Chapter 1

1.1 Introduction

The most important property characterizing the electrical properties of a crystalline solid

is the electrical resistance R. The resistance R offered by a conductor to the flow of

electric charge is found to be directly proportional to the length (1) and inversely

proportional to the area of cross section (A) of the conductor. Therefore R = p (VA)

where p is the proportionality constant called electrical resistivity. Then (I@) is the

electrical conductivity o. The basic unit of conductivity is Siemens (S) (or mho).

Electrical conductivity is an inherent property of most materials, and ranges from

extremely conductive materials like metals to very non-conductive materials like plastics

or glass. The actual cause of the resistivity must be sought as a deviation from the

periodicity of the potential in which the electrons move. It is on this concept that the

modem theory of conductivity is based. Deviations from the periodicity of the potential

causing resistivity may be due to (a) lattice vibration (b) lattice defects such as vacancies,

interstitials and dislocations (c) foreign impurity atoms and (d) boundaries. It is

interesting to note that Wein in 1913, before the development of wave mechanics, put

forward the hypothesis that the resistivity in pure metals was due to thermal vibrations of

the atoms in the lattice. The justification of this idea had to await the development of the

band theory [l]. In metals, electrons act as moving charge carriers. The electrical

conductivity of a metal is defined by

I, = o E,

Where I, is the current density resulting from an applied electric field Ex in the x-

direction. In the case of anisotropic solids, the conductivity depends on direction and o

becomes a tensor.

1.1.1 Anisotropic Conduction Electrical transport is described by the electrical conductivity tensor (oij ) in the relation

or its inverse, the resistivity tensor (p,,) , in

Both o and p are second rank tensors. Let us consider the electrical conductivity in

which an electric field E gives rise to a current I. In general, the current vector will not

have the same direction as the electric vector. Thus, assuming a linear relationship

between cause and effect, one may write for the current components relative to an

arbitrarily chosen Cartesian coordinate system

I x =~xxEx+oxyEy+~xzEz

I, =o,E,+o,E,+o,E,

I, =o,E,+o,E,+o,E,

Where the quantities o, are components of the conductivity tensor. It has been shown by

Onsager that the tensor is symmetric, i.e., oij = oji. Making use of this symmetry property

and multiplying the expressions for I,, I, and I, by k , E , and Ex, one obtains upon adding

I,E,+ IyEy + IIEl = oxx~2x + ~ ~ y y E ~ ~ + ~ ~ E ~ ~ +2oXyExEy +20pEyEz + 2omEzE,

The right-hand side represents a quadratic surface; by choosing the coordinates along the

principal axes of this surface, the mixed terms disappear and one obtains the new

coordinate system

I, = CJ~E,; Iy = ozEy; I ,= 03Ez

where 01, oz and o3 are the principal conductivities . Thus the electrical properties of any

crystal, whatever low symmetry it may possess, may be characterised by three

conductivities 01, 02, 0 3 or by three specific resistivities pl, p2, p3. Note that I and E have

the same direction only when the applied field falls along any one of the three principal

axes of the crystal.

In cubic crystals the three quantities are equal and the specific resistivity does not

vary with direction. In hexagonal, trigonal and tetragonal crystals the resistivity depends

only on the angle 9 between the direction in which p is measured and the hexagonal,

trigonal, or tetragonal axis, since in those crystal two of the three quantities P I , p2;p3 are

equal. One finds

P($) = plsin2($) + P II cos2(41) 1.1

where I and 1 I refer to directions perpendicular and parallel to the axis.

1.2 Dielectrics Dielectrics are material with very few electrons to take part in normal electric

conductivity. Dielectric materials have interesting electric properties because of the

ability of an electric field to polarize the material to create electric dipoles. A dipole is an

entity in which equal positive and negative charges are separated by a small distance.

The electric dipole is given by

p = qdl where q is the charge and dl is the distance.

1.2.1 Measurement of Dielectric Tensor

The si~nplest arrangement for measuring the components of the dielectric tensor occurs

in the conventional thin parallel plate condenser. Here the electric field direction is

dictated by geometry to the normal to the plates and the depolarization field is also in

this direction. Hence the measurement essentially involves only a single direction.

Maintaining the imposed electric field in the presence of the depolarization field, for

example by fixing the potential on the capacitor requires additional charges on the plates,

and thus the dielectric slab increases the capacitance. The ratio of the capacitances of the

system with either dielectric or vacuum between the plates is given by the ratio of the

charge densities at constant voltage

where E0 is the applied electric field, D. is the component of electric displacement

vector(D) along E0 and E" is the appropriate diagonal element of (E ,, ) in a coordinate

system one of whose axes lines up with the electric field .Obviously D. is larger than

€ OE" .

If the direction of the field relative to crystal axes is known, E. can be expressed

in terms of the components of the dielectric tensor in the clystal system by using the

transformations. If the crystal axes are not known, we may choose an arbitrary system of

coordinates to specify the dielectric tensor. In that case the slab directions for

independent measurements could be along the x, y and z- axes and some where in each

of the xy, the yz and the xz planes of this system. Once the schemes of the coefficients

are known, it may then be examined for additional symmetry by transforming it to other

frames of reference.

1.3 Conductivity Measurement Theory

1.3.1 Theory of DC conductivity As detailed earlier the resistance R is defined by R = p (11A) where p is the

proportionality constant called electrical resistivity. Then (llp) is the electrical

conductivity o. Alternately o may be defined as the proportionality constant in the

linear relation of current density 3 with applied electric field gradient (C) such that

j =oC 1.3

The magnitude of the electrical conductivity is determined by

i) The density of the charge carriers (n) i.e. the number of charge carriers per unit

volume,

ii) The charge on the camer represented by '1' when electrons are charge caniers,

iii) The average drift velocity of the carriers per unit electric field (p).

