103

Chapter V

Impedance Spectroscopy

104

5.1 Introduction

A.C. impedance methods are widely used to characterize and study electrical

materials. It is now well known that impedance spectroscopy is a powerful technique

in the investigation of the electrochemical properties of dielectric materials. The

contribution of various processes such as electrode reactions at the electrode-sample

interface and in dielectrics the migration of charge carriers through grains and across

grain boundaries can all be separated out in the frequency domain of measurement.

The HP4294 impedance analyzer was used to measure impedance. In impedance

measurements, the HP4294 can measure eleven parameters-the absolute value of

impedance (Z), absolute admittance(Y), phase angle (θ) etc.

In this chapter detailed impedance studies are done on Sr1-xCaxBi4Ti4O15 [x =

0, 0.1, 0.2, 0.4, 0.6 and 0.8] prepared by both solid state and mechanical milling

method. An effort is made to explain the discrete mechanisms involved in the

conduction process through Impedance studies.

5.2 Results

5.2.1 Variation of real and imaginary part of impedance Z* with frequency

Figures 5.1(a-f) show the variation of real part of impedance (Z') as a function

of frequency at different temperatures for the samples prepared via the solid state

method SSC00, SSC01, SSC02, SSC04, SSC06 and SSC08 in higher temperature

range 500-800 oC depending upon the transition temperature. From the plots we see

the curves merge into a single value at higher frequencies. At higher temperatures, the

frequency at which all these curves merge and shift to higher frequencies. As the

105

Figure 5.1 Z' Vs Frequency at different temperatures for (a) SSC00 (b) SSC01

(c)SSC02 (d) SSC04

Figure 5.1 Z' Vs Frequency at different temperatures for (e) SSC06 (f) SSC08

Figure 5.2 Z' Vs Frequency at different temperatures for (a) MMC00 (b) MMC01

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temperature increases the Z' magnitude decreases. The same pattern is followed for

samples prepared through mechanical milling route as shown in figure 5.2 (a and b)

for MMC00 and MMC01 respectively.

Figures 5.3(a-f) depictsthe variation of Z'', the imaginary part of impedance as

a function of frequency at different temperatures for SSC00, SSC01, SSC02, SSC04,

SSC06 and SSC08 respectively. At low frequencies, the dispersion in Z'' values is not

much, but as one goes to higher frequencies, especially when the temperature is not

very high, peaks are observed. These peaks tend to broaden with increasing

temperature. With increase in the temperature, the peak maximum appears to shift

toward higher frequencies. Similar behavior is observed for mechanically milled

samples MMC00 and MMC01 as shown in figure 5.4(a and b) respectively.

5.2.2 Temperature Variation of real part of impedance Z*

Figures 5.5(a-f) shows the variation of real part of impedance (Z') with the

temperature at different frequencies. Initially the (Z') values do not change

appreciably with temperature up to 250-275 ˚C and above that temperature range, at

lower frequencies Z' values increase rapidly with temperature, attain a maximum

value at a particular temperature then show a decrease. Sharp peaks are observed

between 400 and 500˚C at lower frequencies becoming board with increasing

frequency. Also, there is a shift in peak positions towards higher temperature, as one

goes to higher frequencies. The figures shown here are plotted at four discrete

frequencies which are 1 kHz, 10 kHz, 100 kHz, 500 kHz. At higher frequencies the

variation in Z' values with temperature is not applicable. The increase in Z' values

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Figure 5.3 Z'' Vs Frequency at different temperatures for (a) SSC00 (b) SSC01

(c) SSC02 (d) SSC04

Figure 5.3 Z'' Vs Frequency at different temperatures for (e) SSC06 (f) SSC08

Figure 5.4 Z'' Vs Frequency at different temperatures for (a) MMC00 (b) MMC01

Figure 5.5 Z ' Vs Temperature at different frequencies for (a) SSC00 (b) SSC01

(c) SSC02 (d) SSC04

Figure 5.5 Z ' Vs Temperature at different frequencies for (e) SSC06 (f) SSC08

Figure 5.6 Z ' Vs Temperature at different frequencies for (a) MMC00 (b) MMC01

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with temperature suggests the space charge effects in the sample. Similar behavior of

Z' with temperature at different frequencies is also observed for mechanically milled

samples MMC00 and MMC01 shown in figure 5.6 (a and b). One peak is observed in

all the samples. Peaks are visible only at low frequencies. All these curves merge at

higher temperatures. We can infer that Z' is independent of frequency at higher

temperatures.

