Moulson A.J., Herbert J.M. Electroceramics: Materials, Properties, Applications

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¼ 4rf

ð6:59Þ

where r is the density of the material.

The other ‘33’ parameters can be calculated from the measured values of

, s

and e

and using Eqs (6.33), (6.39), (6.40) and (6.42).

A more common geometry is a thin disc of diameter d electroded over both

faces and poled in a direction perpendicular to the faces. The resonance on which

attention is focused is that of a radial mode, excited through the piezoelectric

eﬀect across the thickness of the disc. In this case the route from the resonant

frequencies to the coeﬃcients d and g is the same as in the case of the rod,

although the expressions are more complex.

Figure 6.5 shows the typical frequency response for such a disc. The lower

frequency responses are radial resonances and the higher frequency response the

thickness mode resonance.

The planar coupling coeﬃcient k

is related to the parallel and series resonant

frequencies by

1  k

¼ f



, J

 f



ð6:60Þ

where J

and J

are Bessel functions and n is Poisson’s ratio. The relationship

between k

and ( f

 f

)/f

is shown in Fig. 6.6 for the case in which v ¼ 0:3.

Fortunately, the curve is very insensitive to v, and in the range 0.275n50.35 the

352 PIEZOELECTRIC CERAMICS

Fig. 6.5 A typical ‘impedance–frequency’ characteristic for a piezoelectric disc specimen

having diameter  25 mm and thickness  1 mm and poled through the thickness.

TEAMFLY

Team-Fly

diﬀerences cannot easily be discerned; for the common piezoelectric ceramics

0.285v50.32.

With an assumed value for n k

can be calculated from

1  v

ð6:61Þ

Alternatively, using Eq. (6.62), k

may be determined directly from the

resonance and antiresonance frequencies of a bar poled across its thickness and

using Eq. (6.62)

/ð1  k

Þ¼ðp/2Þð f

Þtan fðp/2Þ½ð f

 f

Þ/f

g ð6:62Þ

The measured k

and k

values, together with Eq. (6.61), then permit the

calculation of n.

The compliance, s

can be calculated using

ð1  n

Þr

ð6:63Þ

where Z

is a root of an equation involving the Bessel functions and the Poisson

ratio; it has a value of approximately 2. Then (cf. Eq. (6.39))

¼ k

ðe

1=2

ð6:64Þ

and (cf. Eq. (6.48))

ð6:65Þ

PARAMETER S FOR PIEZOELECTRIC CERAMICS AND THEIR MEASUREMENT 353

Fig. 6.6 The planar coupling coeﬃcient k

as a function of ðf

 f

Þ=f

If the minimum impedance jZ

j at resonance is known, the mechanical Q

factor Q

can be obtained from the following equation:

¼ 2pf

jðC

þ CÞ

 f

 4pDf jZ

j(C

þ CÞ

ð6:66Þ

The dielectric Q factor is the reciprocal of the dielectric dissipation factor tan d

(see Section 2.7.2).

Figure 6.5 shows the thickness resonances from which the coupling coeﬃcient

) can be calculated. This usually has a value in the range 0.45 to 0.5, that is

much smaller than typical k

values which lie in the range 0.65–0.75. The reason

for this is that in the case of a disc the thickness (‘longitudinal’) vibrations are in

eﬀect partially clamped since they must be accompanied by planar distortions.

The longitudinal coupling coeﬃcient k

is measured on more appropriately

dimensioned rods (see Eq. (6.57)) to minimize this clamping.

6.3 General Characteristics and Fabrication

of ‘PZT’

6.3.1 E¡ects of domains

The ﬁrst piezoceramic to be developed commercially was BaTiO

, the model

ferroelectric discussed earlier (see Section 2.7.3). By the 1950s the solid solution

system Pb(Ti,Zr)O

(PZT), which also has the perovskite structure, was found to

be ferroelectric and PZT compositions are now the most widely exploited of all

piezoelectric ceramics. The following outline description of their properties and

fabrication introduces important ideas for the following discussion of the

tailoring of piezoceramics, including PZT, for speciﬁc applications. It is assumed

that the reader has studied Sections 2.3 and 2.7.3.

The Pb(Zr

17x

phase diagram [1] is shown in Fig. 6.7. The morphotropic

phase boundary (MPB) deﬁnes the composition at which there is an abrupt

structural change, the composition being almost independent of temperature.