Now the charge carriers can be either ions (cationslanions) or electrons or holes.

In the former case the conductivity is said to be ionic while in the latter it is called

electronic. Further the nature of the chemical bonding plays a virtual role in determining

f i ~ o y of Conductivity and&Lctnis 7

the type of conductivity. In primarily ionic crystals the conductivity may be ionic (e.g.

Sodium chloride).

Migration of ions does not occur to any appreciable extent in most ionic and

covalent solids such as oxides and halides. Rather the atoms tend to be essentially fixed

on their lattice sites and can move only on crystal defects. At high temperatures, where

defect concentration become quite large and atoms have a lot of thermal energy, the

conductivity become appreciable for e.g. the conductivity of NaCl at -800°C (just below

its melting point) is -10"ohm~'cm~' where as at room temperature pure NaCl is an

insulator with a conductivity much less than 10~'~ohm"cm~'

In contrast there exists a small group of solids called variously- solid electrolytes,

fast ion conductors and superionic conductors in which one of the ions can move quite

easily. Such materials have rather special crystal structures in that there are open tunnels

or layers through which the mobile ions may move. There is currently a great interest in

studying the properties of solid electrolytes, developing new ones and extending their

applications in solid-state electrochemical devices.

In metals and broadband semiconductors, electrons and holes carry the electric

current. However in ionic crystals the electrical conductivity is due to the diffusion of

ions thorough the lattice. That means there is a transport of charge and mass. The

diffusion process is facilitated by the presence of defects such as Schotlky and Frenkel

defects. The conductivity of most ionic crystal is low and it is only near the melting

point that it raises rapidly.

1.4 DC conductivity measurement Methods In this section, method of obtaining conductivity data on solids is described. The use of

conductivity measurements in research work is very important, and many excellent

accounts of various measurement techniques are already available [2, 31. Though such

accounts are available, detailed discussion is out of the scope of this thesis and hence all

the methods are only briefly outlined. Various methods have been used to measure

conductivity properties [4], [ S ] , [ 6 ] , [7]. Among the methods to be discussed are

Ohmmeter and Voltmeter-ammeter measurements, potential probe measurements, high

frequency loss measurements, as well as spreading resistance and other specialized

techniques. For the measurement of ac conductivity Impedance Spectroscopy (IS) can be

used. For the measurement of dc conductivity several methods are adopted and discussed

below. Method of measuring dielectric constant is detailed later.

1.4.1 General Methods of Measurement In this section some of the methods widely used in solid-state physics for conductivity

measurement, without considering specifically the nature of the sample or other factors

are discussed. Conductivity is defined in terms of Figl.1, where the conductivity (3 is

given by a = I L IVA

where I is the current measured in

ampere , the voltage V in volts, area

A in cm2 and the length L of the

sample in centimeters.

It is implied in the above

definition that the direct measurement Fig 1.1 Arrangement for resistance

measurement by the V 1 method of the current and voltage, together

with accurate measurement of the

dimensions, suffice to determine the conductivity to any desired degree of accuracy. This

is true only if the number of carriers is sufficiently large that the thermal variations in the

carrier density are negligible. Unless this is so, there will be random fluctuations in the

conductivity as a function of time, and only the time average can be specified accurately.

Such problems are important particularly for weakly conducting semiconductors and

insulators.

1.4.2 Ohmmeter and Voltmeter -Ammeter Measurements

The simplest method of measuring conductivity is to follow the definition above,

measure the voltage drop across a sample and current through the sample, and use the

above equation.

1.4.3 Potential Probe Method

When the contacts at the ends of the sample have an appreciable resistance, as is true of

many metal -semiconductor and semiconductor-semiconductor contacts the simple V-I

method is subject to serious errors. In potential probe method, in which two extra

electrodes are used to eliminate errors due to contact resistance. The method of potential

probe is the most widely used method for conductivity measurements on metals and

semiconductors. In this method, the potential drop is measured across the probes, and the

probe distance D replaces the sample length L and becomes o = IDNDA. The potential

probe method is most widely used for dc measurements; it can also be used with ac and

is often used in connection with ac Hall effect measurements.

1.4.4 Spreading Resistance Method

A specialized but sometimes useful technique is the use of the spreading resistance of a

single small contact. If one of the contacts to the specimen is a fine whisker, the

resistance at that contact will determine at that value read on an ohm-meter, and the other

contact can be of almost any variety, as long as the area is large and the contact

resistance not unduly high.

Depending upon the nature of the whisker, the surface, and the material of the

specimen, the contact may, (a) a circular contact, (b) a hemispherical contact, or (c)

irregular shape. In the first two cases the resistivity is determined from the measured

resistance R and the known diameter D of the contacts [S]. The conductivity is then of

the form o = 11 p =1/2DR and o = l/(xDR) for the first two cases.

The advantage of the spreading resistance method includes simplicity and great

spatial resolving power, since the effective volume under measurement is a cube order d3

in dimensions. Thus it is useful for testing homogeneity and evaluating surface layers.

1.4.5 Four Point Probe Methods

In electronic conductors most common method is four-point measurement with a direct

current (DC). Its advantages are that the resistance of measurement wires and the contact

resistance do not play a role and the measurement system is simple. However, it ideally

requires samples that are long compared to their cross section.

