47

CHAPTER 3

3.1 Nonlinear dielectrics:

Nonlinear dielectrics are an important class of crystalline ceramics, which can

exhibit very large dielectric constant (r>1000), due to spontaneous alignment or

polarization of electric dipoles. The spontaneous alignment of electric dipoles results

in a crystallographic phase transformation below a critical temperature, Tc. The

electric dipoles are ordered parallel to each other within the crystal in the regions

called domains. When an electric field is applied, the domains can switch from one

direction of spontaneous alignment to another. This gives rise to very large changes in

polarization and r. Hence, the name nonlinear dielectrics [1]. This nonlinear electric

polarization is analogous to the nonlinear magnetic behaviour of ferromagnetic

materials. Consequently, certain nonlinear dielectrics are also called ferroelectrics

even though they do not contain iron. Because of a strong electro-mechanical

coupling, these materials have widespread use as pressure transducers, ultrasonic

cleaners, [2, 3] loudspeakers, gas igniters [4], relays, and in a variety of electro-optical

applications [5-7].

Piezoelectricity is the property of a crystal to exhibit electric polarity when

subjected to a stress, that is, when a compressive stress is applied a charge will flow in

one direction in a measuring circuit. A tensile stress causes charge to flow in opposite

direction. Conversely, if an electric field is applied, a piezoelectric crystal will stretch

or compress depending on orientation of the field with the polarization in the crystal,

which is shown in Figure 3.1. Out of the 20 piezoelectric classes of crystals, 10 have a

unique polar axis; an axis which shows properties at one end different than the other.

48

Crystals in these 10 classes are called polar crystals because they are spontaneously

polarized. The spontaneous polarization is usually compensated through an external

or internal conductivity or twinning (domain formation).

The magnitude of the spontaneous polarization depends on temperature.

Consequently, if a change of temperature is imposed on the crystal, an electric charge

is induced on the crystal faces perpendicular to the polar axis. This is called the

pyroelectric effect. Each of the 10 classes of polar crystals is pyroelectric.

∆P

charge development

Δ

Charge

development

(a) Pressure (b) Temperature

Figure 3.1: Applied stress causes (a) Piezoelectric effect; (b) Pyroelectric effect.

Ferroelectric crystals are also pyroelectric. However, ferroelectric crystals are

only those crystals for which the spontaneous polarization can be reversed by applying

an electric field. Thus, a ferroelectric material is a spontaneously polarized material

with reversible polarization.

ΔT

49

3.2 Ferroelectricity – A phenomenon:

Ferroelectricity is a phenomenon, which was discovered in 1921. The name

refers in certain magnetic analogies, though it is somewhat misleading, as it has no

connection with iron (ferrum) at all. Ferroelectricity has also been called Seignette

electricity, or Seignette. Rochelle salt (RS) was the first material found to show

ferroelectric properties such as a spontaneous polarization on cooling below the Curie

point, ferroelectric domains and a ferroelectric hysteresis loop.

3.3 Structural origin of the ferroelectric state:

A large class of ferroelectric crystals is made up of mixed oxides containing

corner sharing octahedral of O2-

ions schematically shown in Figure 3.2 (Perovskite

structure e.g. CaTiO3). Inside each octahedron is a cation Bb+

where ‘b’ represents

valency varies from 3 to 6. The space between the octahedra is occupied by Aa+

ions

where ‘a’ denotes valency varies from 1 to 3. In prototype forms, the geometric

centers of Aa+

, Bb+

and O2-

ions coincide, giving rise to a non-polar lattice. When

polarized, the A and B ions are displaced from their geometric centres with respect to

the O2-

ions, to give a net polarity to the lattice. These displacements occur due to the

changes in the lattice structure when phase transitions take place as the temperature is

changed.

The formation of dipoles by the displacement of ions will not lead to

spontaneous polarization if a compensation pattern of dipoles are formed which give

net zero dipole moment.

50

Figure 3.2. The perovskite structure of CaTiO3.

51

3.3.1 Perovskites:

Perovskites are a large family of crystalline ceramics that derive their name

from a specific mineral known as perovskite (CaTiO3) due to their crystalline

structure. The mineral perovskite, which was first described in the 1830s by the

geologist Gustav Rose, who named it after the famous Russian mineralogist Count

Lev Aleksevich Von Perovski, typically exhibits a crystal lattice that appears cubic,

though it is actually orthorhombic in symmetry due to a slight distortion of the

structure.

Members of the class of ceramics dubbed perovskites all exhibit a structure

that is similar to the mineral of the same name. The characteristic chemical formula of

a perovskite ceramic is ABO3, with A-atom exhibiting a+2

charge and the B-atom

exhibiting b+4

charge. The atoms of the unusual material are generally arranged so that

12-coordinated A-atoms mark the corners of a cube, octahedral oxygen (O)-ions are

featured on the faces of that cube, and tetrahedral B-ions are located in the centre of

the structure.

The existence of this dipole is accountable for the ferroelectric attributes

exhibited by barium titanate. This compound as well as other familiar perovskites,

such as CaTiO3 and SrTiO3, achieves impressive dielectric constants, which makes

them well suited for use in capacitors, components in electric circuits that temporarily

store energy. The capacity of these devices can be greatly increased through the

inclusion of a solid dielectric material such as perovskite.

The spontaneous alignment of dipoles, which occurs at the onset of

ferroelectricity, is often associated with a crystallographic phase change from a

52

centrosymmetric, non-polar lattice to a noncentrosymmetric polar lattice. Barium

titanate (BaTiO3) is an excellent example to illustrate the structural changes that occur

when a crystal changes from a nonferroelectric (paraelectric) to a ferroelectric state.

