most of them without ferroelectric characterization provided. This is mainly ascribed to the difficulty in obtaining a satisfactory ferroelectric measurement result. Figure 6 shows the electrical polarization hysteresis loops (P-E loops) of YST110-4 film. The ferroelectric type hysteresis loop was observed. It is obvious that the P-E loop is significantly different for different layer-number films, as well as for differ‐ ent substrate orientations. For the same STO orientation, the P-E loops were improved as the layer number increase, and the film fabricated on (110) STO shows improved properties com‐ pared to the film on (100) STO with the same layer number. As for the same deposition time for the four samples, it is found that more layers of SnTiO3 in the YST system shows much better ferroelectric properties. It is suggested that the SnTiO3 phase can exist only in ultra-thin thick‐ ness and can be stabilized by YMnO3 sub-layers. That is to say, SnTiO3 is indeed a ferroelectric materials but it is a great challenge [36] to obtain stable single phase SnTiO3 films that is with good ferroelectric performance (it will be tried in our future works). Although the electric charge in the interface may affect its ferroelectric properties, the improved properties of YST110-4 proved that its effect was not significant. While the P-E loops were not improved in

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Although well developed P-E loops were observed for YST100-8 and YST110-4, the loops still show the rounded features, indicating that contributions to the hysteresis loop from movable charges was significant. In addition, the P-E loops of the four films have revealed

The fatigue properties were shown in Figure 7 for YST110-4. The polarization measurement was at 21V, the switching voltage was 14V and the frequency was 5k Hz. The remnant po‐

parisons, YST110-4 sample shows the comparable ferroelectric properties with YST100-8.

) and they do not exhibit saturation.

read/write cycles for YST110-4. From all the com‐

YST111-4, all the observations showed the anisotropy of the films.

**Figure 6.** Ferroelectric polarization loop of YST films. [31]

larization decreased about 40% after 107

that the polarization is rather small (just a few μC/cm2

**Figure 4.** XRD patterns for the five films of YST100-2, YST100-4. [31]

**Figure 5.** XRD patterns of calculated SnTiO3 and YMnO3 targets. [33]

Many reports about the YMnO3 films can be found but most of them without ferroelectric characterization provided. This is mainly ascribed to the difficulty in obtaining a satisfactory ferroelectric measurement result. Figure 6 shows the electrical polarization hysteresis loops (P-E loops) of YST110-4 film. The ferroelectric type hysteresis loop was observed. It is obvious that the P-E loop is significantly different for different layer-number films, as well as for differ‐ ent substrate orientations. For the same STO orientation, the P-E loops were improved as the layer number increase, and the film fabricated on (110) STO shows improved properties com‐ pared to the film on (100) STO with the same layer number. As for the same deposition time for the four samples, it is found that more layers of SnTiO3 in the YST system shows much better ferroelectric properties. It is suggested that the SnTiO3 phase can exist only in ultra-thin thick‐ ness and can be stabilized by YMnO3 sub-layers. That is to say, SnTiO3 is indeed a ferroelectric materials but it is a great challenge [36] to obtain stable single phase SnTiO3 films that is with good ferroelectric performance (it will be tried in our future works). Although the electric charge in the interface may affect its ferroelectric properties, the improved properties of YST110-4 proved that its effect was not significant. While the P-E loops were not improved in YST111-4, all the observations showed the anisotropy of the films.

**Figure 6.** Ferroelectric polarization loop of YST films. [31]

**Figure 4.** XRD patterns for the five films of YST100-2, YST100-4. [31]

200 Advances in Ferroelectrics

**Figure 5.** XRD patterns of calculated SnTiO3 and YMnO3 targets. [33]

Although well developed P-E loops were observed for YST100-8 and YST110-4, the loops still show the rounded features, indicating that contributions to the hysteresis loop from movable charges was significant. In addition, the P-E loops of the four films have revealed that the polarization is rather small (just a few μC/cm2 ) and they do not exhibit saturation. The fatigue properties were shown in Figure 7 for YST110-4. The polarization measurement was at 21V, the switching voltage was 14V and the frequency was 5k Hz. The remnant po‐ larization decreased about 40% after 107 read/write cycles for YST110-4. From all the com‐ parisons, YST110-4 sample shows the comparable ferroelectric properties with YST100-8.

**Figure 7.** Fatigue measurement of YST110-4 film. [31]

Figure 8 shows TEM results of the YST110-4 film. It clearly shows four layers and one layer which is close to the substrate seems have some reaction with the substrate. The other three layers are homogeneous and we could see the domain structures through this image. The surface of the film was studied using AFM as shown in Figure 9. As expected, Figure 10 displays contrast over the polarized square after poled by positive and negative 10 V volt‐ age, due to the different phases of the PFM response for the up and down domains. The ob‐ vious change of the contrast in YST110-4 film confirms that the polarization reversal is indeed possible and that the film is ferroelectric at room temperature.

**Figure 9.** AFM topography image of YST110-4 film.

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**Figure 10.** PFM image of YST110-4 film after poled using ± 10 V voltage

**Figure 8.** TEM image of YST110-4 film.

**Figure 9.** AFM topography image of YST110-4 film.

**Figure 7.** Fatigue measurement of YST110-4 film. [31]

202 Advances in Ferroelectrics

**Figure 8.** TEM image of YST110-4 film.

Figure 8 shows TEM results of the YST110-4 film. It clearly shows four layers and one layer which is close to the substrate seems have some reaction with the substrate. The other three layers are homogeneous and we could see the domain structures through this image. The surface of the film was studied using AFM as shown in Figure 9. As expected, Figure 10 displays contrast over the polarized square after poled by positive and negative 10 V volt‐ age, due to the different phases of the PFM response for the up and down domains. The ob‐ vious change of the contrast in YST110-4 film confirms that the polarization reversal is

indeed possible and that the film is ferroelectric at room temperature.

**Figure 10.** PFM image of YST110-4 film after poled using ± 10 V voltage

As seen in Figure 11, when a direct current voltage up to 10 V was applied on the samples, the sample exhibited ''butterfly'' loop. The loops were not symmetrical due to the asymme‐ try of the upper (tip) and bottom (substrate) electrodes. In addition, the substrate may also affect the d33. According to the equation d33=Δl/V, where Δl is the displacement, the effective d33 could be calculated. At the voltage of 10 V, both samples show the maximum effective d33of about 6.21 pm/V for YST110-4 sample.

**Acknowledgements**

**Author details**

Xiaolin Wang3

**References**

6694.

U3.13.1.

Hongyang Zhao1,2\*, Hideo Kimura1

vation Campus, Fairy Meadow, Australia

als. Nature 2006; 442: 759-765.

The authors gratefully acknowledge Dr. Minoru Osada, Dr. Kazuya Terabe in NIMS and Prof. H. R. Zeng in Shanghai Institute of Ceramics for their experimental help and discus‐

, Lei Guo1

Ferroelectric Domain Imaging Multiferroic Films Using Piezoresponse Force Microscopy

, Zhenxiang Cheng3

and

http://dx.doi.org/10.5772/52519

205

sions. Part of this work was supported by JST ALCA and MEXT GRENE.

\*Address all correspondence to: zhao.hongyang@nims.go.jp

1 National Institute for Materials Science, Sengen 1-2-1, Tsukuba, Japan

, Qiwen Yao1

2 Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai, China

cou-pled magnetic and electric domains.Nature 2002; 419: 818.

single-crystal BiFeO3. Phys. Rev. B 2008; 77: 144403.

SnTiO3.Jpn. J. Appl. Phys. 2008; 47 (9), 7735-7739.

3 Institute for Superconducting and Electronics Materials, University of Wollongong, Inno‐

[1] Hill N. A., Why are there so few magnetic ferroelectrics? J. Phys. Chem. B 2000; 104:

[2] Fiebig M., T. Lottermoser, D. Frohlich, A. V. Goltsev, R. V. Pisarev, Observation of

[3] Eerenstein W., Mathur N. D. and Scott J.F., Multiferroic and magnetoelectric materi‐

[4] Singh M. K., Prellier W., Singh M. P., Katiyar R. S., Scott J. F., Spin-glass transition in

[5] J. Das, Y.Y. Song, M.Z. Wu, Electric-field control of ferromagnetic resonance in mon‐ olithic BaFe12O19–Ba0.5Sr0.5TiO3heterostructures. J. Appl. Phys. 2010; 108: 043911.

[6] Konishi Y., Ohasawa M., Yonezawa Y., Tanimura Y., Chikyow T., Wakisaka T., Koi‐ numa H., Miyamoto A., Kubo M., Sasata K., Mater. Res. Soc. Symp. Proc. 2003; 748

[7] Uratani Y., Shishidou T., Oguchi T., First-Principles Study of Lead-Free Piezoelectric

**Figure 11.** Butterfly and local piezoresponse hysteresis loops of YST110-4 film.

### **4. Conclusions**

The novel multiferroic films of BFM and multilayered YST were successfully produced us‐ ing PLD method. Both of them were characterized using PFM method. The special or excel‐ lent properties can often be found in the metastable materials. Through calculation, s4ix structures are presumed in BFM with two magnetic configurations of G-AFM and FM. In YST films, the metastable SnTiO3 phase was obtained. The improved ferroelectric properties were observed through increasing the layer numbers and more SnTiO3 phase was stabilized, moreover, the change in contrast after bias is applied indicates a change in polarization di‐ rection and hence ferroelectric switching. Although it is a great challenge in obtaining the SnTiO3 thin films, we believe that through the optimization of fabrication process and condi‐ tions, the single phase SnTiO3 or multilayer films, or composite materials containing SnTiO3 could become a new generation lead-free piezoelectric/ferroelectric material. For a thorough understanding of the mechanisms of the films, knowledge of the domain structures is a pre‐ requisite, which is of crucial importance to increase and tune the functionality of multiferro‐ ic films.

### **Acknowledgements**

As seen in Figure 11, when a direct current voltage up to 10 V was applied on the samples, the sample exhibited ''butterfly'' loop. The loops were not symmetrical due to the asymme‐ try of the upper (tip) and bottom (substrate) electrodes. In addition, the substrate may also affect the d33. According to the equation d33=Δl/V, where Δl is the displacement, the effective d33 could be calculated. At the voltage of 10 V, both samples show the maximum effective

The novel multiferroic films of BFM and multilayered YST were successfully produced us‐ ing PLD method. Both of them were characterized using PFM method. The special or excel‐ lent properties can often be found in the metastable materials. Through calculation, s4ix structures are presumed in BFM with two magnetic configurations of G-AFM and FM. In YST films, the metastable SnTiO3 phase was obtained. The improved ferroelectric properties were observed through increasing the layer numbers and more SnTiO3 phase was stabilized, moreover, the change in contrast after bias is applied indicates a change in polarization di‐ rection and hence ferroelectric switching. Although it is a great challenge in obtaining the SnTiO3 thin films, we believe that through the optimization of fabrication process and condi‐ tions, the single phase SnTiO3 or multilayer films, or composite materials containing SnTiO3 could become a new generation lead-free piezoelectric/ferroelectric material. For a thorough understanding of the mechanisms of the films, knowledge of the domain structures is a pre‐ requisite, which is of crucial importance to increase and tune the functionality of multiferro‐

d33of about 6.21 pm/V for YST110-4 sample.

204 Advances in Ferroelectrics

**Figure 11.** Butterfly and local piezoresponse hysteresis loops of YST110-4 film.

**4. Conclusions**

ic films.

The authors gratefully acknowledge Dr. Minoru Osada, Dr. Kazuya Terabe in NIMS and Prof. H. R. Zeng in Shanghai Institute of Ceramics for their experimental help and discus‐ sions. Part of this work was supported by JST ALCA and MEXT GRENE.

### **Author details**

Hongyang Zhao1,2\*, Hideo Kimura1 , Qiwen Yao1 , Lei Guo1 , Zhenxiang Cheng3 and Xiaolin Wang3

\*Address all correspondence to: zhao.hongyang@nims.go.jp

1 National Institute for Materials Science, Sengen 1-2-1, Tsukuba, Japan

2 Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai, China

3 Institute for Superconducting and Electronics Materials, University of Wollongong, Inno‐ vation Campus, Fairy Meadow, Australia

### **References**


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M., Ramesh R., Epitaxial BiFeO3 Multiferroic Thin Film Heterostructures. Science

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[26] Spaldin N. A., Cheong S. W., Ramesh R.,Multiferroics: Past,present, and future.

[27] T. Atou, H. Chiba, K. Ohoyama, Y. Yamaguchi, Y. Syono, Structure determination of ferromagnetic perovskite BiMnO3. J. Solid State Chem. 1999: 145(2) 639-642.

[28] T. Kimura, S. Kawamoto, I. Yamada, M. Azuma, M. Takano, Y. Tokura, Magnetoca‐ pacitance effect in multiferroic BiMnO3, Phys. Rev. B 2003: 67 (R), 180401-180404.

[29] Zhao, H.Y.; Kimura, H.; Cheng, Z.X.; Wang, X.L. &Nishida, T. Room temperature multiferroic properties of Nd:BiFeO3/Bi2FeMnO6 bilayered films.Appl. Phys. Lett.

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[32] ZhaoH Y, Kimura H.; Cheng Z.X.; Wang X.L., New multiferroic materials: Bi2FeM‐

[33] Zhao H Y, Kimura H.; Cheng Z.X.; Wang X.L., Yao Q W, Osada M, Li B W, Nishida T A new multiferroicheterostructure of YMnO3/SnTiO3+x. Scrip. Mater. 2011; 65:

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[8] Ha J. Y., Lin L. W., Jeong D. Y., Yoon S. J., Choi J. W., Improved Figure of Merit of (Ba,Sr)TiO3-Based Ceramics by Sn Substitution.Jpn. J. Appl. Phys. 2009; 48 011402. [9] Suzuki S., Takeda T., Ando A., Takagi H., Ferroelectric phase transition in Sn2+ ions

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[19] Catalan G., Bea H., Fusil S., Bibes M., Paruch P., Barthelemy A., Scott J. F., Fractal Di‐ mension and Size Scaling of Domains in Thin Films of Multiferroic BiFeO3. Phy. Rev.

[20] Kalinin S V, Gruverman A and Bonnell D A Quantitative analysis of nanoscale switching in SrBi2Ta2O9 thin films by piezoresponse force microscopy Appl. Phys.

[21] Tybell T, Paruch P, Giamarchi T, Triscone J-M Domain wall creep in epitaxial ferro‐

[22] Catalan G., Scott J. F., Adv. Mater. Physics and Applications of Bismuth Ferrite.2009;

[23] Wang J., Neaton B. J., Zheng H., Nagarajan V., Ogale S. B., Liu B., Viehland D., Vai‐ thyanathan V., Schlom D. G.,. Waghmare U. V, Spaldi N. A.n, Rabe K. M., Wutting

electric Pb(Zr0.2Ti0.8)O3 thin films 2002 Phys. Rev. Lett. 89 097601

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21, 2643.


**Chapter 10**

**Gelcasting of Ferroelectric Ceramics:**

Additional information is available at the end of the chapter

fine structure, which are required for various devices.

Dong Guo and Kai Cai

http://dx.doi.org/10.5772/53995

**1. Introduction**

1996; Setter et al., 2000).

**Doping Effect and Further Development**

Ferroelectric ceramics may be seen as the most important type of ferroelectric materials, which have been used in a wide spectrum of electrical and microelectronic devices, includ‐ ing underwater transducers, micro-pumps and valves, ultrasonic motors, thermal sensors, probes for medical imaging and non-destructive testing, and accelerometers, etc (Cross LE,

Ferroelectric ceramics used in the devices have various shapes. A certain shape formation technique is required to make ceramics with the desired shapes. Dry pressing is the most commonly used ceramic forming technique. In this technique, dry powders containing or‐ ganic binder are filled into a solid mold, then dry ceramic green bodies with the shape of the mold cavity are formed under mechanical or hydraulic compacting presses selected for the necessary force and powder fill depth. The pressure is around several tens of MPa or higher. Therefore, large ceramic parts require a much higher compacting force. If the ceramic parts are unable to have pressure transmit suitably for a uniform pressed density then isostatic pressing may be used. One of the most serious disadvantages of dry pressing lies in the dif‐ ficulty in fabricating high quality large and complex-shaped ceramics or ceramics with a

To resolve the problems associated with the conventional dry pressing, new wet forming techniques, such as gelcasting (Omatete et al., 1991), electrophoretic casting (Biesheuvel et al., 1999), hydrolysis assisted solidification (Novak et al., 2002), and direct coagulation cast‐ ing (Graule et al., 1996), etc., are becoming increasingly attractive for advanced ceramic ma‐ terials. Since in these techniques the ceramic powders are dispersed in a liquid medium and thoroughly mixed, wet forming techniques have the advantages of reducing some structure defects that are difficult to remove in dry pressed ceramic parts. Among the many wet form‐

> © 2013 Guo and Cai; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

### **Gelcasting of Ferroelectric Ceramics: Doping Effect and Further Development**

Dong Guo and Kai Cai

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53995

### **1. Introduction**

Ferroelectric ceramics may be seen as the most important type of ferroelectric materials, which have been used in a wide spectrum of electrical and microelectronic devices, includ‐ ing underwater transducers, micro-pumps and valves, ultrasonic motors, thermal sensors, probes for medical imaging and non-destructive testing, and accelerometers, etc (Cross LE, 1996; Setter et al., 2000).

Ferroelectric ceramics used in the devices have various shapes. A certain shape formation technique is required to make ceramics with the desired shapes. Dry pressing is the most commonly used ceramic forming technique. In this technique, dry powders containing or‐ ganic binder are filled into a solid mold, then dry ceramic green bodies with the shape of the mold cavity are formed under mechanical or hydraulic compacting presses selected for the necessary force and powder fill depth. The pressure is around several tens of MPa or higher. Therefore, large ceramic parts require a much higher compacting force. If the ceramic parts are unable to have pressure transmit suitably for a uniform pressed density then isostatic pressing may be used. One of the most serious disadvantages of dry pressing lies in the dif‐ ficulty in fabricating high quality large and complex-shaped ceramics or ceramics with a fine structure, which are required for various devices.

To resolve the problems associated with the conventional dry pressing, new wet forming techniques, such as gelcasting (Omatete et al., 1991), electrophoretic casting (Biesheuvel et al., 1999), hydrolysis assisted solidification (Novak et al., 2002), and direct coagulation cast‐ ing (Graule et al., 1996), etc., are becoming increasingly attractive for advanced ceramic ma‐ terials. Since in these techniques the ceramic powders are dispersed in a liquid medium and thoroughly mixed, wet forming techniques have the advantages of reducing some structure defects that are difficult to remove in dry pressed ceramic parts. Among the many wet form‐

© 2013 Guo and Cai; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ing techniques, aqueous gelcasting represents the latest improvements. In the technique, a high solids loading slurry obtained by dispersing the ceramic powders in the pre-mixed sol‐ ution containing monomer and cross-linker is cast in a mold of the desired shape. When heated, the monomer and cross-linker polymerize to form a three-dimensional network structure, thus the slurry is solidified *in situ* and green bodies of the desired shape are ob‐ tained, which consists mainly of ceramic powders with a low polymer content. Gelcasting may be seen as a milestone in fabricating complex-shaped ceramic parts since it has initiated a new branch of research in ceramic processing due to its intriguing properties of near-netshape forming, high green strength, and low binder concentration, etc. As a pressure free method, it can also be used in fabricating large ceramic components that have simple shapes. For example, to make a ceramic disc with a diameter of 20 cm by dry pressing, a load of hundreds of tons is required. In contrast, such a ceramic disc can be easily made by gelcasting via a simple ring-shaped mold at much less cost. Furthermore, gelcasting may be developed to fabricate complex-shaped (Cai et al., 2003) or fine-structured (Guo et al., 2003) ceramic parts that are rather difficult or even impossible to be formed by the conventional dry pressing method.

**2. Colloidal chemistry and rheological properties of PZT suspensions**

around 60-80o

the green ceramic body is obtained.

**Figure 1.** The flowchart of gelcasting

We first give a short introduction the gelcasting technique. The details are out of the focus of this chapter, and interested readers can refer to other papers. According to the liquid medi‐ um used, there are two types of gelcasting systems: aqueous and nonaqueous systems. In aqueous gelcasting deionized water is used as the medium, and acrylamide (C2H3CONH2) may be seen as a prototype monomer. Generally, N,N-methylenebisacrylamide ((C2H3CONH)2CH2, MBAM) N,N,N0,N0-Tetramethylethylenediamine (TEMED) and (NH4)2S2O8 are used as the cross-linker, the catalyst and the initiator, respectively. The gel‐ casting process of PZT is similar to that of previous studies. The flowchart of the gelcasting is shown in Figure 1. First, the PZT powders are added in the premix solution containing AM and MBAM and thoroughly mixed. After addition of initiator and catalyst, the slurry is de-aired in vacuum, then the slurry is cast into the mold with desired shape and heated

Gelcasting of Ferroelectric Ceramics: Processing, Doping Effect of Additives and Further Development

http://dx.doi.org/10.5772/53995

211

C in a oven for several hours for polymerizatoin and drying. After demolding

Since its invention in 1991 gelcasting has been widely used for the fabrication of structural ceramics, including Al2O3 (Young et al., 1991), SiC (Zhou et al., 2000), and Si3N4 (Huang et al., 2000), etc. Later, it was applied to a number of functional ceramics in a number of ways including LaGaO3 (Zha et al., 2001), ZnO (Bell et al., 2004), and Sr0.5Ba0.5Nb2O6 (Chen et al., 2006), etc. The first journal report about the application of gelcasting to piezoelectric ceram‐ ics appeared at 2002 (Guo et al., 2003). Compared to structural ceramics, much less research about gelcasting of functional ceramics has been conducted so far.

Among the many ferroelectric ceramic materials, lead zirconate titanate (PZT) is the most widely used one owing to their superior piezoelectric, pyroelectric and dielectric properties. A fascinating feature of multicomponent ferroelectric ceramics is that their electrical proper‐ ties can be modified by doping with acceptors and donors (Shaw et al., 2000). As a result, a series of PZT materials with tailored properties are commercially available. Unfortunately, this feature also leads to the problem of high sensitivity of the electrical properties of PZT to composition. In addition, PZT is commonly used with a composition close to the morpho‐ tropic phase boundary (MPB) at a Zr/Ti ratio of about 52/48, where properties such as piezo‐ electric coefficients, dielectric permittivity, and coupling factors are maximized and thus may be more sensitive to the composition (Noheda et al., 2006). On the other hand, since shape formation in gelcasting is achieved through *in situ* polymerization, organic additives are used in the premix solution. Also, addition of commercial surfactants that may have a complicated composition is indispensable, because successful fabrication of ceramics via gel‐ casting or other colloidal methods requires to prepare high solids loading ceramic slurry with still a low viscosity. Consequently, in order to apply gelcasting to the formation of fer‐ roelectric ceramics such as PZT, it is necessary to remove the possible influence of the im‐ purities introduced by the various additives on the electrical performance of the final products. This makes the problems more complicated than that of structural ceramics.

### **2. Colloidal chemistry and rheological properties of PZT suspensions**

We first give a short introduction the gelcasting technique. The details are out of the focus of this chapter, and interested readers can refer to other papers. According to the liquid medi‐ um used, there are two types of gelcasting systems: aqueous and nonaqueous systems. In aqueous gelcasting deionized water is used as the medium, and acrylamide (C2H3CONH2) may be seen as a prototype monomer. Generally, N,N-methylenebisacrylamide ((C2H3CONH)2CH2, MBAM) N,N,N0,N0-Tetramethylethylenediamine (TEMED) and (NH4)2S2O8 are used as the cross-linker, the catalyst and the initiator, respectively. The gel‐ casting process of PZT is similar to that of previous studies. The flowchart of the gelcasting is shown in Figure 1. First, the PZT powders are added in the premix solution containing AM and MBAM and thoroughly mixed. After addition of initiator and catalyst, the slurry is de-aired in vacuum, then the slurry is cast into the mold with desired shape and heated around 60-80o C in a oven for several hours for polymerizatoin and drying. After demolding the green ceramic body is obtained.

**Figure 1.** The flowchart of gelcasting

ing techniques, aqueous gelcasting represents the latest improvements. In the technique, a high solids loading slurry obtained by dispersing the ceramic powders in the pre-mixed sol‐ ution containing monomer and cross-linker is cast in a mold of the desired shape. When heated, the monomer and cross-linker polymerize to form a three-dimensional network structure, thus the slurry is solidified *in situ* and green bodies of the desired shape are ob‐ tained, which consists mainly of ceramic powders with a low polymer content. Gelcasting may be seen as a milestone in fabricating complex-shaped ceramic parts since it has initiated a new branch of research in ceramic processing due to its intriguing properties of near-netshape forming, high green strength, and low binder concentration, etc. As a pressure free method, it can also be used in fabricating large ceramic components that have simple shapes. For example, to make a ceramic disc with a diameter of 20 cm by dry pressing, a load of hundreds of tons is required. In contrast, such a ceramic disc can be easily made by gelcasting via a simple ring-shaped mold at much less cost. Furthermore, gelcasting may be developed to fabricate complex-shaped (Cai et al., 2003) or fine-structured (Guo et al., 2003) ceramic parts that are rather difficult or even impossible to be formed by the conventional

Since its invention in 1991 gelcasting has been widely used for the fabrication of structural ceramics, including Al2O3 (Young et al., 1991), SiC (Zhou et al., 2000), and Si3N4 (Huang et al., 2000), etc. Later, it was applied to a number of functional ceramics in a number of ways including LaGaO3 (Zha et al., 2001), ZnO (Bell et al., 2004), and Sr0.5Ba0.5Nb2O6 (Chen et al., 2006), etc. The first journal report about the application of gelcasting to piezoelectric ceram‐ ics appeared at 2002 (Guo et al., 2003). Compared to structural ceramics, much less research

Among the many ferroelectric ceramic materials, lead zirconate titanate (PZT) is the most widely used one owing to their superior piezoelectric, pyroelectric and dielectric properties. A fascinating feature of multicomponent ferroelectric ceramics is that their electrical proper‐ ties can be modified by doping with acceptors and donors (Shaw et al., 2000). As a result, a series of PZT materials with tailored properties are commercially available. Unfortunately, this feature also leads to the problem of high sensitivity of the electrical properties of PZT to composition. In addition, PZT is commonly used with a composition close to the morpho‐ tropic phase boundary (MPB) at a Zr/Ti ratio of about 52/48, where properties such as piezo‐ electric coefficients, dielectric permittivity, and coupling factors are maximized and thus may be more sensitive to the composition (Noheda et al., 2006). On the other hand, since shape formation in gelcasting is achieved through *in situ* polymerization, organic additives are used in the premix solution. Also, addition of commercial surfactants that may have a complicated composition is indispensable, because successful fabrication of ceramics via gel‐ casting or other colloidal methods requires to prepare high solids loading ceramic slurry with still a low viscosity. Consequently, in order to apply gelcasting to the formation of fer‐ roelectric ceramics such as PZT, it is necessary to remove the possible influence of the im‐ purities introduced by the various additives on the electrical performance of the final products. This makes the problems more complicated than that of structural ceramics.

about gelcasting of functional ceramics has been conducted so far.

dry pressing method.

210 Advances in Ferroelectrics

The colloidal and rheological properties of the PZT suspension are important issues that should be addressed first. Homogenous dispersion of the powder in the premix solution and stability of the suspension are determined mainly by attractive and repulsive forces be‐ tween the particles in the system. The former generally arises from the van der Waals forces while the latter can arise from electrostatic repulsion or steric repulsion of surfactant materi‐ als absorbed on the particle surfaces (Israelachvili, 1992). Magnitude of the van der Waals forces is mainly determined by the nature of the particle surface and the solvent. While the repulsive forces can be modified over a wide range by dispersants.

typical polyelectrolyte dispersants and a widely used organic surfactant triammonium cit‐ rate (TAC). The details are listed in Table 1. Electrostatic repulsion is dependent on the zeta potential (ζ) of the powders. The higher this value with the same polarity, the stronger the electrostatic repulsion between the particles. When close to the isoelectric point (IEP), the particles tend to flocculate. Zeta potentials of various PZT aqueous suspensions (0.06 vol % solids) at different pH values are shown in Figure 2. The zeta potential of pure PZT suspen‐ sion changes from 33 mV at pH = 1.7 to - 35.1 mV at pH = 11.9 with an IEP at about pH = 7.2, suggesting that neutral environment is disadvantageous for good dispersion. Addition of monomer only slightly decreases the relative value of the potential and has little effect on IEP, indicating that the uncharged AM molecules screen the charge of the PZT particles. With the addition of TAC, JN281 and SGA the IEP is moved to pH = 2.5, 2.1 and 2.3, respec‐ tively. In the range of neutral environment to pH = 12, the zeta potential is almost constant. The higher absolute potential values of suspensions with JN281 and SGA than that with TAC imply that polyelectrolyte dispersants are more effective as far as the electrostatic re‐

Gelcasting of Ferroelectric Ceramics: Processing, Doping Effect of Additives and Further Development

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213

Figure 3 shows the viscosity as a function of the shear rate for different PZT slurries solids loading. Adjustment of pH values to either acid or basic conditions has little effect, while addition of dispersants can greatly decrease the viscosity. The high viscosities at the begin‐ ning indicate a 'Bingham' type behavior, which is followed by a shear thinning at low shear rates. Shear-thinning behavior can be attributed to a certain kind of rearrangement of the relative spatial disposition of the particles. For concentrated suspensions of hard solid parti‐ cles in Newtonian liquids, a flow-induced layered structure has been verified (Ackerson, 1990). Such a structure can provide a low resistance of the particle movement between dif‐ ferent layers under the shear flow. For the systems containing dispersants, when the shear rates increase to a critical value (γc) a shear-thickening behavior appears, indicating that the flow-induced structure is destroyed. It is clear that the polyelectrolyte dispersants are much more effective than TAC. In addition, PZT slurries in the premix solution have almost the same viscosity values as those in the pure water, suggesting that addition of AM (15 wt.%) has little effect on the viscosity. These and the zeta potential results indicate that the polye‐

lectrolyte dispersants work by both electrostatic and steric stabilization mechanism.

Generally, there is a complex nonlinear relationship between viscosity and solids volume fraction, which is closely related to many factors, including the continuous phase viscosity, particle-size distribution, and particle shape, etc. Influence of solids loading on the apparent viscosity is shown in Figure 4. At low solids loading the slurries show a low viscosity. The continuous addition of particles finally three-dimensional contact throughout the suspen‐ sion, making flow impossible. The particular solid phase volume at which this happens is called the maximum packing fraction Φm (Barnes & Hutton, 1989). Before adding dispersant the PZT suspension has a measurable viscosity at a solids loading slightly smaller than 32 vol.%. Adding a little more PZT powders leads to a 'solidified' slurry whose viscosity is im‐ possible to be measured by the normal rheometer. The Φ<sup>m</sup> is thus determined to be 32 vol.%. Adding 0.8 and 1.5 wt.% of TAC increase Φm to about 47 and 53 vol.%, respectively. The lower Φm for a TAC concentration of 2.2 wt.% indicates that excess dispersant is harmful.

pulsion is concerned.



**Figure 2.** Effect of dispersants on the zeta potentials of the PZT suspensions at different pH values.

Polyelectrolyte dispersants are well known to be effective for various ceramic slurries due to both electrostatic and steric repulsions of the macromolecules. Here we show the effects of typical polyelectrolyte dispersants and a widely used organic surfactant triammonium cit‐ rate (TAC). The details are listed in Table 1. Electrostatic repulsion is dependent on the zeta potential (ζ) of the powders. The higher this value with the same polarity, the stronger the electrostatic repulsion between the particles. When close to the isoelectric point (IEP), the particles tend to flocculate. Zeta potentials of various PZT aqueous suspensions (0.06 vol % solids) at different pH values are shown in Figure 2. The zeta potential of pure PZT suspen‐ sion changes from 33 mV at pH = 1.7 to - 35.1 mV at pH = 11.9 with an IEP at about pH = 7.2, suggesting that neutral environment is disadvantageous for good dispersion. Addition of monomer only slightly decreases the relative value of the potential and has little effect on IEP, indicating that the uncharged AM molecules screen the charge of the PZT particles. With the addition of TAC, JN281 and SGA the IEP is moved to pH = 2.5, 2.1 and 2.3, respec‐ tively. In the range of neutral environment to pH = 12, the zeta potential is almost constant. The higher absolute potential values of suspensions with JN281 and SGA than that with TAC imply that polyelectrolyte dispersants are more effective as far as the electrostatic re‐ pulsion is concerned.

The colloidal and rheological properties of the PZT suspension are important issues that should be addressed first. Homogenous dispersion of the powder in the premix solution and stability of the suspension are determined mainly by attractive and repulsive forces be‐ tween the particles in the system. The former generally arises from the van der Waals forces while the latter can arise from electrostatic repulsion or steric repulsion of surfactant materi‐ als absorbed on the particle surfaces (Israelachvili, 1992). Magnitude of the van der Waals forces is mainly determined by the nature of the particle surface and the solvent. While the

repulsive forces can be modified over a wide range by dispersants.

Producer Beijing Chemical

212 Advances in Ferroelectrics

Reagent Co.

**Table 1.** Composition and producer of the dispersants

Composition Triammonium citrate Poly (acrylic acid),

**Commercial Name TAC JN281 SP2 SGA**

Beijing Pinbao Chemical Co.

NH4 + salt solution

**Figure 2.** Effect of dispersants on the zeta potentials of the PZT suspensions at different pH values.

Polyelectrolyte dispersants are well known to be effective for various ceramic slurries due to both electrostatic and steric repulsions of the macromolecules. Here we show the effects of

Beijing Tongfang Chemical

Poly (acrylic acid-co-maleic acid), Na+ salt solution

Beijing Tongfang Chemical Co.

Poly (acrylic acid), Na + salt solution

Co.

Figure 3 shows the viscosity as a function of the shear rate for different PZT slurries solids loading. Adjustment of pH values to either acid or basic conditions has little effect, while addition of dispersants can greatly decrease the viscosity. The high viscosities at the begin‐ ning indicate a 'Bingham' type behavior, which is followed by a shear thinning at low shear rates. Shear-thinning behavior can be attributed to a certain kind of rearrangement of the relative spatial disposition of the particles. For concentrated suspensions of hard solid parti‐ cles in Newtonian liquids, a flow-induced layered structure has been verified (Ackerson, 1990). Such a structure can provide a low resistance of the particle movement between dif‐ ferent layers under the shear flow. For the systems containing dispersants, when the shear rates increase to a critical value (γc) a shear-thickening behavior appears, indicating that the flow-induced structure is destroyed. It is clear that the polyelectrolyte dispersants are much more effective than TAC. In addition, PZT slurries in the premix solution have almost the same viscosity values as those in the pure water, suggesting that addition of AM (15 wt.%) has little effect on the viscosity. These and the zeta potential results indicate that the polye‐ lectrolyte dispersants work by both electrostatic and steric stabilization mechanism.

Generally, there is a complex nonlinear relationship between viscosity and solids volume fraction, which is closely related to many factors, including the continuous phase viscosity, particle-size distribution, and particle shape, etc. Influence of solids loading on the apparent viscosity is shown in Figure 4. At low solids loading the slurries show a low viscosity. The continuous addition of particles finally three-dimensional contact throughout the suspen‐ sion, making flow impossible. The particular solid phase volume at which this happens is called the maximum packing fraction Φm (Barnes & Hutton, 1989). Before adding dispersant the PZT suspension has a measurable viscosity at a solids loading slightly smaller than 32 vol.%. Adding a little more PZT powders leads to a 'solidified' slurry whose viscosity is im‐ possible to be measured by the normal rheometer. The Φ<sup>m</sup> is thus determined to be 32 vol.%. Adding 0.8 and 1.5 wt.% of TAC increase Φm to about 47 and 53 vol.%, respectively. The lower Φm for a TAC concentration of 2.2 wt.% indicates that excess dispersant is harmful. The relationship between the viscosity and the solids volume fraction for monodispersed suspension can be explained by the Krieger–Dougherty (K–D) model:

[ ] (1 / ) *<sup>m</sup>*

where η and ηo are the viscosity of the suspension and the solvent, respectively. The true volume fraction of the powder in the suspension is represented by f. The intrinsic viscosity [η] is a function of particle geometry; a value of 2.5 is suitable for spherical particles. A maxi‐ mum packing fraction of 0.63±0.002 is suitable for random close packing at low shear rate. The K–D model curve with [η] = 2.5 and Φm=0.63 is plotted in the Figure. Although the ex‐ perimental curves are somewhat deviated from the K–D model, they still show a similar shape: low viscosity at low solids loading and sharp increase at high solids loading. The dis‐ crepancy may partly be ascribed to the inhomogeneity of the particle size. In addition, parti‐ cle flocculation will lead to a lower Φm because the flocs themselves are not close-packed (Starov et al., 2002) and they can trap part of the liquid phase, thus, leading to higher 'effec‐ tive phase volume' and viscosity than those of the primary particles. The results clearly

h f

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*o*

show the remarkable effect of dispersants in getting low viscosity ceramic slurry.

**3. Electrical characterization and analysis of the doping effects**

The piezoelectricity of PZT type materials originates from the displacement of Zr and Ti sublattices and the electrical properties of the materials can be dramastically affected by doping atoms. There are primarily two types of dopants for PZT, i.e. the donor type (soft type), and the acceptor type (hard type) (Jaffe et al., 1971). The former is mainly caused by substitution of higher valence ions for the A site Pb or B site Zr and Ti, and correspondingly higher piezoelectric coefficient (*d*33), planar electromechanical coupling factor (*Kp*), relative permittivity (*εr*), loss tangent (*tgδ*), *Pr* values and a lower mechanical quality factor (*Qm*) val‐ ue are obtained. The latter, which is caused by substitution of lower valence ions for the A

**Sample Average d33 ε<sup>r</sup> tgδ KP**

Dry pressed 469 pC/N 1430 0.0311 0.621

G-TAC 468 pC/N 1369 0.0325 0.633

G-JN281 479 pC/N 1352 0.0401 0.675

G-SP2 403 pC/N 1285 0.0250 0.591

G-SGA 330 pC/N 1244 0.0202 0.541

**Table 2.** Comparison of densities and some electrical parameters of dry pressed and gelcast PZT samples.

**3.1. Electrical characterization for identifying the doping type**

or B site atoms, has contrary effects.

h h ff

**Figure 3.** Influence of dispersants and pH value on the viscosity of the PZT suspensions

**Figure 4.** Influence of solids loading on the viscosities of the PZT suspensions

Gelcasting of Ferroelectric Ceramics: Processing, Doping Effect of Additives and Further Development http://dx.doi.org/10.5772/53995 215

$$
\mu \eta = \eta\_o (1 - \phi \,/\, \phi)^{-\|\eta\|\_{\eta}} \tag{1}
$$

where η and ηo are the viscosity of the suspension and the solvent, respectively. The true volume fraction of the powder in the suspension is represented by f. The intrinsic viscosity [η] is a function of particle geometry; a value of 2.5 is suitable for spherical particles. A maxi‐ mum packing fraction of 0.63±0.002 is suitable for random close packing at low shear rate. The K–D model curve with [η] = 2.5 and Φm=0.63 is plotted in the Figure. Although the ex‐ perimental curves are somewhat deviated from the K–D model, they still show a similar shape: low viscosity at low solids loading and sharp increase at high solids loading. The dis‐ crepancy may partly be ascribed to the inhomogeneity of the particle size. In addition, parti‐ cle flocculation will lead to a lower Φm because the flocs themselves are not close-packed (Starov et al., 2002) and they can trap part of the liquid phase, thus, leading to higher 'effec‐ tive phase volume' and viscosity than those of the primary particles. The results clearly show the remarkable effect of dispersants in getting low viscosity ceramic slurry.

### **3. Electrical characterization and analysis of the doping effects**

### **3.1. Electrical characterization for identifying the doping type**

The relationship between the viscosity and the solids volume fraction for monodispersed

suspension can be explained by the Krieger–Dougherty (K–D) model:

214 Advances in Ferroelectrics

**Figure 3.** Influence of dispersants and pH value on the viscosity of the PZT suspensions

**Figure 4.** Influence of solids loading on the viscosities of the PZT suspensions

The piezoelectricity of PZT type materials originates from the displacement of Zr and Ti sublattices and the electrical properties of the materials can be dramastically affected by doping atoms. There are primarily two types of dopants for PZT, i.e. the donor type (soft type), and the acceptor type (hard type) (Jaffe et al., 1971). The former is mainly caused by substitution of higher valence ions for the A site Pb or B site Zr and Ti, and correspondingly higher piezoelectric coefficient (*d*33), planar electromechanical coupling factor (*Kp*), relative permittivity (*εr*), loss tangent (*tgδ*), *Pr* values and a lower mechanical quality factor (*Qm*) val‐ ue are obtained. The latter, which is caused by substitution of lower valence ions for the A or B site atoms, has contrary effects.


**Table 2.** Comparison of densities and some electrical parameters of dry pressed and gelcast PZT samples.

Some electrical parameters of different soft-doped PZT samples are compared in Table 2, where G-TAC, G-JN281, G-SP2 and G-SGA represent the best available gelcast PZT sam‐ ples with TAC, JN281, SP2 and SGA as the dispersants, respectively. The *ε* and *tgδ* are 1 kHz data under room temperature. The dry pressed sample is obtained under a pressure of ~ 80MPa. G-TAC shows similar electrical properties with those of the dry pressed one. This indicates that the organic species used in gelcasting, including the dispersant TAC, the monomer and the cross-linker, etc., have almost no doping effects. Compared to the dry pressed sample, G-SP2 and G-SGA show higher *Qm* and decreased *d33*, *Kp*, *ε*, *tgδ* and *Pr* values, while change of these parameters for G-JN281 is to the contrary. Thus, the data in reveal that SP2 and SGA induced evident 'hard' doped characteristics, while JN281 in‐ duced 'soft' doped characteristics. Except for the dispersant species, the samples were pre‐ pared under the same conditions. Hence, the different properties should be mainly attributed to the dispersants used. A 'fingerpint' of hard doping effect is the increased *Qm* (Damjanovic, 1998). Since Na+ is the main metal cation in SP2 and SGA, we neglect other difference and check the correlation between Na concentration (inspected by X-ray Fluo‐ rescence) and *Qm*. As shown in Figure 5, *Qm* roughly shows an increasing trend with Na mole fraction except for G-JN281. Then we intentionally introduced Na into the G-JN281 by adding NaOH to the corresponding slurry to increase its pH value to ~13. As expect‐ ed, a higher *Qm* is indeed obtained after adding Na to G-JN281. These reveal that the Na+ ion introduced by the dispersants has a stronge doping effect for gelcast PZT. From the valence and diameter (1.02Å) of Na+ , we deduce that it should substitute for the A site Pb2+ (1.18Å) as an acceptor dopant.

**3.2. Complex impedance spectra**

range of 300~540 o

equation (5):

are compared in Table 3.

Figure 6(a), 6(b), 6(c) and 6(d) show the complex impedance spectra in the temperature

ly. The spectra of G-TAC is very similar with those of the dry pressed sample and thus not shown. The complex impedance spectra of all samples show only one Cole-Cole semicircle crossing the origin, which can be assigned with an equivalent circuit composed of a simple

> <sup>1</sup> , 1( ) *<sup>n</sup> <sup>R</sup> <sup>Z</sup> i RC* w

lationship about the real (Z′) and imaginary (Z″) parts of the impedance can be derived:

ated with the intrinsic ferroelectric bulk (or grain) response.

s s= - =-

shows an Arrhenius type behavior that can be described by equation (4):

s s

where the parameter *n* characterizes the distribution width of the relaxation times around a mean value τ0=RC (West et al., 1997). The semicircles of the samples show no obvious depression, i.e. the depression angle β (=nπ/2) between the real axis and the line from the high frequency in‐ tercept to the centre of the circle is close to 0. Thus, n is close to 0, indicating a debye-like behavior with a single relaxation time (West, 1997; Cao, 1990). Then based on equation (2) the following re‐

The resistance R derived from the diameter of the semicircle, and the capacitance C can be derived from the angular frequency ω at the peak of the circular arc based on the relation‐ ship ωRC=1. The C values of the samples are in the order of 10-10-10-9 F, which can be associ‐

Using the R values, the logarithm of conductivity of different samples as a function of recip‐ rocal temperature is plotted in Figure 6(e). The conductivity σ of the ferroelectric bulk phase

> exp( ), *<sup>s</sup> o*

*E kT*

where *σ0*, *Eg*, *k* and *T* are the pre-exponential factor, the activation energy, the Boltzmann's constant and the absolute temperature, respectively. The logarithm of equation (4) gives

where *ρT* is the resistivity. Based on the sample geometry, the *Eg* values associated with the intrinsic bulk phase conduction of the samples were derived by using equation (5). The data

<sup>1</sup> ln ln ( ) 2.30259log *<sup>s</sup> T o T E k T*

,

r

parallel RC element with a impedance that can be expressed as equation (2):

C of the dry pressed sample, G-JN281, G-JN281+Na and G-SP2, respective‐

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( ) ( ) <sup>2</sup> -R 2 R 2 2 2 *Z Z* ¢ ¢¢ + = . (3)


(5)

**Figure 5.** Illustration of the relationship between Na concentration (mole percentage in total metal elements) and Qm of various PZT samples.

### **3.2. Complex impedance spectra**

Some electrical parameters of different soft-doped PZT samples are compared in Table 2, where G-TAC, G-JN281, G-SP2 and G-SGA represent the best available gelcast PZT sam‐ ples with TAC, JN281, SP2 and SGA as the dispersants, respectively. The *ε* and *tgδ* are 1 kHz data under room temperature. The dry pressed sample is obtained under a pressure of ~ 80MPa. G-TAC shows similar electrical properties with those of the dry pressed one. This indicates that the organic species used in gelcasting, including the dispersant TAC, the monomer and the cross-linker, etc., have almost no doping effects. Compared to the dry pressed sample, G-SP2 and G-SGA show higher *Qm* and decreased *d33*, *Kp*, *ε*, *tgδ* and *Pr* values, while change of these parameters for G-JN281 is to the contrary. Thus, the data in reveal that SP2 and SGA induced evident 'hard' doped characteristics, while JN281 in‐ duced 'soft' doped characteristics. Except for the dispersant species, the samples were pre‐ pared under the same conditions. Hence, the different properties should be mainly attributed to the dispersants used. A 'fingerpint' of hard doping effect is the increased *Qm* (Damjanovic, 1998). Since Na+ is the main metal cation in SP2 and SGA, we neglect other difference and check the correlation between Na concentration (inspected by X-ray Fluo‐ rescence) and *Qm*. As shown in Figure 5, *Qm* roughly shows an increasing trend with Na mole fraction except for G-JN281. Then we intentionally introduced Na into the G-JN281 by adding NaOH to the corresponding slurry to increase its pH value to ~13. As expect‐ ed, a higher *Qm* is indeed obtained after adding Na to G-JN281. These reveal that the Na+ ion introduced by the dispersants has a stronge doping effect for gelcast PZT. From the

**Figure 5.** Illustration of the relationship between Na concentration (mole percentage in total metal elements) and Qm

, we deduce that it should substitute for the A site

valence and diameter (1.02Å) of Na+

Pb2+ (1.18Å) as an acceptor dopant.

216 Advances in Ferroelectrics

of various PZT samples.

Figure 6(a), 6(b), 6(c) and 6(d) show the complex impedance spectra in the temperature range of 300~540 o C of the dry pressed sample, G-JN281, G-JN281+Na and G-SP2, respective‐ ly. The spectra of G-TAC is very similar with those of the dry pressed sample and thus not shown. The complex impedance spectra of all samples show only one Cole-Cole semicircle crossing the origin, which can be assigned with an equivalent circuit composed of a simple parallel RC element with a impedance that can be expressed as equation (2):

$$Z = \frac{\mathcal{R}}{1 + (i\alpha \mathcal{R} \mathcal{C})^{1-n}},\tag{2}$$

where the parameter *n* characterizes the distribution width of the relaxation times around a mean value τ0=RC (West et al., 1997). The semicircles of the samples show no obvious depression, i.e. the depression angle β (=nπ/2) between the real axis and the line from the high frequency in‐ tercept to the centre of the circle is close to 0. Thus, n is close to 0, indicating a debye-like behavior with a single relaxation time (West, 1997; Cao, 1990). Then based on equation (2) the following re‐ lationship about the real (Z′) and imaginary (Z″) parts of the impedance can be derived:

$$\left(\left(Z\text{--}\mathbb{R}f2\right)^{2} + Z^{\*2} = \left(\mathbb{R}f2\right)^{2}.\tag{3}$$

The resistance R derived from the diameter of the semicircle, and the capacitance C can be derived from the angular frequency ω at the peak of the circular arc based on the relation‐ ship ωRC=1. The C values of the samples are in the order of 10-10-10-9 F, which can be associ‐ ated with the intrinsic ferroelectric bulk (or grain) response.

Using the R values, the logarithm of conductivity of different samples as a function of recip‐ rocal temperature is plotted in Figure 6(e). The conductivity σ of the ferroelectric bulk phase shows an Arrhenius type behavior that can be described by equation (4):

$$
\sigma = \sigma\_o \exp(\frac{-E\_s}{kT}),
\tag{4}
$$

where *σ0*, *Eg*, *k* and *T* are the pre-exponential factor, the activation energy, the Boltzmann's constant and the absolute temperature, respectively. The logarithm of equation (4) gives equation (5):

$$
\ln \sigma\_T = \ln \sigma\_o - \frac{E\_s}{k} (\frac{1}{T}) = -2.30259 \log \rho\_{T\_s} \tag{5}
$$

where *ρT* is the resistivity. Based on the sample geometry, the *Eg* values associated with the intrinsic bulk phase conduction of the samples were derived by using equation (5). The data are compared in Table 3.

**Figure 7.** (a) Complex impedance spectra of G-SGA. The inset shows the magnified 375oC and 418oC curves. (b) Arrhe‐ nius plot of the ferroelectric bulk phase resistivity of the sample derived from the grain response of its impedance

Gelcasting of Ferroelectric Ceramics: Processing, Doping Effect of Additives and Further Development

The spectra of G-SGA shown in Figure 7(a), particularly the high temperature curves shown in the inset, consist of two semicircles, which can be assigned with an equivalent circuit composed of two parallel RC elements. This indicates the much different 'electrical micro‐ structure'14 of G-SGA, which has the highest Na concentration. The capacitance obtained by fitting the first semicircle can also be attributed to the ferroelectric bulk response, while the second semicircle may be attributed to the grain boundary response due to the higher capac‐ itance. Actually, one semicircle spectrum may also appear if different responses in a sample overlap as a result of their similar relaxation time. A closer check of Figure 6(d) indicates that the one semicircle impedance spectra of G-SP2 have a slightly higher depression angle than other spectra in Figure 6, consistent with its higher Na concentration (see Figure 5). These imply that the grain boundary response is gradually magnified with increasing Na concentration. The Arrhenius plot of the conductivity of the bulk phase response of G-SGA

The loss of PbO through its volatility causes oxygen vacancies in PZT, leading to a p-type

charge compensation. Oxygen vacancies are the only lattice defects in the perovskite oxides that have a significant mobility, and the conductivity should be improved by the oxygen va‐ cancy conduction mechanism via hopping of atoms in the oxygen octahedral network (Ray‐ mond & Smyth, 1996). This well explains the lower *Eg* data of G-SP2 and G-SGA than that of the dry pressed sample (Table 3), revealing again the hard doping effect of the Na-contain‐ ing dispersants. The relatively higher activation energy of G-JN281 implies the presence of a certain donor-type impurity in G-JN281. Addition of NaOH into G-JN281 leads to a lower *Eg*

ions replace Pb2+ ions and more oxygen vacancies are simultaneously created for

incorporated by the dispersants can

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219

Na O=2Na '+O +V 2 pb o o (6)

is shown in Figure 7 (b). The derived *Eg* value is also listed in Table 3.

conductivity (Barranco et al., 1999). The acceptor Na+

substitute for Pb2+, which can be expressed as following:

value and further confirms the doping effect of Na.

spectra.

Thus, Na+

**Figure 6.** Complex impedance spectra of different PZT samples. (a) Dry pressed. (b). G-JN281. (c) G-JN281+Na. (d) G-SP2. (e) Arrhenius plot for the ferroelectric bulk resistivity of the samples derived from the impedance spectra.


**Table 3.** Comparison of the conductivity activation energies of different PZT samples.

**Figure 7.** (a) Complex impedance spectra of G-SGA. The inset shows the magnified 375oC and 418oC curves. (b) Arrhe‐ nius plot of the ferroelectric bulk phase resistivity of the sample derived from the grain response of its impedance spectra.

The spectra of G-SGA shown in Figure 7(a), particularly the high temperature curves shown in the inset, consist of two semicircles, which can be assigned with an equivalent circuit composed of two parallel RC elements. This indicates the much different 'electrical micro‐ structure'14 of G-SGA, which has the highest Na concentration. The capacitance obtained by fitting the first semicircle can also be attributed to the ferroelectric bulk response, while the second semicircle may be attributed to the grain boundary response due to the higher capac‐ itance. Actually, one semicircle spectrum may also appear if different responses in a sample overlap as a result of their similar relaxation time. A closer check of Figure 6(d) indicates that the one semicircle impedance spectra of G-SP2 have a slightly higher depression angle than other spectra in Figure 6, consistent with its higher Na concentration (see Figure 5). These imply that the grain boundary response is gradually magnified with increasing Na concentration. The Arrhenius plot of the conductivity of the bulk phase response of G-SGA is shown in Figure 7 (b). The derived *Eg* value is also listed in Table 3.

The loss of PbO through its volatility causes oxygen vacancies in PZT, leading to a p-type conductivity (Barranco et al., 1999). The acceptor Na+ incorporated by the dispersants can substitute for Pb2+, which can be expressed as following:

$$\text{Na}\_2\text{O} \text{=} 2\text{Na}\_{\text{pb}}\text{ }' + \text{O}\_o + \text{V}\_o \tag{6}$$

Thus, Na+ ions replace Pb2+ ions and more oxygen vacancies are simultaneously created for charge compensation. Oxygen vacancies are the only lattice defects in the perovskite oxides that have a significant mobility, and the conductivity should be improved by the oxygen va‐ cancy conduction mechanism via hopping of atoms in the oxygen octahedral network (Ray‐ mond & Smyth, 1996). This well explains the lower *Eg* data of G-SP2 and G-SGA than that of the dry pressed sample (Table 3), revealing again the hard doping effect of the Na-contain‐ ing dispersants. The relatively higher activation energy of G-JN281 implies the presence of a certain donor-type impurity in G-JN281. Addition of NaOH into G-JN281 leads to a lower *Eg* value and further confirms the doping effect of Na.

**Figure 6.** Complex impedance spectra of different PZT samples. (a) Dry pressed. (b). G-JN281. (c) G-JN281+Na. (d) G-SP2. (e) Arrhenius plot for the ferroelectric bulk resistivity of the samples derived from the impedance spectra.

**Sample Dry pressed G-JN281 G-JN281+Na G-SP2 G-SGA** *Eg* 1.24 eV 1.48 eV 1.14 eV 1.08 eV 1.07 eV

**Table 3.** Comparison of the conductivity activation energies of different PZT samples.

218 Advances in Ferroelectrics

### **3.3. Ferroelectric hysteresis loops**

The ferroelectric (polarization-electric field) hysteresis loops of the PZT samples are shown in Figure 8. In Figure 8(a), G-SP2 and G-SGA show remnant polarization (*Pr*) of 32.7 μC/cm2 and 16.7 μC/cm2 , respectively. Both values are smaller than the value of 39.8 μC/cm2 of the dry pressed sample. The difference can also be interpreted by increased oxygen vacancies due to doping of Na. As afore mentioned, oxygen vacancies can move in the oxygen octahe‐ dral network. This may lead to a low stability of the 2Napb'-Vo defect dipoles. The defect di‐ poles tend to orient themselves along the polarization direction, resulting in stabilized ferroelectric domains. A stabilized domain wall structure in turn give rise to more difficult poling and depoling (switching) and a smaller *Pr* (Warren, et al., 1996; Lambeck & Jonker, 1978). The round open loop of G-SP2 implies a higher leakage current. The hysteresis loops of G-JN281 and G-JN281+Na in Figure 8 (b) also clearly demonstrates the effect of Na: de‐ creased *Pr* and a loop with a rounder shape.

**Figure 9.** XRD patterns of different PZT samples.

As shown in Figure 10, although sintered under the same procedure, the samples exhibit rather different morphologies. SEM image of G-JN281+Na is not illustrated since it is very similar to that of G-JN281. Dry pressed sample shows an intergranular fracture surface and relative uniform grains with a diameter of 3~5 μm. G-JN281 shows a morphology very simi‐ lar to that of the dry pressed one. G-SP2 also shows a basically intergranular fracture sur‐ face, but the grains are much larger with a diameter of 6~10 μm. Much different from other samples, G-SGA shows a transgranular fracture surface with the largest grains with a diam‐ eter of about 12 μm. Such a transgranular growth may result from chemical inhomogeneity and presence of intergranular phases or expanded grain boundary region. This is consistent

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In summary, electrical and structural characterization indicate that the Na+ ion, which is the main cation in many widely used commercial dispersants (Xu et al., 1996; Tomasik et al., 2003), shows detrimental hard doping effects, leading to deteriorated electrical performance. Also, the impurity species introduced by dispersants or other additives may have a compli‐ cated influence on the electrical properties and microstructure of gelcast PZT samples. The results indicate that the specific doping effect, e.g. the change in electrical performance by the additives during processing, is a critical issue that should be paid special attention when

with the two semicircle impedance spectra of G-SGA shown in Figure 7 (a).

applying gelcasting to the formation of electronic ceramics.

**Figure 8.** Ferroelectric hysteresis loops of the PZT samples.

### **4. Microstructural characterization**

The XRD spectra of All samples (Figure 9) show typical perovskite structures (Soares et al., 2000; Hammer et al., 1998). The XRD patterns of G-TAC, G-SP2 and G-SGA are very similar to that of the dry pressed sample. While the peaks of the XRD pattern of G-JN281 shift very slightly to higher diffraction angles, indicating a contracted lattice cell. Considering the soft doped characteristics of G-JN281, the change in its XRD pattern may again be attributed to a certain unknown donor impurity ion introduced by the dispersant.

**Figure 9.** XRD patterns of different PZT samples.

**3.3. Ferroelectric hysteresis loops**

creased *Pr* and a loop with a rounder shape.

**Figure 8.** Ferroelectric hysteresis loops of the PZT samples.

**4. Microstructural characterization**

and 16.7 μC/cm2

220 Advances in Ferroelectrics

The ferroelectric (polarization-electric field) hysteresis loops of the PZT samples are shown in Figure 8. In Figure 8(a), G-SP2 and G-SGA show remnant polarization (*Pr*) of 32.7 μC/cm2

dry pressed sample. The difference can also be interpreted by increased oxygen vacancies due to doping of Na. As afore mentioned, oxygen vacancies can move in the oxygen octahe‐ dral network. This may lead to a low stability of the 2Napb'-Vo defect dipoles. The defect di‐ poles tend to orient themselves along the polarization direction, resulting in stabilized ferroelectric domains. A stabilized domain wall structure in turn give rise to more difficult poling and depoling (switching) and a smaller *Pr* (Warren, et al., 1996; Lambeck & Jonker, 1978). The round open loop of G-SP2 implies a higher leakage current. The hysteresis loops of G-JN281 and G-JN281+Na in Figure 8 (b) also clearly demonstrates the effect of Na: de‐

The XRD spectra of All samples (Figure 9) show typical perovskite structures (Soares et al., 2000; Hammer et al., 1998). The XRD patterns of G-TAC, G-SP2 and G-SGA are very similar to that of the dry pressed sample. While the peaks of the XRD pattern of G-JN281 shift very slightly to higher diffraction angles, indicating a contracted lattice cell. Considering the soft doped characteristics of G-JN281, the change in its XRD pattern may again be attributed to a

certain unknown donor impurity ion introduced by the dispersant.

, respectively. Both values are smaller than the value of 39.8 μC/cm2

of the

As shown in Figure 10, although sintered under the same procedure, the samples exhibit rather different morphologies. SEM image of G-JN281+Na is not illustrated since it is very similar to that of G-JN281. Dry pressed sample shows an intergranular fracture surface and relative uniform grains with a diameter of 3~5 μm. G-JN281 shows a morphology very simi‐ lar to that of the dry pressed one. G-SP2 also shows a basically intergranular fracture sur‐ face, but the grains are much larger with a diameter of 6~10 μm. Much different from other samples, G-SGA shows a transgranular fracture surface with the largest grains with a diam‐ eter of about 12 μm. Such a transgranular growth may result from chemical inhomogeneity and presence of intergranular phases or expanded grain boundary region. This is consistent with the two semicircle impedance spectra of G-SGA shown in Figure 7 (a).

In summary, electrical and structural characterization indicate that the Na+ ion, which is the main cation in many widely used commercial dispersants (Xu et al., 1996; Tomasik et al., 2003), shows detrimental hard doping effects, leading to deteriorated electrical performance. Also, the impurity species introduced by dispersants or other additives may have a compli‐ cated influence on the electrical properties and microstructure of gelcast PZT samples. The results indicate that the specific doping effect, e.g. the change in electrical performance by the additives during processing, is a critical issue that should be paid special attention when applying gelcasting to the formation of electronic ceramics.

**5.1. Factors affecting fracture and crack growth**

Fracture or crack growth results from the competition of the strength of the gelcast body and the stress developed during gelation and drying. Many factors can affect fracture or crack growth, including the monomer concentration, monomer/cross-linker ratio, initiator concen‐ tration, initiator/catalyst ratio, gelation temperature and drying condition (humidity), etc (Ma et al., 2006). It is easy to be understood that too low a monomer concentration is insufficient to keep the three dimensional polymer network structure. A suitable monomer/cross-linker ratio is necessary to keep a strong gelcast body with still controllable stresses. Compared to other factors, initiator and catalyst seem to be more critical in controlling the gelcast body strength and stresses, since they can determine the 'joints' of the gel network. As shown in Figure 11. A slight decrease of the (NH4)2S2O8 concentration from the normal level causes a seriously broken green body. In addition to use of optimized premix solution, drying is a complicated process

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Fracture or crack growth may also appear after sintering (Zheng et al., 2008), even if the shapes of the ceramics are well preserved after drying. During sintering the macromolecular gel network is destroyed and the stresses developed due to material densification may give

that should be carried out under contolled humidity (Barati et al., 2003).

rise to fracture or cracks in the ceramic body with decreased strength.

**Figure 11.** A fractured PZT-4 type green body caused by a lower initiator concentration

**Figure 10.** SEM images of the fracture surfaces of different PZT samples. (a) Dry pressed sample; (b) G-JN281; (c) G-SP2; (d) G-SGA.

