Zuo-Guang. Ye Advanced Dielectric Piezoelectric and Ferroelectric Materials: Synthesis, Characterisation and Applications

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Handbook of dielectric, piezoelectric and ferroelectric materials300

BaTiO

ceramics, the dielectric constant increased with decreasing grain

size and at a grain size of 800 nm, the dielectric constant reached a maximum

value of 5000.

They also depicted the relationship between the grain size

and domain size, and the domain size of BaTiO

ceramics with a grain size

of 800 nm was 150 nm. This result suggested that for domain sizes above

150 nm, the dielectric and piezoelectric properties can increase with decreasing

domain sizes, while for the domain size below 150 nm, the both properties

can decrease with decreasing domain sizes.

To induce the fine domain size around 150 nm, a combination of patterned

electrode and uniaxial stress-field can be used in the new poling system

above T

. We consider that domain wall engineering includes a universal

concept of ultra-high domain wall contribution to piezoelectricity and any

techniques to induce fine domain size of about 150 nm in the [001]

oriented

orthorhombic crystals. It is expected that the development of the domain

wall engineering will allow us to create new lead-free piezoelectrics with

ultra-high piezoelectric properties.

10.10 Conclusions and future trends

Various engineered domain configurations were induced in BaTiO

single

crystals, and their piezoelectric properties were investigated as a function of

the crystal structure, crystallographic orientation and domain size (domain

wall density). As a result, BaTiO

crystals with the orthorhombic mm2 phase

showed the highest piezoelectric properties among three kinds of BaTiO

crystals of the tetragonal 4mm, orthorhombic mm2 and rhombohedral 3m

phases. On the other hand, BaTiO

crystals oriented along the [001]

direction

always exhibited larger piezoelectric properties than those oriented along the

[111]

direction. The highest piezoelectric properties (d

of 500 pC/N and

of 85%) were obtained for the [001]

poled orthorhombic BaTiO

crystals,

and these values were much larger than those of the conventional PZT ceramics.

The domain size dependence of the piezoelectric properties was also discussed,

and the result revealed that the piezoelectric properties were strongly dependent

on the domain sizes (domain wall density), i.e. the piezoelectric properties

significantly increased with decreasing domain size. The calculated d

the [111]

oriented tetragonal BaTiO

single-domain crystal was –62 pC/N,

while the measured value of the [111]

poled tetragonal BaTiO

crystal with

a domain size of 3 µm was –243.2 pC/N, i.e. a four-fold increase. When the

much finer domain size (below 1 µm) can be induced in the [001]

poled

orthorhombic BaTiO

crystals, the significantly enhanced piezoelectric

properties can be expected. However, BaTiO

crystals have tetragonal 4mm

symmetry at room temperature and the desirable orthorhombic mm2 phase is

stable below 5 °C. Therefore, ferroelectric single crystals with an orthorhombic

mm2 phase at room temperature are highly desirable, and [001]

poled

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Domain wall engineering in piezoelectric crystals 301

orthorhombic KNbO

single crystals with domain sizes below 1 µm will be

one of the promising candidates for much enhanced piezoelectric properties.

When application of the domain wall engineering is being considered in

the future, single crystals have three disadvantages over other piezoelectric

materials. One is very high cost, and this cost can limit applications. Second

is very weak mechanical strength, and this weakness also reduces application.

Third is a intrinsic difficulty to induce very fine domain sizes below 3 µm.

On the other hand, normal piezoelectric ceramics is completely opposite to

the single crystals, and the disadvantage of normal ceramics is lack of

crystallographic orientation. Therefore, we must combine the advantages of

single crystals and ceramics. I believe that this dream material is grain-

oriented ceramics along the engineered domain configuration. This has several

advantages, especially for very fine domain sizes. It is well known that it is

easy to induce fine domain sizes below 100 nm in piezoelectric ceramics.

The application of domain wall engineering for grain-oriented ceramics along

engineered domain direction can lead to super-piezoelectrics with ultra-high

piezoelectric properties over d

of 1000 pC/N and k

of 90%.