The Ti ions of BaTiO3 are surrounded by six oxygen ions in an octahedral

configuration (Figure 3.3). The octahedral coordination is expected from the radius

ratio of 0.468. All crystals possessing the TiO6 configuration have dielectric constant,

as a result of a large dispersion stemming from infrared vibrations. Since a regular

TiO6 octahedron has a center of symmetry, the six Ti–O dipole moments cancel result

only a unilateral -displacement of the positively charged Ti4+

ion against its negatively

charged O2-

surroundings. Ferroelectricity requires the coupling of such displacements

and the dipole moments associated with the displacements.

For TiO2, each oxygen ion has to be coupled to three Ti ions if each Ti is

surrounded by six oxygen. In rutile, brookite, and anatase (three crystal modifications

of TiO2) the TiO6 octahedra are grouped in various compensating arrays by sharing

two, three, and four edges respectively with their neighbors. Consequently, all the Ti-

O dipole moments cancel and none of TiO2 crystal forms are ferroelectric. Thus, in

BaTiO3, the Ba and O ions form a face centred cubic (fcc) lattice with Ti ions fitting

into octahedral interstices (Figures 3.3). The characteristic feature of the Ba, Pb, and

Sr titanate is that the large size of Ba, Pb, and Sr ions increases the size of the cell of

the fcc BaO3 structure so that the Ti atom is at the lower edge of stability in the

octahedral interstices. There are consequently minimum energy positions for the Ti

atom, which is off-center and can therefore give rise to permanent electric dipoles.

53

At high temperature T>Tc, the thermal energy is sufficient to allow the Ti4+

atoms to move randomly from one position to another, so there is no fixed symmetry.

Figure 3.4 (a) Tetragonal perovskite structure below Tc and

(b) Cubic structure above Tc

Ti4+

54

The open octahedral site allows the Ti atom to develop a large dipole moment

in an applied field, but there is no spontaneous alignment of the dipoles. In this

symmetric configuration, the material is paraelectric (i.e., no net dipole moment when

E = 0).

When the temperature is lowered below Tc, the position of the Ti ion and the

octahedral structure changes from cubic to tetragonal symmetry with the Ti ion in an

off-center position corresponding to a permanent dipole. These dipoles are ordered,

giving a domain structure with a net spontaneous polarization within the domains.

3.3.2 Hysteresis:

The result of the spontaneous polarization of a ferroelectric at Tc is the

appearance of very high and a hysteresis loop for polarization. The hysteresis loop

is due to the presence of crystallographic domains within which there is complete

alignment of electric dipoles.

At low field strengths in unpolarized (also called virgin) material, the

polarization P is initially reversible and is nearly linear with the applied field. The

slope gives ′i, the initial dielectric constant (equations 3.3.2.1 and 3.3.2.2). The value

of i will be similar to r of the cubic phase.

tan α = E

P (3.3.2.1)

tan α = (’i –1) 0 (3.3.2.2)

At higher field strengths, polarization increases considerably as a result of the

switching of the ferroelectric domains. The polarization switches with the applied

field by means of domain boundaries while moving through the crystal. This change in

55

polarization is small due to saturation polarization; i.e., all the domains of like

orientation are aligned with the field. Extrapolation of the high field E curve at E = 0

gives Ps, the saturation polarization, corresponding to the spontaneous polarization

with all the dipole aligned in parallel.

When the applied field continues to be applied at values greater than required

achieving Ps, the polarization continues to increase, but only proportional to ΄i. This

is because all of the domains are oriented parallel to each other. However, the

individual TiO6 polarizable units can continue to be distorted increasing the unit

polarization. When E is cut off, P does not go to zero but remains at a finite value,

called the remanent polarization, Pr. This is due to the oriented domains being unable

to return to their random state without an additional energy input by an oppositely

directed field. The strength of E required returning P to zero is the coercive field, Ec

as shown in Figure 3.4.

Figure 3.4: Typical Hysteresis loop

56

There is a substantial effect of temperature on the shape of the hysteresis loop.

At low temperature, a loop become fatter and Ec increases corresponding to a large

energy required for reorientation of domain walls; i.e., the domain configuration is

frozen in. As the temperature increases, Ec decreases until at Tc no hysteresis remains

and ′ is single valued at a value characteristic of the paraelectric phase.

3.3.3 Ferroelectric Domains:

A domain is a region in a crystal where the polarization Ps is homogeneous,

i.e., in the same direction, separated by a domain wall from a neighboring region of

different direction of Ps. Ps may be slightly different in neighboring domains because

of ferro-strictive effects. Electro-striction is the change in physical dimensions and

shape of a material due to application of an electric field.

In order to determine domain configurations (Figure 3.5) it is necessary to

characterize the bulk of the domain and the conditions controlling wall thickness. The

internal field opposing the applied field Ea, is called the depolarizing field Ed (Figure

3.6). Equilibrium wall thickness is determined by two opposing tendencies. Dipoles

align with others in the bulk of the domain, which tends to force domain walls to be

thin. Neighboring dipoles want to align parallel to each other, which tends to force the

walls to be thick.

The dipoles, which are not in one of the two bulk orientations, contribute a

positive energy (+A). Also, the tendency for dipoles to be mutually parallel is

described as energy (+B 2), where is a small angle of departure from the mutually

parallel condition.

57

A

L

D

Figure 3.5: Bulk domain geometry

..

