Contents

Cover

Title page

Copyright page

Dedication

Background of this Book

Acknowledgments

Chapter 1: Introduction to Immittance Spectroscopy

1.1 Basic Definition and Background
1.2 Scope and Limitation
1.3 Applications of the Immittance Studies to Various Material Systems
1.4 Concept of the Linear Circuit Elements: Resistance, Capacitance, and Inductance
1.5 Concept of Impedance, Admittance, Complex Capacitance, and Modulus
1.6 Immittance Functions
1.7 Series Resonant Circuit
1.8 Parallel Resonant Circuit
1.9 Capacitance and Inductance in Alternating Current
Problems
References

Chapter 2: Basics of Solid State Devices and Materials

2.1 Overview of the Fundamentals of Physical Electronics
2.2 Basics of Semiconductors
2.3 Single-Crystal and Polycrystal Materials
2.4 SCSJ and MPCHPH Systems
2.5 Representation of the Competing Phenomena
2.6 Effect of Normalization of the Electrical Parameters
Problems
References

Chapter 3: Dielectric Representation and Operative Mechanisms

3.1 Dielectric Constant of Materials: Single Crystals and Polycrystals
3.2 Dielectric Behavior of Materials: Single Crystals and Polycrystals
3.3 Origin of Frequency Dependence
3.4 Effect of Polarization
3.5 Equivalent Circuit Representation of the Mechanisms and Processes
3.6 Defects and Traps
3.7 Point Defects and Stoichiometric Defects
3.8 Leaky Systems
Problems
References

Chapter 4: Ideal Equivalent Circuits and Models

4.1 Concept of Equivalent Circuit
4.2 Simple and Basic Circuits in Complex Planes: R, C, R-C Series, and R-C Parallel
4.3 Debye Circuits: Single Relaxation
4.4 Duality of the Equivalent Circuits: Multiple Circuits for a Single Plane
4.5 Duality of Equivalent Circuits between Z*- and M*-Planes for Relaxations without Intercept
4.6 Duality of Equivalent Circuits between Y*- and C*-Planes for Relaxations without Intercept
4.7 Duality of Equivalent Circuits for Simultaneous Z*-, Y*-, C*-, and M*-Planes’ Relaxations
4.8 Proposition of Equivalent Circuit: Polycrystalline Grains and Grain Boundaries
Problems
References

Chapter 5: Debye and Non-Debye Relaxations

5.1 Ideal Systems
5.2 Non-Ideal Systems
5.3 Non-Ideal Systems Implying Distributed Time Constants
5.4 D-C Representation, Depression Parameter, and Equivalent Circuit: Conventional Domain
5.5 Depression Parameter Based on ωτ_peak = 1: Complex Domain
5.6 Optimization of Z_HF: Complex Domain
5.7 Depression Parameter β Based on ωτ_peak = 1
5.8 Feature of the Depression Parameter β Based on ωτ ≠ 1
5.9 Analysis of the Havriliak-Negami Representation
5.10 Geometrical Interpretation of H-N Relaxation at the Limiting Case
5.11 Extraction of the Relaxation Time τ and the H-N Depression Parameters α and β
5.12 Checking Generalized Depression Parameter β when α is Real
5.13 Checking Generalized Depression Parameter α when β is Real
5.14 Effect of α and β on the H-N Distribution Function
5.15 Meaning of the Depression Parameters α and β
5.16 Relaxation Function with Respect to the Depression Parameters α and β
Problems
References

Chapter 6: Modeling and Interpretation of the Data

6.1 Equivalent Circuit Model for the Single Complex Plane (SCP) Representation
6.2 Models and Circuits
6.3 Nonconventional Circuits
6.4 Multiple Equivalent Circuits for Multiple Relaxations in a Single Complex Plane
6.5 Single Equivalent Circuit for Multiple Complex Planes
6.6 Equivalent Circuit for Resonance
6.7 Single Equivalent Circuit from Z*- and M*-Planes
6.8 Temperature and Bias Dependence of the Equivalent Circuit Modeling
6.9 Equivalent Circuit: Zinc Oxide (ZnO) Based Varistors
6.10 Equivalent Circuit: Lithium Niobate (LiNbO₃) Single Crystal
6.11 Equivalent Circuit: Polycrystalline Yttria (Y₂O₃)
6.12 Equivalent Circuit: Polycrystalline Calcium Zirconate (CaZrO₃)
6.13 Equivalent Circuit: Polycrystalline Calcium Stannate (CaSnO₃)
6.14 Equivalent Circuit: Polycrystalline Titanium Dioxide (TiO₂)
6.15 Equivalent Circuit: Multi-Layered Thermoelectric Device (Alternate SiO₂/SiO₂+Ge Thin-Film)
6.16 Equivalent Circuit: Polycrystalline Tungsten Oxide (WO₃)
6.17 Equivalent Circuit: Biological Material – E. Coli Bacteria
Problems
References