10.11 Acknowledgements

I would like to thank Mr O. Nakao of Fujikura Ltd for preparing the TSSG-

grown BaTiO

single crystals with excellent chemical quality. I also would

like to thank Dr S.-E. ‘Eagle’ Park, Dr T. R. Shrout and Dr L. E. Cross of

MRL, Pennsylvania State University, for their helpful suggestion and many

discussions about the engineered domain configurations. Moreover, I would

like to thank Dr Y. Ishibashi of Aichi-shukutoku University, Dr D. Damjanovic

of EPFL, Dr A. J. Bell of University of Leeds and Dr L. E. Cross of Pennsylvania

State University for their helpful discussions about the domain wall contribution

to the piezoelectric properties. I would like to thank Dr J. Erhart and Dr J.

Fousek of ICPR, Technical University of Liberec, for their helpful discussions

about the analysis of the domain configuration and calculation of the d

surface. This study was partially supported by (1) a Grant-in-Aid for Scientific

Research (11555164 and 16656201) from the Ministry of Education, Culture,

Sports, Science, and Technology, Japan, (2) TEPCO Research Foundation,

(3) the Japan Securities Scholarship Foundation, (4) Toray Science and

Technology Grant, (5) the Kurata Memorial Hitachi Science and Technology

Foundation, (6) the Electro-Mechanic Technology Advanced Foundation,

(7) the Tokuyama Science Foundation and (8) the Yazaki Memorial Foundation

for Science and Technology.

10.12 References

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Handbook of dielectric, piezoelectric and ferroelectric materials302

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Domain wall engineering in piezoelectric crystals 303

20. Wada S, Suzuki S, Noma T, Suzuki T, Osada M, Kakihana M, Park E-E, Cross L E,

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304

11.1 Introduction

Enhancement of the piezoelectric response and its anisotropy in perovskite

crystals has been a central point of the research in ferroelectrics since the

rediscovery of the large electromechanical activity in relaxor ferroelectric

single crystals in 1997 (Kuwata et al., 1982; Park and Shrout, 1997). Facilitated

polarization rotation, specifically in the presence of spontaneously formed or

field-induced monoclinic phases, has been proposed in the literature as the

origin of the high electromechanical response (Park and Shrout, 1997; Noheda

et al., 1999; Fu and Cohen, 2000; Guo et al., 2000). The high dielectric,

piezoelectric and elastic properties in the morphotropic phase boundary region

of solid solutions have been interpreted in terms of an isotropic free energy

profile by Ishibashi and co-workers. They associated the isotropic free energy

profile with the large permittivity in directions perpendicular to the polar

axis, the latter being equivalent to the propensity of the material for strong

polarization rotation (Ishibashi and Iwata, 1998, 1999a,b; Ishibashi, 2002;

Carl and Häerdtl, 1971). Results of ab-initio calculations by Fu and Cohen

(2000) have indicated that in barium titanate (BaTiO

) crystals the field-

induced polarization rotation is accompanied by flattening of a free energy

profile in the field region where piezoelectric response is maximized. Subsequent

studies have demonstrated that a free energy instability is indeed the

thermodynamic origin of the enhanced piezoelectric response in simple

perovskites, regardless of whether it is induced by external pressure, electric

field, composition or temperature (Wu and Cohen, 2005; Budimir et al., 2006)

and whether it is accompanied by polarization rotation or polarization elongation

(Budimir et al., 2006). The recently discovered enhancement of the piezoelectric

response in some relaxor ferroelectrics near an electric field-induced critical

point (Kutnjak et al., 2006) falls in the same category of thermodynamic

phenomena. The same is probably the case for the adaptive phase mechanism

discussed in Jin et al., (2003) and piezoelectric enhancement by domain

density engineering discovered by Wada and co-workers (2004, 2005).

