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

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as they rotate in an anticlockwise sense with angular frequency o, maintaining a

constant phase diﬀerence, in this case 908.

If there is to be a net extraction of power from the source, there must be a

component I

of I in phase with U, as shown in Fig. 2.31; I

leads to power loss,

whereas the capacitative component I

does not. Therefore the time average

power loss is



PP ¼

UIdt

sinðotÞI

cosðot  dÞdt

ð2:94Þ

Integrating Eq. (2.94) gives



PP ¼

sind ð2:95Þ

Since I

¼ I

=cosd and I

¼ oU



PP ¼

tand ¼

oCtan d ð2:96Þ

ﬃﬃﬃ

and I

ﬃﬃﬃ

being respectively the rms voltage and rms current:

62 ELEMENTARY SOLID STATE SCIENCE

Fig. 2.31 Phasor diagram for a real capacitor.

Fig. 2.30 Phasor diagram for a perfect capacitor.

TEAMFLY

Team-Fly

It can be seen from Eq. (2.95) that sin d (or cos f) represents the fraction of the

current–voltage product that is dissipated as heat. It is termed the ‘power factor’.

From Eq. (2.96) the ‘dissipation factor’ tand is the fraction of the product of the

capacitive current (the component 908 out of phase with the voltage) and the

voltage, dissipated as heat. In most cases of interest d is small enough for

sind  tan d.

Eq. (2.96) can be put in terms of dielectric material parameters by substituting

h for U

, e

A=h for C and V for Ah (see Fig. 2.27), leading to the equation for

the dissipated power density in the dielectric:



tand ð2:97Þ

tand is termed the ‘loss factor’ of the dielectric and oe

tand is the ‘dielectric

(or a.c.) conductivity’:

a:c:

¼ oe

tand ð2:98Þ

The complex permittivity

The behaviour of a.c. circuits can conveniently be analysed using complex

quantities. The method makes use of the identity

expðjyÞ¼cos y þ jsin y where j ¼

ð1Þð2:99Þ

Thus U ¼ U

expðjotÞ can represent a complex sinusoidal voltage. The time

diﬀerential of U is given by

UU ¼ joU

expðjotÞ¼joU

where the j multiplier indicates that

UU is 908 in advance of U. In linear equations

U ¼ U

expðjotÞ can be used to represent U

cosðotÞ, although more correctly it

should be represented by U

RefexpðjotÞg where Re stands for ‘real part of’. The

latter form must be used in higher-order equations.

The method can be demonstrated by repeating the preceding analysis for a

lossy dielectric. The instantaneous charge on a ‘lossless’ vacuum capacitor C

Q ¼ UC

and

I ¼

QQ ¼

UUC

¼ joC

U ð2:100Þ

If the capacitor is now ﬁlled with a lossy dielectric, its eﬀect can be taken into

account by introducing a complex relative permittivity e

* ¼ e

 je@

, where e

and

are respectively the real and imaginary parts of the relative permittivity. It

follows that

CHARGE DISPLACEMENT PROCESSES 63

I ¼ joe

¼ joe

U þ oe@

ð2:101Þ

The current is made up of two components, one capacitive and ‘lossless’ and the

other in phase with U and ‘lossy’, as indicated in Fig. 2.32. It can be seen from

the ﬁgure that

¼ tand ð2:102Þ

The current I

in phase with U can be written

¼ oe@

so that the current density is given by

¼ oe@

E ¼ oe@E ð2:103Þ

It follows that the average dissipated power density is given by



tand ð2:104Þ

which is identical in meaning with Eq. (2.97).

Frequency and temperature dependence of dielectric properties

Resonance effects Because charges have inertia, polarization does not occur

instantaneously with the application of an electric field. In the case of atomic and

ionic polarization, the electrons and ions behave, to a first approximation, as

64 ELEMENTARY SOLID STATE SCIENCE

Fig. 2.32 Capacitative and ‘loss’ components of total current I.

though bound to equilibrium positions by linear springs so that the restoring force is

proportional to displacement x. Because the polarization process is accompanied by

energy dissipation, a damping factor g is included in the equation of motion. If the

damping (resistive) force is assumed to be proportional to the velocity

xx of the

moving charged particle, the equation becomes

þ mg

xx þ mo

x ¼ QE

expðjotÞð2:105Þ

in which m, Q and o

are respectively the mass, charge and natural angular

frequency of the particle and E

expðjotÞ is the sinusoidal forcing ﬁeld. The

appropriate ﬁeld is of course the local ﬁeld and only in the case of a gas would

this be approximately equal to the applied ﬁeld. For the present, attention is

restricted to a gas consisting of N hydrogen-like atoms per unit volume, for

which Q ¼e.

