Abstract

Electronic ceramics are technological materials having a vast variety of applications such as actuators and sensors, computer memories, electrically controlled microwave tuning devices for RADAR, etc. and are playing key role in electronics industry today. An electronic ceramic component can be visualised as grain-grain boundary-electrode system. Impedance spectroscopy is being widely used to separate out contributions of these to the overall property of a ceramic. This involves equivalent circuit models. To facilitate development of suitable equivalent circuit models and obtain values of the components, some most useful circuits with their simulated behaviour are presented. Steps highly useful in the modelling process are summarised. The procedure of impedance spectroscopy is illustrated by analysing the impedance data of the ceramic system BaFexTi1-xO3 (x = 0.05) containing two phases.

Keywords: electronic ceramics, impedance spectroscopy, equivalent circuit models, CPE, CNLS, grain-boundaries, ceramic-electrode interface

### 1. Introduction

Ceramics are inorganic non-metallic solids that have been processed and shaped by heating at high temperatures. Modern ceramics include oxides, nitrides, carbides, etc. and constitute a large fraction of technologically useful materials at present. Ceramics, whose electrical, magnetic, or optical properties are used in devices are broadly termed as electro-ceramics and are being heavily used as electrical insulators, TV baluns, mobile antennas and speakers, substrates for electronic circuits, computer memories, magnetic recording heads, high-temperature heating elements, cryogenic sensors, microwave tuning devices for RADAR applications, etc. [1–3]. The as-prepared ceramics are usually in powder form that are processed and shaped for device applications by subjecting to suitable sintering procedure. A ceramic piece thus consists of small crystallites called grains that are joined together in random orientations. The joining region called the grain boundary has, due to mismatch, strained bonds. Properties of grain boundary, therefore, are different from those of grains and highly depend upon the processing variables such as heating/cooling rates, presence of external fields, atmosphere, etc. By changing these variables and starting ingredients, behaviour of grains and grain boundary

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*Ceramic Materials - Synthesis, Characterization, Applications and Recycling*

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may be altered. It is the interplay between the grain and grain boundary behaviour that bestows ceramics with some very useful properties. Ceramics prepared by controlled crystallisation of glass are called glass-ceramics where the grain boundary consists of the uncrystallised glass [1, 4, 5]. The understanding and controlling of this interplay with the help of processing variables, additives, ingredients, reduction in grain size dwelling in nanometre range or compositing, with a view to develop desired, unforeseen and technologically useful properties, is the subject of intensive research activities at present. For device applications, a ceramic piece is usually connected to some other system through electrodes. Thus, an electronic component may be treated as grain-grain boundary-electrode system where the overall observed properties would have contributions from all these [1]. In order to develop ceramics having desired and reproducible properties, these individual contributions must be separated out. For this, impedance spectroscopy is a very useful technique [6]. It not only helps in developing the understanding of the processes but also provides with an equivalent circuit model for the system that may be used for simulation purposes [1, 6]. Impedance spectroscopic studies proceed with the measurement of electrical impedance as function of frequency and interpretation of the data by using suitable equivalent circuit model that seems appropriate to represent various charge transfer processes in the system. The values of the components used in the model are then obtained by using a least-squares procedure to fit the experimental impedance data with that simulated by using the chosen model. The choice of the most suitable model is a difficult process. Even if a model has been conceived, estimating the values of the components of the model, so much required for the least-squares programs, is not always straightforward. This poses some inconvenience to the worker. Developing a suitable model and obtaining these estimates are greatly facilitated by comparing the experimental plots with those simulated for various models and considering different charge transfer processes thought to be possibly present in the system. In what follows, basics of impedance spectroscopy and some very useful models with their simulated behaviour are presented. Analysis of impedance spectroscopic data on ceramic system BaFexTi1-xO3 (x = 0.05) containing two phases is also presented as illustration. The focus is how to choose a model and how to get an estimate of the values of the components used in the model.

biology, medical diagnostics, agriculture, dairy and fruit production ([9] and refer-

The electrical behaviour of a system can be expressed in terms of interrelated

Z\* = M<sup>0</sup> +jM00), C0 being the capacitance of the empty cell used to house the sample. These are broadly termed as immittance functions and are conveniently used to develop equivalent circuit models for the sample-electrode system [6, 8, 9]. A charge transfer process would have a certain time constant and would respond in the corresponding frequency region. A parallel RC circuit (Figure 1(a)) possesses one time constant RC and is thus conveniently used to represent one charge-