Enhancement of piezoelectric properties in

perovskite crystals by thermally,

compositionally, electric field and

stress-induced instabilities

D DAMJANOVIC, M DAVIS and M BUDIMIR,

Swiss Federal Institute of Technology – EPFL, Switzerland

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Enhancement of piezoelectric properties in perovskite crystals 305

In this chapter we discuss the piezoelectric response and its anisotropy in

monodomain single crystal perovskites as a function of temperature,

composition variation (morphotropic phase boundary), and external electric

field and mechanical stress biases. Using Landau–Ginzburg–Devonshire (LGD)

thermodynamic theory it is possible to show that the common feature for

different enhancements of the piezoelectric effect is anisotropic flattening of

a free energy profile; the enhancement is largest near phase transition points

where the system is in metastable states. The second section reviews the

deformation of crystals under electric and mechanical fields. Monoclinic

deformations obtained from parent tetragonal, orthorhombic and rhombohedral

phases are discussed and polarization rotation is contrasted to polarization

elongation. The treatment of the piezoelectric enhancement and its anisotropy

within the LGD formalism is discussed in the third section. This section also

discusses the instabilities in a free energy induced by temperature, composition,

electric fields and mechanical stresses and their effect on piezoelectric

properties. Some general conclusion and perspectives are given in the fourth

section.

11.2 Deformation of perovskite crystals under

external fields

11.2.1 Polarization rotation and monoclinic phases

In this section the deformation of tetragonal, orthorhombic and rhombohedral

crystals by external fields is considered in terms of polarization rotation and

polarization extension (or contraction). Polarization rotation (Fu and Cohen,

2000) in the presence of monoclinic phases (Noheda, 2002; Noheda and

Cox, 2006), particularly in the morphotropic phase boundary region of solid

solutions, has been widely considered as the main mechanism of the large

piezoelectric response along non-polar axes in complex solid solutions, such

as (1–x)Pb(Zn

1/3

2/3

–xPbTiO

(PZN–xPT), (1–x)Pb(Mg

1/3

2/3

–

xPbTiO

(PMN–xPT) and Pb(Zr, Ti)O

(PZT). In the monoclinic phases, the

polarization is not fixed along a crystallographic axis but is allowed by

symmetry to lie anywhere within a single mirror plane. Thus, the M

)

and M

monoclinic states (see Fig. 11.1), with [101]

and [010]

mirror

planes, respectively, form ‘structural bridges’ (Noheda and Cox, 2006) between

the 〈111〉

, 〈101〉

and 〈001〉

polar directions of the 3m rhombohedral (R),

mm2 orthorhombic (O) and 4mm tetragonal (T) phases typically observed in

perovskite ferroelectrics; the subscript ‘C’ means the direction is defined

with respect to the cubic axes of the high-temperature, parent phase.

As it appears that there is some confusion in the literature with respect to

the role of a monoclinic phase and polarization rotation in enhanced

electromechanical properties , it is perhaps useful to emphasize at this stage

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Handbook of dielectric, piezoelectric and ferroelectric materials306

the following points which will be elaborated further in the text. Firstly, the

fact that the polar vector can lie anywhere within the mirror plane of a

monoclinic phase does not necessarily imply facility of the polarization

rotation within this plane or enhanced electromechanical response. Secondly,

the propensity for polarization rotation under external field, as will be discussed

below, can be described in terms of the dielectric susceptibility perpendicular

to the polar axis and piezoelectric shear coefficients; large electromechanical

response along off-polar directions is equivalent to propensity for large

polarization rotation. Both have the same microscopic origin and can be

phenomenologically described by a thermodynamic instability. Thirdly, it

has been shown by Bell (2006) that extension of the usual sixth order LGD

function in PZT to include monoclinic phase by adding eighth order terms

reveals additional minima in the free energy profile that decrease the barrier

between rhombohedral and tetragonal phases. The resulting flatness of the

free energy profile is accompanied by a large dielectric susceptibility and an

enhanced piezoelectric response (and thus facilitated polarization rotation),

regardless of whether the associated monoclinic phase exists spontaneously

or is induced by external fields. Fourthly, the large piezoelectric effect in

relaxor ferroelectrics and enhanced electromechanical response of simple

perovskites along non-polar axes are essentially weak field effects. This

means that the polarization does not have to rotate by a large angle for a

crystal to exhibit a large piezoelectric effect. However, the crystal must

show propensity for the large polarization rotation; this propensity is determined

[001]

[111]

[101]

11.1

Mirror planes of the monoclinic phases recently suggested at

the morphotropic phase boundaries of PZT, PMN–

PT and PZN–

PT.