5.2.3 Variation of imaginary part of impedance (Z'') and imaginary part of

electric modulus (M'') with frequency

A practical problem frequently encountered while interpreting complex

impedance and admittance diagrams is the rapid variations of absolute magnitudes

with frequency, which makes it difficult to represent the high and low frequency

components on the same diagram . Thus in such situations it is convenient to use

logarithmic co-ordinate plots (Ishida et al 1964) which naturally accommodate a wide

range of values as has been done in this work. It may be shown that the peak height of

the Z'' peak is proportional to R and at the peak maxima, the equation ωRC=1 holds

good. This is true in the case of M'' plots as well. In an ideal material, Z'' and M'' peak

at the same frequency and the shapes of the peaks are identical with that predicted by

Debye’s theory for dielectric loss. In order to understand that non-Debye behavior and

to account for the dispersion in conductivity, impedance and modulus spectrum at

different temperatures are plotted.

In the case of the solid state sintered sample SSC00, Figures 5.7 (a, b, c and d)

show the variation of Z'' and M'' Vs. Log frequency at temperatures of 500, 550 600

and 650 OC respectively and figure 5.8 (a-d) shows the corresponding variations for

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Figure 5.7 Z'' - M'' Vs Frequency at (a) 500oC (b) 550

oC (c) 600

oC and (d) 650

oC for

SSC00

Figure 5.8 Z'' - M'' Vs Frequency at (a) 5000C (b) 550

0C (c) 600

0C and (d) 650

0C for

MMC00

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the Mechanically Milled sample (MMC00). From these figures, the broad inferences

that can be made are the following:

a) Z'' max and M'' max at any given temperature do not occur at the same

frequency.

b) The modulus spectra are broader than the impedance spectrum which shows

that the materials behavior is a deviation from the Debye behavior and goes

towards the non- Debye behavior

c) The general shape of the spectrum at different temperatures remains

unchanged.

d) Both M'' and Z'' decrease with increasing temperature.

e) M'' peaks at a higher frequency than Z'' for any given temperature.

5.2.4 Cole - Cole plots

The Z' and Z'' values can be plotted as distributed functions in the form of a Cole-Cole

distribution. Cole- Cole plots give a coarse indication of the nature of dielectric

response of the sample and give an idea about the DC resistance associated with the

bulk or grain boundaries, thereby facilitating subsequent analysis of the proper

dielectric response. It gives a rapid means of finger printing of dielectric data.

Therefore, Cole-Cole diagrams have been drawn. Z'' values taken at different

frequencies are plotted against Z' values at corresponding frequencies at different

temperatures. It is generally assumed that such diagrams recorded are more or less

semi-circular as seen in the Figure 5.9(a-f). The electric resistivity of the sample at the

111

Figure 5.9 Z '' Vs Z ' at different Temperatures for (a) SSC00 (b) SSC01(c)

SSC02 (d) SSC04

112

Figure 5.9 Z '' Vs Z ' at different Temperatures for (e) SSC06 (f) SSC08

Figure 5.10 Z '' Vs Z ' at different Temperatures for (a) MMC00 (b) MMC01

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particular temperature is expressed by the semi-circle diameter and the resistivity

maxima corresponds to the relative frequency

ω = 1/RC. ---------- (5.1)

In case of semicircles, the distribution of experimental points Z is supposed to obey a

law similar to the Cole-Cole distribution (cole and cole, 1941) despite the fact that

impedance as a function of frequency is not strictly analogous to dielectric

susceptibility. Z' values and the corresponding Z'' values are plotted at various

temperatures on complex impedance planes. At higher temperatures Z'-Z'' plots have

been with fair amount of closeness drawn as semi-circles as shown in the Figure

5.9(a-f) for samples SSC00, SSC01, SSC02, SSC04, SSC06 and SSC08 respectively,

Mechanically milled samples MMC00 and MMC01 also exhibit similar behavior as

shown in figure 5.10(a-b) respectively.

5.2.5 Electrical measurements

5.2.5.1 D.C. Conductivity measurements

The sintered samples were used for conductivity measurements, conductivity

being studied under both static and dynamic conditions at various temperatures

varying from room temperature to 800oC. Figure 5-11 [a-d] shows the Arrhenius plot

for conductivity (DC) measurements in the ranges specified above for the samples

SSC00, SSC02, SSC06 and SSC08 respectively. The samples used for this experiment

were unpoled one. Initially, the conductivity of the sample decreased from room

temperature up to about 170oC and above that temperature, the conductivity

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Figure 5.11 log D.C. conductivity Vs 1000/Temperature at different frequencies for

(a) SSC00 (b) SSC02 (c) SSC06 (d) SSC08

115

monotonously increased with temperature. The activation energies have been

determined in various temperature regions using the equation.

