4. Experimental setup, measurements and results

Development of equivalent circuit model for representing the ceramic system BaFexTi1-xO3 (x = 0.05), prepared in our laboratory, is now described as an illustration. Sample was prepared by solid state synthesis method by taking BaCO3 (Merck 99.5%), Fe2O3 (Merck 99.5%) and TiO2 (Merck 99.5%) in appropriate amounts, mixing in acetone medium for 6 hours and calcining at 1100°C for 6 hours. The calcined powder was mixed with small amount of PVA binder and pressed into disclike (dia 12 mm, thickness 1.5 mm) pellets using uniaxial hydraulic press with 60 kN pressure. These pellets were sintered in an electrical furnace (Lenton, made in Germany) where, first, binder was removed by raising the temperature to 500°C and holding for 2 hours and then increasing the temperature to 1250°C at 5°C/min and maintaining there for 10 hours followed by cooling it to room temperature. Xray diffraction analysis revealed that the sample contained tetragonal and hexagonal phases in equal amounts. Impedance measurements were carried out as function of frequency (20 Hz to 1 MHz) at temperatures from 300 to 650 K. For this, the pellets were polished using emery papers of grade 1/0 and 2/0 and electroded on both sides using silver paste and cured at 600°C for 15 minutes.

The sample holder used to house the sample is schematically shown in Figure 7(a). The spectroscopic and complex plane plots for M<sup>0</sup> and M<sup>00</sup> are shown in Figure 7(b,c). The corresponding Z plots are not shown for brevity. The way an equivalent circuit model representing the data was developed is now described. A quick look at the M<sup>00</sup> vs. M<sup>0</sup> plot shown in Figure 7(b) reveals that there is no shift in the graph as well as no steeply rising high-frequency branch. Similar behaviour was seen in the Z plots also. Therefore, following the tips presented in Section 2, presence of series resistance or capacitance is ruled out. The plot is not a clear semicircular arc but is slightly depressed indicating the presence of more than one charge transfer processes in the system. As the sample contains two phases, we have a system comprising two types of grains, grain boundary and contact electrode interface. Therefore, an equivalent circuit model comprising four parallel RC's connected in series, where two RC's say R1C1 and R2C2 represent the two phases, R3C3 represents the grain boundary and R4C4 correspond to the sample-electrode interface, seems to be a plausible model. If we assume that R1C1 < R2C2 < R3C3 < R4C4 then, since electrode responses appear at low frequencies [1], R1C1 and R2C2 may be assigned to grains, R3C3 to grain boundary and R4C4 to electrode interface. The individual contributions from these RC's may be depicted by drawing tentative arcs (as shown by dotted lines in Figure 7(b)) making intercepts on the M<sup>0</sup> axis at C0/ C4, C0/C4 + C0/C3, C0/C4 + C0/C3 + C0/C2 and C0/C4 + C0/C3 + C0/C2 + C0/C1. As the plot is very symmetric, the intercepts and hence C's may be taken to be equal. By noting the values of intercepts from the graph, taking the value of C0 as 0.6832 � <sup>10</sup>�12, noting the frequencies where the tentative arcs would peak and using the relation ωRC = 1, the values of R' and C's are estimated as R1 = 362 Ω, R2 = 1087 <sup>Ω</sup>, R3 = 32,613 <sup>Ω</sup>, R4 = 9944 <sup>Ω</sup> and C1 = C2 = C3 = C4 = 4.88 � <sup>10</sup>�<sup>10</sup> F. By using these as initial guesses, the values of the components were obtained accurately by running the CNLS program IMPSPEC.BAS developed by one of the authors [22] and being regularly used by us. These values were R1 = (0.77 � 0.01) kΩ, C1 = (0.38 � 0.01) nF, R2 = (3.43 � 0.02) kΩ, C2 = (0.44 � 0.01) nF, R3 = (10.87 � 0.09) kΩ, C3 = (1.08 � 0.06) nF, R4 = (15.36 � 0.26) kΩ and C4 = (5.81 � 0.09) nF. The M<sup>0</sup> and M<sup>00</sup> values corresponding to the fitted RC's are also shown in Figure 7(c).

Use of different models for impedance spectroscopy of few other ceramics is briefly described now. Analysis of data for SrTiO3 borosilicate glass ceramics having

and the CPE behaves like an ideal register of value 1/A0 [6]. The values of Z<sup>0</sup> and Z<sup>00</sup>

/(R1 + R2), (c) M″/(C0/C1 + C0/C2 vs M<sup>0</sup>

[1/(R1 + R2)] and (e) <sup>ɛ</sup>″/[(1/C0){C1(R1/(R1 + R2))<sup>2</sup> + C2(R2/(R1 + R2))<sup>2</sup>

(a) Equivalent circuit model containing series combination of parallel R1C1, parallel R2C2 and CPE.