The polar axes of the monoclinic phases are not fixed but can lie

anywhere within their mirror plane between the limiting directions of

the rhombohedral (

), orthorhombic (O) and tetragonal (T) phases.

The notation is that after Vanderbilt and Cohen (2001). Note that

and

planes are equivalent in the pseudocubic presentation.

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Enhancement of piezoelectric properties in perovskite crystals 307

by the large dielectric susceptibility perpendicular to the polar axis and large

shear piezoelectric coefficient. By modifying the crystal’s free energy, certain

configurations of strong external fields may, however, be helpful in making

these coefficients particularly large.

We first consider the resultant rotation of the polar vector in a 4mm

tetragonal, mm2 orthorhombic or 3m rhombohedral crystal when an electric

field E is applied in a non-polar 〈001〉

, 〈101〉

or 〈111〉

type direction.

Importantly, these are the directions relevant to actuation of domain engineered

crystals (Bell, 2001). In order to do this, it is useful to define the direction of

the applied field with respect to the principal axes of the crystal (x

, x

)

for which the dielectric susceptibility matrix is diagonal. In all cases, x

the polar axis, or the direction of spontaneous polarization P. For a 4mm

tetragonal (T) crystal, the principal axes are parallel to the crystallographic

axes of the high-temperature cubic phase. Thus, the principal axes (x

, x

)

of the tetragonal phase are [100]

≡ [100]

, [010]

≡ [010]

and [001]

≡

[001]

, respectively; with respect to these axes, the crystal has non-zero

dielectric susceptibilities χ

= χ

and χ

. The subscript ‘T’ means a direction

defined with respect to the principal axes of the tetragonal phase; likewise

‘O’ and ‘R’ will mean the same for the orthorhombic and rhombohedral

phases, respectively. Furthermore, when applying electric field we shall assume

a stress-free crystal.

For an mm2 orthorhombic (O) crystal, the principal axes (x

, x

) are

rotated with respect to the cubic axes by 45°; respectively, they lie along

[100]

≡ [10

]

, [010]

≡ [010]

and [001]

≡ [101]

, directions. With

respect to these rotated axes, the crystal has three non-zero dielectric

susceptibilities, χ

, χ

and χ

Finally, for a 3m rhombohedral (R) crystal, the principal axes are chosen

such that x

is perpendicular to a mirror plane (Nye, 1985). Thus, the axes

are commonly chosen to lie along [100]

≡ [1

, [010]

≡ [11

]

and

[001]

≡ [111]

, the latter being the polar axis (Zhang et al., 2003a; Zhang

and Cao, 2004). Note that the first two of these identities are exact only in

the pseudocubic approximation. As will be discussed later, the principal axes

might also be chosen to be rotated by 180° about the polar axis.

In the general case the polarization change, ∆P

, and mechanical deformation

of the crystal, ∆S

, induced by the field component E

are described by the

usual piezoelectric and dielectric relations (Nye, 1985):

∆S

= d

11.1

∆P

= ε

11.2

With the polarization vector at zero field and in the crystallographic coordinate

system defined as (0,0,P

), the polarization rotation angle θ under electric

field applied in an arbitrary direction is given by:

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Handbook of dielectric, piezoelectric and ferroelectric materials308

tan =

()+ ( )

0111

0222

3 0 33 3

εχ εχ

εχ

∆∆

∆

11.3

The field is applied on a free crystal so that the dielectric susceptibilities are

taken at zero (constant) stress, i.e.

. For simplicity, the stress subscript σ

is omitted. In crystals of high symmetries such as 4mm, mm2 and 3m, the

deformation described by Eq. [11.1] corresponds to extension (contraction)

when the field is applied along the polar direction, e.g. ∆S

= d

. When

the field is applied perpendicular to polarization, for example

in a tetragonal

4mm crystal, the deformation is of shear type, e.g. ∆S

= d

. Voigt notation

is used throughout the text (Nye, 1985).

We first consider application of an electric field E to a 4mm tetragonal

crystal along a 〈110〉

type direction. Without loss of generality, we let the

polar vector P lie along [001]

≡ [001]

and apply a field along [101]

(≡

[101]

) in the pseudocubic approximation); this geometry is shown in Fig.