Solving Eq. (2.105) and ignoring the transient term yields

xðtÞ¼

expðjotÞ

mfðo

 o

Þþjgog

ð2:106Þ

Since exðtÞ is the induced dipole moment per atom, the complex polarization

P* is given by

P* ¼ NðeÞxðtÞð2:107Þ

and

e 1



ðo

 o

Þþjgo



ð2:108Þ

so that

r 1

¼ 1 þ



ðo

 o

Þþjgo



ð2:109Þ

where the asterisks indicate complex quantities. Because the resonances of

electrons in atoms and of ions in crystals occur at optical frequencies (about

1

and 10

1

respectively), the susceptibility and permittivity are written

with the additional subscript 1 to distinguish them from values measured at

lower frequencies. It follows from Eq. (2.109), by equating real and imaginary

parts, that

r 1

 1 ¼



 o

ðo

 o

þ g



ð2:110Þ

r 1



ðo

 o

þ g



ð2:111Þ

which are shown graphically in Fig. 2.33.

CHARGE DISPLACEMENT PROCESSES 65

As already mentioned, the above analysis would be valid for a gas; for a solid a

properly calculated local ﬁeld would have to be used in Eq. (2.105). Fortunately,

doing this does not change the general forms of Eqs (2.110) and (2.111) but leads

only to a shift in o

. Furthermore, because the restoring forces are sensibly

independent of temperature, so too are the resonance curves.

Relaxation effects Polarization processes occur in ceramics for which the

damped, forced harmonic motion approach is inappropriate. For example,

because of the random structure of glass the potential energy of a cation moving

through a glass can be shown schematically as in Fig. 2.34. The application of an

alternating electric field causes ions to diffuse over several atomic distances, over

a length such as b for example, surmounting the smaller energy barriers en route.

It therefore takes a considerable time for the new charge distribution to establish

itself after the application of the field.

In contrast to the atomic and ionic polarization processes, the diﬀusional

polarization and depolarization processes are relatively slow and strongly

temperature dependent. Temperature-activated diﬀusional polarization

processes also occur in ionic crystals and can involve ionic migration and

changes in the orientation of defect complexes.

Figure 2.35 illustrates how, on the application of a ﬁeld and following the

initial instantaneous atomic and ionic polarization, the slow diﬀusional

polarization P

approaches its ﬁnal static value P

. It is assumed that at time

t the polarization P

ðtÞ develops at a rate proportional to P

 P

ðtÞ:

 P

ðtÞg ð2:112Þ

66 ELEMENTARY SOLID STATE SCIENCE

Fig. 2.33 Variation in e

r 1

and e

r 1

with frequency close to a resonance frequency o

in which 1=t is a proportionality constant. Integrating Eq. (2.112) with the initial

condition P

¼ 0 when t ¼ 0 gives

¼ P



1  exp







ð2:113Þ

where t is a relaxation time.

The extension of this analysis to account for alternating applied ﬁelds is not

straightforward. P

will now depend upon the instantaneous value of the

applied ﬁeld and so will be time dependent. Furthermore, because the local ﬁeld

is a function of both position and time, its estimation would be extremely

diﬃcult. However, to make progress, if it is assumed that the polarizing ﬁeld is

E* ¼ E

expðjotÞ, it can be shown that Eq. (2.112) is modiﬁed to

CHARGE DISPLACEMENT PROCESSES 67

Fig. 2.34 Schematic one-dimensional representation of the variation in the electrostatic

potential in a glass.

Fig. 2.35 Development of polarization by slow diﬀusional processes; P

and P

are the

‘instantaneous’ atomic and ionic polarization processes capable of responding to very high

frequency (1) ﬁelds.

 e

Þe

E*  P

ðtÞg ð2:114Þ

in which e

is the value of the relative permittivity measured at low frequencies

or with a static ﬁeld applied. Eq. (2.114) can be integrated to give

¼ Cexp







 e

1 þ jot

E* ð2:115Þ

If the transient C expðt=tÞ is neglected, Eq. (2.115) leads to

¼ e

r 1

 e

1 þ jot

ð2:116Þ

or, equating the real and imaginary parts, to

 e

r 1

 e

1 þ o

ð2:117Þ

and

¼ðe

 e

1 þ o

ð2:118Þ

Eqs (2.117) and (2.118), which are known as the Debye equations, are shown

graphically in Fig. 2.36; the relaxation frequency is o

¼ 1=t. Because the

polarization occurs by the same temperature-activated diﬀusional processes

which give rise to d.c. conductivity (see Eq. (2.68)), t depends on temperature

through an exponential factor:

t ¼ t

exp





ð2:119Þ

Figure 2.37 shows the dielectric dispersion and absorption curves for a

common soda lime silica glass and their general form is seen to be consistent with

the predictions (Fig. 2.36 and Eqs (2.117) and (2.118)).