### **5. Development of gelcasting for special-shaped PZT ceramics**

Application of gelcasting to ferroelectric ceramics is not a mechanical copy of the technique to a different powder material. In addition to the demonstrated doping effect of the necessa‐ ry additives, there are many other specific issues deserving further investigation. The many devices require the ferroelectric ceramic components to have various specific shapes (Scott, 2007). The shapes of the gelcast bodies are formed during gelation and drying process with the confinement of the molds. The two most critical issues about shape formation that have attracted more and more attention are probably the fracture (or crack growth) and deforma‐ tion of the ceramics. No doubt that the former should be avoided, however, what is interest‐ ing is that the latter may even be used to form shapes that are difficult to formed by the molds. The two issues are briefly discussed in the following.

### **5.1. Factors affecting fracture and crack growth**

**Figure 10.** SEM images of the fracture surfaces of different PZT samples. (a) Dry pressed sample; (b) G-JN281; (c) G-

Application of gelcasting to ferroelectric ceramics is not a mechanical copy of the technique to a different powder material. In addition to the demonstrated doping effect of the necessa‐ ry additives, there are many other specific issues deserving further investigation. The many devices require the ferroelectric ceramic components to have various specific shapes (Scott, 2007). The shapes of the gelcast bodies are formed during gelation and drying process with the confinement of the molds. The two most critical issues about shape formation that have attracted more and more attention are probably the fracture (or crack growth) and deforma‐ tion of the ceramics. No doubt that the former should be avoided, however, what is interest‐ ing is that the latter may even be used to form shapes that are difficult to formed by the

**5. Development of gelcasting for special-shaped PZT ceramics**

molds. The two issues are briefly discussed in the following.

SP2; (d) G-SGA.

222 Advances in Ferroelectrics

Fracture or crack growth results from the competition of the strength of the gelcast body and the stress developed during gelation and drying. Many factors can affect fracture or crack growth, including the monomer concentration, monomer/cross-linker ratio, initiator concen‐ tration, initiator/catalyst ratio, gelation temperature and drying condition (humidity), etc (Ma et al., 2006). It is easy to be understood that too low a monomer concentration is insufficient to keep the three dimensional polymer network structure. A suitable monomer/cross-linker ratio is necessary to keep a strong gelcast body with still controllable stresses. Compared to other factors, initiator and catalyst seem to be more critical in controlling the gelcast body strength and stresses, since they can determine the 'joints' of the gel network. As shown in Figure 11. A slight decrease of the (NH4)2S2O8 concentration from the normal level causes a seriously broken green body. In addition to use of optimized premix solution, drying is a complicated process that should be carried out under contolled humidity (Barati et al., 2003).

Fracture or crack growth may also appear after sintering (Zheng et al., 2008), even if the shapes of the ceramics are well preserved after drying. During sintering the macromolecular gel network is destroyed and the stresses developed due to material densification may give rise to fracture or cracks in the ceramic body with decreased strength.

**Figure 11.** A fractured PZT-4 type green body caused by a lower initiator concentration

### **5.2. Deformation controllable gelcasting**

Factors that affect crack growth discussed in the forgoing section also affect deformation. In most cases, deformation should be kept as smaller as possible. However, well controlled defor‐ mation can also be used to fabricate certain devices that use specially-shaped PZT as the active components. Here we show two examples: spherical PZT shell vibrator and PZT minitube.

**5.3. Fabrication of hollow spherical PZT shell**

show in Figure 13.

**6. Conclusions**

doping ions like K+

Thin-walled hollow sphere Omnidirectional Transducer has been used in hydrophones for many years (Li et al., 2010), which uses a hollow spherical PZT shell as the active compo‐ nent. Hollow spherical PZT shell is generally fabricated by Cold isostatic pressing, which re‐ quires complicated facilities. We show here that gelcasting can be developed to fabricate such Hollow spherical PZT shells by using a ball-shaped removable polymeric mold. First, the polymeric ball mold is fixed in a normal metal mold with a large spherical cavity, then the aqueous PZT premix slurry is poured into the mold. After gelation and drying, the whole mold containing the PZT green body and polymeric ball is removed by a careful ther‐ mal treatment. The most critical step in this method is thermal treatment, which solidifies the PZT slurry and melt the polymeric ball in due time. A PZT-4 hollow spherical shell with a focal length of ~ 200 mm, a diameter of ~ 120 mm and a wall thickness of 2 mm is also

Gelcasting of Ferroelectric Ceramics: Processing, Doping Effect of Additives and Further Development

http://dx.doi.org/10.5772/53995

225

**Figure 13.** Illustration of special-shaped gelcast PZT ceramics: spherical PZT plate and hollow spherical PZT shell.

bining gelcasting and Selective Laser Sintering (Guo, 2003), etc.

samples with those of the dry pressed PZT one suggests that the Na+

In addition to the above mentioned advanced techniques, other gelcasting based ceramic fabrication or processing techniques have been developed in the authors' group, including spatter laser drilling techniques (Guo, 2004), and Rapid Prototyping of PZT bodies by com‐

Comparison of various electrical properties and microstructures of various gelcast soft PZT

cation in many widely used commercial dispersants, shows detrimental hard doping effects, leading to deteriorated electrical performance. The conclusion may be transferable to other

the doping ions introduced by dispersants, the performance of the glecast PZT sample may

, which is also contained in many commercial dispersants. Also due to

ion, which is the main

As an effective non-invasive surgical tool, High intensity Focused sound (HIFU) has been used for the treatment of human tissues such as liver, kidney, breast, uterus and pancreas, and it is receiving growing interest (Wan, et al., 2008; Davies et al., 1998). As shown in Fig‐ ure 12, some therapeutic HIFU transducers require to use spherical piezoelectric PZT shells as the active components (Saletes et al., 2011). Conventionally, such a spherical plate is fabri‐ cated by grinding both sides of a cylindrical plate. The mechanical method is time-consum‐ ing and wastes a lot of PZT material. Gelcasting of flat PZT plates generally uses a cylindrical ring mold and a flat bottom plate. During drying, the upper surface of the gelcast body is exposed to atmosphere while the other side is still berried until the whole body sol‐ idifies. Then we found an interesting phenomenon: the upper part of the gelcast body might dry and shrink first and a spherical plate was formed. By using suitable premix solution and under well controlled temperature and humidity, the plate edge bends up and a rather ideal spherically shaped PZT green plate can be formed. After sintering a spherical PZT plate was formed, which only needs to be grinded slightly to produce a homogenous spherical vibra‐ tor. Furthermore, because a lower humidity and a higher temperature can give rise to a larg‐ er deformation, the curvature radius can be controlled in a certain range under well controlled experimental conditions. So far the smallest focal length obtained for a plate with a diameter of 10 cm without further machining is around 15 cm. A PZT-8 plate with a diam‐ eter of 12 cm is shown in Figure 13.

**Figure 12.** Schematic illustration of the therapeutic effect of a HIFU transducer

### **5.3. Fabrication of hollow spherical PZT shell**

**5.2. Deformation controllable gelcasting**

224 Advances in Ferroelectrics

eter of 12 cm is shown in Figure 13.

**Figure 12.** Schematic illustration of the therapeutic effect of a HIFU transducer

Factors that affect crack growth discussed in the forgoing section also affect deformation. In most cases, deformation should be kept as smaller as possible. However, well controlled defor‐ mation can also be used to fabricate certain devices that use specially-shaped PZT as the active components. Here we show two examples: spherical PZT shell vibrator and PZT minitube.

As an effective non-invasive surgical tool, High intensity Focused sound (HIFU) has been used for the treatment of human tissues such as liver, kidney, breast, uterus and pancreas, and it is receiving growing interest (Wan, et al., 2008; Davies et al., 1998). As shown in Fig‐ ure 12, some therapeutic HIFU transducers require to use spherical piezoelectric PZT shells as the active components (Saletes et al., 2011). Conventionally, such a spherical plate is fabri‐ cated by grinding both sides of a cylindrical plate. The mechanical method is time-consum‐ ing and wastes a lot of PZT material. Gelcasting of flat PZT plates generally uses a cylindrical ring mold and a flat bottom plate. During drying, the upper surface of the gelcast body is exposed to atmosphere while the other side is still berried until the whole body sol‐ idifies. Then we found an interesting phenomenon: the upper part of the gelcast body might dry and shrink first and a spherical plate was formed. By using suitable premix solution and under well controlled temperature and humidity, the plate edge bends up and a rather ideal spherically shaped PZT green plate can be formed. After sintering a spherical PZT plate was formed, which only needs to be grinded slightly to produce a homogenous spherical vibra‐ tor. Furthermore, because a lower humidity and a higher temperature can give rise to a larg‐ er deformation, the curvature radius can be controlled in a certain range under well controlled experimental conditions. So far the smallest focal length obtained for a plate with a diameter of 10 cm without further machining is around 15 cm. A PZT-8 plate with a diam‐

Thin-walled hollow sphere Omnidirectional Transducer has been used in hydrophones for many years (Li et al., 2010), which uses a hollow spherical PZT shell as the active compo‐ nent. Hollow spherical PZT shell is generally fabricated by Cold isostatic pressing, which re‐ quires complicated facilities. We show here that gelcasting can be developed to fabricate such Hollow spherical PZT shells by using a ball-shaped removable polymeric mold. First, the polymeric ball mold is fixed in a normal metal mold with a large spherical cavity, then the aqueous PZT premix slurry is poured into the mold. After gelation and drying, the whole mold containing the PZT green body and polymeric ball is removed by a careful ther‐ mal treatment. The most critical step in this method is thermal treatment, which solidifies the PZT slurry and melt the polymeric ball in due time. A PZT-4 hollow spherical shell with a focal length of ~ 200 mm, a diameter of ~ 120 mm and a wall thickness of 2 mm is also show in Figure 13.

**Figure 13.** Illustration of special-shaped gelcast PZT ceramics: spherical PZT plate and hollow spherical PZT shell.

In addition to the above mentioned advanced techniques, other gelcasting based ceramic fabrication or processing techniques have been developed in the authors' group, including spatter laser drilling techniques (Guo, 2004), and Rapid Prototyping of PZT bodies by com‐ bining gelcasting and Selective Laser Sintering (Guo, 2003), etc.

### **6. Conclusions**

Comparison of various electrical properties and microstructures of various gelcast soft PZT samples with those of the dry pressed PZT one suggests that the Na+ ion, which is the main cation in many widely used commercial dispersants, shows detrimental hard doping effects, leading to deteriorated electrical performance. The conclusion may be transferable to other doping ions like K+ , which is also contained in many commercial dispersants. Also due to the doping ions introduced by dispersants, the performance of the glecast PZT sample may also be improved as well. Considering that dispersants are indispensible in getting concen‐ trated low viscosity ceramic slurries, the possible doping effect of metal ions or impurities introduced by the dispersants or other additives should be generally considered when ap‐ plying gelcasting to forming multicomponent electronic ceramic materials whose electrical properties are sensitive to the composition.

[8] Guo, D.; Li, LT.; Gui, ZL.; Nan, CW. (2003). B-Solid state materials for advanced tech‐

Gelcasting of Ferroelectric Ceramics: Processing, Doping Effect of Additives and Further Development

[9] Young, A.C.; Omatete, O.O.; Janney, M. A.; Menchhofer, P.A.(1991). Gelcasting of

[10] Zhou, LJ.; Huang, Y.; Xie, ZP.; (2000). Gelcasting of concentrated aqueous silicon car‐

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[13] Bell, NS.; Voigt, JA.; Tuttle, BA. (2004). Colloidal processing of chemically prepared zinc oxide varistors. Part II: Near-net-shape forming and fired electrical properties.

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In addition, we also demonstrate some advanced gelcasting techniques, including deforma‐ tion controllable gelcasting of spherical PZT disc and gelcasting based hollow spherical PZT fabrication technique, etc. Application of gelcasting to PZT is not a mechanical copy of the technique to a different powder material. The results show that successful application of gel‐ casting to ferroelectric ceramics is not a mechanical copy of the technique to a different pow‐ der material.

### **Author details**

Dong Guo1 and Kai Cai2

1 Institute of Acoustics, Chinese Academy of Sciences, Beijing, China

2 Beijing Center for Chemical and Physical Analysis, Beijing Municipal Science and Technol‐ ogy Research Institute, Beijing, China

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[8] Guo, D.; Li, LT.; Gui, ZL.; Nan, CW. (2003). B-Solid state materials for advanced tech‐ nology. *Materials Science and Engineering*, 99, 25-28

also be improved as well. Considering that dispersants are indispensible in getting concen‐ trated low viscosity ceramic slurries, the possible doping effect of metal ions or impurities introduced by the dispersants or other additives should be generally considered when ap‐ plying gelcasting to forming multicomponent electronic ceramic materials whose electrical

In addition, we also demonstrate some advanced gelcasting techniques, including deforma‐ tion controllable gelcasting of spherical PZT disc and gelcasting based hollow spherical PZT fabrication technique, etc. Application of gelcasting to PZT is not a mechanical copy of the technique to a different powder material. The results show that successful application of gel‐ casting to ferroelectric ceramics is not a mechanical copy of the technique to a different pow‐

2 Beijing Center for Chemical and Physical Analysis, Beijing Municipal Science and Technol‐

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**Chapter 11**

**Electronic Ferroelectricity in II-VI Semiconductor ZnO**

II-VI semiconductor Zinc oxide (ZnO) is a well-known electronic material [1-3]. Because of large piezoelectric constant and electromechanical-coupling constant, ZnO has been applied to ultrasonic transducer, SAW filter, gas sensor *etc* [3-5]. In addition, ZnO has been used widely as pigment and UV cut cosmetics extensively, so it is a safe material for our living environment, compared with heavy metals used in semiconducting process and materials science. Recently, ZnO has been studied as a material for the solar cell, transparent conduc‐ tors formed on liquid crystal displays, and the blue laser [6]. Then, *p*-type ZnO was found

Ferroelectricity is recognized to appear mainly by the delicate balance between a long-range dipole-dipole interaction and a short-range interaction. When electrons should be well local‐ ized in dielectrics, the electronic distribution in the unit cell is determined by atomic posi‐ tions of constituent ions. Slight distortion of electronic distribution due to structural changes in dielectrics gives a rise of dipole moments. From this point of view, the ferroelectric phase transition should be understood as a structural phase transition from the paraelectric phase with high symmetry to the ferroelectric phase with low symmetry. The atomic displace‐ ments are generally 0.01~0.1 Å, which are small compared with Bohr radius ( 0.53 Å). There‐ fore ferroelectrics are considered as a group of materials which are sensitive to structural changes. In this sense, the ferroelectric phase transition is classified into one of the structural

where H(phonon) and H(electron) are due to motions of ion cores and valence electrons, and H(electron-phonon) presents interactions between ions and valence electrons. In dielec‐

and reproduction in any medium, provided the original work is properly cited.

© 2013 Onodera and Takesada; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

Akira Onodera and Masaki Takesada

http://dx.doi.org/10.5772/52304

**1. Introduction**

by Joseph *et al* [7].

phase transitions.

The basic Hamiltonian is generally given as

H= H(phonon) + H(electron) + H(electron-phonon),

Additional information is available at the end of the chapter

## **Electronic Ferroelectricity in II-VI Semiconductor ZnO**

Akira Onodera and Masaki Takesada

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52304

### **1. Introduction**

II-VI semiconductor Zinc oxide (ZnO) is a well-known electronic material [1-3]. Because of large piezoelectric constant and electromechanical-coupling constant, ZnO has been applied to ultrasonic transducer, SAW filter, gas sensor *etc* [3-5]. In addition, ZnO has been used widely as pigment and UV cut cosmetics extensively, so it is a safe material for our living environment, compared with heavy metals used in semiconducting process and materials science. Recently, ZnO has been studied as a material for the solar cell, transparent conduc‐ tors formed on liquid crystal displays, and the blue laser [6]. Then, *p*-type ZnO was found by Joseph *et al* [7].

Ferroelectricity is recognized to appear mainly by the delicate balance between a long-range dipole-dipole interaction and a short-range interaction. When electrons should be well local‐ ized in dielectrics, the electronic distribution in the unit cell is determined by atomic posi‐ tions of constituent ions. Slight distortion of electronic distribution due to structural changes in dielectrics gives a rise of dipole moments. From this point of view, the ferroelectric phase transition should be understood as a structural phase transition from the paraelectric phase with high symmetry to the ferroelectric phase with low symmetry. The atomic displace‐ ments are generally 0.01~0.1 Å, which are small compared with Bohr radius ( 0.53 Å). There‐ fore ferroelectrics are considered as a group of materials which are sensitive to structural changes. In this sense, the ferroelectric phase transition is classified into one of the structural phase transitions.

The basic Hamiltonian is generally given as

H= H(phonon) + H(electron) + H(electron-phonon),

where H(phonon) and H(electron) are due to motions of ion cores and valence electrons, and H(electron-phonon) presents interactions between ions and valence electrons. In dielec‐

tric materials, the contribution of electron systems is usually omitted since the band gap is generally wide. However, it should be necessary to consider the electron-phonon interaction in the case of ferroelectric semiconductors, since the correlation energy of dipole-dipole in‐ teraction (about 0.2 eV) is comparable with band-gap energy in some narrow-gap and widegap semiconductors. For this type of compounds, the electronic contribution due to bond charges and conduction electrons should play a key role and must be taken into account for understanding the nature of ferroelectricity. For long time, the importance of electronic con‐ tribution has been pointed out in the field of ferroelectrics, since the simple superposition of electronic polarizability does not hold in many ferroelectric substances. Recently, the novel ferroelectricity was discovered in narrow-gap and wide-gap semiconductors such as Pb1 *<sup>x</sup>*Ge*x*Te [8], Cd1-*x*Zn*x*Te [9] and Zn1-*x*Li*x*O [10].The appearance of ferroelectricity is primarily due to electronic origin in Zn1-*x*Li*x*O. Although the ferroelectric phase transition accompanies with structural distortions in usual ferroelectrics, only small structural changes of the order of 10-3 Å are observed in Zn1-*x*Li*x*O. The change in *d-p* hybridization caused by Li-substitu‐ tion is responsible for the novel ferroelectricity and dielectric properties. In this chapter, we summarized recent works mainly onZnO.

**Figure 1.** Crystal structure of ZnO (Wurtzite structure).

larger than that of LMTO. It is due to the LDA gap error [21].

Electronic structure of ZnO has been studied by LMTO (linear muffin-tin orbital) method [19] and by LAPW (linearized augmented-plane-wave) method using LDA (local density approximation). Band structure and the density of states is shown in Figs. 2 and 3 [20]. Around -17 eV, there are two bands originating from the O 2*s*-states. The narrow bands be‐ tween -6 and -4 eV consist mainly of the Zn 3*d*-orbitals, and moderately dispersive bands from -4 to 0 eV consist mainly of the O 2*p*-orbitals. Figure 4 shows significant *d-p* hybridiza‐ tion. The 3*d* derived bands split into two groups, leading to double-peak structure in DOS (density of states). The lower peak is characterized by a strong *d-p* hybridization. The sharp upper peak between -4.8 and -4.2 eV has strong Zn 3*d* character and the hybridization with the O 2*p*-state is very small. Band gap is 0.77 eV. Discrepancy with the experimental value is

Electronic Ferroelectricity in II-VI Semiconductor ZnO

http://dx.doi.org/10.5772/52304

233

**3. Electronic structure**

**Figure 2.** Band structure of ZnO using LAPW.

### **2. Zinc oxide**

The crystal Structure of ZnO is wurtzite-type (P63mc) as shown in Fig. 1, which belongs to hexagonal system. It does not have the center of symmetry, and is polar along the *c*-axis. Although it has been pointed out that the structure of pure ZnO has the possibility to ex‐ hibit ferroelectricity, the polarization switching does not observed until its melting point (1975o C) because of large activation energy accompanied by dipole switching process. The lattice constants are *a* = 3.249858 Å, *c*= 5.206619 Å at room temperature (298 K) [11]. Zn and O ions are bonded tetrahedrally and form ZnO4 groups. This tetrahedron is not perfect: the apical bond length of Zn-O is 1.992 Å (parallel to the *c*-axis) and the basal one is 1.973 Å. Bond length along the *c*-axis is longer than other three bonds by 0.96%: the ZnO4tetrahedra distort along the *c*-axis, which result in dipole moment. According to recent first-principles studies, it was reported that the hybridization between the Zn 3*d-*electron and the O 2*p*electron plays an important role for dielectric properties of ZnO [12, 13].The energy gap *E*<sup>g</sup> is 3.44 eV [14]. Although stoichiometric ZnO is an insulator (intrinsic semiconductor), it ex‐ hibits *n*-type conductivity because of excess Zn atoms. The resistivity is about 300 Ωcm, which, however, changes drastically by doping various impurities. By doping of trivalent ions, such as Al3+, In3+, Ga3+, it reduces to the order of 10-4Ωcm and shows the conductivity near that of metals [15].

Although *p*-type conductivity is expected by doping of monovalent Li+ ions, ZnO becomes an insulator and the resistivity increases to as much as 1010Ωcm [16-18]. It is remarkable that the change in resistivity reaches over the order of 1014, which covers from metal-like to insu‐ lating region. It suggests that the physical nature in this material changes drastically by a little amount of dopant.

**Figure 1.** Crystal structure of ZnO (Wurtzite structure).

### **3. Electronic structure**

tric materials, the contribution of electron systems is usually omitted since the band gap is generally wide. However, it should be necessary to consider the electron-phonon interaction in the case of ferroelectric semiconductors, since the correlation energy of dipole-dipole in‐ teraction (about 0.2 eV) is comparable with band-gap energy in some narrow-gap and widegap semiconductors. For this type of compounds, the electronic contribution due to bond charges and conduction electrons should play a key role and must be taken into account for understanding the nature of ferroelectricity. For long time, the importance of electronic con‐ tribution has been pointed out in the field of ferroelectrics, since the simple superposition of electronic polarizability does not hold in many ferroelectric substances. Recently, the novel ferroelectricity was discovered in narrow-gap and wide-gap semiconductors such as Pb1 *<sup>x</sup>*Ge*x*Te [8], Cd1-*x*Zn*x*Te [9] and Zn1-*x*Li*x*O [10].The appearance of ferroelectricity is primarily due to electronic origin in Zn1-*x*Li*x*O. Although the ferroelectric phase transition accompanies with structural distortions in usual ferroelectrics, only small structural changes of the order of 10-3 Å are observed in Zn1-*x*Li*x*O. The change in *d-p* hybridization caused by Li-substitu‐ tion is responsible for the novel ferroelectricity and dielectric properties. In this chapter, we

The crystal Structure of ZnO is wurtzite-type (P63mc) as shown in Fig. 1, which belongs to hexagonal system. It does not have the center of symmetry, and is polar along the *c*-axis. Although it has been pointed out that the structure of pure ZnO has the possibility to ex‐ hibit ferroelectricity, the polarization switching does not observed until its melting point

Although *p*-type conductivity is expected by doping of monovalent Li+ ions, ZnO becomes an insulator and the resistivity increases to as much as 1010Ωcm [16-18]. It is remarkable that the change in resistivity reaches over the order of 1014, which covers from metal-like to insu‐ lating region. It suggests that the physical nature in this material changes drastically by a

C) because of large activation energy accompanied by dipole switching process. The lattice constants are *a* = 3.249858 Å, *c*= 5.206619 Å at room temperature (298 K) [11]. Zn and O ions are bonded tetrahedrally and form ZnO4 groups. This tetrahedron is not perfect: the apical bond length of Zn-O is 1.992 Å (parallel to the *c*-axis) and the basal one is 1.973 Å. Bond length along the *c*-axis is longer than other three bonds by 0.96%: the ZnO4tetrahedra distort along the *c*-axis, which result in dipole moment. According to recent first-principles studies, it was reported that the hybridization between the Zn 3*d-*electron and the O 2*p*electron plays an important role for dielectric properties of ZnO [12, 13].The energy gap *E*<sup>g</sup> is 3.44 eV [14]. Although stoichiometric ZnO is an insulator (intrinsic semiconductor), it ex‐ hibits *n*-type conductivity because of excess Zn atoms. The resistivity is about 300 Ωcm, which, however, changes drastically by doping various impurities. By doping of trivalent ions, such as Al3+, In3+, Ga3+, it reduces to the order of 10-4Ωcm and shows the conductivity

summarized recent works mainly onZnO.

**2. Zinc oxide**

232 Advances in Ferroelectrics

near that of metals [15].

little amount of dopant.

(1975o

Electronic structure of ZnO has been studied by LMTO (linear muffin-tin orbital) method [19] and by LAPW (linearized augmented-plane-wave) method using LDA (local density approximation). Band structure and the density of states is shown in Figs. 2 and 3 [20]. Around -17 eV, there are two bands originating from the O 2*s*-states. The narrow bands be‐ tween -6 and -4 eV consist mainly of the Zn 3*d*-orbitals, and moderately dispersive bands from -4 to 0 eV consist mainly of the O 2*p*-orbitals. Figure 4 shows significant *d-p* hybridiza‐ tion. The 3*d* derived bands split into two groups, leading to double-peak structure in DOS (density of states). The lower peak is characterized by a strong *d-p* hybridization. The sharp upper peak between -4.8 and -4.2 eV has strong Zn 3*d* character and the hybridization with the O 2*p*-state is very small. Band gap is 0.77 eV. Discrepancy with the experimental value is larger than that of LMTO. It is due to the LDA gap error [21].

**Figure 2.** Band structure of ZnO using LAPW.

of Pb ion by 0.8 Å and behave as *off-center ions* because of the ionic size-mismatch between Pb ion (the ionic radius: 1.2 Å) and Ge ion (the ionic radius: 0.73 Å) [24]. The direction of displacement is the eight equivalent [111] directions, which is spatially vacant and polar in the rock-salt structure. Above *T*c, the ions shift toward any one of these directions at ran‐ dom, but its displacement would be ordered along the trigonal axis below *T*c. The ordering of the *off-center ion* triggers the softening of TO phonon mode and induces rhombohedral distortion in the way such as ‧‧‧Pb2+-Te2-‧‧‧Pb2+-Te2-‧‧‧Pb2+-Te2-‧‧‧or ‧‧‧Te2--Pb2+‧‧‧Te2-- Pb2+‧‧‧ Te2--Pb2+‧‧‧. This structural ordering generates spontaneous polarization. However, in Pb1 *<sup>x</sup>*Ge*x*Te, the *D-E* hysteresis loop has not been observed, while it is a direct evidence of the ferroelectricity. Whereas traditional dielectric measurements are performed on a parallelplate capacitor, it is difficult to measure dielectric constant of such lossy Pb1-*x*Ge*x*Te because of current leaks. Therefore dielectric constant is determined by the *C-V* measurement or the optical reflectiveity, which are commonly used in semiconductors. The large dielectric

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**Figure 4.** Crystal structures of Pb1-*x*Ge*x*Te, Cd1-*x*Zn*x*Te and Zn1-*x*Li*x*O. The lower figures are plots along the polar [111]

directions for Pb1-*x*Ge*x*Te and Cd1-*x*Zn*x*T, and polar [0 0 1] direction for Zn1-*x*Li*x*O.

anomaly and soft mode of Pb1-*x*Ge*x*Te suggest the ferroelectric activity.

**Figure 3.** Density of states of ZnO using LAPW.

### **4. Ferroelectricity in binary semiconductors**

Recently, the novel ferroelectricity was found in Li-doped ZnO [10, 22, 23], although pure ZnO has not shown any evidence of ferroelectricity. Generally representative ferroelectrics have complicate crystal structures such as Rochelle salt (KNaC4H4O6 4H2O), TGS (tri-glycine sulfate, (NH3CH2COOH)3H2SO4) and BaTiO3. Since ZnO is a simple binary compound, it is very convenient to study the microscopic mechanism and its electronic contribution for the appearance of ferroelectricity. Besides ZnO, IV-VI narrow-gap semiconductor Pb1-*x*Ge*x*Te and II-VI wide-gap semiconductor Cd1-*x*Zn*x*Te, are known as materials of binary crystals ac‐ companying ferroelectricity.

### **4.1. IV-VI Narrow-gap semiconductor**

Among IV-VI semiconductor, Pb1-*x*Ge*x*Te has been investigated extensively about its ferroe‐ lectricity [8]. Pb1-*x*Ge*x*Te has a NaCl (rock-salt) type structure at room temperature. Although any ferroelectric activities have not been observed in pure PbTe and GeTe crystals, the ferro‐ electric phase transition is induced only in solid solution Pb1-*x*Ge*x*Te. In the low temperature phase of Pb1-*x*Ge*x*Te, the crystal becomes rhombohedral and exhibits ferroelectric activity. In the case of *x* = 0.003, it shows a large dielectric anomaly of the order of 103 at T = 100 K as shown in Fig. 5 and the softening of TO mode was observed.

The energy gap of Pb1-*x*Ge*x*Te is no more than 0.3 eV, which is 3000 K in temperature. This value is comparable to the energy of the Lorentz field of dielectrics, (4π/3)*P*. Therefore, con‐ duction electrons can couple strongly with phonons in this solid solution. The electron-pho‐ non interaction decreases the frequency of TO phonon mode which results in the ferroelectric phase transition. In Pb1-*x*Ge*x*Te, doped Ge ions displace from the center position of Pb ion by 0.8 Å and behave as *off-center ions* because of the ionic size-mismatch between Pb ion (the ionic radius: 1.2 Å) and Ge ion (the ionic radius: 0.73 Å) [24]. The direction of displacement is the eight equivalent [111] directions, which is spatially vacant and polar in the rock-salt structure. Above *T*c, the ions shift toward any one of these directions at ran‐ dom, but its displacement would be ordered along the trigonal axis below *T*c. The ordering of the *off-center ion* triggers the softening of TO phonon mode and induces rhombohedral distortion in the way such as ‧‧‧Pb2+-Te2-‧‧‧Pb2+-Te2-‧‧‧Pb2+-Te2-‧‧‧or ‧‧‧Te2--Pb2+‧‧‧Te2-- Pb2+‧‧‧ Te2--Pb2+‧‧‧. This structural ordering generates spontaneous polarization. However, in Pb1 *<sup>x</sup>*Ge*x*Te, the *D-E* hysteresis loop has not been observed, while it is a direct evidence of the ferroelectricity. Whereas traditional dielectric measurements are performed on a parallelplate capacitor, it is difficult to measure dielectric constant of such lossy Pb1-*x*Ge*x*Te because of current leaks. Therefore dielectric constant is determined by the *C-V* measurement or the optical reflectiveity, which are commonly used in semiconductors. The large dielectric anomaly and soft mode of Pb1-*x*Ge*x*Te suggest the ferroelectric activity.

**Figure 3.** Density of states of ZnO using LAPW.

234 Advances in Ferroelectrics

companying ferroelectricity.

**4.1. IV-VI Narrow-gap semiconductor**

**4. Ferroelectricity in binary semiconductors**

shown in Fig. 5 and the softening of TO mode was observed.

Recently, the novel ferroelectricity was found in Li-doped ZnO [10, 22, 23], although pure ZnO has not shown any evidence of ferroelectricity. Generally representative ferroelectrics have complicate crystal structures such as Rochelle salt (KNaC4H4O6 4H2O), TGS (tri-glycine sulfate, (NH3CH2COOH)3H2SO4) and BaTiO3. Since ZnO is a simple binary compound, it is very convenient to study the microscopic mechanism and its electronic contribution for the appearance of ferroelectricity. Besides ZnO, IV-VI narrow-gap semiconductor Pb1-*x*Ge*x*Te and II-VI wide-gap semiconductor Cd1-*x*Zn*x*Te, are known as materials of binary crystals ac‐

Among IV-VI semiconductor, Pb1-*x*Ge*x*Te has been investigated extensively about its ferroe‐ lectricity [8]. Pb1-*x*Ge*x*Te has a NaCl (rock-salt) type structure at room temperature. Although any ferroelectric activities have not been observed in pure PbTe and GeTe crystals, the ferro‐ electric phase transition is induced only in solid solution Pb1-*x*Ge*x*Te. In the low temperature phase of Pb1-*x*Ge*x*Te, the crystal becomes rhombohedral and exhibits ferroelectric activity. In the case of *x* = 0.003, it shows a large dielectric anomaly of the order of 103 at T = 100 K as

The energy gap of Pb1-*x*Ge*x*Te is no more than 0.3 eV, which is 3000 K in temperature. This value is comparable to the energy of the Lorentz field of dielectrics, (4π/3)*P*. Therefore, con‐ duction electrons can couple strongly with phonons in this solid solution. The electron-pho‐ non interaction decreases the frequency of TO phonon mode which results in the ferroelectric phase transition. In Pb1-*x*Ge*x*Te, doped Ge ions displace from the center position

**Figure 4.** Crystal structures of Pb1-*x*Ge*x*Te, Cd1-*x*Zn*x*Te and Zn1-*x*Li*x*O. The lower figures are plots along the polar [111] directions for Pb1-*x*Ge*x*Te and Cd1-*x*Zn*x*T, and polar [0 0 1] direction for Zn1-*x*Li*x*O.

to be driven in different way from that of Pb1-*x*Ge*x*Te, because of the different dielectric prop‐ erties described above. Furthermore, the band gap is much larger than that of the narrowgap ferroelectric semiconductors, which suggests that the electron-phonon coupling is not

Dielectric constants of Zn1-*x*Li*x*O ceramics with *x*=0.09 show an anomaly at 470 K (*T*c) (Fig. 7), although a high purity of ZnO does not show any anomaly from 20 K to 700 K [27]. The peak value of dielectric anomaly is 21 ( *x*=0.1), which is the same order with Cd1-*x*Zn*x*Te (ε ~ 50), but much smaller than ordinal ferroelectrics by 2~4 orders. This means that the dipole-dipole cor‐ relation accompanied with this ferroelectric phase transition is not so large in Zn1-*x*Li*x*O. In the measurement of *D-E* hysteresis loop, a small and clear hysteresis curve was observed. The

that the preferred orientation of the *c*-axis, the direction of *P*s of the samples varies depending on samples. Naturally, powder of ZnO has egg-shaped grains which are elongated along the *c*-axis due to the anisotropy of elastic constants. Therefore, the orientation of the ceramic sam‐ ple depends on the condition of the pressing process in the sample preparation. After the cor‐ rection of the preferred orientation by using X-ray diffraction, the value of spontaneous

tion temperature, *T*c, depends on the Li concentration. The phase diagram between *T*c and *x* is shown in Fig. 9. As temperature increases, *T*c becomes higher,which reminds us the phase di‐

to 0.59 μC/cm<sup>2</sup>

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as shown in Fig. 8 [28, 29]. The transi‐

. This is due to

so strong in Cd1-*x*Zn*x*Te solid solutions.

**Figure 6.** Temperature dependence of dielectric constant in Cd1-*x*Zn*x*Te (*x*=0.1).

spontaneous polarization varies by samples from 0.05 μC/cm<sup>2</sup>

agram of quantum ferroelectrics such as KTa1-*x*Nb*x*O3 and Sr1-*x*Ca*x*TiO3.

polarizations of each sample converges to 0.9 μC/cm<sup>2</sup>

**4.3. Zn1-***x***Li***x***O**

**Figure 5.** Temperature dependence of dielectric constant in Pb1-*x*Ge*x*Te (*x*=0.003).

### **4.2. II-VI Wide-gap semiconductor Cd1-***x***Zn***x***Te**

Among many wide-gap semiconductors, the ferroelectricity of Cd1-*x*Zn*x*Te was discovered by R. Weil *et al* in 1988 [9, 25]. The *E*g of Cd1-*x*Zn*x*Te is 1.53 eV. It is a wide-gap semiconductor unlike the narrow-gap IV-VI Pb1-*x*Ge*x*Te. The crystal structure is cubic zinc-blende-type. Cd(Zn) ion is surrounded by four Te ions tetrahedrally. The center of symmetry in this com‐ pound does not exist. When *x*=0.1, the crystal exhibits a dielectric anomaly at 393 K as shoen in Fig. 6. The peak value of dielectric constant (εpeak) is only 50, which is smaller by about two orders than that of BaTiO3 (~14000) and that of Pb1-*x*Ge*x*Te (~1000). Because of the large *E*g and the resistivity (~kΩ), a ferroelectric *D-E* hysteresis loop was successfully observed in the low-temperature phase. The direction of spontaneous polarization is along the apex of a tetrahedron, [111], which is reported as *P*s= 0.0035 μC/cm2 in their first paper, and 5 μC/cm2 in the second paper. Because the ionic radius of Zn ion (0.83 Å) is smaller than that of Cd ion (1.03 Å), Zn ion also locates at the *off-center* position which deviates from the center of tetra‐ hedra toward the apex. According to the result of EXAFS (X-ray absorption fine structure), it shifts by 0.04 Å [26]. It is considered that the ordering of *off-center ions* causes rhombohedral distortion of the cubic lattice by about 0.01 Å, and induces ferroelectricity.

In Cd1-*x*Zn*x*Te, the soft mode has not been observed and the dielectric anomaly is small. These dielectric properties are different from those found in Pb1-*x*Ge*x*Te, which are summar‐ ized in Table 1. These facts are consistent each other when one consider the LST (Lyddane-Sachs-Teller) relation (ωLO2 /ωTO2 = ε/ε∞). In Pb1-*x*Ge*x*Te, the existence of soft mode is responsible for large dielectric anomaly. Although the behavior of *off-center ions* plays an im‐ portant role in this ferroelectricity like Pb1-*x*Ge*x*Te, the occurrence of phase transition seems to be driven in different way from that of Pb1-*x*Ge*x*Te, because of the different dielectric prop‐ erties described above. Furthermore, the band gap is much larger than that of the narrowgap ferroelectric semiconductors, which suggests that the electron-phonon coupling is not so strong in Cd1-*x*Zn*x*Te solid solutions.

**Figure 6.** Temperature dependence of dielectric constant in Cd1-*x*Zn*x*Te (*x*=0.1).

### **4.3. Zn1-***x***Li***x***O**

**Figure 5.** Temperature dependence of dielectric constant in Pb1-*x*Ge*x*Te (*x*=0.003).

tetrahedron, [111], which is reported as *P*s= 0.0035 μC/cm2

/ωTO<sup>2</sup>

Sachs-Teller) relation (ωLO<sup>2</sup>

distortion of the cubic lattice by about 0.01 Å, and induces ferroelectricity.

Among many wide-gap semiconductors, the ferroelectricity of Cd1-*x*Zn*x*Te was discovered by R. Weil *et al* in 1988 [9, 25]. The *E*g of Cd1-*x*Zn*x*Te is 1.53 eV. It is a wide-gap semiconductor unlike the narrow-gap IV-VI Pb1-*x*Ge*x*Te. The crystal structure is cubic zinc-blende-type. Cd(Zn) ion is surrounded by four Te ions tetrahedrally. The center of symmetry in this com‐ pound does not exist. When *x*=0.1, the crystal exhibits a dielectric anomaly at 393 K as shoen in Fig. 6. The peak value of dielectric constant (εpeak) is only 50, which is smaller by about two orders than that of BaTiO3 (~14000) and that of Pb1-*x*Ge*x*Te (~1000). Because of the large *E*g and the resistivity (~kΩ), a ferroelectric *D-E* hysteresis loop was successfully observed in the low-temperature phase. The direction of spontaneous polarization is along the apex of a

in the second paper. Because the ionic radius of Zn ion (0.83 Å) is smaller than that of Cd ion (1.03 Å), Zn ion also locates at the *off-center* position which deviates from the center of tetra‐ hedra toward the apex. According to the result of EXAFS (X-ray absorption fine structure), it shifts by 0.04 Å [26]. It is considered that the ordering of *off-center ions* causes rhombohedral

In Cd1-*x*Zn*x*Te, the soft mode has not been observed and the dielectric anomaly is small. These dielectric properties are different from those found in Pb1-*x*Ge*x*Te, which are summar‐ ized in Table 1. These facts are consistent each other when one consider the LST (Lyddane-

responsible for large dielectric anomaly. Although the behavior of *off-center ions* plays an im‐ portant role in this ferroelectricity like Pb1-*x*Ge*x*Te, the occurrence of phase transition seems

in their first paper, and 5 μC/cm2

= ε/ε∞). In Pb1-*x*Ge*x*Te, the existence of soft mode is

**4.2. II-VI Wide-gap semiconductor Cd1-***x***Zn***x***Te**

236 Advances in Ferroelectrics

Dielectric constants of Zn1-*x*Li*x*O ceramics with *x*=0.09 show an anomaly at 470 K (*T*c) (Fig. 7), although a high purity of ZnO does not show any anomaly from 20 K to 700 K [27]. The peak value of dielectric anomaly is 21 ( *x*=0.1), which is the same order with Cd1-*x*Zn*x*Te (ε ~ 50), but much smaller than ordinal ferroelectrics by 2~4 orders. This means that the dipole-dipole cor‐ relation accompanied with this ferroelectric phase transition is not so large in Zn1-*x*Li*x*O. In the measurement of *D-E* hysteresis loop, a small and clear hysteresis curve was observed. The spontaneous polarization varies by samples from 0.05 μC/cm<sup>2</sup> to 0.59 μC/cm<sup>2</sup> . This is due to that the preferred orientation of the *c*-axis, the direction of *P*s of the samples varies depending on samples. Naturally, powder of ZnO has egg-shaped grains which are elongated along the *c*-axis due to the anisotropy of elastic constants. Therefore, the orientation of the ceramic sam‐ ple depends on the condition of the pressing process in the sample preparation. After the cor‐ rection of the preferred orientation by using X-ray diffraction, the value of spontaneous polarizations of each sample converges to 0.9 μC/cm<sup>2</sup> as shown in Fig. 8 [28, 29]. The transi‐ tion temperature, *T*c, depends on the Li concentration. The phase diagram between *T*c and *x* is shown in Fig. 9. As temperature increases, *T*c becomes higher,which reminds us the phase di‐ agram of quantum ferroelectrics such as KTa1-*x*Nb*x*O3 and Sr1-*x*Ca*x*TiO3.

**Figure 9.** The *Tc*-*x* phase diagram of Zn1-*x*Li*x*O.

Crystal Structure of Paraelectric Phase

Ferroelectric Phase

Specific heat anomaly is small at the ferroelectric-paraelectric phase transition temperature. The transition entropy Δ*S* is almost 0. If Li ions which substitute Zn ions occupy the off-center position and behave an order-disorder motion, Δ*S* must be about *R*ln 2. If the nature of the tran‐ sition is displacive type, Δ*S*≏0. The observed thermal behavior suggests displacive nature, but, however, no soft mode was observed in Raman scattering measurement [30]. The small dielec‐ tric anomaly and the absence of soft mode are consistent each other, when we consider the LSTrelation. Relaxation mode corresponding to an order-disorder motion has not been found also by Raman scattering [31]. These peculiar dielectric properties of Zn1-*x*Li*x*O is resemble to those

**Pb1-xGexTe Cd1-xZnxTe Zn1-xLixO**

((*F*4 - 3*m*))

Zinc-blende

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Rhombohedral ((*R*3*m*))

Wurtzite ((*P*63*mc*))

Wurtzite ((*P*63*mc*))

of Cd1-*<sup>x</sup>*Zn*x*Te. These evidences suggests a new type of ferroelectric phase transition.

Rock-salt (NaCl) ((*Fm*3*m*))

Rhombohedral ((*R*3*m*))

**Table 1.** Dielectric properties of binary ferroelectric semiconductors.

*T*c (K) (*x*=0.1) 200 390 470 *E*g (eV) 0.3 1.53 3.2 ρ (Ωm) 10 103 1010 *P*s (μC/cm2) - 5 0.9 Soft mode ○ × × εpeak 103 50 21

**Figure 7.** Temperature dependence of dielectric constant of Zn1-*x*Li*x*O ceramics (*x*= 0.09).

**Figure 8.** Temperature dependence of spontaneous polarization of Zn1-*x*Li*x*O ceramics(*x* = 0.06). The inset is *D-E*hytere‐ sis loop observed below (left) and above *T*c (right).

**Figure 9.** The *Tc*-*x* phase diagram of Zn1-*x*Li*x*O.

**Figure 7.** Temperature dependence of dielectric constant of Zn1-*x*Li*x*O ceramics (*x*= 0.09).

**Figure 8.** Temperature dependence of spontaneous polarization of Zn1-*x*Li*x*O ceramics(*x* = 0.06). The inset is *D-E*hytere‐

sis loop observed below (left) and above *T*c (right).

238 Advances in Ferroelectrics

Specific heat anomaly is small at the ferroelectric-paraelectric phase transition temperature. The transition entropy Δ*S* is almost 0. If Li ions which substitute Zn ions occupy the off-center position and behave an order-disorder motion, Δ*S* must be about *R*ln 2. If the nature of the tran‐ sition is displacive type, Δ*S*≏0. The observed thermal behavior suggests displacive nature, but, however, no soft mode was observed in Raman scattering measurement [30]. The small dielec‐ tric anomaly and the absence of soft mode are consistent each other, when we consider the LSTrelation. Relaxation mode corresponding to an order-disorder motion has not been found also by Raman scattering [31]. These peculiar dielectric properties of Zn1-*x*Li*x*O is resemble to those of Cd1-*<sup>x</sup>*Zn*x*Te. These evidences suggests a new type of ferroelectric phase transition.


**Table 1.** Dielectric properties of binary ferroelectric semiconductors.

### **5. Electronic ferroelectricity in ZnO**

It is considered that the replacement of host Zn atoms by substitutional Li atoms plays an primary role for the appearance of ferroelectricity in ZnO. The problem is the effect of Lidoping. Here we consider the following two models [32].

**Atom** *x y z B* **Occupation** Zn 1/3 2/3 0.3821(3) 0.52 0.91 Li 1/3 2/3 03821(3) 0.31 0.09 O 1/3 2/3 0 0.40 1.0

The effect of dopants on *T*c was studied based on the two models mentioned above. If the ionic size-mismatch between the substituted and the host ions is important for ferroelectrici‐ ty, the introduction of Be ions should be more effective than Li and Mg doping. The ionic

Mg- and Be-doped ZnO ceramics [32]. A dielectric anomaly of Zn0.9Be0.1O was found at 496 K, which is very similar to those observed in Zn1-*x*Li*x*O. However, Zn1-*x*Mg*x*O samples show a clear decrease of *T*c. The dielectric anomaly in 30% Mg-doped ZnO found at 345 K is 150 K lower than the Li-doped one. The concentration dependence of *T*c of Be- and Mg-doped ZnO is summarized in Fig. 10. The series of dielectric measurements show that the introduction

It is considered that the appearance of ferroelectricity in ZnO is primarily due to electronic origin. The change in *d-p* hybridization caused by Li-substitution is responsible for this nov‐

, Be2+ and Mg2+ ions are 0.60 Å, 0.3 Å and 0.65 Å, respectively. If the changes in

2*s*<sup>2</sup> 2*p*<sup>6</sup>

). Dielectric constant measurements were done on

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) should play a different role

**Table 2.** Positional (*x,y,z*) and isotropic thermal (*B*) parameters in Zn1-*x*Li*x*O at 293K.

electronic configuration are important, the Mg2+ ion (1*s*<sup>2</sup>

**Figure 10.** *T*c*vs*. molar concentrations(*x*)of dopants, Li, Be and Mg in ZnO.

and Be2+ ions (1*s*<sup>2</sup>

of Mg2+ suppresses *T*c, while isoelectronic Be2+ shows almost the same *T*c.

**5.2. Effect of Be and Mg dopants on** *T***<sup>c</sup>**

el ferroelectricity and dielectric properties.

radii of Li+

from the isoelectronic Li+

### **i.** Structural size-mismatch model

Pure ZnO is polar along the *c*-axis and has dipole moments in crystals. Because of the sizemismatch between Zn ion (the ionic radius: 0.74 Å) and Li ion (the ionic radius: 0.60 Å), sub‐ stituted Li ions displace from the Zn positions. These displacements may force to induce extra dipole moments locally. It is considered that these local dipoles couple with dipoles of the mother ZnO crystal and the structural ordering of local dipoles triggers to induce a fer‐ roelectric phase transition.

**ii.** Electronic model

According to the first-principle study by Corso *et al* [10], the hybridization between Zn 3*d* electron and O 2*p* electron plays an important role in dielectric properties of ZnO. As the Li atom has no *d*-electrons, it is considered that the partial replacement of Zn ions by Li ions changes the nature of d the *d*-*p* hybridization. Therefore this doping is considered to induce local extra dipole moments. These extra local dipoles couple with parent dipoles and induce a new type of ferroelectricity.

To clarify which model is effective in the appearance of ferroelectricity in ZnO, the follow‐ ing structural and dielectric measurements were performed.

### **5.1. Rietveld analysis of structural changes in Zn1-***x***Li***x***O ceramics**

Structural changes associated with Li-substitution were studied by X-ray powder diffraction [33]. Ceramic samples of Zn1-*x*Li*x*O prepared by using a SPS (spark plasma sintering) method were used in this experiment. The nominal value of *x* is 0.1 for the first sample. The *T*c of the sample was 470 K which is determined by dielectric constant measurements. The systematic check of the absence of possible reflections (*h-k*=3*n* and *l*= odd for (*hkl*), *n* = integer) suggests that the space group of the ferroelectric phase still remains *P63mc*, which is same as pure ZnO. The obtained pattern was analyzed by the Rietveld method including the Li concentra‐ tion, *x*, as a refinable parameter. All parameters refined are shown in Table 2. The lattice constants are *a* = 3.2487(1) and *c* = 5.2050(1) Å at 293 K. The actual Li concentration *x* is deter‐ mined to be 0.09. The final discrepancy factors were *Rwp*=7.4%, *Rp*=5.4% and *RI* =5.9%. The Zn-O bond lengths are 1.9887 Å along the *c*-axis and 1.9735Å in the basal plane, respectively. This means that the bond length along the *c*-axis shows a decrease by an amount of 0.003 Å by Li-substitution in Li-doped ZnO. This lattice distortion is the order of 10-3 Å only, while the displacement of Ti ions is the order of 0.1 Å in well-known ferroelectric BaTiO3. In this phase transition, the crystal symmetry does not change associated with Li-substitution and structural change is considerably small compared with the structural phase transition in typ‐ ical ferroelectrics.


**Table 2.** Positional (*x,y,z*) and isotropic thermal (*B*) parameters in Zn1-*x*Li*x*O at 293K.

### **5.2. Effect of Be and Mg dopants on** *T***<sup>c</sup>**

**5. Electronic ferroelectricity in ZnO**

**i.** Structural size-mismatch model

roelectric phase transition. **ii.** Electronic model

240 Advances in Ferroelectrics

a new type of ferroelectricity.

ical ferroelectrics.

doping. Here we consider the following two models [32].

ing structural and dielectric measurements were performed.

**5.1. Rietveld analysis of structural changes in Zn1-***x***Li***x***O ceramics**

mined to be 0.09. The final discrepancy factors were *Rwp*=7.4%, *Rp*=5.4% and *RI*

It is considered that the replacement of host Zn atoms by substitutional Li atoms plays an primary role for the appearance of ferroelectricity in ZnO. The problem is the effect of Li-

Pure ZnO is polar along the *c*-axis and has dipole moments in crystals. Because of the sizemismatch between Zn ion (the ionic radius: 0.74 Å) and Li ion (the ionic radius: 0.60 Å), sub‐ stituted Li ions displace from the Zn positions. These displacements may force to induce extra dipole moments locally. It is considered that these local dipoles couple with dipoles of the mother ZnO crystal and the structural ordering of local dipoles triggers to induce a fer‐

According to the first-principle study by Corso *et al* [10], the hybridization between Zn 3*d* electron and O 2*p* electron plays an important role in dielectric properties of ZnO. As the Li atom has no *d*-electrons, it is considered that the partial replacement of Zn ions by Li ions changes the nature of d the *d*-*p* hybridization. Therefore this doping is considered to induce local extra dipole moments. These extra local dipoles couple with parent dipoles and induce

To clarify which model is effective in the appearance of ferroelectricity in ZnO, the follow‐

Structural changes associated with Li-substitution were studied by X-ray powder diffraction [33]. Ceramic samples of Zn1-*x*Li*x*O prepared by using a SPS (spark plasma sintering) method were used in this experiment. The nominal value of *x* is 0.1 for the first sample. The *T*c of the sample was 470 K which is determined by dielectric constant measurements. The systematic check of the absence of possible reflections (*h-k*=3*n* and *l*= odd for (*hkl*), *n* = integer) suggests that the space group of the ferroelectric phase still remains *P63mc*, which is same as pure ZnO. The obtained pattern was analyzed by the Rietveld method including the Li concentra‐ tion, *x*, as a refinable parameter. All parameters refined are shown in Table 2. The lattice constants are *a* = 3.2487(1) and *c* = 5.2050(1) Å at 293 K. The actual Li concentration *x* is deter‐

O bond lengths are 1.9887 Å along the *c*-axis and 1.9735Å in the basal plane, respectively. This means that the bond length along the *c*-axis shows a decrease by an amount of 0.003 Å by Li-substitution in Li-doped ZnO. This lattice distortion is the order of 10-3 Å only, while the displacement of Ti ions is the order of 0.1 Å in well-known ferroelectric BaTiO3. In this phase transition, the crystal symmetry does not change associated with Li-substitution and structural change is considerably small compared with the structural phase transition in typ‐

=5.9%. The Zn-

The effect of dopants on *T*c was studied based on the two models mentioned above. If the ionic size-mismatch between the substituted and the host ions is important for ferroelectrici‐ ty, the introduction of Be ions should be more effective than Li and Mg doping. The ionic radii of Li+ , Be2+ and Mg2+ ions are 0.60 Å, 0.3 Å and 0.65 Å, respectively. If the changes in electronic configuration are important, the Mg2+ ion (1*s*<sup>2</sup> 2*s*<sup>2</sup> 2*p*<sup>6</sup> ) should play a different role from the isoelectronic Li+ and Be2+ ions (1*s*<sup>2</sup> ). Dielectric constant measurements were done on Mg- and Be-doped ZnO ceramics [32]. A dielectric anomaly of Zn0.9Be0.1O was found at 496 K, which is very similar to those observed in Zn1-*x*Li*x*O. However, Zn1-*x*Mg*x*O samples show a clear decrease of *T*c. The dielectric anomaly in 30% Mg-doped ZnO found at 345 K is 150 K lower than the Li-doped one. The concentration dependence of *T*c of Be- and Mg-doped ZnO is summarized in Fig. 10. The series of dielectric measurements show that the introduction of Mg2+ suppresses *T*c, while isoelectronic Be2+ shows almost the same *T*c.

It is considered that the appearance of ferroelectricity in ZnO is primarily due to electronic origin. The change in *d-p* hybridization caused by Li-substitution is responsible for this nov‐ el ferroelectricity and dielectric properties.

**Figure 10.** *T*c*vs*. molar concentrations(*x*)of dopants, Li, Be and Mg in ZnO.

In the next section, we shall study the change in electronic distribution, especially in the na‐ ture of *d-p* hybridization by Li-doping directly by X-ray diffraction, and discuss new elec‐ tronic ferroelectricity in ZnO.

thermal vibration. The space group is confirmed to be P63mc for both pure and Li-doped crystals. The Lorentz and polarization, thermal diffuse scattering (TDS) (the elastic constants used in TDS collection are C11=20.96, C33=21.09, C12=12.10, C13=10.51, C44=4.243 [1011dyn/cm2

absorption (spherical) and extinction (anisotropic, type I) corrections were made. Crystal

**ZnO Zn1-xLixO**

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3157 2533 1296 2377

structures were refined using a full-matrix least squares program (RADIEL) [35].

Temperature 293 K 19 K 293 K 19 K X-ray Radiation AgKα AgKα MoKα AgKα (sinθ/λ)max 1.36 1.36 1.00 1.27

μ*R* 1.91 2.84 7.67 3.96 *R*(*F*) (%) 2.61 3.84 3.05 3.62 *RW*(*F*) (%) 3.38 4.86 3.90 4.54 S 1.00 1.06 1.09 1.01

**6.1. Crystal structures of single crystals of ZnO and Li-doped ZnO at 293 K**

Accurate electron densities of single crystals of paraelectricZnO and ferroelectric Li-doped ZnO at 19 K were analyzed by the maximum-entropy method (MEM) [36]. The MEM analy‐ ses were performed using the MEED (maximum-entropy electron density) program [37].

The final positional parameters and thermal factors at 293 K are given with their estimated standard deviation in Table 4 [38]. The final discrepancy factors are *R*(*F*)=2.61 %, *Rw*(*F*)=3.38 % for ZnO, and *R*(*F*)=3.05 %, *Rw*(*F*)=3.90 % for Zn1-*x*Li*x*O. The lattice constants *a* and *c* of Zn1 *<sup>x</sup>*Li*x*O become smaller than those of pure ZnO by 0.002 ~ 0.003 Å. Lattice distortion in Zn1 *<sup>x</sup>*Li*x*O is only the order of 10-3 Å along the polar axis, which is consistent with that of ceramic sample measured by thermal expansion [39]. These changes are regarded as the changes as‐ sociated with ferroelectric phase transition because Li-doped ZnO is ferroelectric phase and pure ZnO is paraelectric phase. In BaTiO3, the lattice constants change by 0.02 Å. In ZnO, changes in the lattice constants are smaller than that of BaTiO3 by one order. The thermal factors *U*33 of Zn and O atoms in Zn1-*x*Li*x*O are smaller than *U*11 by 25-35 %, although those values in pure ZnO are almost same. This implies that the thermal vibration in Zn1-*x*Li*x*O is suppressed along the *c*-axis. But in ZnO, the effect from bonding electron is also included in the thermal factors since strong covalence exists. The effects of thermal vibration and bond‐ ing electron cannot be distinguished at 297 K. The Li concentration *x*, the positional parame‐ ter *u* and isotropic thermal factor *U* of Li are refined after all the other parameters are determined. Two cases are assumed for the Li position: the first is the interstitial position

Number of Reflections (|Fo| "/> 3σ |Fo|)

**Table 3.** Experimental data.

],

243

### **6. X-ray study of electronic density distribution**

X-ray diffraction measurements were performed for single crystals of pure ZnO and Zn1 *<sup>x</sup>*Li*x*O at 297 K and 19 K in order to investigate the changes in the crystal structure and the electronic density distribution by Li-substitution in detail. Single crystal of pure ZnO was prepared by the hydrothermal method [34]. The content of excess Zn ions of obtained single crystal was 1.7 ppm and color was light yellowish (Fig. 11). In this hydrothermal method, it is rather difficult to add a large amount of Li ions into single crystal. In order to dope Li ions, several *c*-plate samples (0.16 mm thick) were annealed at an atmosphere of Li ions at 920 K for 24 hours. The light yellowish color of as-grown single crystal became transparent after this doping as shown in Fig. 12. Li concentration *x* was measured by chemical analysis and it is confirmed that *x* = 0.082~0.086.

**Figure 11.** Single crystal of as grown ZnO.

**Figure 12.** Single crystal of pure (left) and Li-doped ZnO (right).

Table 3 shows an experimental data of X-ray diffraction performed at room temperature (293 K) and low temperature (19 K) by using a He-gas closed-cycle cryostat (Cryogenic RC-110) mounted on a Huber off-center four-circle diffractometer to reduce the effect of thermal vibration. The space group is confirmed to be P63mc for both pure and Li-doped crystals. The Lorentz and polarization, thermal diffuse scattering (TDS) (the elastic constants used in TDS collection are C11=20.96, C33=21.09, C12=12.10, C13=10.51, C44=4.243 [1011dyn/cm2 ], absorption (spherical) and extinction (anisotropic, type I) corrections were made. Crystal structures were refined using a full-matrix least squares program (RADIEL) [35].


**Table 3.** Experimental data.

In the next section, we shall study the change in electronic distribution, especially in the na‐ ture of *d-p* hybridization by Li-doping directly by X-ray diffraction, and discuss new elec‐

X-ray diffraction measurements were performed for single crystals of pure ZnO and Zn1 *<sup>x</sup>*Li*x*O at 297 K and 19 K in order to investigate the changes in the crystal structure and the electronic density distribution by Li-substitution in detail. Single crystal of pure ZnO was prepared by the hydrothermal method [34]. The content of excess Zn ions of obtained single crystal was 1.7 ppm and color was light yellowish (Fig. 11). In this hydrothermal method, it is rather difficult to add a large amount of Li ions into single crystal. In order to dope Li ions, several *c*-plate samples (0.16 mm thick) were annealed at an atmosphere of Li ions at 920 K for 24 hours. The light yellowish color of as-grown single crystal became transparent after this doping as shown in Fig. 12. Li concentration *x* was measured by chemical analysis

Table 3 shows an experimental data of X-ray diffraction performed at room temperature (293 K) and low temperature (19 K) by using a He-gas closed-cycle cryostat (Cryogenic RC-110) mounted on a Huber off-center four-circle diffractometer to reduce the effect of

tronic ferroelectricity in ZnO.

242 Advances in Ferroelectrics

and it is confirmed that *x* = 0.082~0.086.

**Figure 11.** Single crystal of as grown ZnO.

**Figure 12.** Single crystal of pure (left) and Li-doped ZnO (right).

**6. X-ray study of electronic density distribution**

Accurate electron densities of single crystals of paraelectricZnO and ferroelectric Li-doped ZnO at 19 K were analyzed by the maximum-entropy method (MEM) [36]. The MEM analy‐ ses were performed using the MEED (maximum-entropy electron density) program [37].

### **6.1. Crystal structures of single crystals of ZnO and Li-doped ZnO at 293 K**

The final positional parameters and thermal factors at 293 K are given with their estimated standard deviation in Table 4 [38]. The final discrepancy factors are *R*(*F*)=2.61 %, *Rw*(*F*)=3.38 % for ZnO, and *R*(*F*)=3.05 %, *Rw*(*F*)=3.90 % for Zn1-*x*Li*x*O. The lattice constants *a* and *c* of Zn1 *<sup>x</sup>*Li*x*O become smaller than those of pure ZnO by 0.002 ~ 0.003 Å. Lattice distortion in Zn1 *<sup>x</sup>*Li*x*O is only the order of 10-3 Å along the polar axis, which is consistent with that of ceramic sample measured by thermal expansion [39]. These changes are regarded as the changes as‐ sociated with ferroelectric phase transition because Li-doped ZnO is ferroelectric phase and pure ZnO is paraelectric phase. In BaTiO3, the lattice constants change by 0.02 Å. In ZnO, changes in the lattice constants are smaller than that of BaTiO3 by one order. The thermal factors *U*33 of Zn and O atoms in Zn1-*x*Li*x*O are smaller than *U*11 by 25-35 %, although those values in pure ZnO are almost same. This implies that the thermal vibration in Zn1-*x*Li*x*O is suppressed along the *c*-axis. But in ZnO, the effect from bonding electron is also included in the thermal factors since strong covalence exists. The effects of thermal vibration and bond‐ ing electron cannot be distinguished at 297 K. The Li concentration *x*, the positional parame‐ ter *u* and isotropic thermal factor *U* of Li are refined after all the other parameters are determined. Two cases are assumed for the Li position: the first is the interstitial position and the second is host Zn position. The discrepancy factor became larger when Li locates at interstitial Li position than Zn position. Therefore, present results support that Li substitutes Zn. The precise refinement shows that Li locates at the off-center position from the Zn posi‐ tion by 0.02 Å in Zn1-*x*Li*x*O along the *c*-axis.