For rapid routine measurements, particularly for semiconductors, the four-probe

method is used. In this method, all the current and potential probes are point contacts,

usually mounted on a special holder. This arrangement permits the rapid testing of

resistivity and conductivity by simple application of the four points to the specimen, and,

within limits, the results are independent of the size and shape of the specimen. In this

arrangement, the simplest case is to have the distance between all four points equal. If

the area of contact of each point is considered to be small, then the contact area doest not

enter the calculations, and the conductivity is given by o = Il2xVs where s is the

distance between points

1.4.6 Electrometer Methods

Electrometers are very useful because of the importance of low current potential

measurement. An electrometer is defined as a voltage-measuring instrument whose

movements are derived from electrostatic forces. These are in wide use in conductivity

measurements because of their high impedance range, good stability and convenience. A

number of problems are associated with measurements at high impedance level. They are

(1) Shielding and interference. This problem can be minimized by careful circuitry,

by enclosing leads in shielded cable, and if necessary, by enclosing the measuring

equipment in a shielded box

(2) Noise due to fluctuations in both contacts and bulk is of considerable importance

in high impedance measurements. In fact noise often determine the limits of impedance

above which measurements cannot be made satisfactorily.

(3) Both high resistivity semiconductors and insulators may be influenced by water

vapor or other atmospherically induced surface effects. To minimize directly the

contribution of surface leakage in a conductivity measurement guard ring may used.

Both the collecting electrode and the guard ring are effectively at ground potential, the

potential difference being only the potential drop in the grid leak or other resistors used

for current measurements. This technique is suitable not only for actual surface

conduction, but the influence of conducting skins or channels can be minimized by this

method, provided their thickness is small compared to the width of the guard ring.

1.5 Theory of Dielectrics and AC conductivity

1.5.1 Dielectric materials Dielectric materials are basically insulators having the property of storing and dissipating

electrical energy when subjected to electromagnetic field. The energy storing property

leads to the fabrication of most important constituents of electrical circuit known as

capacitors. The dielectric phenomenon arises from the interaction of electric field with

different charged particle such as electrons, ions protons and electron shells, which goes

to constitute the dielectric material. All dielectric can be subdivided into two-polar and

non-polar dielectrics.

A dc field behaviour, however provides an information about the nature of charge

carriers, their mobility, conduction mechanism etc. whereas studies in ac field provides

an information into the electrical nature of the molecular or atomic species, which

constitutes the dielectric materials.

Dielectric spectroscopy measures the dielectric permittivity as a function of

frequency and temperature. It can be applied to all non-conducting materials. The

frequency range extends over nearly 18 orders in magnitude: from the pHz to the THz

range close to the infrared region. Dielectric spectroscopy or Impedance spectroscopy

(IS) is sensitive to dipolar species as well as localized charges in a material, it determines

their strength, their kinetics and their interactions. Thus, dielectric spectroscopy is a

powerful tool for the electrical characterisation of nonconducting or semiconducting

materials.

Conducting an IS experiment on a solid electrolyte system allows a number of

characteristics to be determined. The characteristics of interest will depend on the nature

of the system. If the system is a single clystal the bulk electrical property will be of

interest. In polycrystalline form grain boundary and defect structures will form an

interesting part of the spectrum. As IS is a study of interfaces, it is not only possible to

gain some insight into the structure of the material but also know how the material is

behaving at the surface.

The most commonly used technique for the measurement of dielectric properties

was the measure of impedance in the frequency domain. For this type of experiment a

single voltage is applied at varying frequencies and the resulting phase shift and

amplitude of current are recorded. Phase and amplitude can be directly converted into the

real and imaginary components for analysis. The frequencies covered in a single

experiment can range from lmHz to lMHz, but are defined by the components of

interest in the system. This is because in solid state systems all phenomenon have a

characteristic capacitance and hence relaxation time. Such capacitances are shown in

Table 1.1 -- -

~haractzslic responsible -_ Capacitance (F)l<2

10'" Bulk 10'"-10" Grain boundary 10.10-10-~ Bulk fernelecttic 1 o-~- 10" Susface layer 1 0"- 1 0.' Sample-electrode interface

1 od Electrochemical reaction

Table 1.1, Capacitance values with possible characteristics for a given capacitance.

1.5.2 General Theory If a monochromatic signal

~ ( t ) =V, sin (at) 1.4

where v is the signal frequency d2n, is applied to a cell, a steady current

I (t) = I,(at + 0) 1.5

where 0 is the phase difference between current and voltage, can be measured. In the case

of a purely resistive component 0 will be zero. The definition of conventional impedance

is;

Z(o) = v(t)/I(t) 1.6

giving the magnitude as;

I z(W) I = v dIm(o) 1.7

and the phase angle as:8(w)

As impedance allows for differences in phase it is a more general term than resistance

and after the concept was introduced in the 1880's it was developed in terms of vector

diagrams. As the impedance;

q o ) =Z'+cpZ" 1.8

is a vector quantity it may be plotted on plane with either rectangular or planar

coordinates. A representation of such a plot can be found in Figure 1.2

In Figure 1-2 the rectangular co-ordinates

are;

R e ( Z ) = Z ' = 121 cos( 0) 1.9

Ti;::: and;I,(Z)=Z = 121 sin( 0) 1.10

with the associated phase angle of;

0 = tan -' (Z"/Z') 1 . 1 I : x axis, real !Zi 2 o giving a modulus of: z'

Figure 1.2. Impedance Z plotted as 1 Z 1 =[(z,)~ + (2rr)2]"Z 1.12 rectangular and polar coordinates

From this an Argand diagram can be drawn

in polar form, Z to be expressed as:

Z(o) = I Z 1 exp 00) 1.13

As the laws relating charge to potential do not discriminate between ionic or electronic

conductors it is natural to try and express complex ionic situations with a more simple

electronic representation. For this reason analogies are often drawn in the form of

equivalent circuits where an equivalent circuit is made to represent the IS spectrum

produced.

The equivalent circuit representing the dielectric response of the sample, consists

of an RC element. This 'RC' element is characterized by a relaxation time, 7, which is a

product of R and C. It is also possible to express RC in terms of a,,,. This expression is;

&RC= l 1.14

where a,,,, is the frequency of maximum loss. This allows the calculation for values of

the R and C components.