Chapter 7: Data-Handling and Analyzing Criteria

7.1 Acquisition of the Immittance Data
7.2 Lumped Parameter/Complex Plane Analysis (LP/CPA)
7.3 Spectroscopic Analysis (SA)
7.4 Bode Plane Analysis (BPA)
7.5 Misrepresentation of the Measured Data
7.6 Misinterpretation of the Bode Plot: Equivalent Circuit
Problems
References

Chapter 8: Liquid Systems

8.1 Non-Crystalline Systems: Liquids
8.2 Warburg and Faradaic Impedances
8.3 Constant Phase Element (CPE)
8.4 Biological Liquid: E. Coli Bacteria
Problems
References

Chapter 9: Case Studies

9.1 Analysis of the Measured Data: Aspects of Data-Handling/Analyzing Criteria
9.2 Case 1: Proper Physical Geometrical Factors
9.3 Case 2: Improper Normalization
9.4 Case 3: Effect of Electrode and Lead Wire
9.5 Case 4: Identification of Contributions to the Terminal Immittance
9.6 Case 5: Use of Proper Unit
9.7 Case 6: Demonstration of the Invalid Plot
9.8 Case 7: Obscuring Frequency Dependence
9.9 Case 8: Misnomer Nomenclature for the Complex Plane Plot
9.10 Case 9: Extraction of Equivalent Circuit from the Straight Line or the Non-Relaxation Curve
Problems
References

Chapter 10: Analysis of the Complicated Mott-Schottky Behavior

10.1 Capacitance – Voltage (C-V) Measurement
10.2 The Mott-Schottky Plot
10.3 Arbitrary Measurement Frequency and Construction of the Deceiving Mott-Schottky Plot
10.4 Frequency-Independent Representation
10.5 Extraction of the Device-Related Parameters
Problems
References

Chapter 11: Analysis of the Measured Data

11.1 Introduction and Background of the Immittance Data Analysis
11.2 Measurement of the Immittance Data and Complex Plane Analysis
11.3 Nonlinear Least Squares Estimation
11.4 Complex Nonlinear Least Squares (CNLS) Fitting of the Data
11.5 Graphical User Interface Implementation of the Nonlinear Least Square Procedures: Implementation of CNLS using MATLAB
11.6 Effect of Fitting Procedure, Measurement Noise, and Solution Algorithm on the Estimated Parameters
11.7 Case Studies: CNLS Fitting of the Measured Data in the Complex Planes
11.8 Summary
Problems
References

Chapter 12: Appendices

12.1 Appendix – A: Sample Input Data for the R-C Parallel Circuit
12.2 Appendix – B: R-C Parallel Circuit Data Analysis Output in Z*-Plane
12.3 Appendix – C: R-C Parallel Circuit Data Analysis Output in M^*-Plane
12.4 Appendix – D: Lithium Niobate Crystal Data Analysis Output in C*-Plane
12.5 Appendix – E: Multilayer Junction Thermoelectric Device Data Analysis Output in Y*-Plane

Index

End User License Agreement

List of Illustrations

Chapter 1

Figure 1.1 List of material systems as DUT and effects to be diagnosed by the immittance measurements.
Figure 1.2 Circuit symbol for the passive elements: resistor (R), capacitor (C), and inductor (L).
Figure 1.3 The dc power source provides the same voltage to the parallel plate capacitor in vacuum on left, and to the parallel-plate capacitor with dielectric ε_r on right.
Figure 1.4 The capacitor in vacuum is at the top and the capacitor with dielectric is at the bottom. Both capacitors possess the same charges. The capacitor in vacuum requires high voltage whereas the capacitor with dielectric requires low voltage to achieve the same charges.
Figure 1.5 Voltage – current [V(t) – I(t)] response: (a) for a resistor R showing φ = 0°, (b) for a capacitor C showing φ = - 90°, and (c) for an inductor L showing φ = +90°.
Figure 1.6 Usual passive circuit elements with current-voltage relation and their impedance response.
Figure 1.7 Phase shift between voltage and current for a DUT containing passive elements.
Figure 1.8 Choice of measured forms of the ac small-signal electrical data acquisition and four complex plane formalisms along with their inter-convertible relationships.
Figure 1.9 Summary of the four basic immittance functions along with conversions without normalization in (a), and with normalization in (b).
Figure 1.10 Simple circuits with corresponding frequently used parameters.

