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

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Handbook of dielectric, piezoelectric and ferroelectric materials310

axes are chosen; this has led to a certain amount of confusion in calculations

of the piezoelectric coefficient along general non-polar directions in

rhombohedral PMN–xPT (Damjanovic et al., 2003ab; Zhang et al., 2003b;

Zhang and Cao, 2004). However, as has been discussed elsewhere (Davis,

2006), the second principal axis x

can be seen to constitute a secondary

polar axis like that found in 32 rhombohedral quartz (Newnham, 2005). For

the present purposes, we assume (as justified elsewhere; Davis, 2006) that

the secondary polar axis will align itself as close as possible to the applied

field. Thus, as shown in Fig. 11.4(a), we choose x

to lie along

[]

112

The same geometry is shown with respect to the principal axes, within the

pseudocubic approximation, in Fig. 11.4(b). As shown, there is no component

of electric field along the x

axis. Therefore, the polar vector will rotate

(a) (b)

[100]

[010]

[01√2]

= [111]

[001]

[101]

[100]

[010]

[001]

[101]

= [001]

[100]

[010]

[001]

[101]

[111]

= [01√2]

[010]

[100]

[001]

[101]

[001]

11.3

Application of an electric field to an

mm2

orthorhombic

ferroelectric with spontaneous polarization along the [101]

direction.

In (a), the electric field is applied along the [001]

direction; the same

situation is shown with respect to the principal axes of the

orthorhombic crystal in (b). In (c), the electric field is applied along

the [111]

direction; the same situation is shown with respect to the

principal axes of the orthorhombic crystal in (d). Under field, the

resultant symmetry will be

and

in (a) and (c), respectively.

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Enhancement of piezoelectric properties in perovskite crystals 311

within the

(110)

monoclinic plane towards the direction of applied

field. The zero-field rhombohedral symmetry is broken and M

monoclinic

symmetry results.

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

this time, along the [110]

≡

[01 2 ]

direction. Again we assume that the

secondary polar axis will align itself as close as possible to the field such

that, as shown in Fig. 11.4(c), x

is now along

[]

112

. The same geometry

is shown with respect to the new principal axes in Fig. 11.4(d). Again, there

is no component of field along the x

principal axis and the polar vector will

rotate within the M

monoclinic plane shown. The zero-field rhombohedral

(a) (b)

(d)

[100]

= [110]

[010]

= [112]

[01√2]

= [110]

[001]

[111]

(c)

[112]

[110]

= [01√2]

[111]

[001]

[112]

[110]

[111]

[001]

= [0√21]

[100]

= [110]

[010]

= [112]

[0√21]

= [001]

[001]

= [111]

11.4

Application of an electric field to a

rhombohedral

ferroelectric with spontaneous polarization along the [111]

direction.

In (a), the electric field is applied along the [001]

direction; the same

situation is shown with respect to the principal axes of the

rhombohedral crystal in (b). In (c), the electric field is applied along

the [110]

direction; the same situation is shown with respect to the

principal axes of the rhombohedral crystal in (d). Under field, the

resultant symmetry will be

and

in (a) and (c), respectively.

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Handbook of dielectric, piezoelectric and ferroelectric materials312

symmetry is broken, even upon application of an infinitesimally small field,

and monoclinic M

symmetry results.

In summary, M

, M

or M

monoclinic symmetry can be expected whenever

a field (even infinitesimally small) is applied to a rhombohedral, orthorhombic

or tetragonal crystal in a non-polar 〈001〉

, 〈101〉

or 〈111〉

type direction.