A typical IS spectrum for a polycrystalline superionic conductor consists of a

series of semi-circles. Each of the semi-circles represents a separate process. After the

capacitance has been calculated for the semi-circle i t is possible to assign each of the

semi-circles to a process. The capacitance is calculated using the maximum frequency

(C) of the semi-circle and the intercept of the x -axis as the resistance (R). The assigning

of a process to each semi-circle is a derivative of;

C = &'@All 1.15

where A is the area of the sample, 1 is the length between electrodes, &o is the

permittivity of free space and E' is the permittivity of the sample. If the sample geometry

is held to be unity (i.e. IIA = lcm-') and the permittivity is in the range 10, then a

capacitance value of F is expected for a bulk semi-circle.

In order to assign

)I processes or features to any

I ( further semi-circles, it is

Groin boundary necessary to consider the

structure of a polycrystalline ...- ceramic. Factors affecting the

spectrum arise from the .. . interaction of grain boundary

R b Z'. Mohm R,+%b

with the bulk and the relative

amounts of grain boundary and Figure 1. 3. A typical I S spectrum for a polycrystalline

ceramic, displaying two structural features grain regions.

Figure 1-3 shows a typical IS spectrum for a polycrystalline ceramic. Such a result can

be used to determine;

1. the relative values of bulk and grain boundary resistance

2. the degree of sintering and make an assessment of the microstructure

3. and the values of capacitance and resistance for each mechanism

If the IS measurements are taken at a series of temperatores then an Arrhenius style plot

can be made. From this type of plot it is possible to determine the activation energy of

the reaction. It is also possible to assess how different mechanisms interact and at what

temperature different mechanisms become favourable.

Features that are not resolved semi-circles can be seen in an IS spectrum. These

are indicative of charge transfer mechanisms and in some cases spikes or poorly resolved

semi-circles are seen. A 4S0spike is often assigned to charge transfer at the

electrode/electrolyte interface and is termed a Warburg spike. Small poorly resolved

semicircles can be associated with ion transfer from either the atmosphere or electrodes

with the ceramic. A common situation where this can occur is the oxygen transfer into

oxygen conducting solid electrolytes.

1.5.3 Behavior of dielectrics in time varying field If a time varying field is applied to a dielectric, the polarization of the dielecmc will

takes place under this field with frequency. However, due to the presence of damping

and frictional forces in the solid, during the interaction of charges with electric field, the

polarization will lag behind the electric field E by delta, hence permittivity of a

dielectric in an ac field will be a complex quantity &*

Complex permittivity E* = E'- j &"where &'and &" are the real and imaginary part of &*

Now we consider the behavior of a dielectric in a periodic field. The polarization

may not follow the field variation. Then displacement due to polarization may persist

even when the field is stopped. This gives rise to a decay time to attain the equilibrium

and the phenomenon is called the Debye relaxation. The decay time is called relaxation

time.

The dielectric constant can be expressed as

Now equating real and imaginary part leads to

E" ( E O - E , ) W ~ t a n 6 = - = 1.19 E' E o + & 2 2

rnw T

Equation 1.17, 1.18 and 1.19 are known as Debye 's equations.

As o w 0 , &'(a)+&, static dielectric constant and when w=, &'(a)<, dielectric

constant at optical frequency i.e. the real part of dielectric constant has two limits, one at

lower frequency (static field ) and the other at optical frequency as denoted by E, and E,,

At o =O and o = =, E" becomes 0 and E' becomes & and E, respectively. Further when

ot = 1, E" attain maximum value. It is seen that E' continuously falls with the increase of

w , particularly very rapidly at o = l k . The variation of E" with o shows a broad peak,

which is due to relaxation. This arises from the rotational behavior of the dipoles in solid

or liquid where polarized dipole can't rotate freely as in gaseous state. The behavior of E'

and E" is generally in accordance with the Debye's equation. This phenomenon can be

more effectively shown in a graphical form by Cole-Cole diagram [9].

In the derivation of Debye's equation it has been assumed that the dielectric has

average single relaxation time. A single relaxation time implies that the charged particle

or an ion has two equilibrium position separated by a distance d. In many cases dipoles

have more than two equilibrium positions. And in such cases it is not possible to assign

an average relaxation time unlike the Debye relaxation cases [lo]

1.5.4 Complex plane analysis

I A typical impedance plot for a

z . 1 / polycrystalline (e.g. P-A120,) sample will

1 / be shown by a series of semi-circles as shown earlier. These semi-circles are

rarely defined as perfect semicircles with

the center on the real axis. Three

common phenomena can affect the

resulting impedance plot, as shown in Figure 1.4 Impedance plot for a depressed semi-

circle Figure 1.4

The most common of these is the apparent shift to the right of the semi-circle the

high frequency intercept is not 0,O. Fortunately this is a simple anomaly to correct and is

done so by giving the sample impedance equal to that defined by the difference in

intercepts. The real resistance for reaction (R.) is R, - R, where RO is the resistance at

OHz and FL is the resistance at -Hz

Another common deviation is arc depression. This is when the centre of the semi-

circle does not lie on the real axis. This is caused by the presence of a constant phase

element (CPE) and leads to the presence of multiple relaxation times for the system.

These multiple relaxation times are distributed around the mean, T,=&.', where m is

maximum relaxation time. The depression of the curve can be related to the mean

relaxation time by 8 and so the depression can be compensated. The third common affect

to change the simple IS spectra are the overlap of semi-circles. If the grainfgrain, grain

boundarylelectrode relaxation frequency dependencies are close, resolution of such semi-

circles may not be possible. The data points that are produced react to form points of

consbuctive interference and are shifted. It is common for only a small region of data

points, usually to one side, to be affected. It is possible to extrapolate data into these

areas and produce meaningful results.