Ceramic Materials ‐ Synthesis, Characterization, Applications and Recycling

R2 1 þ ð Þ ωC2R2

ωC2R2 R2 1 þ ð Þ ωC2R2

A model comprising series combination of parallel R1-CPE1 and parallel R2-CPE2 is one of the models very widely used to represent the behaviour of a ceramic when the impedance plots have two depressed arcs. The reader is referred to [9] where

<sup>2</sup> þ

<sup>2</sup> þ

1 A0ωψ 

}] for R2/R1 = 2, R2C2/R1C1 = 1, 5, 10, 100, A0 = 6 � <sup>10</sup>–<sup>7</sup> and

1 A0ωψ  cos

sin ψπ 2 

ψπ 2 

/(C0/C1 + C0/C2, (d) Y″/[1/

}] vs ɛ<sup>0</sup>

(9)

/[(1/C0)

(10)

for the model shown in Figure 6(a) are given as [9, 21].

<sup>2</sup> þ

<sup>2</sup> þ

<sup>Z</sup><sup>0</sup> <sup>¼</sup> R1

Figure 6.

θ = 40.

88

(R1 + R2)] vs Y<sup>0</sup>

Normalized (b) Z″/(R1 + R2) vs Z<sup>0</sup>

{C1(R1/(R1 + R2))2 + C2(R2/(R1 + R2))<sup>2</sup>

<sup>Z</sup><sup>00</sup> <sup>¼</sup> <sup>ω</sup>C1R1 R1 1 þ ð Þ ωC1R1

this has been discussed by the authors in detail.

1 þ ð Þ ωC1R1

Ceramic Materials ‐ Synthesis, Characterization, Applications and Recycling

For separating out contact contributions, impedance spectroscopic studies by repeating the measurements with changed electrodes have been reported [36]. Measurements by using samples of different thicknesses might also be useful. Various sources of errors in impedance measurements have been discussed in [6].

Impedance Spectroscopy: A Powerful Technique for Study of Electronic Ceramics

DOI: http://dx.doi.org/10.5772/intechopen.81398

Some equivalent circuit models most useful for impedance spectroscopic studies of electronic ceramics and their simulated immittance behaviour are discussed. In order to facilitate prompt development of equivalent circuit models, few extremely helpful steps have been summarised. A comparison of the experimental plots with simulated ones provides a clue for inclusion of certain lumped components in the model, e.g. a right shift in the Z<sup>00</sup> vs. Z<sup>0</sup> plot indicates presence of series resistance and a shift in M<sup>00</sup> vs. M<sup>0</sup> plot indicates presence of series capacitance in the model. The models are not unique. The most appropriate model may be arrived at by looking at the immittance plots in more than one formalisms (such as Z and M ) for all the experimental parameters such as composition, temperature etc, taken together, and considering the possibilities of processes present/dormant, emerging or dominating in the system as some variables such as temperature are altered. The procedure of impedance spectroscopic modelling is illustrated by analysing the impedance data on the ceramic system BaFexTi1-xO3 (x = 0.05) containing two

The financial support received by one of the authors (SP) from IIT(BHU) in the

, Om Parkash<sup>1</sup> and Lakshman Pandey<sup>2</sup>

form of Teaching Assistantship is gratefully acknowledged.

\*, Devendra Kumar<sup>1</sup>

1 Department of Ceramic Engineering, IIT(BHU), Varanasi, India

\*Address all correspondence to: spandey.rs.cer12@itbhu.ac.in

provided the original work is properly cited.

2 Department of Physics and Electronics, Rani Durgavati University, Jabalpur, India

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

5. Conclusion

phases.

Acknowledgements

Author details

Shukdev Pandey<sup>1</sup>

91

#### Figure 7.

(a) Experimental Setup for Impedance Measurements. Experimental and fitted values of (b) M″ vs M<sup>0</sup> for BaFexTi1–xO3 (x = 0.05) at 600 K and (c) M<sup>0</sup> , M″ vs log f for BaFexTi1–xO3 x = 0.05 at 600 K.

steeply rising low-frequency branch in Z<sup>00</sup> vs. Z<sup>0</sup> plots has been done by Pandey et al. [17]. Ba1-xLaxTi1-xCoxO3 (x ≤ 0.20) ceramic showed a shifted arc in Z" vs. Z' plot and was analysed by using equivalent circuit model involving a series resistance [19]. In this system, presence of two PTCR components, one due to grain boundary as usual and the other in grains was observed by impedance spectroscopy. Study of semiconducting BaTiO3 was carried out by using a model containing series resistance [18]. Ceramic system Ca1-xYxTi1-xCoxO3 could be represented by an equivalent circuit model containing two parallel RC's and a CPE connected in series [21]. Models useful for representing magnetic ceramic are not covered in this paper. A detailed analysis of models involving inductive component L is available in [11] where magnetic ceramic YIG has been studied by using impedance spectroscopy.

While using the impedance spectroscopy, it is important to keep in mind that the observed impedance values are those what the impedance analyser sees at its input points. These values include the contributions from the connecting leads, cables, sample holder and electrodes. It has been found that the electrode-sample contact behaviour changes with the nature of the sample [15]. The sample-electrode contact effects have been studied in detail in [23–35]. A method has been proposed for removing the sample holder contributions [23], which involves doing the complete impedance spectroscopic analysis without the sample, finding an empirical functional relation for the impedance behaviour and then subtracting this from the results obtained with sample placed in the sample holder. Presently available analysers provide with a lead correction step before an actual measurement is done. For separating out contact contributions, impedance spectroscopic studies by repeating the measurements with changed electrodes have been reported [36]. Measurements by using samples of different thicknesses might also be useful. Various sources of errors in impedance measurements have been discussed in [6].