As soon as the electric field is applied, polarization rotation occurs which

breaks the higher-order, zero-field symmetry. The remaining symmetry element

is a single mirror plane such that the resultant symmetry is monoclinic M

or M

. Therefore, field-induced monoclinic symmetries can be expected

in truly zero-field rhombohedral, orthorhombic or tetragonal crystals; this is

a simple consequence of external field configuration and the intrinsic symmetry

of the R, O and T phases. As mentioned above, monoclinic phases can also

occur spontaneously (without field assistance) especially in the morphotropic

phase boundary region of the solid solutions (Noheda and Cox, 2006). More

on field-induced monoclinic distortion and field-induced phases can be found

in Davis et al. (2006) and Davis (2007). Most often, the field-induced

polarization path is more complex than suggested by the simplified pictures

given above. Polarization rotation between two end members (e.g. R → T

via M

plane), for example, may be continuous at weak fields only with

polarization jumping discontinuously at a critical field through a first order-

like phase transition to the other end member (Davis et al., 2006). Even more

complex behavior with polarization moving, e.g. via R–M

–M

–T path with

multiple first order-like phase transitions has been observed experimentally

(Noheda et al., 2001; Davis et al., 2006) and predicted by first principles

calculations (Bellaiche et al., 2001).

In their ferroelectric phase, the crystals’ polarization can be modified by

stress, the main effect being through the piezoelectric coupling. In the simplest

case of 4mm tetragonal material, the tensile or compressive stresses applied

along or perpendicular to the polar axis have the same role as electric field

bias along the polarization direction, resulting in extension or contraction of

polarization. Shear stresses have a similar role as electric fields applied

perpendicular to polarization and will result in polarization rotation. We take

as an example a tetragonal crystal in short circuit condition with polarization

along the x

axis. Shear stress σ

≡ σ

, σ

creates charge ∆P

on (100)

plane through piezoelectric effect ∆P

= d

, where d

is the shear

piezoelectric coefficient (Nye, 1985). This effectively rotates the polarization

vector from (0,0,P

) to (∆P

, 0, P

), that is from [001]

toward [101]

within

plane; the resulting symmetry is monoclinic. Clearly, the larger the shear

piezoelectric coefficient the larger the polarization rotation. Similar structure–

polarization rotation relations can be derived for other symmetries and stress

configurations. It is implied in the above discussion that the crystal also

deforms elastically, the deformation being described by:

, where

is elastic compliance at short circuit condition.

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Enhancement of piezoelectric properties in perovskite crystals 313

11.2.2 Polarization rotation vs. polarization extension:

‘rotator’ and ‘extender’ ferroelectrics and domain

engineering

We have seen in the previous section that the extent of polarization rotation

under an applied electric field or mechanical stress is related to the shear

deformation of the crystal lattice, as determined by the piezoelectric shear

coefficients such as d

and

. In contrast, the relative extension (contraction)

of the polar vector under an applied field is related to the collinear piezoelectric

effect, as quantified by the longitudinal piezoelectric coefficient such as

The piezoelectric coefficients are in turn related to the dielectric susceptibilities

through thermodynamic phenomenological relations d ∝ χQP

(Haun et al.,

1989a,b) where Q are electrostrictive coefficients. These relations are

consequence of the coupling of spontaneous strain, S

, and spontaneous

polarization P

: S

(Haun et al., 1989a,b). As may be expected from

the discussion above, the piezoelectric shear coefficients are related to dielectric

susceptibilities perpendicular to polarization axis, χ

and χ

, while the

piezoelectric longitudinal and transverse coefficients are related to the

longitudinal dielectric susceptibility χ

. In tetragonal 4mm crystals for example,

= 2ε

and d

= ε

In a crystal with 4mm, mm2 or 3m symmetries and with polarization

expressed in the crystallographic coordinate system, P = (0,0,P

), an application

of electric field E

or stress σ

along the polar direction will lead to a

lengthening (or contraction) of the polar vector by an amount ∆P = (0,0,∆P

)

where ∆P

= ε

or ∆P

= d

. Because the dielectric susceptibility

matrix is diagonal, there is no change in the perpendicular components of the

polar vector (no polarization rotation). To avoid confusion between expressions

given in the cubic crystallographic system, as is often made in the LGD

approach, and the crystallographic coordinate system, we shall in the remaining

text discuss the tetragonal 4mm phase only, where these expressions are

identical. For a more general case, the reader can consult Davis et al. (2007).

Using the phenomenological relation d

= 2ε

(Haun et al., 1989a,b;

Budimir et al., 2003), the relative extension of the polar vector expressed in

terms of the electric field and stress is:

∆P

0333

33 3

= =

εχ

11.4

∆P

33 3

11.5

that is, it is proportional to d

and χ

. The polarization rotation angle θ is

zero.

In contrast, if an electric field is applied perpendicular to the polar vector,

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Handbook of dielectric, piezoelectric and ferroelectric materials314

the polar vector will rotate away from its original position by some angle

θ.

For the special case of an electric field applied along the principal x

axis or

a shear stress σ

, there will be a change in polarization given by

∆P = (∆P

,0,0) where ∆P

= ε

or ∆P

= d

The change of the polarization in terms of the electric field is given by:

∆PE

1 0 11 1

344

= = εχ

11.6

while the relative change also defines the angle of polarization rotation:

tan = = =

0111

∆

εχ

11.7

tan = =

15 5

∆

11.8

That is, rotation of the polar vector is directly proportional to the piezoelectric

shear coefficient d

and dielectric susceptibility perpendicular to polarization

. Similar relations are valid for mm2 and 3m point groups (Budimir et al.,

2003; Davis et al., 2007).

In summary, polarization extension is strongest in materials with a large

longitudinal coefficient d

, which in turn is due to a large χ

. Polarization

rotation is strongest in materials with a large shear coefficient d

(or d

)

due to a large transverse susceptibility χ

. Therefore, when an electric field

is applied in some arbitrary direction, the relative contributions to the

piezoelectric response from polarization extension (due to any component of

field along the polar axis) and polarization rotation (due to any perpendicular

component) will be described by the ratios d

and χ

/χ

, the piezoelectric

and dielectric anisotropy factors, respectively. The two are related via

thermodynamic phenomenological equations d ∝ χQP.

Let us consider as examples BaTiO

and PbTiO

monodomain single

crystals and estimate numerically polarization rotation and anisotropy factors.

Numerical values are taken from phenomenological calculations (Budimir et

al., 2003). In PbTiO

, at 298K, with d

= 79 pC/N, d

= 56pC/N, χ

= 66,

= 124, P

= 0.75 C/m

and the simplest stress (or electric field) configuration

with only σ

(or E

) or σ

(or E

) applied, the relative polarization extension,

Eqs [11.4 and 11.5], is ∆P

= 1.05 × 10

–10

and 88 ε

, while the

relative polarization rotation, Eqs [11.6 and 11.7], is ∆P

= 0.74 × 10

–10

and 165 ε

. In BaTiO

, at the same conditions and with d

= 99pC/N,

= 457pC/N, χ

= 192, χ

= 3300 and P

= 0.265C/m

, the relative

polarization extension is ∆P

= 3.7 × 10

–10

and 724 ε

while the

relative polarization rotation is ∆P

= 17.2 × 10

–10

and 12452 ε

Two conclusions can be drawn from these values. First, at 298K, BaTiO

is in

all directions dielectrically and thus piezoelectrically much softer than

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Enhancement of piezoelectric properties in perovskite crystals 315

PbTiO

. Second, propensity of BaTiO

to polarization rotation is much higher

than for polarization extension (contraction). This can be clearly seen when

comparing the dielectric and piezoelectric anisotropy factors d

= 0.71

and χ

/χ

= 1.87 for PbTiO

and d

= 4.59 and χ

/χ

= 17.1 for

BaTiO

. Numerical examples for a large number of other crystals are given

in Davis et al. (2007). Before discussing origins of this large difference in

BaTiO

and PbTiO

, we shall first show how these results are related to high

or enhanced electromechanical activity along the non-polar axis and to the

concept of domain engineering.

Consider for example orientation dependence of the longitudinal d

piezoelectric coefficient. For crystals with symmetry 4mm this dependence

is given by:

()θ

= cosθ[(d

+ d

) sin

θ + d

cosθ] 11.9

where the asterisk denotes coefficient measured along a direction defined by

the Euler angle θ (θ defines the angle away from the polar axis) and d

are

coefficients in the crystallographic coordinate system. Equations for mm2

and 3m symmetries as well as definition of Euler angles used can be found

in Damjanovic et al. (2006). It can be easily shown (Damjanovic et al.,

2002) that in crystals with 4mm symmetry the maximal

()θ

can appear

along a non-polar axis (i.e. θ ≠ 0°) only if:

–

11.10

The number 3/2 – Q

is slightly less than 2 for most compounds (1.9 in

BaTiO

and 1.8 in PbTiO

) and is assumed to be temperature independent.