σ = ------------- (5.2)

Where σo is the dc conductivity, E is the activation energy, T is the absolute

temperature and K is the Boltzmann constant.

Table 5.2 gives the activation energies calculated in two different regions. The

conductivity plot is divided into two regions broadly and the activation energy was

calculated from the slope of the plot in each region was calculated by the method of

least squares. DC conductivity value does not show any linearity with doping

concentration. It has been observed for a particular sample SSC06, the D.C

conductivity is low. Figure 5-12(a) shows the Arrhenius plot for conductivity (DC)

measurements for mechanically milled samples MMC00.

5.2.5.2 Variation of A.C. Conductivity with Temperature

The values of AC conduction were calculated from the admittance data, using the

equations

Y = (Z*)-1

= Y' + jY''-------------------------5.3

σ' = Y'(d/A) -------------------------------------5.4

Where σ' is the real part of a.c. conduction having units of (ohms – cm)-1

and Y' is the

real part of admittance, d being the thickness and A the surface area of the pellet. The

values of conductivity at temperatures just above room temperature and up to about

150oC show a small decrease, after which they begin to increase, both with

temperature and frequency.

The log σ'a.c Vs. /T plot shown in figure 5-13(a-f) indicates the behavior of the

samples with frequency of the applied A.C. and temperature. At low frequencies, the

conductivity increases gradually with temperature but at high frequencies, the

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Figure 5.12 log D.C. conductivity Vs 1000/Temperature at different frequencies for

(a) MMC00

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Figure 5.13 log A.C. conductivity Vs 1000/Temperature at different frequencies for

(a) SSC00 (b) SSC01(c) SSC02 (d) SSC04

Figure 5.13 log A.C. conductivity Vs 1000/Temperature at different frequencies for

(e) MMC00 (f) MMC01

Figure 5.14 log A.C. conductivity Vs 1000/Temperature at different frequencies for

(a) MMC00 (b) MMC01

conductivity values do not vary appreciably with the temperature up to 225oC.

Thereafter the conductivity increases rapidly with the temperature. This is a common

feature in many ceramic compounds. Further, from this figure it may also be inferred

that in the low temperature region, the conductivity variation at different frequencies

is less dependent on temperature. Also the difference in frequency dependence of

conductivity is more evident at low temperatures similar behavior is observed for

mechanically milled samples MMC00 and MMC01 as shown in figure 5.14 (a and b)

respectively.

5.2.5.3 Variation of A.C. Conductivity with frequency as a function of

Temperature

Figure 5.15(a-f) shows the A.C conductivity plotted for samples SSC00,

SSC01, SSC02, SSC04, SSC06 and SSC08 at various temperatures respectively. For

the corresponding frequency ranges, the overall values of conductivity appear to be

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higher by about an order. Similar behavior is observed of mechanically milled

samples MMC00 and MMC01 as shown in figure 5.16 (a and b) The conductivity

shows an increase with frequency and such dependence can be expressed as

σ' = A ωn-------------------------- (5.5)

In this expression A is a constant, the angular frequency (=2 and n is the

exponent (Yootarou and Minorou, 1973). The frequency dependence reported for the

activation energy in the literature is often contradictory. According to Kuznetkova

(1970) and Haberey (1968), for example, the activation energies are frequency

dependent whereas, according to Volger (1954) they are not. Accordingly, the

frequency dependence of activation energy poses an interesting question and a

detailed analysis on this aspect is worthwhile. In addition to the reasons mentioned

earlier in this chapter vis-à-vis the importance and relevance of detailed studies on

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Figure 5.15 log A.C. conductivity Vs Frequency at different Temperatures for

(a) SSC00 (b)SSC01(c) SSC02 (d) SSC04

Figure 5.15 log A.C. conductivity Vs Frequency at different Temperatures for

(e) SSC06 (f) SSC08

Figure 5.16 log A.C. conductivity Vs Frequency at different Temperatures for

(a) MMC00 (b) MMC00

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electrical conductivity in such materials, yet another reason for such a study is that

very often one tends to miss any possible dielectric anomalies if the conductivity

values were too high. For example Smolenskii et al. (1961) could not record dielectric

anomalies in this compound despite the fact that it was possible to trace hysteresis

loops.

5.3 Discussion

Impedance spectroscopy plays a vital role in analyzing the electrical properties

of low conductivity materials and electroceramics. It separate out the grain and grain

boundary effects, useful in determining the space charge polarization and its

relaxation mechanisms by appropriately assigning different values of capacitance and

resistance to the grain boundary and grain effects. Different contributions to the

conductivity can be calculated by the impedance analysis.