**Table 4.** Crystal structure of ZnO and Zn1-*x*Li*x*O at 293 K.(The form of thermal factors are exp[-2π2(*U*11*a*\*2*h*2 + *U*22*b*\*2*k*2 + *U*33c\*2*l* 2 + 2*U*12*a*\**b*\**hk* + 2*U*23*b*\**c*\**kl* + 2*U*31*c*\**a*\**lh*)] for anisotropic (Zn and O) and exp(-8π<sup>2</sup>*U*eqsin2θ/λ) for isotropic (Li) atoms.)

The bond lengths and angles of ZnO4 group are shown in Table 5. The Zn-O bond length does not change in the basal plane, but becomes short along the *c*-axis by an amount of 0.007 Å by Li-substitution.

**Figure 13.** The charge density maps of (a) ZnO and (b) Zn1-*x*Li*x*O at 293 K in the (110) plane obtained by Fourier analysis.

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**Figure 14.** The difference fourier maps of charge densities of (a) ZnO and (b) Zn1-*x*Li*x*O at 293 K in the (110) plane with a contour increment of 0.2 *e*/ Å 3. Bluish cold color means negative charge density and reddish warm region is positive

charge density.


**Table 5.** Bond lengths and angles in ZnO and Zn1-*x*Li*x*O at 293 K.

The distributions of electronic density around Zn and O atoms at 293 K are obtained by us‐ ing Fourier analysis (Fig. 13). The difference Fourier maps between observed distribution ρobs and calculated distribution ρcal in the (110) plane are shown in Fig. 14. Since the bonding electron is not distributed spherically in ZnO, the section of the bonding electron appears in the difference map. It is seen that there are large negative distributions (blue region) around Zn atom along the *c*-axis in Zn1-*x*Li*x*O. This suggests that the core electrons disappear from the Zn atom. The broadening of map due to anharmonic thermal vibrations is also appreci‐ ated. The positive electronic density is observed near the O atom. It corresponds to the anti‐ bonding orbital of O 2*p*-electron.

and the second is host Zn position. The discrepancy factor became larger when Li locates at interstitial Li position than Zn position. Therefore, present results support that Li substitutes Zn. The precise refinement shows that Li locates at the off-center position from the Zn posi‐

> **ZnO (a=3.2489(1) Å, c=5.2049(3) Å) at 293 K** *x y z U* <sup>11</sup> *U* <sup>33</sup> *U* eq Occupation

> **Zn1-xLixO (a=3.2467(3) Å, c=5.2032(1) Å) at 293 K** *x y z U* <sup>11</sup> *U* <sup>33</sup> *U* eq Occupation

**Table 4.** Crystal structure of ZnO and Zn1-*x*Li*x*O at 293 K.(The form of thermal factors are exp[-2π2(*U*11*a*\*2*h*2 + *U*22*b*\*2*k*2 +

The bond lengths and angles of ZnO4 group are shown in Table 5. The Zn-O bond length does not change in the basal plane, but becomes short along the *c*-axis by an amount of 0.007

> ZnO 1.986(1) Å 1.975(1) Å 108.20(3)o 110.71(3)o Zn1-xLi*x*O 1.979(1) Å 1.975(1) Å 108.37(4)o 110.55(4)o

The distributions of electronic density around Zn and O atoms at 293 K are obtained by us‐ ing Fourier analysis (Fig. 13). The difference Fourier maps between observed distribution ρobs and calculated distribution ρcal in the (110) plane are shown in Fig. 14. Since the bonding electron is not distributed spherically in ZnO, the section of the bonding electron appears in the difference map. It is seen that there are large negative distributions (blue region) around Zn atom along the *c*-axis in Zn1-*x*Li*x*O. This suggests that the core electrons disappear from the Zn atom. The broadening of map due to anharmonic thermal vibrations is also appreci‐ ated. The positive electronic density is observed near the O atom. It corresponds to the anti‐

**Bond length Bond Angle** Apical Basal Apical Basal

2 + 2*U*12*a*\**b*\**hk* + 2*U*23*b*\**c*\**kl* + 2*U*31*c*\**a*\**lh*)] for anisotropic (Zn and O) and exp(-8π<sup>2</sup>*U*eqsin2θ/λ) for isotropic (Li)

Zn 1/3 2/3 0.3815(1) 0.0086(1) 0.0088(1) 0.0087(1) 1.0 O 1/3 2/3 0 0.0085(1) 0.0088(1) 0.0086(1) 1.0

Zn 1/3 2/3 0.3804(3) 0.0075(1) 0.0055(1) 0.0068(1) 0.914 Li 1/3 2/3 0.376(36) 0.0036(48) 0.086 O 1/3 2/3 0 0.0088(3) 0.0057(41) 0.0078(3) 0.957

tion by 0.02 Å in Zn1-*x*Li*x*O along the *c*-axis.

*U*33c\*2*l*

244 Advances in Ferroelectrics

atoms.)

Å by Li-substitution.

**Table 5.** Bond lengths and angles in ZnO and Zn1-*x*Li*x*O at 293 K.

bonding orbital of O 2*p*-electron.

**Figure 13.** The charge density maps of (a) ZnO and (b) Zn1-*x*Li*x*O at 293 K in the (110) plane obtained by Fourier analysis.

**Figure 14.** The difference fourier maps of charge densities of (a) ZnO and (b) Zn1-*x*Li*x*O at 293 K in the (110) plane with a contour increment of 0.2 *e*/ Å 3. Bluish cold color means negative charge density and reddish warm region is positive charge density.

### **6.2. Crystal structures at 19 K**

Crystal structures at low temperature are shown in Table 6 [38]. The final discrepancy fac‐ tors are *R*(*F*)=3.84 %, *Rw*(*F*)=4.86 % for ZnO, and *R*(*F*)=3.62 %, *Rw*(*F*)=4.54 % for Zn1-*x*Li*x*O. Li ion shifts from the Zn position by 0.08 Å at 19 K, which is four times larger than that at r. t.. The Zn-O bond lengths decrease by an amount of 0.002 Å along both basal and apical axes in Zn1-*x*Li*x*O, while their bond angles are almost the same in both crystals (Table 7 and Fig. 15). The Fourier and difference maps of electronic distribution are shown in Figs. 16 and 17. Comparing with the result of 293 K, bonding electrons are clearly observed around the cen‐ ter of Zn-O bond in both crystals. In Zn1-*x*Li*x*O, negative distribution is observed around Zn atom, whose shape corresponds to Zn-3*d*<sup>z</sup> 2 -orbital. It suggests the disappearance of 3*d*-elec‐ trons from Zn atom in Zn1-*x*Li*x*O. The positive density near O atoms corresponds to the 2*p*orbital of O atom.

(a) Pure ZnO (b) Li-doped ZnO

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**Figure 16.** The charge density maps of (a) ZnO and (b) Zn1-*x*Li*x*O at 19 K in the (110) plane obtained by Fourier analysis.

**Figure 15.** Structural Changes of ZnO4 group at 19 K. The shift of Li ion is 0.08 Å from the Zn position.


**Table 6.** Crystal structure of ZnO and Zn1-*x*Li*x*O at 19 K.(The form of thermal factors are exp[-2π2(*U*11*a*\*2*h*2 + *U*22*b*\*2*k*2 + *U*33c\*2*l* 2 + 2*U*12*a*\**b*\**hk* + 2*U*23*b*\**c*\**kl* + 2*U*31*c*\**a*\**lh*)] for anisotropic (Zn and O) and exp(-8π<sup>2</sup>*U*eqsin2θ/λ) for isotropic (Li) atoms.)


**Table 7.** Bond lengths and angles in ZnO and Zn1-*x*Li*x*O at 19 K.

**Figure 15.** Structural Changes of ZnO4 group at 19 K. The shift of Li ion is 0.08 Å from the Zn position.

**6.2. Crystal structures at 19 K**

246 Advances in Ferroelectrics

atom, whose shape corresponds to Zn-3*d*<sup>z</sup>

**Table 7.** Bond lengths and angles in ZnO and Zn1-*x*Li*x*O at 19 K.

orbital of O atom.

*U*33c\*2*l*

atoms.)

Crystal structures at low temperature are shown in Table 6 [38]. The final discrepancy fac‐ tors are *R*(*F*)=3.84 %, *Rw*(*F*)=4.86 % for ZnO, and *R*(*F*)=3.62 %, *Rw*(*F*)=4.54 % for Zn1-*x*Li*x*O. Li ion shifts from the Zn position by 0.08 Å at 19 K, which is four times larger than that at r. t.. The Zn-O bond lengths decrease by an amount of 0.002 Å along both basal and apical axes in Zn1-*x*Li*x*O, while their bond angles are almost the same in both crystals (Table 7 and Fig. 15). The Fourier and difference maps of electronic distribution are shown in Figs. 16 and 17. Comparing with the result of 293 K, bonding electrons are clearly observed around the cen‐ ter of Zn-O bond in both crystals. In Zn1-*x*Li*x*O, negative distribution is observed around Zn

2

**ZnO (a=3.2465(8) Å, c=5.2030(19) Å) at 19 K**

Zn 1/3 2/3 0.3812(1) 0.0033(1) 0.0032(1) 1.0

O 1/3 2/3 0 0.0041(1) 0.0048(2) 1.0

**Zn1-xLixO (a=3.2436(5) Å, c=5.1983(30) Å) at 19 K**

trons from Zn atom in Zn1-*x*Li*x*O. The positive density near O atoms corresponds to the 2*p*-

*x y z U* <sup>11</sup> *U* <sup>33</sup> Occupation

Zn 1/3 2/3 0.3811(2) 0.0032(1) 0.0026(1) 0.0030(1) 0.914

Li 1/3 2/3 0.366(12) - - 0.0036(48) 0.086

O 1/3 2/3 0 0.0053(3) 0.0043(3) 0.0050(2) 0.957

**Table 6.** Crystal structure of ZnO and Zn1-*x*Li*x*O at 19 K.(The form of thermal factors are exp[-2π2(*U*11*a*\*2*h*2 + *U*22*b*\*2*k*2 +

2 + 2*U*12*a*\**b*\**hk* + 2*U*23*b*\**c*\**kl* + 2*U*31*c*\**a*\**lh*)] for anisotropic (Zn and O) and exp(-8π<sup>2</sup>*U*eqsin2θ/λ) for isotropic (Li)

ZnO 1.983(2) Å 1.974(1) Å 108.25(4)o 110.66(4)o

Zn1-xLi*x*O 1.981(2) Å 1.972(1) Å 108.27(3)o 110.65(3)o

**0.957 0.957** Apical Basal Apical Basal

*x y z U* <sup>11</sup> *U* <sup>33</sup> *U* eq Occupation


(a)

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)

(b) (c)

**Figure 18.** The electronic Distribution by MEM method. (a) The (110) plane of ZnO. Shaded atoms are included in the plane.(b)The MEM charge density map of ZnO at 19 K in the (110) plane. (c) The MEM charge density map of

The difference of the charge densities between two crystals is calculated by subtracting the MEM charge densities of pure ZnO from those of Zn1-*x*Li*x*O as in Fig. 19 [40]. The values of MEM charge densities of pure ZnO are normalized by multiplying 0.930, because the total charges of two crystals are different. The positive and negative peaks were observed around the O atom. This is due to the shift of O atom from the host position opposite to the Zn atom along the *c*-axis in Zn1-*x*Li*x*O. Four negative peaks are observed around the Zn atom. These peaks correspond to the Zn 3*d*-orbitals and suggest that the 3*d*-electrons disappear from the Zn site, compared with wave functions of Zn 3*d-* and O 2*p*-orbitals of ZnO obtained by DV-Xα calculation in Fig. 20. This result is the same as that calculated by the Fourier synthesis previously. The bonding region around the apical and the basal Zn-O bonds is covered by positive electron distribution. This implies that the Zn 3*d*-electrons transfer to the bonding

Zn0.914Li0.086O at 19 K in the (110) plane. Contours are drawn from 0.4 *e*/ Å 3 at 0.2 e/ Å 3 intervals.

**6.4. Difference between Li-doped ZnO and pure ZnO**

electrons in Zn1-*x*Li*x*O.

**Figure 17.** The difference maps of charge densities of (a) ZnO and (b) Zn1-*x*Li*x*O at 19 K in the (110) plane with a con‐ tour increment of 0.2 *e*/ Å 3. Bluish cold color means negative charge density and reddish warm color region is positive charge density.

### **6.3. Bond electron densities at 19 K**

The discrepancy factors of MEM analysis for pure ZnO at 19 K are *R*(*F*)=1.20 % and *Rw*(*F*)=1.39 % [40]. The electron density map of ZnO in the (110) plane is shown in Fig. 18(b). The covalent character of Zn-O bonds is clearly observed. Charge densities at the center of Zn-O bonds are 0.58 *e*/ Å 3 for the apical bond, and 0.56 *e*/Å 3 for the basal one. The density of bonding electron of the apical bond is larger than that of the basal by 3.6 %. The electron density of O atom distorts toward the Zn atom along the *c*-axis. The electron density of Zn atom elongates to the third nearest O atom.

The final discrepancy factors for the ferroelectric Zn1-*x*Li*x*O at 19 K are *R*(*F*)=0.87 % and *Rw*(*F*)=0.85%. The electron density map of Zn1-*x*Li*x*O in the (110) plane is shown in Fig. 18(c). The covalent character of Zn-O bonds is also seen in Zn1-*x*Li*x*O. Charge densities of the Zn-O at the saddlepoint are 0.46 *e*/ Å <sup>3</sup> for the apical bond, and 0.49 *e*/ Å <sup>3</sup> for the basal one. Each value is smaller than that of pure ZnO. It is considered that this may be due to the decrease of total charge by Li-substitution. The density of bonding electron of the basal bond is larger than that of the apical by 6.5 %, on the contrary to the case of ZnO. The electron density of O atom is distorted anisotropically, similar to the pure ZnO, but the direction is opposite. The extension of Zn atom toward the third nearest O atom, which is observed in ZnO, was not detected in Zn1-*x*Li*x*O.

**Figure 18.** The electronic Distribution by MEM method. (a) The (110) plane of ZnO. Shaded atoms are included in the plane.(b)The MEM charge density map of ZnO at 19 K in the (110) plane. (c) The MEM charge density map of Zn0.914Li0.086O at 19 K in the (110) plane. Contours are drawn from 0.4 *e*/ Å 3 at 0.2 e/ Å 3 intervals.

### **6.4. Difference between Li-doped ZnO and pure ZnO**

**Figure 17.** The difference maps of charge densities of (a) ZnO and (b) Zn1-*x*Li*x*O at 19 K in the (110) plane with a con‐ tour increment of 0.2 *e*/ Å 3. Bluish cold color means negative charge density and reddish warm color region is positive

The discrepancy factors of MEM analysis for pure ZnO at 19 K are *R*(*F*)=1.20 % and *Rw*(*F*)=1.39 % [40]. The electron density map of ZnO in the (110) plane is shown in Fig. 18(b). The covalent character of Zn-O bonds is clearly observed. Charge densities at the center of Zn-O bonds are 0.58 *e*/ Å 3 for the apical bond, and 0.56 *e*/Å 3 for the basal one. The density of bonding electron of the apical bond is larger than that of the basal by 3.6 %. The electron density of O atom distorts toward the Zn atom along the *c*-axis. The electron density of Zn

The final discrepancy factors for the ferroelectric Zn1-*x*Li*x*O at 19 K are *R*(*F*)=0.87 % and *Rw*(*F*)=0.85%. The electron density map of Zn1-*x*Li*x*O in the (110) plane is shown in Fig. 18(c). The covalent character of Zn-O bonds is also seen in Zn1-*x*Li*x*O. Charge densities of the Zn-O

value is smaller than that of pure ZnO. It is considered that this may be due to the decrease of total charge by Li-substitution. The density of bonding electron of the basal bond is larger than that of the apical by 6.5 %, on the contrary to the case of ZnO. The electron density of O atom is distorted anisotropically, similar to the pure ZnO, but the direction is opposite. The extension of Zn atom toward the third nearest O atom, which is observed in ZnO, was not

for the apical bond, and 0.49 *e*/ Å <sup>3</sup> for the basal one. Each

charge density.

248 Advances in Ferroelectrics

**6.3. Bond electron densities at 19 K**

atom elongates to the third nearest O atom.

at the saddlepoint are 0.46 *e*/ Å <sup>3</sup>

detected in Zn1-*x*Li*x*O.

The difference of the charge densities between two crystals is calculated by subtracting the MEM charge densities of pure ZnO from those of Zn1-*x*Li*x*O as in Fig. 19 [40]. The values of MEM charge densities of pure ZnO are normalized by multiplying 0.930, because the total charges of two crystals are different. The positive and negative peaks were observed around the O atom. This is due to the shift of O atom from the host position opposite to the Zn atom along the *c*-axis in Zn1-*x*Li*x*O. Four negative peaks are observed around the Zn atom. These peaks correspond to the Zn 3*d*-orbitals and suggest that the 3*d*-electrons disappear from the Zn site, compared with wave functions of Zn 3*d-* and O 2*p*-orbitals of ZnO obtained by DV-Xα calculation in Fig. 20. This result is the same as that calculated by the Fourier synthesis previously. The bonding region around the apical and the basal Zn-O bonds is covered by positive electron distribution. This implies that the Zn 3*d*-electrons transfer to the bonding electrons in Zn1-*x*Li*x*O.

In Li-doped ZnO, negative peaks around Zn atom corresponding to 3*d*<sup>z</sup>

localized *d-*holes in Li-doped ZnO. It is considered that 3*d*<sup>z</sup>

doped ZnO.

Fig. 21.

**7. Discussion**

served in difference Fourier map, and also in the MEM difference map between the ferro‐ electric phase and the paraelectric phase. These results may be related to the existence of

region and play a role for contribution for the covalence observed from the result of the MEM analysis. Furthermore, the antibonding orbital of 2*p* electron is observed in differ‐ ence Fourier map of Li-doped ZnO. It is suggested that the interaction between *d*-holes and *p*-electrons should be closely related to the appearance of the ferroelectricity in Li-

The ferroelectric phase transition accompanies with structural distortions in usual ferro‐ electrics. Portengen, Ostreich and Sham reported the theory of electronic ferroelectricity [41], which examined possibilities of electronic ferroelectricity, based on the spinless Fali‐ kov-Kimball (FK) model [42] with a *k*-dependence hybridization in Hartree-Fock approxi‐ mation. The FK model introduces two types of electrons, itinerant *d*-electrons and localized *f*-electrons. The valence transition is driven by on-site Coulomb repulsion be‐ tween the *d*-electrons and *f*-electrons. They found that the Coulomb interaction between

value, which breaks the center of symmetry of the crystal and leads to electronic ferroelec‐ tricity in mixed-valent compounds. In this electronic model, the transition involves a change in the electronic structure rather than the structural one. The estimated spontane‐

In the case of Li-doped ZnO, the *d-p* hybridization should be changed by Li-doping. The *d*holes and itinerant electrons should be closely related to the appearance of Li-doped ZnO. It should be further detailed studies would be necessary whether this proposed theory is ap‐

Recently Glinchuk *et al* propose the mechanism of impurities induced ferroelectricity in nonperovskite semiconductor matrices due to indirect dipole interaction via free carriers [43]. They showed that the ferroelectricity in Li-doped ZnO might appear due to indirect interaction of dipoles, formed by off-center impurities, *via* free charge carriers, namely, the Ruderman–Kittel–Kasuya–Yosida (RKKY)-like indirect interaction of impurity dipoles *via* free charge carriers. They estimated that the typical semiconducting concentration of the carriers like 1017 cm−3 is sufficient for the realization of the ferroelectricity. The finite con‐ ductivity does not mean complete destruction of possible ferroelectric order and shows rather many interesting effects. In this theory, they have succeeded to obtain the sponta‐ neous polarization and the phase diagram of *T*c*vs*. impurity molar ratio (*x*) as shown in

itinerant *d*-electrons and the localized *f*-electrons give rise to an excitonic<*d*<sup>+</sup>

ous polarization is of the order of 10 μC/cm2

plicable for the electronic ferroelectricity found in ZnO or not.

type ferroelectrics such as BaTiO3.

<sup>2</sup> orbital were ob‐

251

http://dx.doi.org/10.5772/52304

*f*> expectation

, which is comparable to those in displacive

2 electrons transfer to bonding

Electronic Ferroelectricity in II-VI Semiconductor ZnO

**Figure 19.** The difference map between MEM charge densities of Zn1-*x*Li*x*O and ZnO at 19 K in the (110) plane. Contours are drawn from -5.0 *e*/ Å 3 to 5.0 *e*/ Å 3 at 0.5 *e*/ Å 3 intervals. Shaded area eith reddish color indicates positive charge.

**Figure 20.** Wave functions of Zn 3*d-* and O 2*p*-orbitals of ZnO obtained by DV-Xα calculation. Red lines indicate the positive and Blue lines indicate the negative area.

In Li-doped ZnO, negative peaks around Zn atom corresponding to 3*d*<sup>z</sup> <sup>2</sup> orbital were ob‐ served in difference Fourier map, and also in the MEM difference map between the ferro‐ electric phase and the paraelectric phase. These results may be related to the existence of localized *d-*holes in Li-doped ZnO. It is considered that 3*d*<sup>z</sup> 2 electrons transfer to bonding region and play a role for contribution for the covalence observed from the result of the MEM analysis. Furthermore, the antibonding orbital of 2*p* electron is observed in differ‐ ence Fourier map of Li-doped ZnO. It is suggested that the interaction between *d*-holes and *p*-electrons should be closely related to the appearance of the ferroelectricity in Lidoped ZnO.

### **7. Discussion**

**Figure 19.** The difference map between MEM charge densities of Zn1-*x*Li*x*O and ZnO at 19 K in the (110) plane. Contours are

**Figure 20.** Wave functions of Zn 3*d-* and O 2*p*-orbitals of ZnO obtained by DV-Xα calculation. Red lines indicate the

positive and Blue lines indicate the negative area.

250 Advances in Ferroelectrics

drawn from -5.0 *e*/ Å 3 to 5.0 *e*/ Å 3 at 0.5 *e*/ Å 3 intervals. Shaded area eith reddish color indicates positive charge.

The ferroelectric phase transition accompanies with structural distortions in usual ferro‐ electrics. Portengen, Ostreich and Sham reported the theory of electronic ferroelectricity [41], which examined possibilities of electronic ferroelectricity, based on the spinless Fali‐ kov-Kimball (FK) model [42] with a *k*-dependence hybridization in Hartree-Fock approxi‐ mation. The FK model introduces two types of electrons, itinerant *d*-electrons and localized *f*-electrons. The valence transition is driven by on-site Coulomb repulsion be‐ tween the *d*-electrons and *f*-electrons. They found that the Coulomb interaction between itinerant *d*-electrons and the localized *f*-electrons give rise to an excitonic<*d*<sup>+</sup> *f*> expectation value, which breaks the center of symmetry of the crystal and leads to electronic ferroelec‐ tricity in mixed-valent compounds. In this electronic model, the transition involves a change in the electronic structure rather than the structural one. The estimated spontane‐ ous polarization is of the order of 10 μC/cm2 , which is comparable to those in displacive type ferroelectrics such as BaTiO3.

In the case of Li-doped ZnO, the *d-p* hybridization should be changed by Li-doping. The *d*holes and itinerant electrons should be closely related to the appearance of Li-doped ZnO. It should be further detailed studies would be necessary whether this proposed theory is ap‐ plicable for the electronic ferroelectricity found in ZnO or not.

Recently Glinchuk *et al* propose the mechanism of impurities induced ferroelectricity in nonperovskite semiconductor matrices due to indirect dipole interaction via free carriers [43]. They showed that the ferroelectricity in Li-doped ZnO might appear due to indirect interaction of dipoles, formed by off-center impurities, *via* free charge carriers, namely, the Ruderman–Kittel–Kasuya–Yosida (RKKY)-like indirect interaction of impurity dipoles *via* free charge carriers. They estimated that the typical semiconducting concentration of the carriers like 1017 cm−3 is sufficient for the realization of the ferroelectricity. The finite con‐ ductivity does not mean complete destruction of possible ferroelectric order and shows rather many interesting effects. In this theory, they have succeeded to obtain the sponta‐ neous polarization and the phase diagram of *T*c*vs*. impurity molar ratio (*x*) as shown in Fig. 21.

crystal structures of ferroelectrics. As the crystal structure of ZnO is very simple, the elec‐ tronic contribution could be observed rather easily. It is considered that this result is the first

Electronic Ferroelectricity in II-VI Semiconductor ZnO

http://dx.doi.org/10.5772/52304

253

[1] Klingshirn C F, Meyer B K, Waag A, Hoffmann A and Geurts J. Zinc Oxide From

[2] Yao T (ed.). ZnO Its Most Up-to-date Technology and Application, Perspectives (in

[4] Campbell C. Surface Acoustic Wave Devices and Their Signal Processing Applica‐

[6] Tsukazaki A, Ohtomo A, Onuma T, Ohtani M, Makino T, Sumiya M, Ohtani K, Chi‐ chibu S F, Fuke S, Segawa Y, Ohno H, Koinuma H and Kawasaki M. Nature Materi‐

[8] Bilz H, Bussmann-Holder A, Jantsch W and Vogel P. Dynamical Properties of IV-VI

[11] Onodera A, Tamaki N, Kawamura Y, Sawada T and Yamashita H. Jpn. J Appl Phys

[10] Weil R, Nkum R, Muranevich E and Benguigui L. Phys Rev Lett 1989; 62: 2744.

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example to discuss the electronic contribution for ferroelectricity.

**Author details**

**References**

Akira Onodera and Masaki Takesada

japanese). CMC Books, 2007.

als 2005; 4: 42.

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**Figure 21.** The calculated spontaneous polarization and the phase diagram of *T*c*vs*. impurity molar ratio (*x*) after Glin‐ chuk et al [36].

Experimental results suggest the existence of localized *d*-holes and *p*-electrons in Li-doped ZnO. We consider that the interaction between *d*-holes and *p*-electrons may be related to the appearance of the ferroelectricity in Li-doped ZnO. However, there are many proposals for this peculir novel ferroelectricity in ZnO. Scott and Zubko have pointed out the possibility of a classic electret mechanism for Li-doped ZnO[44]. Tagantsev discussed a Landau theory, where a crystal on cooling from a state with polar symmetry exhibits a maximum of dielec‐ tric permittivity and *D-E* hysteresis loops [45]. He proposed that these ferroelectric like phe‐ nomena corresponds to the case for Li-doped ZnO. Furthermore, the multiferroic behavior has been reported in impurity-doped ZnO [46]. At present, further detailed experiments should be expected to clarify the nature of Li-doped ZnO.

### **8. Conclusion**

Dielectric properties and crystal structures and electron density distributions studied on pure ZnO and Zn1-*<sup>x</sup>* Li*x* O by the precise X-ray diffraction are reviewed in comparison with fer‐ roelectric semiconductors Pb1-*<sup>x</sup>* Ge*<sup>x</sup>* Te and Cd1-*<sup>x</sup>* Zn*<sup>x</sup>* Te. It is considered that the appearance of ferroelectricity in ZnO is primarily due to electronic origin. The change in *d-p* hybridization caused by Li-substitution is considered to be responsible for this novel ferroelectricity and dielectric properties. Although the ferroelectric phase transition accompanies with structural distortions in usual ferroelectrics, the structural changes observed in Zn1-*<sup>x</sup>* Li*x* O are the order of 10-3 Å. The clear change of Zn 3*d*-electron is observed. It suggests Zn 3*d*-electron of Zn1 *x* Li*x* O transfers from Zn atom to the bonding. The positive electronic density is observed near the O atom. It corresponds to the antibonding orbital of O 2*p*-electron. These results suggest the existence of localized *d*-holes and *p*-electrons in Li-doped ZnO. It may be probable that the interaction between *d*-holes and *p*-electrons may be related to the appearance of the fer‐ roelectricity in Li-doped ZnO.

For long time, the importance of electronic contribution has been pointed out in the field of ferroelectrics, since the simple superposition of electronic polarizability does not hold in many ferroelectric substances. Many efforts have been done in vain because of complexity of

crystal structures of ferroelectrics. As the crystal structure of ZnO is very simple, the elec‐ tronic contribution could be observed rather easily. It is considered that this result is the first example to discuss the electronic contribution for ferroelectricity.

### **Author details**

Akira Onodera and Masaki Takesada

Department of Physics, Faculty of Science, Hokkaido University, Sapporo, Japan

### **References**

**Figure 21.** The calculated spontaneous polarization and the phase diagram of *T*c*vs*. impurity molar ratio (*x*) after Glin‐

Experimental results suggest the existence of localized *d*-holes and *p*-electrons in Li-doped ZnO. We consider that the interaction between *d*-holes and *p*-electrons may be related to the appearance of the ferroelectricity in Li-doped ZnO. However, there are many proposals for this peculir novel ferroelectricity in ZnO. Scott and Zubko have pointed out the possibility of a classic electret mechanism for Li-doped ZnO[44]. Tagantsev discussed a Landau theory, where a crystal on cooling from a state with polar symmetry exhibits a maximum of dielec‐ tric permittivity and *D-E* hysteresis loops [45]. He proposed that these ferroelectric like phe‐ nomena corresponds to the case for Li-doped ZnO. Furthermore, the multiferroic behavior has been reported in impurity-doped ZnO [46]. At present, further detailed experiments

Dielectric properties and crystal structures and electron density distributions studied on

Zn*<sup>x</sup>*

Å. The clear change of Zn 3*d*-electron is observed. It suggests Zn 3*d*-electron of Zn1-

O transfers from Zn atom to the bonding. The positive electronic density is observed near the O atom. It corresponds to the antibonding orbital of O 2*p*-electron. These results suggest the existence of localized *d*-holes and *p*-electrons in Li-doped ZnO. It may be probable that the interaction between *d*-holes and *p*-electrons may be related to the appearance of the fer‐

For long time, the importance of electronic contribution has been pointed out in the field of ferroelectrics, since the simple superposition of electronic polarizability does not hold in many ferroelectric substances. Many efforts have been done in vain because of complexity of

ferroelectricity in ZnO is primarily due to electronic origin. The change in *d-p* hybridization caused by Li-substitution is considered to be responsible for this novel ferroelectricity and dielectric properties. Although the ferroelectric phase transition accompanies with structural

O by the precise X-ray diffraction are reviewed in comparison with fer‐

Te. It is considered that the appearance of

Li*x*

O are the order

should be expected to clarify the nature of Li-doped ZnO.

Ge*<sup>x</sup>*

Te and Cd1-*<sup>x</sup>*

distortions in usual ferroelectrics, the structural changes observed in Zn1-*<sup>x</sup>*

chuk et al [36].

252 Advances in Ferroelectrics

**8. Conclusion**

pure ZnO and Zn1-*<sup>x</sup>*

of 10-3

*x* Li*x* Li*x*

roelectric semiconductors Pb1-*<sup>x</sup>*

roelectricity in Li-doped ZnO.


[41] Yoshio K, Onodera A, Satoh H, Sakagami N and Yamashita H. Ferroelectrics 2002;

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[44] Glinchuk D M, Kirichenko V E, Stephanovich A V and Zaulychny Y B. Appl Phys

[45] Scott F J and Zubko P. IEEE Xplore 2005; ISE-12 (12th International Symposium on

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[42] Portengen T, Ostreich Th and Sham J L. Phys Rev 1996; B54: 17452.

[43] Falicov M L and Kimball C J. Phys Rev Lett 1969; 22: 997.

[46] Tagantsev K A. Appl Phys Let 2008; 93: 202905.

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[18] Laudise A R, Kolb D E and Caporaso J A. J Am Ceram Soc 1964; 47: 9.

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[26] Benguigui L, Weil R, Muranevich E, Chack A and Fredj E. J Appl Phys 1993; 74:513. [27] Terauchi H, Yoneda Y, Kasatani H, Sakaue K, Koshiba T, Murakami S, Kuroiwa Y, Noda Y, Sugai S, Nakashima S and Maeda H. Jpn J Appl Phys 1993; 32: 728.

[28] Onodera A, Yoshio K, Satoh H, Yamashita H and Sakagami N. Jpn. J. Phys. Phys.

[29] Onodera A, Tamaki N, Satoh H, Yamashita H and Sakai A. Ceramic Transactions

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**Chapter 12**

**Doping-Induced Ferroelectric Phase Transition and**

**Paraelectric Material Studied by Coherent Phonon**

There has been significant interest in a quantum paraelectric material strontium titanate (SrTiO3), and its lattice dynamics and unusual dielectric character have been extensively studied. In low temperatures, its dielectric constant increases up to about 30 000. The dielec‐ tric constant increases extraordinarily with decreasing temperature while the paraelectric phase is stabilized by quantum fluctuations without any ferroelectric phase transition even below the classical Curie temperature *Tc*=37 K [1]. In SrTiO3, a ferroelectric transition is easi‐ ly induced by a weak perturbation such as an uniaxial stress [2], an isotopic substitution of

SrTiO3 has a perovskite structure as shown in Fig. 1(a) and is known to undergo a structural phase transition at *Tc*=105 K [6]. The cubic (*Oh*) structure above *Tc*, where all phonon modes are Raman forbidden, changes into the tetragonal (*D*4*<sup>h</sup>*) structure below *Tc*, where Raman-al‐ lowed modes of symmetries *A*1*<sup>g</sup>* and *Eg* appear [7]. The phase transition is due to the collapse of the *Γ*25 mode at the *R* point of the high-temperature cubic Brillouin zone. Below *Tc*, the *R* point becomes the *Γ* point of the *D*4*<sup>h</sup>* phase. The phase transition is characterized by the soft‐

The distortion consists of an out-of-phase rotation of adjacent oxygen octahedra in the (100) planes [6]. The order parameter for the phase transition is inferred to be the angle of rotation of the oxygen octahedra. Only a small rotation of the oxygen octahedra is involved for the transition. The rotation angle for the oxygen octahedra varies from ~2° of arc near 0 K down

and reproduction in any medium, provided the original work is properly cited.

© 2013 Kohmoto; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

**Ultraviolet-Illumination Effect in a Quantum**

Additional information is available at the end of the chapter

oxygen 18 for oxygen 16 [3], and an impurity doping [4,5].

ening of phonons at the *R* point and concomitant doubling of the unit cell.

**Spectroscopy**

Toshiro Kohmoto

**1. Introduction**

http://dx.doi.org/10.5772/45744

**Doping-Induced Ferroelectric Phase Transition and Ultraviolet-Illumination Effect in a Quantum Paraelectric Material Studied by Coherent Phonon Spectroscopy**

### Toshiro Kohmoto

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/45744

### **1. Introduction**

There has been significant interest in a quantum paraelectric material strontium titanate (SrTiO3), and its lattice dynamics and unusual dielectric character have been extensively studied. In low temperatures, its dielectric constant increases up to about 30 000. The dielec‐ tric constant increases extraordinarily with decreasing temperature while the paraelectric phase is stabilized by quantum fluctuations without any ferroelectric phase transition even below the classical Curie temperature *Tc*=37 K [1]. In SrTiO3, a ferroelectric transition is easi‐ ly induced by a weak perturbation such as an uniaxial stress [2], an isotopic substitution of oxygen 18 for oxygen 16 [3], and an impurity doping [4,5].

SrTiO3 has a perovskite structure as shown in Fig. 1(a) and is known to undergo a structural phase transition at *Tc*=105 K [6]. The cubic (*Oh*) structure above *Tc*, where all phonon modes are Raman forbidden, changes into the tetragonal (*D*4*<sup>h</sup>*) structure below *Tc*, where Raman-al‐ lowed modes of symmetries *A*1*<sup>g</sup>* and *Eg* appear [7]. The phase transition is due to the collapse of the *Γ*25 mode at the *R* point of the high-temperature cubic Brillouin zone. Below *Tc*, the *R* point becomes the *Γ* point of the *D*4*<sup>h</sup>* phase. The phase transition is characterized by the soft‐ ening of phonons at the *R* point and concomitant doubling of the unit cell.

The distortion consists of an out-of-phase rotation of adjacent oxygen octahedra in the (100) planes [6]. The order parameter for the phase transition is inferred to be the angle of rotation of the oxygen octahedra. Only a small rotation of the oxygen octahedra is involved for the transition. The rotation angle for the oxygen octahedra varies from ~2° of arc near 0 K down

© 2013 Kohmoto; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

to zero at *Tc*=105 K; the transition is second order. At liquid nitrogen temperature, the rota‐ tion angle is about 1. 4° and the linear displacement of the oxygen ions about their high-tem‐ perature equilibrium positions is less than 0.003 nm. This oxygen octahedron motion can be described as a rotation only as a first approximation; the oxygen ions actually remain on the faces of each cube and therefore increase in separation from the titanium. Because the (100) planes are equivalent in the cubic phase, the distortion produces domains below *Tc* in which the [100], [010], or [001] axis becomes the unique tetragonal *c* axis.

Ca2+ ions are not likely to directly polarize the host matrix by an electrostatic mechanism. In‐ stead, the possible role of random fields in inducing the presence of disordered polar clusters

Doping-Induced Ferroelectric Phase Transition and Ultraviolet-Illumination Effect in a Quantum Paraelectric...

http://dx.doi.org/10.5772/45744

259

Recently, a gigantic change in the dielectric constant by an ultraviolet (UV) illumination was discovered [13,14], and a deeper interest has been taken in SrTiO3 again. The origin of the giant dielectric constants, however, has not yet been clarified. In Ca-doped SrTiO3, it was re‐ ported that a UV illumination causes a ferroelectric peak shift of the dielectric constant to‐ ward the lower temperature side [11]. In several ferroelectric materials such as BaTiO3 [15], SbSI [16], and oxygen-isotope-substituted SrTiO3 [17], the *Tc* reduction under a UV illumina‐

The optical information on the dielectric response is usually obtained from the experiments of Raman scattering or infrared spectroscopy. The usefulness of the investigation of low-fre‐ quency dielectric response by observing coherent phonons have also been demonstrated by the time-resolved study of the dynamics of phonons [18] and phonon polaritons [19]. At low frequencies this technique is very sensitive and provides a very good signal-to-noise ratio as compared to the conventional frequency-domain techniques while at higher frequencies a better performance will be achieved by using the conventional techniques. Therefore the co‐ herent phonon spectroscopy and the conventional frequency-domain techniques can be con‐

sidered to be complementary methods for the investigation of the dielectric response.

then a background-free signal of the first-order Raman scattering cannot be observed.

The observed signal of the Raman scattering [20] in SrTiO3 is very weak because the distortion from cubic structure in the low-temperature phase is very small. The intensity of the first-order Raman signal is of the same order of magnitude with many second-order Raman signals, and

Under a UV illumination, SrTiO3 and Ca-doped SrTiO3 show a broadband luminescence in the visible region originated from a relaxed excited state [21]. The coherent phonon spectro‐ scopy is not sensitive to the luminescence and a powerful technique to investigate UV-illu‐ mination effects in paraelectric materials as compared to the Raman-scattering measurement because in Raman-scattering experiments, it is not easy to separate Raman-scattering signals

In the present study, ultrafast polarization spectroscopy is used to observe the coherent opti‐ cal phonons in pure and Ca-doped SrTiO3, which are generated by femtosecond optical pulses through the process of impulsive stimulated Raman scattering [22,23]. Time-depend‐ ent linear birefringence induced by the generated coherent phonons is detected as a change of the polarization of probe pulses. High detection sensitivity of ~10−5 in polarization change has been achieved in our detection system. Damped oscillations of coherent phonons in SrTiO3 were observed below the structural phase-transition temperature (*Tc*=105 K), and temperature dependences of the phonon frequency and the relaxation rate are measured [24]. The mechanism of the phonon relaxation is discussed by using a population decay model, in which an optical phonon decays into two acoustic phonons due to an harmonic

was suggested, which is similar to polar nanoregions in relaxor materials.

tion has been observed.

from the luminescence.

phonon-phonon coupling.

**Figure 1.** (a) Perovskite structure in SrTiO3. (b) Doped Ca ions are substituted for the Sr ions in Ca-doped SrTiO3.

According to the measurement of dielectric constants, Sr1−*<sup>x</sup>*Ca*x*TiO3 undergoes a ferroelectric transition above the critical Ca concentration *xc*=0. 0018, where doped Ca ions are substituted for the Sr ions [8–11] as shown in Fig. 1(b). The cubic structure above the structural phase-tran‐ sition temperature *Tc*1 changes in to the tetragonal structure below *Tc*1 and into the rhombohe‐ dral structure below the ferroelectric transition temperature *Tc*2. The structural phase transition at *Tc*1 is also antiferrodistortive as in pure SrTiO3 at 105 K [8]. As in the case of impurity systems, Li-doped KTaO3 and Nb-doped KTaO3 [4], off-centered impurity ions are supposed. Their po‐ larized dipole moments show a ferroelectric instability below the ferroelectric transition tem‐ perature [4]. In the case of Ca-dopedSrTiO3, a spontaneous polarization occurs along [110] directions in the *c* plane, where the tetragonal (*D*4*<sup>h</sup>*) symmetry is lowered to *C*2*<sup>v</sup>* [8]. With de‐ creasing temperature a ferroelectric ordering process dominates, that is, due to the thermal growth of the polarization clouds surrounding the off-center Ca2+ dipoles [8]. The system be‐ haves like a super paraelectric as the ferroelectric nano-ordered regions contain disordered cluster like regions. The investigation by x-ray and neutron diffractions and first-principles cal‐ culations [12] suggests that polar instabilities originating from the off-center displacements of Ca2+ ions are not likely to directly polarize the host matrix by an electrostatic mechanism. In‐ stead, the possible role of random fields in inducing the presence of disordered polar clusters was suggested, which is similar to polar nanoregions in relaxor materials.

to zero at *Tc*=105 K; the transition is second order. At liquid nitrogen temperature, the rota‐ tion angle is about 1. 4° and the linear displacement of the oxygen ions about their high-tem‐ perature equilibrium positions is less than 0.003 nm. This oxygen octahedron motion can be described as a rotation only as a first approximation; the oxygen ions actually remain on the faces of each cube and therefore increase in separation from the titanium. Because the (100) planes are equivalent in the cubic phase, the distortion produces domains below *Tc* in which

**Figure 1.** (a) Perovskite structure in SrTiO3. (b) Doped Ca ions are substituted for the Sr ions in Ca-doped SrTiO3.

According to the measurement of dielectric constants, Sr1−*<sup>x</sup>*Ca*x*TiO3 undergoes a ferroelectric transition above the critical Ca concentration *xc*=0. 0018, where doped Ca ions are substituted for the Sr ions [8–11] as shown in Fig. 1(b). The cubic structure above the structural phase-tran‐ sition temperature *Tc*1 changes in to the tetragonal structure below *Tc*1 and into the rhombohe‐ dral structure below the ferroelectric transition temperature *Tc*2. The structural phase transition at *Tc*1 is also antiferrodistortive as in pure SrTiO3 at 105 K [8]. As in the case of impurity systems, Li-doped KTaO3 and Nb-doped KTaO3 [4], off-centered impurity ions are supposed. Their po‐ larized dipole moments show a ferroelectric instability below the ferroelectric transition tem‐ perature [4]. In the case of Ca-dopedSrTiO3, a spontaneous polarization occurs along [110] directions in the *c* plane, where the tetragonal (*D*4*<sup>h</sup>*) symmetry is lowered to *C*2*<sup>v</sup>* [8]. With de‐ creasing temperature a ferroelectric ordering process dominates, that is, due to the thermal growth of the polarization clouds surrounding the off-center Ca2+ dipoles [8]. The system be‐ haves like a super paraelectric as the ferroelectric nano-ordered regions contain disordered cluster like regions. The investigation by x-ray and neutron diffractions and first-principles cal‐ culations [12] suggests that polar instabilities originating from the off-center displacements of

the [100], [010], or [001] axis becomes the unique tetragonal *c* axis.

258 Advances in Ferroelectrics

Recently, a gigantic change in the dielectric constant by an ultraviolet (UV) illumination was discovered [13,14], and a deeper interest has been taken in SrTiO3 again. The origin of the giant dielectric constants, however, has not yet been clarified. In Ca-doped SrTiO3, it was re‐ ported that a UV illumination causes a ferroelectric peak shift of the dielectric constant to‐ ward the lower temperature side [11]. In several ferroelectric materials such as BaTiO3 [15], SbSI [16], and oxygen-isotope-substituted SrTiO3 [17], the *Tc* reduction under a UV illumina‐ tion has been observed.

The optical information on the dielectric response is usually obtained from the experiments of Raman scattering or infrared spectroscopy. The usefulness of the investigation of low-fre‐ quency dielectric response by observing coherent phonons have also been demonstrated by the time-resolved study of the dynamics of phonons [18] and phonon polaritons [19]. At low frequencies this technique is very sensitive and provides a very good signal-to-noise ratio as compared to the conventional frequency-domain techniques while at higher frequencies a better performance will be achieved by using the conventional techniques. Therefore the co‐ herent phonon spectroscopy and the conventional frequency-domain techniques can be con‐ sidered to be complementary methods for the investigation of the dielectric response.

The observed signal of the Raman scattering [20] in SrTiO3 is very weak because the distortion from cubic structure in the low-temperature phase is very small. The intensity of the first-order Raman signal is of the same order of magnitude with many second-order Raman signals, and then a background-free signal of the first-order Raman scattering cannot be observed.

Under a UV illumination, SrTiO3 and Ca-doped SrTiO3 show a broadband luminescence in the visible region originated from a relaxed excited state [21]. The coherent phonon spectro‐ scopy is not sensitive to the luminescence and a powerful technique to investigate UV-illu‐ mination effects in paraelectric materials as compared to the Raman-scattering measurement because in Raman-scattering experiments, it is not easy to separate Raman-scattering signals from the luminescence.

In the present study, ultrafast polarization spectroscopy is used to observe the coherent opti‐ cal phonons in pure and Ca-doped SrTiO3, which are generated by femtosecond optical pulses through the process of impulsive stimulated Raman scattering [22,23]. Time-depend‐ ent linear birefringence induced by the generated coherent phonons is detected as a change of the polarization of probe pulses. High detection sensitivity of ~10−5 in polarization change has been achieved in our detection system. Damped oscillations of coherent phonons in SrTiO3 were observed below the structural phase-transition temperature (*Tc*=105 K), and temperature dependences of the phonon frequency and the relaxation rate are measured [24]. The mechanism of the phonon relaxation is discussed by using a population decay model, in which an optical phonon decays into two acoustic phonons due to an harmonic phonon-phonon coupling.

The doping-induced ferroelectric phase transition in Ca-doped SrTiO3 is investigated by ob‐ serving the birefringence and coherent phonons [25]. In the birefringence measurement, the structural and the ferroelectric phase-transition temperatures are examined. In the observa‐ tion of coherent phonons, the soft phonon modes related to the structural (*Tc*1=180 K) and the ferroelectric (*Tc*2=28 K) phase transitions are studied, and their frequencies are obtained from the observed coherent phonon signals. The behavior of the softening toward each phase-transition temperature and the UV-illumination effect on the ferroelectric transition are discussed. In addition to the ferroelectric phase transition at *Tc*2=28 K, another structural deformation at 25 K is found. A shift of the ferroelectric phase-transition temperature under the UV illumination and a decrease in the phonon frequencies after the UV illumination are found. We show the approach in the time domain is very useful for the study of the soft phonon modes and their UV-illumination effect in dielectric materials.

**2.1. Birefringence measurement**

**Figure 3.** Construction of the polarimeter.

by the polarimeter as the signal of the lattice deformation.

**2.2. Observation of coherent phonons**

In the birefringence measurement, the linearly polarized probe beam is provided by a

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The construction of the polarimeter is shown in Fig. 3. The polarimeter [26,27] detects the rotation of polarization plane of a light beam. A linearly-polarized beam is split by a polar‐ ized beam splitter (PBS) and incident on the two photodiodes (PD) whose photocurrents are subtracted at a resistor (R). When the polarized beam splitter is mounted at an angle of 45o to the plane of polarization of the light beam, the two photocurrents cancel. If the plane of polarization rotates, the two currents do not cancel and the voltage appears at the resistor.

In the present experiment, the birefringence generated by the lattice deformation is detected as the change in polarization of the probe beam using a quarterwave plate and the polarime‐ ter. The birefringence generated in the sample changes the linear polarization before trans‐ mission to an elliptical polarization after transmission. The linearly-polarized probe beam is considered to be a superposition of two circularly-polarized components which have the op‐ posite polarizations and the same intensities. The generated birefringence destroys the in‐ tensity balance between the two components. The two circularly-polarized beams are transformed by the quaterwave plate to two linearly-polarized beams whose polarizations are crossed each other, and the unbalance of circular polarization is transformed to the un‐ balance of linear polarization or the rotation of polarization plane. This rotation is detected

Coherent phonons are observed by ultrafast polarization spectroscopy with the pump-probe technique. The experimental setup for coherent phonon spectroscopy is shown in Fig. 4. Coher‐ ent phonons are generated by femtosecond optical pulses through the process of impulsive

Nd:YAG laser (532 nm, cw) and is perpendicular to the (001) surface of the sample.

### **2. Experiment**

The experiments are performed on single crystals of pure SrTiO3 and Ca-doped SrTiO3 with the Ca concentration of *x*=0.011. SrTiO3 was obtained commercially (MTI Corporation) and Ca-doped SrTiO3 was grown by the floating zone method [11]. The thickness of the sample is 1. 0 and 0.5 mm for the pure and Ca-doped crystals, respectively. In Ca-dopedSrTiO3, the structural phase-transition (*Oh→D*4*<sup>h</sup>*) temperature, *Tc*1=180 K, is obtained from the result of the birefringence measurement in section 3. The ferroelectric phase-transition temperature, *Tc*2=28 K, was determined by the measurement of dielectric constants [11]. The value of *x* was determined by the empirical relation of Bednorz and Müller [5].

Schematic diagram of the experiment of polarization spectroscopy is shown in Fig. 2. The change in optical anisotoropy (birefringence) is detected by a polarimeter as the change in the polarization of the probe light (ellipticity). In the birefringence measurement, the bire‐ fringence generated by the lattice deformation is detected by a continuous-wave (cw) probe beam with no pump beam. In the coherent phonon spectroscopy, the transient birefringence due to the coherent phonons generated by a pump pulse is detected by a probe pulse.

**Figure 2.** Schematic diagram of the experiment of polarization spectroscopy.

### **2.1. Birefringence measurement**

The doping-induced ferroelectric phase transition in Ca-doped SrTiO3 is investigated by ob‐ serving the birefringence and coherent phonons [25]. In the birefringence measurement, the structural and the ferroelectric phase-transition temperatures are examined. In the observa‐ tion of coherent phonons, the soft phonon modes related to the structural (*Tc*1=180 K) and the ferroelectric (*Tc*2=28 K) phase transitions are studied, and their frequencies are obtained from the observed coherent phonon signals. The behavior of the softening toward each phase-transition temperature and the UV-illumination effect on the ferroelectric transition are discussed. In addition to the ferroelectric phase transition at *Tc*2=28 K, another structural deformation at 25 K is found. A shift of the ferroelectric phase-transition temperature under the UV illumination and a decrease in the phonon frequencies after the UV illumination are found. We show the approach in the time domain is very useful for the study of the soft

The experiments are performed on single crystals of pure SrTiO3 and Ca-doped SrTiO3 with the Ca concentration of *x*=0.011. SrTiO3 was obtained commercially (MTI Corporation) and Ca-doped SrTiO3 was grown by the floating zone method [11]. The thickness of the sample is 1. 0 and 0.5 mm for the pure and Ca-doped crystals, respectively. In Ca-dopedSrTiO3, the structural phase-transition (*Oh→D*4*<sup>h</sup>*) temperature, *Tc*1=180 K, is obtained from the result of the birefringence measurement in section 3. The ferroelectric phase-transition temperature, *Tc*2=28 K, was determined by the measurement of dielectric constants [11]. The value of *x*

Schematic diagram of the experiment of polarization spectroscopy is shown in Fig. 2. The change in optical anisotoropy (birefringence) is detected by a polarimeter as the change in the polarization of the probe light (ellipticity). In the birefringence measurement, the bire‐ fringence generated by the lattice deformation is detected by a continuous-wave (cw) probe beam with no pump beam. In the coherent phonon spectroscopy, the transient birefringence due to the coherent phonons generated by a pump pulse is detected by a probe pulse.

phonon modes and their UV-illumination effect in dielectric materials.

was determined by the empirical relation of Bednorz and Müller [5].

**Figure 2.** Schematic diagram of the experiment of polarization spectroscopy.

**2. Experiment**

260 Advances in Ferroelectrics

In the birefringence measurement, the linearly polarized probe beam is provided by a Nd:YAG laser (532 nm, cw) and is perpendicular to the (001) surface of the sample.

The construction of the polarimeter is shown in Fig. 3. The polarimeter [26,27] detects the rotation of polarization plane of a light beam. A linearly-polarized beam is split by a polar‐ ized beam splitter (PBS) and incident on the two photodiodes (PD) whose photocurrents are subtracted at a resistor (R). When the polarized beam splitter is mounted at an angle of 45o to the plane of polarization of the light beam, the two photocurrents cancel. If the plane of polarization rotates, the two currents do not cancel and the voltage appears at the resistor.

**Figure 3.** Construction of the polarimeter.

In the present experiment, the birefringence generated by the lattice deformation is detected as the change in polarization of the probe beam using a quarterwave plate and the polarime‐ ter. The birefringence generated in the sample changes the linear polarization before trans‐ mission to an elliptical polarization after transmission. The linearly-polarized probe beam is considered to be a superposition of two circularly-polarized components which have the op‐ posite polarizations and the same intensities. The generated birefringence destroys the in‐ tensity balance between the two components. The two circularly-polarized beams are transformed by the quaterwave plate to two linearly-polarized beams whose polarizations are crossed each other, and the unbalance of circular polarization is transformed to the un‐ balance of linear polarization or the rotation of polarization plane. This rotation is detected by the polarimeter as the signal of the lattice deformation.

### **2.2. Observation of coherent phonons**

Coherent phonons are observed by ultrafast polarization spectroscopy with the pump-probe technique. The experimental setup for coherent phonon spectroscopy is shown in Fig. 4. Coher‐ ent phonons are generated by femtosecond optical pulses through the process of impulsive

stimulated Raman scattering [22,23], and are detected by monitoring the time-dependent ani‐ sotropy of refractive index induced by the pump pulse. The pump pulse is provided by a Ti: sapphire regenerative amplifier whose wavelength, pulse energy, and pulse width at the sam‐ ple are 790 nm, 2 μJ, and 0.2 ps, respectively. The probe pulse is provided by an optical para‐ metric amplifier whose wavelength, pulse energy, and pulse width are 690 nm, 0. 1 μJ, and 0.2 ps, respectively. The repetition rate of the pulses is 1kHz. The linearly polarized pump and probe beams are nearly collinear and perpendicular to the (001) surface of the sample, and are focused on the sample in a temperature-controlled refrigerator. The waist size of the beams at the sample is about 0.5 mm. 3 polarimeter. The birefringence generated in the sample changes the linear polarization 4 before transmission to an elliptical polarization after transmission. The linearly-polarized 5 probe beam is considered to be a superposition of two circularly-polarized components 6 which have the opposite polarizations and the same intensities. The generated birefringence 7 destroys the intensity balance between the two components. The two circularly-polarized 8 beams are transformed by the quaterwave plate to two linearly-polarized beams whose 9 polarizations are crossed each other, and the unbalance of circular polarization is 10 transformed to the unbalance of linear polarization or the rotation of polarization plane. 11 This rotation is detected by the polarimeter as the signal of the lattice deformation. 12 **2.2. Observation of coherent phonons** 

2 detected as the change in polarization of the probe beam using a quarterwave plate and the

5

The source of UV illumination is provided by the second harmonics (380 nm, 3. 3 eV) of the output from another mode-locked Ti: sapphire laser, whose energy is larger than the optical band gap of SrTiO3 (3.2 eV). Since the repetition rate of the UV pulses is 80 MHz, this UV illumination can be considered to be continuous in the present experiment, where the UV-

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Figure 5 shows the temperature dependences of the change in birefringence in SrTiO3 and Ca-doped SrTiO3 between 4.5 and 250 K, where the polarization plane of the probe light is along the [110]and [100]axes. In SrTiO3, a change in birefringence appears below *Tc*=105 K, which is the temperature of the structural phase transition, and is increased as the tempera‐ ture is decreased. In Ca-doped SrTiO3, large changes in birefringence come out at *Tc*1=180 K, which is the temperature of the structural phase transition, and at *Tc*2=28 K, which is the temperature of the ferroelectric phase transition. The change in birefringence is increased as the temperature is decreased from *Tc*1 to *Tc*2, as well as the case in SrTiO3, and shows a in‐ crease in the gradient for both axes around *Tc*2. Below *Tc*2, another kind of lattice distortion is added, and the peaks of Δ*n* due to the competition between the two kinds of lattice distor‐ tion appear. In both SrTiO3 and Ca-dopedSrTiO3, a cusp on the birefringence curve appears around *Tc*=105 K and *Tc*1=180 K, respectively, because of the fluctuation associated with the

**Figure 5.** Temperature dependences of the change in birefringence between 4. 5 and 250 K in SrTiO3 and Ca-doped SrTiO3. The polarization plane of the probe light for SrTiO3 is along the [110] axis (solid circles). That for Ca-doped

SrTiO3 is along the [110] axis (solid squares) and along the [100] axis (open triangles).

illumination effect appearing more than one minute after is studied.

**3. Birefringence measurement**

second-order structural phase transition [28].

38 **Fig. 4.** Experimental setup for coherent phonon spectroscopy. **Figure 4.** Experimental setup for coherent phonon spectroscopy.

39 Coherent phonons are observed by ultrafast polarization spectroscopy with the pump-40 probe technique. The experimental setup for coherent phonon spectroscopy is shown in Fig. 41 4. Coherent phonons are generated by femtosecond optical pulses through the process of 42 impulsive stimulated Raman scattering [22,23], and are detected by monitoring the time-43 dependent anisotropy of refractive index induced by the pump pulse. The pump pulse is 44 provided by a Ti:sapphire regenerative amplifier whose wavelength, pulse energy, and The induced anisotropy of refractive index is detected by the polarimeter with a quarter-wave plate as the polarization change in the probe pulse. The plane of polarization of the probe pulse is tilted by 45° from that of the pump pulse. The two different wavelengths for the pump and probe pulses and pump-cut filters are used to eliminate the leak of the pump light from the in‐ put of the polarimeter. The time evolution of the signal is observed by changing the optical de‐ lay between the pump and probe pulses. The pump pulse is switched on and off shot by shot by using a photoelastic modulator, a quarter-wave plate, and a polarizer, and the output from the polarimeter is lock-in detected to improve the signal-to-noise ratio.

The source of UV illumination is provided by the second harmonics (380 nm, 3. 3 eV) of the output from another mode-locked Ti: sapphire laser, whose energy is larger than the optical band gap of SrTiO3 (3.2 eV). Since the repetition rate of the UV pulses is 80 MHz, this UV illumination can be considered to be continuous in the present experiment, where the UVillumination effect appearing more than one minute after is studied.

### **3. Birefringence measurement**

stimulated Raman scattering [22,23], and are detected by monitoring the time-dependent ani‐ sotropy of refractive index induced by the pump pulse. The pump pulse is provided by a Ti: sapphire regenerative amplifier whose wavelength, pulse energy, and pulse width at the sam‐ ple are 790 nm, 2 μJ, and 0.2 ps, respectively. The probe pulse is provided by an optical para‐ metric amplifier whose wavelength, pulse energy, and pulse width are 690 nm, 0. 1 μJ, and 0.2 ps, respectively. The repetition rate of the pulses is 1kHz. The linearly polarized pump and probe beams are nearly collinear and perpendicular to the (001) surface of the sample, and are focused on the sample in a temperature-controlled refrigerator. The waist size of the beams at

 In the present experiment, the birefringence generated by the lattice deformation is detected as the change in polarization of the probe beam using a quarterwave plate and the polarimeter. The birefringence generated in the sample changes the linear polarization before transmission to an elliptical polarization after transmission. The linearly-polarized probe beam is considered to be a superposition of two circularly-polarized components which have the opposite polarizations and the same intensities. The generated birefringence destroys the intensity balance between the two components. The two circularly-polarized beams are transformed by the quaterwave plate to two linearly-polarized beams whose polarizations are crossed each other, and the unbalance of circular polarization is transformed to the unbalance of linear polarization or the rotation of polarization plane.

11 This rotation is detected by the polarimeter as the signal of the lattice deformation.

Mode-Locked Ti:Sapphire

38 **Fig. 4.** Experimental setup for coherent phonon spectroscopy.

polarimeter is lock-in detected to improve the signal-to-noise ratio.

λ/4

**Figure 4.** Experimental setup for coherent phonon spectroscopy.

 Coherent phonons are observed by ultrafast polarization spectroscopy with the pump- probe technique. The experimental setup for coherent phonon spectroscopy is shown in Fig. 4. Coherent phonons are generated by femtosecond optical pulses through the process of impulsive stimulated Raman scattering [22,23], and are detected by monitoring the time- dependent anisotropy of refractive index induced by the pump pulse. The pump pulse is provided by a Ti:sapphire regenerative amplifier whose wavelength, pulse energy, and

The induced anisotropy of refractive index is detected by the polarimeter with a quarter-wave plate as the polarization change in the probe pulse. The plane of polarization of the probe pulse is tilted by 45° from that of the pump pulse. The two different wavelengths for the pump and probe pulses and pump-cut filters are used to eliminate the leak of the pump light from the in‐ put of the polarimeter. The time evolution of the signal is observed by changing the optical de‐ lay between the pump and probe pulses. The pump pulse is switched on and off shot by shot by using a photoelastic modulator, a quarter-wave plate, and a polarizer, and the output from the

sample

delay

1 kHz, 0.2 ps Ti:Sapphire

Regen. Amp.

probe

45<sup>o</sup>

polarizer

Photoelastic Modulator

690 nm

790 nm

Optical Parametric Amp.

λ/4

pump

5

the sample is about 0.5 mm.