That is the phase boundary between the high temperature rhombohedral and

tetragonal forms is, practically speaking, a vertical line. As shown in Fig. 6.8, the

piezoelectric activity peaks in the region of the MPB composition and

considerable eﬀort has been directed to elucidating the reason(s) for this

technically very important phenomenon.

The current understanding is that the MPB is not a sharp boundary but rather

a temperature-dependent compositional range over which there is a mixture of

tetragonal and monoclinic phases. At room temperature (300 K) the two phases

coexist over the range 0.4554x40.48 [7]. The enhanced piezoelectric activity of

354 PIEZOELECTRIC CERAMICS

the commercial compositions (x  0:48) can be rationalised in terms of the

relatively large ionic displacements associated with stress (electrical or

mechanical) – induced rotation of the monoclinic polar axis [8].

As a ferroelectric perovskite in ceramic form cools through its Curie point it

contracts isotropically since the orientations of its component crystals are

random. However, the individual crystals will have a tendency to assume the

anisotropic shapes required by the orientation of their crystal axes. This tendency

will be counteracted by the isotropic contraction of the cavities they occupy. As a

consequence a complex system of diﬀerently oriented domains that minimizes the

elastic strain energy within the crystals will become established.

The application of a suﬃciently strong ﬁeld will orient the 1808 domains in the

ﬁeld direction, as nearly as the orientation of the crystal axes allows (see

Section 2.7.3). The ﬁeld will also have an orienting eﬀect on 908 domains in

tetragonal material and on 718 and 1098 domains in the rhombohedral form, but

the response will be limited by the strain situation within and between the crystals.

There will be an overall change in the shape of the ceramic body with an expansion

in the ﬁeld direction and a contraction at right angles to it. When the ﬁeld is

removed the strain in some regions will cause the polar orientation to revert to its

previous direction, but a substantial part of the reorientation will be permanent.

An externally applied stress will aﬀect the internal strain and the domain

structures will respond; this process is termed the ferroelastic eﬀect. Compression

will favour polar orientations perpendicular to the stress while tension will

favour a parallel orientation. Thus the polarity conferred by a ﬁeld through 908

domain changes can be reversed by a compressive stress in the ﬁeld direction.

Stress will not aﬀect 1808 domains except in so far as their behaviour may be

coupled with other domain changes.

GENERAL CHARACTERISTICS AND FABRICATION 355

Fig. 6.7 Phase stabilities in the system Pb(Ti

17x

. (After Jaﬀe et al. [1].)

A polar axis can be conferred on the isotropic ceramic by applying a static

ﬁeld of 1–4 MV m

for periods of several minutes at temperatures usually

somewhat above 100 8C when domain alignment occurs. More rapid domain

movement takes place at higher temperatures but the maximum alignment of

908 domains takes some time to be established. The temperature is limited by

the leakage current which can lead to an increase in the internal temperature

and to thermal breakdown (see Section 5.2.2); the ﬁeld is limited by the

breakdown strength. If the applied voltage exceeds about 1 kV it is necessary

to ensure very clean surfaces between the electrodes and to immerse the pieces

to be poled in an insulating oil. Surface breakdown takes place very readily

between electrodes on the surface of a high-permittivity material when it is

exposed to air.

Depoling can be achieved by applying a ﬁeld in the opposite direction to that

used for poling or, in some cases, by applying a high a.c. ﬁeld and gradually

reducing it to zero, but there is a danger of overheating because of the high

dielectric loss at high ﬁelds. Some compositions can be depoled by applying a

compressive stress (10–100 MPa). Complete depoling is achieved by raising the

temperature to well above the Curie point and cooling without a ﬁeld.

Alternating ﬁelds cause domain walls to oscillate. At low ﬁelds the excursions

of 908,718 or 1098 walls result in stress–strain cycles that lead to the conversion

of some electrical energy into heat and therefore contribute to the dielectric loss.

When peak ﬁelds are suﬃcient to reverse the spontaneous polarization the loss

becomes very high, as shown by a marked expansion of the hysteresis loop

(Fig. 6.9).

356 PIEZOELECTRIC CERAMICS

Fig. 6.8 Coupling coeﬃcient k

and permittivity e

values across the PZT compositional

range. (After Jaﬀe et al. [1].)

‘Ageing’ (see Section 2.7.3) aﬀects many of the properties of piezoelectric

ceramics. Most of the piezoelectric coeﬃcients fall by a few per cent per decade

although the frequency constant increases. Ageing can be accelerated and

properties stabilized by heating to temperatures of about 80 8C in the case of

BaTiO

and rather higher for PZT.