+

Ps Ea

Ed

Figure 3.6: Depolarizing field within polarizable material

Then, in magnetic materials, B is very large, since it represents the exchange

energy. However, for the 1800 ferroelectric wall,

= π / n (3.3.3.1)

where n is the wall thickness in number of dipole spacing, so

U = An + Bn (π/n)2 (3.3.3.2)

This energy is a minimum energy with respect to n when

n = π(B/A)1/2 (3.3.3.3)

Therefore, because of the smaller values of B in ferroelectric materials the

walls are much thinner than in magnetic materials, as illustrated in Figure 3.7.

+ + + + +

_ _ _ _ _

+ + + + +

_ _ _ _ _

+ + + + +

_ _ _ _ _

Ps

Ps

Ps

Ps

58

Wall thickness

(a)

Wall

Thickness (b)

Figure 3.7: Wall thickness for (a) ferromagnetic and ferrimagnetic material,

(b) Ferroelectric material.

The significance of the thin wall is that the wall energy Uw will be highly

localized, so that it can greatly exceed kT and will be difficult to move under thermal

energy.

The differences in domain walls also produce differences in the hysteresis loop

of ferroelectric and magnetic materials. For a magnetic material, there is reversible

wall motion at a low applied magnetic field strength H, and irreversible wall motion at

higher H. At a high H, domain-reorientation occurs into an easy direction nearest to

59

the field, and at highest fields, no magnetic vectors are parallel to the field, being

pulled out of easy directions. In contrast, for a ferroelectric material, because of the

very high anisotropies, only easy directions are polarized into domains. Thus, domain

growth is not impeded appreciably and a major part of polarization occurs over a very

small range of field. Therefore, the sides of the hysteresis loop are nearly vertical and

the top is nearly flat. This is true because the dielectric polarization adds only one part

in 1000 to the spontaneous value of polarization.

Coercive field in some ferroelectrics also appears to be nearly non-existent.

The typical switching loop occurs only when the rate of exchange of field lies within

certain limits. At very low frequencies, any field will cause switching, at very high

frequencies the walls will not have time to switch.

3.3.4 Effect of environment on switching and transitions:

Small changes in external conditions (field (E), stress (X), and temperature

(T)) may result in large changes in polarization (P) for ferroelectrics. The large

changes in P occur with changes in temperature and pressure. The “phases” represents

changes in polarization behaviour. When the temperature (T) is above the

ferroelectric transition temperature, Tc, it has no spontaneous polarization. When

T < Tc; P ≠ 0. To demonstrate, this requires measuring P but this is difficult because

charges in crystal faces are quickly compensated, either through the crystal or by the

external circuit. The measurement of P is possible by measuring the change in P

induced by changes in E, X or T. Thus, dP exists as a current in a circuit connected to

electrodes on the crystal, (dP/dt) = i. Thus, ferroelectrics have a distinguishing

response to a changing field.

60

3.3.5 Effect of an electric field on Tc:

The magnitude of applied field E has a large effect on the switching of

domains and also on the temperature of the onset of ferroelectricity, Tc.

3.4: Aging phenomena in ceramic dielectrics:

3.4.1: Aging of the small-signal dielectric constant:

With time, ceramic dielectrics undergo a spontaneous and gradual change in

dielectric constant. This is called aging and takes place even under isothermal and

stress-free conditions. Aging of the dielectric constant is approximately a linear

function of logarithmic time [8]; that is, the capacitance drops by a specific

percentage. Age also affects the dielectric loss and all the electromechanical

parameters associated with ferroelectric ceramic-based devices. Actually logarithmic

time dependent aging is only an approximation: it implies an infinite dielectric

constant at zero time. Following are the additional facts about aging:

i) Aging can be partially reversed by an external stimulus of electrical, mechanical,

or thermal energy. Aging can be completely reversed only by heating and holding

the dielectric above the Curie temperature for an extended amount of time. This is

called de-aging [8].

ii) The dissipation factor also decreases logarithmically as a function of time and

usually at a greater rate than the dielectric constant [9],

iii) Transient aging effects are observed after the application or removal of electrical

or mechanical stresses [10, 11],

iv) Polarization versus field hysteresis loops of barium titanate become constricted

with time into a propeller shape [10],

61

v) Time dependent changes are observable in the x-ray reflections from a single

crystal of BaTiO3, after the application or removal of an electric field [12],

vi) Aging does not occur in the paraelectric phase. The room temperature rates of

barium strontium titanates with Curie temperatures above and below room

temperature: the aging rates drop precipitously with >31% strontium additions.

The aging rate increases as the tetragonal distortion of the perovskite unit cell is

reduced. In electrically poled ceramic, aging decreases not only the r and tan ,

but also the electromechanical coupling factor. The mechanical quality factor and

resonance frequency constant, on the other hand, increases the material appears to

stiffen in its electrical, mechanical, and cross coupled piezoelectric interactions

with time [13],

ix) The aging rates were also found to depend on processing conditions and on the

level and types of impurities in the host BaTiO3 structure.

3.4.2 Aging of the dielectric loss:

Aged material shows a critical threshold of applied field below, which tan is

independent of the field. This threshold is dependent on the presence of dopants and

increases with acceptor concentration.

The dissipative loss in the strong field region is generally attributed to

microhysteresis of domain wall motion, where the domain wall movement results

from domain nucleation and growth. The amount of movement depends on a power

function of the electric field having an exponent above 1. This gives a phase-shifted

polarization that increases more rapidly with the field than the in-phase polarization

62

caused by the intrinsic permittivity of the lattice. As a result, the tan increases with

the applied field.