12 **2.2. Observation of coherent phonons** 

Polarimeter

Lock-in Amp.

Computer

Running Title

262 Advances in Ferroelectrics

Figure 5 shows the temperature dependences of the change in birefringence in SrTiO3 and Ca-doped SrTiO3 between 4.5 and 250 K, where the polarization plane of the probe light is along the [110]and [100]axes. In SrTiO3, a change in birefringence appears below *Tc*=105 K, which is the temperature of the structural phase transition, and is increased as the tempera‐ ture is decreased. In Ca-doped SrTiO3, large changes in birefringence come out at *Tc*1=180 K, which is the temperature of the structural phase transition, and at *Tc*2=28 K, which is the temperature of the ferroelectric phase transition. The change in birefringence is increased as the temperature is decreased from *Tc*1 to *Tc*2, as well as the case in SrTiO3, and shows a in‐ crease in the gradient for both axes around *Tc*2. Below *Tc*2, another kind of lattice distortion is added, and the peaks of Δ*n* due to the competition between the two kinds of lattice distor‐ tion appear. In both SrTiO3 and Ca-dopedSrTiO3, a cusp on the birefringence curve appears around *Tc*=105 K and *Tc*1=180 K, respectively, because of the fluctuation associated with the second-order structural phase transition [28].

**Figure 5.** Temperature dependences of the change in birefringence between 4. 5 and 250 K in SrTiO3 and Ca-doped SrTiO3. The polarization plane of the probe light for SrTiO3 is along the [110] axis (solid circles). That for Ca-doped SrTiO3 is along the [110] axis (solid squares) and along the [100] axis (open triangles).

### **4. Coherent phonon spectroscopy in SrTiO3**

### **4.1. Angular dependence of the coherent phonon signal**

Figure 6 shows the transient birefringence in SrTiO3 observed at 6 K for the 0° pumping, where the polarization direction of the pump pulse is parallel to the [100] axis of the crystal. Vertical axis is the ellipticity *η* in electric-field amplitude of the transmitted probe pulse. At zero delay, a large signal due to the optical Kerr effect, whose width is determined by the pulse width, appears. After that, damped oscillations of coherent phonons are observed as shown in Fig. 6(b), where the vertical axis is enlarged by 50 from that of Fig. 6(a). The change of the polarization for the oscillation amplitude of the coherent phonon signal is 4×10−4 of the electric-field amplitudeof the probe pulse, which corresponds to the change Δ*n*=7×10−8 of the refractive index. In our detection system, polarization change of ~10−5 in the electric field amplitude can be detected.

pumping angles both frequency components coexist in the coherent phonon signal. From the oscillation frequencies the 1.35 THz component is considered to correspond to the *A*1*<sup>g</sup>*

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**Figure 7.** (a) Coherent phonon signals in SrTiO3 observed at 6 K for the 0°, 15°, and 45° pumping, where the angle between the [100] axis of the crystal and the polarization direction of the pump pulse is changed. (b) Fourier trans‐

In addition to the oscillation signal there exists a dc component. The dc component has a maximum amplitude for the 0° pumping and minimum amplitude for the 45° pumping. However, the creation mechanism is not clear at present. In the following we pay attention

The temperature dependence of the coherent phonon signal in SrTiO3 observed for the 0° pumping, which corresponds to the *A*1*<sup>g</sup>* mode, is shown in Fig. 8(a). The oscillation period and the relaxation time of coherent phonons at 10 K are 0.7 and 12 ps. As the temperature is increased, the oscillation period becomes longer and the relaxation time becomes shorter. At

mode, and the 0.4 THz component to the *Eg* mode [7].

form of the coherent phonon signals in (a).

to the oscillation component.

**4.2. Temperature dependence of the coherent phonon signal**

**Figure 6.** Transient birefringence in SrTiO3 observed at 6 K for the 0° pumping. The vertical axis is the ellipticity η in electric-field amplitude of the transmitted probe pulse. (a) A large signal due to the optical Kerr effect appears at zero delay. (b) Damped oscillations of coherent phonons follow after the Kerr signal. The vertical axis of (b) is enlarged by 50 from that of (a).

Angular dependence of the coherent phonon signal in SrTiO3 observed at 6 K is shown in Fig. 7(a), where the angle between the [100] axis of the crystal and the polarization direction of the pump pulse is 0°, 15°, and 45°. The angle between the polarization directions of the pump and probe pulses is fixed to 45°. The 0.7 ps period signal for the 0° pumping disap‐ pears for the 45°pumping, where the 2.3 ps period signal appears. The Fourier transform of the coherent phonon signals in Fig. 7(a) is shown in Fig. 7(b). Oscillation frequency of the signal for the 0° pumping is 1.35 THz, and that for the 45° pumping is 0.4 THz. For other pumping angles both frequency components coexist in the coherent phonon signal. From the oscillation frequencies the 1.35 THz component is considered to correspond to the *A*1*<sup>g</sup>* mode, and the 0.4 THz component to the *Eg* mode [7].

**4. Coherent phonon spectroscopy in SrTiO3**

264 Advances in Ferroelectrics

**4.1. Angular dependence of the coherent phonon signal**

electric field amplitude can be detected.

50 from that of (a).

Figure 6 shows the transient birefringence in SrTiO3 observed at 6 K for the 0° pumping, where the polarization direction of the pump pulse is parallel to the [100] axis of the crystal. Vertical axis is the ellipticity *η* in electric-field amplitude of the transmitted probe pulse. At zero delay, a large signal due to the optical Kerr effect, whose width is determined by the pulse width, appears. After that, damped oscillations of coherent phonons are observed as shown in Fig. 6(b), where the vertical axis is enlarged by 50 from that of Fig. 6(a). The change of the polarization for the oscillation amplitude of the coherent phonon signal is 4×10−4 of the electric-field amplitudeof the probe pulse, which corresponds to the change Δ*n*=7×10−8 of the refractive index. In our detection system, polarization change of ~10−5 in the

**Figure 6.** Transient birefringence in SrTiO3 observed at 6 K for the 0° pumping. The vertical axis is the ellipticity η in electric-field amplitude of the transmitted probe pulse. (a) A large signal due to the optical Kerr effect appears at zero delay. (b) Damped oscillations of coherent phonons follow after the Kerr signal. The vertical axis of (b) is enlarged by

Angular dependence of the coherent phonon signal in SrTiO3 observed at 6 K is shown in Fig. 7(a), where the angle between the [100] axis of the crystal and the polarization direction of the pump pulse is 0°, 15°, and 45°. The angle between the polarization directions of the pump and probe pulses is fixed to 45°. The 0.7 ps period signal for the 0° pumping disap‐ pears for the 45°pumping, where the 2.3 ps period signal appears. The Fourier transform of the coherent phonon signals in Fig. 7(a) is shown in Fig. 7(b). Oscillation frequency of the signal for the 0° pumping is 1.35 THz, and that for the 45° pumping is 0.4 THz. For other

**Figure 7.** (a) Coherent phonon signals in SrTiO3 observed at 6 K for the 0°, 15°, and 45° pumping, where the angle between the [100] axis of the crystal and the polarization direction of the pump pulse is changed. (b) Fourier trans‐ form of the coherent phonon signals in (a).

In addition to the oscillation signal there exists a dc component. The dc component has a maximum amplitude for the 0° pumping and minimum amplitude for the 45° pumping. However, the creation mechanism is not clear at present. In the following we pay attention to the oscillation component.

### **4.2. Temperature dependence of the coherent phonon signal**

The temperature dependence of the coherent phonon signal in SrTiO3 observed for the 0° pumping, which corresponds to the *A*1*<sup>g</sup>* mode, is shown in Fig. 8(a). The oscillation period and the relaxation time of coherent phonons at 10 K are 0.7 and 12 ps. As the temperature is increased, the oscillation period becomes longer and the relaxation time becomes shorter. At *Tc*=105 K, the phase transition point, the oscillation disappears. Above *Tc* no signal of coher‐ ent phonons is observed. The temperature dependence of the coherent phonon signal in SrTiO3observed for the 45° pumping, which corresponds to the *Eg* mode, is shown in Fig. 8(b). The oscillation period and the relaxation time of coherent phonons at 10 K are 2.3 and 45 ps. Similar behavior to that of the *A*1*<sup>g</sup>* mode was observed as the temperature was in‐ creased.

herent phonon signal below *Tc* is shown in Fig. 9. The diamonds are the oscillation frequency for the *A*1*<sup>g</sup>* mode, and the squares are that for the *Eg* mode. As the temperature is increased from 6 K, the oscillation frequencies decrease and approach zero at the phase tran‐ sition temperature *Tc* for both modes. This result is consistent with the temperature depend‐ ence of phonon frequency observed by Raman scattering [7]. The solid curves describe a temperature dependence of the form *ω* ∝(*Tc* - *T* )*n*. The experimental results for the temper‐ ature region between 50 K and *Tc* are explained well by *n*=0.4 for both modes, while those

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**Figure 9.** Temperature dependence of the oscillation frequency in SrTiO3 obtained from the coherent phonon signal below *Tc*. The diamonds are the oscillation frequency for the *A*1*g* mode, and the squares are that for the *Eg* mode. The solid curves describe a temperature dependence of the form ω ∝(*Tc* - *T* )*n*, where *Tc*=105 K and *n*=0.4 for both modes.

The intensity of the first-order Raman-scattering signal in SrTiO3 is very weak and is of the same order of magnitude as many second-order Raman-scattering signals because the dis‐ tortion from the cubic structure in the low-temperature phase is very small. Then observa‐ tion of a background-free signal of first-order Raman scattering is not easy, and information on the relaxation, or the spectral width, is not given in a study of Raman scattering [20]. By the present method of coherent phonon spectroscopy in the time domain, on the other hand, background-free damped oscillations can be observed directly, and the oscillation frequency

The temperature dependence of the relaxation rate in SrTiO3 obtained from the observed co‐ herent phonon signal below *Tc* is shown in Fig. 10. The diamonds in Fig. 10(a) are the relaxa‐ tion rate for the *A*1*<sup>g</sup>* mode and the squares in Fig. 10(b) are that for the *Eg* mode. The

below 40 K deviate from that form.

and the relaxation rate can be obtained accurately.

relaxation rates increase as the temperature is increased.

**4.4. Relaxation rates**

**Figure 8.** Temperature dependence of the coherent phonon signals in SrTiO3 observed (a) for the 0° pumping which corresponds to the *A*1*g* mode and (b) for the 45° pumping which corresponds to the *Eg* mode.

### **4.3. Phonon frequencies**

The observed coherent phonon signal *S*(*t*) is expressed well by the damped oscillation

*S*(*t*)= *A*e-*γ<sup>t</sup>* sin *ωt* , (1)

where *ω* is the oscillation frequency and *γ* is the relaxation rate. This sine-type function is expected for phonons induced by impulsive stimulated Raman scattering [22,23]. The tem‐ perature dependence of the oscillation frequency in SrTiO3 obtained from the observed co‐ herent phonon signal below *Tc* is shown in Fig. 9. The diamonds are the oscillation frequency for the *A*1*<sup>g</sup>* mode, and the squares are that for the *Eg* mode. As the temperature is increased from 6 K, the oscillation frequencies decrease and approach zero at the phase tran‐ sition temperature *Tc* for both modes. This result is consistent with the temperature depend‐ ence of phonon frequency observed by Raman scattering [7]. The solid curves describe a temperature dependence of the form *ω* ∝(*Tc* - *T* )*n*. The experimental results for the temper‐ ature region between 50 K and *Tc* are explained well by *n*=0.4 for both modes, while those below 40 K deviate from that form.

**Figure 9.** Temperature dependence of the oscillation frequency in SrTiO3 obtained from the coherent phonon signal below *Tc*. The diamonds are the oscillation frequency for the *A*1*g* mode, and the squares are that for the *Eg* mode. The solid curves describe a temperature dependence of the form ω ∝(*Tc* - *T* )*n*, where *Tc*=105 K and *n*=0.4 for both modes.

The intensity of the first-order Raman-scattering signal in SrTiO3 is very weak and is of the same order of magnitude as many second-order Raman-scattering signals because the dis‐ tortion from the cubic structure in the low-temperature phase is very small. Then observa‐ tion of a background-free signal of first-order Raman scattering is not easy, and information on the relaxation, or the spectral width, is not given in a study of Raman scattering [20]. By the present method of coherent phonon spectroscopy in the time domain, on the other hand, background-free damped oscillations can be observed directly, and the oscillation frequency and the relaxation rate can be obtained accurately.

### **4.4. Relaxation rates**

*Tc*=105 K, the phase transition point, the oscillation disappears. Above *Tc* no signal of coher‐ ent phonons is observed. The temperature dependence of the coherent phonon signal in SrTiO3observed for the 45° pumping, which corresponds to the *Eg* mode, is shown in Fig. 8(b). The oscillation period and the relaxation time of coherent phonons at 10 K are 2.3 and 45 ps. Similar behavior to that of the *A*1*<sup>g</sup>* mode was observed as the temperature was in‐

**Figure 8.** Temperature dependence of the coherent phonon signals in SrTiO3 observed (a) for the 0° pumping which

where *ω* is the oscillation frequency and *γ* is the relaxation rate. This sine-type function is expected for phonons induced by impulsive stimulated Raman scattering [22,23]. The tem‐ perature dependence of the oscillation frequency in SrTiO3 obtained from the observed co‐

sin *ωt* , (1)

The observed coherent phonon signal *S*(*t*) is expressed well by the damped oscillation

*S*(*t*)= *A*e-*γ<sup>t</sup>*

corresponds to the *A*1*g* mode and (b) for the 45° pumping which corresponds to the *Eg* mode.

**4.3. Phonon frequencies**

creased.

266 Advances in Ferroelectrics

The temperature dependence of the relaxation rate in SrTiO3 obtained from the observed co‐ herent phonon signal below *Tc* is shown in Fig. 10. The diamonds in Fig. 10(a) are the relaxa‐ tion rate for the *A*1*<sup>g</sup>* mode and the squares in Fig. 10(b) are that for the *Eg* mode. The relaxation rates increase as the temperature is increased.

tors. Schematic diagram of this down-conversion process is shown in Fig. 11. The tempera‐

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2

**Figure 11.** Schematic diagram of the down-conversion processe (overtone channel) in the an harmonic decay model of optical phonons. The initial ω<sup>0</sup> optical phonon with wave vector *k*≅ 0 decays into two ω<sup>0</sup> / 2 acoustic phonons, with

In ordinary materials the temperature dependence of the phonon frequency is small, and a theoretical curve with a fixed value of phonon frequency fits the experimental data well. In SrTiO3, however, the phonon frequencies are changed greatly as the temperature is in‐ creased; thus a frequency change has to be considered. The solid curves in Fig. 10 show the theoretical curves including the frequency change obtained from Eq. (2) with *γ*0=8. 0×1010 s−1 for the A1*<sup>g</sup>* mode and 1.5×1010 s−1 for the E*<sup>g</sup>* mode, where the observed phonon frequencies in Fig. 9 are used for each temperature. The broken curves show the theoretical curves in the case of no frequency change. As is seen in Fig. 10, the solid curves explain well the experi‐

Deviation of the experimental data near *Tc* from the solid curve may be caused by the effect of the phase transition. However, the relaxation rate just around the phase transition point cannot be obtained in the present experiment, and the relation between the temperature-de‐

pendent relaxation rate and the structural phase transition is not clear.

opposite wave vectors *k* and –*k*.

mental data.

exp <sup>ℏ</sup>(*ω*<sup>0</sup> / 2) / *kBT* - <sup>1</sup> ) (2)

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269

ture dependence of the relaxation rate *γ* of the coherent phonon is given by [30,31]

where *ω*0 is the frequency of the optical phonon, and *kB* is the Boltzmann constant.

*γ* =*γ*0(1 +

**Figure 10.** Temperature dependence of the relaxation rate in SrTiO3 obtained from the coherent phonon signal below *Tc*. (a) The diamonds are the relaxation rate for the *A*1*g* mode, and (b) the squares are that for the *Eg* mode. The solid curves show the theoretical curves including the frequency change obtained from Eq. (2).

In general, relaxation of coherent phonons is determined by population decay (inelastic scat‐ tering) and pure dephasing (elastic scattering). In metals, pure dephasing due to electronphonon scattering, which depends on the hot electron density, contributes to the phonon relaxation [29]. In dielectric crystals, the relaxation process of the coherent phononis consid‐ ered to be dominated by the population decay due to the anharmonic phonon-phonon cou‐ pling [30-32], rather than pure dephasing. According to the anharmonic decay model [30], the relaxation of optical phonons in the center of the Brillouin zone is considered to occur through two types of decay process, the down-conversion and up-conversion processes. In a down-conversion process, the initial *ω*<sup>0</sup> phonon with wave vector *k*≅ 0 decays into two low‐ er-energy phonons *ω<sup>i</sup>* and *ω<sup>j</sup>* , with opposite wave vectors *k* and −*k*, which belong to the *i* branch and the *j* branch of the phonon. Energy and wave-vector conservation is given by *ω* 0=*ωik*+*ω<sup>j</sup>*−*<sup>k</sup>*. In an up-conversion process, the initial excitation is scattered by a thermal phonon (*ωik*) into a phonon of higher energy (*ωjk*), where *ω*0+*ωik*=*ωjk*. The down-conversion process can be realized either for *i*= *j* (overtone channel), or for *i≠ j* (combination channel), depend‐ ing on the phonon band structure of the material, while the up-conversion process contains only the combination channel and has no overtone channel. The combination channel is less likely, because three frequencies of phonons and three phonon branches have to be con‐ cerned and stringent limitations are imposed by the energy and wave-vector conservation. The overtone channel, on the other hand, is more likely because two (an optical and an acoustic) phonon branches are concerned, and the energy and wave-vector conservation are necessarily satisfied by two acoustic phonons with the same frequency and opposite wave vectors, if the frequency maximum of the acoustic branch is higher than half the frequency of the initial optical phonon.

Here we consider the down-conversion process in which an optical phonon decays into two acoustic phonons with half the frequency of the optical phonon and with opposite wave vec‐ tors. Schematic diagram of this down-conversion process is shown in Fig. 11. The tempera‐ ture dependence of the relaxation rate *γ* of the coherent phonon is given by [30,31]

$$\gamma = \gamma\_0 \left( 1 + \frac{2}{\exp\left[\hbar \left(\omega\_0 / 2\right) / k\_B T\right] \cdot 1} \right) \tag{2}$$

where *ω*0 is the frequency of the optical phonon, and *kB* is the Boltzmann constant.

**Figure 10.** Temperature dependence of the relaxation rate in SrTiO3 obtained from the coherent phonon signal below *Tc*. (a) The diamonds are the relaxation rate for the *A*1*g* mode, and (b) the squares are that for the *Eg* mode. The solid

In general, relaxation of coherent phonons is determined by population decay (inelastic scat‐ tering) and pure dephasing (elastic scattering). In metals, pure dephasing due to electronphonon scattering, which depends on the hot electron density, contributes to the phonon relaxation [29]. In dielectric crystals, the relaxation process of the coherent phononis consid‐ ered to be dominated by the population decay due to the anharmonic phonon-phonon cou‐ pling [30-32], rather than pure dephasing. According to the anharmonic decay model [30], the relaxation of optical phonons in the center of the Brillouin zone is considered to occur through two types of decay process, the down-conversion and up-conversion processes. In a down-conversion process, the initial *ω*<sup>0</sup> phonon with wave vector *k*≅ 0 decays into two low‐

branch and the *j* branch of the phonon. Energy and wave-vector conservation is given by *ω* 0=*ωik*+*ω<sup>j</sup>*−*<sup>k</sup>*. In an up-conversion process, the initial excitation is scattered by a thermal phonon (*ωik*) into a phonon of higher energy (*ωjk*), where *ω*0+*ωik*=*ωjk*. The down-conversion process can be realized either for *i*= *j* (overtone channel), or for *i≠ j* (combination channel), depend‐ ing on the phonon band structure of the material, while the up-conversion process contains only the combination channel and has no overtone channel. The combination channel is less likely, because three frequencies of phonons and three phonon branches have to be con‐ cerned and stringent limitations are imposed by the energy and wave-vector conservation. The overtone channel, on the other hand, is more likely because two (an optical and an acoustic) phonon branches are concerned, and the energy and wave-vector conservation are necessarily satisfied by two acoustic phonons with the same frequency and opposite wave vectors, if the frequency maximum of the acoustic branch is higher than half the frequency

Here we consider the down-conversion process in which an optical phonon decays into two acoustic phonons with half the frequency of the optical phonon and with opposite wave vec‐

, with opposite wave vectors *k* and −*k*, which belong to the *i*

curves show the theoretical curves including the frequency change obtained from Eq. (2).

er-energy phonons *ω<sup>i</sup>*

268 Advances in Ferroelectrics

of the initial optical phonon.

and *ω<sup>j</sup>*

**Figure 11.** Schematic diagram of the down-conversion processe (overtone channel) in the an harmonic decay model of optical phonons. The initial ω<sup>0</sup> optical phonon with wave vector *k*≅ 0 decays into two ω<sup>0</sup> / 2 acoustic phonons, with opposite wave vectors *k* and –*k*.

In ordinary materials the temperature dependence of the phonon frequency is small, and a theoretical curve with a fixed value of phonon frequency fits the experimental data well. In SrTiO3, however, the phonon frequencies are changed greatly as the temperature is in‐ creased; thus a frequency change has to be considered. The solid curves in Fig. 10 show the theoretical curves including the frequency change obtained from Eq. (2) with *γ*0=8. 0×1010 s−1 for the A1*<sup>g</sup>* mode and 1.5×1010 s−1 for the E*<sup>g</sup>* mode, where the observed phonon frequencies in Fig. 9 are used for each temperature. The broken curves show the theoretical curves in the case of no frequency change. As is seen in Fig. 10, the solid curves explain well the experi‐ mental data.

Deviation of the experimental data near *Tc* from the solid curve may be caused by the effect of the phase transition. However, the relaxation rate just around the phase transition point cannot be obtained in the present experiment, and the relation between the temperature-de‐ pendent relaxation rate and the structural phase transition is not clear.

### **5. Coherent phonon spectroscopy in Ca-doped SrTiO3**

### **5.1. Angular dependence of the coherent phonon signal**

The angular dependence of the coherent phonon signal in Ca-doped SrTiO3 observed at 50 K is shown in Fig. 12(a), where the angle between the [100] axis of the crystal and the polarization plane of the pump pulse is 0°, 25°, and 45°. The angle between the polarization planes of the pump and probe pulses is fixed to 45°. Vertical axis is the ellipticity in electric field amplitude of the transmitted probe. At zero delay, a large signal due to the optical Kerr effect, whose width is de‐ termined by the laser-pulse width, appears. After that, damped oscillations of coherent pho‐ nons are observed. For the 0° pumping a 0.5 ps period signal appears while it disappears for the 45° pumping and a 2 ps period signal appears. TheFourier transform of the coherent phonon sig‐ nals in Fig. 12(a) is shown in Fig. 12(b). The oscillation frequency of the signalfor the 0° pumping is 1. 9 THz and that for the 45° pumping is 0.5 THz. For other pumping angles both frequency com‐ ponents coexist in the coherent phonon signal. These phonon modes are the soft modes related to the structural phase transition at *Tc*<sup>1</sup> =180 K. From the oscillation frequencies the 1.9 THz compo‐ nent corresponds to the *A*1*<sup>g</sup>* mode, and the 0.5 THz components to the *Eg* mode, which are as‐ signed from the modes of pure SrTiO3 [7] and from that of Sr1−*<sup>x</sup>* Ca*<sup>x</sup>* TiO3 (*x*=0.007) [8].

**5.2. Temperature dependence of the coherent phonon signal**

signal of coherent phonons is observed.

The temperature dependence of the coherent phonon signal in Ca-doped SrTiO3 observed for the 45° pumping, which corresponds to the E*g* mode, is shown in Fig. 13. At 6 K some oscillation components, which have different frequencies, are superposed. As the tempera‐ ture is increased, the number of the oscillation components is decreased, the oscillation peri‐ od becomes longer and the relaxation time becomes shorter. At *Tc*1=180 K, which is the structural phase-transition temperature obtained from the birefringence measurement, no

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**Figure 13.** Temperature dependence of the coherent phonon signal in Ca-doped SrTiO3 observed for the 45° pumping.

the 25° pumping are shown in Figs. 14 and 15. Figure 14(a) shows the coherent phonon signals

Figure 14(b) shows the Fourier transform of the coherent phonon signals in Fig. 14(a), where the peaks with arrows 1, 2, and 3 correspond to the ferroelectric phonon modes [8]. Figure 15(b) shows the Fourier transform of the coherent phonon signals in Fig. 15(a), where modes 1, 2, and

modes related to the structural phase-transition remain.

observed for

.

and Fig. 15(a) shows those above *Tc*<sup>2</sup>

The temperature dependence of the coherent phonon signal in Ca-doped SrTiO3

below the ferroelectric phase-transition temperature *Tc*<sup>2</sup>

and *Eg*

3 disappear but the *A*1*<sup>g</sup>*

**Figure 12.** (a) Coherent phonon signals in Ca-doped SrTiO3 observed at 50 K for the 0°, 25°, and 45° pumping, where the angle between the [100] axis of the crystal and the polarization direction of the pump pulse is changed. (b) Fourier transform of the coherent phonon signals in (a).

### **5.2. Temperature dependence of the coherent phonon signal**

**5. Coherent phonon spectroscopy in Ca-doped SrTiO3**

The angular dependence of the coherent phonon signal in Ca-doped SrTiO3

shown in Fig. 12(a), where the angle between the [100] axis of the crystal and the polarization plane of the pump pulse is 0°, 25°, and 45°. The angle between the polarization planes of the pump and probe pulses is fixed to 45°. Vertical axis is the ellipticity in electric field amplitude of the transmitted probe. At zero delay, a large signal due to the optical Kerr effect, whose width is de‐ termined by the laser-pulse width, appears. After that, damped oscillations of coherent pho‐ nons are observed. For the 0° pumping a 0.5 ps period signal appears while it disappears for the 45° pumping and a 2 ps period signal appears. TheFourier transform of the coherent phonon sig‐ nals in Fig. 12(a) is shown in Fig. 12(b). The oscillation frequency of the signalfor the 0° pumping is 1. 9 THz and that for the 45° pumping is 0.5 THz. For other pumping angles both frequency com‐ ponents coexist in the coherent phonon signal. These phonon modes are the soft modes related to

mode, and the 0.5 THz components to the *Eg*

[7] and from that of Sr1−*<sup>x</sup>*

**Figure 12.** (a) Coherent phonon signals in Ca-doped SrTiO3 observed at 50 K for the 0°, 25°, and 45° pumping, where the angle between the [100] axis of the crystal and the polarization direction of the pump pulse is changed. (b) Fourier

=180 K. From the oscillation frequencies the 1.9 THz compo‐

(*x*=0.007) [8].

Ca*<sup>x</sup>* TiO3 observed at 50 K is

mode, which are as‐

**5.1. Angular dependence of the coherent phonon signal**

the structural phase transition at *Tc*<sup>1</sup>

signed from the modes of pure SrTiO3

transform of the coherent phonon signals in (a).

nent corresponds to the *A*1*<sup>g</sup>*

270 Advances in Ferroelectrics

The temperature dependence of the coherent phonon signal in Ca-doped SrTiO3 observed for the 45° pumping, which corresponds to the E*g* mode, is shown in Fig. 13. At 6 K some oscillation components, which have different frequencies, are superposed. As the tempera‐ ture is increased, the number of the oscillation components is decreased, the oscillation peri‐ od becomes longer and the relaxation time becomes shorter. At *Tc*1=180 K, which is the structural phase-transition temperature obtained from the birefringence measurement, no signal of coherent phonons is observed.

**Figure 13.** Temperature dependence of the coherent phonon signal in Ca-doped SrTiO3 observed for the 45° pumping.

The temperature dependence of the coherent phonon signal in Ca-doped SrTiO3 observed for the 25° pumping are shown in Figs. 14 and 15. Figure 14(a) shows the coherent phonon signals below the ferroelectric phase-transition temperature *Tc*<sup>2</sup> and Fig. 15(a) shows those above *Tc*<sup>2</sup> . Figure 14(b) shows the Fourier transform of the coherent phonon signals in Fig. 14(a), where the peaks with arrows 1, 2, and 3 correspond to the ferroelectric phonon modes [8]. Figure 15(b) shows the Fourier transform of the coherent phonon signals in Fig. 15(a), where modes 1, 2, and 3 disappear but the *A*1*<sup>g</sup>* and *Eg* modes related to the structural phase-transition remain.

considered that the doubly degenerate *Eg*

nal-to-rhombohedrallattice distortion.

mode is split into two components under the tetrago‐

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Doping-Induced Ferroelectric Phase Transition and Ultraviolet-Illumination Effect in a Quantum Paraelectric...

. (b) Fourier transform of the coherent phonon sig‐

=28 K. The lift of degeneracy for mode

mode has been observed. The

is shown

**Figure 15.** (a) Temperature dependence of the coherent phonon signal in Ca-doped SrTiO3 observed for the 25°

The temperature dependence of the phonon frequencies, which are related to the ferroelec‐

in Fig. 17, where only the frequencies of the reproducible peaks in the Fourier spectrum are plotted. While the lowest peaks at 6 and 16 K in Fig. 14, for example, may be the third mode observed in the Raman experiment [8], the frequencies of the peaks with poor reproducibili‐ ty are not plotted in Figs. 16 and 17. Under dark illumination, two phonon modes 1 and 2 are softened toward about 25 K and degenerate into mode 3, which seems to be softened to‐

three modes do not all split off at the same temperature; the highest-energy component splits off at the ferroelectric transition temperature while the other two split off at the tem‐ perature about 3 K lower. These two temperatures may correspond to the two temperatures, 28 and 25 K, observed in our experiment, and the temperature differences are nearly equal to each other. The existence of the second structural deformation at 25 K is consistent with

suggests that another structural deformation occurs at 25 K. In the Raman-scat‐

tric phase transition, obtained from the observed coherent phonon signals below *Tc*<sup>2</sup>

pumping above the ferroelectric phase-transition temperature *Tc*<sup>2</sup>

ward the ferroelectric phase-transition temperature *Tc*<sup>2</sup>

tering experiment [8], three modes deriving from the TO1

nals in (a).

3 below *Tc*<sup>2</sup>

the Raman-scattering data.

**Figure 14.** (a) Temperature dependence of the coherent phonon signal in Ca-doped SrTiO3 observed for the 25° pump‐ ing below the ferroelectric phase-transition temperature *Tc*2. (b) Fourier transform of the coherent phonon signals in (a).

### **5.3. Phonon frequencies**

Each component of the coherent phonon signal is expressed well by damped oscillations in Eq. (1). The temperature dependence of the oscillation frequencies in Ca-doped SrTiO3 obtained from the observed coherent phonon signals below *Tc*<sup>1</sup> is shown in Fig. 16. The solid circles are the oscillation frequencies for the *A*1*<sup>g</sup>* and *Eg* modes. The triangles are that for modes 1, 2, and 3 which are related to the ferroelectric phase transition. As for the *A*1*<sup>g</sup>* and *Eg* modes, as the temper‐ ature is increased from 6 K, the oscillation frequencies decrease and approach to zero at the structural phase-transition temperature *Tc*<sup>1</sup> for both modes. This result is consistent with the temperature dependence of phonon frequency observed by Raman scattering [7] and coherent phonons in section 4.3 for pure SrTiO3 except for the phase-transition temperature. The broken curves describe a temperature dependence of the form *ω* ∝(*Tc* - *T* )*n*. The experimental results for the temperature region between *Tc*<sup>1</sup> and *Tc*<sup>2</sup> are explained well by *n*=0.5 for both modes. Be‐ low the ferroelectric phase-transition temperature *Tc*<sup>2</sup> , another mode appears at 0.9 THz. It is considered that the doubly degenerate *Eg* mode is split into two components under the tetrago‐ nal-to-rhombohedrallattice distortion.

**Figure 15.** (a) Temperature dependence of the coherent phonon signal in Ca-doped SrTiO3 observed for the 25° pumping above the ferroelectric phase-transition temperature *Tc*<sup>2</sup> . (b) Fourier transform of the coherent phonon sig‐ nals in (a).

**Figure 14.** (a) Temperature dependence of the coherent phonon signal in Ca-doped SrTiO3 observed for the 25° pump‐ ing below the ferroelectric phase-transition temperature *Tc*2. (b) Fourier transform of the coherent phonon signals in (a).

Each component of the coherent phonon signal is expressed well by damped oscillations in Eq.

ature is increased from 6 K, the oscillation frequencies decrease and approach to zero at the

temperature dependence of phonon frequency observed by Raman scattering [7] and coherent

curves describe a temperature dependence of the form *ω* ∝(*Tc* - *T* )*n*. The experimental results

and *Tc*<sup>2</sup>

obtained

modes, as the temper‐

is shown in Fig. 16. The solid circles are

modes. The triangles are that for modes 1, 2, and 3

and *Eg*

for both modes. This result is consistent with the

are explained well by *n*=0.5 for both modes. Be‐

, another mode appears at 0.9 THz. It is

except for the phase-transition temperature. The broken

(1). The temperature dependence of the oscillation frequencies in Ca-doped SrTiO3

and *Eg*

from the observed coherent phonon signals below *Tc*<sup>1</sup>

low the ferroelectric phase-transition temperature *Tc*<sup>2</sup>

which are related to the ferroelectric phase transition. As for the *A*1*<sup>g</sup>*

**5.3. Phonon frequencies**

272 Advances in Ferroelectrics

the oscillation frequencies for the *A*1*<sup>g</sup>*

phonons in section 4.3 for pure SrTiO3

for the temperature region between *Tc*<sup>1</sup>

structural phase-transition temperature *Tc*<sup>1</sup>

The temperature dependence of the phonon frequencies, which are related to the ferroelec‐ tric phase transition, obtained from the observed coherent phonon signals below *Tc*<sup>2</sup> is shown in Fig. 17, where only the frequencies of the reproducible peaks in the Fourier spectrum are plotted. While the lowest peaks at 6 and 16 K in Fig. 14, for example, may be the third mode observed in the Raman experiment [8], the frequencies of the peaks with poor reproducibili‐ ty are not plotted in Figs. 16 and 17. Under dark illumination, two phonon modes 1 and 2 are softened toward about 25 K and degenerate into mode 3, which seems to be softened to‐ ward the ferroelectric phase-transition temperature *Tc*<sup>2</sup> =28 K. The lift of degeneracy for mode 3 below *Tc*<sup>2</sup> suggests that another structural deformation occurs at 25 K. In the Raman-scat‐ tering experiment [8], three modes deriving from the TO1 mode has been observed. The three modes do not all split off at the same temperature; the highest-energy component splits off at the ferroelectric transition temperature while the other two split off at the tem‐ perature about 3 K lower. These two temperatures may correspond to the two temperatures, 28 and 25 K, observed in our experiment, and the temperature differences are nearly equal to each other. The existence of the second structural deformation at 25 K is consistent with the Raman-scattering data.

**5.4. UV-illumination effect**

The theoretical *Tc*<sup>2</sup>

intensity dependence of the *Tc*<sup>2</sup>

illumination intensity of 7 mW/mm2

of the screening length is obtained as

[11], the carrier concentration is *n*=2. 5× 1017

mate the carrier concentration to be *n*=5. 8× 1017

<sup>λ</sup><sup>=</sup> *ne* <sup>2</sup>

where *Tc*<sup>2</sup>

where *aB*

ty of the UV illumination is 7 mW/mm2

dipoles, and the transition temperature is decreased.

∆ *Tc*<sup>2</sup>

In order to examine the UV-illumination effect, two types of measurements, under the UV illumination and after the UV illumination, are carried out. The temperature dependence of the phonon frequencies under the UV illumination is shown in Fig. 17(a), where the intensi‐

Doping-Induced Ferroelectric Phase Transition and Ultraviolet-Illumination Effect in a Quantum Paraelectric...

which the two modes 1 and 2 are softened and degenerate into mode 3, shifts to the lower

Doped Ca ions behave as permanent dipoles and ferroelectric clusters are formed around Ca dipoles with high polarizability of the host crystal. The ferroelectric transition is caused by

is related to the screening of the internal macroscopic field by UV-excited carriers. Under the UV-illumination non equilibrium carriers appear, which are captured by traps and screen the polarization field. The ordering is prevented by the photo excited carriers. Thus, the screening effect due to the UV-excited carriers weakens the Coulomb interaction between

<sup>4</sup>*<sup>π</sup>* <sup>λ</sup><sup>2</sup>

screening length, *a* is the mean separation between the dipoles, and *γ* is the number of the nearest-neighbor dipoles. The expression of λ is presented by λ<sup>=</sup> *ne* <sup>2</sup> / *εε*0*kBT* , where *<sup>n</sup>* is the carrier concentration which depends on the UV-illumination intensity and *ε*is the rela‐ tive dielectric constant. In the present case, the domain is large enough and the dipole-di‐ pole interaction is expressed as a simple Coulomb interaction. The UV-illumination-

(0) is the transition temperature before the UV illumination, λ is the inverse of the

Ca*<sup>x</sup>* TiO3

*<sup>ε</sup>kBT* =1.1×10<sup>7</sup> cm-1

and *E*1*<sup>s</sup>* are Bohr radius and the energy of the hydrogen atom in the 1*s* ground state,

respectively, and we use the dielectric constant *ε*=4 in the visible region. Substituting the value of λ into Eq. (3), the shift of the transition temperature can be estimated to be ∆ *Tc*<sup>2</sup>

This estimation is not inconsistent with the observed values of the temperature shift in the

cm−3 at 3 mW/mm2

cm−3 at 7 mW/mm2

(3). Assuming that the carrier concentration is proportional to the UV-illumination intensity, the fitting result reproduced well the experimental result obtained in the measurement of dielectric constants [11]. Here we estimate the inverse λ of the screening length at the UV-

temperature side. The temperature shift due to the UV-illumination effect is ~3 K.

the ordering of randomly distributed Ca dipoles. The UV-illumination-induced *Tc*<sup>2</sup>

reduction under the UV illumination ∆ *Tc*<sup>2</sup>

reduction for Sr1−*<sup>x</sup>*

*εε*0*kBT* <sup>=</sup> <sup>8</sup>*πnaBE*1*<sup>s</sup>*

*Tc*2(0) =1 - (1 <sup>+</sup> <sup>λ</sup>*<sup>a</sup>* <sup>+</sup> *<sup>γ</sup>*

. As is seen in Fig. 17(a), the temperature, toward

is given by [11]

*a* 2)exp (-λ*a*), (3)

(*x*=0. 011) was analyzed by using Eq.

. From this value we can esti‐

, (4)

. The value of the inverse

~8 K.

. As a result of the measurement of dielectric constant

reduction

275

http://dx.doi.org/10.5772/45744

**Figure 16.** Temperature dependence of the phonon frequencies in Ca-doped SrTiO3 obtained from the coherent pho‐ non signals below *Tc*<sup>1</sup> . The solid circles are the oscillation frequencies for the *A*1*<sup>g</sup>* and *Eg* modes. The broken curves de‐ scribe a temperature dependence of the form ω ∝(*Tc* - *T* )*n*, where *Tc* =180 K and *n*=0.5 for both modes. The triangles are that for modes 1, 2, and 3 related to the ferroelectric phase transition.

**Figure 17.** Temperature dependence of the phonon frequencies (a) under dark (1, 2, 3) and under UV (1',2',3') illumi‐ nation below *Tc*<sup>2</sup> , which correspond the phonon modes related to the ferroelectric phase transition, and (b)under dark (1, 2, 3) and after UV(1'',2'',3'') illumination, where the intensity of the UV illumination is 7 mW/mm2 .

### **5.4. UV-illumination effect**

**Figure 16.** Temperature dependence of the phonon frequencies in Ca-doped SrTiO3 obtained from the coherent pho‐

**Figure 17.** Temperature dependence of the phonon frequencies (a) under dark (1, 2, 3) and under UV (1',2',3') illumi‐

(1, 2, 3) and after UV(1'',2'',3'') illumination, where the intensity of the UV illumination is 7 mW/mm2

, which correspond the phonon modes related to the ferroelectric phase transition, and (b)under dark

modes. The broken curves de‐

.

=180 K and *n*=0.5 for both modes. The triangles

. The solid circles are the oscillation frequencies for the *A*1*<sup>g</sup>* and *Eg*

scribe a temperature dependence of the form ω ∝(*Tc* - *T* )*n*, where *Tc*

are that for modes 1, 2, and 3 related to the ferroelectric phase transition.

non signals below *Tc*<sup>1</sup>

274 Advances in Ferroelectrics

nation below *Tc*<sup>2</sup>

In order to examine the UV-illumination effect, two types of measurements, under the UV illumination and after the UV illumination, are carried out. The temperature dependence of the phonon frequencies under the UV illumination is shown in Fig. 17(a), where the intensi‐ ty of the UV illumination is 7 mW/mm2 . As is seen in Fig. 17(a), the temperature, toward which the two modes 1 and 2 are softened and degenerate into mode 3, shifts to the lower temperature side. The temperature shift due to the UV-illumination effect is ~3 K.

Doped Ca ions behave as permanent dipoles and ferroelectric clusters are formed around Ca dipoles with high polarizability of the host crystal. The ferroelectric transition is caused by the ordering of randomly distributed Ca dipoles. The UV-illumination-induced *Tc*<sup>2</sup> reduction is related to the screening of the internal macroscopic field by UV-excited carriers. Under the UV-illumination non equilibrium carriers appear, which are captured by traps and screen the polarization field. The ordering is prevented by the photo excited carriers. Thus, the screening effect due to the UV-excited carriers weakens the Coulomb interaction between dipoles, and the transition temperature is decreased.

The theoretical *Tc*<sup>2</sup> reduction under the UV illumination ∆ *Tc*<sup>2</sup> is given by [11]

$$\frac{\Delta T\_{c2}}{T\_{c2}(0)} = 1 \cdot \left(1 + \lambda a + \frac{\gamma}{4\pi} \lambda^2 a^2\right) \exp\left(-\lambda a\right),\tag{3}$$

where *Tc*<sup>2</sup> (0) is the transition temperature before the UV illumination, λ is the inverse of the screening length, *a* is the mean separation between the dipoles, and *γ* is the number of the nearest-neighbor dipoles. The expression of λ is presented by λ<sup>=</sup> *ne* <sup>2</sup> / *εε*0*kBT* , where *<sup>n</sup>* is the carrier concentration which depends on the UV-illumination intensity and *ε*is the rela‐ tive dielectric constant. In the present case, the domain is large enough and the dipole-di‐ pole interaction is expressed as a simple Coulomb interaction. The UV-illuminationintensity dependence of the *Tc*<sup>2</sup> reduction for Sr1−*<sup>x</sup>* Ca*<sup>x</sup>* TiO3 (*x*=0. 011) was analyzed by using Eq. (3). Assuming that the carrier concentration is proportional to the UV-illumination intensity, the fitting result reproduced well the experimental result obtained in the measurement of dielectric constants [11]. Here we estimate the inverse λ of the screening length at the UVillumination intensity of 7 mW/mm2 . As a result of the measurement of dielectric constant [11], the carrier concentration is *n*=2. 5× 1017 cm−3 at 3 mW/mm2 . From this value we can esti‐ mate the carrier concentration to be *n*=5. 8× 1017 cm−3 at 7 mW/mm2 . The value of the inverse of the screening length is obtained as

$$\lambda = \sqrt{\frac{ne^2}{\varepsilon \varepsilon\_0 k\_B T}} = \sqrt{\frac{8\pi n a\_b E\_{1s}}{\varepsilon k\_B T}} = 1.1 \times 10^7 \text{ cm}^{-1} \tag{4}$$

where *aB* and *E*1*<sup>s</sup>* are Bohr radius and the energy of the hydrogen atom in the 1*s* ground state, respectively, and we use the dielectric constant *ε*=4 in the visible region. Substituting the value of λ into Eq. (3), the shift of the transition temperature can be estimated to be ∆ *Tc*<sup>2</sup> ~8 K. This estimation is not inconsistent with the observed values of the temperature shift in the experiment of coherent phonons, in which the value of the observed transition temperature shift is ~3 K.

shift is ~3 K. The decrease in the phonon frequencies after the UV illumination suggests the

Doping-Induced Ferroelectric Phase Transition and Ultraviolet-Illumination Effect in a Quantum Paraelectric...

http://dx.doi.org/10.5772/45744

277

It was shown that the coherent phonon spectroscopy in the time domain is a very useful ap‐ proach to study the soft phonon modes and their UV-illumination effect in dielectric materi‐

We would like to thank Dr. Y. Koyama for experimental help and Dr. Y. Yamada and Prof.

[3] M. Itoh, R. Wang, Y. Inaguma, T. Yamaguchi, Y.-J. Shan, and T.Nakamura, Phys.

[8] U. Bianchi, W. Kleemann, and J. G. Bednorz, J. Phys.: Condens. Matter 6, 1229 (1994).

[9] U. Bianchi, J. Dec, W. Kleemann, and J. G. Bednorz, Phys. Rev. B 51, 8737 (1995).

[13] M. Takesada, T. Yagi, M. Itoh, and S. Koshihara, J. Phys. Soc. Jpn. 72, 37 (2003).

UV-illumination effect remains even if the UV illumination is switched off.

K. Tanaka for providing us the samples of Ca-doped SrTiO3.

Graduate School of Science, Kobe University, Japan

Rev. Lett. 82, 3540 (1999).

[1] K. A. Müller and H. Burkard, Phys. Rev. B 19, 3593 (1979).

[4] B. E. Vugmeister and M. P. Glinchuk, Rev. Mod. Phys. 62, 993 (1990).

[7] P. A. Fleury, J. F. Scott, and J. M. Worlock, Phys. Rev. Lett. 21,16 (1968).

[10] Bürgel, W. Kleemann, and U. Bianchi, Phys. Rev. B 53, 5222 (1995).

[11] Y. Yamada and K. Tanaka, J. Phys. Soc. Jpn. 77, 5 (2008). [12] G. Geneste and J.-M. Kiat, Phys. Rev. B 77, 174101 (2008).

[5] J. G. Bednorz and K. A. Müller, Phys. Rev. Lett. 52, 2289 (1984). [6] J. F. Scott, Rev. Mod. Phys. 46, 83 (1974), and references therein.

[2] H. Uwe and T. Sakudo, Phys. Rev. B 13, 271 (1976).

als.

**Acknowledgement**

**Author details**

Toshiro Kohmoto

**References**

The temperature dependence of the phonon frequencies after the UV illumination is shown in Fig. 17(b), where theUV illumination of 7 mW/mm2 is on before the coherent phonon measurement but is off during the measurement. In this case, on the other hand, the shift of the softening temperature for modes 1 and 2 is not clear. The phonon frequencies for modes 1 and 2 are decreased while the coherent phonon signal for mode 3 is not observed. The re‐ laxation time of the UV-illumination induced carriers is on the order of milliseconds below 30 K for SrTiO3 [21]. As is seen in Fig. 17(b), however, it is suggested that the UV-illumina‐ tion effect, frequency decrease for modes 1 and 2 and disappearance of mode-3 signal, re‐ mains for at least several minutes, even if the UV illumination is switched off, although the *Tc*2 shifting effect disappears immediately.

### **6. Summary**

We observed coherent phonons in pure and Ca-doped SrTiO3 using ultrafast polarization spectroscopy to study the ultrafast dynamics of soft phonon modes and their UV-illumina‐ tion effect. Coherent phonons are generated by linearly polarized pump pulses. The timedependent linear birefringence induced by the generated coherent phonons is detected asa change of the polarization of the probe pulses. A high detection sensitivity of ~10−5 in polari‐ zation change, which corresponds to the change ∆ *n*=2× 10−9 of the refractive index for a 1 mm sample, has been achieved in our detection system.

In SrTiO3 , damped oscillations of coherent phonons for the *A*1*<sup>g</sup>* and *Eg* modes, which contrib‐ ute to the structural phase transition at 105 K, were observed. The temperature dependences of the frequency and the relaxation rate of the observed coherent phonons were measured. Softening of the phonon frequencies was observed. The phonon relaxation is explained well by a decay model of a frequency-changing phonon, in which the optical phonon decays into two acoustic phonons due to the anharmonic phonon-phonon coupling.

We observed the temperature dependences of the birefringence and the coherent phonon signal to investigate the doping-induced ferroelectric phase transition in Ca-dopedSrTiO3 with the Ca concentration of *x*=0.011. In the birefringence measurement, it was confirmed that the structural phase-transition temperature is *Tc*<sup>1</sup> =180 K. Coherent phonons were ob‐ served by using ultrafast polarization spectroscopy. The damped oscillations of coherent phonons for the *A*1*<sup>g</sup>* and *Eg* modes, which contribute to the structural phase transition at *Tc*1 =180 K, and for the modes 1, 2, and3, which contribute to the ferroelectric phase transition at *Tc*<sup>2</sup> =28 K, were observed. The phonon frequencies were obtained from the observed signals of coherent phonons, and their softening toward each phase-transition temperature was ob‐ served. Another structural deformation at 25 K was found in addition to the ferroelectric phase transition at *Tc*<sup>2</sup> . It was found that the UV illumination causes the shift of the ferroelec‐ tric phase-transition temperature toward the lower temperature side, and the temperature shift is ~3 K. The decrease in the phonon frequencies after the UV illumination suggests the UV-illumination effect remains even if the UV illumination is switched off.

It was shown that the coherent phonon spectroscopy in the time domain is a very useful ap‐ proach to study the soft phonon modes and their UV-illumination effect in dielectric materi‐ als.

### **Acknowledgement**

experiment of coherent phonons, in which the value of the observed transition temperature

The temperature dependence of the phonon frequencies after the UV illumination is shown

measurement but is off during the measurement. In this case, on the other hand, the shift of the softening temperature for modes 1 and 2 is not clear. The phonon frequencies for modes 1 and 2 are decreased while the coherent phonon signal for mode 3 is not observed. The re‐ laxation time of the UV-illumination induced carriers is on the order of milliseconds below 30 K for SrTiO3 [21]. As is seen in Fig. 17(b), however, it is suggested that the UV-illumina‐ tion effect, frequency decrease for modes 1 and 2 and disappearance of mode-3 signal, re‐ mains for at least several minutes, even if the UV illumination is switched off, although the

spectroscopy to study the ultrafast dynamics of soft phonon modes and their UV-illumina‐ tion effect. Coherent phonons are generated by linearly polarized pump pulses. The timedependent linear birefringence induced by the generated coherent phonons is detected asa

ute to the structural phase transition at 105 K, were observed. The temperature dependences of the frequency and the relaxation rate of the observed coherent phonons were measured. Softening of the phonon frequencies was observed. The phonon relaxation is explained well by a decay model of a frequency-changing phonon, in which the optical phonon decays into

We observed the temperature dependences of the birefringence and the coherent phonon signal to investigate the doping-induced ferroelectric phase transition in Ca-dopedSrTiO3 with the Ca concentration of *x*=0.011. In the birefringence measurement, it was confirmed

served by using ultrafast polarization spectroscopy. The damped oscillations of coherent

=180 K, and for the modes 1, 2, and3, which contribute to the ferroelectric phase transition

tric phase-transition temperature toward the lower temperature side, and the temperature

=28 K, were observed. The phonon frequencies were obtained from the observed signals of coherent phonons, and their softening toward each phase-transition temperature was ob‐ served. Another structural deformation at 25 K was found in addition to the ferroelectric

modes, which contribute to the structural phase transition at

. It was found that the UV illumination causes the shift of the ferroelec‐

, damped oscillations of coherent phonons for the *A*1*<sup>g</sup>* and *Eg* modes, which contrib‐

change of the polarization of the probe pulses. A high detection sensitivity of ~10−5

is on before the coherent phonon

using ultrafast polarization

of the refractive index for a 1

=180 K. Coherent phonons were ob‐

in polari‐

in Fig. 17(b), where theUV illumination of 7 mW/mm2

We observed coherent phonons in pure and Ca-doped SrTiO3

zation change, which corresponds to the change ∆ *n*=2× 10−9

two acoustic phonons due to the anharmonic phonon-phonon coupling.

mm sample, has been achieved in our detection system.

that the structural phase-transition temperature is *Tc*<sup>1</sup>

and *Eg*

shifting effect disappears immediately.

shift is ~3 K.

276 Advances in Ferroelectrics

*Tc*2

**6. Summary**

In SrTiO3

phonons for the *A*1*<sup>g</sup>*

phase transition at *Tc*<sup>2</sup>

*Tc*1

at *Tc*<sup>2</sup>

We would like to thank Dr. Y. Koyama for experimental help and Dr. Y. Yamada and Prof. K. Tanaka for providing us the samples of Ca-doped SrTiO3.

### **Author details**

Toshiro Kohmoto

Graduate School of Science, Kobe University, Japan

### **References**


[14] T. Hasegawa, S. Mouri, Y. Yamada, and K. Tanaka, J. Phys. Soc. Jpn. 72, 41 (2003).

**Chapter 13**

**Raman Scattering Study on the Phase Transition**

Ferroelectric phase transitions are conventionally divided into two types: an order-disorder and a displacive-type.[1] In the former one, which is frequently seen in hydrogen-bond-type ferroelectrics such as KH2PO4 (KDP), local dipole moments *μ*s are randomly distributed be‐ tween opposite directions in the paraelectric phase, leading to zero macroscopic net polariza‐ tion *P* (= Σ*μ*). A spontaneous polarization in the ferroelectric phase is then driven by their ordering through the ferroelectric phase transition. In the later type, in contrast, there are no dipole moments in the paraelectric phase. The spontaneous polarization in the ferroelectric phase stems from relative polar displacement between cationic and anionic sublattices. It is in‐ duced by freezing of a so-called ferroelectric "soft mode", which is known as a strongly anhar‐ monic optical phonon mode at Γ-point in the Brillouin zone (BZ). The displacive-type transition is often found in the ferroelectric oxides as represented by perovskite-type com‐ pounds such as PbTiO3. Since the ferroelectric oxides have been widely applied for the elec‐ tronic devices, such as actuators, sensors, and memories, due to their chemical stability, the large spontaneous polarization, and the relatively high transition temperature, a better under‐ standing of the soft mode behavior would be important from viewpoints of both application and fundamental sciences. Spectroscopic techniques employing light scattering, infrared ab‐ sorption, and neutron scattering have been generally utilized for investigating the dynamics of the soft mode. Among them, Raman scattering has an advantage especially in a low-fre‐ quency region, which is significant to resolve the critical dynamics of the soft mode near the transition temperature. In the present review, we will discuss the soft mode behavior at the ferroelectric phase transition by referring the Raman scattering studies on CdTiO3 and 18Osubstituted SrTiO3. Furthermore, a microscopic origin of the soft mode is figured out from a

viewpoint of local chemical bonds with the help of first-principles calculations.

© 2013 Taniguchi et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Dynamics of Ferroelectric Oxides**

Additional information is available at the end of the chapter

Hiroki Taniguchi, Hiroki Moriwake, Toshirou Yagi and Mitsuru Itoh

http://dx.doi.org/10.5772/52059

**1. Introduction**


## **Raman Scattering Study on the Phase Transition Dynamics of Ferroelectric Oxides**

Hiroki Taniguchi, Hiroki Moriwake, Toshirou Yagi and Mitsuru Itoh

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52059

### **1. Introduction**

[14] T. Hasegawa, S. Mouri, Y. Yamada, and K. Tanaka, J. Phys. Soc. Jpn. 72, 41 (2003).

[18] T. P. Dougherty, G. P. Wiederrecht, K. A. Nelson, M. H. Garrett,H. P. Jensen, and C.

[23] G. A. Garrett, T. F. Albrecht, J. F. Whitaker, and R. Merlin, Phys. Rev. Lett. 77, 3661

[25] Y. Koyama, T. Moriyasu, H. Okamura, Y. Yamada, K. Tanaka, and T. Kohmoto, Phys.

[24] T. Kohmoto, K. Tada, T. Moriyasu, and Y. Fukuda,Phys. Rev. B 74, 064303 (2006).

[26] T. Kohmoto, Y. Fukuda, M. Kunitomo, and K. Isoda, Phys. Rev. B 62, 579 (2000).

[29] K. Watanabe, N. Takagi, and Y. Matsumoto, Phys. Rev. Lett. 92,057401 (2004).

[31] M. Hase, K. Mizoguchi, H. Harima, S. I. Nakashima, and K. Sakai, Phys. Rev. B 58,

[32] M. Hase, K. Ishioka, J. Demsar, K. Ushida, and M. Kitajima, Phys. Rev. B 71, 184301

[15] G. Godefroy, P. Jullien, and L. Cai, Ferroelectrics 13, 309 (1976).

[17] Y. Yamada and K. Tanaka, J. Lumin. 112, 259 (2005).

Warde, Science 258, 770 (1992).

(1996).

278 Advances in Ferroelectrics

5448 (1998).

(2005).

Rev. B 81, 024104 (2010).

[16] S. Ueda, I. Tatsuzaki, and Y. Shindo, Phys. Rev. Lett. 18, 453 (1967).

[19] H. J. Bakker, S. Hunsche, and H. Kurz, Rev. Mod. Phys. 70, 523 (1998).

[21] T. Hasegawa, M. Shirai, and K. Tanaka, J. Lumin. 87-89, 1217 (2000).

[22] Y.-X. Yan, E. B. Gamble, Jr., and K. A. Nelson, J. Chem. Phys.83, 5391 (1985).

[20] W. G. Nilsen and J. G. Skinner, J. Chem. Phys. 48, 2240 (1968).

[27] R. V. Jones, Proc. R. Soc. London, Ser. A 349, 423 (1976).

[28] E. Courtens, Phys. Rev. Lett. 29, 1380 (1972).

[30] F. Vallée, Phys. Rev. B 49, 2460 (1994).

Ferroelectric phase transitions are conventionally divided into two types: an order-disorder and a displacive-type.[1] In the former one, which is frequently seen in hydrogen-bond-type ferroelectrics such as KH2PO4 (KDP), local dipole moments *μ*s are randomly distributed be‐ tween opposite directions in the paraelectric phase, leading to zero macroscopic net polariza‐ tion *P* (= Σ*μ*). A spontaneous polarization in the ferroelectric phase is then driven by their ordering through the ferroelectric phase transition. In the later type, in contrast, there are no dipole moments in the paraelectric phase. The spontaneous polarization in the ferroelectric phase stems from relative polar displacement between cationic and anionic sublattices. It is in‐ duced by freezing of a so-called ferroelectric "soft mode", which is known as a strongly anhar‐ monic optical phonon mode at Γ-point in the Brillouin zone (BZ). The displacive-type transition is often found in the ferroelectric oxides as represented by perovskite-type com‐ pounds such as PbTiO3. Since the ferroelectric oxides have been widely applied for the elec‐ tronic devices, such as actuators, sensors, and memories, due to their chemical stability, the large spontaneous polarization, and the relatively high transition temperature, a better under‐ standing of the soft mode behavior would be important from viewpoints of both application and fundamental sciences. Spectroscopic techniques employing light scattering, infrared ab‐ sorption, and neutron scattering have been generally utilized for investigating the dynamics of the soft mode. Among them, Raman scattering has an advantage especially in a low-fre‐ quency region, which is significant to resolve the critical dynamics of the soft mode near the transition temperature. In the present review, we will discuss the soft mode behavior at the ferroelectric phase transition by referring the Raman scattering studies on CdTiO3 and 18Osubstituted SrTiO3. Furthermore, a microscopic origin of the soft mode is figured out from a viewpoint of local chemical bonds with the help of first-principles calculations.

© 2013 Taniguchi et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **2. Ferroelectric soft mode in the classical scheme**

Since a concept of the soft mode was proposed, many theoretical and experimental studies have been conducted to clarify its dynamics.[2-4] In this section, we discuss the soft mode behavior in the classical scheme.