Impedance Spectroscopy: A Powerful Technique for Study of Electronic Ceramics

transfer process. The impedance of this model circuit is given as

<sup>Z</sup><sup>0</sup> � <sup>R</sup> 2 <sup>2</sup>

<sup>1</sup> <sup>þ</sup> ð Þ <sup>ω</sup>CR <sup>2</sup> , <sup>Z</sup>

As the values of R and C are positive, the Z<sup>00</sup> vs. Z<sup>0</sup> plot will be a semicircle

(a) Equivalent circuit model containing parallel combination of R and C. Normalized (b) Z″/R vs Z<sup>0</sup>

00

<sup>2</sup> <sup>¼</sup> <sup>R</sup> 2 <sup>2</sup>

<sup>þ</sup> <sup>Z</sup>″

(Figure 1(b)). On this semicircle, the peak point occurs at Z<sup>0</sup> = R/2. At this point, Z<sup>00</sup> is equal to R/2 and ωCR =1. Also, Z<sup>0</sup> = R at ω = 0. Thus, if values Z<sup>0</sup> and Z<sup>00</sup> for a sample are experimentally measured in certain frequency range and Z<sup>00</sup> vs. Z<sup>0</sup> plot is found to be a clear semicircular arc, then a parallel RC circuit model may be used to represent the electrical behaviour of the sample, to begin with (for more details see Section 3.1). The value of intercept of the extrapolated arc with the Z<sup>0</sup> axis towards

which is equation of a circle having centre at point (R/2, 0), radius equal to R/2, intercept with Z<sup>0</sup> axis at the point (R,0), and passing through origin in Z<sup>00</sup> vs. Z<sup>0</sup> plot.

<sup>¼</sup> <sup>ω</sup>CR2

<sup>Z</sup><sup>0</sup> <sup>¼</sup> <sup>R</sup>


<sup>1</sup> <sup>þ</sup> ð Þ <sup>ω</sup>CR <sup>2</sup> (1)

(2)

/R and

– j ε00), and modulus (M\* = (ε\*)�<sup>1</sup> = jωC0

ences therein).

Figure 1.

81

(c) M″/(C0/C) vs M<sup>0</sup>

/(C0/C) plots.

functions known as impedance (Z\* = Z0

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

permittivity (ε\* = (j ωC0 Z\*)�<sup>1</sup> = ε<sup>0</sup>

Z<sup>0</sup> and Z<sup>00</sup> satisfy the relation

### 2. Basics of impedance spectroscopy

Impedance spectroscopy essentially involves measurement of real and imaginary parts of electrical impedance (Z\* = Z<sup>0</sup> -j Z00,j= √(�1)) of a system as function of frequency (ω = 2πf) for various parameters of interest such as composition, temperature, etc. The values of Z<sup>0</sup> and Z<sup>00</sup> are plotted as function of frequency (Z<sup>0</sup> , Z<sup>00</sup> vs. log f) and also in complex plane (i.e. Z<sup>00</sup> vs. Z<sup>0</sup> ). These complex plane plots are usually distorted semicircular overlapping arcs. By looking at the shapes of these plots, charge transfer processes present in the system and accessible in the measurement frequency range are inferred. This is greatly facilitated by comparing the experimental plots with those simulated for possible equivalent circuit models for the system. Detailed discussion on impedance spectroscopy and equivalent circuit models has been presented by Barsoukov and Macdonald [6], Jonscher [7], and Pandey et al. [8]. Due to ready availability of versatile impedance analysers working in extended frequency ranges and ease of measurements, impedance spectroscopy has emerged as a very popular and powerful tool in recent years and is being widely used in various fields encompassing materials technology, electrochemistry,

Impedance Spectroscopy: A Powerful Technique for Study of Electronic Ceramics DOI: http://dx.doi.org/10.5772/intechopen.81398

biology, medical diagnostics, agriculture, dairy and fruit production ([9] and references therein).