3.4.3 Aging of relaxor dielectrics:

Relaxor materials are characterized by a low-frequency (< 1 MHz) dispersion

of the complex permittivity. Significant frequency dependence of the temperature of

the peak in permittivity is associated with a diffuse transition between paraelectric and

a low–temperature ferroelectric or antiferroelectric phase. Typical relaxors are PLZT,

Pb3MgNb2O9, and Pb2(Sc,Ta)O6 and Sc and Ta are ordered on the perovskite B- site

[14]. These relaxors have an interesting aging behaviour.

Models for the aging phenomenon in relaxor material [15, 16] have the same

difficulty as those for barium titanate–based dielectrics, that is, the lack of a

universally accepted mechanism for the dielectric constant. It is evident, however, that

the inherent disorder of these materials contributes a low–frequency component to the

dielectric constant that ages much more rapidly than the high-frequency intrinsic

permittivity.

Aging has also been observed in thermally de-poled strontium barium niobate

single crystals, with aging rates ranging from 10% at 1 kHz to 6.2% at 100 kHz [17].

3.5 Special Ferroelectric Compositions:

BaTiO3 is iso-structural with the mineral perovskite (CaTiO3) and so is

referred to as “a perovskite”. Above its Curie point (approximately 1300C) the unit

cell is cubic with the ions arranged as in Figure 3.3(b). Below the Curie point, the

structure is slightly distorted to the tetragonal form with a dipole moment along the c-

63

direction. Other transformations occur at temperatures close to 00C and -80

0C: below

00C the unit cell is orthorhombic with the polar axis parallel to a face diagonal and

below -800C it is rhombohedral with the polar axis along a body diagonal. The

transformations are illustrated in Figure 3.3, and the corresponding changes in the

values of the lattice parameters, the spontaneous polarization and the relative

permittivity are shown in Figures. 3.8(a), (b) and (c).

A consideration of the ion displacements accompanying the cubic-tetragonal

transformation can give insight into how the spontaneous polarization might be

coupled from unit cell to unit cell. X-ray studies have established that in the

tetragonal form, taking the four central (B) oxygen ions in the cubic phase as origin,

the other ions are slightly shifted as shown in Figure 3.3. It is evident that, if the

central Ti4+

ion is closer to one of the O2-

ions marked as A, it will be energetically

favourable for the Ti4+

ion on the opposite side of A, to be located more distantly from

the O2-

ion, thus engendering a similar displacement of all the Ti4+

ions in a particular

column in the same direction. Coupling between neighbouring columns occurs in

BaTiO3 so that all the Ti4+

ions are displaced in the same direction. In contrast, in the

orthorhombic perovskite PbZrO3 the Zr4+

ions in neighboring columns are displaced in

opposite senses so that the overall dipole moment is zero. Such a structure is termed

antiferroelectric if the material shows a Curie point.

64

Figure 3.8: (a) Lattice parameters of BaTiO3 as a function of temperature.

(b) Dielectric constants of BaTiO3 as a function of temperature.

65

Figure 3.8: (c) Crystallographic changes of BaTiO3.

66

3.5.1 Strontium titanate–based dielectrics:

Strontium titanate also has a perovskite structure, but it is cubic and

paraelectric at room temperature. Its dielectric constant at room temperature is about

320, increasing gradually with decreasing temperatures to about 20,000 near 0 K, with

no evidence of a ferroelectric transition. Compositions based on strontium titanate

have been used for high voltage capacitors. For example, the system

{(1-x)(Sr0.5Pb0.25Ca0.25)TiO3+ x(Bi2)3.3TiO3)} was found [18] to have dielectric

constants near 2000 (for x=0.043). These were very stable with applied voltages up to

5 kV/mm. Similar composition based on strontium titanate, but containing

magnesium titanate, has also been described [19].

Strontium titanate forms a continuous series of solid solutions with barium

titanate. These have a high r (>5000) at room temperature, but the peak in the

dielectric constant-temperature curve is normally too sharp for use as a capacitor

material.

3.5.2 Lead titanate–derived dielectrics:

Lead titanate is ferroelectric with a Curie temperature of 4900C. The r at 25

0C

is about 350. Pb(Mg0.5W0.5)O3 ceramic can be sintered near 10000

C and a high r can

be obtained [20]. Dielectrics of this type have been adapted for multiplayer capacitor

applications [21] using the basic composition SrxPb1-xTiO3+y(PbMg0.5W0.5O3), where

0 ≤ x ≤ 0.1 and 0.35 ≤ y ≤ 0.5.

The system PbTiO3-PbZrO3, known as PZT, is widely used for piezoelectric

applications. Modifications of PZT with lanthanum oxide (PLZT) have interesting

67

electro-optic behaviour [22] and some compositions also have very attractive

properties for capacitor applications [23]. A typical composition is based on

Pb0.88La0.12Zr0.70Ti0.30O3 formulated with a small excess of PbO to facilitate sintering.

If such PLZT ceramics are cofired with Pd-Ag electrodes in multilayer capacitors,

silver can react with the ceramic to modify its properties. Silver oxide and other

additives have been used in PLZT to achieve a temperature stable (X7R) dielectric

with r ~ 2000. It also has good DC bias characteristics in multilayer capacitor

applications [24].