11.2(a). The applied field can be resolved into components E

and E

, along

the x

and x

principal axes, respectively. Importantly, since there is no

component of field along the third principal axis, x

, and because the dielectric

susceptibility matrix is diagonal, the polar vector will rotate towards the

applied field within the (010)

plane. When this happens, even at infinitesimally

small fields, the zero-field 4mm symmetry will be broken. By consideration

of all the symmetry elements of the 4mm point group (Nye, 1985), and from

Fig. 11.2(a) it can easily be shown that the only remaining symmetry element

will be the mirror plane parallel to (010)

. In the notation of Vanderbilt and

Cohen (2001), this is the M

monoclinic plane (see Fig. 11.1). The resultant,

field-induced symmetry is monoclinic M

(b)

[001]

= [001]

[100]

[010]

[111]

= [111]

(a)

[100]

[010]

[001]

= [001]

[101]

= [101]

11.2

Application of an electric field to a

4mm

tetragonal ferroelectric

along the (a) [101]

and (b) [111]

directions. Under zero field, the

polar vector lies along [001]

. Under field, the resultant symmetry

will be M

and M

in (a) and (b), respectively.

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Enhancement of piezoelectric properties in perovskite crystals 309

Next, we consider application of an electric field E to the same crystal,

this time along 〈111〉

(≡ [111]

in the pseudocubic approximation) (see Fig.

11.2b). In this case, there are three components of electric field along the

three principal axes. Again, the component E

will lead to extension of the

polar vector. In contrast, the other two components, E

and E

, will lead to

rotations of the polar vector towards the x

and x

axes. However, since the

two transverse dielectric susceptibilities are equal for 4mm symmetry (χ

), the polar vector will be constrained to rotate within a plane that bisects

the x

and x

principal axes. That is, it will rotate within a M

monoclinic

plane (see Fig. 11.2b). The zero-field tetragonal symmetry will be broken

under field, with only one mirror plane remaining; the resultant, field-distorted

symmetry is monoclinic M

For an mm2 orthorhombic crystal, we consider application of an electric

field along the non-polar 〈011〉

and 〈111〉

directions. Without loss of

generality, we first apply an electric field E along the [001]

≡

[101]

direction of a crystal whose polar vector P lies along [101]

≡ [001]

. This

geometry is shown with respect to the cubic axes of the high-temperature

phase in Fig. 11.3(a) and with respect to the principal axes of the orthorhombic

phase in Fig. 11.3(b). As shown in Fig. 11.3(b), there will be two non-zero

components of electric field with respect to the principal axes. Because there

is no component of field parallel to the second principal axis, and because

the susceptibility matrix is diagonal, the polar vector will rotate towards the

applied field in a plane perpendicular to x

: that is, a M

monoclinic plane.

The mm2 orthorhombic symmetry will be broken under field, with the only

remaining symmetry element being a (010)

mirror plane; the resultant

symmetry is monoclinic M

. Note that the expression [001]

[101]

correct only in the pseudocubic approximation, although this will not affect

the general conclusion.

Next we consider application of an electric field to the same orthorhombic

crystal along the [111]

≡

[]

01 2

direction; again, this equivalence holds

in the pseudocubic approximation. This geometry is shown with respect to

the cubic axes in Fig. 11.3(c) and with respect to the principal axes of the

orthorhombic phase in Fig. 11.3(d). This time, as shown in Fig. 11.3(d),

there is no component of field along the x

principal axis. As a result, the

polar vector will rotate away from its zero field [101]

≡ [001]

position,

towards the applied field, within a M

monoclinic plane. The resultant field-

induced symmetry will be monoclinic M

Lastly, for a 3m rhombohedral crystal we can consider application of an

electric field along a non-polar 〈001〉

or 〈101〉

direction. Without loss of

generality, we take a crystal with polar vector P along the [111]

≡ [001]

direction and first apply an electric field along [001]

≡

[]

021

. This is

shown with respect to the cubic axes in Fig. 11. 4(a). To complicate matters,

with a rhombohedral crystal we have some freedom in how the principal

WPNL2204