#### **2.1. Theoretical background of the ferroelectric soft mode in the classical scheme**

The potential of the soft mode is approximately described by accounting only a biquadratic phonon-phonon interaction as follows:

$$\mathcal{L}\ U\_{\rm o} = \frac{1}{2}M(\mathbf{s}, \mathbf{0})a\_{\rm bare}^2(\mathbf{s}, \mathbf{0})\boldsymbol{\mathcal{Q}}^2(\mathbf{s}, \mathbf{0}) + \sum\_{\lambda} \sum\_{k} \frac{1}{2}J(\mathbf{s}, \mathbf{0}; \lambda, k)\boldsymbol{\mathcal{Q}}^2(\mathbf{s}, \mathbf{0})\boldsymbol{\mathcal{Q}}^2(\lambda, k) \tag{1}$$

2 B c bare 2 ( ,0; , ) ( ,0)

2 B B c s 2 2 ( ,0; , ) ( ,0; , )

( ,0; , )

l

*Js k k <sup>C</sup>*

w

l

l wl

l

w

electric oxide, CdTiO3, is presented in this section.

w

l

We thus obtain

w

Defining *C* as

where

(LST) relation,

l

it finally comes to Cochran's law

*Js k kT <sup>s</sup>*

l

*<sup>k</sup>* ( ,0) ( , ) ( , )

<sup>=</sup> åå -åå (6)

B 2

l wl= -åå (5)

> l

Raman Scattering Study on the Phase Transition Dynamics of Ferroelectric Oxides

1 2

= - *CT T* ( ) (8)

1


¢ <sup>µ</sup> (10)

é ù <sup>º</sup> ê ú ë û åå (7)

l wl

http://dx.doi.org/10.5772/52059

281

*Ms M k k*

*k k* ( ,0) ( , ) ( , ) ( ,0) ( , ) ( , ) *Js k kT Js k kT*

*<sup>k</sup>* ( ,0) ( , ) ( , )

l wl

1 2

*<sup>k</sup>* ( ,0) ( , ) ( , )

l wl

é ù º - ê ú ë û åå (9)

*Ms M k k*

l

As seen in the above equation, the soft mode decreases its frequency with approaching *T*c, and finally freezes to induce the ferroelectric phase transition. In the second-order phase transition, in particular, the displacement of the soft mode *Q*(*s*,0) corresponds to the polar displacement in the ferroelectric phase. It should be noted that the bare frequency *ω*bare(s,0) of the soft mode is assumed to be imaginary at zero-Kelvin. The softening of the soft mode connects to the divergent increase of the dielectric constant through Lyddane-Sachs-Teller

> 2 s 1

The experimental observation of the typical soft mode behavior in the perovskite-type ferro‐

w

**2.2. Experimental observation of the ferroelectric soft mode in the classical scheme**

e

*Ms M k k*

s c

2 B c bare 2 ( ,0; , ) ( ,0)

*Js k k T s* l

 l*Ms M k k Ms M k k*

where *M*(*s*,0), *ω*bare(*s*,0), *Q*(*s*,0) in the first term are effective mass, bare frequency, and dis‐ placement of the soft mode at the Γ-point. The *J*(*s*,0;*λ*,*k*) in the second term denotes an inter‐ action between the soft mode at the Γ-point and the optical phonon mode *λ* at the wave vector *k*, where the summation runs over all blanches and BZ. The equation can be trans‐ formed by a pseudo-harmonic approximation as

$$\begin{split} \boldsymbol{U}\_{\text{o}} &\approx \frac{1}{2} \boldsymbol{M}(\boldsymbol{s}, \boldsymbol{0}) \boldsymbol{\alpha}\_{\text{bare}}^{2}(\boldsymbol{s}, \boldsymbol{0}) \boldsymbol{Q}^{2}(\boldsymbol{s}, \boldsymbol{0}) + \sum\_{\boldsymbol{\lambda}} \sum\_{\boldsymbol{\lambda}} \frac{1}{2} \boldsymbol{J}(\boldsymbol{s}, \boldsymbol{0}; \boldsymbol{\lambda}, \boldsymbol{k}) \boldsymbol{Q}^{2}(\boldsymbol{s}, \boldsymbol{0}) \left\{ \boldsymbol{Q}^{2}(\boldsymbol{\lambda}, \boldsymbol{k}) \right\} \\ &= \frac{1}{2} \boldsymbol{M}(\boldsymbol{s}, \boldsymbol{0}) \left[ \boldsymbol{\alpha}\_{\text{bare}}^{2}(\boldsymbol{s}, \boldsymbol{0}) + \sum\_{\boldsymbol{\lambda}} \sum\_{\boldsymbol{k}} \frac{\boldsymbol{J}(\boldsymbol{s}, \boldsymbol{0}; \boldsymbol{\lambda}, \boldsymbol{k})}{\boldsymbol{M}(\boldsymbol{s}, \boldsymbol{0})} \left\{ \boldsymbol{Q}^{2}(\boldsymbol{\lambda}, \boldsymbol{k}) \right\} \right] \boldsymbol{Q}^{2}(\boldsymbol{s}, \boldsymbol{0}) \\ &\equiv \frac{1}{2} \boldsymbol{M}(\boldsymbol{s}, \boldsymbol{0}) \boldsymbol{\alpha}\_{\text{s}}^{2} \boldsymbol{Q}^{2}(\boldsymbol{s}, \boldsymbol{0}) \end{split} \tag{2}$$

where the inside of the brackets is redefined by the soft mode frequency *ω*s. In the classical regime, the mean square displacement <*Q*<sup>2</sup> (*λ*,*k*)> is proportional to *k*B*T*, where *k*<sup>B</sup> is a Boltz‐ mann constant. The soft mode frequency is thus expressed as

$$\alpha\_s^2 \equiv \alpha^2(\mathbf{s}, \mathbf{0}) = \alpha\_{\text{bare}}^2(\mathbf{s}, \mathbf{0}) + \sum\_{\lambda} \sum\_{k} \frac{J(\mathbf{s}, \mathbf{0}; \lambda, k)}{M(\mathbf{s}, \mathbf{0})} \frac{k\_\mathbf{B} T}{M(\lambda, k) \alpha^2(\lambda, k)} \tag{3}$$

Since *ω*s is ideally zero at *T*c of the second-order phase transition,

$$\alpha\_s^2 = \alpha\_{\text{bare}}^2(s, 0) + \sum\_{\lambda} \sum\_{k} \frac{J(s, 0; \lambda, k)}{M(s, 0)} \frac{k\_{\text{B}} T\_{\text{c}}}{M(\lambda, k) \alpha^2(\lambda, k)} = 0 \tag{4}$$

leading to

Raman Scattering Study on the Phase Transition Dynamics of Ferroelectric Oxides http://dx.doi.org/10.5772/52059 281

$$\rho o\_{\text{bare}}^2(\mathbf{s}, \mathbf{0}) = -\sum\_{\lambda} \sum\_{k} \frac{J(\mathbf{s}, \mathbf{0}; \lambda, k)}{M(\mathbf{s}, \mathbf{0})} \frac{k\_{\text{B}} T\_{\text{c}}}{M(\lambda, k) o^2(\lambda, k)}\tag{5}$$

We thus obtain

**2. Ferroelectric soft mode in the classical scheme**

behavior in the classical scheme.

280 Advances in Ferroelectrics

phonon-phonon interaction as follows:

0 bare

formed by a pseudo-harmonic approximation as

w

bare

» +

w

2 2 s

mann constant. The soft mode frequency is thus expressed as

 w

Since *ω*s is ideally zero at *T*c of the second-order phase transition,

l

<sup>1</sup> ( ,0) ( ,0) <sup>2</sup>

w

*Ms Q s*

regime, the mean square displacement <*Q*<sup>2</sup>

ww

w w

leading to

0 bare

º

= + w

åå

Since a concept of the soft mode was proposed, many theoretical and experimental studies have been conducted to clarify its dynamics.[2-4] In this section, we discuss the soft mode

The potential of the soft mode is approximately described by accounting only a biquadratic

2 2 2 2

l

l

 l

*<sup>k</sup>* ( ,0) ( , ) ( , )

= +åå <sup>=</sup> (4)

l wl

*Ms M k k*

º= +åå (3)

l

 l

> l

(*λ*,*k*)> is proportional to *k*B*T*, where *k*<sup>B</sup> is a Boltz‐

åå (2)

(1)

1 1 ( ,0) ( ,0) ( ,0) ( ,0; , ) ( ,0) ( , ) 2 2 *<sup>k</sup> U Ms s Q s Js kQ s Q k* l

where *M*(*s*,0), *ω*bare(*s*,0), *Q*(*s*,0) in the first term are effective mass, bare frequency, and dis‐ placement of the soft mode at the Γ-point. The *J*(*s*,0;*λ*,*k*) in the second term denotes an inter‐ action between the soft mode at the Γ-point and the optical phonon mode *λ* at the wave vector *k*, where the summation runs over all blanches and BZ. The equation can be trans‐

2 2 2 2

l

2 2 2

where the inside of the brackets is redefined by the soft mode frequency *ω*s. In the classical

<sup>1</sup> ( ,0; , ) ( ,0) ( ,0) ( , ) ( ,0) <sup>2</sup> ( ,0)

l

é ù

ë û

*Js k Ms s Q k Qs M s*

*U Ms s Q s Js kQ s Q k*

*k*

22 2 B s bare 2 ( ,0; , ) ( ,0) ( ,0)

2 2 B c s bare 2

*Js k kT <sup>s</sup>*

*Js k kT s s* l

( ,0; , ) ( ,0) <sup>0</sup> *<sup>k</sup>* ( ,0) ( , ) ( , )

l

*Ms M k k*

l wl

l

= + ê ú

1 1 ( ,0) ( ,0) ( ,0) ( ,0; , ) ( ,0) ( , ) 2 2

åå

*k*

**2.1. Theoretical background of the ferroelectric soft mode in the classical scheme**

$$\rho o\_{\mathbf{s}}^{2} = \sum\_{\lambda} \sum\_{k} \frac{J(\mathbf{s}, \mathbf{0}; \lambda, k)}{M(\mathbf{s}, \mathbf{0})} \frac{k\_{\mathbf{B}} T}{M(\lambda, k) o^{2}(\lambda, k)} - \sum\_{\lambda} \sum\_{k} \frac{J(\mathbf{s}, \mathbf{0}; \lambda, k)}{M(\mathbf{s}, \mathbf{0})} \frac{k\_{\mathbf{B}} T\_{\mathbf{c}}}{M(\lambda, k) o^{2}(\lambda, k)}\tag{6}$$

Defining *C* as

$$C = \left[\sum\_{\lambda} \sum\_{k} \frac{J(s, 0; \lambda, k)}{M(s, 0)} \frac{k\_{\text{B}}}{M(\lambda, k) o^2(\lambda, k)}\right]^{\sharp 2} \tag{7}$$

it finally comes to Cochran's law

$$
\alpha\_s = C(T - T\_c)^{42} \tag{8}
$$

where

$$T\_{\rm c} \equiv -o\alpha\_{\rm bare}^2(\mathbf{s}, \mathbf{0}) \left[ \sum\_{\lambda} \sum\_{k} \frac{J(\mathbf{s}, \mathbf{0}; \lambda, k)}{M(\mathbf{s}, \mathbf{0})} \frac{k\_{\mathbf{s}}}{M(\lambda, k) o^2(\lambda, k)} \right]^{-1} \tag{9}$$

As seen in the above equation, the soft mode decreases its frequency with approaching *T*c, and finally freezes to induce the ferroelectric phase transition. In the second-order phase transition, in particular, the displacement of the soft mode *Q*(*s*,0) corresponds to the polar displacement in the ferroelectric phase. It should be noted that the bare frequency *ω*bare(s,0) of the soft mode is assumed to be imaginary at zero-Kelvin. The softening of the soft mode connects to the divergent increase of the dielectric constant through Lyddane-Sachs-Teller (LST) relation,

$$
\varepsilon' \propto \frac{1}{o\_s^2} \tag{10}
$$

#### **2.2. Experimental observation of the ferroelectric soft mode in the classical scheme**

The experimental observation of the typical soft mode behavior in the perovskite-type ferro‐ electric oxide, CdTiO3, is presented in this section.

CdTiO3 possesses an orthorhombic *Pnma* structure at room temperature due to (*a*<sup>+</sup> *bb*- )-type octahedral rotations in the Glazer's notation from the prototypical Cubic *Pm*-3*m* structure. The CdTiO3 undergoes ferroelectric phase transition into the orthorhombic *Pna*21 phase around 85 K.[5-7]

0.2×0.1×0.1 mm3

. Since the present samples were twinned, we carefully determined the di‐

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rections of the axes in the observed area by checking the angular dependence of the Raman spectra with the confocal micro-Raman system, whose spatial resolution is around 1 *μ*m (Fig. 1). Here we use laboratory coordinates X, Y, and Z, corresponding respectively to the crystallographic axes of [100], [010], and [001] in the paraelectric phase, to denote the Raman scattering geometries. For example, XZ(Y,Y)-Z-X means that the incident laser polarized parallel to the [010] direction penetrates the sample along the [101] direction, and the [010] polarized scattered light is collected with the backscattering geometry. The scattered light was analyzed by a Jovin-Yvon triple monochromator T64000 with the subtractive dispersion mode, and was finally observed by a liquid-N2-cooled CCD camera. The frequency resolu‐ tion of the present experiment was better than 1 cm-1 The temperature of the sample was

Figure 2 shows the temperature dependence of the Raman spectrum of CdTiO3, which is ob‐ served in Y(ZX,ZX)-Y scattering geometry with several temperatures from 85.0 K to 40.0 K. In the vicinity of *T*c, in particular, spectra were observed with the temperature intervals of 0.5 K. As shown in the figures, the soft mode is seen at around 60 cm-1 at 40 K. It gradually softens toward zero-frxuency as approaching *T*<sup>c</sup> ~ 85 K. The spectral profile of the soft mode

> 2 0 2 22 2 2

where *B*, *ω*0, and *Γ* denote amplitude, harmonic frequency, and a damping constant of the

**Figure 2.** Temperature Dependence of the Raman spectrum of CdTiO3 observed in the Y(ZX,ZX)-Y scattering geome‐

2 22

 w

w

 w

<sup>G</sup> <sup>=</sup> - +G (11)

s 0 = -G (12)

w w

0 ( ) ( ) *<sup>B</sup> <sup>I</sup>*

ww

controlled by an Oxford microstat with a temperature stability of ±0*.*01 K.

is analyzed by a damped-harmonic-oscillator (DHO) model;

try, and from 40K to 85 K on heating.

w

soft mode, respectively. The soft mode frequency *ω*s is defined here as

The confocal micro-Raman scattering is the one of useful techniques to investigate the soft mode dynamics in the displacive-type ferroelectric phase transition due to the following rea‐ son; In general, the complicated domain structure forms across the ferroelectric phase transi‐ tion, where the principle axis of the crystal orients to various directions, which are allowed by the symmetry relation between the paraelectric and the ferroelectric phases. The size of the individual domain generally ranges over several nanometers to microns. When the do‐ mains are smaller than the radiated area of the incident laser, the observed Raman spectrum is composed of signals from differently oriented domains, leading to difficulty in the precise spectral analyses. With application of the confocal micro-Raman scattering, on the other hand, we can selectively observe the spectrum from the single domain region, because its spatial resolution reaches sub-microns not only for lateral but also depth directions.[8] Note that the Raman scattering can observe the soft mode only in the non-centrosymmetric phase due to the selection rule, therefore the critical dynamics of the soft mode is in principle in‐ vestigated in the ferroelectric phase below *T*c.

**Figure 1.** Raman spectra of CdTiO3 at room temperature observed with several scattering geometries. See text for configuration notations, for instance Y(X,X)-Y.

Flux-grown colorless single crystals of CdTiO3 with a rectangular solid shape were used for this study. The two samples have dimensions of approximately 0.3×0.2×0.1 mm3 and 0.2×0.1×0.1 mm3 . Since the present samples were twinned, we carefully determined the di‐ rections of the axes in the observed area by checking the angular dependence of the Raman spectra with the confocal micro-Raman system, whose spatial resolution is around 1 *μ*m (Fig. 1). Here we use laboratory coordinates X, Y, and Z, corresponding respectively to the crystallographic axes of [100], [010], and [001] in the paraelectric phase, to denote the Raman scattering geometries. For example, XZ(Y,Y)-Z-X means that the incident laser polarized parallel to the [010] direction penetrates the sample along the [101] direction, and the [010] polarized scattered light is collected with the backscattering geometry. The scattered light was analyzed by a Jovin-Yvon triple monochromator T64000 with the subtractive dispersion mode, and was finally observed by a liquid-N2-cooled CCD camera. The frequency resolu‐ tion of the present experiment was better than 1 cm-1 The temperature of the sample was controlled by an Oxford microstat with a temperature stability of ±0*.*01 K.

CdTiO3 possesses an orthorhombic *Pnma* structure at room temperature due to (*a*<sup>+</sup>

around 85 K.[5-7]

282 Advances in Ferroelectrics

vestigated in the ferroelectric phase below *T*c.

configuration notations, for instance Y(X,X)-Y.

octahedral rotations in the Glazer's notation from the prototypical Cubic *Pm*-3*m* structure. The CdTiO3 undergoes ferroelectric phase transition into the orthorhombic *Pna*21 phase

The confocal micro-Raman scattering is the one of useful techniques to investigate the soft mode dynamics in the displacive-type ferroelectric phase transition due to the following rea‐ son; In general, the complicated domain structure forms across the ferroelectric phase transi‐ tion, where the principle axis of the crystal orients to various directions, which are allowed by the symmetry relation between the paraelectric and the ferroelectric phases. The size of the individual domain generally ranges over several nanometers to microns. When the do‐ mains are smaller than the radiated area of the incident laser, the observed Raman spectrum is composed of signals from differently oriented domains, leading to difficulty in the precise spectral analyses. With application of the confocal micro-Raman scattering, on the other hand, we can selectively observe the spectrum from the single domain region, because its spatial resolution reaches sub-microns not only for lateral but also depth directions.[8] Note that the Raman scattering can observe the soft mode only in the non-centrosymmetric phase due to the selection rule, therefore the critical dynamics of the soft mode is in principle in‐

**Figure 1.** Raman spectra of CdTiO3 at room temperature observed with several scattering geometries. See text for

Flux-grown colorless single crystals of CdTiO3 with a rectangular solid shape were used for this study. The two samples have dimensions of approximately 0.3×0.2×0.1 mm3 and

*bb*- )-type

> Figure 2 shows the temperature dependence of the Raman spectrum of CdTiO3, which is ob‐ served in Y(ZX,ZX)-Y scattering geometry with several temperatures from 85.0 K to 40.0 K. In the vicinity of *T*c, in particular, spectra were observed with the temperature intervals of 0.5 K. As shown in the figures, the soft mode is seen at around 60 cm-1 at 40 K. It gradually softens toward zero-frxuency as approaching *T*<sup>c</sup> ~ 85 K. The spectral profile of the soft mode is analyzed by a damped-harmonic-oscillator (DHO) model;

$$I(\alpha) = \frac{Boo\_0^2 \Gamma \alpha o}{(o\_0^2 - \alpha^2)^2 + \Gamma^2 \alpha^2} \tag{11}$$

where *B*, *ω*0, and *Γ* denote amplitude, harmonic frequency, and a damping constant of the soft mode, respectively. The soft mode frequency *ω*s is defined here as

**Figure 2.** Temperature Dependence of the Raman spectrum of CdTiO3 observed in the Y(ZX,ZX)-Y scattering geome‐ try, and from 40K to 85 K on heating.

$$
\alpha\_s^2 = \sqrt{\alpha\_0^2 - \Gamma^2} \tag{12}
$$

The temperature dependence of the soft mode frequency and the damping constant, which are determined by the analyses, are plotted by solid circles and open squares in Fig. 3. Syn‐ chronizing with the softening of the soft mode, the damping constant is increased as ap‐ proaching *T*c.

The temperature dependence of the soft mode frequency obeys the Cochran's low (Eq. 1.8) with *C* = 8.5 and *T*<sup>c</sup> = 85 K as expressed by the solid curve in the figure, indicating the typical

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The displacement pattern of the soft mode is obtained by first principle calculations. The cal‐ culations were conducted with a pseudopotential method based on a density functional per‐ turbation theory with norm-conserving pseudopotentials, which was implemented in the CASTEP code.[9] Figure 4a presents phonon dispersion curves of CdTiO3 in the paraelectric *Pnma* phase, which was obtained by the first-principle calculations. As seen in the figure, the soft mode, which is indicated by an imaginary frequency, is observed at Γ-point in Brillouin zone. According to the calculation, a displacement pattern of the Γ-point soft mode is com‐ posed of relative displacement of Ti with respective to the octahedra and an asymmetric stretching of O-Cd-O bond. It has been previously proposed that the hybridization between O-2p and empty Ti-3d orbitals triggers the non-centrosymmetric displacement of Ti thor‐ ough a second-order Jahn-Teller (SOJT) effect to induce the ferroelectricity as exemplified by BaTiO3.[10] The SOJT effect, however, does not provide clear understanding on the role of *A*-site ion. In the case of CdTiO3, in particular, the ferroelectricity disappears when the Cd is replaced by Ca, though the CaTiO3 is an isomorph of CdTiO3. Therefore, the ferroelectricity in CdTiO3 can not be explained only by the *B*-O orbital hybridization. Here, we focus on the asymmetric stretching of O-Cd-O to elucidate the mechanism of ferroelectricity in CdTiO3. The asymmetric stretching in the Γ-point soft mode is schematically illustrated by arrows in the right panel in Fig. 4b, where large and small spheres denote Cd and O ions, respectively. Ti and O ions that do not participate in the O-Cd-O bonds are omitted for simplicity. In a prototypical *Pm*-3*m* structure of the perovskite-type oxide, an *A*-site ion is surrounded by neighboring twelve O ions. In the CdTiO3 of paraelectric *Pnma* structure, however, the num‐ ber of closest oxygen ions for Cd ion is reduced to four due to the octahedral rotations. (Note that the octahedral rotations in the *Pnma* structure are different from each other ac‐ cording to the direction; the rotation is in-phase for a [010] direction whereas those for [101] and [-101] are anti-phase.) As a result, CdTiO3 has three different O-Cd-O bonds along [-101], [101], and [010] directions, whose equilibrium structures are respectively denoted by (1), (2), and (3) in the Fig. 2(b). The result of first-principles calculations indicates that the soft mode displacement is inherently dominated by the asymmetric stretching of the O-Cd-O bond along the [010] direction denoted by (3). By comparing three equilibrium structures for O-Cd-O bonds presented in the right-hand side of Fig. 4b, it is found that the O-Cd-O bond along [010] direction is symmetric with the Cd-O bond lengths of 2.186 Å in the para‐ electric *Pnma* phase, in contrast the other two are composed of non-equivalent bonds having 2.326 and 2.165 Å. Therefore, there is a remaining degree of freedom for further O-Cd-O offcenter ordering due to the covalent bonding along the [010] direction. This remaining degree

displacive-type ferroelectric phase transition on CdTiO3.

of freedom would be the origin of the ferroelectric soft mode.

**3. A role of covalency in the ferroelectric soft mode**

As indicated above, the covalency of the *A*-site ion plays an important role in the ferroelec‐ tricity in addition to that of *B*-site ion. The *A*-site substitution with an isovalent ion is in‐

**Figure 3.** Temperature Dependencies of (a) the soft mode frequency and (b) the damping constant of CdTiO3. The solid line in (a) is calculated by Cochran's law (see text).

**Figure 4.** a) Phonon dispersion in CdTiO3 obtained by the first-principles calculations. (b) The displacement pattern of the soft mode, and configurations of three O-Cd-O bonds along (1) [-101], (2) [101], and (3) [010] directions.

The temperature dependence of the soft mode frequency obeys the Cochran's low (Eq. 1.8) with *C* = 8.5 and *T*<sup>c</sup> = 85 K as expressed by the solid curve in the figure, indicating the typical displacive-type ferroelectric phase transition on CdTiO3.

The temperature dependence of the soft mode frequency and the damping constant, which are determined by the analyses, are plotted by solid circles and open squares in Fig. 3. Syn‐ chronizing with the softening of the soft mode, the damping constant is increased as ap‐

**Figure 3.** Temperature Dependencies of (a) the soft mode frequency and (b) the damping constant of CdTiO3. The

**Figure 4.** a) Phonon dispersion in CdTiO3 obtained by the first-principles calculations. (b) The displacement pattern of the soft mode, and configurations of three O-Cd-O bonds along (1) [-101], (2) [101], and (3) [010] directions.

solid line in (a) is calculated by Cochran's law (see text).

proaching *T*c.

284 Advances in Ferroelectrics

The displacement pattern of the soft mode is obtained by first principle calculations. The cal‐ culations were conducted with a pseudopotential method based on a density functional per‐ turbation theory with norm-conserving pseudopotentials, which was implemented in the CASTEP code.[9] Figure 4a presents phonon dispersion curves of CdTiO3 in the paraelectric *Pnma* phase, which was obtained by the first-principle calculations. As seen in the figure, the soft mode, which is indicated by an imaginary frequency, is observed at Γ-point in Brillouin zone. According to the calculation, a displacement pattern of the Γ-point soft mode is com‐ posed of relative displacement of Ti with respective to the octahedra and an asymmetric stretching of O-Cd-O bond. It has been previously proposed that the hybridization between O-2p and empty Ti-3d orbitals triggers the non-centrosymmetric displacement of Ti thor‐ ough a second-order Jahn-Teller (SOJT) effect to induce the ferroelectricity as exemplified by BaTiO3.[10] The SOJT effect, however, does not provide clear understanding on the role of *A*-site ion. In the case of CdTiO3, in particular, the ferroelectricity disappears when the Cd is replaced by Ca, though the CaTiO3 is an isomorph of CdTiO3. Therefore, the ferroelectricity in CdTiO3 can not be explained only by the *B*-O orbital hybridization. Here, we focus on the asymmetric stretching of O-Cd-O to elucidate the mechanism of ferroelectricity in CdTiO3. The asymmetric stretching in the Γ-point soft mode is schematically illustrated by arrows in the right panel in Fig. 4b, where large and small spheres denote Cd and O ions, respectively. Ti and O ions that do not participate in the O-Cd-O bonds are omitted for simplicity. In a prototypical *Pm*-3*m* structure of the perovskite-type oxide, an *A*-site ion is surrounded by neighboring twelve O ions. In the CdTiO3 of paraelectric *Pnma* structure, however, the num‐ ber of closest oxygen ions for Cd ion is reduced to four due to the octahedral rotations. (Note that the octahedral rotations in the *Pnma* structure are different from each other ac‐ cording to the direction; the rotation is in-phase for a [010] direction whereas those for [101] and [-101] are anti-phase.) As a result, CdTiO3 has three different O-Cd-O bonds along [-101], [101], and [010] directions, whose equilibrium structures are respectively denoted by (1), (2), and (3) in the Fig. 2(b). The result of first-principles calculations indicates that the soft mode displacement is inherently dominated by the asymmetric stretching of the O-Cd-O bond along the [010] direction denoted by (3). By comparing three equilibrium structures for O-Cd-O bonds presented in the right-hand side of Fig. 4b, it is found that the O-Cd-O bond along [010] direction is symmetric with the Cd-O bond lengths of 2.186 Å in the para‐ electric *Pnma* phase, in contrast the other two are composed of non-equivalent bonds having 2.326 and 2.165 Å. Therefore, there is a remaining degree of freedom for further O-Cd-O offcenter ordering due to the covalent bonding along the [010] direction. This remaining degree of freedom would be the origin of the ferroelectric soft mode.

### **3. A role of covalency in the ferroelectric soft mode**

As indicated above, the covalency of the *A*-site ion plays an important role in the ferroelec‐ tricity in addition to that of *B*-site ion. The *A*-site substitution with an isovalent ion is in‐ structive for the better understanding on the mechanism of the ferroelectricity in the perovskite-type oxides. This section is thus devoted to the effect of the isovalent Ca-substitu‐ tion on the ferroelectricity of CdTiO3. [11, 12]

crease with increasing *x* because the dielectric constant near 0 K increases with *x* (see Fig. 5a). The Raman spectra at 4 K presented in the bottom panels in Fig. 6 show that the soft mode softens as *x* increases, indicating a displacive-type phase transition of the CCT-*x* system. This result suggests that the phase transition of CCT-*x* can be discussed in terms of lattice dynamics. Note that, the soft mode become observable at low temperature in CCT-0.05 and -0.07, although it does not undergo the ferroelectric phase transition within a finite temperature range. This is a typical characteristic of precursory softening of the

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**Figure 6.** Contour plots for the temperature dependence of soft mode spectra in CCT-*x* (upper panels). Bottom panels

Figure 7 presents the partial electronic density of states (p-DOS) for CdTiO3 (left) and CaT‐ iO3 (right) obtained by the first-principles calculations, where the *p*-orbital of oxygen, the *s*orbital of Cd and Ca, and the total DOS are denoted by oppositely hatched and blank areas, respectively. As shown in the figure, the *s*-orbital of Cd ion has strong hybridization with the *p*-orbital of oxygen. In marked contrast, the *s*-orbital of Ca has little hybridization, indi‐ cating larger *A*-O covalency in CdTiO3 than CaTiO3. This characteristic substantially agrees with the difference in electronegativity of Cd and Ca ions, where those in Cd and Ca are 1.7 and 1.0, respectively. A calculated charge density distributions around the O-Cd(Ca)-O bonds along the [010] direction are indicated in Fig. 8. As shown in the panels, larger charge density is observed between Cd and O ions in CdTiO3, confirming its strong covalency. In contrast, Ca-O bond is nearly ionic. If the Cd(Ca) bonding is ionic, Cd(Ca) ion in the O-Cd(Ca)-O bond tends to locate a centric position. Therefore, Ca-substitution suppresses the freezing of the asymmetric stretching vibration with the off-centering of Cd(Ca) by decreas‐ ing the covalency. Since the asymmetric stretching of the O-Cd-O bond along [010] direction is suggested to be the origin of the soft mode, Ca-substitution results in the suppression of the softening of the soft mode. The present result confirms that the ferroelectric instability of

soft mode in the quantum paraelectric state.[13, 14]

show the low-frequency spectra observed at ~ 4 K.

CCT-*x* stems from *A*-site covalency.

**Figure 5.** a) Dielectric constants of CCT-*x*. (b) Ca concentration dependence of the ferroelectric phase transition tem‐ perature *T*c. Solid curve denotes Curie–Weiss fit of data points for CCT-0, 0.01, and 0.03.

Ceramics form of Ca-doped CdTiO3, Cd1-*x*Ca*x*TiO3 (CCT-*x*) with *x* = 0, 0.01, 0.03, 0.05, and 0.07, were fabricated by conventional solid state reactions from stoichiometric mixtures of CdO (3N), CaCO3 (4N), and TiO2 (3N). The mixtures were calcined during 24 h in air at 975 ~ 1175ºC. Then the reground powders were pressed at 4 ton/cm2 , and sintered at 1200 ~ 1230ºC. Dielectric measurements were performed by a LCZ meter.

Temperature dependencies of the dielectric constants in CCT-*x* are presented in Fig. 5a. As seen in the figure, the dielectric constant of CCT-0 divergently increases and culminates at the phase transition temperature *T*c ~ 85 K. The *T*c decreases with Ca doping, and the peak transforms into a low-temperature plateau at *x* = 0.05, indicating a quantum paraelectric state as discussed later. The Ca concentration dependence of *T*c, which is estimated from the dielectric peaks, is shown in Fig. 5b. The fit curve with conventional Curie-Weiss law (a red line in the figure) indicates that the ferroelectricity in the CCT-*x* system is suppressed with a value of *x* larger than *x*c = 0.047.

The temperature dependence of the soft mode frequency is presented in Fig. 6 as a func‐ tion of *x*. The soft modes in CCT-*x* increase in frequency and intensity with distance from *T*c on cooling, exhibiting typical behavior of the soft-mode-driven phase transition. Ac‐ cording to the Lyddane–Sachs–Teller relationship (Eq. 1.10), the squared soft mode fre‐ quency *ω*<sup>s</sup> 2 and the dielectric constant *ε'* are inversely proportional as mentioned in the preceding section. Therefore, if the dielectric properties of the CCT-*x* system are governed by the lattice dynamics, the soft mode frequency at the lowest temperature should de‐ crease with increasing *x* because the dielectric constant near 0 K increases with *x* (see Fig. 5a). The Raman spectra at 4 K presented in the bottom panels in Fig. 6 show that the soft mode softens as *x* increases, indicating a displacive-type phase transition of the CCT-*x* system. This result suggests that the phase transition of CCT-*x* can be discussed in terms of lattice dynamics. Note that, the soft mode become observable at low temperature in CCT-0.05 and -0.07, although it does not undergo the ferroelectric phase transition within a finite temperature range. This is a typical characteristic of precursory softening of the soft mode in the quantum paraelectric state.[13, 14]

structive for the better understanding on the mechanism of the ferroelectricity in the perovskite-type oxides. This section is thus devoted to the effect of the isovalent Ca-substitu‐

**Figure 5.** a) Dielectric constants of CCT-*x*. (b) Ca concentration dependence of the ferroelectric phase transition tem‐

Ceramics form of Ca-doped CdTiO3, Cd1-*x*Ca*x*TiO3 (CCT-*x*) with *x* = 0, 0.01, 0.03, 0.05, and 0.07, were fabricated by conventional solid state reactions from stoichiometric mixtures of CdO (3N), CaCO3 (4N), and TiO2 (3N). The mixtures were calcined during 24 h in air at 975

Temperature dependencies of the dielectric constants in CCT-*x* are presented in Fig. 5a. As seen in the figure, the dielectric constant of CCT-0 divergently increases and culminates at the phase transition temperature *T*c ~ 85 K. The *T*c decreases with Ca doping, and the peak transforms into a low-temperature plateau at *x* = 0.05, indicating a quantum paraelectric state as discussed later. The Ca concentration dependence of *T*c, which is estimated from the dielectric peaks, is shown in Fig. 5b. The fit curve with conventional Curie-Weiss law (a red line in the figure) indicates that the ferroelectricity in the CCT-*x* system is suppressed with a

The temperature dependence of the soft mode frequency is presented in Fig. 6 as a func‐ tion of *x*. The soft modes in CCT-*x* increase in frequency and intensity with distance from *T*c on cooling, exhibiting typical behavior of the soft-mode-driven phase transition. Ac‐ cording to the Lyddane–Sachs–Teller relationship (Eq. 1.10), the squared soft mode fre‐

preceding section. Therefore, if the dielectric properties of the CCT-*x* system are governed by the lattice dynamics, the soft mode frequency at the lowest temperature should de‐

2 and the dielectric constant *ε'* are inversely proportional as mentioned in the

, and sintered at 1200 ~

perature *T*c. Solid curve denotes Curie–Weiss fit of data points for CCT-0, 0.01, and 0.03.

~ 1175ºC. Then the reground powders were pressed at 4 ton/cm2

1230ºC. Dielectric measurements were performed by a LCZ meter.

value of *x* larger than *x*c = 0.047.

quency *ω*<sup>s</sup>

tion on the ferroelectricity of CdTiO3. [11, 12]

286 Advances in Ferroelectrics

**Figure 6.** Contour plots for the temperature dependence of soft mode spectra in CCT-*x* (upper panels). Bottom panels show the low-frequency spectra observed at ~ 4 K.

Figure 7 presents the partial electronic density of states (p-DOS) for CdTiO3 (left) and CaT‐ iO3 (right) obtained by the first-principles calculations, where the *p*-orbital of oxygen, the *s*orbital of Cd and Ca, and the total DOS are denoted by oppositely hatched and blank areas, respectively. As shown in the figure, the *s*-orbital of Cd ion has strong hybridization with the *p*-orbital of oxygen. In marked contrast, the *s*-orbital of Ca has little hybridization, indi‐ cating larger *A*-O covalency in CdTiO3 than CaTiO3. This characteristic substantially agrees with the difference in electronegativity of Cd and Ca ions, where those in Cd and Ca are 1.7 and 1.0, respectively. A calculated charge density distributions around the O-Cd(Ca)-O bonds along the [010] direction are indicated in Fig. 8. As shown in the panels, larger charge density is observed between Cd and O ions in CdTiO3, confirming its strong covalency. In contrast, Ca-O bond is nearly ionic. If the Cd(Ca) bonding is ionic, Cd(Ca) ion in the O-Cd(Ca)-O bond tends to locate a centric position. Therefore, Ca-substitution suppresses the freezing of the asymmetric stretching vibration with the off-centering of Cd(Ca) by decreas‐ ing the covalency. Since the asymmetric stretching of the O-Cd-O bond along [010] direction is suggested to be the origin of the soft mode, Ca-substitution results in the suppression of the softening of the soft mode. The present result confirms that the ferroelectric instability of CCT-*x* stems from *A*-site covalency.

nored, the approximation <*Q*<sup>2</sup>

2 2 s bare

w w

As in the classical case,

pressed as

gives

with

modified with quantum statistic treatment as

w w

w w

w

l

(*λ*, *k*)> ~ *k*B*T* is not appropriate. Therefore, Eq. 1.3 should be

Raman Scattering Study on the Phase Transition Dynamics of Ferroelectric Oxides

( ,0; , ) (,) ( ,0) coth *<sup>k</sup>* ( ,0) 2 ( , ) ( , ) 2 *Js k <sup>k</sup> <sup>s</sup>*

At the sufficiently low-temperature, where only the Γ-point soft mode that is the optical phonon with lowest energy can be thermally excited, any other optical phonons are not ef‐ fective in the second term except for λ = *s* at *k* = 0. The soft mode frequency can thus be ex‐

2 2 s

2 2 s

= + <sup>=</sup> h h

2 s

1 1 s c coth 2 2 *T T C T T*

1 2

é ù ¢ ¢ <sup>º</sup> ê ú º º

w

ë û

2 2 1 c

( ,0; ,0) , , coth ( ,0) 2 2 *Js s k T T <sup>C</sup> T T M s k T* w

In this treatment, the temperature dependence of the squared soft mode frequency, which is inversely proportional to the dielectric permittivity, in no longer linear with respect to tem‐ perature, but saturated with the constant value near 0 K. The qualitative behavior of the soft

( ,0; ,0) ( ,0) coth

( ,0; ,0) ( ,0) coth

2 ( ,0) 2 *Js s <sup>s</sup> M s kT*

( ,0; ,0) ( ,0) coth 0 2 ( ,0) 2 *Js s <sup>s</sup> M s kT*

2 ( ,0) 2 *Js s <sup>s</sup> M s kT*

w= - h h

w

w

s bare 2

= +

s bare 2

bare 2

We thus obtain the soft mode frequency at the low-temperature,

w

l

= +åå h h

*M s M k k kT*

s B

h h

s Bc

s Bc

1 2

B s 11

sB c

w

w

æ ö <sup>=</sup> ¢ ¢ - ç ÷ è ø (17)

<sup>h</sup> (18)

w

l wl

B

http://dx.doi.org/10.5772/52059

(13)

289

(14)

(15)

(16)

w l

**Figure 7.** Partial electron density of states (p-DOS) for CdTiO3 (left) and CaTiO3 (right). Blue, red, and black curves rep‐ resent *p* orbital of oxygen ions, *s* orbital of Cd and Ca ions, and total DOS.

**Figure 8.** Cross sections of charge density around 4-coordinated Cd (left) and Ca (right) ions obtained by first principle calculations. Note that the upper and lower cross sections include O–Cd(Ca)–O bonds, which are involved in O– Cd(Ca)–O chains along the *b* and *a* direction, respectively.

### **4. Ferroelectric Soft Mode in the Quantum Scheme**

It has been known that the quantum fluctuation plays a non-negligible role in the phase transition dynamics when the *T*c goes down near 0 K, where the transition is suppressed though the dielectric permittivity reaches to several tens thousand. This effect is known as "quantum paraelectricity", which was first proposed as an origin of the giant dielectric pla‐ teau of SrTiO3 at the low-temperature.[15] Twenty years later, it has been discovered that an isotope substitution with 18O induces the ferroelectricity of SrTiO3, attracting considerable attention of researchers.[16] In this section, the dynamics of quantum para/ferroelectric is discussed from a view point of the soft mode.

### **4.1. Theoretical background of the ferroelectric soft mode in the quantum scheme**

In the classical case as mentioned before, the soft mode frequency can be described as Eq. 1.3. At the low-temperature, where the influence of the quantum fluctuation can not be ig‐ nored, the approximation <*Q*<sup>2</sup> (*λ*, *k*)> ~ *k*B*T* is not appropriate. Therefore, Eq. 1.3 should be modified with quantum statistic treatment as

$$\alpha\_s^2 = \alpha\_{\text{bare}}^2(s, 0) + \sum\_{\lambda} \sum\_{\lambda} \frac{J(s, 0; \lambda, k)}{M(s, 0)} \frac{\hbar}{2M(\lambda, k)o(\lambda, k)} \text{coth}\frac{\hbar o(\lambda, k)}{2k\_{\text{B}}T} \tag{13}$$

At the sufficiently low-temperature, where only the Γ-point soft mode that is the optical phonon with lowest energy can be thermally excited, any other optical phonons are not ef‐ fective in the second term except for λ = *s* at *k* = 0. The soft mode frequency can thus be ex‐ pressed as

$$
\rho \alpha\_{\rm s}^{2} = \alpha \rho\_{\rm bare}^{2} (s, 0) + \frac{J(s, 0; s, 0)}{2M^{2} (s, 0)} \frac{\hbar}{\alpha\_{\rm s}} \coth \frac{\hbar \alpha\_{\rm s}}{2k\_{\rm B}T} \tag{14}
$$

As in the classical case,

$$
\rho \alpha\_{\circ}^{2} = \alpha\_{\text{bare}}^{2}(\text{s}, 0) + \frac{J(\text{s}, 0; \text{s}, 0)}{2M^{2}(\text{s}, 0)} \frac{\hbar}{\alpha\_{\circ}} \coth \frac{\hbar \alpha\_{\circ}}{2k\_{\text{B}}T\_{\text{c}}} = 0 \tag{15}
$$

gives

**Figure 7.** Partial electron density of states (p-DOS) for CdTiO3 (left) and CaTiO3 (right). Blue, red, and black curves rep‐

**Figure 8.** Cross sections of charge density around 4-coordinated Cd (left) and Ca (right) ions obtained by first principle calculations. Note that the upper and lower cross sections include O–Cd(Ca)–O bonds, which are involved in O–

It has been known that the quantum fluctuation plays a non-negligible role in the phase transition dynamics when the *T*c goes down near 0 K, where the transition is suppressed though the dielectric permittivity reaches to several tens thousand. This effect is known as "quantum paraelectricity", which was first proposed as an origin of the giant dielectric pla‐ teau of SrTiO3 at the low-temperature.[15] Twenty years later, it has been discovered that an isotope substitution with 18O induces the ferroelectricity of SrTiO3, attracting considerable attention of researchers.[16] In this section, the dynamics of quantum para/ferroelectric is

**4.1. Theoretical background of the ferroelectric soft mode in the quantum scheme**

In the classical case as mentioned before, the soft mode frequency can be described as Eq. 1.3. At the low-temperature, where the influence of the quantum fluctuation can not be ig‐

resent *p* orbital of oxygen ions, *s* orbital of Cd and Ca ions, and total DOS.

288 Advances in Ferroelectrics

Cd(Ca)–O chains along the *b* and *a* direction, respectively.

discussed from a view point of the soft mode.

**4. Ferroelectric Soft Mode in the Quantum Scheme**

$$\rho o\_{\text{bare}}^2(s,0) = -\frac{J(s,0;s,0)}{2M^2(s,0)} \frac{\hbar}{o o\_s} \coth \frac{\hbar o\_s}{2k\_\text{B}T\_c} \tag{16}$$

We thus obtain the soft mode frequency at the low-temperature,

$$\alpha\_{\rm s} = C' \left( \frac{T\_{\rm l}}{2} \coth \frac{T\_{\rm l}}{2T} - T\_{\rm c}' \right)^{\rm ly} \tag{17}$$

with

$$C' \equiv \left[\frac{J(\mathbf{s}, \mathbf{0}; \mathbf{s}, \mathbf{0})}{M^2(\mathbf{s}, \mathbf{0})} \frac{k\_\mathbf{B}}{\alpha\_\mathbf{s}^2}\right]^{1/2}, \quad T\_\mathbf{i} \equiv \frac{\hbar \alpha\_\mathbf{s}}{k\_\mathbf{B}}, \quad T'\_\mathbf{e} \equiv \frac{T\_\mathbf{i}}{2} \coth \frac{T\_\mathbf{i}}{2T\_\mathbf{e}} \tag{18}$$

In this treatment, the temperature dependence of the squared soft mode frequency, which is inversely proportional to the dielectric permittivity, in no longer linear with respect to tem‐ perature, but saturated with the constant value near 0 K. The qualitative behavior of the soft mode in the quantum para/ferroelectrics is schematically illustrated as a function of *T*<sup>1</sup> with the fixed *T'*c in Fig. 9.

Here we show the Raman scattering study on the soft mode in SrTi(16O1-*<sup>x</sup>*

the cubic coordination of SrTiO3.[13,14,18]

model.

0.32).

as functions of temperature and the isotope substitution rate *x*. The experiments were per‐ formed with the Raman scattering geometries of X(YY)-X, where X, Y, and Z are denoted by

Raman Scattering Study on the Phase Transition Dynamics of Ferroelectric Oxides

Figure 10 presents the temperature dependence of the squared soft mode frequency in STO18-100*x* for 0.23, 0.32, 0.50 0.66, and 0.96, observed in the X(YY)-X scattering geometry, respectively. In *x* = 0.50, 0.66, and 0.96, the soft mode frequency is shown only for the ferro‐ electric phase. As indicated in the figure, the softening of the soft mode in STO18-23 satu‐ rates at the low-temperature region near 0 K, showing excellent agreement with the theory. Since the square of the soft mode frequency is inversely proportional to the dielectric per‐ mittivity as indicated by the LST-relation as mentioned before, it is clear that the dielectric plateau in SrTiO3 stems from the soft mode dynamics. Note that the soft mode is nominally Raman inactive in the centrosymmetric structure as for the paraelectric phase of SrTiO3 due to the selection rule. In the present case, however, the centrosymmetry is locally broken in the low-temperature region of STO18-100*x* (*x* < *x*c ~ 0.32), leading to the observation of the soft mode even in the macroscopic centrosymmetry. The local non-centrosymmetric regions grow with 18O-substitution to activate the soft mode spectrum even in the paraelectric phase as indicated in Fig. 11. A mechanism of such defect-induced Raman process and an expected spectral profile are discussed in detail in Ref. [13] and [14]. The date points for STO18-*x* (*x* < *x*c) in Fig. 10 are obtained by the spectral fitting with the defect-induced Raman scattering

**Figure 11.** Temperature dependencies of the Raman spectra in STO18-*x* (*x* = 0 [STO16: the pure SrTiO3], 0.23, and

With the isotope substitution, softening of the soft mode is enhanced and the phase transi‐ tion takes place above the critical concentration *x*<sup>c</sup> as manifested by the hardening of the soft mode on cooling in the low temperature region. The transition temperature elevates as in‐ creasing *x*. The temperature dependencies of the soft modes in all samples except for that in

18O*x*)3 (STO18-100*x*)

291

http://dx.doi.org/10.5772/52059

**Figure 9.** The schematic illustration describing the variation of the temperature dependence of the soft mode fre‐ quency as a function of *T*1. See text for the detail.

The classical limit is also presented for comparison. Note that the value of *T*<sup>1</sup> is varied with constant intervals. As presented in the figure, the phase transition is completely suppressed when *T*1 is sufficiently large, whereas it recovers as decreasing *T*1. Interestingly, the *T*<sup>1</sup> de‐ pendence of the transition temperature becomes extremely sensitive when the transition temperature is close to 0K, suggesting that the quantum paraelectric-ferroelectric transition can be induced by subtle perturbation such as the isotope substitutions. [17]

**Figure 10.** Temperature dependencies of the squared frequencies of the soft mode in STO18-*x* with various *x*. Solid lines in the figures denote temperature dependencies calculated by Eq. 3.7 with optimized control parameters (see text).

### **4.2. Experimental observation of the ferroelectric soft mode in the quantum scheme**

The isotopically induced phase transition of the quantum paraelectric SrTiO3 is a good ex‐ ample for the quantum paraelectric-ferroelectric phase transition driven by the soft mode. Here we show the Raman scattering study on the soft mode in SrTi(16O1-*<sup>x</sup>* 18O*x*)3 (STO18-100*x*) as functions of temperature and the isotope substitution rate *x*. The experiments were per‐ formed with the Raman scattering geometries of X(YY)-X, where X, Y, and Z are denoted by the cubic coordination of SrTiO3.[13,14,18]

mode in the quantum para/ferroelectrics is schematically illustrated as a function of *T*<sup>1</sup> with

**Figure 9.** The schematic illustration describing the variation of the temperature dependence of the soft mode fre‐

The classical limit is also presented for comparison. Note that the value of *T*<sup>1</sup> is varied with constant intervals. As presented in the figure, the phase transition is completely suppressed when *T*1 is sufficiently large, whereas it recovers as decreasing *T*1. Interestingly, the *T*<sup>1</sup> de‐ pendence of the transition temperature becomes extremely sensitive when the transition temperature is close to 0K, suggesting that the quantum paraelectric-ferroelectric transition

**Figure 10.** Temperature dependencies of the squared frequencies of the soft mode in STO18-*x* with various *x*. Solid lines in the figures denote temperature dependencies calculated by Eq. 3.7 with optimized control parameters (see text).

The isotopically induced phase transition of the quantum paraelectric SrTiO3 is a good ex‐ ample for the quantum paraelectric-ferroelectric phase transition driven by the soft mode.

**4.2. Experimental observation of the ferroelectric soft mode in the quantum scheme**

can be induced by subtle perturbation such as the isotope substitutions. [17]

the fixed *T'*c in Fig. 9.

290 Advances in Ferroelectrics

quency as a function of *T*1. See text for the detail.

Figure 10 presents the temperature dependence of the squared soft mode frequency in STO18-100*x* for 0.23, 0.32, 0.50 0.66, and 0.96, observed in the X(YY)-X scattering geometry, respectively. In *x* = 0.50, 0.66, and 0.96, the soft mode frequency is shown only for the ferro‐ electric phase. As indicated in the figure, the softening of the soft mode in STO18-23 satu‐ rates at the low-temperature region near 0 K, showing excellent agreement with the theory. Since the square of the soft mode frequency is inversely proportional to the dielectric per‐ mittivity as indicated by the LST-relation as mentioned before, it is clear that the dielectric plateau in SrTiO3 stems from the soft mode dynamics. Note that the soft mode is nominally Raman inactive in the centrosymmetric structure as for the paraelectric phase of SrTiO3 due to the selection rule. In the present case, however, the centrosymmetry is locally broken in the low-temperature region of STO18-100*x* (*x* < *x*c ~ 0.32), leading to the observation of the soft mode even in the macroscopic centrosymmetry. The local non-centrosymmetric regions grow with 18O-substitution to activate the soft mode spectrum even in the paraelectric phase as indicated in Fig. 11. A mechanism of such defect-induced Raman process and an expected spectral profile are discussed in detail in Ref. [13] and [14]. The date points for STO18-*x* (*x* < *x*c) in Fig. 10 are obtained by the spectral fitting with the defect-induced Raman scattering model.

**Figure 11.** Temperature dependencies of the Raman spectra in STO18-*x* (*x* = 0 [STO16: the pure SrTiO3], 0.23, and 0.32).

With the isotope substitution, softening of the soft mode is enhanced and the phase transi‐ tion takes place above the critical concentration *x*<sup>c</sup> as manifested by the hardening of the soft mode on cooling in the low temperature region. The transition temperature elevates as in‐ creasing *x*. The temperature dependencies of the soft modes in all samples except for that in STO18-32 were examined by the generalized quantum Curie-Weiss law, which was pro‐ posed in Ref. 19, where the power of Eq. 3.5 is modified by γ/2;

$$
\rho o\_s = C' \left( \frac{T\_1}{2} \coth \frac{T\_1}{2T} - T\_\varsigma' \right)^{\prime 2} \tag{19}
$$

ments and the first-principles calculations on Ca-substituted CdTiO3 clarified that the covalency between constituent cation and oxygen plays an essential role in the origin of the soft mode. We hope the present review serves better understanding on the mechanism of

, Toshirou Yagi3

[1] Line, M. E., & Glass, A. M. (1997). Principles and Applications of Ferroelectrics and

[2] Cochran, W. (1960). Crystal stability and the theory of ferroelectricity. *Adv. Phys.*, 9,

[3] Cochran, W. (1961). Crystal stability and the theory of ferroelectricity part II. Piezo‐

[4] Scott, J. F. (1974). Soft-mode spectroscopy: Experimental studies of structural phase

[5] Shan, Y. J., Mori, H., Imoto, H., & Itoh, M. (2002). Ferroelectric Phase Transition in

[6] Shan, Y. J., Mori, H., Tezuka, K., Imoto, H., & Itoh, M. (2003). Ferroelectric Phase

[7] Moriwake, H., Kuwabara, A., Fisher, C. A. J., Taniguchi, H., Itoh, M., & Tanaka, I. (2011). First-principles calculations of lattice dynamics in CdTiO3 and CaTiO3: Phase

[8] Taniguchi, H., Shan, Y. J., Mori, H., & Itoh, M. (2007). Critical soft-mode dynamics and unusual anticrossing in CdTiO3 studied by Raman scattering. *Phys. Rev. B*, 76,

1 Materials and Science Laboratory, Tokyo Institute of Technology, Yokohama, Japan

2 Nanostructures Research Laboratory, Japan Fine Ceramics Center, Nagoya, Japan

3 Research Institute for Electronic Science, Hokkaido University, Sapporo, Japan

and Mitsuru Itoh1

Raman Scattering Study on the Phase Transition Dynamics of Ferroelectric Oxides

http://dx.doi.org/10.5772/52059

293

displacive-type ferroelectric phase transition.

Related Materials,. Claredon, Oxford,.

electric crystals. *Adv. Phys.*, 10, 401 EOF-420 EOF.

transitions. Rev. Mod. Phys. ., 46, 83 EOF-128 EOF.

stability and ferroelectricity. *Phys. Rev. B*, 84, 104.

Perovskite Oxide CdTiO3. *Ferroelectrics*, 381 EOF-386 EOF.

Transition in CdTiO3 Single Crystal. *Ferroelectrics*, 107 EOF-112 EOF.

Hiroki Taniguchi1\*, Hiroki Moriwake2

\*Address all correspondence to:

**Author details**

**References**

387.

212103.

The *γ* is the one of critical exponents, which characterizes the critical behavior of the sus‐ ceptibility. As seen in the figure, the plots are well reproduced by the fitting with the sys‐ tematic variations of *T*1 and γ as indicated in Fig. 12.

**Figure 12.** a) The *x* dependence of *T*1obtained by the fitting. The dashed line is a guide for the eye. (b) The *x* depend‐ ence of critical exponent γ. In STO18-23, -50, -66, -96, γ is determined by the fitting of the observed soft mode fre‐ quency with Eq. 3.7. In STO18-32, we directly determined γ = 2 from the linear temperature dependence of the soft mode frequency with *T*, as seen in the inset. The solid lines are eye guides.

In STO18-32, the strongly rounded soft mode behavior keeps us from fitting with Eq. 3.7. However, we can determine γ *=* 2 for STO18-32 from the obvious linear temperature de‐ pendence of the soft mode frequency, as seen in the inset of the panel (b). It should be noted here that γ *=* 2 corresponds to the theoretical value for the quantum ferroelectric phase tran‐ sition [20–22], and is caused by the quantum mechanical noncommutativity between kinetic and potential energy. These results show experimentally that the quantum phase transition of STO18-*x* is an ideal soft mode-type transition driven by direct control of the quantum fluctuation with the 18O-substitution. An origin of the strongly rounded softening was sug‐ gested in our previous study to be the nanoscopic phase coexistence between ferroelectric and paraelectric region. (See Ref. 18 for the detail)

### **5. Summary**

In the present review, we overviewed the soft mode behavior in both classical and quantum schemes through Raman scattering experiment in the CdTiO3 and the 18O-substituted SrTiO3. The analyses show excellent agreement qualitatively between the fundamental theo‐ ry and experimental observations. The systematic studies with Raman scattering experi‐ ments and the first-principles calculations on Ca-substituted CdTiO3 clarified that the covalency between constituent cation and oxygen plays an essential role in the origin of the soft mode. We hope the present review serves better understanding on the mechanism of displacive-type ferroelectric phase transition.

### **Author details**

STO18-32 were examined by the generalized quantum Curie-Weiss law, which was pro‐

The *γ* is the one of critical exponents, which characterizes the critical behavior of the sus‐ ceptibility. As seen in the figure, the plots are well reproduced by the fitting with the sys‐

**Figure 12.** a) The *x* dependence of *T*1obtained by the fitting. The dashed line is a guide for the eye. (b) The *x* depend‐ ence of critical exponent γ. In STO18-23, -50, -66, -96, γ is determined by the fitting of the observed soft mode fre‐ quency with Eq. 3.7. In STO18-32, we directly determined γ = 2 from the linear temperature dependence of the soft

In STO18-32, the strongly rounded soft mode behavior keeps us from fitting with Eq. 3.7. However, we can determine γ *=* 2 for STO18-32 from the obvious linear temperature de‐ pendence of the soft mode frequency, as seen in the inset of the panel (b). It should be noted here that γ *=* 2 corresponds to the theoretical value for the quantum ferroelectric phase tran‐ sition [20–22], and is caused by the quantum mechanical noncommutativity between kinetic and potential energy. These results show experimentally that the quantum phase transition of STO18-*x* is an ideal soft mode-type transition driven by direct control of the quantum fluctuation with the 18O-substitution. An origin of the strongly rounded softening was sug‐ gested in our previous study to be the nanoscopic phase coexistence between ferroelectric

In the present review, we overviewed the soft mode behavior in both classical and quantum schemes through Raman scattering experiment in the CdTiO3 and the 18O-substituted SrTiO3. The analyses show excellent agreement qualitatively between the fundamental theo‐ ry and experimental observations. The systematic studies with Raman scattering experi‐

1 1 s c coth 2 2 *T T C T T*

æ ö <sup>=</sup> ¢ ¢ - ç ÷

2

è ø (19)

g

posed in Ref. 19, where the power of Eq. 3.5 is modified by γ/2;

w

tematic variations of *T*1 and γ as indicated in Fig. 12.

mode frequency with *T*, as seen in the inset. The solid lines are eye guides.

and paraelectric region. (See Ref. 18 for the detail)

**5. Summary**

292 Advances in Ferroelectrics

Hiroki Taniguchi1\*, Hiroki Moriwake2 , Toshirou Yagi3 and Mitsuru Itoh1

\*Address all correspondence to:

1 Materials and Science Laboratory, Tokyo Institute of Technology, Yokohama, Japan

2 Nanostructures Research Laboratory, Japan Fine Ceramics Center, Nagoya, Japan

3 Research Institute for Electronic Science, Hokkaido University, Sapporo, Japan

### **References**


[9] Clark, S. J., Segall, M. D., Pickard, C. J., Hasnip, P. J., Probert, M. I. J., Refson, K., & Payne, M. C. (2005). First principles methods using CASTEP. *Z. Kristallogr*, 220, 567.

**Chapter 14**

**Electromechanical Coupling Multiaxial Experimental**

Compared to polycrystalline ferroelectric ceramics such as Pb(Zr1/2Ti1/2)-O3 (PZT), domain en‐ gineered relaxor ferroelectric single crystals Pb(Zn1/3Nb2/3)O3-*x*PbTiO3 (PZN-*x*PT) and Pb(Mg1/3Nb2/3)O3-*x*PbTiO3 (PMN-*x*PT) show greatly enhanced electromechanical properties: the piezoelectric coefficient *d*33 and electrically induced strain of <001> oriented single crystals direction can respectively reach 2500pC/N and 1.7%, or even greater [1],[2]. PZN-*x*PT and PMN-*x*PT with outstanding properties are usually near the morphotropic phase boundary (MPB), separating rhombohedra (R) and tetragonal (T) phases [1], [4], [5]. For example, the MPB is located within a range of *x*=0.275-0.33 for PMN-*x*PT [6], [7]. Recent investigations have also shown that the MPB of PMN-*x*PT and PZN-*x*PT is actually in a multi-phase state at room temperature [8], [9]. e.g., the coexistence of rhombohedra (R) and monoclinic (M) phases at *x*=0.33 or rhombohedra and tetragonal phases at *x*=0.32. Other phases can also exist near MPB under stress and/or electric field loading. Experiment studies [10], [11], [12] and first-princi‐ ples calculations [4], [13]reveal that two different homogeneous polarization rotation path‐ ways are present between rhombohedra and tetragonal phases under an electric field. It is found that three kinds of intermediate monoclinic phases MA, MB and MC are associated with the two pathways [9], [10]. Orthorhombic phase has also been observed between the rhombo‐ hedra and tetragonal phases when PMN-PT is loaded with strong electric field along the <110> direction [9],[14]. Although the mechanism underlying the high performance of these crystals is not completely clear, the existence of the intermediate phases and the associated phase tran‐

> © 2013 Qiang et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

**Pb(Mg1/3Nb2/3)O3- 0.32PbTiO3 Ferroelectric Single**

**and Micro-Constitutive Model Study of**

Wan Qiang, Chen Changqing and Shen Yapeng

sitions are believed to be one of the main reasons [1], [3], [4], [8], [10].

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52540

**1. Introduction**

**Crystal**


**Electromechanical Coupling Multiaxial Experimental and Micro-Constitutive Model Study of Pb(Mg1/3Nb2/3)O3- 0.32PbTiO3 Ferroelectric Single Crystal**

Wan Qiang, Chen Changqing and Shen Yapeng

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52540

### **1. Introduction**

[9] Clark, S. J., Segall, M. D., Pickard, C. J., Hasnip, P. J., Probert, M. I. J., Refson, K., & Payne, M. C. (2005). First principles methods using CASTEP. *Z. Kristallogr*, 220, 567.

[10] Bersuker, I. B. (1995). Recent development of the vibronic theory of ferroelectricity:.

[11] Taniguchi, H., Soon, H. P., Shimizu, T., Moriwake, H., Shan, Y. J., & Itoh, M. (2011). Mechanism for suppression of ferroelectricity in Cd1−xCaxTiO3. *Phys. Rev. B*, 84,

[12] Taniguchi, H., Soon, H. P., Moriwake, H., Shan, Y. J., & Itoh, M. (2012). Effect of Ca-Substitution on CdTiO3 Studied by Raman Scattering and First Principles Calcula‐

[13] Taniguchi, H., Takesada, M., Itoh, M., & Yagi, T. (2004). Effect of oxygen isotope ex‐ change on ferroelectric microregion in SrTiO3 studied by Raman scattering. *J. Phys.*

[14] Taniguchi, H., Takesada, M., Itoh, M., & Yagi, T. (2005). Isotope effect on the softmode dynamics of SrTiO3 studied by Raman scattering. *Phys. Rev. B*, 72, 064111. [15] Müller, K. A., Burkard, H., & Sr Ti, O. (1979). An intrinsic quantum paraelectric be‐

[16] Itoh, M., Wang, R., Inaguma, Y., Yamaguchi, Y., Shan, Y. J., & Nakamura, T. (1999). Ferroelectricity Induced by Oxygen Isotope Exchange in Strontium Titanate Perov‐

[17] Tokunaga, M., & Aikawa, Y. (2010). Isotope-Induced Ferroelectric Phase Transition in Strontium Titanate Only by a Decrease in Zero-Point Vibration Frequency. *J. Phys.*

[18] Taniguchi, H., Itoh, M., & Yagi, T. (2007). Ideal Soft Mode-Type Quantum Phase Transition and Phase Coexistence at Quantum Critical Point in <sup>18</sup>O-Exchanged

[19] Dec, J., & Kleemann, W. (1998). From barrett to generalized quantum curie-weiss

[21] Sondhi, S. L., Girvin, S. M., Carini, J. P., & Shahar, D. (1997). Continuous quantum

[22] Schneider, T., & Stoll, E. F. (1976). Quantum effects in an n-component vector model

[20] Vojta, M. (2003). Quantum phase transitions. *Rep. Prog. Phys.*, 66, 2069.

for structural phase transitions. *Phys. Rev. B*, 13, 1123 EOF-1130 EOF.

phase transitions. *Rev. Mod. Phys.*, 69, 315 EOF-333 EOF.

*Ferroelectrics*, 164, 75.

tions. *Ferroelectrics*, 268 EOF-273 EOF.

*Soc. Jpn.*, 73, 3262 EOF-3265 EOF.

low 4 K. *Phys. Rev. B*, 19, 3593.

*Soc. Jpn.*, 79, 024707.

skite. *Phys. Rev. Lett.*, 82, 3540 EOF-3543 EOF.

SrTiO3. *Phys. Rev. Lett.*, 99, 017602 EOF.

law. *Solid State Commun.*, 106, 695 EOF.

174106.