Figure 5.1(a-f) and 5.2(a, b) shows the magnitude of Z' decreases on increasing

temperature, which signify the increase in AC conductivity. Large value of Z' at low

frequency and temperature shows the effect of polarization. For all the samples it is

observed that, at higher frequencies the value of Z' merges for all temperatures and

which indicates the presence of space charge polarization [Jonscher (1977)]. Due to

the polarization effect there must be a reduction in the barrier properties of the

material with rise in temperature. This may be a reason for the reduction of resistivity

of the material with temperature at higher frequencies.

The imaginary part of the impedance would pass through a maximum in the

spectroscopic plot as shown in 5.3(a-f) and 5.4(a, b). The magnitude of Z'' in a charge

carrier system is frequency dependant. This indicates the presence of space charge

polarization. The height of the peaks in these frequency explicit plots (i.e. the

magnitude of Z'' at the peak) is proportional to the resistance [Sharma et.al.1974]

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Z'' = R (τω / (1+ τ2ω

2)) --------- (5.6)

As expected the magnitude of Z'' at the peak decreased with the increase in

temperature upto the transition temperature at higher frequencies these curves either

merge or appear to merge the merging of all the Z'' curves at the higher frequency end,

indicate the depletion of space charges at those frequencies, since these curves

basically denote the arc losses of the sample. The relaxation process may be due to the

presence of electrons at low temperature and defects at higher temperature [Laha and

Krupanidhi (2003)]. In perovskite oxides, the major mode of charge transport is a

multiple hopping process [Gupta et.al. (1994)].The hopping process takes place across

the potential barriers set up by the lattice structure. However, due to irregularities in

the lattice structure near defect sites, the potential barriers will have different

magnitudes [Victor et.al. (2003)]. In the present work, heterogeneities may have been

caused because of disorder resulting from random occupation of Ca with other ionic

radii and valence sites in A site of the ABO3 lattice of SrBi4Ti4O15 (SBT).

In SBT system each unit cell contains four units of ABO3 structures. Each set

is separated by (Bi2O2)2+

layers. Because of random occupation of equivalent sites by

different ions, there are fluctuations in structure and hence may result in multiple

relaxation phenomenon in these materials.

It can be seen from the plots Z' Vs Temperature shown in the figure 5.5 (a-f)

and 5.6(a, b), Z' attains a maximum value. The observed critical temperature value in

the dielectric plots and the temperature at which Z' attains a maximum value are found

to be approximately same for the pure sample SSC00 and MMC00. The pseudo

perovskite blocks which are stacked between the two (Bi2O2)2+

layers do not have the

same dimensions nether the orientation in case of layered structure compounds. When

compared to the unit cells the perovskite blocks closer to the (Bi2O2)2+

layers are more

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elongated and distorted. The distortion in the unit cells decreases as one move into the

centre of the cell. The rotation of oxygen octahedral about the c-axis in these

perovskites layers is zero at center, as it is compressed on either side and hence can be

thought of to be more stable and less strained than those which are closer to the

(Bi2O2)2+

layers. Finally with the increases in temperature, the displacement of Ti ion

in the present compound is much easier in the cells which are closer to the bismuth

oxide layers rather than the one which is at the centre of the unit cell.

For all the samples prepared by solid state method and mechanically milled

two effects due to microstructural inhomogeneity – grain and grain boundary are

observed in Cole-Cole plots as shown in figures 5.9(a-f) and 5.10 (a,b). Relaxation

mechanism is not identified for the analyzed frequency range. This observation can be

conformed from the SEM taken on the samples as shown in chapter III. These SEM

images showed microstructure comprising of polycrystalline grains in the form of

plates separated from each other by grain boundaries. Using Impedance Spectroscopy

one can separate the resistance related to grains (bulk) and grain boundaries because

each of them has different relaxation times resulting in separate semicircles in the

Complex Impedance plot.

