Comments on the spontaneous strain and polarization of polycrystalline ferroelectric ceramics

Comments on the spontaneous strain and polarization of polycrystalline

ferroelectric ceramics

Jiangyu Li and Kaushik Bhattacharya

Division ofEngineering and Applied Science

California Institute of Technology, Pasadena, CA 91125

ABSTRACT

A framework to calculate the spontaneous strain and polarization of a polycrystalline ferroelectric is presented, and

various applications are discussed.

Keywords: Ferroelectric, spontaneous strain, texture, energy minimization

1. INTRODUCTION

It has recently been recognized that it is possible to obtain large strains through electrostriction of ferroelectric crystals

obtained by polarization rotation ',

by domain switching as a result of applied electric field or mechanical stress 2

However,

large strains usually cannot be obtained in a polycrystalline ferroelectric ceramics due to the incompatibility

between neighboring grains. So the identification of the optimal texture of polycrystals for high-strain actuation is

worthwhile challenge. A related problem is the poling of the piezoelectric ceramic PZT, which is a solid solution of Lead

Titanate and Lead Zirconate. The Titanium rich compositions where the material is tetragonal, and the rhombohedral rich

compositions where the material is rhombohedral are both very difficult to pole, while the material is dramatically easy to

pole at the 'morphotropic phase boundary'. An understanding ofthis difference is also desirable.

This work examines the macroscopic behaviors of ferroelectric solids in terms of the crystal systems and texture, using

the homogenization theory and energy minimization following the framework used by Bhattacharya and Kobn to study

polycrystaiiine shape-memory alloys. The spontaneous strain and polarization for single crystals is characterized, and the

optimal texture for high-actuation strain for polycrystals is identified. The optimal electro-niechanical property of PZT at

morphotropic phase boundary is also explained. This paper summarizes the results, and the reader is refër:ed to Li and

Ltaary4 for details.

2. EFFECTWE THEORY FOR FERROELECTRIC CERAMICS

For a ferroelectric crystal Q subject to an applied traction t on part of its boundary )2,

the

displacement u and

polarization p ofthe ferroelectric are those that minimize the potential energy

(u,p)=$!Vp.AVp+W(x,e[uJ,p)_E0 .pdx— Jt0

•udS+

f2IVbI2

ô)2

where the electric potential q is determined by solving Maxwell equation in all space subject to appropriate boundary

conditions, and E0 is the electric field in the absence of the ferroelectric. Briefly, the first term is the exchange energy or the

energy needed to form domain walls, the second is the stored energy density which encodes the information about

crystallography and texture, the third term represents the applied electric field, the fourth the applied mechanical loads and

finally the last term is the electrostatic energy created due to the polarization distribution. Because of symmetry, the stored

energy density exhibits a multi-well structure characterized by the set K =

u{(e,

p()} on which W(x,e, p) =

where

is the transformation strain and p( is the spontaneous polarization; see Fig. 1. It is first postulated that the exchange

energy is negligible, and this is reasonable in large specimens. Second, a homogenization theory has been developed for

periodic ferroelectrics based on the theoretical framework of F-convergence .

This

theory tells us that if the grain size is

small compared to the size of the specimen, one may replace the spatially inhomogeneous term W in the potential energy

Smart Structures and Materials 2001: Active Materials: Behavior and Mechanics, Christopher S. Lynch,

Downloaded From: http://proceedings.spiedigitallibrary.org/ on 01/26/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx

with an effective energy density W(e,

which

are spatially homogeneous. W(e, p) is the effective energy of a polycrystal

when the average strain is e and the average polarization is p. The theory also provides a formula for this effective energy;

this formula is unfortunately not explicit but involves a minimization problem.

This effective energy W(e, p) can be obtained in two steps; see Fig. 1. First, a mesoscopic energy W(e, p) may be

obtained for a single crystal, and then this can be homogenized to obtain the effective energy of the polycrystal. The

mesoscopic energy is zero on a set Zs which identifies all the possible average spontaneous strain and polarization that a

single crystal can display as it forms different domain patterns. The effective energy is zero on a set Z" which identifies all

the possible average spontaneous strain and polarization that a polycrystalline ceramic can display as the grains form

different domain patterns. The rest ofthe paper discusses estimates ofthese sets.

ZS)

(e, p)

Figure 1. The effective energy and zero energy set ofsingle crystal and polycrystal.

3. SPONTANEOUS STRAIN AND POLARIZATION

The set Zs of spontatneous strains and polarizations of a single crystal correspond to the average strain and polarization

as it forms different domain patterns. It has been wn7 that when the ferroelectric wells are pair-wise compatible, (e°, p°)

is a spontaneous strains and polarizations for a single crystal ifand only if

e0 =

1eW,

p0 1(2f, —

1)p',

where 0 and

and

0 f,

1 .

tetragonal crystal, the spontaneous strains of single crystal have fixed trace

and zero off-diagonal elements. The magnitude of the diagonal elements is also bounded. These strains lie in a reduced 2-

dimensional strain space. In rhombohedral crystals, the spontaneous strain must have fixed and identical diagonal elements,

and their off-diagonal elements are bounded. They lie in a reduced 3-dimensional strain space. hi ferroelectric with

tetragonal-rhombohedral morphotropic phase boundary (MPB), such as PZT, or Pb(ZrTi1)O3, around x=O.52, the tetragonal

and rhombohedral variants are compatible with each other if restriction on lattice parameters are satisfied, and the

spontaneous strains in ferroelectric with MPB have fixed trace, and their off-diagonal elements are bounded. They lie in the

5-dimensional strain space. So crystal at MPB has much larger set of spontaneous strain than that of tetragonal and

rhombohedral ferroelectrics. This has important implication on electro-mechanical properties of PZT ceramics. It is recently

discovered that a previous unreported monoclinic phase exist in PZT at MPB 8,

9 where

not all the variants are pair-wise

compatible. We can derive an inner set of spontaneous strain, which lies in the 5-dimiensional strain space, with the trace

fixed, and off-diagonal elements bounded.

(e, p)

Proc. SPIE Vol. 4333

Downloaded From: http://proceedings.spiedigitallibrary.org/ on 01/26/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx