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

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Handbook of dielectric, piezoelectric and ferroelectric materials320

character that is reflected in the maximum of

()θ

along the polar axis

(Budimir et al., 2003).

It is easy to understand why the proximity of a phase transition qualitatively

changes the behavior of BaTiO

from ‘extender’ to ‘rotator’ and leads to

anisotropic flattening of the free energy. The polarization of BaTiO

orthorhombic phase lies along [101]

. When the crystal in its tetragonal

phase approaches the tetragonal–orthorhombic phase transition temperature

it softens along the direction perpendicular to the polar axis (χ

and d

become high) anticipating the new direction of polarization. Close to the

tetragonal-paraelectric phase transition temperature the crystal behaves as

‘extender’, softening along the polar axis (χ

and d

become high) anticipating

disappearance of the polarization in the cubic phase. These behaviors are

analogous to the Curie law in the permittivity above ferroelectric–paraelectric

phase transition temperature where crystal becomes soft along the future

–0.2

0.2

(C/m

)

0.2

–0.2

(C/m

)

∆

(MJ/m

)

11.6

The Gibbs free energy ∆

for BaTiO

at 298K calculated from

Eq. [11.11]. The solid line shows ∆

for variable

and for

= 0.26C/m

, i.e. at the zero-field value of the spontaneous

polarization at this temperature. This energy profile indicates

propensity of the crystal for polarization rotation within

plane.

The dashed line shows ∆

for variable

and for

= 0. This energy

profile indicates propensity of the crystal for polarization extension.

C indicates point of the cubic phase (

= 0,

= 0), which is not

stable at this temperature. See also Budimir

et al.

(2005) and

Damjanovic (2005).

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Enhancement of piezoelectric properties in perovskite crystals 321

(b) BaTiO

– ‘extender’ at 378K

∆

(MJ/m

)

(a) BaTiO

– ‘rotator’ at 298K

∆

(MJ/m

)

∆

(

= 0,

)

∆

(

, =

)

(C/m

)

–2

–1

–0.2 –0.1 0.2 0.4

∆

(

= 0,

)

∆

(

, =

)

(C/m

)

–0.4 –0.2 0.2 0.4

0.5

1.5

[010]

[001]

[100]

100

(pC/N)

100

(pC/N)

400

300

200

100

(pC/N)

100

100100

100

(pC/N)

[100]

[001]

[010]

11.7

Profiles of ∆

(

) corresponding to profiles marked in Fig. 11.6 and orientation dependence of

()θ

for

BaTiO

at 298 K and 378 K: (a) at 298 K, close to the tetragonal–orthorhombic phase transition temperature; (b) at 378K,

closer to the ferroelectric –paraelectric phase transition temperature. See text and caption of Fig. 11.6 for details.

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Handbook of dielectric, piezoelectric and ferroelectric materials322

polar direction (Budimir et al., 2003) tens of degrees before the actual phase

transition takes place.

11.3.3 Compositionally induced phase transitions:

morphotropic phase boundary

A detailed discussion of the behavior of tetragonal and rhombohedral PZT at

each side of the morphotropic phase boundary (MPB) is given in Damjanovic

(2005). The phenomenological equation [11.11] cannot describe the monoclinic

phase (Noheda et al., 1999) and the discussion is limited to compositions

some 10% away from the MPB at Zr/Ti ratio 52/48. For description of the

monoclinic phase, see Bell (2006) and Vanderbilt and Cohen (2001).

Thermodynamically, the MPB behaves just like thermally induced phase

transitions discussed in the previous section. At the MPB the polarization

changes orientation from [111]

on the rhombohedral side to [001]

on the

tetragonal side. In agreement with general behavior of oxygen octahedral

perovskites (Section 11.2.2 and Davis et al., 2007) the material changes

character from ‘extender’ on the tetragonal side to ‘rotator’ on the rhombohedral

side of the MPB.

This behavior is accompanied with the anisotropic flattening of the Gibbs

free energy as compositions approach the MPB from either side, as shown in

Fig. 11.8 for two rhombohedral compositions. On the rhombohedral side, i.e.

in ‘rotator’ compositions, the free energy is flattest along non-polar directions,

with propensity for polarization rotation under weak fields being stronger

within M

than within M

plane (M

plane not examined) and propensity for

polarization extension (i.e. change along polar axis) being the weakest. On

the tetragonal side (not shown) the material is softest along polar axis in all

compositions. As expected, and clearly seen from Fig. 11.8, the anisotropy

increases on approaching the MPB. The above discussion is valid for mono

domain single crystals. In ceramics, the complex electro-elastic boundary

condition at each grain must be taken into account. The present behaviour

may be qualitatively different in ceramics having the same nominal composition

as crystals.

11.3.4 Electric field and stress-induced enhancement of

the piezoelectric properties

Discussions in Sections 11.3.2 and 11.3.3 show that thermally and

compositionally induced enhancements of the piezoelectric properties are

correlated with increase in free energy instability as phase transitions are

approached. Phase transitions can be induced by electric fields and stresses

(Davis et al., 2006) and, thermodynamically, the same qualitative effect on

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Enhancement of piezoelectric properties in perovskite crystals 323

piezoelectric anisotropy is obtained as with thermally and compositionally

induced instabilities.

In this section we discuss a special type of instability, which does not

include change of the crystal structure. In ferroelectric materials the direction/

orientation of the spontaneous polarization can be switched by electric field

and/or stress; this is a defining property of ferroelectric crystals (Lines and

Glass, 1979). It is easily shown using Eq. [11.11] that near and just below the

60/40 PZT

(a)

90/10 PZT

(b)

∆

(MJ/m

)

–10

–0.2 0.2

/√3

(C/m

)

∆

[R]

∆

[MA]

∆

[MB]

/√3

(C/m

)

∆

(MJ/m

)

–10

–0.2 0.2

11.8

The Gibbs free energy as a function of polarization for two

rhombohedral PZT compositions: (a) Pb(Zr

0.6

0.4

(PZT 60/40) and

(b) Pb(Zr

0.9

0.1

(PZT 90/10). The composition PZT 60/40, which is

closer to the morphotropic phase boundary (at Zr/Ti ratio 52/48),

has flatter profiles and higher permittivities and piezoelectric

properties than PZT 90/10. In both compositions profiles

∆G

[M ]

and

∆G

[M ]

are flatter and energy barrier lower along the directions

away from the polar axis than the profile along the polar axis

(represented by )

[R]

∆G

, indicating ‘rotator’ character. For details, see

Damjanovic, 2005).

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Handbook of dielectric, piezoelectric and ferroelectric materials324

switching field the crystal softens and that its properties may reach high

values (Budimir et al., 2004, 2005). The thermodynamic description for

electric field or stress-induced switching is similar to that for thermally

induced phase transitions. The most important issue with thermodynamically

treated switching of ferroelectric crystals is that the thermodynamic switching

fields are one or more orders of magnitude higher than experimental switching

fields. The main reason for this is that the LGD thermodynamics (e.g. Eg.

[11.11]) treats perfect crystals while a real crystal contains defects that can

serve as nucleation centers for domain reversal (Gerra et al., 2005). In addition,

as mentioned earlier, the field dependence of thermodynamic and material

coefficients in [11.11] is ignored. Nevertheless, the treatment used in this

section gives a qualitative picture how external fields may enhance

electromechanical response of ferroelectric crystals.

Different field and stress configurations can be used for such a study, as

indicated in the first sections of this chapter. For the sake of simplicity, it is

convenient to choose an electric field applied antiparallel to spontaneous

polarization direction (E

), and compressive stress applied along polar axis

(σ

). In a ferroelectric with 4mm structure a sufficiently high antiparallel

electric field will cause polarization switching by 180° while a high compressive

stress will switch polarization by 90°. As a model material we choose PbTiO

because it does not undergo multiple phase transitions like BaTiO

. This

simplifies the analysis by avoiding interference between effects of thermally

induced phase transitions and those induced by external fields. The analysis

also holds for the tetragonal PZT compositions that are sufficiently far from

the MPB (Zr/Ti ratio < 40/60).

At zero external field PbTiO

is a strong ‘extender’ ferroelectric.

Thermodynamic switching fields are high: the absolute value of coercive

compressive stress is about –1.95GPa while the coercive electric field is

about –160MV. Application of the compressive stress and antiparallel electric

field decreases polarization and increases permittivity of the monodomain

single crystal (Budimir et al., 2004, 2005). Consequently, the piezoelectric

response increases. It is assumed that electrostrictive coefficients are

independent of stresses and electric fields below coercive values. The results

for χ

, χ

, d

and d

are shown together with anisotropy ratios in Fig.

11.9. There is a striking difference in the behavior of the crystal subjected to

compressive stress and antiparallel field: while stress increases the anisotropy

ratios, changing the character of the crystal from ‘extender’ to ‘rotator’ (see

Fig. 11.10), the electric field reinforces the ‘extender’ nature of PbTiO

Importantly, in both cases fields and stresses close to the threshold values for

switching lead to a huge enhancement of the piezoelectric effect. Thus, the

piezoelectric effect may be enhanced by both colinear effects (polarization

contraction) and by polarization rotation.

Despite this qualitative difference, the thermodynamic origin of the

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Enhancement of piezoelectric properties in perovskite crystals 325

Electric field

(MV/m)

–200 –150 –100 –50 0

Electric field

(MV/m)

–200 –150 –100 –50 0

Electric field

(MV/m)

–200 –150 –100 –50 0

Stress σ

(GPa)

–0.5 0–1–1.5–2

Stress σ

(GPa)

–0.5 0–1–1.5–2

Stress σ

(GPa)

–0.5 0–1–1.5–2

0.3

0.2

0.4

0.5

0.6

0.7

0.8

0.5

1.0

1.5

2.0

2.5

3.0

Piezolectric anisotropy Piezolectric anisotropy

Dielectric anisotropy

Piezoelectric coefficients (pC/N)

Relative dielectric susceptibility

Dielectric anisotropy

Piezoelectric coefficients (pC/N)

Relative dielectric susceptibility

3000

2500

2000

1500

1000

500

700

600

500

400

300

200

100

1000

800

600

400

200

600

500

400

300

200

100

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

/χ

11.9

Dielectric susceptibilities and piezoelectric coefficients in PbTiO

at 298K as a function of compressive stress (upper

row) and antiparallel electric field (lower row) applied along the polar axis. Both the stress and electric field enhance the

properties. However, the enhancement is anisotropic and the anisotropy ratios increase with stress but decrease with

electric field. Thus, under extreme compressive stresses close to coercive stress PbTiO

behaves as a ‘rotator’ while

close to switching electric fields it behaves as an ‘extender’. See Figs 11.10 and 11.11.

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Handbook of dielectric, piezoelectric and ferroelectric materials326

enhancement is the same in the two cases. Figure 11.11 shows that for both

enhancement types the free energy becomes less stable (its profile flatter) as

the coercive fields are approached: this results in the enhancement of the

piezoelectric effect and dielectric susceptibility. The instability in the free

energy is, however, anisotropic: at compressive stress σ

, the instability is

higher along the directions perpendicular to the polarization axis; at antiparallel

electric field E

, the instability is higher along the polar axis. This anisotropy

in the free energy instability is responsible for the different character of the

enhancement. In the stress case, the sample becomes ‘rotator’ and the

enhancement is by polarization rotation. In contrast, the electric field reinforces

the ‘extender’ character and the enhancement mechanism is by polarization

contraction (collinear effect).

One can intuitively understand this qualitative difference by considering

polarization switching as a phase transition. In the case of switching by

stress, polarization changes direction by 90°. This is analogous to a phase

transition between two crystal phases in which polarization changes orientation.

Examples are thermally induced tetragonal–orthorhombic phase transition

in BaTiO

and compositionally induced tetragonal–orthorhombic phase

transition at MPB in PZT. In both cases, the character of the crystal changes

from ‘extender’ to ‘rotator’ as polarization changes orientation (or even

while approaching the phase transition point), and the main enhancement of

the piezoelectric effect is by polarization rotation. In the case of the electric

field, the switching is by 180°. The polarization changes direction (by going

through zero value) but not orientation. This case is analogous to ferroelectric–

paraelectric phase transitions, where polarization diminishes and eventually

(θ) (pC/N)

4002000

(θ) (pC/N)

400

200

0MV/m

–150MV/m

[001]

–1.78GPa

8006004002000

(θ) (pC/N)

800

600

400

200

0GPa

11.10

Orientation dependence of

()θ

in PbTiO

at 298K at (a) zero

and compressive stress and (b) zero and antiparallel electric field

close to switching values. The crystal behaves as a ‘rotator’ under

compressive stress and as an ‘extender’ under antiparallel field. See

Fig. 11.9 and 11.11.

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Enhancement of piezoelectric properties in perovskite crystals 327

becomes zero. In both BaTiO

and PbTiO

, for example, their ‘extender’

character becomes stronger as it is heated toward the Curie temperature.

11.4 Final remarks

By analyzing the free energy of simple perovskite ferroelectrics it is possible

to qualitatively understand the origins of the piezoelectric effect enhancement.

The enhancement is associated with free energy instabilities (reduction of

the energy barrier between states) in a vicinity of phase transitions and

critical points; the nature of the phase transition (thermally induced, a

morphotropic phase boundary, polarization switching) is not important. The

(a)

(b)

∆

(MJ/m

) ∆

(MJ/m

)

150

100

–50

0.5 1.5–0.5–1.5

0MV/m

–150

MV/m

–150

MV/m

(C/m

)

0.5 1–0.5–1

150

100

–50

(C/m

)

0MV/m

∆

(MJ/m

) ∆

(MJ/m

)

–11

0GPa

–40

–20

–1.78GPa

0GPa

(C/m

)

(C/m

)

–1.78GPa

1.50.5–0.5–1.5

–60

–40

–20

11.11

The Gibbs free energy for PbTiO

at 298K, for free crystal and

at compressive stress and electric field close to the switching values:

(a) as a function of

at zero and near-switching compressive stress;

(b) as a function of

and

at zero and near-switching

compressive stress; (c) as a function of

at zero and near-switching

antiparallel electric field and (d) as a function of

and

zero and near-switching antiparallel electric field. See Figs 11.9 and

11.10. Note metastable state at antiparallel field.

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Handbook of dielectric, piezoelectric and ferroelectric materials328

anisotropy of the enhancement is closely related to the anisotropy of the free

energy instability. When the instability is stronger along non-polar directions,

the enhancement is due to polarization rotation and crystal can be classified

as a ‘rotator’. When the instability is stronger along the polar axis, the

enhancement is due to collinear effects (polarization extension/contraction)

and the crystal can be classified as an ‘extender’.

The present analysis (as well as other approaches discussed in the literature

(Fu and Cohen, 2000; Kutnjak et al., 2006) does not say anything about the

origin of the large, zero-field piezoelectric effect in relaxor-ferroelectrics; it

only suggests a possible mechanism for its enhancement. The large piezoelectric

response of relaxor ferroelectric single crystals could be related to intrinsic

disorder in these materials (chemical disorder, polarization fluctuations on

nanoscopic scale) that could lead to a flat energy profile. However, ab-initio

studies (Wu and Cohen, 2005) and this work (the case of the huge field-

induced piezoelectric effect in PbTiO

) have shown that disorder is not

essential for a high piezoelectric effect.

The present analysis could be, in principle, useful in explaining the gradual

enhancement of piezoelectric effect in crystals with increasingly denser domain

structure (Wada et al., 2005). In domain engineered crystals, the polarization

on each side of some domain walls appear to be oriented head-to-head or

tail-to-tail. Whether these domain walls are charged or neutralized by the

free carriers needs to be verified experimentally. Thermodynamic theory

suggests that charged walls should not be stable (Ishibashi and Salje, 2004),

however, it is not clear whether a small fraction of the depolarizing field can

still be tolerated. If so, then a residual depolarizing field could have the same

effect on the polarization within a domain as an antiparallel field discussed

here: it would enhance the piezoelectric effect. The enhancing effect would

be stronger in crystals with denser domains. Very rough calculations show

that the presence of a small amount of charge on a domain wall (a few

percent of the total, uncompensated charge of a head-to-head/tail-to-tail domain

wall) could be sufficient to account for the enhancement of the piezoelectric

effect in domain engineered BaTiO

crystals. The same can be argued about

the stress in the domain wall region. Clearly, a rigorous, quantitative model

needs to be developed to verify this conjecture. An alternative model in

which domain walls broaden by the small driving fields is proposed by Rao

and Wang (2007).

Finally, a different approach is needed to uncover atomistic mechanisms

leading to a large electromechanical response. The formidable difficulties in

understanding ferroelectric materials are well illustrated on the example of

BaTiO

, where atomistic details of the phase transition behavior have only

recently been elucidated (Zalar et al., 2003). The theoretical ab-initio studies

are perhaps the most promising approach and have led in the past decade to

an enormous progress in understanding microscopic mechanisms operating

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Enhancement of piezoelectric properties in perovskite crystals 329

in ferroelectric materials (Cohen, 1992; Bellaiche and Vanderbilt, 1999;

Bellaiche et al., 2000; Fu and Cohen, 2000; Vanderbilt, 2004) With exception

of BaTiO

and KNbO

, most of the work on understanding enhanced

piezoelectric response in ferroelectrics has so far been carried out on lead-

based materials. The challenge is to extend the present efforts to lead-free

materials and find practical materials with piezoelectric properties comparable

to those in PZT and relaxor-ferroelectrics. The current work on lead-free

materials based on modified (K,Na)NbO

is hopefully only a beginning of

broader, more concentrated and systematic investigations.

11.5 References

Amin, A., Cross, L. E. & Newnham, R. E. (1989) Sign notation in ferroelectric free

energy function. Ferroelectrics, 99, 145–148.

Bell, A. J. (2001) Phenomenologically derived electric field–temperature phase diagrams

and piezoelectric coefficients for single crystal barium titanate under fields along

different axis. J. Appl. Phys., 89, 3907–3914.

Bell, A. J. (2006) Factors influencing the piezoelectric behaviour of PZT and other

‘morphotropic phase boundary’ ferroelectrics. J. Mat. Sci., 41, 13–25.

Bellaiche, L. & Vanderbilt, D. (1999) Intrinsic piezoelectric response in perovskite alloys:

PMN vs PT. Phys. Rev. Lett., 83, 1347–1350.

Bellaiche, L., Garcia, A. & Vanderbilt, D. (2000) Finite-temperature properties of

Pb(Zr

1–x

alloys from first principles. Phys. Rev. Lett., 84, 5427–5430.

Bellaiche, L., Garcia, A. & Vanderbilt, D. (2001) Electric-field induced polarization paths

in Pb(Zr

1–x

alloys. Phys. Rev. B, 64, 060103.

Budimir, M., Damjanovic, D. & Setter, N. (2003) Piezoelectric anisotropy-phase transition

relations in perovskite single crystals. J. Appl. Phys., 94, 6753–6761.

Budimir, M., Damjanovic, D. & Setter, N. (2004) Large enhancement of the piezoelectric

response in perovskite crystals by electric bias field antiparallel to polarization. Appl.

Phys. Lett., 85, 2890–2892.

Budimir, M., Damjanovic, D. & Setter, N. (2005) Enhancement of the piezoelectric

response of tetragonal perovskite single crystals by uniaxial stress applied along the

polar axis: a free energy approach. Phys. Rev. B, 72, 064107.

Budimir, M., Damjanovic, D. & Setter, N. (2006) Piezoelectric response and free-energy

instability in the perovskite crystals BaTiO

, PbTiO

, and Pb(Zr,Ti)O

. Phys. Rev. B,

73, 174106

Cohen, R. E. (1992) Origin of ferroelectricity in perovskite oxides. Nature, 358, 136–

138.

Damjanovic, D. (2005) Contributions to the piezoelectric effect in ferroelectric single

crystals and ceramics. J. Am. Ceram. Soc., 88, 2663–2676.

Damjanovic, D., Brem, F. & Setter, N. (2002) Crystal orientation dependence of the

piezoelectric d

coefficient in tetragonal BaTiO

as a function of temperature. Appl.

Phys. Lett., 80, 652–654.

Damjanovic, D., Budimir, M., Davis, M. & Setter, N. (2003a) Erratum: ‘Monodomain

versus polydomain piezoelectric response of 0.67 Pb(Mg

1/3

2/3

–0.33PbTiO

single

crystals along nonpolar directions’ [Appl. Phys. Lett. 83, 527 (2003)]. Appl. Phys.

Lett., 83, 2490.

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