Figure 5.17 show the fitted and experimental curve for the sample SSC00 and

MMC00 respectively at 500oC. Good agreement between the experimental and fitted

curve is observed for all the compositions. The calculated errors between the

measured and fitted data are in the range 3- 10% over the entire frequency range,

showing that the fitted results are dependable. The equivalent circuit consists of a

Figure 5.17 Experimental and fitted curves for samples SSC00 and MMC00 at 500oC

123

124

series array of RC circuits in series with a resistor. One RC circuit represents grain

effects and the other represents grain boundaries. Let (Rb, Rgb) and (Cb, Cgb) be the

resistances and capacitances of grains and grain boundaries respectively then the

impedance Z* for the equivalent circuit is [Sinclair et.al. (1989]

Z* = Z' –jZ'' = + -------- (5.7)

Where Z' = + -------- (5.8)

Z'' = R b ] + Rgb ] ------ (5.9)

Based on Equation 5.7, the response peaks of the grains and grain boundaries are

located at 1/ (2πRbCb) and 1/ (2πRgbCgb) respectively. In general, the peak frequency

for grain boundaries is lower than that for grains due to their large resistance and

capacitance compared with those of grains [Sinclair et.al. (2002)]. Hence in the

impedance spectra, the lower frequency response is attributed to the grain boundaries

and the higher one to the grains.

The variation of grain and grain-boundary resistance evaluated from the

semicircular arc in the impedance spectrum has been expressed in the temperature

range 500-650oC shown in figure 5.18(a, b) (SSC00 and SSC01) and 5.19(a, b)

(MMC00 and MMC01). The resistance of grains is smaller than that of grain

boundaries in all samples. This indicates a relatively large grain boundary

contribution to the total resistivity. The reason may be that close to the grain

boundaries, the transport properties of the materials are controlled by defects,

Figure 5.18 Grain and Grain boundary resistance Vs 1000/Temperature at different

frequencies for (a) SSC00 (b) SSC01

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Figure 5.19 Grain and Grain boundary Resistance Vs 1000/Temperature at

different frequencies for (a) MMC00 (b) MMC01

126

expected to be present in higher concentration than in grains, leading to an additional

contribution to the impedance. The internal space charge created at the grain

boundaries may lead to a significant increase in the concentration of charge effects.

Perovskite oxides contain long ranged charge interactions. The less ordered

arrangement of defect in the grain boundaries causes irregular columbic potential

fields which discourage migrations of charge carriers across the grain boundaries.

This may be the reason for high resistance values of grain boundaries [Iguchia et.al.

(2004]. Figures 5.18(a, b) and 5.19(a, b) shows drop in the grain resistance (Rb) and

grain boundary resistance (Rgb) with increase in temperature in all the samples.

Therefore, the variation of grain and grain boundary resistance with

temperature follows the Arrhenius relation

(R = Ro e (ε/KT)

)

where ε being activation energy for conduction. It is observed that there is an increase

in the values of grain and grain boundary resistances with increase in the doping

concentration of Ca in both solid state samples and mechanically milled samples. This

shows that the conduction in the grain and grain boundary is affected by the dopant

concentration. The incorporation into the SBT crystal structure is expected to increase

the complexity. The exact mechanism for such a change is not yet known.

Table 5.1 shows the activation energies for the conduction of grain and grain

boundary calculated for solid state and mechanically milled samples, respectively.

These activation energies are in accordance with the values reported in the literature

by Octavia Alvarez-Fregosoa (1997). Grain boundary conduction dominates over the

grain conduction as shown in the Table 5.1. This shows that the conduction

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Table 5.1 Grain and Grain boundary Activation energies

Sample

Activation Energy for

Conduction (eV)

Temperature (500 – 650oC)

Activation Energy for

Relaxation (eV)

Temperature (500 – 650oC)

E (Grain)

E (Grain

boundary)

ε (Grain) ε (Grain

Boundary)

SSC00 1.01 1.28 1.05 1.25

SSC01 1.05 1.31 1.10 1.28

SSC02 1.12 1.40 1.15 1.34

SSC04 1.18 1.45 1.22 1.38

SSC06 1.26 1.22 1.29 1.35

SSC08 1.24 1.29 1.30 1.27

MMC00 0.95 1.02 1.01 1.08

MMC01 0.98 1.05 1.06 1.12

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mechanism is basically dominated by grain boundary conduction and the hopping

electrons are created due to oxygen vacancies [Sambasiva Rao et.al. (2006)]. Bismuth

oxide compounds are expected to lose oxygen during sintering at high temperature as

per reaction:

O0 = O2 + V0'' + 2e- ------- (5.10)