The electrical behaviour of a system can be expressed in terms of interrelated functions known as impedance (Z\* = Z0 -j Z00), admittance (Y\* = (Z\*)�<sup>1</sup> = Y<sup>0</sup> +jY00), permittivity (ε\* = (j ωC0 Z\*)�<sup>1</sup> = ε<sup>0</sup> – j ε00), and modulus (M\* = (ε\*)�<sup>1</sup> = jωC0 Z\* = M<sup>0</sup> +jM00), C0 being the capacitance of the empty cell used to house the sample. These are broadly termed as immittance functions and are conveniently used to develop equivalent circuit models for the sample-electrode system [6, 8, 9]. A charge transfer process would have a certain time constant and would respond in the corresponding frequency region. A parallel RC circuit (Figure 1(a)) possesses one time constant RC and is thus conveniently used to represent one chargetransfer process. The impedance of this model circuit is given as

$$Z' = \frac{\text{R}}{\text{1} + \text{(aCR)}^2}, Z' = \frac{a \text{CR}^2}{\text{1} + \text{(aCR)}^2} \tag{1}$$

Z<sup>0</sup> and Z<sup>00</sup> satisfy the relation

may be altered. It is the interplay between the grain and grain boundary behaviour that bestows ceramics with some very useful properties. Ceramics prepared by controlled crystallisation of glass are called glass-ceramics where the grain boundary consists of the uncrystallised glass [1, 4, 5]. The understanding and controlling of this interplay with the help of processing variables, additives, ingredients, reduction in grain size dwelling in nanometre range or compositing, with a view to develop desired, unforeseen and technologically useful properties, is the subject of intensive research activities at present. For device applications, a ceramic piece is usually connected to some other system through electrodes. Thus, an electronic component may be treated as grain-grain boundary-electrode system where the overall observed properties would have contributions from all these [1]. In order to develop ceramics having desired and reproducible properties, these individual contributions must be separated out. For this, impedance spectroscopy is a very useful technique [6]. It not only helps in developing the understanding of the processes but also provides with an equivalent circuit model for the system that may be used for simulation purposes [1, 6]. Impedance spectroscopic studies proceed with the measurement of electrical impedance as function of frequency and interpretation of the data by using suitable equivalent circuit model that seems appropriate to represent various charge transfer processes in the system. The values of the components used in the model are then obtained by using a least-squares procedure to fit the experimental impedance data with that simulated by using the chosen model. The choice of the most suitable model is a difficult process. Even if a model has been conceived, estimating the values of the components of the model, so much required for the least-squares programs, is not always straightforward. This poses some inconvenience to the worker. Developing a suitable model and obtaining these estimates are greatly facilitated by comparing the experimental plots with those simulated for various models and considering different charge transfer processes thought to be possibly present in the system. In what follows, basics of impedance spectroscopy and some very useful models with their simulated behaviour are presented. Analysis of impedance spectroscopic data on ceramic system BaFexTi1-xO3 (x = 0.05) containing two phases is also presented as illustration. The focus is how to choose a model and how to get an estimate of the values of

Ceramic Materials ‐ Synthesis, Characterization, Applications and Recycling

Impedance spectroscopy essentially involves measurement of real and imaginary

frequency (ω = 2πf) for various parameters of interest such as composition, temperature, etc. The values of Z<sup>0</sup> and Z<sup>00</sup> are plotted as function of frequency (Z<sup>0</sup>

usually distorted semicircular overlapping arcs. By looking at the shapes of these plots, charge transfer processes present in the system and accessible in the measurement frequency range are inferred. This is greatly facilitated by comparing the experimental plots with those simulated for possible equivalent circuit models for the system. Detailed discussion on impedance spectroscopy and equivalent circuit models has been presented by Barsoukov and Macdonald [6], Jonscher [7], and Pandey et al. [8]. Due to ready availability of versatile impedance analysers working in extended frequency ranges and ease of measurements, impedance spectroscopy has emerged as a very popular and powerful tool in recent years and is being widely

used in various fields encompassing materials technology, electrochemistry,


). These complex plane plots are

, Z<sup>00</sup>

the components used in the model.

2. Basics of impedance spectroscopy

vs. log f) and also in complex plane (i.e. Z<sup>00</sup> vs. Z<sup>0</sup>

parts of electrical impedance (Z\* = Z<sup>0</sup>

80

$$\left(\mathbf{Z}' - \frac{\mathbf{R}}{2}\right)^2 + \mathbf{Z}''^2 = \left(\frac{\mathbf{R}}{2}\right)^2\tag{2}$$

which is equation of a circle having centre at point (R/2, 0), radius equal to R/2, intercept with Z<sup>0</sup> axis at the point (R,0), and passing through origin in Z<sup>00</sup> vs. Z<sup>0</sup> plot. As the values of R and C are positive, the Z<sup>00</sup> vs. Z<sup>0</sup> plot will be a semicircle (Figure 1(b)). On this semicircle, the peak point occurs at Z<sup>0</sup> = R/2. At this point, Z<sup>00</sup> is equal to R/2 and ωCR =1. Also, Z<sup>0</sup> = R at ω = 0. Thus, if values Z<sup>0</sup> and Z<sup>00</sup> for a sample are experimentally measured in certain frequency range and Z<sup>00</sup> vs. Z<sup>0</sup> plot is found to be a clear semicircular arc, then a parallel RC circuit model may be used to represent the electrical behaviour of the sample, to begin with (for more details see Section 3.1). The value of intercept of the extrapolated arc with the Z<sup>0</sup> axis towards