Chapter 2

Figure 2.1 Distinguishing metal, semiconductor, and insulator using the band gap concept.
Figure 2.2 The context of Fermi-level of a semiconductor at absolute zero (0 °K) using the band gap concept.
Figure 2.3 The properties of Fermi-Dirac distribution function.
Figure 2.4 The Fermi-Dirac distribution functions at different temperatures.
Figure 2.5 Electrons at the conduction band as temperature increases.
Figure 2.6 Energy band diagram for: (a) n-type semiconductor and (b) p-type semiconductor.
Figure 2.7 Schematic illustration of the lumped representation incorporating simultaneously operative competing phenomena in the electrically active MPCHPH systems.
Figure 2.8 Representation of the multi-phase/multi-component heterogeneous polycrystalline hybrid (MPCHPH) system showing physical regions and operative mechanisms.

Chapter 3

Figure 3.1 Representation of a dielectric material: (a) possessing dielectric constant ε_r, conductivity σ or resistivity ρ, thickness d, and electrode area A; (b) equivalent circuit of the same dielectric with time constant τ having resistance R_P and capacitance C_P.
Figure 3.2 Electrical response in the form of admittance of a dielectric material.
Figure 3.3 Depletion region in the p-n junction where far away from the junction is the bulk region.
Figure 3.4 Interfacial polarization at (a) electrodes; and at (b) crystallites or grains.
Figure 3.5 Frequency dependence of polarizability or dielectric constant showing several contributing mechanisms.
Figure 3.6 Equivalent circuit model for potential operative mechanism and process both at zero bias and under bias occurring within the DUT.
Figure 3.7 Band diagram of a semiconductor showing conduction band edge E_C, valence band edge E_V, donor energy level E_D, acceptor energy level E_A with E_T as the deep trap energy level for charge trapping center, recombination center, and generation center for the specific energy level E_D.
Figure 3.8 Band diagram of a semiconductor showing conduction band edge E_C, valence band edge E_V, donor energy level E_D, acceptor energy level E_A, and E_T is the deep trap energy level. Charge trapping center, recombination center, and generation center are shown from a specific energy level E_T.
Figure 3.9 Processes of emission and capture for a single trap center E_T.
Figure 3.10 Representation of a covalent semiconductor where O is the Si atom in the crystal having (a) a regular vacancy involving four non-relaxed dangling bonds; and (b) a relaxed vacancy with no dangling bond.
Figure 3.11 Notations of point defects for the crystal and polycrystal systems.

Chapter 4

Figure 4.1 Thévenin’s equivalent circuit with voltage source V_OC is shown at the left and Norton’s (current source) equivalent circuit with the current source I_SC at the right. The impedance R_eq is the same in both cases and the source values are related to each other by V_OC = R_eq I_SC.
Figure 4.2 Representation of the resistance R or conductance G in four complex planes: Z*, Y*, C*, and M* where an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 4.3 Representation of a capacitor C in four complex planes: Z*, Y*, C*, and M* where an arrow indicates direction of the increasing measurement angular frequency ω.
Figure 4.4 Simplistic representation of one R-C series circuit in four complex planes: Z*, Y*, C*, and M* where τ_S is the time constant and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 4.5 Simplistic representation of one R-C parallel circuit in four complex planes: Z*, Y*, C*, and M* where τ_P is the time constant and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 4.6 Simplistic equivalent circuit for single relaxation without the intercept in four complex planes where τ is the time constant in the respective complex plane and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 4.7 Simplistic equivalent circuit for single relaxation with intercept in each of the four complex planes where τ is the time constant in the respective complex plane and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 4.8 Simplistic equivalent circuit for two relaxations with intercept in each of the four complex planes where τ is the time constant in the respective complex plane and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 4.9 Various R-C combinations representing Voigt, Maxwell, and Ladder type circuits in the Z*-plane (= R + j X) showing two relaxations with an intercept where τ is the time constant and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 4.10 Duality between two relaxations in Z*-plane and one relaxation in M*-plane without intercept where τ is the time constant and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 4.11 Duality between two relaxations in M*-plane and one relaxation in Z*-plane without intercept where τ is the time constant and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 4.12 Duality between two relaxations in Y*-plane and one relaxation in C*-plane without intercept where τ is the time constant and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 4.13 Duality of equivalent circuits between two relaxations in C*-plane and one relaxation in Y*-plane without intercept where τ is the time constant and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 4.14 Conditions on magnitude of resistance and capacitance to obtain duality with another complex plane providing best separation of multiple time constant systems or DUTs.
Figure 4.15 Idealized polycrystals in (a) show the equivalent circuit in (b) where grains and grain boundaries are distinguishable, while idealized polycrystals in (c) show the equivalent circuit in (d) where grains and grain boundaries are not distinguishable, and for a random orientation of grains in (e) shows the equivalent circuit in (f).
Figure 4.16 Systematic progression of idealized layered grains shown in (a) is modeled in (b) while the eventual circuit is displayed in (c).
Figure 4.17 Two-layered polycrystalline material system of (a) possessing two distinct time constants shown in (b).