Doubly charged oxygen vacancies are considered to be the mobile charges in

perovskite ferroelectrics and play an important role in the conduction process. Oxygen

vacancies move and reach the electrodes and get trapped. Hence, space charges can be

observed as a slope change in log conductivity Vs 1000/Temperature plots, as

observed in the present measurement. These defects form barrier layers at the grain-

grain boundaries interface. Cooling of sample is followed after sintering in ceramic

method resulting in re-oxidation. Due to the falling temperature and insufficient time

available for diffusion of oxygen to the bulk material re-oxidation is limited to the

surface and grain boundaries. This makes the grain boundaries insulating as compared

to grain, which still remain semiconductor. This results in difference between grain

and grain boundary resistance [James et.al. (1996)]. It is observed from the table 5.1

that the activation energy for conduction for grain increased with increase in Ca

concentration for grain in both solid state and mechanically milled samples

The variation of grain (Cb) and grain boundary (Cgb) capacitance with temperature in

the range 500- 650 oC is shown in figure5.20 (a,b) and 5.21 (a,b). The formation of

barrier layers at the grain- grain boundary interface due to the defects formed during

the high temperature sintering of the samples is the cause for Cb > Cgb. It can be seen

from the plots that the capacitance increases with temperature and show a peak

Figure 5.20 Grain and Grain boundary Capacitance Vs 1000/Temperature at different

129

Frequencies for (a) SSC00 (b) SSC01

Figure 5.21 Grain and Grain boundary Capacitance Vs 1000/Temperature at different

frequencies for (a) MMC00 (b) MMC01

130

around transition temperature and there after decreases for pure SBT. We can infer

that the capacitance is associated with the net polarization of the ferroelectric domains.

These ferroelectric domains are formed when individual atomic displacements are

linked cooperatively. At the dielectric transition, the polarization of individual dipole

starts to increase as it approaches transition temperature and is optimized when the

Curie temperature is reached. This domain polarization is reversible under the

influence of electric field as evidence by the maximum in Cb and Cgb at transition

temperature. Above Tc, domains break up leading to a decrease in the net polarization.

This explains the decrease in capacitance beyond Tc.The grain and grain boundary

relaxation time are evaluated using the relation

τ =

The thermal variation of relaxation time for grain and grain boundary follows a

variation of Arrhenius type as shown in the Figure 5.22 (a, b) and 5.23 (a, b).

Relaxation time decreases with the increase of temperature (in the range 500 – 650

oC). Activation energy of relaxation for grain (εb) and grain boundary (εgb) are listed

in Table 5.1. The observed activation energy values explain that the grain boundaries

require higher activation energies for hoping across them than the grains. This

indicated possible grain boundary barrier formation against electron conduction. One

can conclude that grain boundaries resistances are the dominating character for the

overall electrical behavior of the material. The conduction through the grain boundary

is the motion of electrons, since ionic motion across the grain boundary might be

easier than the more tightly packed grains. Inside the grains, the low value for the

activation energy is due to the electron being trapped in shallow potential wells or

oxygen vacancies. Therefore it can be conclude that inside the grains, electrons and

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Figure 5.22 Grain and Grain boundary log ( ) Vs 1000/Temperature for (a) SSC00

(b) SSC01

Figure 5.23 Grain and Grain boundary log ( ) Vs 1000/Temperature for (a) MMC00

(b) MMC01

132

oxygen vacancies dominate the conduction, and electronic conduction is the

contributor in the grain boundary region [Victor et.al. (2003)].

As can be seen from Table 5.1, for a given sample, the activation energy grain for

conduction and activation energy for grain relaxation (εg) are approximately equal.

Similarly, the activation energy for grain boundary conduction (Egb) is approximately

equal. This indicates that the charged carriers involved in conduction and relaxation

mechanisms are the same. These observed low values of activation energy in ceramics

compared to other perovskites may be due to the presence of charges carriers inside

the grains and same electronic charge created due to the use of silver electrodes at

elevated temperatures [Sinclair et.al. 1996].

For conduction a material must possess two features: a pathway to mediate

the migration of charge and presence of charge carriers. For many years AC

conduction has been studied in disordered solids such as amorphous semiconductors,

glasses, polymers compounds. All disordered solids show similar AC behavior

whether the conduction is electronic, ionic [Owen (1997), Jonscher (1983),

Elliott(1990) and Van Staveren et.al.(1991)]. The simplest and most common

explanation for conductivity is the existence of one or the other kind of

inhomogienities in the solid. This assumption is the fact that a strong frequency

dispersion of the conductivity is observed only in disordered solids. These

inhomogienities may be of microscopic or a macroscopic scale. In this, conduction is

explained based on hopping mechanism. In hopping mechanism we assume the

inhomogienities on the atomic scale are randomly varying jump frequencies for the

carrier.

The thermal plots of AC conductivity [figures 5.15(a-f) and 5.16(a, b)] are fitted to

the general conduction activation mechanism given in equation (5.3). AC conductivity

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activation energies for all the samples at different frequencies are reported in table 5.2.