Similar relations can be derived for other point groups (Davis et al., 2007).

The condition Eq. [11.10] is clearly related to polarization rotation and extension

Eqs [11.4 and 11.8] and piezoelectric anisotropy factors introduced above. If

()θ

is plotted for BaTiO

and PbTiO

(Figure 11.5), one finds that for

lead titanate with a small d

and small anisotropy factors (d

= 0.71 and

/χ

= 1.87)

()θ

exhibits a maximum along the polar axis; BaTiO

with a large d

and large anisotropy factors (d

= 4.59 and χ

/χ

17.1) exhibits a maximum

()θ

along a non-polar axis.

Crystals for which

()θ

has a maximal value along the polar axis, the

polarization extension (contraction) is stronger than polarization rotation

and are called ‘extender’ ferroelectrics; crystals in which polarization rotation

is stronger are called ‘rotator’ ferroelectrics (Davis et al., 2007). Whether a

crystal is a ‘rotator’ or ‘extender’ may depend on temperature, the presence

of ferroelectric–ferroelectric phase transitions, and external fields. It turns

out that most crystals with rhombohedral and orthorhombic structure are

‘rotators’ while those with tetragonal structure tend to be ‘extenders’. This

general behavior appears to be related to anisotropy and electrostrictive

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Handbook of dielectric, piezoelectric and ferroelectric materials316

properties of oxygen octahedra, the main building block of many oxide

ferroelectrics (Yamada, 1972; Davis et al., 2007). Giant piezoelectric response

off-polar axis in relaxor–ferroelectrics, PMN–xPT and PZN–xPT, is due to

exceptionally strong ‘rotator’ characteristics and, in particular, a large

piezoelectric shear coefficients. In PMN–30PT, for example, the piezoelectric

and dielectric anisotropy factors are d

≈ 21 and χ

/χ

≈ 6.2, with d

≈ 4100 pC/N, d

≈ 190pC/N, χ

≈ 3950 and χ

≈ 640 (Zhang et al.,

2003b).

In compounds that change symmetry with temperature, such as BaTiO

the ‘rotator/extender’ character is a function of temperature. Interestingly,

BaTiO

changes its character within its tetragonal phase: it is an ‘extender’

close to the ferroelectric–paraelectric phase transition temperature and is a

‘rotator’ close to tetragonal-orthorhombic phase transition temperature, see

Fig. 11.5 and Budimir et al. (2003). Lead titanate, which is tetragonal

throughout the temperature range of its ferroelectric phase, is a strong

‘extender’. However, as shall be shown later, its ‘extender’ character can be

broken by suitably applied external fields. On the other hand, external fields

or temperature can increase ‘extender’ character of certain crystals and thus

strongly enhance their piezoelectric response along the polar axis. This is

also the case for PbTiO

and will be discussed in the following sections. This

result hints that the polarization rotation is not the only way the piezoelectric

response may be increased. This leads us to the next section in which we

shall use the LGD thermodynamic approach to look for a deeper, common

origin of the enhanced piezoelectric response of ferroelectric crystals. It will

be shown that the thermodynamic origin of the enhancement lies in the

instability of a free energy function, regardless whether this instability is

PbTiO

at 298K

‘extender’ ferroelectric

(b)

BaTiO

at 298K

‘rotator’ ferroelectric

(a)

[010]

[100]

[001]

(pC/N)

[100]

[001]

[010]

100

(pC/N)

100

(pC/N)

100

(pC/N)

11.5

Orientation dependence of

()θ

for (a) BaTiO

and (b) PbTiO

at 298 K. The maximum

()θ

for BaTiO

is along a non-polar axis,

and it is along polar axis [001] for PbTiO

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Enhancement of piezoelectric properties in perovskite crystals 317

caused by composition variation at a morphotropic phase boundary,

temperature-induced phase transitions, or electric field and stress-induced

metastable states. It will be further shown that whether the enhancement

happens by polarization rotation or by polarization extension (contraction) is

determined by the anisotropy of the free energy profile.

