CHAPTER-5 COMPLEX IMPEDANCE SPECTROSCOPYshodhganga.inflibnet.ac.in/bitstream/10603/48048/15/15_chapter 5.pdf · Chapter 5 Complex impedance spectroscopy 99 ... x=0.0 400 1.09 x10

CHAPTER-5

COMPLEX

IMPEDANCE

SPECTROSCOPY

Chapter 5 Complex impedance spectroscopy

99

Chapter 5

Complex impedance spectroscopy

5.1 Introduction

The frequency dependent measurement of dielectric parameters of dielectric/ferroelectric

ceramics, ionic solids etc., has some limitations on getting sufficient information regarding

the proper characterization of their electrical microstructure. The limitations can be

overcome by analyzing frequency-dependent electrical/dielectric data of the materials using

the complex impedance technique [225, 242, 243].The first and most significant attempt in

this regard was made by Cole and Cole [244], who made data analysis of real system by

plotting the real and imaginary part of complex permittivity (ε) of dielectric materials in the

complex plane, known as Cole-Cole plot. There after a lot of progress has been made in

utilizing complex plane plots and frequency explicit plots of different parameters like

complex permittivity (ε) [245-247], complex impedance (Z) [248-250], complex

admittance (Y), complex electric modulus (M) [251-253] and loss tangent (tan δ), to

explain the dielectric behavior and electrical conductivity of materials. Complex impedance

spectroscopy (CIS) is a helpful technique used for investigation, characterization of the

electrical and electrochemical properties of polycrystalline materials in relation to their

microstructure. Polycrystalline ceramics are inhomogeneous materials constituted by grains

separated by grain boundaries. This method ensures proper separation among the bulk,

grain, grain boundaries and electrode-interface properties. Some micro-structural properties

of the material (i.e., charge transport, charge diffusion at the interface within the cell,

dielectric relaxation) can also be investigated. In this chapter, the structure- electrical

properties of all the studied compounds have been explored through the complex electric

impedance formalism.

5.2 Experimental

Some important electrical properties of the proposed materials were studied by the

impedance measurement procedure using a computer-controlled PSM 1735: N4L


100

impedance analyzer in a wide temperature (room temperature-500oC) and frequency (1-

1000 kHz) ranges in air atmosphere.

5.3 Complex impedance

Ferroelectric ceramics are in general electrically heterogeneous. For characterization of

these materials a proper relation between microstructure and electrical properties is

essential. CIS technique is used for simultaneous electrical and dielectric characterization

of samples. In impedance spectroscopy the impedance data are generally plotted in

complex plane. The variation of real with imaginary part of the impedance is known as

Nyquist plots. Similar to impedance spectroscopy, the modulus data plotted in complex

plane are used to represent the response of dielectric systems [254]. Polycrystalline

materials generally show inter-granular or grain-boundary impedance and capacitance.

From micro-structural point of view, a ceramic sample is composed of both grains and

grain boundaries which exhibit different resistivity and dielectric permittivity [255]. In

order to establish a relation between the microstructure and electrical properties, a ‘brick

layer model’ [256] was proposed. In this model grains are assumed to be of cubic-shaped,

and grain boundaries to exist as flat layers between grains. The advantage of adopting this

is the determination of grain-boundary conductivity without detailed micro-structural and

electrical information. To analyze the impedance spectra, data usually are modeled by an

ideal equivalent circuit consisting of a resistor R and a capacitor C. The experimental

impedance data points measured were fitted using software Zswimpwin, with an equivalent

circuit.

A circuit consists of a series collection of two sub-circuits (consisting of a resistor and

capacitor connected in parallel), one representing grain effect and the other representing

grain boundaries. If Rg, Rgb are the resistances of grains and grain boundary and Cg, Cgb are

the capacitances of grains and grain boundaries respectively, the complex impedance for

the equivalent circuit is given by:

Z*(ω) =

1

𝑅𝑔−1+𝑖𝜔𝑐𝑔

+ 1

𝑅𝑔𝑏−1+𝑖𝜔𝑐𝑔𝑏

= Z' (ω) - iZ'' (ω) ………………………… (1)

Z' (ω) = 𝑅𝑔

1+(𝜔𝑅𝑔𝐶𝑔)2 +

𝑅𝑔𝑏

1+(𝜔𝑅𝑔𝑏 𝐶𝑔𝑏 )2 …………………………….. (2)


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Z'' (ω) =Rg 𝜔𝑅𝑔𝐶𝑔

1+(𝜔𝑅𝑔𝐶𝑔)2 + Rgb

𝜔𝑅𝑔𝑏 𝐶𝑔𝑏

1+(𝜔𝑅𝑔𝑏 𝐶𝑔𝑏 )2 ……………………. (3)

Based on the above equations, the response peaks of the grains and grain boundaries are

represented by 1/ (2πRgCg) and 1/ (2πRgbCgb) respectively, and the peak values are

proportional to the associated resistance. Therefore, in the impedance spectra (Nyquist

plot), the higher frequency response corresponds to the grains and the lower one to the

grain boundaries [257]. The high- frequency semicircle is due to bulk effect that is the

parallel combination of bulk resistance (Rb) and bulk capacitance (Cb) along with a constant

phase element (CPE). The admittance Y of CPE is normally expressed as,

Y (CPE) = Ao (jω)n = A ω

n + j B ω

n……………………………………….(4)

Where A=AoCos (nπ/2), B = AoSin (nπ/2), Ao gives the magnitude of dispersion and 0 ≤ n

≤ 1 [258]. For ideal capacitor the vaue of n=1 and for ideal resistor n=0.