3.5.3 Other Titanates:

Some capacitor applications require that dielectric constant be very insensitive

to changes in temperature (e.g., < 30 ppm/0C). For these requirements, materials with

a lower r must be used. Compositions based on the system BaO-Nd2O3-TiO2 provide

good temperature stability with a r of about 60. A typical composition might contain

12-20 mol% BaO, 12-20 mol% Nd2O3 and 60-70 mol% TiO2. Sometimes a mixture of

rare earth oxides has been used instead of neodymium oxide alone [25]. Addition of

Bi2O3 to this system can increase the r close to hundred [26,27] Composition of this

type has also been modified by adding fluxes or glass forming oxides. They lower the

sintering temperature and permit cofiring with 70% Ag-30% Pd electrodes in

multiplayer capacitors [28, 29].

Calcium titanate has dielectric properties similar to those of strontium titanate,

except that its dielectric constant is about 50% lower. It is used mainly for

temperature-compensating capacitors for which a linear temperature dependence of

68

the dielectric constant with predetermined slope is required. The calcium titanate is

usually blended with SrTiO3 and other additives, such as La2O3 or Bi2O3 [29].

Magnesium titanate has the ilmenite crystal structure and exhibits a low

dielectric constant (about 20). However, this dielectric constant has a positive

temperature coefficient so it can be used to adjust the temperature dependence of

negative thermal efficient materials, such as CaTiO3 and SrTiO3. These compositions

often have good high-frequency performance [30]. The high frequency performance of

multilayer capacitors is also strongly influenced by the conductivity of the electrodes.

Magnesium titanate dielectrics have therefore been applied with copper electrodes;

together with appropriate no reducible fluxes [31].

3.5.4 Niobates and related relaxor dielectrics:

In recent years many ceramics based on lead niobate, lead titanate, or lead

tungstate have been investigated for use in multilayer capacitor applications. The

properties of most compositions based on BaTiO3 do not vary greatly with frequency

above about 500 Hz until the GHz range is reached. However, certain ferroelectrics

known as “relaxors”, show a pronounced change in permittivity with frequency at

temperatures near Curie point. Above Tc, the r-T relation does not follow the Curie-

Weiss law but is almost linear, thus giving a broad temperature range with r near its

maximum value. As with other ferroelectrics, tan is more below Tc and then falls

rapidly. Some of these systems, studied originally by Russian workers in the early

1960s, are ferroelectric, have peak dielectric constants as high as 20,000 (Table 3.1),

and sinter at below 10000C. This combination of properties makes them very

attractive for use in multilayer capacitors with silver electrodes. Materials with high r

69

and Tc near 250C (e.g., PbMg1/3Nb2/3)03 [32, 33], or slightly above room temperature

Pb(Fe1/2Nb1/2)O3 or Pb(Zn1/3Nb2/3)O3 of particular interest for capacitor applications

because they can be used with only minor modifications. Also, additions of lead

titanate moves the Curie point to higher temperatures and increase the r. These

materials are generally referred to as relaxors because the temperature at which the r

peaks and the magnitude of the peak usually depends on frequency.

In addition, the system Pb(Fe1/2nb1/2)O3-Pb(Fe2/3W1/3)O3 with 30-35 mol% lead

iron tungstate has r = 21,000 at 250C and sintering temperatures as low as 920

0C

[34]. Addition of lead manganese niobate to this system decreases the dielectric

losses. Excess niobium oxide improves its mechanical strength, and added lead zinc

niobate aids densification. An even higher dielectric constant (34,000) has been

reported for lead iron niobate with 18 mole% lead iron tungstate and 2% barium

copper tungstate [35]; the firing temperature was 9000 C. As a result, Relaxor

dielectrics can be used to make multilayer ceramic capacitors with both high

volumetric efficiency and high reliability [36].

Table 3.1: Relaxor dielectric ceramics

Relaxor Dielectric

ceramic

Curie

temperature (0C)

ε r

Pb (Mg1/3Nb2/3)O3 -12 15,000 Pb (Zn1/3Nb2/3)O3 140 22,000 Pb (Ni1/3Nb2/3)O3 -120 4,000 Pb (Co1/3Nb2/3)O3 -70 6,000 Pb (Fe1/2Nb1/2)O3 114 12,000 Pb (Mg1/3Ta2/3)O3 -98 7,000 Pb (Ni1/3Ta2/3)O3 -180 2,5000

Pb (Co1/3Ta2/3)O3 -140 4,000

Pb (Fe2/3W1/3)O3 -30 10,000

Pb (Fe2/3W1/3)O3 -95 9,000

Pb (Mg1/2W1/2)O3 39 250

Pb(Co1/2W1/2)O3 32 250

70

3.6 (a) Ferroelectric theory:

Ferroelectricity is the reversible spontaneous alignment of electric dipoles by

their mutual interaction. Ferroelectricity occurs due to the local field E′ increasing in

proportion to the polarization which is increased by the aligning of dipoles in a

parallel array with the field. The alignment is spontaneous at a temperature Tc, where

the randomizing effect of thermal energy kT is overcome.