294 Advances in Ferroelectrics

Compared to polycrystalline ferroelectric ceramics such as Pb(Zr1/2Ti1/2)-O3 (PZT), domain en‐ gineered relaxor ferroelectric single crystals Pb(Zn1/3Nb2/3)O3-*x*PbTiO3 (PZN-*x*PT) and Pb(Mg1/3Nb2/3)O3-*x*PbTiO3 (PMN-*x*PT) show greatly enhanced electromechanical properties: the piezoelectric coefficient *d*33 and electrically induced strain of <001> oriented single crystals direction can respectively reach 2500pC/N and 1.7%, or even greater [1],[2]. PZN-*x*PT and PMN-*x*PT with outstanding properties are usually near the morphotropic phase boundary (MPB), separating rhombohedra (R) and tetragonal (T) phases [1], [4], [5]. For example, the MPB is located within a range of *x*=0.275-0.33 for PMN-*x*PT [6], [7]. Recent investigations have also shown that the MPB of PMN-*x*PT and PZN-*x*PT is actually in a multi-phase state at room temperature [8], [9]. e.g., the coexistence of rhombohedra (R) and monoclinic (M) phases at *x*=0.33 or rhombohedra and tetragonal phases at *x*=0.32. Other phases can also exist near MPB under stress and/or electric field loading. Experiment studies [10], [11], [12] and first-princi‐ ples calculations [4], [13]reveal that two different homogeneous polarization rotation path‐ ways are present between rhombohedra and tetragonal phases under an electric field. It is found that three kinds of intermediate monoclinic phases MA, MB and MC are associated with the two pathways [9], [10]. Orthorhombic phase has also been observed between the rhombo‐ hedra and tetragonal phases when PMN-PT is loaded with strong electric field along the <110> direction [9],[14]. Although the mechanism underlying the high performance of these crystals is not completely clear, the existence of the intermediate phases and the associated phase tran‐ sitions are believed to be one of the main reasons [1], [3], [4], [8], [10].

© 2013 Qiang et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Note that PZN-PT and PMN-PT single crystals usually experience electric and/or mechani‐ cal loading during their in-service life. It has been shown that externally applied loading has significant effect on the properties of these crystals [1], [5]-[28]. A number of studies have focused on the loading induced behavior of <001> and <110> oriented anisotropic PZN-*x*PT and PMN-*x*PT crystals [1], [5], [9], [15]-[19]. It is has been shown loading in the form of elec‐ tric field [1]-[21], and stress[21]-[26], can lead to polarization switching and phase transi‐ tions, which changes the crystal phase and domain structure of these single crystals, and hence dramatically alter their electromechanical properties. A mature level understanding of their responses to electrical, mechanical and temperature loading condition is thus essen‐ tial to fulfill the applications of these crystals.

**2. Experimental methodology**

and vice versa (also labeled as type 2, ).

R

O T

and all numbers and notations refer to a quasi-cubic unit cell.

54.70 E3

[001] s

At the test room temperature, PMN-0.32PT single crystals used in this study are of mor‐ photropic composition, and in the rhombohedral phase, very close to MPB. The pseudocubic {001}, {011} and {111} directions of these crystals are determined by x-ray diffraction (XRD). Pellet-like specimens of dimensions 5×5×3mm3 are then cut from these crystals, with the normal of the 5×5mm2 major specimen surfaces along the pseu‐ do-cubic {001}, {011} or {111} direction. All specimens are electroded with silver on the 5×5mm major surfaces and poled along the {001}, {011} and {111} orientations (i.e., the specimen thickness direction) under a field of 1.5kV/mm. Note that there are eight possi‐ ble dipole orientations along the body diagonal directions of unpoled PMN-0.32PT sin‐ gle crystals (i.e., the <111> direction). When an electric poling field is applied to the crystals along the {001} direction, a multi-domain structure can be produced, comprising four degenerate states. For {011} direction poled single crystals, the number of degener‐ ate states is two. The single crystals can be poled into a single domain state when they are poled along the {111} direction. The states of {001}, {011} and {111} poled PMN-0.32PT single crystals are sketched in Fig. 1, where solid arrows refer to the do‐ main states induced by poling (also labeled as type 1 in Figs. 1b and 1c), dotted arrows represent possible domains switched from type 1 domains upon loading or unloading,

Electromechanical Coupling Multiaxial Experimental and Micro-Constitutive Model Study of...

(a) (b) (c)

**Figure 1.** (a) Multidomain rhombohedral crystal obtained by poling along the {001} orientation. The solid line arrows show possible directions of the polarization vector in a fully {001} poled rhombohedral crystal, the dashed line arrows show possible directions of the polarization vector in orthorhombic and tetragonal phases crystal. (b) Two domain rhombohedral crystal obtained by {011} poling. The solid line arrows indicate type 1 domains in a fully {011} poled crystal and dashed line arrows indicate possible type 2 domains under compressive stress. (c) Monodomain rhombo‐ hedral crystal poled along the {111} orientation. The solid and dashed line arrows have similar means to (b). The hol‐ low arrows in Fig. a ~ Fig. c show the electric field and stress loading. The small rhombohedral distortion is neglected

E3

[011] <sup>s</sup>

s

http://dx.doi.org/10.5772/52540

297

type 1

type 2

[111] E3

35.50

type 1

type 2

710

Rhombohedral phase of PZN-*x*PT(0<*x*<0.1) and PMN-*x*PT(0<*x*<0.35) exhibit excellent electro‐ mechanical properties along {001} and {110} orientations compared to the {111}(cubic cell ref‐ erence) spontaneous polarization direction. A number of studies have focused on the loading induced behavior of {001}, {011} and {111} oriented PZN-*x*PT and PMN-*x*PT single crystals. It has been shown that loadings in the form of electric field and stress can lead to polarization rotation and phase transition, which change the crystal phase and domain structure of these single crystals, and hence dramatically alter their electromechanical prop‐ erties. When an electric field is applied along the {001} direction of PZN-PT, polarization ro‐ tation occurs from {111} towards {001} via either MA or MB, depending the composition (e.g., the rotation is R-MA-T for PMN-4.5PT). For {110}-oriented PMN-PT, polarization rotates from {111} to {110} via MB when electric field is along {110} and ends up with an orthorhom‐ bic phase when the electric field exceeds a critical value. These single crystals can be polar‐ ized into a single domain state with {111} oriented electric field. Most available studies on the effect of bias stress on the crystal behavior focus only on uni-polar electric field loading. However, it has been shown that the uni-polar and bi-polar responses of these crystals can be very different.

A number of studies have focused on experimental, but the constitutive model of ferroelec‐ tric single is absent. Huber [29] and Bhattacharya[30], [31] et.al established a model based on the micromechanical method, but the simulation is not well compare to experimental result.

At present, a mature model to explain the stress-strain behavior of ferroelectric single is ab‐ sent all along. In this study, electric field induced "butterfly" curves and polarization loops for a set of compressive bias stress of {001}, {011} and {111} poled PMN-0.32PT single crystals will be explored by systematical experiment study. The effects of the compressive bias stress on the material properties along these three crystallographic directions of PMN-0.32PT sin‐ gle crystals will be quantified. The underlying mechanisms for the observed feature will be explained in terms of phase transformation or domain switching,depending on the crystallo‐ graphic direction.The stress-strain curves along <001> crystallographic direction of ferroelec‐ tric single crystals BaTiO3 will be calculated in the first principle method to validate polarization rotation model. Finally, Based on the experimental phase transformation mech‐ anism of ferroelectric single crystal, a constitutive model of ferroelectric single crystal is pro‐ posed based on micromechanical method. This constitutive model is facility and high computational efficiency.

### **2. Experimental methodology**

Note that PZN-PT and PMN-PT single crystals usually experience electric and/or mechani‐ cal loading during their in-service life. It has been shown that externally applied loading has significant effect on the properties of these crystals [1], [5]-[28]. A number of studies have focused on the loading induced behavior of <001> and <110> oriented anisotropic PZN-*x*PT and PMN-*x*PT crystals [1], [5], [9], [15]-[19]. It is has been shown loading in the form of elec‐ tric field [1]-[21], and stress[21]-[26], can lead to polarization switching and phase transi‐ tions, which changes the crystal phase and domain structure of these single crystals, and hence dramatically alter their electromechanical properties. A mature level understanding of their responses to electrical, mechanical and temperature loading condition is thus essen‐

Rhombohedral phase of PZN-*x*PT(0<*x*<0.1) and PMN-*x*PT(0<*x*<0.35) exhibit excellent electro‐ mechanical properties along {001} and {110} orientations compared to the {111}(cubic cell ref‐ erence) spontaneous polarization direction. A number of studies have focused on the loading induced behavior of {001}, {011} and {111} oriented PZN-*x*PT and PMN-*x*PT single crystals. It has been shown that loadings in the form of electric field and stress can lead to polarization rotation and phase transition, which change the crystal phase and domain structure of these single crystals, and hence dramatically alter their electromechanical prop‐ erties. When an electric field is applied along the {001} direction of PZN-PT, polarization ro‐ tation occurs from {111} towards {001} via either MA or MB, depending the composition (e.g., the rotation is R-MA-T for PMN-4.5PT). For {110}-oriented PMN-PT, polarization rotates from {111} to {110} via MB when electric field is along {110} and ends up with an orthorhom‐ bic phase when the electric field exceeds a critical value. These single crystals can be polar‐ ized into a single domain state with {111} oriented electric field. Most available studies on the effect of bias stress on the crystal behavior focus only on uni-polar electric field loading. However, it has been shown that the uni-polar and bi-polar responses of these crystals can

A number of studies have focused on experimental, but the constitutive model of ferroelec‐ tric single is absent. Huber [29] and Bhattacharya[30], [31] et.al established a model based on the micromechanical method, but the simulation is not well compare to experimental result.

At present, a mature model to explain the stress-strain behavior of ferroelectric single is ab‐ sent all along. In this study, electric field induced "butterfly" curves and polarization loops for a set of compressive bias stress of {001}, {011} and {111} poled PMN-0.32PT single crystals will be explored by systematical experiment study. The effects of the compressive bias stress on the material properties along these three crystallographic directions of PMN-0.32PT sin‐ gle crystals will be quantified. The underlying mechanisms for the observed feature will be explained in terms of phase transformation or domain switching,depending on the crystallo‐ graphic direction.The stress-strain curves along <001> crystallographic direction of ferroelec‐ tric single crystals BaTiO3 will be calculated in the first principle method to validate polarization rotation model. Finally, Based on the experimental phase transformation mech‐ anism of ferroelectric single crystal, a constitutive model of ferroelectric single crystal is pro‐ posed based on micromechanical method. This constitutive model is facility and high

tial to fulfill the applications of these crystals.

be very different.

296 Advances in Ferroelectrics

computational efficiency.

At the test room temperature, PMN-0.32PT single crystals used in this study are of mor‐ photropic composition, and in the rhombohedral phase, very close to MPB. The pseudocubic {001}, {011} and {111} directions of these crystals are determined by x-ray diffraction (XRD). Pellet-like specimens of dimensions 5×5×3mm3 are then cut from these crystals, with the normal of the 5×5mm2 major specimen surfaces along the pseu‐ do-cubic {001}, {011} or {111} direction. All specimens are electroded with silver on the 5×5mm major surfaces and poled along the {001}, {011} and {111} orientations (i.e., the specimen thickness direction) under a field of 1.5kV/mm. Note that there are eight possi‐ ble dipole orientations along the body diagonal directions of unpoled PMN-0.32PT sin‐ gle crystals (i.e., the <111> direction). When an electric poling field is applied to the crystals along the {001} direction, a multi-domain structure can be produced, comprising four degenerate states. For {011} direction poled single crystals, the number of degener‐ ate states is two. The single crystals can be poled into a single domain state when they are poled along the {111} direction. The states of {001}, {011} and {111} poled PMN-0.32PT single crystals are sketched in Fig. 1, where solid arrows refer to the do‐ main states induced by poling (also labeled as type 1 in Figs. 1b and 1c), dotted arrows represent possible domains switched from type 1 domains upon loading or unloading, and vice versa (also labeled as type 2, ).

**Figure 1.** (a) Multidomain rhombohedral crystal obtained by poling along the {001} orientation. The solid line arrows show possible directions of the polarization vector in a fully {001} poled rhombohedral crystal, the dashed line arrows show possible directions of the polarization vector in orthorhombic and tetragonal phases crystal. (b) Two domain rhombohedral crystal obtained by {011} poling. The solid line arrows indicate type 1 domains in a fully {011} poled crystal and dashed line arrows indicate possible type 2 domains under compressive stress. (c) Monodomain rhombo‐ hedral crystal poled along the {111} orientation. The solid and dashed line arrows have similar means to (b). The hol‐ low arrows in Fig. a ~ Fig. c show the electric field and stress loading. The small rhombohedral distortion is neglected and all numbers and notations refer to a quasi-cubic unit cell.

Since the focus of this study is to explore the effect of bias stress on the electromechani‐ cal properties of PMN-0.32PT single crystals along different crystallographic directions, experimental setup is adapted from Ref. [31] (Fig.2) to allow simultaneously imposing uniaxial stress and electric field to the specimen along the thickness direction. Mechani‐ cal load is applied by a servo-hydraulic materials test system (MTS) and electric field is applied to the specimen using a high voltage power amplifier. Once the specimen is placed in the fixture, a compressive bias stress with magnitude of at least 0.4MPa is maintained throughout the test to ensure electrical contact. Stress controlled loading in‐ stead of displacement controlled loading is adopted during the test, so that the speci‐ men is not clamped but is free to move longitudinally when electric field *E*3 and mechanical stress *σ*33 loading are applied, where subscript 3 refers to the thickness direc‐ tion of the samples, corresponding to the {001}, {011} and {111} direction of {001}, {011} and {111} oriented PMN-0.32PT single crystals, respectively. During test, polarization *P*<sup>3</sup> (or electric displacement) is measured using a modified Sawyer-Tower bridge, and the deformation (i.e., the strain) is monitored by two pairs of strain gauges (in total four strain gauges used) mounted on the four 5×3mm2 surfaces: one pair placed on two op‐ posite 5×3mm2 surfaces is applied to measure the longitudinal normal strain *ε*33 along the {001}, {011} and {111} direction and another pair to measure the transverse normal strain *ε*11 in the direction perpendicular to the {001}, {011} and {111} direction respective‐ ly. Output of the strain gauges during deformation is recorded by a computer through a multiple-channel analog-to-digital (AD) converter.

Force

Electromechanical Coupling Multiaxial Experimental and Micro-Constitutive Model Study of...

Steel

Steel ball

http://dx.doi.org/10.5772/52540

299

Strain gauges

> Oil bath

are 0.247, 0.324 and 0.395C/m2

, *d*<sup>33</sup> <sup>011</sup>

,

and *d*<sup>33</sup> <sup>111</sup>

High voltage

Ground

**Figure 2.** The sketch of experimental set

**3. Experimental results and discussion**

3a and 3b, one can calculate that *Pr* <sup>001</sup>

and *Ec* <sup>111</sup>

*Ec* <sup>001</sup>

, *Ec* <sup>011</sup>

Ethoxyline

Brass

Oil

Brass

Ethoxyline

**3.1. Crystallographic dependence of electric behavior and piezoelectric properties**

The measured electric field induced polarization hysteresis loops and butterfly curves for {001}, {011} and {111} oriented poled PMN-0.32PT single crystals without stress loading are shown in Figs. 3a and 3b. The remnant polarizations *Pr* (defined as the polarization value at zero electric field), the coercive electric fields *Ec* (defined as the electric field value at zero polarization) and the piezoelectric coefficients *d*33 (defined as *d*33=*Δε*<sup>33</sup> / *ΔE*3 where *ΔE*<sup>3</sup> is limited between –0.05 and +0.05 kV/mm, namely the slops of the *ε*<sup>33</sup> −*E*3curves as the electric field passes through zero) depend strongly on the crystallographic orientation. From Figs.

, *Pr* <sup>011</sup>

are 0.255, 0.298 and 0.216kV/mm, and *d*<sup>33</sup> <sup>001</sup>

are 1828, 1049 and 200pC/N, respectively. It is noticed that there are eight possible polariza‐

and *Pr* <sup>111</sup>

The first set of tests is performed for electric field loading of triangular wave form of magnitude 0.5kV/mm and frequency 0.02Hz, free of stress loading. The low frequency is chosen to mimic quasistatic electric loading, which is of particular interest in this study [32]. Unless stated otherwise, this loading frequency for the electric field is used throughout the following test. The second set of tests consists of mechanical loading upon short circuited samples. The samples are compressed to –40MPa and unloaded to –0.4MPa at loading and unloading rate of 5MPa/min, followed by an electric field which is sufficiently large to remove the residual stress and strain to re-polarize the samples. In the third set of tests, triangular wave form electric field is applied to the samples which are simultaneously subject to co-axial constant compressive stress preload. The magnitude of the preload is varied from test to test and is in the range between 0 and -40MPa. Note that there is a time-dependent effect of the depolarization and strain re‐ sponses under constant compressive stress. To minimize this effect, each electric field loading starts after a holding time of 150 seconds for a new stress preloading. It is found that three cycles of electric field loading and unloading are sufficient to produce stabilized response for each constant prestress, and the results for the last cycle are re‐ ported in the following.

**Figure 2.** The sketch of experimental set

Since the focus of this study is to explore the effect of bias stress on the electromechani‐ cal properties of PMN-0.32PT single crystals along different crystallographic directions, experimental setup is adapted from Ref. [31] (Fig.2) to allow simultaneously imposing uniaxial stress and electric field to the specimen along the thickness direction. Mechani‐ cal load is applied by a servo-hydraulic materials test system (MTS) and electric field is applied to the specimen using a high voltage power amplifier. Once the specimen is placed in the fixture, a compressive bias stress with magnitude of at least 0.4MPa is maintained throughout the test to ensure electrical contact. Stress controlled loading in‐ stead of displacement controlled loading is adopted during the test, so that the speci‐ men is not clamped but is free to move longitudinally when electric field *E*3 and mechanical stress *σ*33 loading are applied, where subscript 3 refers to the thickness direc‐ tion of the samples, corresponding to the {001}, {011} and {111} direction of {001}, {011} and {111} oriented PMN-0.32PT single crystals, respectively. During test, polarization *P*<sup>3</sup> (or electric displacement) is measured using a modified Sawyer-Tower bridge, and the deformation (i.e., the strain) is monitored by two pairs of strain gauges (in total four strain gauges used) mounted on the four 5×3mm2 surfaces: one pair placed on two op‐ posite 5×3mm2 surfaces is applied to measure the longitudinal normal strain *ε*33 along the {001}, {011} and {111} direction and another pair to measure the transverse normal strain *ε*11 in the direction perpendicular to the {001}, {011} and {111} direction respective‐ ly. Output of the strain gauges during deformation is recorded by a computer through a

The first set of tests is performed for electric field loading of triangular wave form of magnitude 0.5kV/mm and frequency 0.02Hz, free of stress loading. The low frequency is chosen to mimic quasistatic electric loading, which is of particular interest in this study [32]. Unless stated otherwise, this loading frequency for the electric field is used throughout the following test. The second set of tests consists of mechanical loading upon short circuited samples. The samples are compressed to –40MPa and unloaded to –0.4MPa at loading and unloading rate of 5MPa/min, followed by an electric field which is sufficiently large to remove the residual stress and strain to re-polarize the samples. In the third set of tests, triangular wave form electric field is applied to the samples which are simultaneously subject to co-axial constant compressive stress preload. The magnitude of the preload is varied from test to test and is in the range between 0 and -40MPa. Note that there is a time-dependent effect of the depolarization and strain re‐ sponses under constant compressive stress. To minimize this effect, each electric field loading starts after a holding time of 150 seconds for a new stress preloading. It is found that three cycles of electric field loading and unloading are sufficient to produce stabilized response for each constant prestress, and the results for the last cycle are re‐

multiple-channel analog-to-digital (AD) converter.

ported in the following.

298 Advances in Ferroelectrics

### **3. Experimental results and discussion**

### **3.1. Crystallographic dependence of electric behavior and piezoelectric properties**

The measured electric field induced polarization hysteresis loops and butterfly curves for {001}, {011} and {111} oriented poled PMN-0.32PT single crystals without stress loading are shown in Figs. 3a and 3b. The remnant polarizations *Pr* (defined as the polarization value at zero electric field), the coercive electric fields *Ec* (defined as the electric field value at zero polarization) and the piezoelectric coefficients *d*33 (defined as *d*33=*Δε*<sup>33</sup> / *ΔE*3 where *ΔE*<sup>3</sup> is limited between –0.05 and +0.05 kV/mm, namely the slops of the *ε*<sup>33</sup> −*E*3curves as the electric field passes through zero) depend strongly on the crystallographic orientation. From Figs. 3a and 3b, one can calculate that *Pr* <sup>001</sup> , *Pr* <sup>011</sup> and *Pr* <sup>111</sup> are 0.247, 0.324 and 0.395C/m2 , *Ec* <sup>001</sup> , *Ec* <sup>011</sup> and *Ec* <sup>111</sup> are 0.255, 0.298 and 0.216kV/mm, and *d*<sup>33</sup> <sup>001</sup> , *d*<sup>33</sup> <sup>011</sup> and *d*<sup>33</sup> <sup>111</sup> are 1828, 1049 and 200pC/N, respectively. It is noticed that there are eight possible polariza‐ tion orientations along the pseudo-cubic {111} for un-poled rhombohedral PMN-0.32PT sin‐ gle crystals. Upon poling, the dipoles switch as close as possible to the applied electric field direction: For {001} poled crystals, there are four equivalent polar vectors along the {111} ori‐ entation, with an inclined angle of -54.7º from the poling field (Fig. 1a); For {011} poled crys‐ tals, there are two equivalent polar vectors along the {111} direction (labeled as type 1 in Fig. 1b); For {111} poled crystals, there is one polar vector along {111} (type 1 in Fig. 1c). Accord‐ ing to the domain configurations in Fig. 1, the remnant polarizations *Pr* <sup>001</sup> and *Pr* <sup>011</sup> are

approximately related to *Pr* <sup>111</sup> by *Pr* <sup>001</sup> <sup>=</sup>*Pr* <sup>111</sup> / <sup>3</sup> and *Pr* <sup>011</sup> <sup>=</sup> <sup>2</sup>*Pr* <sup>111</sup> / <sup>3</sup>, respective‐ ly. By taking the measured value for *Pr* <sup>111</sup> (0.395C/m2 ), *Pr* <sup>001</sup> and *Pr* <sup>011</sup> are predicted to be 0.228 and 0.322C/m2 , respectively, which are very close to the measured ones (*Pr* <sup>001</sup> =0.227C/m2 and *Pr* <sup>011</sup> =0.324C/m2 ). Therefore, the measured results are consistent with the domain configurations shown in Fig. 1.

It is also seen from Figs. 3a and 3b that the coercive field is lowest for the {111} oriented crys‐ tals, and becomes successively higher for {001} and {110} orientations. This is same to Ref.25 except for {110} orientation. This trend of coercive field is due to two reasons: One is due to reorientation driving force being proportional to the component of electric field aligned with the rhombohedral direction; The other one is due to the domain switching process. In {111} oriented PMN-0.32PT single crystals, there are two types of domains (shown as type 1 and type 2 in Fig.1c). When the electric field is decreased from 0.5kV/mm to –0.05kV/mm, the strain first decreases linearly (see Fig.3b). When the electric field is decreased further, the type 1 domain switches to type 2 domains, leading to abrupt displacement change. When the electric field exceeds the coercive field –0.216kV/mm, the type 2 domains switch back to type 1 domain, recovering the deformation. In type 2 domain state, three equivalent polar vectors with an angle of 71º from the {111} direction can coexist and are separated by do‐ main walls across which the normal components of electric displacement and displacement jump are zero. Ideally, this type of domain walls has no associated local stress or electric field. So the existing of type 2 domain state and the largest component of electric field along polarization direction induces the lowest coercive field in {111} orientation poled crystal. In the {011} orientation crystals, there are also two types of domains (i.e., type 1 and type 2 do‐ main in Fig. 1b), the domain switching process is similar to {111}-poled crystals. In the type 2 domain state, however, the four possible polar vectors are perpendicular to the applied elec‐ tric field (Fig. 1b). It is thus difficult to switch type 2 domain to type 1 domain only by ap‐ plying electric field. Both the type 2 domain state and the smallest component of *E* contribute to the largest coercive field in {011} oriented crystals. In {001} oriented crystals, there will be no associated local stress or electric field, similar to type 2 domain state of {111} poled crystals. This feature again renders domain switching easy. On the other hand, the do‐ main structure of {001} poled crystals is stable [4] and the component of electric field is smaller, giving the coercive field higher than that of {111} poled crystals but lower than that of {011} oriented crystals (see, Figs. 3a and 3b).

**Figure 3.** Electric field induced polarization and strain responses for {001}, {011} and {111}-oriented crystals of PMN-0.32PT: (a) *P*<sup>3</sup> − *E*3 curves and (b) ε<sup>33</sup> − *E*3 curves. (c) Relationship between rhombohedral (solid lines) and cubic (dashed lines) coordinate systems. Subscripts *r* and *c* denote directions with respect to the rhombohedral and cubic

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Some researchers attribute the high piezoelectric coefficients along {001} and {011} oriented ferroelectric single crystals to the engineered domain state [4, 11]. It has also reported that, however, the piezoelectric coefficient along the {001} direction of single crystals with monodomain structure is comparable to that of crystals with multi-domain structure [17], imply‐ ing the origins of the high piezoelectric constants of PMN-0.32PT single crystals may not be due to the engineered domain state. Instead, it could be due to the effect of crystal lattice properties. To further explore this issue, we follow Ref. [17] to calculate the piezoelectric co‐

along an arbitrary direction in a mono-domain crystal. For a direction defined

<sup>33</sup> of PMN-0.32PT in the

coordinate systems, respectively. (d) Orientational dependence of piezoelectric coefficient *d* \*

polar plane.

efficients *d* \*

*ij*

Electromechanical Coupling Multiaxial Experimental and Micro-Constitutive Model Study of... http://dx.doi.org/10.5772/52540 301

tion orientations along the pseudo-cubic {111} for un-poled rhombohedral PMN-0.32PT sin‐ gle crystals. Upon poling, the dipoles switch as close as possible to the applied electric field direction: For {001} poled crystals, there are four equivalent polar vectors along the {111} ori‐ entation, with an inclined angle of -54.7º from the poling field (Fig. 1a); For {011} poled crys‐ tals, there are two equivalent polar vectors along the {111} direction (labeled as type 1 in Fig. 1b); For {111} poled crystals, there is one polar vector along {111} (type 1 in Fig. 1c). Accord‐

<sup>=</sup>*Pr* <sup>111</sup> / <sup>3</sup> and *Pr* <sup>011</sup>

), *Pr* <sup>001</sup>

). Therefore, the measured results are consistent with the

, respectively, which are very close to the measured ones (*Pr* <sup>001</sup>

and *Pr* <sup>011</sup>

(0.395C/m2

It is also seen from Figs. 3a and 3b that the coercive field is lowest for the {111} oriented crys‐ tals, and becomes successively higher for {001} and {110} orientations. This is same to Ref.25 except for {110} orientation. This trend of coercive field is due to two reasons: One is due to reorientation driving force being proportional to the component of electric field aligned with the rhombohedral direction; The other one is due to the domain switching process. In {111} oriented PMN-0.32PT single crystals, there are two types of domains (shown as type 1 and type 2 in Fig.1c). When the electric field is decreased from 0.5kV/mm to –0.05kV/mm, the strain first decreases linearly (see Fig.3b). When the electric field is decreased further, the type 1 domain switches to type 2 domains, leading to abrupt displacement change. When the electric field exceeds the coercive field –0.216kV/mm, the type 2 domains switch back to type 1 domain, recovering the deformation. In type 2 domain state, three equivalent polar vectors with an angle of 71º from the {111} direction can coexist and are separated by do‐ main walls across which the normal components of electric displacement and displacement jump are zero. Ideally, this type of domain walls has no associated local stress or electric field. So the existing of type 2 domain state and the largest component of electric field along polarization direction induces the lowest coercive field in {111} orientation poled crystal. In the {011} orientation crystals, there are also two types of domains (i.e., type 1 and type 2 do‐ main in Fig. 1b), the domain switching process is similar to {111}-poled crystals. In the type 2 domain state, however, the four possible polar vectors are perpendicular to the applied elec‐ tric field (Fig. 1b). It is thus difficult to switch type 2 domain to type 1 domain only by ap‐ plying electric field. Both the type 2 domain state and the smallest component of *E* contribute to the largest coercive field in {011} oriented crystals. In {001} oriented crystals, there will be no associated local stress or electric field, similar to type 2 domain state of {111} poled crystals. This feature again renders domain switching easy. On the other hand, the do‐ main structure of {001} poled crystals is stable [4] and the component of electric field is smaller, giving the coercive field higher than that of {111} poled crystals but lower than that

and *Pr* <sup>011</sup>

are predicted to

<sup>=</sup> <sup>2</sup>*Pr* <sup>111</sup> / <sup>3</sup>, respective‐

are

ing to the domain configurations in Fig. 1, the remnant polarizations *Pr* <sup>001</sup>

by *Pr* <sup>001</sup>

approximately related to *Pr* <sup>111</sup>

and *Pr* <sup>011</sup>

domain configurations shown in Fig. 1.

of {011} oriented crystals (see, Figs. 3a and 3b).

be 0.228 and 0.322C/m2

=0.227C/m2

300 Advances in Ferroelectrics

ly. By taking the measured value for *Pr* <sup>111</sup>

=0.324C/m2

**Figure 3.** Electric field induced polarization and strain responses for {001}, {011} and {111}-oriented crystals of PMN-0.32PT: (a) *P*<sup>3</sup> − *E*3 curves and (b) ε<sup>33</sup> − *E*3 curves. (c) Relationship between rhombohedral (solid lines) and cubic (dashed lines) coordinate systems. Subscripts *r* and *c* denote directions with respect to the rhombohedral and cubic coordinate systems, respectively. (d) Orientational dependence of piezoelectric coefficient *d* \* <sup>33</sup> of PMN-0.32PT in the polar plane.

Some researchers attribute the high piezoelectric coefficients along {001} and {011} oriented ferroelectric single crystals to the engineered domain state [4, 11]. It has also reported that, however, the piezoelectric coefficient along the {001} direction of single crystals with monodomain structure is comparable to that of crystals with multi-domain structure [17], imply‐ ing the origins of the high piezoelectric constants of PMN-0.32PT single crystals may not be due to the engineered domain state. Instead, it could be due to the effect of crystal lattice properties. To further explore this issue, we follow Ref. [17] to calculate the piezoelectric co‐ efficients *d* \* *ij* along an arbitrary direction in a mono-domain crystal. For a direction defined by the Euler angles (*φ*, *θ*, *ψ*) (see Fig. 3c), *d* \* *ij* is related to *dij* (measured along the principal crystallographic axes) by,

$$\boldsymbol{d}^{\uparrow} \,\_{33} \{ \boldsymbol{\phi}, \boldsymbol{\theta} \} = \boldsymbol{d}\_{33} \cos^{3} \boldsymbol{\theta} + (\boldsymbol{d}\_{15} + \boldsymbol{d}\_{31}) \cos \boldsymbol{\theta} \sin^{2} \boldsymbol{\theta} - \boldsymbol{d}\_{22} \cos \boldsymbol{\phi} \sin^{3} \boldsymbol{\theta} (\cos^{2} \boldsymbol{\phi} - 3 \sin^{2} \boldsymbol{\phi}) \tag{1}$$

however, there is only domain switching (i.e., switching between type 1 and type 2 do‐ mains) and no phase transformation occurs. During unloading, the *σ*<sup>33</sup> −*ε*<sup>33</sup> curves and *σ*<sup>33</sup> − *P*<sup>3</sup> curves of {011} and {111} show linear response since almost no domain switches back

Electromechanical Coupling Multiaxial Experimental and Micro-Constitutive Model Study of...


¯11>, <11

¯1>, and <1

¯1

¯1>. Since the <001>

s33(MPa)

**Figure 4.** Compressive loading and unloading stress cycles induced polarization and strain responses for {001}, {011}

The stress cycles of {001} poled crystals can be explained by the polarization vector rota‐ tion mechanism sketched in Fig. 5 as follows. In general two most possible mechanisms responsible for the observed features of {001} poled crystals in Figs.4 are domain switch‐ ing and polarization rotation associated phase transformation. Recall that the PMN-0.32PT single crystals considered here are in the R phase close to MPB (although in reality there could also be M or T phase in these crystals near MPB [8], [9] the R phase is nevertheless the dominant phase). It is noted that that, upon application of a field along the <001> po‐ ling axis of R phase domain engineered PMN-0.32PT single crystals, only four of the eight

components of these four polar vectors are completely equivalent, each domain wall can‐ not move under an external electric field along the <001> direction owing to the equiva‐ lent domain wall energies [32]. In other words, no ferroelectric domain switching is possible. Equally, no ferroelastic domain switching is expected when a compressive stress is applied along the <001> axis. (For PMN-PT systems with orthorhombic 4O <001> do‐ main engineered structure [33], 60º switching from <011> to <110> would be driven by an applied stress. But, this is not what we are considering). Such facts partly rationalize our hypothesis that the underlying mechanism associated with the observed behavior of the Rphased PMN-0.32PT single crystals considered in this paper is phase transformation. Nev‐ ertheless, it should be emphasized that since no in situ diffraction observation is conducted to properly identify the stress induced phase transformation in PMN-0.32PT single crystals, discussions in this paper on phase transition are solely inferred from the

[001] [011] [111]


D3(C/m<sup>2</sup>

)

[001] [011] [111]

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303

(Figs. 4a and 4b).


s33(MPa)


e33(microstrain)

polarization orientations are possible, i.e., <111>, <1

measured stress induced strain and polarization curves.

and {111}-oriented crystals of PMN-0.32PT: (a) σ<sup>33</sup> −ε33 curves and (b) σ<sup>33</sup> − *P*3 curves.

With *d*33=200pC/N taken from our measured value and *d*15=4100, *d*31=–90 and *d*22=1340pC/N, *d* \* 33 111 (0, *θ*) for PMN-0.32PT single crystals can be calculated and is shown in Fig. 3d. Note that in the case of *ϕ* =0 *θ*=-35.5º and -54.7º correspond to {011} and {001} direction, respec‐ tively. *d* \* 33 011 =1471pC/N and *d* \* 33 001 =2311pC/N can be inferred from Figs. 3d, which are not far away from our measured values (*d* <sup>33</sup> <sup>011</sup> =1049 pC/N and *<sup>d</sup>* <sup>33</sup> <sup>001</sup> =1828 pC/N). The small discrepancy between the predictions and measurements is believed to be due to the fact that the predictions are based on ideal mono-domain single crystals while the material parameters used in Eq. (1) are actually from less ideal mono-domain crystals (in fact, they are more or less multi-domain crystals). Nevertheless, we can conclude that the dominant contribution to the large {001} and {011} piezoelectric response should be the crystal aniso‐ tropy other than engineered domain state.

### **3.2. Crystallographic dependence of stress induced strain and polarization responses**

Figure 4 shows the measured *σ*<sup>33</sup> −*ε*33 and *σ*<sup>33</sup> − *P*3 curves for {001}, {011} and {111} oriented short circuited samples. The results indicate obvious crystallographic anisotropy in stress in‐ ducing responses. From domain switching viewpoint, there should be no significant defor‐ mation in the thickness dimension of {001} poled crystal samples under stress loading, except for the elastic strain. However the longitudinal strain of {001} is contractive with a maximum magnitude of 0.3% under –40MPa. The contractive strain of {001} oriented crys‐ tals is about twice as much as those of {011} (about –0.18%) and {111} (about –0.17%) orient‐ ed crystals under a loading of –40MPa (Fig. 4a). This abnormal behavior of {001}-oriented crystals lies in that the mechanism underlying the stress induced response of {001}-oriented crystals is the R to O and T phase transition (see, Fig. 1a) rather than domain switching for {011} and {111} oriented crystals: the lattice distortion due to phase transition induces large deformation in the thickness dimension of {001} poled crystals. In {011} and {111}- poled crystals, on the contrary, the type 2 multi-domain state induced by compressive stress is the stable and preferred state (Figs. 1b and 1c), and can form more easily by domain switching than phase transformation. Therefore, the stress induced strain and polarization curves in {011} and {111} orientation crystals are similar to those of ferroelectric polycrystals of which domain switching is also the dominant deformation mechanism (Figs. 4a and 4b). During the unloading of {001} oriented crystals (corresponding to the returning to R phase of the unstable O phase), the *σ*<sup>33</sup> −*ε*33 and *σ*<sup>33</sup> − *P*3 curves show obvious nonlinear behavior. Note that the remnant strain and polarization at the end of unloading are attributed to the stable T phase which does not switches back to the R phase. For {011} and {111}-oriented crystals, however, there is only domain switching (i.e., switching between type 1 and type 2 do‐ mains) and no phase transformation occurs. During unloading, the *σ*<sup>33</sup> −*ε*<sup>33</sup> curves and *σ*<sup>33</sup> − *P*<sup>3</sup> curves of {011} and {111} show linear response since almost no domain switches back (Figs. 4a and 4b).

by the Euler angles (*φ*, *θ*, *ψ*) (see Fig. 3c), *d* \*

 q

=1471pC/N and *d* \*

not far away from our measured values (*d*

tropy other than engineered domain state.

= ++ -

33 001

crystallographic axes) by,

302 Advances in Ferroelectrics

fq

33 011

*d* \* 33 111

tively. *d* \*

*ij*

\* 3 2 32 2 <sup>33</sup> <sup>33</sup> 15 31 <sup>22</sup> *d d dd* ( , ) cos ( )cos sin cos sin (cos 3sin )

qq

With *d*33=200pC/N taken from our measured value and *d*15=4100, *d*31=–90 and *d*22=1340pC/N,

that in the case of *ϕ* =0 *θ*=-35.5º and -54.7º correspond to {011} and {001} direction, respec‐

small discrepancy between the predictions and measurements is believed to be due to the fact that the predictions are based on ideal mono-domain single crystals while the material parameters used in Eq. (1) are actually from less ideal mono-domain crystals (in fact, they are more or less multi-domain crystals). Nevertheless, we can conclude that the dominant contribution to the large {001} and {011} piezoelectric response should be the crystal aniso‐

**3.2. Crystallographic dependence of stress induced strain and polarization responses**

Figure 4 shows the measured *σ*<sup>33</sup> −*ε*33 and *σ*<sup>33</sup> − *P*3 curves for {001}, {011} and {111} oriented short circuited samples. The results indicate obvious crystallographic anisotropy in stress in‐ ducing responses. From domain switching viewpoint, there should be no significant defor‐ mation in the thickness dimension of {001} poled crystal samples under stress loading, except for the elastic strain. However the longitudinal strain of {001} is contractive with a maximum magnitude of 0.3% under –40MPa. The contractive strain of {001} oriented crys‐ tals is about twice as much as those of {011} (about –0.18%) and {111} (about –0.17%) orient‐ ed crystals under a loading of –40MPa (Fig. 4a). This abnormal behavior of {001}-oriented crystals lies in that the mechanism underlying the stress induced response of {001}-oriented crystals is the R to O and T phase transition (see, Fig. 1a) rather than domain switching for {011} and {111} oriented crystals: the lattice distortion due to phase transition induces large deformation in the thickness dimension of {001} poled crystals. In {011} and {111}- poled crystals, on the contrary, the type 2 multi-domain state induced by compressive stress is the stable and preferred state (Figs. 1b and 1c), and can form more easily by domain switching than phase transformation. Therefore, the stress induced strain and polarization curves in {011} and {111} orientation crystals are similar to those of ferroelectric polycrystals of which domain switching is also the dominant deformation mechanism (Figs. 4a and 4b). During the unloading of {001} oriented crystals (corresponding to the returning to R phase of the unstable O phase), the *σ*<sup>33</sup> −*ε*33 and *σ*<sup>33</sup> − *P*3 curves show obvious nonlinear behavior. Note that the remnant strain and polarization at the end of unloading are attributed to the stable T phase which does not switches back to the R phase. For {011} and {111}-oriented crystals,

(0, *θ*) for PMN-0.32PT single crystals can be calculated and is shown in Fig. 3d. Note

is related to *dij*

 fq

<sup>33</sup> <sup>011</sup> =1049 pC/N and *<sup>d</sup>*

(measured along the principal

 f

<sup>33</sup> <sup>001</sup> =1828 pC/N). The

 f

=2311pC/N can be inferred from Figs. 3d, which are

*d* - (1)

**Figure 4.** Compressive loading and unloading stress cycles induced polarization and strain responses for {001}, {011} and {111}-oriented crystals of PMN-0.32PT: (a) σ<sup>33</sup> −ε33 curves and (b) σ<sup>33</sup> − *P*3 curves.

The stress cycles of {001} poled crystals can be explained by the polarization vector rota‐ tion mechanism sketched in Fig. 5 as follows. In general two most possible mechanisms responsible for the observed features of {001} poled crystals in Figs.4 are domain switch‐ ing and polarization rotation associated phase transformation. Recall that the PMN-0.32PT single crystals considered here are in the R phase close to MPB (although in reality there could also be M or T phase in these crystals near MPB [8], [9] the R phase is nevertheless the dominant phase). It is noted that that, upon application of a field along the <001> po‐ ling axis of R phase domain engineered PMN-0.32PT single crystals, only four of the eight polarization orientations are possible, i.e., <111>, <1 ¯11>, <11 ¯1>, and <1 ¯1 ¯1>. Since the <001> components of these four polar vectors are completely equivalent, each domain wall can‐ not move under an external electric field along the <001> direction owing to the equiva‐ lent domain wall energies [32]. In other words, no ferroelectric domain switching is possible. Equally, no ferroelastic domain switching is expected when a compressive stress is applied along the <001> axis. (For PMN-PT systems with orthorhombic 4O <001> do‐ main engineered structure [33], 60º switching from <011> to <110> would be driven by an applied stress. But, this is not what we are considering). Such facts partly rationalize our hypothesis that the underlying mechanism associated with the observed behavior of the Rphased PMN-0.32PT single crystals considered in this paper is phase transformation. Nev‐ ertheless, it should be emphasized that since no in situ diffraction observation is conducted to properly identify the stress induced phase transformation in PMN-0.32PT single crystals, discussions in this paper on phase transition are solely inferred from the measured stress induced strain and polarization curves.

and mechanical loading history [9], [14]). Upon unloading, the O phase switches back to R phase when the compression lower than -10MPa (Figs. 4, and 5(d)). On the other hand T phase is a stable phase. It will not switch back to R phase upon unloading, leading to rem‐ nant strain of about -0.07% for and remnant polarization of about -0.13C/m2 at the end of the -40MPa stress cycle (Figs. 4(a) and 4(b), Fig. 5(e)). After the stress cycle, a unipolar electric field of 0.5kV/mm in magnitude is applied to re-pole the single crystal to its initial state, i.e., the polarization vector completely switches back. As a result, the remnant strain and polari‐

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305

zation diminish to zero at the end of the re-polarization process (Figs. 5(a) and 5(e)).

**3.3. Crystallographic dependence of electric field induced behavior at constant bias**

The *ε*<sup>33</sup> −*E*3 "butterfly" curves and *P*<sup>3</sup> −*E*3 hysteresis loops of {001}, {011} and {111} poled crystals under different constant compressive bias stresses are shown in Fig. 6. Depend‐ ence of the electric coercive *Ec*, remnant polarization *Pr*, dielectric permittivity *χ*33, piezo‐ electric coefficient *d*33 and aggregate strain *Δε* for {001}, {011} and {111} oriented crystals on the compressive bias stress is summarized in Fig. 7. The aggregate strain *Δε* is defined as the difference between the maximum and minimum strain for a complete responsive butterfly curves. Similar to *d*33, *χ*33 are calculated by *ΔP*<sup>3</sup> / *ΔE*3 where *ΔP*3 is the polariza‐ tion difference between –0.05 and +0.05kV/mm. The calculated *χ*33 within such a small field range is almost equal to the slope of *P*<sup>3</sup> −*E*3 hysteresis loops when the electric field passes through zero. The calculated *χ*33 includes both the reversible (intrinsic dielectric property) and irreversible (extrinsic domain switching and phase transformation related property) contributions of the material, which is generally higher than the permittivity

The influence of the preloaded compressive stress on the aggregate strain *Δε*, remnant po‐ larization *Pr* and piezoelectric coefficient *d*33 seems to be similar for {001}, {011} and {111} oriented crystals: The remnant polarization *Pr* decreases with increasing the magnitude of the compressive prestress; The aggregate strain *Δε* and piezoelectric coefficient *d*33 first in‐ crease and then decrease with the magnitude of the prestress increasing. As suggested earli‐ er, however, the underlying mechanisms for the electromechanical behavior of {001} {011}, and {111} oriented crystals are different. The change of the dielectric permittivity *χ*<sup>33</sup> {001} oriented crystals under compressive stress is similar to that of *χ*<sup>33</sup> induced by temperature: near the phase transformation temperature there is a peak in the *χ*33 curves, As is shown in Fig. 7c, there is a small peak near –6MPa and a big peak near –20MPa in *χ*<sup>33</sup> curves for {001} oriented crystals. This may be due to R-M (near –6MPa) and M-R and M-O phase transfor‐ mation (near –20MPa). On the other hand, there is no obvious peak in *χ*<sup>33</sup> curves for {011} and {111}-oriented crystals (Fig.7c). The change of *χ*33 curves induced by compressive stress and the aforementioned stress induced strain and polarization responses are consistent with the hypothesis that there is phase transformation for {001}-orientated crystals under com‐

**compressive stress**

pressive stress.

measured by a dynamic method [32].

**Figure 5.** Schematic drawing of the polarization rotation from R to T and O phases due to <001> direction compres‐ sion: (a) the initial state, (b) polarization starts to rotate from R to O and T via two pathways MA and MC, (c) R phase completely transformed, (d) O phase starts to switch back to R phase upon unloading, (e) O phase completely switch‐ ed back to R phase. The states (a) to (e) correspond to the stress levels marked in Figs. 2(a) and 2(b). At the initial poled state (a), PMN-0.32PT possesses four equivalent <111>-oriented polarizations vector close to <001> direction though only one polarization rotation processes is shown here.

Fig. 5 illustrates the polarization rotation mechanism of PMN-0.32PT single crystals under stress loading and unloading cycle. The polarization vector states shown in Figs. 5(a)-5(e) correspond to the stress levels marked on the curves in Figs. 4 for the -40MPa stress cycle. In the initial poled state, PMN-0.32PT single crystals possess four equivalent <111> polariza‐ tions, with only one shown in Fig. 5a for the sake of clarity. Upon loading, polarization vec‐ tor starts to rotate from R to O and T phases through intermediate phases MA and MC, when the compressive stress exceeds about 15MPa in magnitude (Fig. 5(b)). This gives a mixture of R and T phases and remarkable augment in the polarization and strain change around *σ*<sup>33</sup> =-20MPa (Figs. 4). When the presence of an electric field along <110> direction, polarization rotation only occurs from R to O under compression along <001> direction. In the absence of electric field along <110>, however, almost all the polarization vectors may switch to T and O phases when the compression exceed 30MPa in magnitude (Fig. 5(c)). When the magni‐ tude of the compressive loading is further increased, there is no more polarization rotation and the PMN-0.32PT single crystal shows linear response in Figs. 4. It is noted that O phase is usually unstable (its free energy balance between R and T phases depends on the electric and mechanical loading history [9], [14]). Upon unloading, the O phase switches back to R phase when the compression lower than -10MPa (Figs. 4, and 5(d)). On the other hand T phase is a stable phase. It will not switch back to R phase upon unloading, leading to rem‐ nant strain of about -0.07% for and remnant polarization of about -0.13C/m2 at the end of the -40MPa stress cycle (Figs. 4(a) and 4(b), Fig. 5(e)). After the stress cycle, a unipolar electric field of 0.5kV/mm in magnitude is applied to re-pole the single crystal to its initial state, i.e., the polarization vector completely switches back. As a result, the remnant strain and polari‐ zation diminish to zero at the end of the re-polarization process (Figs. 5(a) and 5(e)).

*R*

304 Advances in Ferroelectrics

*T O*

only one polarization rotation processes is shown here.

*MC <sup>T</sup>*

*R*

*T O*

*O*

s33

*MC*

( ) *a* ( ) *b* ( ) *c*

( ) *e* ( ) *d*

**Figure 5.** Schematic drawing of the polarization rotation from R to T and O phases due to <001> direction compres‐ sion: (a) the initial state, (b) polarization starts to rotate from R to O and T via two pathways MA and MC, (c) R phase completely transformed, (d) O phase starts to switch back to R phase upon unloading, (e) O phase completely switch‐ ed back to R phase. The states (a) to (e) correspond to the stress levels marked in Figs. 2(a) and 2(b). At the initial poled state (a), PMN-0.32PT possesses four equivalent <111>-oriented polarizations vector close to <001> direction though

Fig. 5 illustrates the polarization rotation mechanism of PMN-0.32PT single crystals under stress loading and unloading cycle. The polarization vector states shown in Figs. 5(a)-5(e) correspond to the stress levels marked on the curves in Figs. 4 for the -40MPa stress cycle. In the initial poled state, PMN-0.32PT single crystals possess four equivalent <111> polariza‐ tions, with only one shown in Fig. 5a for the sake of clarity. Upon loading, polarization vec‐ tor starts to rotate from R to O and T phases through intermediate phases MA and MC, when the compressive stress exceeds about 15MPa in magnitude (Fig. 5(b)). This gives a mixture of R and T phases and remarkable augment in the polarization and strain change around *σ*<sup>33</sup> =-20MPa (Figs. 4). When the presence of an electric field along <110> direction, polarization rotation only occurs from R to O under compression along <001> direction. In the absence of electric field along <110>, however, almost all the polarization vectors may switch to T and O phases when the compression exceed 30MPa in magnitude (Fig. 5(c)). When the magni‐ tude of the compressive loading is further increased, there is no more polarization rotation and the PMN-0.32PT single crystal shows linear response in Figs. 4. It is noted that O phase is usually unstable (its free energy balance between R and T phases depends on the electric

*M <sup>A</sup>*

*R*

*<sup>R</sup> <sup>R</sup>*

*MC*

*T O*

*T O*

s33

*M <sup>A</sup>*

### **3.3. Crystallographic dependence of electric field induced behavior at constant bias compressive stress**

The *ε*<sup>33</sup> −*E*3 "butterfly" curves and *P*<sup>3</sup> −*E*3 hysteresis loops of {001}, {011} and {111} poled crystals under different constant compressive bias stresses are shown in Fig. 6. Depend‐ ence of the electric coercive *Ec*, remnant polarization *Pr*, dielectric permittivity *χ*33, piezo‐ electric coefficient *d*33 and aggregate strain *Δε* for {001}, {011} and {111} oriented crystals on the compressive bias stress is summarized in Fig. 7. The aggregate strain *Δε* is defined as the difference between the maximum and minimum strain for a complete responsive butterfly curves. Similar to *d*33, *χ*33 are calculated by *ΔP*<sup>3</sup> / *ΔE*3 where *ΔP*3 is the polariza‐ tion difference between –0.05 and +0.05kV/mm. The calculated *χ*33 within such a small field range is almost equal to the slope of *P*<sup>3</sup> −*E*3 hysteresis loops when the electric field passes through zero. The calculated *χ*33 includes both the reversible (intrinsic dielectric property) and irreversible (extrinsic domain switching and phase transformation related property) contributions of the material, which is generally higher than the permittivity measured by a dynamic method [32].

The influence of the preloaded compressive stress on the aggregate strain *Δε*, remnant po‐ larization *Pr* and piezoelectric coefficient *d*33 seems to be similar for {001}, {011} and {111} oriented crystals: The remnant polarization *Pr* decreases with increasing the magnitude of the compressive prestress; The aggregate strain *Δε* and piezoelectric coefficient *d*33 first in‐ crease and then decrease with the magnitude of the prestress increasing. As suggested earli‐ er, however, the underlying mechanisms for the electromechanical behavior of {001} {011}, and {111} oriented crystals are different. The change of the dielectric permittivity *χ*<sup>33</sup> {001} oriented crystals under compressive stress is similar to that of *χ*<sup>33</sup> induced by temperature: near the phase transformation temperature there is a peak in the *χ*33 curves, As is shown in Fig. 7c, there is a small peak near –6MPa and a big peak near –20MPa in *χ*<sup>33</sup> curves for {001} oriented crystals. This may be due to R-M (near –6MPa) and M-R and M-O phase transfor‐ mation (near –20MPa). On the other hand, there is no obvious peak in *χ*<sup>33</sup> curves for {011} and {111}-oriented crystals (Fig.7c). The change of *χ*33 curves induced by compressive stress and the aforementioned stress induced strain and polarization responses are consistent with the hypothesis that there is phase transformation for {001}-orientated crystals under com‐ pressive stress.

As suggested by Fu et al. [7], for {001} oriented crystal origins of large aggregate strain and high piezoelectric coefficient at a moderate compressive bias stress (i.e., around -20MPa for the single crystals considered here) may be attributed to the stress induced intermediate states between rhombohedral and tetragonal phases. Under a bias stress of about -20MPa, PMN-0.32PT single crystals, after a phase transformation, are in a state of monoclinic phase which has a larger *c* / *a* (*c* and *a* are lattice parameters) and a smaller polarization component along the field direction than those of rhombohedral phase [20], implying electric field in‐ duced greater aggregate strain and piezoelectric coefficient (Figs. 7d and 7e). With the mag‐ nitude of the compressive bias stress increased further (i.e., *σ*33=-30 and -40MPa), the {001} oriented single crystals are in the state of a mixture of orthorhombic and tetragonal phases. Although O and T phases also have large *c* to *a* ratio, the presence of a large compression has an opposite effect (i.e., preventing larger deformation induced by electric field) and re‐ sults in small aggregate strain (Fig.7e).

As is noted, the polarization rotation introduced by compression gives rise to *θ* in *d*<sup>33</sup> \* (*ϕ*, *θ*) in the range between -54.7º and -90º (the angle between loading and polarization direction of T or O phase) under -20MPa, and results in larger *d*<sup>33</sup> \* in accordance with Eq. (1) (see in Fig. 7d). When the magnitude of the compressive bias stress increases further, *θ* approaches 90º and *d*<sup>33</sup> \* becomes smaller. Meanwhile, the remnant polarization *Pr* and coercive field *Ec* de‐ crease monotonically with the applied compressive bias stress because of the decreased component polarization along the {001} direction under compression (Fig.7b).

Under zero stress, the initial state of {011}-oriented crystals is of multi-domain with two equivalent polarization directions (Fig.1b). For {111}-oriented crystals, the initial state is of mono-domain with polarization direction in the {111} direction (Fig.1c). When the applied electric field decreases from 0.5kV/mm to –*Ec*, the strain and polarization decrease linearly. Upon approaching –*Ec*, domain state switches to four polar domain state for {011} oriented crystals (type 2 in Fig. 1b) and three polar domain state for {111} oriented crystals (type 2 in Fig. 1c), giving the jumps in strain and polarization responses. When electric field exceeds – *Ec*, domain complete second switching, from four polar domain state to two polar domain state for {011} oriented crystals and from three polar domain state to mono-domain state for {111} oriented crystals. When the electric field reaches -0.5kV/mm, the strain and polariza‐ tion change linearly again (Fig. 7).

Note that compressive stress can induce domain switching in {011} and {111}-orientated crystals. When compressive stress is superimposed on the samples, the shapes of *ε*<sup>33</sup> −*E*3 and *P*<sup>3</sup> −*E*3 curves are different from those under the zero stress state (Fig.6c-f). Due to compres‐ sion inducing depolarization, the remnant polarization decreases in {011} and {111} oriented crystals (Fig. 7b). A low compressive stress (e.g., -5MPa for {011}, -7MPa for {111} ) can lead to type 1 to type 2 domain switching. As a result, more domains take part in switching dur‐ ing the electric field loading cycle and larger aggregate strain*Δε* than under zero compres‐ sive stress is observed. This partly explains the observed features in Fig. 7e.

12




**Figure.6** Electric field induced




33 (microstrain)

0

0.5

[011]

33 (microstrain)

(c)


> -30 -40

33 (microstrain)

(a)


[001]



> -0.4 -6 -14






P3 (C/m2

(b) for {001}-oriented; (c) and (d) for {011}-oriented, (e) and (f) for {111}-oriented. **Figure 6.** Electric field induced ε<sup>33</sup> <sup>−</sup> *<sup>E</sup>*3 and *P*<sup>3</sup> <sup>−</sup> *<sup>E</sup>*3curves at different compressive bias stresses: (a) and (b) for {001}-

)

(f)

0

0.2

0.4

P3 (C/m2

)

(d)

P3 (C/m2

)

(b)

Electromechanical Coupling Multiaxial Experimental and Micro-Constitutive Model Study of...



E3(kV/mm)


E3(kV/mm)

33 3 *E* and *P*3 3 *E* curves at different compressive bias stresses: (a) and

(kV/mm)


http://dx.doi.org/10.5772/52540

[001]


[011]

[111]

E3

(kV/mm)

E3



oriented; (c) and (d) for {011}-oriented, (e) and (f) for {111}-oriented.

E3(kV/mm)

(kV/mm)

E3

(e) [111]

Electromechanical Coupling Multiaxial Experimental and Micro-Constitutive Model Study of... http://dx.doi.org/10.5772/52540 307

As suggested by Fu et al. [7], for {001} oriented crystal origins of large aggregate strain and high piezoelectric coefficient at a moderate compressive bias stress (i.e., around -20MPa for the single crystals considered here) may be attributed to the stress induced intermediate states between rhombohedral and tetragonal phases. Under a bias stress of about -20MPa, PMN-0.32PT single crystals, after a phase transformation, are in a state of monoclinic phase which has a larger *c* / *a* (*c* and *a* are lattice parameters) and a smaller polarization component along the field direction than those of rhombohedral phase [20], implying electric field in‐ duced greater aggregate strain and piezoelectric coefficient (Figs. 7d and 7e). With the mag‐ nitude of the compressive bias stress increased further (i.e., *σ*33=-30 and -40MPa), the {001} oriented single crystals are in the state of a mixture of orthorhombic and tetragonal phases. Although O and T phases also have large *c* to *a* ratio, the presence of a large compression has an opposite effect (i.e., preventing larger deformation induced by electric field) and re‐

As is noted, the polarization rotation introduced by compression gives rise to *θ* in *d*<sup>33</sup>

in the range between -54.7º and -90º (the angle between loading and polarization direction of

7d). When the magnitude of the compressive bias stress increases further, *θ* approaches 90º

crease monotonically with the applied compressive bias stress because of the decreased

Under zero stress, the initial state of {011}-oriented crystals is of multi-domain with two equivalent polarization directions (Fig.1b). For {111}-oriented crystals, the initial state is of mono-domain with polarization direction in the {111} direction (Fig.1c). When the applied electric field decreases from 0.5kV/mm to –*Ec*, the strain and polarization decrease linearly. Upon approaching –*Ec*, domain state switches to four polar domain state for {011} oriented crystals (type 2 in Fig. 1b) and three polar domain state for {111} oriented crystals (type 2 in Fig. 1c), giving the jumps in strain and polarization responses. When electric field exceeds – *Ec*, domain complete second switching, from four polar domain state to two polar domain state for {011} oriented crystals and from three polar domain state to mono-domain state for {111} oriented crystals. When the electric field reaches -0.5kV/mm, the strain and polariza‐

Note that compressive stress can induce domain switching in {011} and {111}-orientated crystals. When compressive stress is superimposed on the samples, the shapes of *ε*<sup>33</sup> −*E*3 and *P*<sup>3</sup> −*E*3 curves are different from those under the zero stress state (Fig.6c-f). Due to compres‐ sion inducing depolarization, the remnant polarization decreases in {011} and {111} oriented crystals (Fig. 7b). A low compressive stress (e.g., -5MPa for {011}, -7MPa for {111} ) can lead to type 1 to type 2 domain switching. As a result, more domains take part in switching dur‐ ing the electric field loading cycle and larger aggregate strain*Δε* than under zero compres‐

sive stress is observed. This partly explains the observed features in Fig. 7e.

component polarization along the {001} direction under compression (Fig.7b).

\* becomes smaller. Meanwhile, the remnant polarization *Pr* and coercive field *Ec* de‐

\* (*ϕ*, *θ*)

\* in accordance with Eq. (1) (see in Fig.

sults in small aggregate strain (Fig.7e).

tion change linearly again (Fig. 7).

and *d*<sup>33</sup>

306 Advances in Ferroelectrics

T or O phase) under -20MPa, and results in larger *d*<sup>33</sup>

(b) for {001}-oriented; (c) and (d) for {011}-oriented, (e) and (f) for {111}-oriented. **Figure 6.** Electric field induced ε<sup>33</sup> <sup>−</sup> *<sup>E</sup>*3 and *P*<sup>3</sup> <sup>−</sup> *<sup>E</sup>*3curves at different compressive bias stresses: (a) and (b) for {001} oriented; (c) and (d) for {011}-oriented, (e) and (f) for {111}-oriented.

33 3 *E* and *P*3 3 *E* curves at different compressive bias stresses: (a) and

12

**Figure.6** Electric field induced

**4. The first principle calculation of stress-strain**

0 5 10 15 20 25 30 35 40

**4 The first principle calculation of stress-strain**


ient in calculation and it keeps the similar ferroelectric to PMN-PT.

calculation and it keeps the similar ferroelectric to PMN-PT.

on the first result. Each increment of strain in this paper is 0.5%, untill 4%.

constant of R phase BaTiO3 ferroelectric single crystal is 4.001

14

**Figure 8.** The cell calculation model of BaTiO3 in R phase

[100]

[010]

[001]

PMN-PT and BaTiO3 have the similar ABO3 structure, so they have the similar ferroelectric properties. There is only Ti4+ particle in B site of BaTiO3, however, there is not only Ti4+ but also minim Mn4+ and Ni2+ particle in B site of PMN-PT. So the single cell of BaTiO3 is conven‐

piezoelectric constant 33 *d* , (d) relative dielectric constant 33/0, and (e) and aggregate strain <sup>33</sup>

**Figure.7** Effect of compressive bias stress on (a) the coercive field *Ec* , (b) remnant polarization *Pr* , (c)

Electromechanical Coupling Multiaxial Experimental and Micro-Constitutive Model Study of...

PMN-PT and BaTiO3 have the similar ABO3 structure, so they have the similar ferroelectric properties. There is only Ti4+ particle in B site of BaTiO3, however, there is not only Ti4+ but also minim Mn4+ and Ni2+ particle in B site of PMN-PT. So the single cell of BaTiO3 is convenient in

In this paper, we calculate a single cell of BaTiO3, the single cell should be the smallest periodic reduplicate cell, it includes one Ti4+ particle, three O2- particles and one Ba2+ particle(Fig.8). It is suggested that the initialized state of BaTiO3 ferroelectric single crystal is R phase after {001} oriented polarization. Refer to literature**Error! Reference source not found.**, the crystal lattice

single cell is shown in table 1.The loading along {001} direction is carried out through application increasing strain by degrees. Stress and other parameters under each strain level is calculated by VASP. Calculation under each strain level include two steps. Firstly, the particle coordinate of single cell under each strain level is calcluated by first principle molecular dynamic method. Secondly, Stress and other parameters are obtained through relaxation that is

*PR*

*c* / 2

2- O1

<sup>4</sup> Ti

*b* / 2

2- O2

2-

.