In order to analyze and interpret experimental data, it is essential to have an equivalent

circuit model that gives more information of the electrical properties. The circuit model

provides (i) the kind of impedances and their arrangement (series/parallel) in the sample,

(ii) confirmation of the experimental data to see the consistency of experimental value with

the proposed circuit, and (iii) compares the temperature dependence of the resistance and

capacitance values to that of simulated values [242]. According to Debye’s model, a

material having single relaxation time gives rise to a semicircle whose centre lies on Z' axis

but for non-Debye type of relaxation, the centre lies below Z' axis. The complex impedance

in such case is given by

Z*(ω) = Z'(ω) - iZ'' (ω) = R / [1+ (iω/ωo)1-α

]……………………………………(5)

Where α represents the magnitude of departure of the electrical response from ideal

condition and this can be found out from the location of the centre of the semicircles. The

value of α increases with rise in temperature. If α approaches 0 then Eqn. 5 gives rise to

Debye’s formalism. But in practice, an ideal Debye-like response is not generally realized.

Instead of getting a perfect semicircle, depressed semicircles are observed with their center

lying below the real axis. A non-ideal Debye type behavior is represented by introducing a


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constant phase element (CPE) with resistors and capacitors. A CPE has impedance which is

given by [259],

Z*CPE = [Ao(jω)

n]

-1 ………………………………………… (6)

Where Ao= A/cos(nπ/2), j=√-1. A and n are independent of frequency but depend on

temperature. Thus the CPE impedance has a Joncher’s power law dependence.

Figure 5.1 (i) Figure 5.1 (ii)

Figure 5.1 (iii) Figure 5.1 (iv)

Fig. 5.1 (i) and (ii) represent the ideal type of RC circuits for bulk and grain- boundary

contributions, whereas the actual data modeled circuits are shown in Fig. 5.1 (iii) and (iv)

with constant phase element (CPE) in parallel with RC-circuit.

5.3.1 Nyquist Plots

5.3.1 (a) (Bi1-xLix)(Fe1-xNbx)O3 (BLFN):

The complex impedance spectra (Nyquist plots) of (Bi1-xLix)(Fe1-xNbx)O3 (x=0.0-0.5) at

selected temperatures (400, 420, 440 and 460oC) are shown in Fig. 5.2. At lower

temperatures (< 200oC) these plots represent a straight line parallel to Y-axis indicating

insulating behavior of the samples at low temperatures. It is observed that the slope of the

lines decreases as temperature increases and then gradually bend towards real (Z') axis.


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Above 200oC, the tendency of formation of semicircular arcs is seen. The intercept of the

semicircular arc along Z' axis gives the value of bulk and grain boundary resistance which

decreases on increasing temperature (given in Table 5.1) showing increase in conductivity.

This shows negative temperature coefficient of resistance (NTCR) property of the material

which is normal behavior of semiconductors. These plots suggest that the dielectric

relaxation is of non-Debye type.

Fig. 5.2: Complex impedance spectra of (Bi1-xLix)(Fe1-xNbx)O3 at selected temperatures.

On increasing temperature the plots consist of depressed semicircular arcs. At high

temperatures two semicircular arcs could be seen which means at low frequencies a small

tail appears. The effect of temperature on the impedance behavior of the samples is clearly

visible from these plots. This additional semicircular arc may be attributed to the inter or

intra granular (bulk and grain boundary) phenomenon. The two semicircles in the figures

represent two types of relaxations: one corresponding to the grain (high frequency range)


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and the other grain boundary (low frequency range). In BFO, above 340oC, a second semi-

circular arc starts to appear showing the grain boundary effect whereas this effect in

x=0.1and x=0.3 is seen above 400oC.Thus the electrical properties of all the samples above

400oC can be represented by two parallel RC series connected in series. For all the samples

except x=0.2, the bulk property of the material dominates in the total value of the electrical

response. For x=0.2 it is clear that the low frequency peaks are with high Rgb values

attributing to the insulating grain boundaries and oxidized insulating surface layers [260].

More depressed semicircular arcs are observed for x=0.4. The values of Rb, Rgb, Cb and Cgb

at different temperatures are compared in Table 5.1. The bulk resistance of BFO is least

suggesting the most cation defects or oxygen vacancies present in the grains [261].

Table 5.1: Comparison of impedance parameters-Rb, Rgb, Cb and Cgb at different

temperatures of (Bi1-xLix)(Fe1-xNbx)O3.

Temperature

(oC)

Rb (Ω) Cb (F) Rgb (Ω) Cgb (F)

