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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
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