Figure 1.

(a) Equivalent circuit model containing parallel combination of R and C. Normalized (b) Z″/R vs Z<sup>0</sup> /R and (c) M″/(C0/C) vs M<sup>0</sup> /(C0/C) plots.

low-frequency side will give the value of R. The value of C can be obtained by noting the frequency where the arc peaks and using the relation ωCR =1.

For this model, M<sup>0</sup> and M<sup>00</sup> are given as

$$\mathbf{M}' = \mathsf{oC}\_0 \left( \frac{\mathsf{oC} \mathbf{R}^2}{\mathbf{1} + \mathsf{o}^2 \mathbf{C}^2 \mathbf{R}^2} \right), \mathbf{M}'' = \mathsf{oC}\_0 \left( \frac{\mathbf{R}}{\mathbf{1} + \mathsf{o}^2 \mathbf{C}^2 \mathbf{R}^2} \right) \tag{3}$$

which satisfy the relation

$$\left(\text{M}' - \frac{\text{C}\_0}{2\text{ C}}\right)^2 + \text{M}''^2 = \left(\frac{\text{C}\_0}{2\text{ C}}\right)^2\tag{4}$$

2. If plots of both Z00 vs. Z0 and M00 vs. M0 are clear semicircles passing through origin, then the system may be represented by a model having one parallel RC

3. Appearance of a shift towards right in M<sup>00</sup> vs. M<sup>0</sup> plot (here, the arc traverses from left low-frequency side to right high-frequency side) and steeply rising branch at low-frequency side in the Z<sup>00</sup> vs. Z<sup>0</sup> plot (here, the arc traverses from right to left) indicates the presence of series capacitance C in the equivalent

4.Appearance of a shift towards right in Z<sup>00</sup> vs. Z<sup>0</sup> plot and steeply rising branch at high-frequency side in M<sup>00</sup> vs. M<sup>0</sup> plot indicates the presence of series

5. Appearance of two clear semicircular arcs in Z<sup>00</sup> vs. Z<sup>0</sup> plot or M<sup>00</sup> vs. M<sup>0</sup> plot indicates presence of two processes and equivalent circuit would contain two parallel RC's (R1C1 and R2C2). A depressed looking semicircular arc in the Z or M plots would indicate the presence of at least two processes having ratio of

6. If the experimental Z<sup>00</sup> vs. Z<sup>0</sup> plot shows a semicircular arc with changed sign of Z<sup>00</sup> in the whole frequency range (i.e. arc appears in fourth quadrant traversing from left to right), then a model with parallel R-L circuit may be used [11].

7. A cross over from positive values of Z<sup>00</sup> to negative values or vice versa within the overall frequency range covered in the experiment indicates a situation of resonance and inclusion of R, C and L in the model would be needed

8. Presence of a linear portion in the immittance plots would indicate the presence of series CPE in the model. Depressed looking immittance plots indicate that CPE connected in parallel may be included in the model [6, 9,

Detailed analysis of various models involving CPE is available in [6, 9]. It is believed that CPE represents distribution in properties. Therefore, when the data is well represented by a model involving CPE, presence of distribution in certain properties of ceramics is inferred. It may be mentioned that capacitor C and inductor L are used to represent the storage of electrical and magnetic forms of energy in a system. At frequencies much below resonance, the situation may be approximately modelled by considering RC's only. Similarly, a mechanically vibrating system can be analogously represented by an RLC circuit. Therefore, equivalent circuits having combinations of R, L and C may be used [14, 16] for piezoelectric ceramics. In what follows, some model circuits found very useful in study of ceramics are briefly presented. Emphasis is given to Z<sup>00</sup> vs. Z<sup>0</sup> and M<sup>00</sup> vs. M<sup>0</sup> plots as

3.1 Model 1: parallel combinations R1C1 and R2C2 connected in series

Figure 2(a) and the values for Z<sup>0</sup> and Z<sup>00</sup> are given in Eq.(5):

Parallel combinations R1C1 and R2C2 connected in series [8] are shown in

and presence of one process may be inferred [6, 8].