Polarization rotation is used for increasing the piezoelectric response of

some crystals by so-called domain engineering (see Chapter 10). According

to the definition of Bell (2001), a domain engineered crystal is one which

has been poled by the application of a sufficiently high field along one of the

possible polar axes of the crystal other than the zero-field polar axis, creating

a set of domains in which the polarizations are oriented such that their angles

to the poling direction are minimized. In a perovskite material there are

therefore three possible sets of poling directions 〈111〉

, 〈101〉

and 〈001〉

if monoclinic phases are ignored (Wada et al., 1998, 1999, 2004, 2005 Park

et al., 1999; Park and Shrout, 1997; Erhart and Cao, 2003, Erhart, 2004;

Davis et al., 2005). Obviously, domain engineering can be used to enhance

(increase) piezoelectric properties only in ‘rotator’ ferroelectrics; indeed, the

domain engineered crystals with the largest enhancement of the piezoelectric

properties are strong ‘rotators’ such as PMN–xPT and PZN–xPT (Park and

Shrout, 1997). In contrast, ‘extender’ ferroelectrics such as PbTiO

, where

the zero-field piezoelectric coefficient is highest along the polar axis, will

not profit from domain engineering. Thus,

for example, can be enhanced

only by domain-engineering in ‘rotator’ ferroelectrics. As already mentioned,

this can be changed under external bias fields (electric or mechanical).

Since poling along the zero-field non-polar direction creates domain walls

in the crystals, the question is posed whether enhanced piezoelectric response

is due to the presence of domain walls or is related to piezoelectric anisotropy

(that is, their intrinsic ‘rotator’ character). It can be shown that in some

domain engineered crystals the enhancement is largely accomplished by

their ‘rotator’ character (Damjanovic et al., 2003b; Zhang et al., 2003b).

However, there is strong evidence that increased density of domain walls in

domain engineered crystals can further enhance this intrinsic effect (Wada et

al., 2004, 2005). The effect seems to be related to the broadening of the

domain walls by the driving field (Rao and Wang, 2007).

11.3 Anisotropy of a free energy and piezoelectric

enhancement

In this section we discuss enhancement of the piezoelectric effect in terms of

the crystal’s free energy. It will be shown that the enhancement may be

induced by proximity of thermally induced phase transitions, by presence of

a morphotropic phase boundary, and by special configurations of external

electric field and stresses. In each case, the origin of the piezoelectric (and

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Handbook of dielectric, piezoelectric and ferroelectric materials318

dielectric) enhancement can be traced to instabilities or metastable states of

the free energy; term ‘flattening of a free energy profile’ is used to describe

associated reduction of the energy barrier between states. A dominant ‘rotator’

or ‘extender’ characteristic of a crystal is a consequence of the anisotropic

flattening of the free energy.

11.3.1 Gibbs free energy

In the framework of the Landau–Ginzburg–Devonshire phenomenological

model (Devonshire, 1949, 1951), the Gibbs free energy ∆G of a ferroelectric

crystal with symmetry 4mm, mm2 or 3m can be expressed in the coordinate

system of the parent m3m cubic paraelectric phase as (Haun et al., 1989a;

Bell, 2001):

∆α αG PPP PPP = ( + + ) + ( + )

+ ( + + ) + ( + + )

111

ααPP PP PP P P P

+ [ ( ) + ( + ) + ( + )]

112

α PP P PP P PP P+

+ ( ) –

(+ + )

123

α σσσPPP s

– ( + + ) –

(+ + )

12 23 13

ssσσ σσ σσ σ σ σ

– ( + + ) – [ ( + )

11 1

12 1

QP P PQ PPσσσ σ

+ ( + ) + ( + )

σσPP PP

]

– Q

(σ

+ σ

)

– E

11.11

where αs are dielectric stiffnesses, P

the polarization, E

the electric field,

stress components in Voight notation,

elastic compliances at

constant polarization, and Q

electrostrictive constants. Under zero field

conditions, in the tetragonal phase P

= P

= 0, P

≠ 0, in the orthorhombic

PP P

= 0; = 0≠

and in the rhombohedral phase

PPP

= = 0≠

. The

spontaneous polarization is calculated from the stability condition ∂G/∂P

= 0.