x=0.0 400 1.09 x103 4.42 x10

13 1.25 x10

1 1.00 x10

20

420 2.26 x102 5.83 x10

19 4.45 x10

2 6.03 x10

10

440 9.67 x101 1.00 x10

20 2.77 x10

2 5.47 x10

10

460 7.50 x101 1.00 x10

20 3.45 x10

2 3.70 x10

10

x=0.1 400 1.63 x105 2.12 x10

10 - -

420 7.61 x104 2.10 x10

10 2.95 x10

4 1.25 x10

8

440 4.88 x104 2.07 x10

10 5.49 x10

4 2.58 x10

8

460 2.94 x104 2.07 x10

10 3.29 x10

3 4.21 x10

7

x=0.2 400 6.74 x104 2.13 x10

10 8.38 x10

4 3.80 x10

9

420 4.30 x104 3.58 x10

10 1.40 x10

4 5.13 x10

10

440 2.90 x104 3.70 x10

10 9.10 x10

3 4.09 x10

10

460 1.00 x104 4.18 x10

10 6.56 x10

3 4.35 x10

10

x=0.3 400 4.54 x104 2.64 x10

10 - -

420 2.58 x104 2.29 x10

10 9.86 x10

3 5.93 x10

8

440 11.45 x103 4.03 x10

11 1.14 x10

4 3.86 x10

10

460 9.29 x103 1.08 x10

09 8.40 x10

3 3.71 x10

10

x=0.4 400 5.03 x103 1.72 x10

10 1.55 x10

3 5.39 x10

10

420 3.45 x103 1.61 x10

11 1.09 x10

3 5.21 x10

10

440 2.54 x103 1.63 x10

10 7.72 x10

2 5.38 x10

10

460 2.02 x103 1.52 x10

10 5.19 x10

3 6.14 x10

10

x=0.5 400 4.91 x104 2.16 x10

10 1.35 x10

4 9.69 x10

10

420 2.71 x104 2.74 x10

10 1.01 x10

4 9.31 x10

10

440 1.94 x104 2.21 x10

10 8.19 x10

3 8.43 x10

10

460 1.23 x104 2.33 x10

10 6.18 x10

3 7.14 x10

10


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5.3.1 (b) (Bi1-xNax)(Fe1-xNbx)O3 (BNFN):

The complex impedance spectra (Nyquist plots) of the (Bi1-xNax)(Fe1-xNbx)O3 (x=0.0-0.5)

at selected temperatures (300, 320, 340 and 360oC) are shown in Fig. 5.3. For x=0.0 (BFO)

incomplete single semicircular arcs appear up to 300oC. At 340

oC, a second semi-circular

arc starts to appear showing the grain boundary activity. As shown in the figure, above

340oC the impedance of BFO decreases by several orders which have also been given in

Table 5.2. At elevated temperatures though the second semicircle appears, it is poorly

resolved. For x=0.1 above 280oC these arcs take the shape of depressed semicircles. At

more elevated temperatures (i.e., above 340oC) poorly resolved second semicircular tails

starts to appear.

Fig. 5.3: Complex impedance spectra of (Bi1-xNax)(Fe1-xNbx)O3 at selected temperatures.

For x=0.2 the second semicircular arc starts to appear at relatively lower temperature

(280oC onwards) indicating increased conductivity at this concentration. For x=0.3 the


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second semicircular arc starts appearing 290oC onwards. This suggests that for x=0.2 and

0.3 grain boundary resistance becomes dominant at lower temperatures. For x=0.4 grain

boundary effect is seen from 320oC onwards. It is seen that for this compound with rise in

temperature the bulk resistances decreases gradually, and the grain boundary resistance

increases. For x=0.5 the grain boundary contribution to the overall impedance is seen from

320oC onwards. For all other compounds the grain and grain boundary resistances are

found to be decreasing with increase in temperature. Above 320oC (i.e., at higher

temperatures) it is seen that grain boundary resistance increases with increase in NaNbO3

content up to x=0.4 but decreases for x=0.5.

Table 5.2: Comparison of impedance parameters -Rb, Rgb, Cb and Cgb at different

temperatures of (Bi1-xNax)(Fe1-xNbx)O3.

Temperature

(oC)


x=0.0 300 6.33x105 2.229x10

-10 - -

320 9.68x104 2.183x10

-10 - -

340 1.098x104 3.268x10

-10 9.199x10

3 6.8x10

-10

360 4.503x103 2.799x10

-10 4.031x10

3 7.25x10

-10

x=0.1 300 1.079x105 1.602x10

-10 - -

320 4.497x104 1.566x10

-10 - -

340 1.214x104 1.487x10

-10 743 1.154x10

-9

360 3.08x103 1.0x10

-10 311.8 1.064x10

-9

x=0.2 300 4.448x104 4.508x10

-10 1.616x10

4 4.188x10

-10

320 1.926x104 2.281x10

-10 5.53x10

3 1.547x10-8

340 1.234x104 9.991x10

-10 5.943x10

3 3.878x10

-10

360 9.033x103 2.394x10

-10 1.882x10

3 1.922x10-8

x=0.3 300 4.741x104 4.047x10

-10 3.267x10

4 3.736x10

-10

320 3.844x104 1.955x10

-10 5.241x10

3 2.341x10

-8

340 2.065x104 3.322x10

-10 9.830x10

3 4.83x10

-11

360 1.645x104 1.971x10

-10 1.797x10

3 2.553x10

-8

x=0.4 300 1.159X105 1.774x10

-10 - -

320 7.269x104 1.482x10

-10 9163 1.621x10

-8

340 6.719x104 1.405x10

-10 10170 1.353x10

-8

360 3.616x104 1.396x10

-10 37480 3.03x10

-10

x=0.5 300 8.887x104 2.43x10

-10 - -

320 4.059x104 5.113x10

-10 2.63x10

4 4.638x10

-10

340 3.77x104 4.268x10

-10 1.563x10

4 5.676x10

-10

360 1.965x104 3.536x10

-10 5902 7.263x10

-10


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5.3.1 (c) (Bi1-xKx)(Fe1-xNbx)O3 (BKFN):

Fig. 5.4 shows the complex impedance spectra of (Bi1-xKx)(Fe1-xNbx)O3 (x=0.0-0.5) at

selected temperatures (300, 320, 340 and 360oC). On increasing temperature the semicircles

make smaller intercepts on the real Z’ axis showing decrease in impedance and supporting

NTCR behavior which has been clearly given in Table 5.3. With increase in temperature

the peak maxima of the plots decreases and the frequency shifts towards higher frequency

side. The poly-dispersive nature of dielectric relaxation can be explained using complex

impedance plots.