309

http://dx.doi.org/10.5772/52540

A , the coordinate of particle in

In this paper, we calculate a single cell of BaTiO3, the single cell should be the smallest peri‐ odic reduplicate cell, it includes one Ti4+ particle, three O2- particles and one Ba2+ particle (Fig.8). It is suggested that the initialized state of BaTiO3 ferroelectric single crystal is R phase after {001} oriented polarization. Refer to literature [34], the crystal lattice constant of

R phase BaTiO3 ferroelectric single crystal is 4.001A, the coordinate of particle in single cell is shown in table 1.The loading along {001} direction is carried out through application in‐ creasing strain by degrees. Stress and other parameters under each strain level is calculated by VASP. Calculation under each strain level include two steps. Firstly, the particle coordi‐ nate of single cell under each strain level is calcluated by first principle molecular dynamic method. Secondly, Stress and other parameters are obtained through relaxation that is based

based on the first result. Each increment of strain in this paper is 0.5%, untill 4%.

2- O3

*a* / 2

**Figure.8** The cell calculation model of BaTiO3 in R phase

<sup>2</sup> Ba

**4.1. Calculation methodology**

**4.1 Calculation methodology** 

[001] [011] [111]

0

0.5

1

**33**(microstrain)

(e)

1.5

2

**Figure 7.** Effect of compressive bias stress on (a) the coercive field *Ec*, (b) remnant polarization *Pr*, (c) piezoelectric con‐ stant *d*33, (d) relative dielectric constant *X*33/*X*0, and (e) and aggregate strainΔε33.

.

### **4. The first principle calculation of stress-strain** -33(MPa)

0 5 10 15 20 25 30 35 40

### **4.1. Calculation methodology Figure.7** Effect of compressive bias stress on (a) the coercive field *Ec* , (b) remnant polarization *Pr* , (c) piezoelectric constant 33 *d* , (d) relative dielectric constant 33/0, and (e) and aggregate strain <sup>33</sup>

[001] [011]

0

0.5

1

**33**(microstrain)

(e)

1.5

2

0.05

0

0.5

1

De**33**(microstrain)

(e)

1.5

2

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40


stant *d*33, (d) relative dielectric constant *X*33/*X*0, and (e) and aggregate strainΔε33.

[001] [011] [111] -s33(MPa)

[001] [011]

[111]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

1000

2000

3000

d33(pC/N)

**Figure 7.** Effect of compressive bias stress on (a) the coercive field *Ec*, (b) remnant polarization *Pr*, (c) piezoelectric con‐

(d)

4000

5000

Pr(C/mm

)

(b)

2

0 5 10 15 20 25 30 35 40


[001] [011] [111]

0 5 10 15 20 25 30 35 40


[001] [011] [111]

0.1

0.15

Ec(kV/mm)

(a)

0.2

0.25

0.3

308 Advances in Ferroelectrics

PMN-PT and BaTiO3 have the similar ABO3 structure, so they have the similar ferroelectric properties. There is only Ti4+ particle in B site of BaTiO3, however, there is not only Ti4+ but also minim Mn4+ and Ni2+ particle in B site of PMN-PT. So the single cell of BaTiO3 is conven‐ ient in calculation and it keeps the similar ferroelectric to PMN-PT. **4 The first principle calculation of stress-strain 4.1 Calculation methodology**  PMN-PT and BaTiO3 have the similar ABO3 structure, so they have the similar ferroelectric

In this paper, we calculate a single cell of BaTiO3, the single cell should be the smallest peri‐ odic reduplicate cell, it includes one Ti4+ particle, three O2- particles and one Ba2+ particle (Fig.8). It is suggested that the initialized state of BaTiO3 ferroelectric single crystal is R phase after {001} oriented polarization. Refer to literature [34], the crystal lattice constant of properties. There is only Ti4+ particle in B site of BaTiO3, however, there is not only Ti4+ but also minim Mn4+ and Ni2+ particle in B site of PMN-PT. So the single cell of BaTiO3 is convenient in calculation and it keeps the similar ferroelectric to PMN-PT.

R phase BaTiO3 ferroelectric single crystal is 4.001A, the coordinate of particle in single cell is shown in table 1.The loading along {001} direction is carried out through application in‐ creasing strain by degrees. Stress and other parameters under each strain level is calculated by VASP. Calculation under each strain level include two steps. Firstly, the particle coordi‐ nate of single cell under each strain level is calcluated by first principle molecular dynamic method. Secondly, Stress and other parameters are obtained through relaxation that is based on the first result. Each increment of strain in this paper is 0.5%, untill 4%. In this paper, we calculate a single cell of BaTiO3, the single cell should be the smallest periodic reduplicate cell, it includes one Ti4+ particle, three O2- particles and one Ba2+ particle(Fig.8). It is suggested that the initialized state of BaTiO3 ferroelectric single crystal is R phase after {001} oriented polarization. Refer to literature**Error! Reference source not found.**, the crystal lattice constant of R phase BaTiO3 ferroelectric single crystal is 4.001 A , the coordinate of particle in single cell is shown in table 1.The loading along {001} direction is carried out through application increasing strain by degrees. Stress and other parameters under each strain level is

based on the first result. Each increment of strain in this paper is 0.5%, untill 4%.

calculated by VASP. Calculation under each strain level include two steps. Firstly, the particle coordinate of single cell under each strain level is calcluated by first principle molecular dynamic method. Secondly, Stress and other parameters are obtained through relaxation that is

**Figure.8** The cell calculation model of BaTiO3 in R phase

**Figure 8.** The cell calculation model of BaTiO3 in R phase

14

### **4.2. Discussion**

In order to validate that the method a mentioned in this paper is correct, we calculate the elasticity constant *C*33 of T phase BaTiO3 using the method mentioned before. Fig.9 shows that the *C*33 is 180GPa, which is approach to the experimental value 189± 8Gpa reported in refer[34]. This means that the method used in this paper is trusty.

(a)



b

Electromechanical Coupling Multiaxial Experimental and Micro-Constitutive Model Study of...

A

a

http://dx.doi.org/10.5772/52540

311

<sup>4</sup><sup>+</sup> Ti


BaTiO <sup>3</sup>


e<sup>33</sup> (%)



c

]001[ *PO*

*c*

<sup>2</sup><sup>+</sup> Ba

*a*

]100[

]010[

**5. The viscoplastic model of ferroelection single**

different polarization states during loading.

**5.1. The viscoplastic model**

*b*

*PR*


s33

(GPa)



0

(b)

**Figure 10.** (a)The stress-strain curve of {001} oriented BaTiO3 ferroelectric single crystal calculated in first principle cal‐ culation, the scatter point is calculation value and the broken line fit the scatter point.(b) The sketch map of polariza‐ tion rotation during the calculation. The "a,b,c" letter in (a) is correspond to the "a,b,c" letter in (b), the letter indicates

From the experimental analysis, it is known that the <001>-oriented PMN-0.32PT single crystal undergo the R→M→T and R→M→O phase transformation under the compression. In order to establish a compact model and keep the essence of experiment, R→O phase

**Figure 9.** The stress-train curve of BaTiO3 ferrorlctric single crystal obtained by the method amentioned in this paper, the {001} oriented BaTiO3 ferroelectric single crystal is in T phase.

Stress-strain curve of {001} orientated R phase BaTiO3 calculated in first principle method is shown in Fig.10(a). Fig.10(a) shows that the result has simlar nonlinear behavior to the ex‐ perimental result of PMN-0.32PT during the loading that sketched in Fig.4(a), namely, there is obvious "a,b,c" steps during loading. We know that the nonlinear behavior of PMN-0.32PT shown in Fig.4(a)should be polarization rotation (R→M→O and R→M→T). The R→M→O is corresponding to the processing of polarization vector *PR* switching to *PO*, which is shown in Fig.10(b). table 2 is calculated coordinate of particle in BaTiO3, the coordi‐ nate is correspond to point A in fig.10(a). Table 3 is coordinate of particle in O phase BaTiO3 ferroelectric single crystal from refer [35]. Data in table 2 are equal to those in table 3, which indicates that "A" point in fig.10(a) should be O phase. With increasing strain, the coordi‐ nates of particle are unchangeable after "A", this indicates BaTiO3 ferroelectric single crystal is stable in O phase after "A". From the first principle calculation, we testify that the PMN-0.32PT ferroelectric single crystal undergoes polarization rotation(R→M→O), this prove the polarization rotation model is reasonable.

Electromechanical Coupling Multiaxial Experimental and Micro-Constitutive Model Study of... http://dx.doi.org/10.5772/52540 311

**Figure 10.** (a)The stress-strain curve of {001} oriented BaTiO3 ferroelectric single crystal calculated in first principle cal‐ culation, the scatter point is calculation value and the broken line fit the scatter point.(b) The sketch map of polariza‐ tion rotation during the calculation. The "a,b,c" letter in (a) is correspond to the "a,b,c" letter in (b), the letter indicates different polarization states during loading.

### **5. The viscoplastic model of ferroelection single**

#### **5.1. The viscoplastic model**

**4.2. Discussion**

310 Advances in Ferroelectrics

In order to validate that the method a mentioned in this paper is correct, we calculate the elasticity constant *C*33 of T phase BaTiO3 using the method mentioned before. Fig.9 shows that the *C*33 is 180GPa, which is approach to the experimental value 189± 8Gpa reported in


**Figure 9.** The stress-train curve of BaTiO3 ferrorlctric single crystal obtained by the method amentioned in this paper,

Stress-strain curve of {001} orientated R phase BaTiO3 calculated in first principle method is shown in Fig.10(a). Fig.10(a) shows that the result has simlar nonlinear behavior to the ex‐ perimental result of PMN-0.32PT during the loading that sketched in Fig.4(a), namely, there is obvious "a,b,c" steps during loading. We know that the nonlinear behavior of PMN-0.32PT shown in Fig.4(a)should be polarization rotation (R→M→O and R→M→T). The R→M→O is corresponding to the processing of polarization vector *PR* switching to *PO*, which is shown in Fig.10(b). table 2 is calculated coordinate of particle in BaTiO3, the coordi‐ nate is correspond to point A in fig.10(a). Table 3 is coordinate of particle in O phase BaTiO3 ferroelectric single crystal from refer [35]. Data in table 2 are equal to those in table 3, which indicates that "A" point in fig.10(a) should be O phase. With increasing strain, the coordi‐ nates of particle are unchangeable after "A", this indicates BaTiO3 ferroelectric single crystal is stable in O phase after "A". From the first principle calculation, we testify that the PMN-0.32PT ferroelectric single crystal undergoes polarization rotation(R→M→O), this

e33(%)

BaTiO3

refer[34]. This means that the method used in this paper is trusty.


the {001} oriented BaTiO3 ferroelectric single crystal is in T phase.

prove the polarization rotation model is reasonable.

s33

(GPa)

From the experimental analysis, it is known that the <001>-oriented PMN-0.32PT single crystal undergo the R→M→T and R→M→O phase transformation under the compression. In order to establish a compact model and keep the essence of experiment, R→O phase transformation of PMN-0.32PT is considered in this study. Polarization rotation cause bulk deformation and shear deformation associated with the slip planes. It is assumed that the corresponding deformation associated with the slip planes is shear dominated, a feature similar to that of the multi-slip system of the crystal plasticity. This similarity renders possi‐ ble to use crystal plasticity models to describe the transformation deformation of PMN-0.32PT single crystal.

The poled ferroelectric single is four domain state, the polarization vector is along <111>, <1 ¯ 11>, <11 ¯1> and <1 ¯1 ¯1> respectively. In this study, it is assumed that the polarization vector of R phase can switch to the polarization vector of O phase, such as <111> vector switches to <001>, the vector also can switch back from O phase to R phase. The polarization rotation has an analogy to crystal plasticity slip, so we suggest there is eight slip systems in PMN-0.32PT ferroelectric single crystal. According to the crystalgraphic theory, we suggest ferroelectric single crystal has possible 8 variants; transformation of variants can be charac‐ terized by the habit planes illustrated in Fig.11. where n denotes the unit normal to the habit plane and s refers to the direction of transformation.

In general, a criterion (i.e., the phase transformation criterion) exists for the phase transfor‐ mation of ferroelectric single crystal, and the material is assumed to undergo phase transfor‐ mation when at least one of the 8 variants satisfies the phase transformation criterion. This is detailed as follows. Upon loading, the condition to produce R→O polarization rotation on a specified habit plane is that the driving force G of that plane reaches the critical value G0O. The driving force is composed of the chemical driving force Gchem and mechanical driving force Gmech[36]

$$\mathbf{G} = \mathbf{G}\_{chem} + \mathbf{G}\_{mech} = \mathbf{G}\_{0O} \tag{2}$$

3

, 1 *a a*

=

 as

<sup>1</sup> ( )

being the unit normal to the habit plane and shear direction to the variant

Electromechanical Coupling Multiaxial Experimental and Micro-Constitutive Model Study of...

*a aa aa*

3

*a*

1

*i E Es* =

<sup>0</sup> ( ) *G TT chem* = b

0

*i i*

The chemical driving force in Eqs. (2) and (3) is assumed to be a linear function of the tem‐

where *T* and *T*0 denote the temperature in the single crystal and the equilibrium tempera‐ ture respectively, and *β* is the stress-temperature coefficient. Note that the equilibrium tem‐ perature *T*0 is defined as the average of the starting temperature of O phase transformation

<sup>1</sup> ( )

When applying the rate independent crystal theory based model, Eqs. (2) and (3), to simu‐ late the behavior of ferroelectric single crystals, one of the most computationally consuming tasks is to determine the set of instantaneously active transformation systems among the 8 possible variants at crystal level. This determination is usually achieved by an iterative pro‐ cedure and must be carried out at each loading step, requiring extensive computation. Note that in the crystal theory of plasticity, a similar problem exists whilst in a rate dependent viscoplastic version of crystal theory of plasticity, determination of the set of active transfor‐ mation systems is not necessary. As a result, computation effort can be reduced significant‐ ly. Following this idea, a viscoplastic version of Eqs. (21) and (32) are proposed and employed in this study. In the viscoplastic crystal model for PMN-0.32PT ferroelectric single

*i j*

t

2

a

where the Schmid factor is defined as follow

"*a*", respectively. The resolved stress *E <sup>a</sup>*

and that of R phase transformation, that is

with *mi*

tensor *Ei*

perature

and *ni*

*ij ij*

<sup>=</sup> å (5)

http://dx.doi.org/10.5772/52540

313

*ij i j j i* = + *mn mn* (6)

of the variant "*a*" is related to the electrical field

<sup>=</sup> å (7)

(8)

<sup>2</sup> *s s T OT* = + (9)

A similar condition holds for reverse transformation from O→R polarization rotation with a critical value G0A

$$\mathbf{G} = \mathbf{G}\_{chem} + \mathbf{G}\_{mech} = \mathbf{G}\_{0A} \tag{3}$$

In Eqs. (2) and (3), the mechanical driving force can be expressed by

$$G\_{mech}^{r} = \tau^a \boldsymbol{\gamma}^\* + E^a \boldsymbol{P}^\* \tag{4}$$

Where *τ <sup>a</sup>* and *E <sup>a</sup>* are the resolved stress on the "*a*" transformation system, and *γ* \* and *P* \* denote the associated transformation strain and transformation polarization. Following the crystal theory of plasticity, the resolved stress *τ <sup>a</sup>* of the variant "*a*" is related to the stress tensor *σij* and the Schmid factor *aij a* by

Electromechanical Coupling Multiaxial Experimental and Micro-Constitutive Model Study of... http://dx.doi.org/10.5772/52540 313

$$
\pi^a = \sum\_{i,j=1}^3 \alpha\_{ij}^a \sigma\_{ij} \tag{5}
$$

where the Schmid factor is defined as follow

transformation of PMN-0.32PT is considered in this study. Polarization rotation cause bulk deformation and shear deformation associated with the slip planes. It is assumed that the corresponding deformation associated with the slip planes is shear dominated, a feature similar to that of the multi-slip system of the crystal plasticity. This similarity renders possi‐ ble to use crystal plasticity models to describe the transformation deformation of

The poled ferroelectric single is four domain state, the polarization vector is along <111>, <1

R phase can switch to the polarization vector of O phase, such as <111> vector switches to <001>, the vector also can switch back from O phase to R phase. The polarization rotation has an analogy to crystal plasticity slip, so we suggest there is eight slip systems in PMN-0.32PT ferroelectric single crystal. According to the crystalgraphic theory, we suggest ferroelectric single crystal has possible 8 variants; transformation of variants can be charac‐ terized by the habit planes illustrated in Fig.11. where n denotes the unit normal to the habit

In general, a criterion (i.e., the phase transformation criterion) exists for the phase transfor‐ mation of ferroelectric single crystal, and the material is assumed to undergo phase transfor‐ mation when at least one of the 8 variants satisfies the phase transformation criterion. This is detailed as follows. Upon loading, the condition to produce R→O polarization rotation on a specified habit plane is that the driving force G of that plane reaches the critical value G0O. The driving force is composed of the chemical driving force Gchem and mechanical driving

A similar condition holds for reverse transformation from O→R polarization rotation with a

*r aa* \* *G EP mech* t g

are the resolved stress on the "*a*" transformation system, and *γ* \*

denote the associated transformation strain and transformation polarization. Following the

In Eqs. (2) and (3), the mechanical driving force can be expressed by

*a* by

¯1> respectively. In this study, it is assumed that the polarization vector of

*GG G G chem mech O*<sup>0</sup> =+= (2)

*GG G G chem mech A*<sup>0</sup> =+= (3)

\* = + (4)

of the variant "*a*" is related to the stress

¯

and *P* \*

PMN-0.32PT single crystal.

¯1> and <1

¯1

plane and s refers to the direction of transformation.

11>, <11

312 Advances in Ferroelectrics

force Gmech[36]

critical value G0A

Where *τ <sup>a</sup>*

tensor *σij*

and *E <sup>a</sup>*

crystal theory of plasticity, the resolved stress *τ <sup>a</sup>*

and the Schmid factor *aij*

$$\alpha\_{ij}^a = \frac{1}{2} (m\_i^a n\_j^a + m\_j^a n\_i^a) \tag{6}$$

with *mi* and *ni* being the unit normal to the habit plane and shear direction to the variant "*a*", respectively. The resolved stress *E <sup>a</sup>* of the variant "*a*" is related to the electrical field tensor *Ei*

$$E^a = \sum\_{i=1}^3 E\_i s\_i \tag{7}$$

The chemical driving force in Eqs. (2) and (3) is assumed to be a linear function of the tem‐ perature

$$G\_{chem} = \mathcal{B}(T - T\_0) \tag{8}$$

where *T* and *T*0 denote the temperature in the single crystal and the equilibrium tempera‐ ture respectively, and *β* is the stress-temperature coefficient. Note that the equilibrium tem‐ perature *T*0 is defined as the average of the starting temperature of O phase transformation and that of R phase transformation, that is

$$T\_0 = \frac{1}{2} (O\_s + T\_s) \tag{9}$$

When applying the rate independent crystal theory based model, Eqs. (2) and (3), to simu‐ late the behavior of ferroelectric single crystals, one of the most computationally consuming tasks is to determine the set of instantaneously active transformation systems among the 8 possible variants at crystal level. This determination is usually achieved by an iterative pro‐ cedure and must be carried out at each loading step, requiring extensive computation. Note that in the crystal theory of plasticity, a similar problem exists whilst in a rate dependent viscoplastic version of crystal theory of plasticity, determination of the set of active transfor‐ mation systems is not necessary. As a result, computation effort can be reduced significant‐ ly. Following this idea, a viscoplastic version of Eqs. (21) and (32) are proposed and employed in this study. In the viscoplastic crystal model for PMN-0.32PT ferroelectric single crystals all transformation systems are assumed to be instantaneously active of varying ex‐ tent, which is governed by a rate dependent viscoplastic law. In this paper, the phase trans‐ formation of variant "*a*" is assumed to comply with the following power law of viscoplasticity,

$$\dot{f}^a = \dot{f}\_0 \left| \frac{\mathbf{G}^a}{\mathbf{G}\_0} \right|^{\frac{1}{m} - 1} \frac{\mathbf{G}^a}{\mathbf{G}\_0} \left| \frac{\mathbf{c}}{\mathbf{c}\_0} \right|^{1/k} \tag{10}$$

*e E ij ijkl ij ij kij k d Cd*

 e

Increment of the total strain *dεij* can then expressed as the sum of an elastic component *dεij*

*e E tr ij ij ij ij dddd* eee

To validate the model, the corresponding calculation result is compared to experimental stress-strain curve. The material parameters used in this study are as follow [37, 38]. Materi‐

*<sup>R</sup>* =9.2GPa, *C*<sup>12</sup>

1 / *m*=11.5, 1 / *k* =1.1. Fig.12 shows the model predicted and experimental measured re‐ sponse. Result in fig.8 shows that stress-strain response of PMN-0.32PT can be predicted by

**Figure 11.** The comparison between ferroelectric phase transformation and plastic slip of single crystal, *n* is direction

*<sup>O</sup>* =40GPa*C*<sup>44</sup>

 e

= = *d d dE* (15)

Electromechanical Coupling Multiaxial Experimental and Micro-Constitutive Model Study of...

*<sup>E</sup>* and a transformation induced component *dεij*

=++ (16)

*<sup>R</sup>* =10.3GPa, *C*<sup>44</sup>

, *β* =0.004MPa/K, 1 / *m*=11, 1 / *k* =1.0. *T* =403, *T*<sup>0</sup> =298K. Material parame‐

*<sup>R</sup>* =6.9GPa. *γ* \*

*<sup>O</sup>* =28GPa. *G*0*<sup>O</sup>* =0.276MJ/m<sup>3</sup> *<sup>β</sup>* =0.039MPa/K,

*e* , 315

*tr* ,

http://dx.doi.org/10.5772/52540

=0.0033,

es

*<sup>O</sup>* =38GPa, *C*<sup>12</sup>

the developed constitutive model, with quantitative agreement.

normal to slip surface, *s* is direction along phase transformation.

a electrical field induced component *dεij*

al parameters of R phase are *C*<sup>11</sup>

**5.2. Numerical results**

*<sup>G</sup>*0*<sup>R</sup>* =0.5073MJ/m<sup>3</sup>

ters of O phase are *C*<sup>11</sup>

where *f* ˙ *a* is phase fraction transformation rate of variant "*a*", *f* ˙ <sup>0</sup> is reference phase fraction transformation rate, *G <sup>a</sup>* is the driving force of variant "*a*", *G*0 is the critical driving force, ex‐ ponents k and m are material parameters dictating the rate effect, *c* depends upon the phase fraction, *c*0 the reference value at initial state.

The phase fraction of variant "*a*" is calculated as,

$$
\xi^a = f^a \wr f\_0 \tag{11}
$$

Summation over all possible variants provides the phase fraction for the whole ferroelectric single crystal, that is

$$
\xi = \sum\_{a=1}^{8} \xi^a \tag{12}
$$

The incremental form of Eq. (12) can be written as

$$d\mathfrak{E}^a = df^a \wr f\_0 \tag{13}$$

The transformation strain tensor *εij tr* , which is associated with *d f <sup>a</sup>* , can be obtained as

$$d\boldsymbol{\omega}\_{ij}^{tr} = \sum\_{r=1}^{8} \boldsymbol{\alpha}\_{ij}^{a} \boldsymbol{\upgamma}^{\*} \boldsymbol{d} f^{a} \tag{14}$$

where *αij* is Schmid factor. *dεij tr* is increment of phase train. *γ* \* is the maximal strain transfor‐ mation during loading. *d f <sup>a</sup>* is increment of phase fraction.

Elastic strain and electrical field induced strain during loading can expressed as

$$d\boldsymbol{\varepsilon}\_{ij}^{\boldsymbol{\varepsilon}} = \mathbb{C}\_{ijkl} d\boldsymbol{\sigma}\_{ij} \qquad \qquad \qquad d\boldsymbol{\varepsilon}\_{ij}^{\boldsymbol{E}} = \boldsymbol{d}\_{klj} d\boldsymbol{E}\_{k} \tag{15}$$

Increment of the total strain *dεij* can then expressed as the sum of an elastic component *dεij e* , a electrical field induced component *dεij <sup>E</sup>* and a transformation induced component *dεij tr* ,

$$d\varepsilon\_{i\dot{\jmath}} = d\varepsilon\_{i\dot{\jmath}}^{\varepsilon} + d\varepsilon\_{i\dot{\jmath}}^{E} + d\varepsilon\_{i\dot{\jmath}}^{tr} \tag{16}$$

### **5.2. Numerical results**

crystals all transformation systems are assumed to be instantaneously active of varying ex‐ tent, which is governed by a rate dependent viscoplastic law. In this paper, the phase trans‐ formation of variant "*a*" is assumed to comply with the following power law of

<sup>1</sup> <sup>1</sup> 1/

= & & (10)

= *f f* (11)

<sup>=</sup> å (12)

= (13)

<sup>=</sup> å (14)

, can be obtained as

is the maximal strain transfor‐

is the driving force of variant "*a*", *G*0 is the critical driving force, ex‐

˙ <sup>0</sup> is reference phase fraction

0 00 *<sup>k</sup> a a <sup>m</sup> <sup>a</sup> G Gc f f G Gc* -

ponents k and m are material parameters dictating the rate effect, *c* depends upon the phase

<sup>0</sup> / *a a*

8

1 *a*

=

 x

*a* x

<sup>0</sup> / *a a d df f* x

<sup>8</sup> \* 1 *tr a a ij ij r*

*tr* is increment of phase train. *γ* \*

 ag*df*

is increment of phase fraction.

Elastic strain and electrical field induced strain during loading can expressed as

=

, which is associated with *d f <sup>a</sup>*

*tr*

*d*e

Summation over all possible variants provides the phase fraction for the whole ferroelectric

x

0

is phase fraction transformation rate of variant "*a*", *f*

viscoplasticity,

314 Advances in Ferroelectrics

where *f* ˙ *a*

transformation rate, *G <sup>a</sup>*

single crystal, that is

where *αij*

fraction, *c*0 the reference value at initial state.

The phase fraction of variant "*a*" is calculated as,

The incremental form of Eq. (12) can be written as

The transformation strain tensor *εij*

is Schmid factor. *dεij*

mation during loading. *d f <sup>a</sup>*

To validate the model, the corresponding calculation result is compared to experimental stress-strain curve. The material parameters used in this study are as follow [37, 38]. Materi‐ al parameters of R phase are *C*<sup>11</sup> *<sup>R</sup>* =9.2GPa, *C*<sup>12</sup> *<sup>R</sup>* =10.3GPa, *C*<sup>44</sup> *<sup>R</sup>* =6.9GPa. *γ* \* =0.0033, *<sup>G</sup>*0*<sup>R</sup>* =0.5073MJ/m<sup>3</sup> , *β* =0.004MPa/K, 1 / *m*=11, 1 / *k* =1.0. *T* =403, *T*<sup>0</sup> =298K. Material parame‐ ters of O phase are *C*<sup>11</sup> *<sup>O</sup>* =38GPa, *C*<sup>12</sup> *<sup>O</sup>* =40GPa*C*<sup>44</sup> *<sup>O</sup>* =28GPa. *G*0*<sup>O</sup>* =0.276MJ/m<sup>3</sup> *<sup>β</sup>* =0.039MPa/K, 1 / *m*=11.5, 1 / *k* =1.1. Fig.12 shows the model predicted and experimental measured re‐ sponse. Result in fig.8 shows that stress-strain response of PMN-0.32PT can be predicted by the developed constitutive model, with quantitative agreement.

**Figure 11.** The comparison between ferroelectric phase transformation and plastic slip of single crystal, *n* is direction normal to slip surface, *s* is direction along phase transformation.

**100 010 001**

Electromechanical Coupling Multiaxial Experimental and Micro-Constitutive Model Study of...

0 0.5144 0.0162 0.523 0.4857

0 0 0.523 0.5 0.5

http://dx.doi.org/10.5772/52540

317

Ba2+ O1 2- O2 2- O3 2- Ti4+

oriented PMN-0.32PT.

**6. Conclusion and discussion**

0 0.5144 0.5 0.0162 0.4857

**Table 3.** The coordinate of particle in O phase BaTiO3 ferroelectric single crystal cell from refer.

In this paper, Stress induced strain and polarization, and electric field induced "butter‐ fly" curves and polarization loops for a set of compressive bias stress for {001}, {011} and {111} poled PMN-0.32PT single crystals are experimentally explored. Obtained re‐ sults indicate that high piezoelectric responses of PMN-0.32PT single crystals are con‐ trolled by the anisotropy of the crystals and the multi-domain structure (i.e., engineered domain structure) has a relatively minor effect. Analysis shows that in all three direc‐ tions the electric field induced aggregate strain *Δε* and piezoelectric constant *d*33 in‐ crease with increasing the magnitude of the compressive bias stress. However, when the magnitude of the compressive bias stress is further increased the electric field induce *Δε* and *d*33 decrease. As a result, an optimized compressive bias stress exists for the purpose of enhancing the electromechanical properties of {001} and {111} oriented PMN-0.32PT single crystals. These results have apparent importance in the design of ac‐ tuators and sensors using PMN-0.32PT single crystals. It is found that the observed stress induced strain and polarization in {001}-oriented PMN-0.32PT can be described by a polarization rotation mechanism, i.e., polarization rotates from rhombohedral (R) to orthohombic (O) and tetragonal (T) phases through the intermediate Monoclinic (M) phase during loading, and O to R transition during unloading. However, domain switching is believed to be the main mechanism dictating the electromechanical behav‐ ior of {011} and {111} oriented PMN-0.32PT single crystals. polarization rotation model is developed to explain the observed behaviors of PMN-0.32PT. The stress-strain curve along <001> crystallographic direction of ferroelectric single crystal BaTiO3 is calculated with the first principle method. Obtained results show that the R→M→O polarization rotation (phase transformation) takes place in rhombohedral BaTiO3 ferroelectric single crystals under compression, which is consistent with the polarization rotation model. Based on the polarization rotation model, a constitutive model of PMN-0.32PT is pro‐ posed based on micromechanical model. It is shown that the developed model can faithfully capture the key characteristic of the observed constitutive behavior of <001>

**Figure 12.** The comparison between experimental and simulation stress-strain curve along <001> direction of ferro‐ electric single crystal.


**Table 1.** The coordinate of particle in R phase BaTiO3 ferroelectric single crystal cell [34]


**Table 2.** The calculated coordinate that is correspond to point A in fig.6(a)


**Table 3.** The coordinate of particle in O phase BaTiO3 ferroelectric single crystal cell from refer.

### **6. Conclusion and discussion**



)

**100 010 001**

0 0.5133 0.0192 0.5133 0.489

**100 010 001**

0 0.514 0.016 0.521 0.4846

0 0.0192 0.5133 0.5133 0.489

0 0.01 0.521 0.499 0.499

e33(10-3

**Figure 12.** The comparison between experimental and simulation stress-strain curve along <001> direction of ferro‐

0 0.5133 0.5133 0.0192 0.489

**Table 1.** The coordinate of particle in R phase BaTiO3 ferroelectric single crystal cell [34]

0 0.514 0.514 0.016 0.4846

**Table 2.** The calculated coordinate that is correspond to point A in fig.6(a)

Experiment

Model



s33

electric single crystal.

316 Advances in Ferroelectrics

Ba2+ O1 2- O2 2- O3 2- Ti4+

Ba2+ O1 2- O2 2- O3 2- Ti4+

(MPa)




0

In this paper, Stress induced strain and polarization, and electric field induced "butter‐ fly" curves and polarization loops for a set of compressive bias stress for {001}, {011} and {111} poled PMN-0.32PT single crystals are experimentally explored. Obtained re‐ sults indicate that high piezoelectric responses of PMN-0.32PT single crystals are con‐ trolled by the anisotropy of the crystals and the multi-domain structure (i.e., engineered domain structure) has a relatively minor effect. Analysis shows that in all three direc‐ tions the electric field induced aggregate strain *Δε* and piezoelectric constant *d*33 in‐ crease with increasing the magnitude of the compressive bias stress. However, when the magnitude of the compressive bias stress is further increased the electric field induce *Δε* and *d*33 decrease. As a result, an optimized compressive bias stress exists for the purpose of enhancing the electromechanical properties of {001} and {111} oriented PMN-0.32PT single crystals. These results have apparent importance in the design of ac‐ tuators and sensors using PMN-0.32PT single crystals. It is found that the observed stress induced strain and polarization in {001}-oriented PMN-0.32PT can be described by a polarization rotation mechanism, i.e., polarization rotates from rhombohedral (R) to orthohombic (O) and tetragonal (T) phases through the intermediate Monoclinic (M) phase during loading, and O to R transition during unloading. However, domain switching is believed to be the main mechanism dictating the electromechanical behav‐ ior of {011} and {111} oriented PMN-0.32PT single crystals. polarization rotation model is developed to explain the observed behaviors of PMN-0.32PT. The stress-strain curve along <001> crystallographic direction of ferroelectric single crystal BaTiO3 is calculated with the first principle method. Obtained results show that the R→M→O polarization rotation (phase transformation) takes place in rhombohedral BaTiO3 ferroelectric single crystals under compression, which is consistent with the polarization rotation model. Based on the polarization rotation model, a constitutive model of PMN-0.32PT is pro‐ posed based on micromechanical model. It is shown that the developed model can faithfully capture the key characteristic of the observed constitutive behavior of <001> oriented PMN-0.32PT.

### **Acknowledgments**

The authors are grateful for the financial supported by the Natural Science Foundation of China (No. 10425210,10802081) and the Ministry of Education of China.

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### **Author details**

Wan Qiang1 , Chen Changqing2 and Shen Yapeng2

1 Institute of Structural Mechanics, China Academy of Engineering Physics, Mianyang, Si‐ chuan, China

2 Key Laboratory, School of Astronautics and Aeronautics, Xi'an Jiaotong University, Xi'an, China

### **References**


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**Acknowledgments**

318 Advances in Ferroelectrics

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The authors are grateful for the financial supported by the Natural Science Foundation of

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2 Key Laboratory, School of Astronautics and Aeronautics, Xi'an Jiaotong University, Xi'an,

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[34] Ghosez P. Gonze X. Michenaud JP. First-principles characterization of the four phas‐

[35] Shuo D. Yong Z. Long LY. Elastic and Piezoelectric Properties of Ce:BaTiO3 Single

[36] Feng ZY. Tan OK. Zhu WG. et al. Aging-induced giant recoverable electrostrain in Fe-doped 0.62Pb(Mg1/3Nb2/3)O3-0.38PbTiO3 single crystals. Applied Physics Letters

sidual hysteresis. Journal of Applied Physics 2001:89,1826-1829.

tages and limitations. Journal of Applied Physics 2001:90,2479-2484

2001:89,1820-1824.

320 Advances in Ferroelectrics

2001:86,3891-3895.

plied Physics 2002:92,7690-7696

crystal. Acta Materialia 2001:49,2993-2999.

Solid State Communications 2005:133,311-314.

es of barium titanate. Ferroelectrics 1999:220,1-15.

2008:92, 142910-142914.

Crystals.Chinese Physics Letter 2005:22(7),1790-1792.

PbZr0.52Ti0.48O3. Physics Review B 2004:69,020105-020108.

barium titanate. Applied Physics Letter 2000:77,1698-1670.

materials. Philosophical Magazine Part B 2001:81(12),2021-2054.

[38] Desheng F. Takahiro A. Hiroki T. Ferroelectricity and electromechanical coupling in (1− x)AgNbO3-xNaNbO3 solid solutions. Applied Physics Letter 2011:99,012904-012907.

**Chapter 15**

**'Universal' Synthesis of PZT (1-X)/X Submicrometric**

**Structures Using Highly Stable Colloidal Dispersions:**

The synthesis of nanostructured ferroelectric materials has been a subject of increasing interest for more than a decade due to their possible applications as nonvolatile/dynamic random ac‐ cess memories (NVRAMs/DRAMs), tunnel effect capacitors for high frequency microwave ap‐ plications, infrared detectors, micro-electromechanical systems (MEMs) and electro-optical modulators among others. These materials, when fully integrated into appropriately designed micro-systems, are promising candidates for robotics sensing and for future medical proce‐ dures requiring an in situ, real time and nondestructive monitoring with very high sensitivity.

As a logical consequence, the quest for nanostructured ferroelectric materials has also led to modify the synthesis routes usually considered for obtaining conventional bulk materials and thin films. One of those routes, the sol–gel method, overcomes the inherent limitations of the conventional powders-based synthesis routes when dealing with molecular homogeneity and, therefore, it has been extensively studied and applied in different scenarios in order to obtain not only thin films, nanotubes or nanorods, but also submicron grains due to the dis‐ tinctive dielectric and ferro/piezoelectric features that are associated with size related effects when average grain size is well below 1 μm [1]-[5]and because of the potential use of these

One of the backbones of the ferroelectrics industry is the Lead zirconate titanate [PZT: Pb(Zr1−*<sup>x</sup>*Ti*x*)O3] ceramic system, a well-known ABO3 perovskite with a wide range of industri‐ al applications since the early 1950s of the 20th century. Its dielectric and electromechanical properties have made possible to develop and implement a myriad of devices such as under‐

and reproduction in any medium, provided the original work is properly cited.

© 2013 Suárez-Gómez et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**A Bottom-Up Approach**

F. Calderón-Piñar

**1. Introduction**

http://dx.doi.org/10.5772/51996

materials in a plethora of nanodevices [6].

A. Suárez-Gómez, J.M. Saniger-Blesa and

Additional information is available at the end of the chapter

## **'Universal' Synthesis of PZT (1-X)/X Submicrometric Structures Using Highly Stable Colloidal Dispersions: A Bottom-Up Approach**

A. Suárez-Gómez, J.M. Saniger-Blesa and

F. Calderón-Piñar

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51996

### **1. Introduction**

The synthesis of nanostructured ferroelectric materials has been a subject of increasing interest for more than a decade due to their possible applications as nonvolatile/dynamic random ac‐ cess memories (NVRAMs/DRAMs), tunnel effect capacitors for high frequency microwave ap‐ plications, infrared detectors, micro-electromechanical systems (MEMs) and electro-optical modulators among others. These materials, when fully integrated into appropriately designed micro-systems, are promising candidates for robotics sensing and for future medical proce‐ dures requiring an in situ, real time and nondestructive monitoring with very high sensitivity.

As a logical consequence, the quest for nanostructured ferroelectric materials has also led to modify the synthesis routes usually considered for obtaining conventional bulk materials and thin films. One of those routes, the sol–gel method, overcomes the inherent limitations of the conventional powders-based synthesis routes when dealing with molecular homogeneity and, therefore, it has been extensively studied and applied in different scenarios in order to obtain not only thin films, nanotubes or nanorods, but also submicron grains due to the dis‐ tinctive dielectric and ferro/piezoelectric features that are associated with size related effects when average grain size is well below 1 μm [1]-[5]and because of the potential use of these materials in a plethora of nanodevices [6].

One of the backbones of the ferroelectrics industry is the Lead zirconate titanate [PZT: Pb(Zr1−*<sup>x</sup>*Ti*x*)O3] ceramic system, a well-known ABO3 perovskite with a wide range of industri‐ al applications since the early 1950s of the 20th century. Its dielectric and electromechanical properties have made possible to develop and implement a myriad of devices such as under‐

© 2013 Suárez-Gómez et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

water sonar systems, sensors, actuators, accelerometers, ultrasonic equipment, imaging devi‐ ces, microphones and multiple active and passive damping systems for the car industry.

x)/x solutions were: (1) lead(II) acetate trihydrate (Pb(OAc)2∙3H2O, Mallinckrodt Baker, Inc., 99.8% pure), (2) glacial acetic acid (HOAc, Mallinckrodt Baker, Inc., 99.7% pure), (3) acetyla‐ cetone (AcacH, Sigma-Aldrich Co., 99% pure), (4) 2-methoxyethanol (2-MOE, Mallinckrodt Baker, Inc., 100% pure), (5) zirconium(IV) propoxide (Zr(OPr)4, Sigma-Aldrich Co., 70 wt.%

'Universal' Synthesis of PZT (1-X)/X Submicrometric Structures Using Highly Stable Colloidal Dispersions...

First, lead acetate was dissolved in acetic acid with a 1:3 molar ratio while stirred and re‐ fluxed at 115 °C during 3 h for dehydration and homogeneity purposes. After this step, a thick transparent solution was obtained which will be referred hereafter as solution A. On a separate process, stoichiometric amounts of zirconium and titanium propoxides were mixed with acetylacetone on a 1:2 molar ratio in order to avoid fast hydrolysis of reactants. This mixture was stirred and refluxed at 90 °C during 4 h forming a clear yellow solution refer‐ red hereafter as solution B. When this solution cooled down, the precipitation of several nee‐ dle shaped crystals was verified. In order to keep stoichiometry unaffected, we then repeated the solution B procedure to isolate and characterize some of those crystals by sin‐

**Figure 1.** Flow chart depicting the basic experimental procedure followed in order to obtain a 0.4M PZT (1-x)/x sol–

Solutions A and B were then mixed together as appropriate amounts of solvent (2-MOE) were slowly added for complete dilution of the precipitated crystals and for controlling the PZT concentration on the final solution. Afterwards, a light yellow solution was obtained af‐

It must be stated here that we devoted our work to several Zr:Ti molar ratios that are com‐ monly used in practical applications: *(i)* PZT 25/75, *(ii)* PZT 53/47, *(iii)* PZT 60/40, *(iv)* PZT 80/20 and *(v)* PZT 95/05. Besides, we also tried to cover a relatively wide concentration range for every PZT (1-x)/x sol-gel precursor under study: from 0.05 M to 0.4 M. Due to these facts, we will only focus on the discussion of representative samples instead of discussing irrele‐ vant data of samples that showed no significant discrepancies between each other. As de‐ picted in Figure 1, this section will analyze the synthesis of a 0.4 M PZT 53/47 precursor

Pr)4, Sigma-Aldrich Co., 97% pure).

http://dx.doi.org/10.5772/51996

325

in 1-propanol), and (6) titanium(IV) propoxide (Ti(O<sup>i</sup>

gle crystal XRD.

gel based precursor.

solution.

ter stirring for 24 h at room temperature.

The sol–gel synthesis of PZT ceramics has evolved a lot since it was first reported in the mid-1980s of the last century [7]-[9]; thermal treatments, precursors, additives, diluents and stabilizers have been incorporated and/or modified in order to achieve less hydrolysable, more stable, compounds to suit a specific need. In the case of the synthesis of thin films, nanotubes or nanorods, sol–gel routes have been used in many different ways: from the well-known spin-coating method to the electrophoretic deposition on a given substrate [10] or the insertion on nanoporous templates [11]. Submicron and nanosized PZT particles are also important in the fabrication of highly dense bulk ceramics which, even nowadays, com‐ prise almost the entirety of the electroceramics market. In this particular case, several works have been published in which PZT powders are synthesized first via sol–gel and then put to sinter for densification [12]-[15]. In this procedure, sintering temperatures tend to lower and densification attains notably high values mostly due to a higher Gibb's free energy per unit surface area while some material properties strongly dependent on grain size are also en‐ hanced [16].

Accordingly, in this chapter we will be devoted to analyze the feasibility of a 'customized' syn‐ thesis of PbZr1-xTixO3 [PZT (1-x)/x] nano/submicrometric structures by using a sol-gel based colloidal dispersion as a precursor solution. This study will be done on the basis of a 'bottomup' approach as it will take into account (i) Synthesis route, (ii) Properties of colloidal disper‐ sions and (iii) Final crystallization. We will try to thoroughly illustrate every synthesis step while paying special attention to the physicochemical depiction of some phenomena that not always are sufficiently described or explained. It has to be pointed out that most of the main procedures, discussions and results contained herein could be easily extrapolated to a wide range of materials, not exclusively PZT-based ones.

### **2. Sol-gel based synthesis route**

The complexity of the intermediate reactions, one of the few handicaps of the sol–gel meth‐ od, makes almost mandatory a step by step study of the synthesis method. The chemical re‐ activity of precursors is a well-known key feature determining the nature of the intermediate organic ligands and the control of the hydrolysis rate in the final sol–gel. Sever‐ al studies have been made for particular synthesis routes both theoretical and experimental and many possible reaction pathways have been proposed and dissected. Particularly, the chemistry of metal alkoxides has been intensively studied by Sanchez et al. [17] and Sayer and coworker [18]. It is our purpose here to analyze the different reaction steps and inter‐ mediates involved in the PZT (1-x)/x sol–gel synthesis using propoxides as starting metal alkoxides.

### **2.1. Synthesis**

The followed sol-gel route was the acetic acid, acetylacetone and 2-methoxyethanol pro‐ poxy-based sol-gel method as illustrated in Figure 1. Starting reagents for the sol-gel PZT (1x)/x solutions were: (1) lead(II) acetate trihydrate (Pb(OAc)2∙3H2O, Mallinckrodt Baker, Inc., 99.8% pure), (2) glacial acetic acid (HOAc, Mallinckrodt Baker, Inc., 99.7% pure), (3) acetyla‐ cetone (AcacH, Sigma-Aldrich Co., 99% pure), (4) 2-methoxyethanol (2-MOE, Mallinckrodt Baker, Inc., 100% pure), (5) zirconium(IV) propoxide (Zr(OPr)4, Sigma-Aldrich Co., 70 wt.% in 1-propanol), and (6) titanium(IV) propoxide (Ti(O<sup>i</sup> Pr)4, Sigma-Aldrich Co., 97% pure).

water sonar systems, sensors, actuators, accelerometers, ultrasonic equipment, imaging devi‐ ces, microphones and multiple active and passive damping systems for the car industry.

The sol–gel synthesis of PZT ceramics has evolved a lot since it was first reported in the mid-1980s of the last century [7]-[9]; thermal treatments, precursors, additives, diluents and stabilizers have been incorporated and/or modified in order to achieve less hydrolysable, more stable, compounds to suit a specific need. In the case of the synthesis of thin films, nanotubes or nanorods, sol–gel routes have been used in many different ways: from the well-known spin-coating method to the electrophoretic deposition on a given substrate [10] or the insertion on nanoporous templates [11]. Submicron and nanosized PZT particles are also important in the fabrication of highly dense bulk ceramics which, even nowadays, com‐ prise almost the entirety of the electroceramics market. In this particular case, several works have been published in which PZT powders are synthesized first via sol–gel and then put to sinter for densification [12]-[15]. In this procedure, sintering temperatures tend to lower and densification attains notably high values mostly due to a higher Gibb's free energy per unit surface area while some material properties strongly dependent on grain size are also en‐

Accordingly, in this chapter we will be devoted to analyze the feasibility of a 'customized' syn‐ thesis of PbZr1-xTixO3 [PZT (1-x)/x] nano/submicrometric structures by using a sol-gel based colloidal dispersion as a precursor solution. This study will be done on the basis of a 'bottomup' approach as it will take into account (i) Synthesis route, (ii) Properties of colloidal disper‐ sions and (iii) Final crystallization. We will try to thoroughly illustrate every synthesis step while paying special attention to the physicochemical depiction of some phenomena that not always are sufficiently described or explained. It has to be pointed out that most of the main procedures, discussions and results contained herein could be easily extrapolated to a wide

The complexity of the intermediate reactions, one of the few handicaps of the sol–gel meth‐ od, makes almost mandatory a step by step study of the synthesis method. The chemical re‐ activity of precursors is a well-known key feature determining the nature of the intermediate organic ligands and the control of the hydrolysis rate in the final sol–gel. Sever‐ al studies have been made for particular synthesis routes both theoretical and experimental and many possible reaction pathways have been proposed and dissected. Particularly, the chemistry of metal alkoxides has been intensively studied by Sanchez et al. [17] and Sayer and coworker [18]. It is our purpose here to analyze the different reaction steps and inter‐ mediates involved in the PZT (1-x)/x sol–gel synthesis using propoxides as starting metal

The followed sol-gel route was the acetic acid, acetylacetone and 2-methoxyethanol pro‐ poxy-based sol-gel method as illustrated in Figure 1. Starting reagents for the sol-gel PZT (1-

hanced [16].

324 Advances in Ferroelectrics

alkoxides.

**2.1. Synthesis**

range of materials, not exclusively PZT-based ones.

**2. Sol-gel based synthesis route**

First, lead acetate was dissolved in acetic acid with a 1:3 molar ratio while stirred and re‐ fluxed at 115 °C during 3 h for dehydration and homogeneity purposes. After this step, a thick transparent solution was obtained which will be referred hereafter as solution A. On a separate process, stoichiometric amounts of zirconium and titanium propoxides were mixed with acetylacetone on a 1:2 molar ratio in order to avoid fast hydrolysis of reactants. This mixture was stirred and refluxed at 90 °C during 4 h forming a clear yellow solution refer‐ red hereafter as solution B. When this solution cooled down, the precipitation of several nee‐ dle shaped crystals was verified. In order to keep stoichiometry unaffected, we then repeated the solution B procedure to isolate and characterize some of those crystals by sin‐ gle crystal XRD.

**Figure 1.** Flow chart depicting the basic experimental procedure followed in order to obtain a 0.4M PZT (1-x)/x sol– gel based precursor.

Solutions A and B were then mixed together as appropriate amounts of solvent (2-MOE) were slowly added for complete dilution of the precipitated crystals and for controlling the PZT concentration on the final solution. Afterwards, a light yellow solution was obtained af‐ ter stirring for 24 h at room temperature.

It must be stated here that we devoted our work to several Zr:Ti molar ratios that are com‐ monly used in practical applications: *(i)* PZT 25/75, *(ii)* PZT 53/47, *(iii)* PZT 60/40, *(iv)* PZT 80/20 and *(v)* PZT 95/05. Besides, we also tried to cover a relatively wide concentration range for every PZT (1-x)/x sol-gel precursor under study: from 0.05 M to 0.4 M. Due to these facts, we will only focus on the discussion of representative samples instead of discussing irrele‐ vant data of samples that showed no significant discrepancies between each other. As de‐ picted in Figure 1, this section will analyze the synthesis of a 0.4 M PZT 53/47 precursor solution.

### **2.2. Experimental techniques**

Every reactant and intermediate product was analyzed using FT-IR and Raman spectroscop‐ ies. As was described earlier, single crystal XRD was also carried out for the solution B acic‐ ular precipitates.

cm-1 (RA), 1716 cm-1 (IRA) and 1757 cm-1 (IRA) for this molecule. Those peaks correspond, respectively, to the τ(C-C=O), ν(C-C), δS(CH3), δA(CH3), ν(C=O), ν(C=O)dimer and ν(C=O)mono‐ mer vibrations [20],[21]. The clear splitting detected for the C=O dimeric and monomeric stretch vibrations somewhat evidences the presence of some small amounts of water in the acidic medium that, for the purpose of our study, will not be taken into consideration.

'Universal' Synthesis of PZT (1-X)/X Submicrometric Structures Using Highly Stable Colloidal Dispersions...

http://dx.doi.org/10.5772/51996

327

**Figure 2.** Infrared and Raman spectra of the reactants involved in solution A formation.

to incorrect conclusions [20],[22].

On the other hand, the lead(II) acetate trihydrate spectra also feature some representative peaks located at 216 cm-1 (RA), 615-617 cm-1 (RA, IRA), 665 cm-1 (IRA), 934-935 cm-1 (RA, IRA), 1342-1346 cm-1 (RA, IRA), 1417-1420 cm-1 (RA, IRA) and 1541-1543 cm-1 (RA, IRA) re‐ lated to the ν(Pb-O), ρ(COO), δS(COO), νS(C-C), δS(CH3), νS(C-O) and νA(C=O) vibrational modes, respectively [20],[22]. It is well known that the acetate ligands can complex a metal ion in three different ways (monodentate, bidentate chelating and bridged) and that, un‐ fortunately, none of these can be uniquely identified by symmetry considerations. Tradition‐ ally, the various types of bonding have been identified by the magnitude of the difference between symmetric νS(C-O) and asymmetric νA(C=O) vibrations. In our case, this difference is 122 cm-1 indicating a bidentate chelating coordination for the acetate-metal complex that has been widely accepted for lead(II) acetate even though the Δν criterion has led sometimes

Solution A vibrational spectra shown in Figure 2 evidences the expected dilution of lead(II) acetate in acetic acid: there are no new vibrational modes and a strong band overlapping is seen. There is some band shift due to the overlapping and is worth to notice the weakening

### *2.2.1. Raman spectroscopy*

Raman characterization was made on an Almega XR Dispersive Raman spectrometer equip‐ ped with an Olympus microscope (BX51). An Olympus 10x objective (N.A. = 0.25) was used both for focusing the laser on the sample, with a spot size ∼ 5 μm, and collecting the scat‐ tered light in a 180° backscattering configuration. The scattered light was detected by a CCD detector, thermoelectrically cooled to -50 °C. The spectrometer used a grating (675 lines/mm) to resolve the scattered radiation and a notch filter to block the Rayleigh light. The pinhole of the monochromator was set at 25μm. The Raman spectra were accumulated over 25 s with a resolution of ∼ 4 cm-1 in the 100-2000 cm-1 interval. The excitation source was a 532 nm radiation from a Nd:YVO4 laser (frequency-doubled) and the incident power at the sam‐ ple was of ∼ 8 mW.

### *2.2.2. FT-IR spectroscopy*

FT-IR analysis was carried out on a Thermo Nicolet Nexus 670 FT-IR in transmission mode with a resolution of ∼ 4 cm-1 in the 400-2000 cm-1 interval. The excitation source was a Heli‐ um-Neon laser light incident on a KBr compact target containing ∼ 0.5 % in weight of the sample of interest.

### *2.2.3. Single crystal XRD*

Single crystal XRD experiments were carried out for selected specimens. A Bruker SMART APEX CCD-based X-ray three circle diffractometer was employed for crystal screening, unit cell determination and data collection. A Van Guard 40x microscope was used to identify suitable samples and the goniometer was controlled using the SMART software suite. The X-ray radiation employed was generated from a Mo sealed X-ray tube (Kα= 0.70173 Å with a potential of 50 kV and a current of 30 mA) and filtered with a graphite monochromator in the parallel mode (175 mm collimator with 0.5mm pinholes).

### **2.3. Solution A**

Figure 2 shows, from bottom to top, the IR and Raman spectra of starting acetic acid and lead acetate as well as the final product, solution A, after stirring and refluxing. There are several features in common between these spectra due to the organic nature of ligands. Basi‐ cally, bands corresponding to the CH2 and CH3 groups in the 1300-1400 cm-1 interval as well as in the low frequency range [19].

The glacial acetic acid spectra reveal several representative peaks found at 619 cm-1 (RA: Raman active), 889-893 cm-1 (RA, IRA: Infrared active), 1294 cm-1 (IRA), 1410 cm-1 (IRA), 1668 cm-1 (RA), 1716 cm-1 (IRA) and 1757 cm-1 (IRA) for this molecule. Those peaks correspond, respectively, to the τ(C-C=O), ν(C-C), δS(CH3), δA(CH3), ν(C=O), ν(C=O)dimer and ν(C=O)mono‐ mer vibrations [20],[21]. The clear splitting detected for the C=O dimeric and monomeric stretch vibrations somewhat evidences the presence of some small amounts of water in the acidic medium that, for the purpose of our study, will not be taken into consideration.

**2.2. Experimental techniques**

ular precipitates.

326 Advances in Ferroelectrics

*2.2.1. Raman spectroscopy*

ple was of ∼ 8 mW.

sample of interest.

**2.3. Solution A**

as in the low frequency range [19].

*2.2.3. Single crystal XRD*

*2.2.2. FT-IR spectroscopy*

Every reactant and intermediate product was analyzed using FT-IR and Raman spectroscop‐ ies. As was described earlier, single crystal XRD was also carried out for the solution B acic‐

Raman characterization was made on an Almega XR Dispersive Raman spectrometer equip‐ ped with an Olympus microscope (BX51). An Olympus 10x objective (N.A. = 0.25) was used both for focusing the laser on the sample, with a spot size ∼ 5 μm, and collecting the scat‐ tered light in a 180° backscattering configuration. The scattered light was detected by a CCD detector, thermoelectrically cooled to -50 °C. The spectrometer used a grating (675 lines/mm) to resolve the scattered radiation and a notch filter to block the Rayleigh light. The pinhole of the monochromator was set at 25μm. The Raman spectra were accumulated over 25 s with a resolution of ∼ 4 cm-1 in the 100-2000 cm-1 interval. The excitation source was a 532 nm radiation from a Nd:YVO4 laser (frequency-doubled) and the incident power at the sam‐

FT-IR analysis was carried out on a Thermo Nicolet Nexus 670 FT-IR in transmission mode with a resolution of ∼ 4 cm-1 in the 400-2000 cm-1 interval. The excitation source was a Heli‐ um-Neon laser light incident on a KBr compact target containing ∼ 0.5 % in weight of the

Single crystal XRD experiments were carried out for selected specimens. A Bruker SMART APEX CCD-based X-ray three circle diffractometer was employed for crystal screening, unit cell determination and data collection. A Van Guard 40x microscope was used to identify suitable samples and the goniometer was controlled using the SMART software suite. The X-ray radiation employed was generated from a Mo sealed X-ray tube (Kα= 0.70173 Å with a potential of 50 kV and a current of 30 mA) and filtered with a graphite monochromator in

Figure 2 shows, from bottom to top, the IR and Raman spectra of starting acetic acid and lead acetate as well as the final product, solution A, after stirring and refluxing. There are several features in common between these spectra due to the organic nature of ligands. Basi‐ cally, bands corresponding to the CH2 and CH3 groups in the 1300-1400 cm-1 interval as well

The glacial acetic acid spectra reveal several representative peaks found at 619 cm-1 (RA: Raman active), 889-893 cm-1 (RA, IRA: Infrared active), 1294 cm-1 (IRA), 1410 cm-1 (IRA), 1668

the parallel mode (175 mm collimator with 0.5mm pinholes).

**Figure 2.** Infrared and Raman spectra of the reactants involved in solution A formation.

On the other hand, the lead(II) acetate trihydrate spectra also feature some representative peaks located at 216 cm-1 (RA), 615-617 cm-1 (RA, IRA), 665 cm-1 (IRA), 934-935 cm-1 (RA, IRA), 1342-1346 cm-1 (RA, IRA), 1417-1420 cm-1 (RA, IRA) and 1541-1543 cm-1 (RA, IRA) re‐ lated to the ν(Pb-O), ρ(COO), δS(COO), νS(C-C), δS(CH3), νS(C-O) and νA(C=O) vibrational modes, respectively [20],[22]. It is well known that the acetate ligands can complex a metal ion in three different ways (monodentate, bidentate chelating and bridged) and that, un‐ fortunately, none of these can be uniquely identified by symmetry considerations. Tradition‐ ally, the various types of bonding have been identified by the magnitude of the difference between symmetric νS(C-O) and asymmetric νA(C=O) vibrations. In our case, this difference is 122 cm-1 indicating a bidentate chelating coordination for the acetate-metal complex that has been widely accepted for lead(II) acetate even though the Δν criterion has led sometimes to incorrect conclusions [20],[22].

Solution A vibrational spectra shown in Figure 2 evidences the expected dilution of lead(II) acetate in acetic acid: there are no new vibrational modes and a strong band overlapping is seen. There is some band shift due to the overlapping and is worth to notice the weakening of the lead acetate Raman active Pb-O band at 216 cm-1 when in solution. Unfortunately, a noticeable fluorescence in the Raman spectrum could not be avoided and this fact made very difficult to carry on an appropriate analysis with useful data.

Nevertheless, and according to our results, the formation of Solution A could be fairly de‐ scribed by the following equation:

$$\begin{aligned} \text{Pb}\left(\text{OAc}\right)\_1 \cdot 3H\_1O\_{\binom{n}{4}} + 3\text{HOAc}\_{\binom{n+4}{4}} \xrightarrow[t \to \text{blue}]{\text{T} \rightarrow \text{HOAc}} \text{Pb}^{2+}\_{\binom{n}{4}} + 3\text{a}H\_1O^+\_{\binom{n}{4}} + \left(2 + 3\text{a}\right)\text{OAc}^-\_{\binom{n}{4}} + \text{anode} \\ + 3\left(1 - \text{a}\right)\text{HOAc}\_{\binom{n}{4}} \end{aligned} \tag{1}$$

including zone photographs, and no super cell or erroneous reflections were observed. A search performed on the Cambridge Structural Database and updates using the program Conquest afforded 77 coincidences within 1% of the longest length of a monoclinic C-cen‐ tered cell whose lattice parameters and reported structure are shown in Figure 4(a). Two en‐ tries, those with ACACZR and ZZZADD CCD reference code, in addition, revealed chemical coincidences (both in composition and stoichiometry) for the compound tetra‐

'Universal' Synthesis of PZT (1-X)/X Submicrometric Structures Using Highly Stable Colloidal Dispersions...

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329

**Figure 3.** Infrared spectra of the reactants involved in solution B formation. The IR spectra for the resultant precipitat‐

According to these data, the Zr(Acac)4 structure was simulated using the Accelrys Materials Studio 3D visualizer environment [25]as shown in Figure 4(b). In this case, the acetylacetone reaction with the Zr propoxide results in the formation of an oxo cluster where the metallic atom changes its coordination number from 4 to 8 which is the highest possible value for zirconium. Metallic cations are now bonded to the acetylacetonate chelating ligand giving

In the case of the titanium propoxide reaction, and taking into account our experimental conditions, it is not a bad assumption to consider the formation of a fully chelated organo‐ metallic complex, just as it was described for zirconium. Given the maximum coordination number of 6 for titanium, a Ti(Acac)2(OPr)2 compound will be the most likely to expect as is also confirmed by earlier reports that consider the multiple chelation routes for titanium

kis(acetylacetonate-O,O')-zirconium(IV) or Zr(Acac)4.

ed single crystals are also shown.

rise to a less hydrolysable organic complex.

Where α is the HOAc dissociation degree under our experimental conditions and where we have also assumed, on the simplest approach, neglectable losses due to evaporation of H2O and HOAc during the whole process.

### **2.4. Solution B**

FT-IR spectra for Solution B precursors, Figure 3, showed some features like the ones dis‐ cussed above. A Raman spectra based analysis for these compounds could not be completed due to the strong fluorescence exhibited by the zirconium and titanium alkoxides at our fixed operating laser wavelength.

The titanium propoxide IR spectrum exhibit sharp bands at 1377-1464 cm-1 corresponding, respectively, to the stretching and bending vibrations of the aliphatic CH3 groups. A distinc‐ tive single peak at 1011 cm-1 corresponds to the propoxy- Ti-O-C vibration and a similar be‐ havior is found for zirconium propoxide with Zr-O-C vibrations located at 1142 cm-1[18].