It has been experimentally demonstrated, for a wide range of glass types that

values of the activation energy E

derived from (2.119) are in close agreement

with those obtained from d.c. conductivity measurements and so support the

model as outlined. The various polarization processes which lead to dielectric

dispersion and attendant energy dissipation are summarized in Fig. 2.38.

In conclusion, it is opportune to mention the relationship between the

refractive index n and e

for a non-magnetic dielectric (m

¼ 1Þ, i.e.

¼ n

ð2:120Þ

which is a consequence of Maxwell’s electromagnetic theory. It is only valid, of

course, when the same polarization processes occur during the measurement of

both e

and n. This would be so for elemental solids such as diamond and silicon

68 ELEMENTARY SOLID STATE SCIENCE

CHARGE DISPLACEMENT PROCESSES 69

Fig. 2.36 Variation in permittivity with frequency for a dielectric showing ‘Debye’ relaxation.

Fig. 2.37 Permittivity dispersion and dielectric loss for a glass: 18 Na

O10CaO72SiO

(after

Taylor, H.E., J. Soc. Glass Tech., 43, 124T, 1959. Also see Rawson, H. Properties and

Applications of Glass. Elsevier, Amsterdam, p. 266, 1980).

for which there is only a single polarization process, i.e. atomic, across the

frequency spectrum. In the case of water, however, optical refraction measured

at 10

Hz, say, involves only atomic polarization, whereas during the

measurement of e

, usually at a much lower frequency of typically 10

Hz,

both atomic and dipolar polarization processes are occurring and therefore

Eq. (2.120) would not be expected to apply. Illustrative data are given in

Table 2.6.

70 ELEMENTARY SOLID STATE SCIENCE

Fig. 2.38 Variation of e

and e

with frequency. Space charge and dipolar polarizations are

relaxation processes and are strongly temperature dependent; ionic and electronic

polarizations are resonance processes and sensibly temperature independent. Over critical

frequency ranges energy dissipation is a maximum as shown by peaks in e

ðoÞ:

Table 2.6 Relationship between e

and n

Diamond 5.68 2.42 5.85

Germanium 16 4.09 16.73

NaCl 5.9 1.54 2.37

O 80 1.33 1.77

2.7.3 Barium titanate ^ the proto type ferroelectric ceramic

Barium titanate, the ﬁrst ceramic material in which ferroelectric behaviour was

observed, is the ideal model for a discussion of the phenomenon from the point

of view of crystal structure and microstructure; see also [10] and [11].

BaTiO

is isostructural with the mineral perovskite (CaTiO

) and so is referred

to as ‘a perovskite’. The generalized perovskite structure ABO

, is visualized as

based on a cubic close-packed assembly (see Fig. 2.1) of composition AO

with

the A-ion cordinated with 12 oxygen ions and the B-ion in the octahedral

interstices (see Fig. 2.39). A consideration of the geometry shows that for a

prefect ﬁt the following relationship between the ionic radii holds [12].

þ R

¼ 2ðR

þ R

Þð2:121Þ

For the many compounds having the perovskite structure the relationship will

not hold exactly because of small variations in the sizes of the A and B ions.

Therefore, to allow for this Eq. (2.121) is written

þ R

¼ t 2ðR

þ R

Þð2:122Þ

in which ‘t’ is termed the ‘tolerance factor’ with a value typically in the range

0.955t51.06. In the case of SrTiO

t ¼ 1. When t 6¼ 1 then small lattice

distortions (the octahedra tilt) occur in order to minimize lattice energy. These

distortions have a signiﬁcant eﬀect on dielectric properties.

Above its Curie point (approximately 130 8C) the unit cell is cubic with the

ions arranged as in Fig. 2.39. Below the Curie point the structure is slightly

distorted to the tetragonal form with a dipole moment along the c direction.

Other transformations occur at temperatures close to 0 8C and 80 8C: below

0 8C the unit cell is orthorhombic with the polar axis parallel to a face diagonal

and below 80 8C it is rhombohedral with the polar axis along a body diagonal.

CHARGE DISPLACEMENT PROCESSES 71

Fig. 2.39 The unit cell of BaTiO