Chapter 5

Figure 5.1 (a) Parallel equivalent circuit containing a resistor R_P and a capacitor C_P. (b) Equivalent circuit comprising of a resistor R in series with the parallel R_P-C_P combination.
Figure 5.2 Ideal Debye response in the impedance (Z*) plane via the semicircular response using equation (5.7) showing the intercept on the left side of it where the center is on the x-axis and the measurement frequency in the anti-clockwise direction.
Figure 5.3 Two possible relaxation situations in the M*-plane representation of the parallel R_P-C_P combination in series with R where τ is the time constant and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 5.4 (a) Series equivalent circuit containing a resistor R_S and a capacitor C_S. (b) Equivalent circuit comprising of a resistor R in parallel with the series R_S-C_S combination.
Figure 5.5 Ideal Debye response in the Y*-plane via the semicircular response using equation (5.2) showing the intercept on the left side of it where the center is on the x-axis and the measurement frequency in the anti-clockwise direction.
Figure 5.6 Two possible relaxation situations in the C*-plane representation of the series R_S-C_S combination in parallel with R where τ is the time constant and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 5.7 Reproduction of complex conductivity (σ′ versus σ″) plot and complex permittivity (ε′ versus ε″) plot corresponding to four equivalent circuits from the classic paper of Grant [7].
Figure 5.8 Influence of the dc conductivity σ on the dielectric relaxation presented in the permittivity plane plot for the equivalent circuit shown in the inset where curves 1 through 4 represent different dc conductivity values and curve 0 represents perfect insulator having σ₁ = 0.
Figure 5.9 Schematic illustration of a depressed semicircular relaxation in the Z*-plane showing how the electrical parameters are extracted using the complex plane analysis where θ is the depression angle, h is the depression parameter, and U_i and V_I are the chords of the semicircular loci.
Figure 5.10 Generic representation of the relaxation curves in complex plane for the relaxation function F*(ω) given in equation (5.16): (1) Debye formalism with depression angle θ = 0° or α = 1 and β = 1, (2) non-Debye Cole-Cole formalism with depression angle θ = 9° or α = 0.9 and β = 1, (3) Davidson-Cole formalism with α = 1 and β = 0.5, (4) Havriliak-Negami formalism with α = 0.85 and β = 0.45; and (5) Havriliak-Negami formalism with α = 0.45 and β = 0.85.
Figure 5.11 (a) Schematic illustration of a depressed semicircular relaxation in the ε*-plane showing how the electrical parameters are extracted using the complex plane analysis where θ is the depression angle, h is the depression parameter, and U_i and V_I are the chords of the semicircular loci; (b) chord fitting verification via Log(U_i / V_i) versus Log(ω) plot; (c) ideal Debye circuit presuming Debye relaxation in (a); (d) Cole-Cole proposed circuit termed as non-Debye circuit for the relaxation in (a); (e) Debye resistor (resistivity) of (c) replaced by Cole-Cole (non-Debye) frequency-independent impedance.
Figure 5.12 The dc voltage dependence of the depression angle θ_im obtained in the impedance (Z*) plane semicircular relaxation for the well-formed commercial ZnO-Bi₂O₃-based varistor.
Figure 5.13 Fitting procedure of the semicircle in the C*-plane in (a) and Z*-plane in (b) where angular frequency ω increases in the anti-clockwise direction showing chord ratio U_i/V_i versus ω in the log-log scale in (c) and in (d), respectively, yielding depression parameter α, depression angle θ (= α π/2), peak angular frequency ω_peak (= 2 π f_peak), and time constant τ_peak.
Figure 5.14 Fitting procedure of the semicircle in the Y*-plane in (a) and M*-plane in (b) where angular frequency ω increases in the clockwise direction showing chord ratio U_i/V_i versus ω in the log-log scale in (c) and in (d), respectively, yielding depression parameter α, depression angle θ (=απ/2), peak angular frequency ω_peak (= 2 π f_peak), and time constant τ_peak.