The acetylacetone vibrational spectrum shows the main features attributed to the most prob‐ able staggered conformation of this compound. This conformation has some typical weak IR active vibrations in the low frequency range, as it is shown. Modes at 509, 554 and 640 cm-1 correspond to the in plane ring deformation (Δ ring), out of plane ring deformation (Γ ring) and Δ ring + ρ(CH3) modes, respectively [23].

As described above, when reaction took place, some crystals precipitated short after solution B reached room temperature; IR spectra of these single crystals are also shown in Figure 3. Unlike solution A, a reaction is now verified by the shifting and/or reinforcement of bands associated to reactants vibrations. A discussion of several mechanisms for this kind of reac‐ tion has been reviewed by several authors taking into account, primarily, the mixing condi‐ tions, the reactivity of metal alkoxides and the Acac/Alkoxides molar ratio [17],[18],[24].

### *2.4.1. Single crystal XRD characterization*

At this stage, a suitable colorless parallelepiped 0.356 mm × 0.162 mm × 0.066 mm was chos‐ en from a representative sample of crystals of the same habit. After the determination of a suitable cell, it was refined by nonlinear least squares and Brava is lattice procedures. The unit cell was then verified by examination of the (h k l) overlays on several frames of data, including zone photographs, and no super cell or erroneous reflections were observed. A search performed on the Cambridge Structural Database and updates using the program Conquest afforded 77 coincidences within 1% of the longest length of a monoclinic C-cen‐ tered cell whose lattice parameters and reported structure are shown in Figure 4(a). Two en‐ tries, those with ACACZR and ZZZADD CCD reference code, in addition, revealed chemical coincidences (both in composition and stoichiometry) for the compound tetra‐ kis(acetylacetonate-O,O')-zirconium(IV) or Zr(Acac)4.

of the lead acetate Raman active Pb-O band at 216 cm-1 when in solution. Unfortunately, a noticeable fluorescence in the Raman spectrum could not be avoided and this fact made

Nevertheless, and according to our results, the formation of Solution A could be fairly de‐

*S Aq Aq Aq Aq t hrs Aq*

× + + ++ +

Where α is the HOAc dissociation degree under our experimental conditions and where we have also assumed, on the simplest approach, neglectable losses due to evaporation of H2O

FT-IR spectra for Solution B precursors, Figure 3, showed some features like the ones dis‐ cussed above. A Raman spectra based analysis for these compounds could not be completed due to the strong fluorescence exhibited by the zirconium and titanium alkoxides at our

The titanium propoxide IR spectrum exhibit sharp bands at 1377-1464 cm-1 corresponding, respectively, to the stretching and bending vibrations of the aliphatic CH3 groups. A distinc‐ tive single peak at 1011 cm-1 corresponds to the propoxy- Ti-O-C vibration and a similar be‐ havior is found for zirconium propoxide with Zr-O-C vibrations located at 1142 cm-1[18].

The acetylacetone vibrational spectrum shows the main features attributed to the most prob‐ able staggered conformation of this compound. This conformation has some typical weak IR active vibrations in the low frequency range, as it is shown. Modes at 509, 554 and 640 cm-1 correspond to the in plane ring deformation (Δ ring), out of plane ring deformation (Γ ring)

As described above, when reaction took place, some crystals precipitated short after solution B reached room temperature; IR spectra of these single crystals are also shown in Figure 3. Unlike solution A, a reaction is now verified by the shifting and/or reinforcement of bands associated to reactants vibrations. A discussion of several mechanisms for this kind of reac‐ tion has been reviewed by several authors taking into account, primarily, the mixing condi‐ tions, the reactivity of metal alkoxides and the Acac/Alkoxides molar ratio [17],[18],[24].

At this stage, a suitable colorless parallelepiped 0.356 mm × 0.162 mm × 0.066 mm was chos‐ en from a representative sample of crystals of the same habit. After the determination of a suitable cell, it was refined by nonlinear least squares and Brava is lattice procedures. The unit cell was then verified by examination of the (h k l) overlays on several frames of data,

a

° + + -

 a

(1)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

*HOAc*

3 3 3 23

*T C*

*Pb OAc H O HOAc Pb H O OAc*

a

2 2 3 ~3 .

3 1

+ - ¾¾¾®

~ 115 2

very difficult to carry on an appropriate analysis with useful data.

scribed by the following equation:

and HOAc during the whole process.

fixed operating laser wavelength.

and Δ ring + ρ(CH3) modes, respectively [23].

*2.4.1. Single crystal XRD characterization*

**2.4. Solution B**

328 Advances in Ferroelectrics

**Figure 3.** Infrared spectra of the reactants involved in solution B formation. The IR spectra for the resultant precipitat‐ ed single crystals are also shown.

According to these data, the Zr(Acac)4 structure was simulated using the Accelrys Materials Studio 3D visualizer environment [25]as shown in Figure 4(b). In this case, the acetylacetone reaction with the Zr propoxide results in the formation of an oxo cluster where the metallic atom changes its coordination number from 4 to 8 which is the highest possible value for zirconium. Metallic cations are now bonded to the acetylacetonate chelating ligand giving rise to a less hydrolysable organic complex.

In the case of the titanium propoxide reaction, and taking into account our experimental conditions, it is not a bad assumption to consider the formation of a fully chelated organo‐ metallic complex, just as it was described for zirconium. Given the maximum coordination number of 6 for titanium, a Ti(Acac)2(OPr)2 compound will be the most likely to expect as is also confirmed by earlier reports that consider the multiple chelation routes for titanium propoxide [18],[20]. Figure 4(c) shows a very simple Materials Studio 3D modeling for the most probable configuration of Ti(Acac)2(OPr)2 as determined by the Forcite package [25] on a single step relaxation and energy minimization routine.

Now, it is obvious that Δ = 1.06 implying an alkoxides: acacH molar ratio of 1:3.06 for full che‐ lation of metallic centers. Equation [3] can now be written for any given Zr:Ti ratio (1-x)/x:

( ) ( ) ( ) ( ) ( ) ( )

*i i T C i*


~4 . 4 4

+ - +

**3. Some properties of the PZT precursors**

are not reversible, generally follows after some time.

processing.

ready discussed above.

and, therefore, very stable.

(1 ) Pr Pr 2(2 ) 2(2 ) Pr

*x Zr O xTi O x AcacH x HO*

(1 ) Pr

*x Zr Acac xTi Acac O*

( ) ( ) ( ) ( ) ( )

This equation represents an extension for the initially proposed sol-gel based route in order to obtain fully chelated and no hydrolysable precursor solutions for PZT based ferroelectric materials. It allows us to synthesize PZT at any Zr:Ti ratio while maximizing solution stabili‐ ty and offering a noticeable repetitiveness for both small and large scale manufacturing and

On the following section we will analyze some features of different sol-gel based PZT (1-x)/x precursors when synthesized using the universal 1:2(2-x) alkoxides: acacH molar ratio al‐

In the final step of the synthesis, when mixing Solutions A and B with 2-methoxyethanol, the dissolved lead acetate complex can react with 2-MOE forming a very stable acetate–me‐ thoxyethoxy lead complex [27] that, along with the chelated metal complexes already formed, may lead to turn this final solution into a hydrophobic sol, poorly hydrolysable

Stability issues tend to be crucial when using sol-gel based precursors in research, small scale applications and industry. Therefore, it is very important to keep control over some parameters that affects the solution stability and, most of all, the average particles size. Par‐ ticularly, particles size can be an indicator of some undesirable processes that could be tak‐ ing place in the sol: aggregation, flocculation and sedimentation; aggregation, even though is a reversible process, is a good indicator of instability since the other two processes, which

In this respect, the aging time dependences of some physical parameters directly related to the stability of colloidal dispersions must be explored and discussed. Two of the most im‐ portant parameters that should be taken into account are pH and the average particles size. The first one is determinant for fixing the thickness of the ionic layers surrounding any giv‐ en charged colloidal particle (Stern layers) while affecting, at the same time, the Zeta poten‐ tial, the electrophoretic mobility and the aggregation mechanisms as well as the apparent hydrodynamic particle size. As a consequence, the average particles size is the final result of the conjugate action of all the physicochemical variables hardly mentioned before. It is also the definitive experimental variable on which any post processing technique should be

4 2 2

*S Aq*

*Aq Aq Aq Aq t hrs*

'Universal' Synthesis of PZT (1-X)/X Submicrometric Structures Using Highly Stable Colloidal Dispersions...

~90

¾¾¾®

°

(4)

331

http://dx.doi.org/10.5772/51996

*i*

Accordingly, bands located at 1280, 1370-1440 and 1530-1595 cm-1 in the crystals spectrum, shown in Figure 3, can be assigned to the ν(C-CH3:Acac), δ(CH3:Acac) and ν(C-C) + ν(C-O:Acac) vibrational modes, respectively [26].

**Figure 4.** Fully chelated metal-acetylacetonate complexes. (a) Zr(Acac)4 as reported by single crystal XRD charateriza‐ tion, (b) Zr(Acac)4 structure simulated by Materials Studio (MS) according to its crystallographic data and (c) Ti(Acac)2(OPr)2 simulated by MS according to its most probable configuration.

### *2.4.2. Chemical reaction*

As seen before, solution B can be thought as the resulting chelated metal complexes mixed with residual isopropanol. In this case, an appropriate chemical equation for the reaction could be:

$$\begin{aligned} &0.53Zr \Big(\text{O}^{i}\Pr\Big)\_{4\text{(A)}} + 0.47\text{Ti} \Big(\text{O}^{i}\Pr\Big)\_{4\text{(A)}} + 2A\text{acch} \mathbf{H}\_{\{\text{Aq}\}} \xrightarrow[t\to\text{4m}]{\text{T}\cdot\text{spt}\heartseq} 2\text{HO}^{i}\Pr\_{\{\text{Aq}\}} + yZr \Big(\text{Aac}\Big)\_{4\text{(A)}} \\ &+ \Big(1-2y\Big) \text{Ti} \Big(\text{Aac}\Big)\_{2} \Big(\text{O}^{i}\Pr\Big)\_{2\text{(Aq)}} + \Big(0.53-y\Big) \text{Zr} \Big(\text{O}^{i}\Pr\Big)\_{4\text{(Aq)}} + \Big(2y-0.53\Big) \text{Ti} \Big(\text{O}^{i}\Pr\Big)\_{4\text{(Aq)}} \end{aligned} \tag{2}$$

where we have assumed that all the acetylacetone reacts with the alkoxides thus forming the chelated organometallic complexes. As we see, there is still a fraction of nonchelated reac‐ tants due to the insufficient amount of acetylacetone needed for that purpose. Let us find now the exact amount of acetylacetone required for obtaining fully chelated organometallic complexes without any further byproducts. We could rewrite equation [2] as:

$$\begin{aligned} \text{(0.53Zr)} \left(\text{O}^{\text{i}}\text{Pr}\right)\_{\text{4\text{\textdegree{}}\text{(A\text{\textdegree{}})}} &+ 0.477\text{i}\text{\textdegree{}(\text{O}^{\text{i}}\text{Pr})}\_{\text{4\text{\textdegree{}(A\text{\textdegree{}})}} + (\text{Z} + \text{A})\text{AcacH}\_{\text{(A\text{\textdegree{}})}} \xrightarrow[\text{I}\text{\text{--}4\text{km}.}]{\text{T}\text{-}40\text{°C}} (\text{Z} + \text{A})\text{HO}^{\text{i}}\text{Pr}\_{\text{\textdegree{}(A\text{\textdegree{}})}} + \\ &+ 0.53Zr\text{(A\text{\textdegree{}}A\text{\textdegree{}})}\_{\text{4\text{\textdegree{}}\text{(}\text{})} + 0.477\text{i}\text{\textdegree{}(A\text{\textdegree{}}A\text{\textdegree{}})} \text{\textdegree{}(\text{O}^{\text{i}}\text{Pr})\_{\text{2\textdegree{}(A\text{\textdegree{}})}} \end{aligned} \tag{3}$$

Now, it is obvious that Δ = 1.06 implying an alkoxides: acacH molar ratio of 1:3.06 for full che‐ lation of metallic centers. Equation [3] can now be written for any given Zr:Ti ratio (1-x)/x:

$$\begin{aligned} \text{Tr}(\mathbf{1} - \mathbf{x}) \text{Zr} \Big( \mathbf{O}^{\dagger} \text{Pr} \Big)\_{\mathbf{4}\_{\{\!4\_{\emptyset}\}}} + \text{xlTi} \Big( \mathbf{O}^{\dagger} \text{Pr} \Big)\_{\mathbf{4}\_{\{\!4\_{\emptyset}\}}} + \text{2(2 - x)} \text{AacarH}\_{\{\!4\_{\emptyset}\}} \xrightarrow[\text{t-4km.}]{\text{T} \rightarrow \text{00} \text{°C}} \text{2(2 - x)} \text{HO}^{\dagger} \text{Pr}\_{\{\!4\_{\emptyset}\}} + \text{2(2 - x)} \text{H} \text{O}^{\dagger} \text{Ar} \\ + \text{(1 - x)} \text{Zr} \Big( \text{Aacar} \Big)\_{\mathbf{4}\_{\{\!4\}}} + \text{xlTi} \Big( \text{Aacac} \Big)\_{\mathbf{2}} \Big( \mathbf{O}^{\dagger} \text{Pr} \Big)\_{\mathbf{2}\{\!4\_{\emptyset}\}} \end{aligned} \tag{4}$$

This equation represents an extension for the initially proposed sol-gel based route in order to obtain fully chelated and no hydrolysable precursor solutions for PZT based ferroelectric materials. It allows us to synthesize PZT at any Zr:Ti ratio while maximizing solution stabili‐ ty and offering a noticeable repetitiveness for both small and large scale manufacturing and processing.

On the following section we will analyze some features of different sol-gel based PZT (1-x)/x precursors when synthesized using the universal 1:2(2-x) alkoxides: acacH molar ratio al‐ ready discussed above.

### **3. Some properties of the PZT precursors**

propoxide [18],[20]. Figure 4(c) shows a very simple Materials Studio 3D modeling for the most probable configuration of Ti(Acac)2(OPr)2 as determined by the Forcite package [25] on

Accordingly, bands located at 1280, 1370-1440 and 1530-1595 cm-1 in the crystals spectrum, shown in Figure 3, can be assigned to the ν(C-CH3:Acac), δ(CH3:Acac) and ν(C-C) + ν(C-

**Figure 4.** Fully chelated metal-acetylacetonate complexes. (a) Zr(Acac)4 as reported by single crystal XRD charateriza‐ tion, (b) Zr(Acac)4 structure simulated by Materials Studio (MS) according to its crystallographic data and (c)

As seen before, solution B can be thought as the resulting chelated metal complexes mixed with residual isopropanol. In this case, an appropriate chemical equation for the reaction

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

~4 . 4 4 4

where we have assumed that all the acetylacetone reacts with the alkoxides thus forming the chelated organometallic complexes. As we see, there is still a fraction of nonchelated reac‐ tants due to the insufficient amount of acetylacetone needed for that purpose. Let us find now the exact amount of acetylacetone required for obtaining fully chelated organometallic

( ) ( ) ( ) ( ) ( ) ( )

*i i T C i*

( ) ( ) ( ) ( ) ( )

4 2 2

*S Aq*

*Aq Aq Aq Aq t hrs*

+ + +D +D +

*Aq t hrs Aq <sup>S</sup> Aq Aq ii i*

2 24 4

~90

¾¾¾®

° + + + +

*Aq Aq Aq*

~90

¾¾¾®

°

*i*

(2)

(3)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2 Pr 0.53 Pr 2 0.53 Pr

*y Ti Acac O y Zr O y Ti O*

*Zr O Ti O AcacH HO yZr Acac*

a single step relaxation and energy minimization routine.

Ti(Acac)2(OPr)2 simulated by MS according to its most probable configuration.

0.53 Pr 0.47 Pr 2 2 Pr

+ - + - + -

*i i T C i*

complexes without any further byproducts. We could rewrite equation [2] as:

~4 . 4 4

+ +

0.53 Pr 0.47 Pr (2 ) (2 ) Pr

*Zr O Ti O AcacH HO*

0.53 0.47 Pr

*Zr Acac Ti Acac O*

*2.4.2. Chemical reaction*

330 Advances in Ferroelectrics

could be:

O:Acac) vibrational modes, respectively [26].

In the final step of the synthesis, when mixing Solutions A and B with 2-methoxyethanol, the dissolved lead acetate complex can react with 2-MOE forming a very stable acetate–me‐ thoxyethoxy lead complex [27] that, along with the chelated metal complexes already formed, may lead to turn this final solution into a hydrophobic sol, poorly hydrolysable and, therefore, very stable.

Stability issues tend to be crucial when using sol-gel based precursors in research, small scale applications and industry. Therefore, it is very important to keep control over some parameters that affects the solution stability and, most of all, the average particles size. Par‐ ticularly, particles size can be an indicator of some undesirable processes that could be tak‐ ing place in the sol: aggregation, flocculation and sedimentation; aggregation, even though is a reversible process, is a good indicator of instability since the other two processes, which are not reversible, generally follows after some time.

In this respect, the aging time dependences of some physical parameters directly related to the stability of colloidal dispersions must be explored and discussed. Two of the most im‐ portant parameters that should be taken into account are pH and the average particles size. The first one is determinant for fixing the thickness of the ionic layers surrounding any giv‐ en charged colloidal particle (Stern layers) while affecting, at the same time, the Zeta poten‐ tial, the electrophoretic mobility and the aggregation mechanisms as well as the apparent hydrodynamic particle size. As a consequence, the average particles size is the final result of the conjugate action of all the physicochemical variables hardly mentioned before. It is also the definitive experimental variable on which any post processing technique should be based on as it explicitly defines the size range of the so called "building blocks" for bottomup studies or applications.

fore, (c) to the small amount of alcohol that could be formed afterwards. After 4 months of aging, no noticeable changes in pH values were detected. Moreover, the stability of these solutions could be eye inspected by verifying neither the absence of sedimentation nor pre‐

'Universal' Synthesis of PZT (1-X)/X Submicrometric Structures Using Highly Stable Colloidal Dispersions...

http://dx.doi.org/10.5772/51996

333

**Figure 5.** a) pH vs. concentration dependencies for the PZT 53/47 samples understudy measured at different time in‐ tervals. Semi-log fittings to guide the eye are also drawn. (b) Aging effects on pH for the same samples. Three distinc‐

Particles size measurements were carried out by means of the dynamic light scattering (DLS) technique in a Zetasizer Nano ZS90 manufactured by Malvern Instruments Ltd. equipped with a HeNe laser source. Approximately 1.5 ml of as synthesized sols were stor‐ ed in polystyrene cells (DTS0012 cells) provided by the same manufacturer and then size distributions curves were recorded from the very first day until 125 days after synthesis. For this purpose, several solvent parameters were needed for further processing of dispersed light intensity, namely viscosity (η2-MOE ~ 1.5410 cP), dielectric constant (ε2-MOE ~ 16.9) and re‐ fraction index (n2-MOE ~ 1.33) [32]. After each measurement, a nonlinear least squares fitting (NLLSF) was made to the experimental data according to a log-normal distribution function:

> 1 ln 0 2

b

p b

b b

size intervals where unimodal distribution curves were measured.

(, ) exp ln <sup>2</sup> *<sup>i</sup> <sup>x</sup> f x x*

=+ -

3 2 2

b

are fitting parameters. It must be stressed that this fitting was carried out only for

1 1

2

(5)

b

é ù æ ö ê ú ç ÷ ê ú è ø ë û

tive stages were detected and are highlighted in the graph.

**3.2. Mean particle size**

Where *β<sup>I</sup>*

*3.2.1. Experimental technique*

cipitation of single particles or aggregates after almost 1 year of stocking.

Currently, there are no known extensive reports in the literature directly concerned to the study of the time dependence of the average particles size or pH in a PZT precursor sol. It must be noted that, in the case of magnetic nanoparticles, some studies have been carried out [28]-[30]and all of them stress the strong correlation between the size of the colloidal particle and the final properties of the conceived structure, whether it will be nano- or not.

### **3.1. pH**

For this study, we synthesized PZT (1-x)/x precursor solutions by using the same route de‐ scribed in 2.1 with the only difference being the propoxides: acacH molar ratio. In this case, and for the rest of our text, that ratio will be 1:2(2-x) for full chelation of the metal alkoxides.

As it was stated earlier, we will avoid again showing and discussing irrelevant data of sam‐ ples that showed no significant discrepancies between each other and, because of this, this section will analyze the pH vs. Concentration vs. aging time behaviors of several PZT 53/47 precursor solutions with concentrations ranging from 0.05 M to 0.37 M in a time interval of up to 4 months of stocking.

As can be seen in Figure 5, the acidity/basicity of the solutions clearly shows a tendency with both sols concentration and aging time. Figure 5(a) shows the measured pH vs concen‐ tration for the studied sols at several aging steps and, as expected, the more concentrated sample implies the more acidic medium which is consistent with the chemical reaction pro‐ posed for sols synthesis.

Figure 5(b), on the other hand, shows the measured pH vs. time behavior. As a general ten‐ dency, sols pH dependency with aging time can be divided in three stages which are high‐ lighted in the graph: (i) pH increases notably in the first days after which (ii) it decreases to values somehow close to the initial ones. In this moment, (iii) pH starts to rise again but with a slower time gradient than on stage (i). In our opinion, for the understanding of the pH vs. aging time behavior, we have to take into account the coexistence of different com‐ petitive processes right after sols were prepared. In this way, a qualitative description could be done as follows [31]: (i) Remnant unreacted acetic acid evaporates during the first days implying a decrement on the acidity of sols. At this point, (ii) particles are less positively charged and are able to aggregate via condensation reactions followed by the formation of small portions of alcohol (1-propanol) and thus implying a more acidic environment as seen in Figure 5. The number of polyanions per aggregate chain must be limited, however, by the high chemical stability of the chelated metal complexes and that is why the aggregation process does not imply polymerization and/or gelation as it has been reported for more hy‐ drophilic sols. Shortly after condensation rate vanishes, (iii) it is possible for the residual al‐ cohols to evaporate as solutions age thus allowing pH to rise slowly, as seen in Figure 5 for the more aged solutions.

At this scenario, however, hydrolysis is not expected due to (a) the complete chelation of the metal complexes, (b) to the short lengths of the already formed polymeric chains and, there‐ fore, (c) to the small amount of alcohol that could be formed afterwards. After 4 months of aging, no noticeable changes in pH values were detected. Moreover, the stability of these solutions could be eye inspected by verifying neither the absence of sedimentation nor pre‐ cipitation of single particles or aggregates after almost 1 year of stocking.

**Figure 5.** a) pH vs. concentration dependencies for the PZT 53/47 samples understudy measured at different time in‐ tervals. Semi-log fittings to guide the eye are also drawn. (b) Aging effects on pH for the same samples. Three distinc‐ tive stages were detected and are highlighted in the graph.

### **3.2. Mean particle size**

based on as it explicitly defines the size range of the so called "building blocks" for bottom-

Currently, there are no known extensive reports in the literature directly concerned to the study of the time dependence of the average particles size or pH in a PZT precursor sol. It must be noted that, in the case of magnetic nanoparticles, some studies have been carried out [28]-[30]and all of them stress the strong correlation between the size of the colloidal particle and the final properties of the conceived structure, whether it will be nano- or not.

For this study, we synthesized PZT (1-x)/x precursor solutions by using the same route de‐ scribed in 2.1 with the only difference being the propoxides: acacH molar ratio. In this case, and for the rest of our text, that ratio will be 1:2(2-x) for full chelation of the metal alkoxides. As it was stated earlier, we will avoid again showing and discussing irrelevant data of sam‐ ples that showed no significant discrepancies between each other and, because of this, this section will analyze the pH vs. Concentration vs. aging time behaviors of several PZT 53/47 precursor solutions with concentrations ranging from 0.05 M to 0.37 M in a time interval of

As can be seen in Figure 5, the acidity/basicity of the solutions clearly shows a tendency with both sols concentration and aging time. Figure 5(a) shows the measured pH vs concen‐ tration for the studied sols at several aging steps and, as expected, the more concentrated sample implies the more acidic medium which is consistent with the chemical reaction pro‐

Figure 5(b), on the other hand, shows the measured pH vs. time behavior. As a general ten‐ dency, sols pH dependency with aging time can be divided in three stages which are high‐ lighted in the graph: (i) pH increases notably in the first days after which (ii) it decreases to values somehow close to the initial ones. In this moment, (iii) pH starts to rise again but with a slower time gradient than on stage (i). In our opinion, for the understanding of the pH vs. aging time behavior, we have to take into account the coexistence of different com‐ petitive processes right after sols were prepared. In this way, a qualitative description could be done as follows [31]: (i) Remnant unreacted acetic acid evaporates during the first days implying a decrement on the acidity of sols. At this point, (ii) particles are less positively charged and are able to aggregate via condensation reactions followed by the formation of small portions of alcohol (1-propanol) and thus implying a more acidic environment as seen in Figure 5. The number of polyanions per aggregate chain must be limited, however, by the high chemical stability of the chelated metal complexes and that is why the aggregation process does not imply polymerization and/or gelation as it has been reported for more hy‐ drophilic sols. Shortly after condensation rate vanishes, (iii) it is possible for the residual al‐ cohols to evaporate as solutions age thus allowing pH to rise slowly, as seen in Figure 5 for

At this scenario, however, hydrolysis is not expected due to (a) the complete chelation of the metal complexes, (b) to the short lengths of the already formed polymeric chains and, there‐

up studies or applications.

332 Advances in Ferroelectrics

up to 4 months of stocking.

posed for sols synthesis.

the more aged solutions.

**3.1. pH**

### *3.2.1. Experimental technique*

Particles size measurements were carried out by means of the dynamic light scattering (DLS) technique in a Zetasizer Nano ZS90 manufactured by Malvern Instruments Ltd. equipped with a HeNe laser source. Approximately 1.5 ml of as synthesized sols were stor‐ ed in polystyrene cells (DTS0012 cells) provided by the same manufacturer and then size distributions curves were recorded from the very first day until 125 days after synthesis. For this purpose, several solvent parameters were needed for further processing of dispersed light intensity, namely viscosity (η2-MOE ~ 1.5410 cP), dielectric constant (ε2-MOE ~ 16.9) and re‐ fraction index (n2-MOE ~ 1.33) [32]. After each measurement, a nonlinear least squares fitting (NLLSF) was made to the experimental data according to a log-normal distribution function:

$$f\_{\ln}(\mathbf{x}, \boldsymbol{\beta}\_{i}) = \boldsymbol{\beta}\_{0} + \frac{\boldsymbol{\beta}\_{i}}{\sqrt{\pi \boldsymbol{\beta}\_{2}}} \frac{1}{\boldsymbol{x}} \exp\left[-\frac{1}{2\boldsymbol{\beta}\_{2}^{2}} \ln\left(\frac{\boldsymbol{x}}{\boldsymbol{\beta}\_{3}}\right)^{2}\right] \tag{5}$$

Where *β<sup>I</sup>* are fitting parameters. It must be stressed that this fitting was carried out only for size intervals where unimodal distribution curves were measured.

### *3.2.2. Concentration dependence*

Figure 6 shows the results obtained for particles size distribution for two concentrations un‐ der study (a PZT 53/47 precursor) after different time lapses that are also shown in the figure.

Zr(Acac)4 in 2-MOE, (III) to a dilution process while mixing A+ B, or (IV) to a combined ef‐ fect of all of the above. Must be noted that the less acidic solution (C = 0.20M) possesses, at the same time, the less unimodal distribution function which is an expected result in con‐

'Universal' Synthesis of PZT (1-X)/X Submicrometric Structures Using Highly Stable Colloidal Dispersions...

http://dx.doi.org/10.5772/51996

335

Thus, it seems plausible to assume solution B as a dispersion containing the Ti(Acac)2(OPr)2 compound at molecular level and submicrometric aggregates of Zr(Acac)4 that do not pre‐ cipitate; these are redissolved afterwards when mixed with the acetate rich solution A and 2- MOE. After stirring the resultant sol for one day, the size of some percentage of the Zr(Acac)4 aggregates still ranges between 100 and 1000 nm; after 20 days, however, size is well below 10 nm. From these results, and somehow confirming our previous discussion, we can also see that the less concentrated solution features a considerable amount of parti‐ cles with sizes well above 50nm even after 100 days of stocking. For this reason, we will fo‐ cus our attention from now on in precursor solutions with 0.35 M aged during the first 30-35 days, a time period that, according to our results, seems to be critical in the colloidal stability

As it was said before, all samples under study featured a noticeable stabilization when aged for about 1 month. Figure 7 shows this aging behavior for two PZT (1-x)/x precursor solu‐ tions concentrated at 0.35 M. In all cases, it could be observed a similar time evolution as the

**Figure 7.** Aging behavior, as depicted by the time evolution of particles size distribution curves, for two PZT (1-x)/x

On the other hand, Figure 8 shows the whole dataset of the measured mean particles size for the synthesized solutions. Fitting curves shown here were determined by considering a time

in the van Dongen–Ernst approximation, describes appropriately a nongelling system of dis‐

*1/(1 -λ)+ C2* that, according to the classical Smoluchowski theory

precursor solutions concentrated at 0.35 M: (a) PZT 25/75 and (b) PZT 60/40.

cordance to the relationship between ionic strength and particles size.

of as synthesized sols.

*3.2.3. Aging time dependence*

one described earlier for Figure 6.

dependence given by *<d>= C1t*

According to what is shown, an appropriate discussion relaying on the acidity/particles size dependence cannot be easily established: oscillations in the pH values, see Figure 5, during aging are not followed by oscillations in the mean particles size values. Even though this feature is not fully understood, it may be due to the short lengths of the oligomeric chains present in the solution and to the weak sensitivity of the completely chelated metal com‐ plexes to the ionic strength of the solvent medium. Another possible explanation could be given in terms of the irreversibility associated to the formation of these chains when the sol‐ vent medium is not acidic enough to break the corresponding bonds and/or coordinations.

As can be seen, as prepared sols featured multimodal distributions with some small percent of particles that could even have sizes higher than 1μm. As solutions are aged, these distri‐ butions tend to be unimodal featuring a mean particles size well below 10 nm. We consider that this is a remarkable result of this work as it shows that there is no rule of thumb closely related to the need of using fresh, as prepared solutions for synthesizing several kinds of nanodimensional systems by means of electrophoretic deposition(EPD) or simple template immersion, [33]-[36]. Moreover, in this case we could have chosen solutions aged for 20 days or more for bulk, thin films or nanostructures synthesis due to the better homogeneity re‐ garding particles size distribution in both cases under study.

**Figure 6.** Particles size distribution functions for the analysed PZT 53/47 precursor solutions and for several aging times in the whole measurement range. Log-normal fittings for the more populated size intervals are also depicted.

This behavior could be explained, in principle, on the basis of the chemical reaction which was described for obtaining what we called solution B, see Section 2.4. As we stated, by the end of this step we observed the crystallization of the compound Zr(Acac)4. Such crystalliza‐ tion does not occur after mixing solutions A and B in the presence of 2-MOE which can be attributed (I) to the higher acidity of the final A+B+2-MOE solution, (II) to the solubility of Zr(Acac)4 in 2-MOE, (III) to a dilution process while mixing A+ B, or (IV) to a combined ef‐ fect of all of the above. Must be noted that the less acidic solution (C = 0.20M) possesses, at the same time, the less unimodal distribution function which is an expected result in con‐ cordance to the relationship between ionic strength and particles size.

Thus, it seems plausible to assume solution B as a dispersion containing the Ti(Acac)2(OPr)2 compound at molecular level and submicrometric aggregates of Zr(Acac)4 that do not pre‐ cipitate; these are redissolved afterwards when mixed with the acetate rich solution A and 2- MOE. After stirring the resultant sol for one day, the size of some percentage of the Zr(Acac)4 aggregates still ranges between 100 and 1000 nm; after 20 days, however, size is well below 10 nm. From these results, and somehow confirming our previous discussion, we can also see that the less concentrated solution features a considerable amount of parti‐ cles with sizes well above 50nm even after 100 days of stocking. For this reason, we will fo‐ cus our attention from now on in precursor solutions with 0.35 M aged during the first 30-35 days, a time period that, according to our results, seems to be critical in the colloidal stability of as synthesized sols.

### *3.2.3. Aging time dependence*

*3.2.2. Concentration dependence*

334 Advances in Ferroelectrics

Figure 6 shows the results obtained for particles size distribution for two concentrations un‐ der study (a PZT 53/47 precursor) after different time lapses that are also shown in the figure. According to what is shown, an appropriate discussion relaying on the acidity/particles size dependence cannot be easily established: oscillations in the pH values, see Figure 5, during aging are not followed by oscillations in the mean particles size values. Even though this feature is not fully understood, it may be due to the short lengths of the oligomeric chains present in the solution and to the weak sensitivity of the completely chelated metal com‐ plexes to the ionic strength of the solvent medium. Another possible explanation could be given in terms of the irreversibility associated to the formation of these chains when the sol‐ vent medium is not acidic enough to break the corresponding bonds and/or coordinations. As can be seen, as prepared sols featured multimodal distributions with some small percent of particles that could even have sizes higher than 1μm. As solutions are aged, these distri‐ butions tend to be unimodal featuring a mean particles size well below 10 nm. We consider that this is a remarkable result of this work as it shows that there is no rule of thumb closely related to the need of using fresh, as prepared solutions for synthesizing several kinds of nanodimensional systems by means of electrophoretic deposition(EPD) or simple template immersion, [33]-[36]. Moreover, in this case we could have chosen solutions aged for 20 days or more for bulk, thin films or nanostructures synthesis due to the better homogeneity re‐

**Figure 6.** Particles size distribution functions for the analysed PZT 53/47 precursor solutions and for several aging times in the whole measurement range. Log-normal fittings for the more populated size intervals are also depicted.

This behavior could be explained, in principle, on the basis of the chemical reaction which was described for obtaining what we called solution B, see Section 2.4. As we stated, by the end of this step we observed the crystallization of the compound Zr(Acac)4. Such crystalliza‐ tion does not occur after mixing solutions A and B in the presence of 2-MOE which can be attributed (I) to the higher acidity of the final A+B+2-MOE solution, (II) to the solubility of

garding particles size distribution in both cases under study.

As it was said before, all samples under study featured a noticeable stabilization when aged for about 1 month. Figure 7 shows this aging behavior for two PZT (1-x)/x precursor solu‐ tions concentrated at 0.35 M. In all cases, it could be observed a similar time evolution as the one described earlier for Figure 6.

**Figure 7.** Aging behavior, as depicted by the time evolution of particles size distribution curves, for two PZT (1-x)/x precursor solutions concentrated at 0.35 M: (a) PZT 25/75 and (b) PZT 60/40.

On the other hand, Figure 8 shows the whole dataset of the measured mean particles size for the synthesized solutions. Fitting curves shown here were determined by considering a time dependence given by *<d>= C1t 1/(1 -λ)+ C2* that, according to the classical Smoluchowski theory in the van Dongen–Ernst approximation, describes appropriately a nongelling system of dis‐ persed clusters. Moreover, in the same approximation, the small values of *λ*(*λ<<1*) are usual‐ ly associated with a Brownian diffusion limited aggregation process vastly dominated by collisions between larger clusters with smaller ones [31].

Our samples were also characterized by the High Resolution Transmission Electron Micro‐ scopy (HR-TEM) technique. A few drops of the synthesized sols were deposited on copper grids, evaporated afterwards and put under a JEOL JEM2200 microscope with Omega filter and a spherical aberration corrector; some of the obtained images are shown in Figure 9 for a PZT 95/05 precursor solution. As expected, particles are likely to coalesce as the solvent evaporated prior to characterization, see Figure 9(a) and (b), and some of those nanoparti‐ cles are shown in higher magnification images in Figure 9(c) and (d). Most of the features regarding particle size that were discussed earlier were corroborated by means of this tech‐

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At this point, and considering what has been discussed until now, it must be highlighted that a rigorous control of the chelation rate, pH and aging time could give us a chance to "tune" the average nanoparticles size and/or the colloidal stability as desired. By looking back at Figure 5 through Figure 8 one may feel free to choose to explore several "working points" according to our research or technological needs for every (1-x)/x Zr:Ti ratio in the

**Figure 9.** HR-TEMimages at different magnifications for a PZT 95/05 precursor solution, with C = 0.35 M and aged for

nique for all the precursor solutions.

PZT system.

2 months.

Another view of the aging process could be illustrated by means of the dimensionless nor‐ malized distribution curves, not shown here. In that representation, the broadening of the distribution curves with aging is an indicator of very likely aggregation processes and some‐ how will help to complete the kinetic analysis that we have been through in this section.

In a way of summarizing our results, it could be said that, just after synthesis, several popu‐ lations of particles were detected until homogenized a few days later; then particles tend to grow very slowly with time according to an almost linear law (corresponding to that de‐ scribing a nongelling system) and aggregation is expected. Moreover, these cases fit very well in the Brownian diffusion limited aggregation type where smaller clusters stick to big‐ ger ones when they collide. The slow but evident increase in the mean cluster size, as well as the noticeable broadening in the distribution peaks, must warn us about an undesirable and irreversible precipitation process for, hypothetically, t → ∞.

The high stability shown by our samples even after 4months of stocking and the small val‐ ues for mean particles size (well below 10 nm) are good indicators that the complete chela‐ tion of the organometallic compounds plays a key role in keeping short, and poorly reactive, oligomeric chains. An aggregation process dominated exclusively by Brownian motion is highly desirable when solution stability needs to be maximized.

**Figure 8.** Mean particles size vs. aging time dependence for the systems under study. Fitting curves obeying a *t1/(1*  λ)scaling law are depicted.

Our samples were also characterized by the High Resolution Transmission Electron Micro‐ scopy (HR-TEM) technique. A few drops of the synthesized sols were deposited on copper grids, evaporated afterwards and put under a JEOL JEM2200 microscope with Omega filter and a spherical aberration corrector; some of the obtained images are shown in Figure 9 for a PZT 95/05 precursor solution. As expected, particles are likely to coalesce as the solvent evaporated prior to characterization, see Figure 9(a) and (b), and some of those nanoparti‐ cles are shown in higher magnification images in Figure 9(c) and (d). Most of the features regarding particle size that were discussed earlier were corroborated by means of this tech‐ nique for all the precursor solutions.

persed clusters. Moreover, in the same approximation, the small values of *λ*(*λ<<1*) are usual‐ ly associated with a Brownian diffusion limited aggregation process vastly dominated by

Another view of the aging process could be illustrated by means of the dimensionless nor‐ malized distribution curves, not shown here. In that representation, the broadening of the distribution curves with aging is an indicator of very likely aggregation processes and some‐ how will help to complete the kinetic analysis that we have been through in this section.

In a way of summarizing our results, it could be said that, just after synthesis, several popu‐ lations of particles were detected until homogenized a few days later; then particles tend to grow very slowly with time according to an almost linear law (corresponding to that de‐ scribing a nongelling system) and aggregation is expected. Moreover, these cases fit very well in the Brownian diffusion limited aggregation type where smaller clusters stick to big‐ ger ones when they collide. The slow but evident increase in the mean cluster size, as well as the noticeable broadening in the distribution peaks, must warn us about an undesirable and

The high stability shown by our samples even after 4months of stocking and the small val‐ ues for mean particles size (well below 10 nm) are good indicators that the complete chela‐ tion of the organometallic compounds plays a key role in keeping short, and poorly reactive, oligomeric chains. An aggregation process dominated exclusively by Brownian motion is

**Figure 8.** Mean particles size vs. aging time dependence for the systems under study. Fitting curves obeying a *t1/(1 -*

collisions between larger clusters with smaller ones [31].

336 Advances in Ferroelectrics

irreversible precipitation process for, hypothetically, t → ∞.

highly desirable when solution stability needs to be maximized.

λ)scaling law are depicted.

At this point, and considering what has been discussed until now, it must be highlighted that a rigorous control of the chelation rate, pH and aging time could give us a chance to "tune" the average nanoparticles size and/or the colloidal stability as desired. By looking back at Figure 5 through Figure 8 one may feel free to choose to explore several "working points" according to our research or technological needs for every (1-x)/x Zr:Ti ratio in the PZT system.

**Figure 9.** HR-TEMimages at different magnifications for a PZT 95/05 precursor solution, with C = 0.35 M and aged for 2 months.

This, in fact, is a powerful tool provided by the physicochemical phenomena and mecha‐ nisms previously discussed and, generally speaking, could be extended to other material systems of technological interest at a moderate cost.

### **4. Crystallization route**

The final A + B + 2-MOE solutions were dried at 100°C for several days and, after that, they were thermally treated in order to analyze the phase evolution from the amorphous PZT sol-gel network to the expected final Perovskite structure. Due to fundamental similarities among the different samples we will be discussing only the case where Zr:Ti ratio is 53/47, concentrated at 0.35 M and aged for 2 months.

### **4.1. Experimental techniques**

Crystallization was monitored by means of FT-IR and Raman vibrational spectroscopies, see Sections 2.2.1 and 2.2.2, and by X-Ray Diffraction (XRD) and Scanning Electron Microscopy (SEM) techniques after treating the samples for 12 hours at certain temperatures that were previously chosen. For the sake of clarity, we decided to divide the crystallization study in two temperature intervals: (i) 100 ≤ T ≤ 510° C and (ii) 550 ≤ T ≤ 900° C. At this point, it is im‐ portant to say that, for powders heated at 850° C and 900° C, the treatment was carried out for just 2 hours due to the high volatility of lead for 850° C and beyond.

**Figure 10.** Phase transformations for 100 °C≤ T≤ 510 °C as registered by Raman (a) and FT-IR (b) vibrational spectros‐ copies; in this case, remnant vibrations due to the presence of acetate and chelated metal complexes are highlighted.

are almost the same and, on the other hand, Raman spectra featured no signal given the strong presence of π-bonded organic species. Afterwards, IR spectra revealed the almost complete decomposition of acetates and a decrement of Metal:Acac complexes while any no‐

assigned to a PZT 53/47 started to show up. After treating at this temperature, the FT-IR spectra reveals the formation and definition of the A1(3TO) vibrational mode (~600 cm-1)

Another view of the whole process in this temperature range could be given, as seen in Fig‐ ure 11, by means of XRD characterization. XRD patterns clearly illustrate the phase transfor‐ mation exhibited by an amorphous material turning into crystalline; as for the Raman

These maxima are usually associated to an intermediate metastable phase belonging to the Fluorite crystal system (F) which is mainly characterized by the spatial disarrangement of the oxygen atoms, vacancies and metal cations while coexisting with some carbonates and/or oxides that still remain in the material. Afterwards, when carbonates and oxides re‐ act, a new metastable phase is formed but this new arrangement tends to be more ordered than the previous fluorite. This intermediate phase is considered as belonging to the Pyro‐ chlore crystal system (P or Pyr) and consisting of a 2 x 2 x 2 ordered fluorite cell with oxygen and metallic vacancies or, from another point of view, consisting of a 2 x 2 x 2 non stoichio‐ metric Perovskite (Per) cell. This rigorous differentiation between Fluorite and Pyrochlore is not always considered in literature and it can be subtle sometimes, especially when working

ticeable Raman signal is still absent. Just after T = 450°

which is also characteristic for this compound [38]-[41].

spectra, crystallinity starts to get noticed at T = 450°

existence of well defined peaks near 28°

C, IR spectra showed no variation; the intensity ratios between the detected modes

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C the active Raman modes normally

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C. However, these patterns revealed the

that disappear almost entirely at 510°C.

Until 225°

Scanning Electron Microscopy (SEM) was carried out on fine ground powders in a Leica Cambridge Stereoscan 440 microscope and the X-Ray powder diffraction patterns were re‐ corded over a 20-60° 2θ range on a Bruker D8 Advance diffractometer with filtered CuKα radiation. In this technique, the identified phases were indexed by comparing the resulting diffraction patterns with those of similar compositions reported in the International Union of Crystallography (IUCr) JCPDS-ICDD database.

### **4.2. 100 ≤ T ≤ 510 °C**

In this temperature range the material goes through very noticeable phase transformations; the initial amorphous powder starts to show some crystallinity for about 500° C, right after the combustion of the remaining organic species. Figure 10 shows these first crystallization stages as recorded by the Raman (a) and FT-IR (b) vibrational spectroscopies. The vibration‐ al modes associated to the remnant organic compounds coexisting after heat treating at 100 °C are also highlighted in this graph, Figure 10(b), and it must be noticed that they corre‐ spond, basically, to the lead acetate and to the completely chelated organometallic com‐ plexes. More explicitly, numbered modes in Figure 2(b) have been assigned to: (1)Pb(OAc)2:ρ(COO), (2)Pb(OAc)2:δS(COO), (3)Pb(OAc)2:νS(C-C), (4)Ti(OPr)2(Acac)2:Ti-Acac, (5)Zr(Acac)4:Zr-Acac,(6)Pb(OAc)2:δS(CH3), (7)Ti(OPr)2(Acac)2/Zr(Acac)4:δ(CH3:Acac) + Pb(OAc)2:νS(CO)and (8)Ti(OPr)2(Acac)2/Zr(Acac)4:[ν(C-C) + ν(C-O:Acac)] + Pb(OAc)2:νA(C=O)[26],[37].

This, in fact, is a powerful tool provided by the physicochemical phenomena and mecha‐ nisms previously discussed and, generally speaking, could be extended to other material

The final A + B + 2-MOE solutions were dried at 100°C for several days and, after that, they were thermally treated in order to analyze the phase evolution from the amorphous PZT sol-gel network to the expected final Perovskite structure. Due to fundamental similarities among the different samples we will be discussing only the case where Zr:Ti ratio is 53/47,

Crystallization was monitored by means of FT-IR and Raman vibrational spectroscopies, see Sections 2.2.1 and 2.2.2, and by X-Ray Diffraction (XRD) and Scanning Electron Microscopy (SEM) techniques after treating the samples for 12 hours at certain temperatures that were previously chosen. For the sake of clarity, we decided to divide the crystallization study in

Scanning Electron Microscopy (SEM) was carried out on fine ground powders in a Leica Cambridge Stereoscan 440 microscope and the X-Ray powder diffraction patterns were re‐ corded over a 20-60° 2θ range on a Bruker D8 Advance diffractometer with filtered CuKα radiation. In this technique, the identified phases were indexed by comparing the resulting diffraction patterns with those of similar compositions reported in the International Union

In this temperature range the material goes through very noticeable phase transformations;

the combustion of the remaining organic species. Figure 10 shows these first crystallization stages as recorded by the Raman (a) and FT-IR (b) vibrational spectroscopies. The vibration‐ al modes associated to the remnant organic compounds coexisting after heat treating at 100 °C are also highlighted in this graph, Figure 10(b), and it must be noticed that they corre‐ spond, basically, to the lead acetate and to the completely chelated organometallic com‐ plexes. More explicitly, numbered modes in Figure 2(b) have been assigned to: (1)Pb(OAc)2:ρ(COO), (2)Pb(OAc)2:δS(COO), (3)Pb(OAc)2:νS(C-C), (4)Ti(OPr)2(Acac)2:Ti-Acac, (5)Zr(Acac)4:Zr-Acac,(6)Pb(OAc)2:δS(CH3), (7)Ti(OPr)2(Acac)2/Zr(Acac)4:δ(CH3:Acac) + Pb(OAc)2:νS(CO)and (8)Ti(OPr)2(Acac)2/Zr(Acac)4:[ν(C-C) + ν(C-O:Acac)] +

the initial amorphous powder starts to show some crystallinity for about 500°

C and (ii) 550 ≤ T ≤ 900°

C and beyond.

C and 900°

C. At this point, it is im‐

C, right after

C, the treatment was carried out for

systems of technological interest at a moderate cost.

concentrated at 0.35 M and aged for 2 months.

two temperature intervals: (i) 100 ≤ T ≤ 510°

portant to say that, for powders heated at 850°

just 2 hours due to the high volatility of lead for 850°

of Crystallography (IUCr) JCPDS-ICDD database.

**4. Crystallization route**

338 Advances in Ferroelectrics

**4.1. Experimental techniques**

**4.2. 100 ≤ T ≤ 510 °C**

Pb(OAc)2:νA(C=O)[26],[37].

**Figure 10.** Phase transformations for 100 °C≤ T≤ 510 °C as registered by Raman (a) and FT-IR (b) vibrational spectros‐ copies; in this case, remnant vibrations due to the presence of acetate and chelated metal complexes are highlighted.

Until 225° C, IR spectra showed no variation; the intensity ratios between the detected modes are almost the same and, on the other hand, Raman spectra featured no signal given the strong presence of π-bonded organic species. Afterwards, IR spectra revealed the almost complete decomposition of acetates and a decrement of Metal:Acac complexes while any no‐ ticeable Raman signal is still absent. Just after T = 450° C the active Raman modes normally assigned to a PZT 53/47 started to show up. After treating at this temperature, the FT-IR spectra reveals the formation and definition of the A1(3TO) vibrational mode (~600 cm-1) which is also characteristic for this compound [38]-[41].

Another view of the whole process in this temperature range could be given, as seen in Fig‐ ure 11, by means of XRD characterization. XRD patterns clearly illustrate the phase transfor‐ mation exhibited by an amorphous material turning into crystalline; as for the Raman spectra, crystallinity starts to get noticed at T = 450° C. However, these patterns revealed the existence of well defined peaks near 28° that disappear almost entirely at 510°C.

These maxima are usually associated to an intermediate metastable phase belonging to the Fluorite crystal system (F) which is mainly characterized by the spatial disarrangement of the oxygen atoms, vacancies and metal cations while coexisting with some carbonates and/or oxides that still remain in the material. Afterwards, when carbonates and oxides re‐ act, a new metastable phase is formed but this new arrangement tends to be more ordered than the previous fluorite. This intermediate phase is considered as belonging to the Pyro‐ chlore crystal system (P or Pyr) and consisting of a 2 x 2 x 2 ordered fluorite cell with oxygen and metallic vacancies or, from another point of view, consisting of a 2 x 2 x 2 non stoichio‐ metric Perovskite (Per) cell. This rigorous differentiation between Fluorite and Pyrochlore is not always considered in literature and it can be subtle sometimes, especially when working with a well known compound as PZT is. Anyway, in Figure 11 we have denoted by F (Fluo‐ rite) and P (Pyrochlore) the diffraction maxima associated with each of these phases and ac‐ cording to the indexed cubic structures reported in reference [42] (a = 5.25 Å) and in reference [43] (a = 10.48Å), respectively.

Up to this point, we have seen the almost complete elimination of organic ligands with heating as well as the amorphous/crystalline phase transformation going through two intermediate metastable phases. In the next subsection we will carry on a similar study for higher tempera‐ tures while exploring the formation of pure Perovskite phase with morphological quality.

**Figure 12.** Phase transformations for 550 °C≤ T≤ 900 °C as registered by Raman (a) and FT-IR (b) vibrational spectros‐

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**Figure 13.** Thermal evolution of the powders under study when 550 °C ≤ T ≤ 900 °C as evidenced by XRD patterns in

the 20 - 60 deg. 2θ interval. Regions where the most intense Pyr peak showed up are denoted by P.

copies.

**Figure 11.** Thermal evolution of PZT 53/47 precursor powders when100 °C≤ T≤ 510 °C as recorded by XRD patterns.

### **4.3. 550≤ T ≤ 900 °C**

Figure 12 shows Raman (b) and FT-IR (b) spectra for heat treated powders in the 550 – 900 °C range. Unlike the previously studied temperature range, vibrational spectra did not show drastic or very noticeable changes. The FT-IR spectra features the only IR active band for aPZT-R3m in the 400-1500 cm-1 range (A1(3TO)); this band, associated to the extensional vi‐ brations of the Perovskite BO6 octahedra, shifts slightly to higher frequencies while gets nar‐ rower as temperature increases. This shifting suggests a more compact octahedral structure while a narrower band could be the evidence of a better 'environment' around the octahe‐ dron or, in other words, a better local stoichiometry [44] that seems to be minimum for T = 800° C, an indicative of an optimum crystallization. Raman spectra featured vibrational modes that tend to define better as temperature increases.

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with a well known compound as PZT is. Anyway, in Figure 11 we have denoted by F (Fluo‐ rite) and P (Pyrochlore) the diffraction maxima associated with each of these phases and ac‐ cording to the indexed cubic structures reported in reference [42] (a = 5.25 Å) and in

Up to this point, we have seen the almost complete elimination of organic ligands with heating as well as the amorphous/crystalline phase transformation going through two intermediate metastable phases. In the next subsection we will carry on a similar study for higher tempera‐ tures while exploring the formation of pure Perovskite phase with morphological quality.

**Figure 11.** Thermal evolution of PZT 53/47 precursor powders when100 °C≤ T≤ 510 °C as recorded by XRD patterns.

Figure 12 shows Raman (b) and FT-IR (b) spectra for heat treated powders in the 550 – 900 °C range. Unlike the previously studied temperature range, vibrational spectra did not show drastic or very noticeable changes. The FT-IR spectra features the only IR active band for aPZT-R3m in the 400-1500 cm-1 range (A1(3TO)); this band, associated to the extensional vi‐ brations of the Perovskite BO6 octahedra, shifts slightly to higher frequencies while gets nar‐ rower as temperature increases. This shifting suggests a more compact octahedral structure while a narrower band could be the evidence of a better 'environment' around the octahe‐ dron or, in other words, a better local stoichiometry [44] that seems to be minimum for T =

C, an indicative of an optimum crystallization. Raman spectra featured vibrational

modes that tend to define better as temperature increases.

reference [43] (a = 10.48Å), respectively.

340 Advances in Ferroelectrics

**4.3. 550≤ T ≤ 900 °C**

800°

**Figure 12.** Phase transformations for 550 °C≤ T≤ 900 °C as registered by Raman (a) and FT-IR (b) vibrational spectros‐ copies.

**Figure 13.** Thermal evolution of the powders under study when 550 °C ≤ T ≤ 900 °C as evidenced by XRD patterns in the 20 - 60 deg. 2θ interval. Regions where the most intense Pyr peak showed up are denoted by P.

XRD patterns depicted in Figure 13 allow us to notice the temperature evolution of the dis‐ tinctive Perovskite diffraction peaks. Peaks indexing has been done according to a PZT 53/47, FR(HT); sym: R3m. Besides, regions where the most intense Pyr peak showed up (~28.5° ) are denoted by P; the Pyr phase is present in a small percentage (%Pyr ~ 1 %) except for powders treated at T = 800°C which must be chosen as the appropriate crystallization tem‐ perature for this material. The final Perovskite phase homogenizes and tends to be predomi‐ nant as temperature increases until lead losses become noticeable.

On the other hand, Figure 15 shows SEM micrographs of crystallized PZT (1-x)/x powdered samples. As illustrated, a granular structure of fair morphological uniformity is found in all cases along with a remarkable submicrometric average grain size. Another interesting fea‐ ture is the low porosity found in all compositions, somehow resembling the granular struc‐ ture of sintered bulk samples, which certainly favors the formation of a high density

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As it was said at the beginning of this chapter, this kind of sub-micron granular structures, or nanoceramics, allows the technological exploitation of dielectric and ferro/piezoelectric size related features. It is not shown here, but that very same samples have been resynthe‐ sized a couple of times with starting reactants being bought to different companies and final results showed almost the same granular and morphological quality. Repetitiveness is also a bonus when working with chemical routes of synthesis that tend to be less straightforward,

**Figure 15.** SEM micrographs showing the morphological quality of the PZT (1-x)/x powders when crystallized at 800

°C: (a) PZT 25/75, (b) PZT 60/40, (c) PZT 80/20 and (d) PZT 95/05.

ferroelectric material.

even though cheaper, than physical ones.

The rest of the PZT compositions that were chosen for this study, PZT (1-x)/x with C = 0.35 M and 2 months aged, were also treated at 800 °C in order to obtain the appropiate Perov‐ skite phase; Figure 14 shows the corresponding XRD patterns for each one of them. As seen in the graph, all samples attained a perfect crystallization in the corresponding phase and the proper pattern indexing as well as the phase indentification are also shown.

**Figure 14.** XRD patterns for PZT (1-x)/x under study: (a) PZT 25/75, (b) PZT 60/40, (c) PZT 80/20 and (d) PZT 95/05.

On the other hand, Figure 15 shows SEM micrographs of crystallized PZT (1-x)/x powdered samples. As illustrated, a granular structure of fair morphological uniformity is found in all cases along with a remarkable submicrometric average grain size. Another interesting fea‐ ture is the low porosity found in all compositions, somehow resembling the granular struc‐ ture of sintered bulk samples, which certainly favors the formation of a high density ferroelectric material.

XRD patterns depicted in Figure 13 allow us to notice the temperature evolution of the dis‐ tinctive Perovskite diffraction peaks. Peaks indexing has been done according to a PZT 53/47, FR(HT); sym: R3m. Besides, regions where the most intense Pyr peak showed up (~28.5°

are denoted by P; the Pyr phase is present in a small percentage (%Pyr ~ 1 %) except for powders treated at T = 800°C which must be chosen as the appropriate crystallization tem‐ perature for this material. The final Perovskite phase homogenizes and tends to be predomi‐

The rest of the PZT compositions that were chosen for this study, PZT (1-x)/x with C = 0.35 M and 2 months aged, were also treated at 800 °C in order to obtain the appropiate Perov‐ skite phase; Figure 14 shows the corresponding XRD patterns for each one of them. As seen in the graph, all samples attained a perfect crystallization in the corresponding phase and

**Figure 14.** XRD patterns for PZT (1-x)/x under study: (a) PZT 25/75, (b) PZT 60/40, (c) PZT 80/20 and (d) PZT 95/05.

the proper pattern indexing as well as the phase indentification are also shown.

nant as temperature increases until lead losses become noticeable.

342 Advances in Ferroelectrics

)

As it was said at the beginning of this chapter, this kind of sub-micron granular structures, or nanoceramics, allows the technological exploitation of dielectric and ferro/piezoelectric size related features. It is not shown here, but that very same samples have been resynthe‐ sized a couple of times with starting reactants being bought to different companies and final results showed almost the same granular and morphological quality. Repetitiveness is also a bonus when working with chemical routes of synthesis that tend to be less straightforward, even though cheaper, than physical ones.

**Figure 15.** SEM micrographs showing the morphological quality of the PZT (1-x)/x powders when crystallized at 800 °C: (a) PZT 25/75, (b) PZT 60/40, (c) PZT 80/20 and (d) PZT 95/05.

### **5. Chapter remarks**

Through this chapter, we tried to expose some relevant results directly concerned with a fea‐ sible 'universal' controlled synthesis of nano/submicrometric grain-sized PZT (1-x)/x piezo‐ electric structures that, technically speaking, potentially enhance the performance of current commercially bulk based devices by exploiting the size effects related phenomena that arise, as is commonly accepted, for grain sizes below 1 μm. Under the light of this study, the tenta‐ tive bottom-up "design" of any desired nano/submicrometric PZT (1-x)/x structure seems to be a plausible and successful task that, methodologically at least, could be expanded to more complex materials systems.

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'Universal' Synthesis of PZT (1-X)/X Submicrometric Structures Using Highly Stable Colloidal Dispersions...

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[8] Schneller T, Waser R. Chemical Solution Deposition of Ferroelectric Thin Films: State

[9] Majumder SB, Bhaskar S, Katiyar RS. Critical issues in sol-gel derived ferroelectric

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[13] Lakeman CDE, Payne DA. Processing Effects in the Sol–Gel Preparation of PZT Dried Gels, Powders, and Ferroelectric Thin Layers. Journal of the American Ceramic

[14] Wang FP, Yu YJ, Jiang ZH, Zhao LC. Synthesis of Pb1−xEux(Zr0.52Ti0.48)O3 nanopow‐ ders by a modified sol–gel process using zirconium oxynitrate source. Materials

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Further reading regarding the work that has been shown here can be found in references [45]-[48]. On the other hand, references [49]-[60] will also provide the reader with very re‐ cent research papers in this field that undoubtedly extend the applicability and versatility of what has been discussed.

### **Author details**


2 CCADET-UNAM, Cd. Universitaria, Coyoacán, México D.F., México

3 Fac. de Física/IMRE, San Lázaro y L, Univ. de la Habana, Habana , Cuba

### **References**


[6] Wang ZL. The new field of nanopiezotronics. Materials Today 2007;10(5) 20-28.

**5. Chapter remarks**

344 Advances in Ferroelectrics

complex materials systems.

what has been discussed.

**Author details**

A. Suárez-Gómez1

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Further reading regarding the work that has been shown here can be found in references [45]-[48]. On the other hand, references [49]-[60] will also provide the reader with very re‐ cent research papers in this field that undoubtedly extend the applicability and versatility of

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**Chapter 16**

**Phase Transitions, Dielectric and Ferroelectric**

Ferroelectric perovskites based on Na0.5Bi0.5TiO3 (NBT) are considered among the most promising lead-free candidate materials to substitute Pb(Zr*1-x*Ti*x*)O3 (PZT) in devices de‐ signed to respect standards and environmental laws. Taking into account the toxicity of lead-based systems, there are numerous lead-free piezoelectric materials under investigation in worldwide spread laboratories for replacing PZT in future devices. Constant efforts are

Solid-solution systems based on lead-free perovskites like Na0.5K0.5NbO3 (NKN), BaTiO3 (BT), Na0.5Bi0.5TiO3 (NBT) or bismuth layered-structured SrBi2Ta2O9 (SBT), and SrBi2Nb2O9 (SBN) are considered as viable alternatives for replacing lead-based materials. For example, (K,Na)NbO3–LiTaO3–LiSbO3 alkaline niobate ceramics exhibit a *d <sup>33</sup>* piezoelectric coefficient up to 416 pC/N together with Curie temperature Tc around 526 K, as reported by Saito *et al* [1]. Sodium/bismuth titanate (NBT) belongs to the bismuth-based perovskites in which the A-site atom is replaced. The crystalline structure, phase transitions and physical properties have been intensively studied since the discovery of the material in 1960 by Smolensky *et al* [2]. NBT has a relatively high depolarization temperature, Td = 470 K, high remanent polari‐

high value of the coercive field and high electrical conductivity, NBT cannot be easily polar‐ ized, therefore different A-site substitutions have been attempted to avoid this drawback.

and reproduction in any medium, provided the original work is properly cited.

and piezoelectric coefficient d33 = 125 pC/N [3]. However, owing to the

© 2013 Scarisoreanu et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

made to find viable replacements for all these materials containing harmful elements.

**Properties of Lead-free NBT-BT Thin Films**

N. D. Scarisoreanu, R. Birjega, A. Andrei, M. Dinescu,

Additional information is available at the end of the chapter

F. Craciun and C. Galassi

http://dx.doi.org/10.5772/ 52395

**1. Introduction**

zation, 38 *μ*C/cm2