Fig. 5.4: Complex impedance spectra of (Bi1-xKx)(Fe1-xNbx)O3 at selected temperatures.

The low-frequency arcs at high temperatures are due to the presence of grain boundary that

is due to a parallel combination of grain boundary resistance (Rgb) and grain boundary

capacitance (Cgb). It is found that the measured and fitted data are in good agreement along

with an equivalent circuit. For x=0.1 the grain boundary effect is seen at 340oC and above


108

similar to that of BFO (x=0.0). For x=0.2 depressed semicircles with a tendency of

formation of second semicircle is clearly seen. However, the decreasing value of impedance

for some compounds (x=0.2 and 0.4) as shown in Figure 5.4 indicates the increase in

conductivity of the compounds. Moreover, for these two compounds well developed

semicircular arcs (starting at 260oC) indicate the contribution of bulk to the electrical

property of the compounds and the grain boundary effect for were seen at 300 and 320oC

respectively. The decrease in impedance value for these two compositions may be due to

their increased grain size which can be seen in the SEM micrograph. For x=0.1, 0.3 and 0.5

bulk resistances decreases with increase in KNbO3 content.

Table 5.3: Comparison of impedance parameters -Rb, Rgb, Cb and Cgb at different

temperatures of (Bi1-xKx)(Fe1-xNbx)O3.

Temperature

(oC)


x=0.0 300 6.33x105 2.229x10

-10 - -

320 9.68x104 2.183x10

-10 - -

340 1.098x104 3.268x10

-10 9.199x10

3 6.8x10

-10

360 4.503x103 2.799x10

-10 4.031x10

3 7.25x10

-10

x=0.1 300 1.706x105 1.39x10

-10 - -

320 1.139x105 1.432x10

-10 - -

340 3.986x104 2.201x10

-10 2.489x10

4 4.348x10

-10

360 2.374x104 1.311x10

-10 1.344 x10

3 1.108x10

-7

x=0.2 300 2.185x104 4.105x10

-10 1.132X10

4 4.786X10

-10

320 2.763x104 3.948x10

-10 7.377 x10

3 5.659x10

-10

340 1.173x104 2.321x10

-10 3.086 x10

3 1.219x10

-7

360 3.921 x103 3.737x10

-10 1.111 x10

3 6.796x10

-10

x=0.3 300 1.543x105 1.906x10

-10 - -

320 9.869x104 1.879x10

-10 - -

340 5.464x104 1.693x10

-10 - -

360 2.398x104 2.163x10

-10 1.534x10

4 7.055x10

-10

x=0.4 300 1.786x104 1.825x10

-10 - -

320 8.538x103 2.651x10

-10 5.539 x10

3 6.586x10

-10

340 7.449 x103 2.794x10

-10 5.432 x10

3 5.973x10

-10

360 6.616 x103 2.698x10

-10 4.682 x10

3 6.405x10

-10

x=0.5 300 4.21x104 1.156x10

-10 - -

320 2.736x104 1.073x10

-10 - -

340 2.022x104 1.113x10

-10 - -

360 1.194x104 1.301x10

-10 3.850 x10

3 6.303X10

-10


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5.3.1 (d) Ceramic-polymer composites:

The ionic conductivity has been determined from ac impedance analysis. Fig. 5.5 exhibits a

typical impedance spectrum of pure PVDF, BFOP, BLFNP, BNFNP and BKFNP

composites at different temperatures in the frequency range of 1 kHz-1 MHz. At room

temperature these plots represent straight lines parallel to the ordinate indicating a high

order insulating behavior of the samples. With increase in temperature the curves show a

tendency to bend towards the abscissa to form semicircles with their centers lying below

real axis. This indicates a distribution of relaxation time with a deviation from ideal Debye-

type behavior. It can be clearly noticed that the values of Rg decreases with rise in

temperature for all samples which indicates the NTCR character of the samples.

Fig. 5.5: Complex impedance plots of pure PVDF, BFOP (BFO+PVDF), BLFNP

(BLFN+PVDF), BNFNP (BNFN+PVDF) and BKFNP (BKFN+PVDF) composites at

different temperatures.

It is seen that the Rg value is maximum for BLFNP. By fitting the impedance response with

one given by an appropriate equivalent circuit we can obtain information about the resistive


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and capacitive responses of the components. The component in the complex impedance

spectra can be assigned to a RC-parallel circuit response which indicates the contribution of

grains in pure PVDF as well as its composites. No other relaxation mechanism such as

grain boundary or electrode effect could be identified in the samples studied in this

frequency range. PVDF shows more insulating nature as compared to other compounds. At

120oC there is a tendency of forming a semicircular arc for all the composites.

5.3.2 Variation of Z' with frequency


The temperature and frequency dependent ac conductivity of the materials can be explained

in terms of the variation of the real part of impedance with frequency. Fig. 5.6 shows the

variation of Z' with frequency at different temperatures (400, 420, 440 and 460oC) of (Bi1-

xLix)(Fe1-xNbx)O3 (x=0.0-0.5).

Fig. 5.6: Variation of Z' with frequency of (Bi1-xLix)(Fe1-xNbx)O3 at selected temperatures.