Ageing eﬀects are known to be signiﬁcantly changed when the concentration

of vacant oxygen sites is increased either by doping or by heating in mildly

reducing atmospheres. It has been suggested that dipoles are formed between

negatively charged defects (e.g. Co

on Ti

sites) and the positively charged

V€

, and that these are aligned in the polar direction by movement of the V €

as a

result of the combined action of the local ﬁeld associated with P

and thermal

energy. The dipoles then provide an internal ﬁeld stabilizing the domain

conﬁguration thereby reducing ageing rate.

An alternative mechanism is based on the local stresses that are caused by the

development of axial anisotropy within the grains as they cool below T

. Lower-

energy domain conﬁgurations may be generated by phonon action at room

temperature and might be assisted by a high concentration of V

. Another

GENERAL CHARACTERISTICS AND FABRICATION 357

Fig. 6.9 P–E loops with increasing peak ﬁeld strengths at 50 Hz for a ceramic ferroelectric.

suggestion has been that defects such as V

may diﬀuse to the domain walls and

stabilize them. The ﬁrst of these mechanisms, dipole formation and rotation, is

assumed in the next section.

6.3.2 E¡ects of aliovalent substituents

The eﬀect of substituents is a complex matter but, with caution, a number of

important generalizations can be made regarding aliovalent substituents in

perovskites.

Donor dopants, i.e. those of higher charge than that of the ions they replace,

are compensated by cation vacancies; acceptors, i.e. dopants of lower charge

than that of the replaced ions, are compensated by oxygen vacancies. Each

dopant type tends to suppress the vacancy type that the other promotes. The

common dopants in perovskite-type ceramics are listed in Table 6.1. The eﬀects

of aliovalent substituents are discussed in Section 2.6.2.

The signiﬁcant diﬀerence between oxygen vacancies and cation vacancies in

perovskite-type structures is the higher mobility of the former. Cations and

cation vacancies tend to be separated by oxygen ions so that there is a

considerable energy barrier to be overcome before the ion and its vacancy can be

interchanged. Oxygen ions, however, form a continuous lattice structure so that

oxygen vacancies have oxygen ion neighbours with which they can easily

exchange.

Typical concentrations of dopants (0.05–5 at.%) must result in the formation

of dipolar pairs between an appreciable fraction of the dopant ions and the

vacancies, e.g. 2La_

–V

or 2Fe

3þ

–V€

. Donor–cation vacancy combinations can

be assumed to have a stable orientation so that their initially random state is

unaﬀected by spontaneous polarization or applied ﬁelds. Acceptor–oxygen

vacancy combinations are likely to be less stable and thermally activated

reorientation may take place in the presence of local or applied ﬁelds. The

dipoles, once oriented in a common direction, will provide a ﬁeld stabilizing the

domain structure. A reduction in permittivity, dielectric and mechanical loss and

an increase in the coercive ﬁeld will result from the inhibition of wall movement.

Since the compliance is aﬀected by the elastic movement of 908 walls under

stress, it will also be reduced by domain stabilization.

358 PIEZOELECTRIC CERAMICS

Table 6.1 Common aliovalent substituents

A-site donors La

,Bi

,Nd

B-site donors Nb

,Ta

,Sb

A-site acceptors K

,Rb

B-site acceptors Co

,Fe

,Sc

,Ga

,Cr

,Mn

,Mg

,Cu

Donor doping in PZT would be expected to reduce the concentration of oxygen

vacancies, leading to a reduction in the concentration of domain-stabilizing defect

pairs and so to lower ageing rates. The resulting increase in wall mobility causes

the observed increases in permittivity, dielectric losses, elastic compliance and

coupling coeﬃcients, and reductions in mechanical Q and coercivity.

The introduction of oxygen vacancies through acceptor doping also leads to a

slight reduction in unit-cell size, which tends to reinforce the eﬀects referred to

above.

Some indication of the real complexities is aﬀorded by the diﬀerent responses

of BaTiO

and PZT to the addition of donors such as La

(see Section 2.6.2). In

PZT PbO lost by volatilization during sintering can be replaced in the crystal by

. For example, if the excess positive charge of the La

is balanced by lead

site vacancies,

0.01La

þ PbðTi,ZrÞO

!ðPb

0:97

0:02

0:01

ÞðTi,ZrÞO

þ 0.03PbO "

Thus the lanthanum dopant is vacancy compensated and no electronic charge

carriers are generated. As shown in Fig. 6.10, La substitution results in a marked

increase in the resistivity of PZT.