Figure 5.15 The generic impedance representation obtained from the Davidson-Cole relation given in equation (5.26) for the depression parameter β = 0.4. [Imaginary part shows positive.]
Figure 5.16 Energy band diagram for the back-to-back Schottky barrier device in the polycrystalline semiconductors where grain-to-grain junction provides variation in the relaxation processes associated with the defect states in conjunction with the barrier layer relaxation.
Figure 5.17 The product ω Í τ is displayed as a function of the imaginary part of the impedance for β = 0.40 showing ωτ_peak = 208 and Z″ = ImZ* = 310. Inset 1 shows β = 0.85 with ωτ_peak = 1.14 and Z″ = ImZ* = 0.464; and inset 2 shows β = 0.5 with ωτ_peak = 1.74 and Z″ = ImZ* = 0.353.
Figure 5.18 The variation of the D-C depression parameter β as a function of the product ω × τ. Inset 1 shows variation of F″ as a function of the product ω × τ; and inset 2 shows the relation between β and F″.
Figure 5.19 (a) Havriliak-Negami representation of the relaxation function F*(ω) per equation (5.75a) where measurement frequency runs from right to the left. (b) Havriliak-Negami representation of the relaxation function F*(ω) with α = 1, β = 0.6 that corresponds to F′_peak = F′_ωτ=1497 = 0.5843 and F″_peak = F″_ωτ=1.497 = 0.3905 along with F′_ωτ=1 = 0.7237, F″_ωτ=1 = 0.3888. (c) Havriliak-Negami representation of the relaxation function F*(ω) with α = 0.999, β = 0.6 that corresponds to F′_peak = F′_ωτ=1.498 = 0.5842 and F″_peak = F″_ωτ=1.498 = 0.3899 along with F′_ωτ=1 = 0.7236, F″_ωτ=1 = 0.3682. (d) Havriliak-Negami representation of the relaxation function F*(ω) with α = 0.995, β = 0.6 that corresponds to F′_peak = F′_ωτ=1.501 = 0.5841 and F″_peak = F″_ωτ=1.501 = 0.3877 along with F′_ωτ=1 = 0.7229, F″_ωτ=1 = 0.3662.
Figure 5.20 Havriliak-Negami representation of the Ni-Cd battery via impedance plot⁴¹ for (a): α = 1 and β = 0.6; (b): α = 0.999 and β = 0.6; and (c): α = 0.995 and β = 0.6.
Figure 5.21 H-N response for tan(βγ) versus angular frequency ω of the Ni-Cd battery data with α = 0.995 and β = 0.6. Maximum theoretical value of tan(βγ_max) obtained as 1.3628, while experimental value shows 1.2449. Inset shows ω = 4.9390 rad/sec corresponding to τ = 0.2025 sec.
Figure 5.22 D-C response for tan(βγ) versus angular frequency ω of the Ni-Cd battery data with α = 1.0 and β = 0.6. Maximum theoretical value of tan(βγ_max) obtained as 1.3764, while experimental value shows 1.2449. Inset shows ω = 4.9766 rad/sec corresponding to τ = 0.2009 sec.
Figure 5.23 The product ω × τ is displayed as a function of the imaginary part of the relaxation function F*(ω) for (1) Debye response with α = 1 and β = 1 yielding (ωτ)_peak = 1; (2) Non-Debye (i.e., Cole-Cole) response with α = 0.5 and β = 1 yielding (ωτ)_peak = 1; (3) Davidson-Cole response with α = 1 and β = 0.6 yielding (ωτ)_peak = 1.497; and (4) Havriliak-Negami response with α = 0.5 and β = 0.6 yielding (ωτ)_peak = 2.637. Vertical dotted line indicates ωτ = 1.
Figure 5.24 The product ω × τ is displayed as a function of the imaginary part of Z*(ω) for curve 1: Davidson-Cole response with α = 1.0 and β = 0.6 yielding (ωτ)_peak = 1.4970; curve 2: Havriliak-Negami response with α = 0.999 and β = 0.6 yielding (ωτ)_peak = 1.4980; and curve 3: Havriliak-Negami response with α = 0.995 and β = 0.6 yielding (ωτ)_peak = 1.5010.
Figure 5.25 The parameter β versus (ωτ)_peak for various values of α.
Figure 5.26 The parameter α versus (ωτ)_peak for various values of β.
Figure 5.27 Effect of β on the H-N distribution function for α = 0.5. For τ = τ_HN.
Figure 5.28 Effect of α on the HN distribution function for β = 0.6.
Figure 5.29 Effect of α and β on the difference between expected value of ln(τ) and ln(τ_HN) for α = 0.5 and β = 0.6.
Figure 5.30 Effect of α and β on the skewness of the H-N distribution function for α = 0.5 and β = 0.6.
Figure 5.31 Imaginary part of C-C immittance function (F”) versus log(ωτ) for different values of the depression parameter α.
Figure 5.34 Imaginary part of H-N immittance function (F”) versus log(ωτ) for β = 0.30, 0.65 and 1 for fixed value of α = 0.65 displaying the peak-values of log(ωτ).