1.6 Dielectric spectra analysis Equivalent circuit analysis is the first step of the dielectric analysis. It can be particularly

helpful in recognizing and separating in-series processes. The second step is the

determination of the relaxation parameters. Each process has to be considered separately

because the Debye equations hold only for a 'single" relaxation. The third step is the

physical interpretation of the experimentally determined dielectric function. This

includes its relation to processes occumng on the molecular scale, such as movements of

certain molecular species, dipole orientation and its stability, phase transitions or phase

separation as well as chemical reactions. Furthermore, the physical interpretation

involves the calculation of quantities, which are related to these processes: e. g. effective

dipole moments, such as pyroelectric and piezoelectric coefficients or the glass transition

temperature. Of course, the third step requires additional information from other

experiments, particularly thermal analysis and X-ray structural analysis.

1.6.1 Equivalent circuit analysis Most electrochemical cells can be described by an equivalent circuit. This is a network of

capacitors and resistors, which give a representation of what is happening within the

system. As the system becomes more complicated the equivalent circuit also becomes

more complicated. A pure resistor, with resistance R, will display a phase angle of 0" and

have Z = R. This is represented by a single distinct point on the x-axis at Z = R. (fig 1.5)

A sample with pure capacitance will display Z = l i d , with a phase angle of 90'. This is

displayed by a series of points on the y-axis. When the two elements of resistor and

capacitor are placed in series, a summing of the two effects is observed. This is shown by

a vertical line at Z = R, where R is the resistance. Taking the case of resistor and

capacitor in parallel as an example we fmd Z' = Rpand Z" = l/wCp. This leads to:

The impedance plots for a series of equivalent circuits are shown in Figure 1.5. These

diagram shows that as the equivalent circuit becomes more complex so does the

impedance plot.

1.6.2 Theoretical aspect of Cole -Cole plot

A dielectric material provides data on permittivity and dielectric loss as a function of

frequency and temperature. These data are conveniently displayed in the form of graphs

for the real and imaginary parts of permittivity as a function of log f. The evaluation of

experimental data is much facilitated by certain graphical methods of displays, which

permit the derivation of parameters by geometrical construction. The most use of these,

methods consist of plotting E" for a certain frequency against E' at the same frequency.

This diagram may be called the complex locus diagram or Argand diagram and was

applied to dielectric by Cole and Cole. It 1s often called as Cole -Cole plot.

7 7 c q ofConductM'y andD2L,ctnis - 19

Equivalent circuit

w

Impedance plot

i t -

Figure 1.5. Equivalent circuits with impedance plots for selected circuits (MacDo~ld.1987 [341).

In a dielectric with single relaxation time the Cole- Cole plot is a semicircle. A

simple evaluation of the Debye's equation shows that the equation between the real and

imaginary part of the dielectric constant is the equation of the circle

The Cole -Cole plot provides therefore an elegant method of finding out whether a

system has a single relaxation time. These plots are also useful for the characterization of

different types of distribution function and are widely applied.

A point on the semicircle defines two vectors u and v by virtue of the construction

U-V = Eo-E, 1.22

u = ~ ' ( ~ 1 - E, + j E"(,,,, and v = ujwt 1.23

the quantities u, v considered as vectors in the complex plane is perpendicular, their

vector sum being the constant real quantity (Q-G). The right angle included by these

vectors is therefore inscribed in a semi circle of diameter (EO-E,). K S Cole and R H Cole

suggested that the permittivity might follow the,empirical equation

where E* is the complex dielectric constant, w =2nf, 70 is the generalized relaxation time

and - is defined as the spreading factor of the actual relaxation time about its mean value

ro, m is a constant and lies 0 5 m < 1. For m =O the above equation reduces to Debye's

equation.

1.6.3 Dielectric relaxation

The microscopic behavior of a dielectric material under an electric filed can be attributed

to the polarization phenomena of the charged particles, which constitutes the dielectrics.

This may an extra nuclear electron, and positively charged nuclei, anion, and cations etc.

The appositively charged species with a reasonable bond between them will form a

dipole which will behave as one unit. These dipoles under an electric field may undergo

special translational or rotational displacement, which will be reflected in macroscopic

behavior of dielectric. The polarization effect will also be dependent on the nature of the

dipole and frequency of the applied field under an ac field, but the polarization of the

dielectric may not follow the field variation. This displacement due to polarization may

persist even when the filed is stopped. This gives rise to a decay time to attain the

equilibrium and the phenomenon is known as relaxation. The decay time is known as

relaxation time.

The Cole - Cole plot have been used to determine the molecular relaxation time T.

The temperature dependence of 7 is influenced by a thermally activated process.

Considerable progress has been made since the early days of Cole- Cole, in utilizing

complex plots to explain the dielectric behavior and electrical conductivity of a wide

range of solid state materials. The dielectric relaxation, which is due to a number of

different polarization mechanisms, is observed under an ac field. The presence of any

dielectric relaxation then corresponds to one or more of the possible polarization

mechanism that occur on a microscopic scale. Each relaxation process may be

characterized by a relaxation time, which describe a decay of polarization with time in a

periodic field.

The macroscopic relaxation time TO can be calculated using the relation ulv =

(wro)(' '"I. The molecular relaxation time TO could be calculated from the equation

7 = [(~E~+E-)/~Eo] 70 1.25

Where EO and E, are the static and optic dielectric constant. If the dielectric relaxation is

related to the thermally activated process the relaxation time should have the form

TO = z,exp(Eo/KT) 1.26

1.6.4 Static and optic dielectric constant

In the equation &*- E, =(~,,-~,)/[l+(im)]'.- 1.27

Q, is the static dielectric constant or dielectric constant at very low frequency. At this

frequency the loss vanishes and dipoles contribute their full share to the polarization mechanism.