GENERAL CHARACTERISTICS AND FABRICATION 359

Fig. 6.10 The eﬀect of lanthanum addition on the resistivity of PZT. (After J.C. Jaﬀe et al. [1].)

Lead lanthanum zirconate titanates (PLZT) containing 3–12 mol.% La and 5–

30 mol.% Ti form a class of ceramics with important dielectric, piezoelectric and

electro-optic properties. They may contain vacancies on B as well as A sites and

have a remarkable facility for changing their polar states under the inﬂuence of

applied ﬁelds.

In contrast, in BaTiO

, in which the volatility of the constituents during

sintering is low, small concentrations (51 mol.%) of donors are compensated by

electrons in the conduction band with an accompanying change of colour from

pale yellow to grey or black. However, higher donor concentrations lead to two

eﬀects: a decrease in crystal size in ceramics sintered to full density, and an

increase in resistivity above that for undoped BaTiO

together with a reversion to

the pale colour. The enhancement of permittivity and piezoelectric coupling

coeﬃcient that occurs with PZT does not occur with donor-doped BaTiO

.It

appears that at higher donor concentrations the formation of cation vacancies

becomes energetically favoured. At present there is no widely accepted

explanation of this very pronounced diﬀerence in behaviour between PZT and

BaTiO

360 PIEZOELECTRIC CERAMICS

Fig. 6.11 Vapour pressure of PbO over lead-containing ceramics: curve A, solid PbO; curve

B, PbZrO

; curve C, Pb(Ti

0.45

0.55

; curve D, PbTiO

6.3.3 Fabrication of PZT

Normal powder technology is used in the fabrication of piezoelectric ceramics.

The highest values of the coeﬃcients are obtained when the composition is near

stoichiometric, the content of ﬂuxing reagents and impurities is minimal, and the

density is as high as possible. Contamination during milling is kept low by using

zirconia-based milling media.

Most of the compositions used at present contain PbO as a major constituent.

Despite its volatility above 800 8C (Fig. 6.11), the PbO must be retained during

sintering at temperatures up to 1300 8C. A.I. Kingon and B. Clark [9] address in

some detail the matter of ‘atmosphere control’ with regard to ﬁring PZT, and the

eﬀect of PbO content on densiﬁcation kinetics. Calcination is usually carried out

in lidded alumina pots. For the ﬁnal sintering the ‘work’ is surrounded by a lead-

rich powder, such as PbZrO

, and again placed in closed saggars. Because of the

limited access to the atmosphere that results and the ease with which PbO is

reduced to metallic lead, all organic constituents must be removed before

sintering by a preliminary ﬁring at about 600 8C in air. The ﬁnal ﬁring is usually

carried out in batch-type electric kilns well ﬁlled with ‘work’. Despite

precautions, there is normally a loss of 2–3% of the initial PbO content which

is compensated by an addition to the starting materials.

An alternative to ﬁring in closed vessels is very rapid ﬁring in which the material,

carried on a moving belt, is exposed to a high peak temperature for a brief period

in a special kiln. This procedure is applicable to thin sheets of material.

A further diﬃculty with compositions containing substantial amounts of ZrO

results from the low reactivity of some grades of the material. The TiO

powders,

developed as pigments, react rapidly with PbO, and the resulting titanates only

take up Zr

ions slowly from unreacted ZrO

. Grades of ZrO

are available

which have not been calcined at high temperatures during their preparation and

which therefore react more rapidly with PbO so that a mixed Pb(Zr,Ti)O

formed at an early stage. The problem can also be alleviated by prolonged

milling during the mixing stage and by a second calcination after milling the

product of an initial ﬁring.

All the constituents of a PZT composition can be precipitated from nitrate

solutions to yield highly homogeneous reactive powders. Such powders can also

be prepared by calcining citrates that contain the A- and B-site ions in a 1:1 ratio.

It is possible to sinter at lower temperatures using these materials and, apart

from the economy in energy, this has the advantage of lessening the loss of PbO

through volatilization. However, sintered bodies adequate for the majority of

piezoelectric applications are obtained by the lower-cost route starting from a

mixture of oxides or carbonates.

Simple shapes are formed by die-pressing, long bodies of uniform section are

formed by extrusion, thin plates are formed by band-casting or calendering, and

large rings and more intricate shapes are formed by slip-casting.

GENERAL CHARACTERISTICS AND FABRICATION 361