The absolute dielectric susceptibilities χ

are defined as χ

= [∂

G/∂P

∂P

]

–1

The piezoelectric coefficients g are determined from g

= ∂

G/∂P

∂σ

and

the d coefficients, which are the only ones considered here, as d

= χ

(Haun et al., 1989a). Coefficients α may be temperature dependent (Wang et

al., 2006). For the present purposes they are taken from Bell (2001) for

BaTiO

, from Haun et al. (1987) for PbTiO

, and from Haun et al. (1989b)

for Pb(Zr,Ti)O

. By definition, negative σ

and E

values have the meaning

of compressive stress and electric field applied antiparallel to crystal axes

(Amin et al., 1989). Note that all coefficients in [11.11] are assumed to be

independent on external fields.

In the description of ferroelectric perovskites it is common to terminate

function [11.11] with the sixth order terms (Mitsui et al., 1976; Lines and

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Enhancement of piezoelectric properties in perovskite crystals 319

Glass, 1979). The need for the theory with higher-order terms has emerged

with the discovery of the monoclinic phase in PZT (Vanderbilt and Cohen,

2001; Bell, 2006) and recent attempts to describe some details of the BaTiO

behavior under pressure (Li et al., 2005; Wang et al., 2006). Atomistic origins

of the strong anharmonicity indicated by the presence of eighth order terms

are probably related to the mixed order–disorder and displacive character of

the paraelectric–ferroelectric phase transition in BaTiO

(Zalar et al., 2003)

and by chemical disorder in PZT (Vanderbilt and Cohen, 2001). For the

present purposes, the sixth order series [11.11] is considered to be sufficient.

For further discussion, the reader is advised to consult articles by Vanderbilt

and Cohen (2001) and Bell (2006).

11.3.2 Thermally induced phase transitions and ‘rotator’

character

Consider now Eq. [11.11] for a tetragonal crystal, such as BaTiO

, and the

influence of incipient orthorhombic phase on its anisotropy. BaTiO

is a

well-known ferroelectric material which on cooling transforms from cubic

m3m phase to tetragonal 4mm phase at ≈ 400K, to orthorhombic mm2 phase

at ≈ 280 K and to rhombohedral 3m phase at ≈ 210K (Ishidate et al., 1997)

With E

= 0, σ

= 0, Eq. [11.11] becomes (Budimir et al., 2005):

∆α α αGPP PP PP = ( + ) + ( + ) + ( )

+ ( + ) + [ + ]

111

112

ααP P PP PP

11.12

which is graphically shown in Fig. 11.6 at 298K.

Two profiles of ∆G(P

, P

) are marked in Fig. 11.6: the solid line represents

∆G(P

, P

= 0.26) and the dashed line ∆G(P

= 0, P

). The flatness of the

former indicates propensity of the crystal for polarization rotation from polar

axis [001]

toward [101]

, i.e. within M

plane (see Fig. 11.2a) while the

flatness of the latter indicates propensity for polarization elongation

(contraction) along the polar axis [001]

. The profiles are extracted for two

temperatures, 298K and 378K, and shown in Fig. 11.7 together with orientation

dependence of

()θ

at each temperature.

At 298K (Fig. 11.7a) close to the tetragonal–orthorhombic phase transition

temperature (≈278K), the free energy profile corresponding to polarization

rotation within the M

plane (solid line) is flatter than along the [001]

direction (dashed line), indicating large anisotropy factors (d

= 4.6, χ

= 17.1) (Budimir et al., 2003). The crystal behaves as a rotator and

()θ

exhibits maximum along a non-polar direction. At 378K (Fig. 11.7b)

closer to the ferroelectric–paraelectric phase transition temperature (≈ 400

K), the free energy profile is flatter along the polar axis. The anisotropy

factors are small (d

= 0.6, χ

/χ

= 2.2), indicating the ‘extender’

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