Activation energies are in accordance with the values reported in the literature. As it

can be seen from the Table 5.2, AC conductivity activation energy increases with

decrease in frequency. Here AC conductivity phenomenon corresponds to the short

range hopping of charge carriers through the trap sites separated by energy barriers of

varied heights. The time constant for transition across a lower energy barrier is small

compared to that across the higher ones. Therefore, the traps with low activation

energy can respond only at high frequencies. Also as it is anticipated the numbers of

traps having higher energies are less than those having lower energies, the AC

conductivity at lower frequencies are expected to be low. This theory predicts a

decrease in activation energy at low temperature with a corresponding increase in the

frequency dependence. At high temperatures the energy distribution of the traps is

more uniform and the variation of the A.C. conductivity with frequency is low. This

makes the A.C. conductivity Vs Frequency curves for different temperature converge

at high frequency indicating that at high frequencies the A.C conduction becomes

almost independent of temperature.

A formalism to investigate the frequency behaviour of conductivity in a

variety of materials is based on the power relation proposed by Jonscher 1977 shown

in equation (5.5). The term contains the A.C dependence. The exponent n is a

function of temperature and frequency and lies between 0 and 1.

The polarization process is characterized by the exponent n.

Usually the interaction between the neighboring dipoles decrease with decrease in

temperature and consequently the exponent n increases.

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Table 5.2 Activation energies for AC and DC conduction in various ceramics

Sample

Activation Energy Activation Energy

D C (eV) A C ( 10kHz) (eV)

350 - 500 oC 500 – 650

oC 350 - 500

oC 500 - 650

oC

SSC00 0.56 0.75 0.41 0.74

SSC01 0.63 0.82 0.56 0.86

SSC02 0.72 0.74 0.62 0.75

SSC04 0.81 0.88 0.53 0.79

SSC06 0.82 1.07 0.49 0.65

SSC08 0.88 1.21 0.65 1.02

MMC00 0.58 0.77 0.45 0.75

MMC01 0.68 0.85 0.59 0.89

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. It can be seen in the frequency variation of AC conductivity plots (figures

5.13(a-f) and 5.14(a,b)). In high frequency regions the electrical conductivity is high.

This dispersion in the conductivity values can be explained on the basis of conduction

mechanisms in a disordered solid. At high frequencies, the electrical conductivity

increased by the hopping of the charge carrier at places with high jump probability.

This increase in the electrical conductivity continues as long as the frequency

of the applied field is lower than the maximum jump frequency in the solid [Macedoa

et.al. (2003)]. As frequency increases, two dispersion regions are found for all

temperatures. Here the results do not follow the simple power law. The following

power relation is used to explain the frequency dependence of AC conductivity.

------- (5.11)

Where n1 and n2 are exponents corresponding to low and high frequency region.

Similarly A1 and A2 are constants corresponding to low and higher frequency regions

slopes drawn to the experimental data in the low and high frequency region will

intersect at a point and the frequency responding to this point is called relaxation

frequency.

A small amount of oxygen loss occurs since the compounds have been

sintered at high temperatures. According to the Kroger-Vink notation [KrÖger and

Vink (1956)]

-------- (5.12) (a)

----------- (5.12) (b)

Where is the oxygen vacancy with two effective negative charges, is single

ionized oxygen vacancy, is doubly ionized oxygen vacancy and is the electrons

released. The electron released in the above reaction may be captured by Ti ions

present in the compounds. The reason for the capture of electrons by Ti is due to its

136

unstable valency (Ti4+

→ Ti3+

). This polaronic conduction of 3d electrons on

Ti3+

ions takes place at low temperatures [upadhyay et.al. (1998)].

Equation 5.5 describes different contributions to conductivity. If the high frequency

dispersion is associated with grains since it is associated with the smaller capacitance

value, the low frequency dispersion is associated with grains boundaries (larger

capacitance value) [Barranco et.al.(1998)]. The temperature at which the grain

boundary resistance dominates over grain resistance is represented by a change in

slope of AC conductivity with frequency. The frequency at which the slope change

takes place is known as the hopping frequency, which corresponds to polaron hopping

of charged carriers [Li et.al (2003)]. The hopping frequency shifts to higher frequency

with temperature. The charged species that have been accumulated at the grain

boundaries have enough energy to jump over the barrier with rise in temperature and

thereby increase the conductivity [Sen et.al. (2007)].

For a single phase material with a homogeneous microstructure, the D.C electrical

conductivity depends on both the concentration and the mobility of charge carriers.