111

It is seen that in the low-frequency region Z' shows sigmoidal variation as a function of

frequency followed by a saturation in the high-frequency region (> 10 kHz) irrespective of

temperature for all the plots. With rise in temperature spreading of the dispersion region in

the high-frequency region is observed. The decrease in value of Z' with increase in

frequency may be due to a slow dynamics relaxation process in the material which may be

attributed to space charges [262]. At higher temperatures in the low-frequency region

plateau is observed which may be related to frequency invariant electrical property of the

materials. At higher frequencies the real part of impedance merges suggesting a possible

release of space charge, and consequently lowering the barrier in the ceramic samples [263,

264]. Z' decreases with rise in temperature. This indicates enhancement of ac conductivity

exhibiting negative temperature coefficient of resistance (NTCR) behavior similar to that of

semiconductors. For x=0.0 a low-frequency dispersion followed by a plateau region is seen

and these curves finally merge above 1000 kHz. But for x=0.2 plateau region was not

observed. The low-frequency dispersion decreases from x=0.1 to x=0.4 but then again

increases for x=0.5. With increasing temperature defects interacted and had a significant

influence on the conducting process.

5.3.2 (b) (Bi1-xNax)(Fe1-xNbx)O3 (BNFN):Fig. 5.7 shows the frequency dependence of Z' at

different temperatures (300, 320, 340 and 360oC) of (Bi1-xLix)(Fe1-xNbx)O3 (x=0.0-0.5).

From the plots it is clear that the value of Z' decreases with rise in both temperature and

frequency. These plots indicate an increase in conduction with temperature (NTCR

behavior). The plateau in the low frequency region indicates the presence of relaxation

process in the materials. More dispersion in low-frequency region is seen for x=0.0 (BFO)

and least for x=0.4. Above 300oC, this low-frequency plateau and shifting of merger of Z'

towards high frequency side are seen. The low-frequency dispersion gradually decreases

with increases in NaNbO3 content up to x=0.2.


112

Fig. 5.7: Variation of Z' with frequency of (Bi1-xNax)(Fe1-xNbx)O3 at selected temperatures.

5.3.2 (b) (Bi1-xKx)(Fe1-xNbx)O3 (BKFN):

Fig. 5.8 shows the variation of Z’ with frequency at different temperatures (300, 320, 340

and 360oC) of (Bi1-xKx)(Fe1-xNbx)O3 (x=0.0-0.5). They show a monotonous decrease of Z’

with rise in frequency and then attains a constant value irrespective of temperature for all

the plots which may possibly be due to increase in the ac conductivity with rise in

frequency. The space charge has less time to relax and so recombination is faster. Hence

the space charge polarization reduces in the high frequency region leading to a merger of

curves at higher frequency. Low frequency plateau and shifting of merger of Z' towards

high frequency side is seen is seen for all the samples above 300oC except for x=0.0.


113

Fig. 5.8: Variation of Z' with frequency of (Bi1-xKx)(Fe1-xNbx)O3 at selected temperatures.

5.3.3 Variation of Z'' with frequency


In order to make a deeper understanding of the space charge effect and the relaxation

processes, the frequency dependence of imaginary part of impedance with frequency at

different temperatures (300, 320, 340 and 360oC) have been studied and the results are

shown in Fig. 5.9 for (Bi1-xLix)(Fe1-xNbx)O3 (x=0.0-0.5). At high temperatures the curves

exhibit peaks. A single peak (Z''max) which is temperature dependent is seen above 10 kHz

for all the samples. These peaks shift towards higher frequencies on increasing temperature

and a broadening in the curves is observed with the decrease in peak height. This

broadening suggests spreading of relaxation time (i.e., the existence of a temperature

dependent electrical relaxation phenomenon in the compound) [265].


114

Fig. 5.9: Variation of Z'' with frequency of (Bi1-xLix)(Fe1-xNbx)O3 at selected temperatures.

It indicates a thermally activated dielectric relaxation process in the materials and shows

that with temperature bulk resistance reduces. But at low temperatures (not shown) these

peaks have not been found. This may be due to the weak current dissipation in the material

or may be beyond the experimental range of frequency [266]. The dispersion curves appear

to merge at higher frequencies irrespective of LiNbO3 content in BFO due to release of

space charges [267, 268]. This plot suggests an enhancement in the net impedance of

LiNbO3 modified BFO thereby increasing the barrier to the mobility of charge carrier in the

materials [261].


The loss spectrum of (Bi1-xNax)(Fe1-xNbx)O3 (x=0.0-0.5) at different temperatures (300,

320, 340 and 360oC) are shown in Fig. 5.10. For x=0.0 no peak appears before 280

oC and


115

it appears in terms of very broad and diffused peak at elevated temperatures. For x=0.2

another small peak appears on low-frequency side at 300oC which gradually becomes

prominent at elevated temperatures. The first peak in the low-frequency region is correlated

to the grain boundary contribution while the second one in the high-frequency region is

correlated with the bulk response. The shifting of the peaks towards higher temperature

may be due to reduction in the bulk resistance.

Fig. 5.10: Variation of Z'' with frequency of (Bi1-xNax)(Fe1-xNbx)O3 at selected

temperatures.

The merger of the entire high-frequency end (>100 kHz) indicates the depletion of space

charges at those frequencies, since these curves basically denote the ac losses of the

samples. But for x=0.4 the curves at high frequency merge at >10 kHz. For x=0.4 the

dispersion in low frequency region is least and maximum for x=0.0.