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

Impedance Spectroscopy: A Powerful Technique for Study of Electronic Ceramics

resistance R in the equivalent circuit [8, 12].

time constants R2C2/R1C1 in the range 1–5 [8].

circuit [17].

[6, 14, 16].

12, 21].

these were found more informative.

3. Equivalent circuit models

83

Eq. (4) indicates that M<sup>00</sup> vs. M<sup>0</sup> plot would be a semicircular arc passing through the origin and having intercept at M<sup>0</sup> = C0/C and centre at (C0/2C, 0) (Figure 1(c)). On this plot, M<sup>00</sup> would peak at frequency where ωCR = 1. Value of M<sup>0</sup> becomes equal to C0/C as ω ! ∞. Thus, by using M<sup>00</sup> vs. M<sup>0</sup> plot of the same experimental data, C can be obtained by noting the high-frequency intercept in the M<sup>00</sup> vs. M<sup>0</sup> plot. The value of R can be obtained by using this value of C and noting the frequency where M<sup>00</sup> peaks. In the M<sup>00</sup> vs. M<sup>0</sup> plot, the arc traverses from left to right whereas in Z<sup>00</sup> vs. Z<sup>0</sup> plot, it traverses from right to left as the frequency is increased.

Choice of a model to represent a system is a difficult process, becoming more so since same behaviour can be simulated by different models [6]. The choice is greatly facilitated by comparing the experimental plots with those simulated for various models and considering the processes that might be present in the system with a preference to simple models to start with. When the complex plane plots display more than one arcs, presence of more polarisation/charge transfer processes is inferred. A general practice is to consider one RC circuit to represent one process and connect more RCs in series to develop an equivalent circuit model. Thus for representing grain-grain boundary-electrode system in a ceramic, a model comprising three parallel RCs connected in series might be taken [1]. It has been found that usually combinations of resistances (R) and capacitances (C) suffice for dielectrics, combinations of R and inductance L suffice for magnetic systems, and combinations of R, L and C suffice for ferro-/piezoelectrics [6–8, 10–20]. Sometimes, it is found that the lumped-component type of models do not yield good fits and their simulated patterns do not show even qualitative resemblance with the experimental plots. In those cases, attempt is made to represent the data by models involving constant phase angle elements (CPE) [6, 9] that are considered to correspond to some sort of distribution in the material properties. The values of the components used in the model are estimated by comparing the experimental plots with the simulated ones. These values are obtained more accurately by fitting the experimental data with complex non-linear least-square (CNLS) procedure using these values as initial guesses. For cases, where it is not possible to decide upon an appropriate model, equivalent circuits models that seem to be most probable to represent the processes thought to be possibly present, or to be dormant but becoming dominant as some variables change, in the system may be considered and the model that yields the lowest value of sum of squares of errors in the CNLS runs may be accepted [6].

In order to facilitate a prompt development of an equivalent circuit model for electronic ceramics, some very important clues have been given in [9] and are summarised below for a ready reference.

1. The experimental data can be represented in any of the formalisms (Z, M, Y and ε) and interpreted. However, analysis by using only one formalism might lead to erroneous conclusions and two or more functions such as Z and M should be used [6, 8].

Impedance Spectroscopy: A Powerful Technique for Study of Electronic Ceramics DOI: http://dx.doi.org/10.5772/intechopen.81398


Detailed analysis of various models involving CPE is available in [6, 9]. It is believed that CPE represents distribution in properties. Therefore, when the data is well represented by a model involving CPE, presence of distribution in certain properties of ceramics is inferred. It may be mentioned that capacitor C and inductor L are used to represent the storage of electrical and magnetic forms of energy in a system. At frequencies much below resonance, the situation may be approximately modelled by considering RC's only. Similarly, a mechanically vibrating system can be analogously represented by an RLC circuit. Therefore, equivalent circuits having combinations of R, L and C may be used [14, 16] for piezoelectric ceramics. In what follows, some model circuits found very useful in study of ceramics are briefly presented. Emphasis is given to Z<sup>00</sup> vs. Z<sup>0</sup> and M<sup>00</sup> vs. M<sup>0</sup> plots as these were found more informative.