The defining equation for the onset of ferroelectricity follows from the

definition of electric polarization

P = (′- 1) 0 E = N α E ′ (3.6a.1)

where

E ′ = E + 03

p and ′ = relative dielectric constant (3.6a.2)

thus

P = N α ( E + 03

p) = N α E +

03PN

( 3.6a.3)

Rearranging yields

P - 03PN

= N α E (3.6a.4)

and

P (1 - 03PN ) = N α E (3.6a.5)

so

P = )3/1( 0

N

EN

(3.6a.6)

71

Since the electric susceptibility, χ, is defined as

χ = ′- 1 = E

P

0 (3.6a.7)

then substituting equation (3.6a.6) into equation (3.6a.7) yields

χ = ′-1 = )3/1(

/

0

0

N

N

(3.6a.8)

Recalling that ’ = / 0, then equation (3.6a.8) is called Clausius-Mosotti equation

when rearranged as

0N

=2

1

(3.6a.9)

From equation (3.6a.8) when 03N →1 (3.6a.10)

Then P , χ , and k′ must go to infinity. It is known that the orientation of a dipole is

inversely proportional to temperature:

α0 = C / k T (3.6a.11)

where C is the Curie constant of a material. If we consider materials where

α0 αe + αa + αi, then a critical temperature Tc will be reached, where

133 00

0

ckT

CNN

(3.6a.12)

Consequently, Tc occurs when the following condition is met:

Tc =03k

NC (3.6a.13)

Below this critical temperature, spontaneous polarization sets in and all the elementary

dipoles have the same orientation.

72

Combining the above equations yields the Curie-Weiss law, which defines the

temperature onset of ferroelectricity:

χ ′- 1 =

0

0

31

N

TN (3.6a.14)

When equation (3.6a.14), the Curie-Weiss law, is combined with the defining equation

for the critical temperature, for example,

03N

T

Tc (3.6a.15)

the following relation is obtained which describes the temperature dependence of the

electric susceptibility of a ferroelectric and the onset of ferroelectric behavior at Tc.

χ =

0

0

31

3/3

N

N=

T

T

TT

c

c

1

/3=

TTT

TT

c

c

/)(

/3

= c

c

TT

T

3

(3.6a.16)

Figure 3.9 shows the linear Curie-Weiss dependence of χ -1 on temperature

above the onset of ferroelectricity for BaSrTiO2.

BaSrTiO2

10

8

1 6

χ

4

2

T

-25 0 Tc 25 50 75

Temperature (

0C)

Figure 3.9: Curie-Weiss law.

73

At the Curie point there is a spontaneous alignment of the dipoles leading to a

discontinuity in the temperature dependence. Values of the Curie constant Tc,

spontaneous polarization, and dielectric constants for various ferroelectric materials

are given in Table 3.2.

Table 3.2: Ferroelectric properties of several materials

Materials Curie

constant

(C)

Curie

temperature

(0)

Spontaneous

polarization

at (T0 C)

Dielectric

constant at (Tc)

Ferroelectric complex crystal structure

Rochelle salt

(NaKC4H406.4H2O)

2.2x102 + 24

0 0.25(23) 5000

(KH2PO4) KDP 3.3x103 -150

0 4.7 10

5 (c-axis),

70 (a-axis)

Ferroelectric: Perovskite –type crystal structure

BaTiO3 1.7x105 +120

0, 5, 90 26 (23) 1600

PbTiO3 1.1x105 +490

0 750(23) -----

KNbO3 2.4x105 +415, 225, -

10

30 4200, 2000,900

Antiferroelectric: Perovskite type crystal structure

PbZrO3 1.6x105 +230

0 ----- 3500

Table 3.3: Additional ferroelectric oxides

Formula Tc (0C) Ps (μC/cm

2) T (

0 C)

LiNbO3 +1210 71 +23

NaNbO3 -200 12.0 -200

KNbO3 +435 30.3 +250

Pb(0.5Sc0.5Nb)O3 +90 3.6 +18

Pb(0.33Mg0.67Nb)O3 -8 24.0 -170

Pb(0.33Zn0.67Nb)O3 +140 24.0 +125

LiTaO3 665 50.0 +25

PbTa2O6 260 10.0 +25

Pb(0.5Fe0.5Ta)O3 -40 28.0 -170

SrBi2Ta2O9 335 5.8 +25

Sm(MoO4)3 197 0.24 +50

Eu2(MoO4)3 180 0.14 +25

Pb5GeO11 178 4.6 +25

SrTeO3 485 3.7 312

74

3.6 (b) Phenomenological Theories:

(i) Devonshire theory:

Devonshire started by assuming that structure could be treated as a strained

cubic crystal. For zero stress the Gibb’s free energy can be expanded in even powers

of the polarization with coefficients, which are functions of the temperature only. He

then showed that in the tetragonal and cubic region where the polarization is along a

fixed axis, the Helmontz free energy F simplifies to

F(P, T) = F (O, T) + A(T-) P2 + B P

4 + CP

6 , (3.6b.1)

Where T is the temperature, P is the polarization, and is the Curie

temperature.

Now above the Curie temperature in the paraelectric state where the follows a

Curie-Weiss law 4 π/(2A(T-)) we may compute the dielectric constant as a function

of P as follows: Since » 1 we may write

E / P = 4π / (-1) ≈ 4π / , (3.6b.2)

And since

∂F / ∂P = E, (3.6b.3)

we have

4π / = 2A (T-) + 4BP2 + 6CP

4 + …. (3.6b.4)

The incremental permittivity ′ is given by

′ = ∂D / ∂E = ∂ (k E) / ∂ E, ( 3.6b.5)

so Eq. (3.6.4) is

4π / ′ = 2A (T-) + 12 BP2 + …. (3.6b.6)

75

Consider the case of small fields. Then we can neglect terms in P4

or greater.