Usually at low temperatures, extrinsic conduction is predominant. The temperature

range 500 – 650 oC where the DC conductivity measurements are done in the present

work corresponds to the intrinsic ionic conduction range and the conduction is

dominated by the intrinsic imperfections [Wu et.al. (2000)]. The conductivity

phenomenon in these materials is due to the movement of oxygen vacancies and their

motion in the lattice and in between the layers. The evaporation of bismuth oxide at

elevated temperatures in layered perovskite would lead to oxygen vacancies [James

et.al. (1999)].

The DC electrical conductivity in all the samples follows the equation (5.3). DC

conductivity plots i.e. variations of DC conductivity with inverse of temperature are

137

shown in figure 5.11(a-d) and 5.12. From the plots, it is observed that conductivity is

found to increase with temperature. The conductivity graph shows that change in

slope occurs around 550oC. Probably this is due to the dielectric transition occurring

at this temperature range. The change in slope of curve will reflect a change in the

conductivity phenomenon in paraelectric and ferroelectric regions. The activation

energy values calculated from the slope of Arrhenius plots are listed in Table 5.2.

Change in activation energy is due to the lattice adjustment at the phase transition

from ferroelectric to paraelectric region. It is observed that the DC activation energy

values at lower temperature regions are low compared to activation energy at higher

temperature regions. This may be due to the high energy required to overcome the

thermal fluctuation by the charge carriers at higher temperatures. At higher

temperatures, the increased activation energies increase vacancy concentration as well

as the motion of these vacancies. Hence the conductivity is high at higher

temperatures.

It is known that DC conduction is due to the movement of oxygen vacancies in

all the layered compounds. In these samples, the conduction is through a-b plane, the

mobility of oxygen vacancies is affected by the distortion of all these planes due to

the various cations present in this layer. The conductivity at lower temperatures in the

sample is due to the loss of oxygen that occurred during the sintering. At higher

temperatures, the increased activation energies increase the vacancy concentration

leading to an increase in motion of these vacancies. The difference in activation

energies between the two regions could be approximately equal to the energy required

for the creation of oxygen vacancies.

A difference in the activation energy for DC conduction and AC conduction is

observed at lower temperature at high frequency (50 kHz) and the activation energy

138

for AC conduction is small when compared to DC conduction. This is expected

because at lower temperature regions, the DC conductivity is due to the mobility of

conducting charges over a long distance rather than reorientation mechanism as in AC

conductivity via dipole formation.

5.4 Conclusions

In view of the above discussions, the following conclusions can be drawn from

the impedance studies done on the system Sr1-xCaxBi4Ti4O15 (x = 0, 0.1, 0.2, 0.4, 0.6

and 0.8) prepared by solid state method and (x = 0 and 0.1) by mechanically milling

1. Variation of Z'' and Z' with frequency shows the effect of space charge

polarization. Shifting of peaks with temperature and frequency indicates the

dependency of space charges on temperature and frequency.

2. Broadening of peaks in Z'' with frequency indicated the presence of multiple

relaxation processes in the material.

3. These relaxation processes can be attributed to the vacancy sites present in

the compound and also to the heterogeneities caused because of disorder

resulting from random occupation of Ca with different ionic radii and

valence sites in A site of the ABO3 lattice of SBT.

4. Impedance analysis explains that the conduction in the present samples

occurs through grain and grain boundaries.

5. Resistance of the grain and grain boundary revealed a large grain boundary

contribution to the total resistivity. It is observed that there is an increase in

the values of grain and grain boundary resistances with increase in the

doping concentration of Ca in both solid state samples and mechanically.

6. Grain boundary conduction dominates over the grain conduction. Hence we

can say that the conduction mechanism is basically dominated by grain

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boundary conduction through hopping electrons created due to oxygen

vacancies. The oxygen vacancies are created during the high temperature

sintering of these compounds.

7. Grain and grain boundary capacitance increases with temperature and shows

a peak around transition temperature. Capacitance again decreases beyond

transition temperature. This behaviour shows that the capacitance is

associated with the polarization of ferroelectric domains.

8. The activation energy for relaxation calculated for grain and grain boundary

show that the grain boundaries required higher activation energies for

hopping than the grains.

9. Charged carriers involved in conduction and relaxation mechanism are the

same as the activation energy for conduction and relaxation are

approximately same for both grain and grain boundary conduction.

10. DC and AC conductivity analysis revealed that conduction occurs by

different mechanism in each case.

11. AC conduction is explained based on hopping mechanism.

12. DC conduction is basically due to the migration of oxygen vacancies in

layered compounds.

140