Chapter 6

Figure 6.1 Three successive relaxations obtained for a single set of ac small-signal immittance data acquired as a function of frequency displayed in a single complex plane represented by three different equivalent circuits: (1) Voigt model containing three R-C parallel circuits in series; (2) Maxwell model containing two R-C series circuits in parallel with another R-C parallel circuit in parallel; and (3) Ladder or Macdonald model displayed in two arrangements where each relaxation occurred for the R-C series circuit.
Figure 6.2 Various combination of R-C series and/or R-C parallel circuit representations in the Z*-plane (= R – j X) where τ is the time constant and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 6.3 Various combination of R-C series and/or R-C parallel circuit representations in the Y*-plane (= G + j B) where τ is the time constant and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 6.4 Various combination of R-C series and/or R-C parallel circuit representations in the C*-plane (= B/ω – j G/ω) where τ is the time constant and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 6.5 Various combination of R-C series and/or R-C parallel circuit representations in the M*-plane (= ωX – j ωR) where τ is the time constant and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 6.6 Various R-C series and/or R-C parallel combination displayed as nonconventional circuits in the complex planes where τ is the time constant and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 6.7 Various R-C series and/or R-C parallel circuits in the complex planes for the nonconventional circuits where τ is the time constant, and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 6.8 Four mathematically possible equivalent circuits for two consecutive distinct relaxations include Voigt, Maxwell, and Ladder configurations where τ is the time constant, and an arrow indicates the direction of the increasing measurement angular frequency ω.
Figure 6.9 Single circuit of (a) satisfies two semicircular relaxations in the Z*-plane of (b) whereas the same circuit satisfies the relaxation of Y*- and C*-planes in (c) and (d), respectively, per Grant [9].
Figure 6.10 Two different resonance circuits can be obtained from the Y*-plane.
Figure 6.11 Z*- and M*-plane plots of four basic equivalent circuits that are sometimes found as dual representation of the same measured electrical data.
Figure 6.12 Equivalent circuit model of the ZnO-based varistor in the frequency range 10^–2 Hz ≤ f ≤ 10⁹ Hz for electric field 0 ≤ E ≤ 80%E_1mA/cm² in the temperature range 26 °C ≤ T ≤ 90 °C where electric field E_1mA/cm² corresponds to the current density 1 mA/cm².
Figure 6.13 Resonance circuit in (a) in the frequency range 10⁶ Hz ≤ f ≤ 10⁹ Hz, and trapping circuit in (b) in the frequency range 10^–2 Hz ≤ f ≤ 13 MHz for ZnO-based varistor for electric field 0 ≤ E ≤ 80%E_1mA/cm² in the temperature range 26 °C ≤ T ≤ 90 °C where electric field E_1mA/cm² corresponds to the current density 1 mA/cm².
Figure 6.14 Equivalent circuit model of ZnO-based varistor at various experimental conditions.
Figure 6.15 The Mott-Schottky plot using equivalent circuit capacitance C_im obtained in the Z*-plane semicircular relaxation for the ZnO-based varistor.
Figure 6.16 Equivalent circuit model corresponds to (a) C*-plane semicircular relaxation and (b) M*-plane semicircular relaxation.