Since E ≈ 4πP so we may write Eq. (3.6.4) as

4π / = 2. A (T-)+ [4B2 E

2 / (4π)

2 ] (3.6b.7)

Now, in an experiment in which we hold k constant and vary only E and T we have

∆T = (2B 2E

2 / A (4π)2 ( 3.6b.8)

Now since we have held k constant, this amounts to holding ′ constant so a similar

equation can be written for ′, i.e.,

∆T = 6B ′2 E2 / A (4π)

2 , (3.6b.9)

The relation of Eq. (3.6.9) was verified by Rupprecht [37] for SrTiO3 with a

large number of measurements over a wide temperature and frequency range as he

reported.[38] Since this was verified using the incremental permittivity ′ (which is

measured using small ac fields ) a better representation of the dielectric nonlinearity

then eq. (3.6.7) is

4π / ′ = 2 A (T-) + [12 B ′ 2 E2 / (4π)

2 ] (3.6b.10)

This could not have been derived from Eq. (3.6.6) but is rather postulated because it

leads to results, which are experimentally verified in large fields. Starting with Eq.

(3.6.10) and assuming constant temperature, we may write

1/ ′ = (1/ 0) + a ′2 E2 , (3.6b.11)

where, a = 12 B / (4π)3 and 0 = 1/ 2A (T- ).

This can be re-expressed as

76

′ / 0 = [ 1- ( ′ / 0 ) +( ′3 / 03) ]

1/2 / [ 1 + a 0

3 E

2]1/2

(3.6b.12)

This form is convenient since by examining the numerator for all values of / 0, vary

slowly and deviate from unity by only 15% at / 0 = 0.58. We can, therefore, assume

the numerator to be unity and absorb the slight difference by changing the value of a

slightly. Thus, Eq. (3.6.12) becomes

′ / 0 ≈ [1+ a 03 E

2 ]

-1/2 (3.6b.13)

This equation has been found to be correct for both SrTiO3, BaTiO3 and

polycrystalline combinations of the two at far above the Curie temperature.

(ii) Cochran theory:

Cochran’s theory [39] is based on the assumption that the ferroelectric phase

transition is the result of instability of crystal lattice with respect to one of the

homogeneous (wave vector g=0) transverse optical mode (ωT). Essentially the theory

is based on the assumption that if the crystal is wholly or partially ionic, lattice

vibrations are accompanied by polarization oscillations of equal frequency, which

create a local field interacting with the ions through long-range coulomb forces. If for

one particular mode of vibration these long-range forces have the magnitude equal and

opposite in sign to the short-range forces, the crystal becomes unstable for this mode.

The r, which is connected to the frequency of the critical mode, becomes large as it

happens at the Curie temperature.

Ferroelectric phase transitions are a special case of structural phase transitions,

and can thus be interpreted in terms of stability of the crystal lattice dynamics. In a

structural phase transition, the order parameter (polarization in case of ferroelectric

77

phase transition) is associated with a lattice vibrational mode that exhibits instability at

the transition temperature. For a second order transition, for example, the frequency

spectrum of the lattice vibrations related to the order parameter is proportional to T-Tc,

so that this mode ‘softens’ (its frequency goes to zero) as the material is cooled

towards Tc. Freezing of the vibrations at Tc gives rise to non-zero order parameter and

the corresponding reduction in symmetry. A soft mode is an optic mode and can be

studied experimentally by infrared spectroscopy and neutron scattering [40]. The

lattice dynamics approach and its recent extension [41] have been very successful in

describing qualitatively ferroelectric phase transitions.

(iii) Lydanne – Sachs -Teller (LST) theory:

A Lydanne-Sachs-Teller (LST) theory [42] gives the relation between the

ferroelectric properties and the thermodynamic properties of the crystals. For g = 0

mode of the diatomic crystal, LST relation gives the ratio of frequencies of the

longitudinal optical (ωδ) and transverse optical (ωT) mode of the infinite wavelength

in terms of the ratio of the static dielectric constant (s) of the crystal to the high

frequency dielectric constant (c ) as :

c

s

T

L2

2

(3.6c.1)

Where ω2 T = 0, Hence, s = ∞, as s is equal to refractive index.

Cochran obtained the more general case in which there are n atoms in the

elementary cell as :

c

s

n

j T

L

22

2

(3.6c.2)

78

This equation produces one essential anomaly needed to explain ferroelectric

transition. In order to have a complete understanding of the ferroelectric behaviour it

is necessary to investigate the temperature dependence of ω2T. In ferroelectrics, s

follows Curie –Weiss law above the transition temperature.

0 s + 4πc / T - Tc, T >Tc (3.6c.3)

Where 0 is the temperature independent part of the dielectric constant. This

equation (3.6c.3) through (3.6c.1) implies that the transverse optical modes of infinite

wavelength have an anomalous temperature dependence given by

(2gT T = Tc), for T > Tc (3.6c.4)

where g depends on short range force constant.

The softening of the transverse optical mode ωT by polarization and the rapid

rise of static dielectric constant offers an explanation for the onset of ferroelectricity.

The temperature dependence of ωT follows Curie-Weiss law and is related to s

through LST relation.

This theory provides an explanation of ferroelectric phase transition in

diatomic crystal. Cochran has used the data on lattice dynamics, in an analysis of some

characteristics of transitions in BaTiO3 and estimated the dielectric constant value

equal to 14,000 and spontaneous polarization equal to 19.5 μc/cm2 for BaTiO3 at the

Curie temperature. The values are very close those obtained experimentally by Merz

[43]. The prediction of absolute value of the frequency ωT of the transverse of optical

mode with wave vector (g) zero for perovskite materials is the most interesting new

result of this theory. Infrared studies by Ballantyne [44] on BaTiO3 above the Curie

79

temperature show that within the experimental error, the results are in agreement with

Cochran theory.