116


The loss spectrum of (Bi1-xKx)(Fe1-xNbx)O3 (x=0.0-0.5) is shown in Fig. 5.11 for selected

temperatures (300, 320, 340 and 360oC). At lower temperatures (≤ 250

oC), the value of Z''

falls monotonically on increasing frequency without any peak in the investigated frequency

range. It indicates that the samples may not relax at lower temperatures due to presence of

polarization field in the lattice. Above 250oC, the broad and asymmetric peaks start to

appear in the low-frequency region.

Fig. 5.11: Variation of Z'' with frequency of (Bi1-xKx)(Fe1-xNbx)O3 at selected temperatures.

For x=0.2 onset of Z’’ peak (Z''max) has been observed above 250oC. But for other

compounds these peaks are seen above 280oC. The peak position shifts to higher

frequencies as temperature increases. This shift occurs at maximum frequency for all the

samples indicating an active conduction associated with dipole reorientation. This offset is

characteristic of high-permittivity systems as well as of localized-conduction electronics


117

due to grains (bulk) and grain-boundary (interface) effects for all the compounds. The value

of Z''max shows a decreasing trend on increasing temperature indicating an increasing loss in

the resistive property of the samples. Moreover the peaks in the Z'' spectra occur in the

region of frequency dispersion in Z' spectra. For x=0.4 the dispersion in low frequency

region is least. The peaks are more sharp and intense in comparison to the rest of the

compounds in this series.

5.4 Complex modulus

Complex modulus analysis is a convenient technique which determines analyzes and

interprets the dynamical aspects of transport phenomena (i.e., parameters such as carrier/ion

hopping rate, conductivity relaxation time etc.). Another advantage of this technique is that

it can discriminate against electrode polarization and grain boundary conduction processes.

The combined analysis of impedance and modulus spectroscopic plots to rationalize the

dielectric properties was suggested by Sinclair and West [242, 269]. Complex impedance

plots are useful in determining the dominant resistance in the sample whereas complex

modulus plots are useful in determining the smallest capacitance. Hence the modulus plots

are used to separate the components with similar resistance but different capacitance. The

Nyquist plot (M'' vs. M') gives rise to a semicircle, and the smallest one corresponds to the

highest capacitance. Also the absence of subsequent semicircles in the modulus plots

neglects the electrode effects [270]. The electrical properties of materials showing a single

circular arc in complex modulus plane are defined by the parallel combination of grain

capacitance (C) and resistance (R). Complex electric modulus can be calculated from the

impedance data using the following relation:

M* (ω) =

1

𝜀∗ = M'(ω) + i M''(ω)

= M∞ 1− exp(−𝑖𝜔𝑡)𝑑𝜙 (𝑡)

𝑑𝑡

∞

0𝑑𝑡

where, M∞ = 1/ε∞, ε∞ is the limiting high-frequency real part of permittivity, and the

function 𝜙(t) is a relaxation function or Kohlrausch-Williams-Watts (KWW) function

[242].


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5.4.1 Nyquist plots


Fig. 5.12 shows the complex modulus spectrum (M' ~ M'') of (Bi1-xLix)(Fe1-xNbx)O3 (x=0.0-

0.5) at selected temperatures (300, 320, 340 and 360oC). These plots do not form exact

semicircles rather they form depressed semicircles with their centers positioned below the

x-axis. This indicates the spreading of relaxation time and hence non-Debye type of

relaxation in these compounds.

Fig. 5.12: complex modulus spectra of (Bi1-xLix)(Fe1-xNbx)O3 at some selected

temperatures.

The Nyquist plots of electric modulus justify the poly-dispersive nature for the dielectric

relaxation at lower frequencies. The appearance of asymmetric semicircular arcs indicates

the electrical relaxation phenomenon in the materials. The plots show a semicircle with a


119

tendency of formation of another semicircle for all the compounds except x=0.4. The

intercept on the real axis indicates the total capacitance contributed by the grain and grain

boundaries. Appearance of single arc for x=0.4 indicates negligible contribution of the

grain boundary effect to the polarization in the temperature range. The grain boundary

effect is more prominent for x=0.2. The modulus loss profiles are collapsed into one master

curve suggesting temperature independent relaxation time (with different mean time

constants).


Fig. 5.13 shows the complex modulus spectrum (M' ~ M'') of (Bi1-xNax)(Fe1-xNbx)O3

(x=0.0-0.5) at selected temperatures (300, 320, 340 and 360oC). For x=0.0 and 0.1 the total

capacitances increases with increase in temperature.

Fig. 5.13: complex modulus spectra of (Bi1-xNax)(Fe1-xNbx)O3 at some selected

temperatures.


120

For these compounds depressed semicircular arcs appear up to 340oC and above this

temperature, they are incomplete at the high frequency side. For x=0.2, 0.3 and 0.5, the

arc/semicircles overlap with each other at various temperatures implying the presence of

relaxation phenomenon in these compounds. For x=0.4 up to 340oC the semicircles overlap

but at 360oC there is a tendency of formation of another semicircle. The intercept of first

semicircle on the real axis gives the capacitance contributed by grain, and the second

semicircle to the contribution from grain boundary.


The complex modulus spectra of (Bi1-xKx)(Fe1-xNbx)O3 (x=0.0-0.5) at some selected

temperatures (300, 320, 340 and 360oC) are shown in Fig. 5.14. For x=0.1 and 0.2

depressed semicircular arcs are seen with intercepts of the arcs on M' axis which decreases

with rise in temperature indicating increase in capacitance.