Figure 6.17 Equivalent circuit model in (a) and in (b) obtained from the C*- and M*-planes, respectively, for the LiNbO₃ single crystal in the temperature range 59 °C ≤ T ≤ 73 °C. These two equivalent circuits represent idealized forms for single semicircular relaxation where the presence of R₂ and R₄ are partly recognized via skewed response attributing to the limited window of data acquisition. Equivalent circuits in (c) and in (d) are the same when the skewed portion of the data representing R₂ (arrow-cut) is eliminated to achieve blocking behavior. Equivalent circuits in (e) and in (f) are the same when the skewed portion of the data representing R₄ (arrow-cut) is eliminated in achieving blocking behavior. Equivalent circuit elements R₂ or R₄ represent dc resistance in respective complex plane.
Figure 6.18 Comparison of equivalent circuit parameters obtained simultaneously from the C*- and M*-planes.
Figure 6.19 (a) Equivalent circuit model corresponding to the two semicircular relaxations in the Z*-plane where R₁-C₁ and R₂-C₂ parallel combinations in series constitute the low-frequency relaxation associating grain-boundaries and high-frequency relaxation associating grains, respectively. (b) Equivalent circuit obtained in the M*-plane where one semicircle was obtained. The series combination of R_Y- C_Y constitutes the low-frequency relaxation. R_dc and C_∞ are the dc resistance and high-frequency limit capacitance, respectively.
Figure 6.20 (a) Equivalent circuit model corresponding to the two semicircular relaxations in the Z*-plane where R₁-C₁ and R₂-C₂ parallel combinations in series constitute the low-frequency relaxation associating grain-boundaries and high-frequency relaxation associating grains, respectively. (b) Equivalent circuit obtained in the M*-plane where one semicircle was obtained. The series combination of R_Y- C_Y constitutes the low-frequency relaxation. R_dc and C_∞ are the dc resistance and high-frequency limit capacitance, respectively.
Figure 6.21 (a) Equivalent circuit model corresponding to the semicircular relaxations in the Z*-plane, (b) equivalent circuit representation of the M*-plane, and (c) the series R-C combination comes from the C*-plane.
Figure 6.22 Equivalent circuit model corresponding to the (a) one relaxation in the Z*-plane for the effect of the Y₂O₃ addition, (b) relaxation in the C*-plane for the effect of the Y₂O₃ addition, and (c) two relaxations in the Z*-plane for the effect of the ZrO₂ addition.
Figure 6.23 Equivalent circuit model of the (a) unbombarded and bombarded (1×10¹⁴ ions/cm²) 200-multilayered thermoelectric devices as obtained in the Z*-plane, (b) bombarded (5×10¹² ions/cm², 1×10¹³ ions/cm², and 5×10¹³ ions/cm²) 200-multilayered thermoelectric devices as obtained in the Z*-plane, (c) bombarded (1×10¹⁴ ions/cm²) 200-multilayered thermoelectric devices in the Z*-plane having dotted resistance R₃₃₃ obtained as the intercept on the left side of the semicircle [the dotted resistance often may be negligible as it is too small to obtain left-intercept for the semicircle upon using CNLS fitting procedure], (d) unbombarded and bombarded (1×10¹⁴ ions/cm²) 200-multilayered thermoelectric devices as obtained in the Y*-plane, (e) bombarded (5×10¹² ions/cm², 1×10¹³ ions/cm², and 5×10¹³ ions/cm²) 200-multilayered thermoelectric devices as obtained in the Y*-plane, (f) bombarded (1×10¹⁴ ions/cm²) 200-multilayered thermoelectric devices in the Y*-plane having dotted resistance R₃₃₃ obtained as the intercept on the left side of the semicircle [the inductance is external to the multilayered device and omitted from the equivalent circuit model].
Figure 6.24 Equivalent circuit model of the polycrystalline tungsten oxide pellet surface showing total resistance R_total = R₁ + R₂ ≈ R₂ since R₁ is detected as too small via the left-intercept i.e., intercept on the left side of the semicircular relaxation in the Z*-plane.
Figure 6.25 Non-Debye equivalent circuit model observed for the sterile BHI or the E. Coli bacteria suspended BHI system. The shunt resistance R₂ for the blocking R₁-C₁ series circuit obtained from the Bode plot in conjunction with the complex plane representation.