The lattice dynamic theory has been successful in describing ferroelectric

phase transitions qualitatively but for quantitative predictions difficulties originate in

model simplification, which is often unavoidable in many body problems. However,

the phase transition sequence, effective charge of ions and polarization can now be

fairly calculated for many pure ferroelectrics, whereas the predictions of piezoelectric

behaviour based on microscopic theories are still not available for any ferroelectrics

[45].

80

References:

1. ε. E. δines and A. ε. Glass, “Principles and applications of ferroelectrics and related materials,” Clarendon press, Oxford (1979).

2. P. N. T. Wells, “Biomedical Ultrasonics”, Academic Press, δondon, 1977.

3. D. A.Christensen, “Ultrasonic Bioinstrumentation”, John Wiley, NY,1988.

4. A. J. εoulson and J. ε. Herbert, “Electroceramics: εaterials, properties and

applications”, Chapman and Hall, δondon, 1990.

5. A. Bandrant, H. Vial, and J. Daval, J. Cryst. Growth, 43, 197 (1978).

6. S. Miyazawa and N. Uchida, Opt. Quant. Electronics, 7, 451 (1975).

7. C. E. Land, J. Am. Ceram. Soc., 72, 2059 (1989).

8. B. H. εarks, “Ceramic dielectric materials”, Electronics, 21(B), 116-120 (1948).

9. M. C. McQuarrie and W.R. Buessem, Bull, Amer. Ceram. Soc., 34(12), 402-406

(1955).

10. G. Shirane and K. Sato, J. Phys. Soc., Japan, 6, 20-26 (1951).

11. J. R. Partington, G.V. Planer, and I.I. Bosewell, Nature, 162, 151 (1948).

12. H.F. Kay, Acta. Crystallogr., 1, 229-237 (1948).

13. W. P. Mason, J. Acoust. Soc. Amer., 27(1), 73-85 (1955).

14. N. Setter, Ph.D Thesis, Pennsylvania State University, 1980.

15. G. Borchhardt, J. Von Cieminski, and G. Schmidt, Phys. Stat. Sol. (a), 59, 749-754

(1980).

16. W. A Schulze, J. V. Biggers, and L.E. Cross, J. Amer. Ceram. Soc., 61(1), 46-49

(1978).

17. T.W. Cline and L.E. Cross, J. Appl. Phys., 49, 4298-4300 (1978).

18. S.Nishigaki, K. Murano, and A. Ohkoshi, J. Amer. Soc., 65, 554 (1982).

19. Y. Sakabe, U.S. Patent 4, 226,735, 1980.

20. N. N. Krainik and A. I. Aaagranovskaya, Sov. Phys. Sol. St., 2, 63 (1960).

21. R. J. Bouchard, U.S. Patent 4, 228, 482, 1980.

22. G. H. Haertling and C.E. Land, J. Amer. Ceram. Soc., 54, 1 (1971).

23. T. V. Biggers and W.A Schulze, Bull. Amer. Ceram. Soc., 53, 809 (1974).

81

24. G. H. Maher, Proc.33rd

Electronic components. Conf., IEEE, Orlando, Florida,

1983, pp. 173-177.

25. R. L. Bolton, Ph.D. Thesis, University of Illinois, 1968.

26. R. R . Roup, U.S. Patent 3, 775,142, (1973).

27. D. Kolar. S. Gaberseek, Z. Stoder, and D. Suvorov, Ferroelectrics, 27, 269 (1980).

28. T. M. Wilson, U.S. Patent 4, 500, 942, 1985.

29. Y. Sakabe and Y. Yoshiko, K.Furukowa, and S. Fujimara, U.S. Patent 3, 951, 873

(1976).

30. N. Kikuchi, T. Yoshioko, K. Furukawa, and S. Fujimara, U.S. Patent 3, 951, 873

(1976).

31. I. Burn, U.S. Patent 4, 308,570 (1981).

32. S. L. Swartz, T. R. Schrout, W. A. Schulze, and L. E. Cross, J. Amer. Ceram. Soc.,

67, 311 (1984).

33. S. Fuziwara, K. Furukawa, N.Kikuchi, O. Iizawa, and H.Tanaka, U. S. Patent 4,

265, 668 (1981).

34. M. Yonezawa, Bull. Amer. Ceram. Soc., 62, 1375 (1983).

35. Y. Yamashita, T.Takashasi, and M Harata, Europeans patent application No. 121,

161 ( 1984).

36. M. Kawamura ,T. Tokumaru, J. Suzuki, K.K. Tsukawaki, and M. Yoshimaga,

N.E.C, Res. Dev., 75 , 47 (1984).

37. G. Rupprecht, Science Report No.1, AFCRC-TN-60-596, Research Div. Raytheon

Company (1960).

38. G. Rupprecht, R. O. Bell, and B. D. Silverman, Phys. Rev., 123, 97 (1961).

39. A. Cochran, Adv. Phys., 9, 387 (1960).

40. G. Maugin, J. Pouget, R. Drouot and B. Collet, Wiley Pub., Chichester (1992).

41. R. E. Cohen, Ferroelectrics, 194, 1 (1996).

42. R. δyddanne, R. Sac’hs and E. Teller, Phys. Rev., 59, 673 (1941).

43. W. J. Mertz, Phys. Rev., 91, 513 (1953).

44. J. M. Ballantyne, Phys. Rev., 136 , 429 (1964).

45. Dragan damjanovic, Rep. Prog. Phys., 61, 1267 (1998).