At very high frequency. the dipoles can no longer follow the local field variation and E'-+E-

where E- is the optic dielectric constant at infinite frequency.

1.6.5 Interpretation of the dielectric behaviour The Debye model is frequently used to explain the dielectric behaviour of materials. A

good description of the experimental results is often obtained by using a distribution of

relaxation times. The dielectric behaviour of materials under an externally applied ac

field has been the focus of numerous papers, in view of its high scientific and

technological importance. Measurements are made in a very wide range of frequencies

and temperatures and for many types of materials. Jonscher [ I l l made a very

comprehensive review on the subject, and presented a model, the many-body model, for

the dielectric spectroscopy [IZ]. This model introduces the idea of correlated states in the

material, arising from interactions between individual dipoles in an interactive system,

which form a narrow half-filled band. The preferred orientations of the system can be

represented by two potential wells, where their relative occupancy determines the total

polarisation, and the application of the external field will excite these states making

transitions between the two wells.

P.Q Mantas [13] proposed three types of transitions, one of them,

corresponding to the classical thermally-excited transition of a single particle from one

well to the other, which is the case involved in the Debye process [ I I]. The other two

transitions correspond to configurational tunneling in which large number of interacting

particles undergoes small adjustments, which collectively give the result of a large

transition of a single particle (sic). The results of this model agree with many

experimental data, since it gives an expression similar to the 'Universal' law, i.e. the

experimental observation that the

depolarisation current i(t) has a

I t - ? power- - law dependence on t; i( t) - t-" ' I ,-,. C- -- - I r n significant that near the

n, -- loss peak frequency, the model gives

a Debye-like behaviour. Fig. 1.6. Equivalent circuit for the dielectric

response of a material with dc conductivity, 0 , The dipoles are However, it is well known assumed to contribute with a pure capacitive port^',, and a resistive one that the ideal Debye model do not

describe the dielectric response of the

majority of the materials, and other approaches have been made to this model, namely,

the assumption that the dipoles have a distribution of relaxation times. The many-body

model predicts a region near the loss peak where the Debye description is observed. It

would be interesting to check if a system with two or more types of dipoles, (i.e. dipoles

with different relaxation times, described by the Debye model), would give a dielectric

behaviour similar to those experimentally found. This case is different from the ones

considering a distribution of relaxation times, since we are not looking for any specific

distribution in only one process, but rather to different processes with specific relaxation

times.

Knowing the variation of e'and E" with frequency, it will be important to see their effect

on the impedance spectroscopy of the material. We introduce here the dc conductivity

(odc), which we assume constant throughout the frequency spectrum, i.e. it does not

interfere with the dipoles. This should be the case of a homogeneous and resistive

material, since we are omitting other types of polarisation, namely the Maxwell Wagner

relaxation behaviour. Assuming that e' is the permittivity value of a perfect capacitance

and e" the conductivity of a resistance, with o,, =two&", any dipole will function as an RC

parallel element, because same voltage is applied to both elements. The equivalent circuit

of such a case is shown in Fig. 1.6.

1.7 Electrode effects on the measurement of dielectric properties Frequency dependent measurements of the ac conductivity have been recognised as an

important tool for the study of ionic transport properties of the solids [14, 15, 16 1. For

this, the conductivity data of solid electrolytes are generally analysed in terms of

equivalent circuits composed of frequency independent resistances and capacitances

[17]. In most of the cases, the ac response cannot be simulated correctly by simple R-c

circuits, e.g. the cases of depressed semicircles in the impedance formalism. To explain

such deviation from the ideal behaviour various empirical functions have been

formulated which are helpful to fit the experimental data [la, 19, 201.

Most dielectric samples used for impedance measurements have some form of

evaporated or otherwise applied metallic electrodes which may be relatively thin, say, of

the order of lpm or less. With samples having appreciable values of conductance and

capacitance this may lead to effects which are not negligible and which may affect the

shape of the characteristics, giving a false impression of the properties of the material in

question [2 11.

In recent years Johnscher's universal law of dielectric response [22, 271 has been

used to model the observed propenies of the solid electrolytes. This approach considers

theories of cooperative migration [23, 241 and when applied for simulation, needs

frequency dependent elements in the equivalent circuits. A R Kulkarni and H S Maiti

[17] studied the electrolyte/electrode interface using various electrodes namely silver,

graphite and lithium. The equivalent circuit parameters have been determined and tested

with the MacDonald's 1251 model. The nature of the plots for graphite electrodes was

exactly similar to that of silver electrodes. Complex impedance plots of lithium

electrodes, however, show only one semicircle passing through the origin and the low

frequency spike is absent. Comparison of the results obtained with the two blocking

electrodes, silver and graphite with those of reversible lithium electrode, indicates that

the low frequency dispersion is completely absent for the lithium electrodes. From the

complex impedance analysis with lithium electrodes, it has been observed that the

resistances are slightly higher than those observed for silver and graphite electrodes.

Since the use of lithium electrodes involved spring loaded thin lithium disks, the contacts

may not be very good as compared to silver and graphite electrodes which adhere

properly to the glass surface. The other possibility might be the oxidation of the lithium

metal at the interface which results in higher contact resistance. Comparison of two

blocking electrodes (silver and graphite) can be made in most straightforward way by

superimposing the results of one electrode over the other.

1.8 Dielectric spectroscopy - Outlook Dielectric spectroscopy is an old experimental tool, which has dramatically developed in

the last two decades. It covers nowadays the extraordinary spectral range from 10" to

10" Hz. This enables researchers to make sound contributions to contemporary problems

in modem physics [26].

The complex dielectric function has its foundation in Maxwell's equations. It

describes within the regime of linear response - the interaction of electromagnetic waves

with matter and reflects by that the underlying molecular mechanisms. It is the intention

of this review to describe briefly the development of the corresponding spectroscopic

technique, i.e. dielectric spectroscopy and to discuss some of its future prospects.