Fig. 5.14: complex modulus spectra of (Bi1-xKx)(Fe1-xNbx)O3 at some selected

temperatures.


121

This indicates the spreading of relaxation time with different mean time constant and non-

Debye type of relaxation in the materials. For x=0.1 there is a tendency of formation of

second semicircle confirming the active role of grain boundary capacitance in the

conduction mechanism. The broadening observed for all the samples in the semicircular

arcs in the complex modulus plots suggest the involvement of both the grain and grain

boundary towards electrical capacitance in the ceramic samples. For x=0.3, 0.4 and 0.5 the

arcs perfectly overlap which indicates several relaxations occurring in these materials.

5.4.2 Variation of M' with frequency


Another formalism of data presentation is the complex electric modulus, M* formalism.

The frequency dependence of real part of electric modulus (M') at selected temperatures

(400, 420, 440, and 460oC) of (Bi1-xLix)(Fe1-xNbx)O3 (x=0.0-0.5) is shown in Fig. 5.15. It is

found that at low frequencies the value of M' is found to be very low (or nearly equal to

zero). A continuous dispersion with increase in frequency is observed and finally these

curves have a tendency to saturate at a maximum asymptotic value designated at M∞ in the

high-frequency region irrespective of temperature. These phenomena can be related to lack

of restoring force governing the mobility of charge carriers under the action of an induced

electric field [271]. But this saturation of M' at high frequency region is not seen for x=0.0

and 0.4. This confirms elimination of electrode effect in the materials. M' is also found to

decrease with the increase in temperature which indicates a temperature dependent

relaxation process in the materials. It is also found that the dispersion region shifts towards

higher frequency side suggesting the long-range mobility of charge carriers. The plateau

region (or its tendency to saturate) observed at higher frequencies suggests about the

frequency invariant electrical properties of the materials.


122

Fig. 5.15: Variation of M' with frequency at selected temperatures of (Bi1-xLix)(Fe1-

xNbx)O3.


Fig. 5.16 shows the frequency response of real part of electric modulus (M') of (Bi1-

xNax)(Fe1-xNbx)O3 (x=0.0-0.5) at different temperatures (300, 320, 340, and 360oC). The

sigmodial increase in the value of M' with frequency approaches ultimately to a value of

M∞ for all temperatures which indicates short-range mobility of carriers (especially ions).

Well dispersed curves are observed in all the compounds except for x=0.4. This dispersive

nature of the compounds implies that a well defined relaxation mechanism occurs over

several decades of frequency at all these temperatures.


123

Fig. 5.16: Variation of M' with frequency at selected temperatures of (Bi1-xNax)(Fe1-

xNbx)O3.


Fig. 5.17 shows the frequency response of real part of electric modulus (M') of (Bi1-

xKx)(Fe1-xNbx)O3 (x=0.0-0.5) at different temperatures (300, 320, 340, and 360oC). The

plots clearly show very low value of M' in the low-frequency region with a continuous

dispersion in the high-frequency region for all temperatures. These curves have a tendency

to saturate at a maximum asymptotic value designated as M∞ in the high-frequency region.

With increase in frequency each ion moves a shorter path of electric field till the electric

field changes so rapidly that the ions only rattle within the confinement of their potential

energy wells. This indicates the long-range mobility of charge carriers [272].


124

Fig. 5.17: Variation of M' with frequency at selected temperatures of (Bi1-xKx)(Fe1-xNbx)O3.

As temperature increases the value of M' decreases and the dispersion region shifts to

higher frequency side indicating a thermally activated relaxation process. Dispersion

decreases with increase in KNbO3 content except for x=0.4. For x=0.3 at very high

frequency the curves tend to merge.

5.4.3 Variation of M'' with frequency


Fig. 5.18 shows the frequency dependence of imaginary part of electric modulus (M”) at

selected temperatures (400, 420, 440, and 460oC) of (Bi1-xNax)(Fe1-xNbx)O3 (x=0.0-0.5).

The modulus spectra exhibit well-resolved asymmetric peaks. On decreasing frequency

these peaks indicate that there is a transition from short range to long-range mobility.


125

Fig. 5.18: Variation of M'' with frequency at selected temperatures of (Bi1-xLix)(Fe1-

xNbx)O3.

The lower-frequency side (below M''max) of the peak represents the range of frequencies in

which the ions are capable of moving long distances from one site to the neighboring site

by hopping. In the high-frequency region (above M''max), the ions are confined to their

potential wells and can execute only localized motion [273, 274]. On increasing the

temperature the peaks shifts towards higher frequencies side confirming the thermally

activated nature of relaxation time. Some of the main reasons for such broadness in the

spectra are: (i) the random orientation of anisotropically conducting species, (ii) the

presence of phases of more than one composition or structure [275] (iii) distribution of

relaxation times due to local defects. For compounds except 0.4, the height of M''max

increases gradually with rise in temperature.


126


Fig. 5.19 shows the frequency dependence of imaginary part of modulus (M'') at various

temperatures (300, 320, 340, and 360oC) of (Bi1-xNax)(Fe1-xNbx)O3 (x=0.0-0.5). These

asymmetric peaks shift towards higher frequency which indicates correlation between

motions of mobile ion charges [266]. The broadening of peak indicates the distribution of

relaxation times indicating relaxation of non- Debye type.

Fig. 5.19: Variation of M'' with frequency at selected temperatures of (Bi1-xNax)(Fe1-

xNbx)O3.

For x=0.1, 0.2 and 0.5 the height of decreases with rise in temperature. The constancy of

peak height in the modulus plot for x=0.3 at different temperatures suggests the invariance

of the dielectric constant and distribution of relaxation times with temperature [276]. The

distribution is due to irregularities in the lattice structure near the defect sites. For


127

compounds except x=0.0 and 0.4, the height of M''max decreases slightly with rise in

temperature.


Fig. 5.20 shows the frequency dependence of imaginary part of electric modulus (M'') of

(Bi1-xKx)(Fe1-xNbx)O3 (x=0.0-0.5). The peaks are clearly resolved and appear at unique

frequency at various temperatures. It is clearly shown that the M''max shifts towards higher

relaxation frequency with the temperatures increases.

Fig. 5.20: Variation of M'' with frequency at selected temperatures of (Bi1-xKx)(Fe1-

xNbx)O3.

This behavior (non-Debye type) suggests that the relaxation process is thermally activated

in which hopping of charge carriers with small polarons is dominated intrinsically [277].

The low-frequency side of the M''max represents the range of frequencies in which charge

carriers can move over a long distance and the high frequency represents localized motion.


128

The region where peak occurs indicates transition from long-range to short-range mobility

with increase in frequency [278]. The peak height in the modulus plot decreases with rise in

temperature for x=0.1, 0.2, 0.3 and 0.4. For x=0.5 the height of M''max increases with rise in

temperature. This type of effect has also been seen in some real ionic conductors [279].

This steady increase in the values of M''max as a function of temperature indicates decrease

in capacitance [280].

5.4.4 Normalization of modulus spectra


Fig. 5.21 shows the normalized plot of (Bi1-xLix)(Fe1-xNbx)O3 (x=0.0-0.5) versus at various

temperatures (400, 420, 440 and 460oC). This plot is known as modulus master curve

which enables us to have an insight into the dielectric process occurring in the materials as

a function of temperature. It is seen that all the curves irrespective of temperature coalesced

into a single master curve. This coincidence indicates temperature independent behavior of

the dynamical processes occurring in the material. This indicates that the distribution

function for relaxation times is nearly temperature independent with non-exponential type

of conductivity relaxation.

These plots may be analyzed in terms of non-exponential decay function or Kohlrauseh–

Williams–Watts (KWW) parameter by the expression:

Φ (t) = exp [(-t/τm) β]; (0 < β < 1)

Where Φ (t) is the time evolution of an electric field, and τm is the characteristic relaxation

time. For an ideal Debye single relaxation, β = 1, which indicates that the interaction

between the ions is maximum [281].

A non-exponential type relaxation suggests the possibility of ion migration that takes place

via hopping [282]. The comparison of impedance and modulus plots are helpful in

rationalizing the bulk response in terms of dielectric (localized) relaxation and conductivity

(non-localized) relaxation process.


129

Fig. 5.21: Plot of M''/ M''max with f/fmax of (Bi1-xLix)(Fe1-xNbx)O3 at selected temperatures.


Fig. 5.22 shows the normalized plot of (Bi1-xNax)(Fe1-xNbx)O3 (x=0.0-0.5) versus at various

temperatures (300, 320, 340 and 360oC). All the peaks collapse into one master curve at

different temperatures suggesting temperature independent distribution of relaxation time.

For x=0.0, 0.1, 0.4 and 0.5 a small deviation from scaling curve both at higher and lower

frequencies have been observed. But for x=0.0 this deviation in the scaling curve is not

observed above 380oC (Figure 5.21). The observed deviation at higher frequency indicates

a change in the dynamic properties of the materials. The low-frequency deviation mostly

originates from some interfacial effects.


130

Fig. 5.22: Plot of M''/ M''max with f/fmax of (Bi1-xNax)(Fe1-xNbx)O3 at selected temperatures.

It indicates that the failure of merging into a single master curve for these compounds may

be due to the change of concentration of charge carriers of the materials [283]. The above

results show that the relaxation dynamics of oxide ions are temperature independent but

they depend on the structure and/or the concentration of charge carriers.


Fig. 5.23 shows scaling behavior of imaginary part of modulus (M'') with frequency at

different temperatures (300, 320, 340 and 360oC) of (Bi1-xNax)(Fe1-xNbx)O3 (x=0.0-0.5). All

the curves superimpose into a single master curve indicating that all the dynamic processes

occur at different frequencies.


131

Fig. 5.23: Plot of M''/ M''max with f/fmax of (Bi1-xKx)(Fe1-xNbx)O3 at selected temperatures.

The dielectric processes occurring in the material can be investigated via these plots [263].

The coincidence of all the peaks at different temperatures exhibits temperature independent

behavior of the dynamic processes occurring in the materials [279]. It is observed that all

the peaks of the pattern appear at unique frequency for different temperatures. Small

deviation from scaling at low and high frequency is seen for x=0.1, 0.3 and 0.4.

5.5 Summary

On the basis of above results the following conclusions have been drawn:

The impedance spectroscopy data provides the contribution of both grain and grain

boundary on the electrical properties of the materials. But for PVDF and ceramic-


132

polymer composites no grain boundary effect was observed in the studied

temperature and frequency range.

The impedance pattern suggests a decrease in bulk resistance with rise in

temperature. Negative temperature coefficient of resistance (NTCR) behavior of the

materials indicates semiconducting nature of the materials.

The equivalent circuit models provide an insight of the structure-property

relationship of materials.

The combined analysis of impedance with modulus spectroscopy provides

important information about the contribution to the relaxation process of different

micro-regions in the poly-crystalline ceramics, such as grains and grain boundaries.

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