Dielectric spectroscopy in the past is characterized by limitations in frequency. Usually

measurements could be carried out over 4-5 decades only. It is shown that within the

limited frequency range of the measurement, time temperature superposition holds. The

data can be described by a generalized relaxation function.

where A& describes the dielectric strength, z is the relaxation time and a and y quantify

the symmetric and asymmetric broadening of the relaxation time distribution function.

For cx = y = 1 the famous Debye function is obtained [27, 28,291

In the last two to three decades, the experimental techniques in dielectric

spectroscopy 130, 311 have strongly developed so that nowadays the whole spectral range

from lo4 to 1012 Hz is readily accessible with automatic spectrometers [32] enabling

measurements with high resolution in tan 6 = E"/E' for samples in all phases of matter. In

summary it may be allowed to state that dielectric spectroscopy is an old but still

developing experimental technique. It has a strong technological impact and a variety of

novel routes that can open exciting new horizon.

1.9 Theory of Phase Transition

1.9.1 Introduction Phase transitions occur when a material undergoes a physical change from one state to

another. The most common examples is the phase changes that water undergoes when it

is heated. As the internal temperature of water rises, water changes from solid (ice) to

liquid and then to vapor (steam). There are several models of a different type of phase

transition, that of one solid phase to another solid phase [33]. Phase transition from one

solid state to another occur when there is a rearrangement of molecules that results in a

change in the material's lattice structure. For example, a phase transition occurs when the

shape of the lattice changes due to mechanical strain or temperature, but the atoms

maintain the same relationship with each other in the cells of the lattice.

A material may have several inherent lattice configurations such as a high-

temperature, high-symmetry phase and a low-temperature, low-symmetry phase. For

example, Ball and James [34] give a detailed description of the transition of Indium-

Thallium from a cubic configuration to a tetragonal configuration. Shape-memory

materials of Cu-Al-Ni in tension [35] undergo just such a reversible structural phase

transformation. This transformation may be triggered by a change in strain or

temperature. The shape-memory effect occurs when a material in its high-symmetry

phase is cooled from a temperature above its transformation temperature T, to a

temperature below T,.

At the lower temperature the material, in its low-symmetry phase, can undergo

large changes in shape that remain until the temperature is raised above T,, at which time

the material returns to the original high-symmetry phase for which there is only one

possible configuration. Physical models of these phenomena remain the subject of

intense research and dispute. The mathematical properties of the models are also widely

studied.

The phase transitions in solids are accompanied by interesting changes in many

of material properties [36]. Measurement of any sensitive property across the phase

transition provides a mean of investigating the transition. Changes in properties at the

phase transition are often of technological interest.

The change of structure at a phase transition in a solid can occur in two quite

distinct ways. First of all, those transitions where the atoms of a solid reconstruct to a

new lattice. Secondly, there are those where a regular lattice is distorted slightly without

disrupting the linkage of networks. This can occur as a result of small displacements in

the lattice position of single atoms or molecular units on one hand or the ordering of

atoms or molecules among various equivalent positions on the other hand. These types of

transitions are symmetry related. The term structural phase transition [SPT] is used to

describe the second type only. Since two different mechanisms are there for the second

type of structural phase transition or the distortive structural transition, they can be

classified as displasive type and order-disorder type.[37]

The symmetry of a crystal normally undergoes a change when a structural phase

transition takes place. For a large class of systems, this change consists of the loss of part

of symmetry elements by a crystal under its transformation from the higher to the lower

temperature phase. In this, the symmetry group of the new phase is subgroup of the old

phase.

In explaining the phase transition the energy of the thermal motion of the lattice

atoms cannot be neglected, as in the calculations of the crystal lattice energy. The reason

is that the phase transition mechanism involves changes in the frequencies of atomic

vibrations in the lattice and sometimes the appearance of an unstable vibration mode at a

particular temperature or pressure. Thus ferroelectric phase transitions are due to the

instability of one of the transverse optical vibrations, i.e. to the appearance of the so-

called soft mode.

Apart from a change in temperature, phase transition may be caused by changes

in pressure, by extemal fields, or by combinations of these effects. This thesis is focused

only on the phase transition in the solid state, especially on crystal, although this concept

is equally well applied in describing solid-liquid and liquid-gas transitions.

P 4 A phase transition is finite change in

the microscopic structure and macro properties

of the medium due to continuous small change Solid h se kr. in the extemal conditions. The changes which

structures undergo phase transitions are

usually changes in the arrangement or the Fig 1.7 Simpl~fied P- T diagram

nature of ordering of the atoms (their centers),

but there are also phase transitions related only to the state of the electron subsystem.

Magnetic transitions, for instance. are associated with a change in spin ordering, while

the transition of some metals to the superconducting state is related to a change in the

type of interaction between conduction electrons and phonons.

The phase equilibrium and phase composition of a substance are usually

characterized by a phase, or state, called phase diagram. The simplest example of a phase

diagram is the P- T diagram [38] where P being the pressure and T is the temperature

(Fig 1.7). Here, each point with coordinates P and T, which is called a figurative point,

characterizes the state of a substance at a given temperature and pressure. In this

diagram, curve T = T(p) separate the possible phases of the substance, which include, in

particular, the gas, liquid, and various crystal phases.

1.9.2 Order of Phase Transition We distinguish phase transition of the first and second order. First-order phase transitions

are accompanied by a jump in such thermodynamic functions like entropy, volume etc.,

and hence the latent heat of transition. Accordingly, the crystal structure also changes

jump-wise. Thus, for the first order phase transitions, curves T = T(p) in the phase

diagram satisfy the Klausius -Clapeyron equation dT/dp = T(AV)IQ where AV is the

volume jump, and Q is the latent heat of transition. In the second order phase transition,

it is the derivatives of the thermodynamic functions that experience a jump. In a second

order transition the crystal structure changes continuously. Since a first order transition

is, irrespective of its structural mechanism, associated with the nucleation process, it is

attended by a temperature hysterisis. This means the non-coincidence of the phase

transition temperatures during heating and cooling, and implies that each first-order

phase transition. For second order phase transition no temperature hysterisis is observed.

For phase transitions of both the first and second order, the crystal symmetry

changes jump-wise at the phase transition point. There is, however, a substantial

difference between the change in symmetry on first and second-order phase transitions.

In second order phase transitions the symmetry of one of the phases is a subgroup of the

symmetry of the other phase, because during the displacement of atoms only some

symmetry elements are lost, while others remains, and they form a subgroup. In most

cases the high temperature phase is the more symmetric, and the low temperature phase

is less symmetric. On first order phase transitions the crystal symmetry generally

changes arbitrarily, and the two phases may have no symmetry elements in common.

Phase transition in condensed matter can basically be interpreted within the scope

of thermodynamical principles, while for critical regions precise knowledge of transition

mechanism is essential. In nature, there are various types of phase transition, which

Ehrenfest classified in terms of a derivative of the thermodynamical potential exhibit a

discontinuous change of the Gibbs potential has attracted many investigation, since the

problem is closely related to a fundamental subject of lattice instability. In the study of

phase transitions the order parameter N(T) is a crucial quantity. Landau formulated a

theory of continuous phase transitions in binary systems based on, a single

thermodynamical variable called the order parameter emerges at T,. Below the transition

temperature T,, it is nonzero and increases on cooling. He proposed that the variation of

the Gibbs potential near T, is expressed by an infinite power series of the order

parameter, implying that ordering is essentially a nonlinear process.

1.9.3 Landau theory of Phase Transition Landau formulated the thermodynamical problem of a continuous phase transition in

binary systems, where the Gibbs potential G(q) is invariant under inversion of the order

parameter i.e., q 4 - 11. The Landau theory is abstract, where q is unspecified, but

invariance of G(q) under inversion is physically significant for a binary system. Namely

G(ll) = '3-q) 1.29

It is realized that inversion symmetry assumed in the landau theory may be

considered as reflection on a mirror plane, which is often significant in anisotropic

crystals, where the order parameter is a vector. Landau proposed that the Gibbs potential

in the absence of an external field or stress can be expressed as an infinite series of q ,

i.e., G(q) =Go + 1/2,4q2 + 1 1 4 ~ q ~ + 116 cv6+ ........ 1.30

Where Go =G(o) =G(Tc). The coefficient in 1.30 A, B, C. are normally smooth functions

of temperature. It is noted that in the expansion of G (q) there is no term in odd power,

owing to the inversion symmetry expressed by 1.29

At temperature close to T,, the magnitude of q is sufficiently small, so that the expansion

in 1.30 can be trimmed and can be written as

G (q) =Go + 1 / 2 ~ q ~ + 1 1 4 ~ ~ ~

The value of q in thermal equilibrium can then be obtained from the equation

Therefore the solution can be either

q = 0

supposing that A>O and B>O, 1.32 is the only solution, since 1.33 is imaginary, and

hence the solution 11 = 0 represents the disordered state above T,. On the other hand, if

A< 0 and B>O 1.33 gives the real solution of nonzero value, hence represents the ordered

phase below Tc. In this context, the phase transition is signified by changing the sign of

the coefficient A, Landau wrote that

A = A'(T-Tc) where A' > 0 1.34

The two solution 1.32 and 1.33,

must be consistent at T =T, for a

continuous transition, for which

it is sufficient to consider A 2 0

for T 2 T,. The coeff~cient B can T < be regarded as absent from G(q)

11 at and above T,, whereas it

signifies the presence of a -11 + 11

positive quadric potential 1 1 4 ~ ~ ~ Fig 1.8: The behaviour of the Gibbs potential

G(q) in the vicinity of Tc below T,. In the vicinity of T,, the

corresponding order parameter in the

low temperature phase can therefore be expressed by

Considering q as a continuous variable, thermodynamical states of the substance

can be specified at minima of the potential curve G(q), as shown schematically in fig 1.8.

Here assuming that G(0) = 0, the parabolic G(q) = 1/2Aq2 (A>O) above T, has the

minimum at q = 0, whereas in double-well potential G(q) =1/2Aq2 +1/4Bq4 (A<O and

B> 0 ) below T, there are two minima at + qo = f (-AIB)"~ related by inversion. These

two minima emerge as the temperature is lowered through T,, shifting their position

symmetrically away from q = 0. From a parabolic q , the transition temperature may be

determined as the intersect To of a linear extrapolation of q2-T plot with the T axis, but

such a To was always found higher than the transition temperature T,.

While the order parameter is a well-accepted concept in a uniformly ordered

phase, it is well known that the Landau theory is inadequate to explain the critical

anomalies. The failure can be attributed to the fact that the theory is not dealing with

spontaneous inhomogeneity due to distributed critical strains in otherwise uniform

crystals. Landau recognized such shortcomings, in his abstract theory, and suggested

including spatial derivatives of the order parameter in the Gibbs potential for an

improved description of phase transitions. In such a revised Landau expansion, an

additional term, called the Lifshitz term that is composed of such derivatives, is known

to be responsible for a modulated structure in crystals. However, it is still not clear in

such a revised theory if critical anomalies can be attributed to a dynamical behaviour of

the order parameter.

Needless to say, phase transitions are phenomena in macroscopic scale. In a non-

critical phase away from T,, thermodynamic properties can be described by the ergodic

average of distributed microscopic variables that correspond to the order parameter.

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