Chapter 7

Figure 7.1 (a) Cole-Cole representation in the ε*-plane displaying semicircular relaxation where θ is the depression angle, α is the depression parameter, U_i and V_I are the chords of the semicircular loci; (b) ideal Debye circuit presuming Debye relaxation for (a); (c) Cole-Cole proposed circuit termed as non-Debye circuit for the relaxation in (a); (d) Debye resistor of (b) replaced by the Cole-Cole (non-Debye) frequency-independent impedance.
Figure 7.2 The dc and ac current density (J) versus electric field (E) showing four distinct regions for ZnO-based varistor.
Figure 7.3 Demonstration of identical results obtained via (a) admittance spectroscopic analysis (SA) and (b) lumped parameter/complex plane analysis (LP/CPA) techniques where reciprocal temperature for the peak 1/T_P is plotted against spectroscopic term [Ln(T_P^–3/2/τ_X)] in (c) and reciprocal of various temperatures 1/T_X is plotted against spectroscopic term [Ln(T_x^–3/2/τ_P)] in (d).
Figure 7.4 (a) Bode plot of absolute impedance versus frequency, (b) Bode plot of phase angle versus frequency, and (c) Z*-plane plot for the R-C parallel circuit.
Figure 7.5 (a) Bode plot of absolute admittance versus frequency, and (b) Y*-plane plot for the R-C parallel circuit.
Figure 7.6 (a) Bode plot of absolute impedance versus frequency, (b) Bode plot of phase angle versus frequency, and (c) Z*-plane plot for the R-C parallel circuit.
Figure 7.7 Bode plot of R-C series and parallel circuits where horizontal line gives the value of R, and slope of the slant line dictates the value of C from the time constant τ.
Figure 7.8 Fitting of two consecutive semicircular relaxations in a complex plane where I* is the immittance parameter: (a) as it appears upon plotting, (b) attempted fitting having significant or finite overlapped region, and (c) correct way of fitting having no overlapped region.
Figure 7.9 (a) Behavior of the total measured data of the E. Coli bacteria suspended in the BHI medium in the Y*-plane displaying negative capacitance via resonance, (b) using only the positive portion of the data in the C*-plane, and (c) Bode plot via Log(|Z|) versus Log(ω) showing dc blocking R-C series circuit.

Chapter 8

Figure 8.1 (a) and (b) Equivalent circuit for electrolysis with two notations, and (c) Bode plot for the circuit in (a) and (b).
Figure 8.2 Warburg postulated model for the R-C series circuit of the electrode-electrolyte interface where both resistance and capacitance vary inversely with the square root of frequency, f, with R_O and C_O are equilibrium values dependent upon electrode material, electrolyte concentration and temperature.
Figure 8.3 Electrode-electrolyte interface via 45° phase shift for the Warburg impedance.
Figure 8.4 Bode plot of the Warburg impedance where the slope for the slant line is 0.5 and the capacitive element is referred to as W.
Figure 8.5 Measured data of the sterile BHI medium and E. Coli bacteria suspended in BHI medium in the Y*-plane for: the sterile BHI medium (•), the E. Coli after 3 hours (Δ), after 16.5 hours (o), and after 24 hours (×).
Figure 8.6 Measured data of the sterile BHI medium and E. Coli bacteria suspended in BHI medium in the C*-plane for: the sterile BHI medium (•), the E. Coli after 3 hours (Δ), after 16.5 hours (o), and after 24 hours (×).
Figure 8.7 Equivalent circuit obtained from the Y*-plane of Figure 6.5.
Figure 8.8 Measured data of the sterile BHI medium and E. Coli bacteria suspended in BHI medium in the C*-plane showing negative and positive domain of the data for: the sterile BHI medium (•), and the E. Coli after 16.5 hours (o).
Figure 8.9 Bode plot representing absolute impedance versus frequency of the sterile BHI medium (•), the E. Coli after 16.5 hours (o).

Chapter 10

Figure 10.1 Frequency-independent Mott-Schottky straight line at any given frequency for the ideal near non-trap controlled metal-semiconductor and p⁺-n junction devices.
Figure 10.2 Frequency-dependent converging Mott-Schottky straight lines at various frequencies for trap-controlled junction device converging at a single built-in-potential φ_i.
Figure 10.3 Frequency-dependent non-converging Mott-Schottky straight lines at various frequencies as always found in a multi-trap controlled ZnO-based varistor.
Figure 10.4 The Frequency-Dependent Mott-Schottky Slope for a series of non-converging straight lines in the C_P^–2 versus V_dc plot observed in ZnO-based varistor.
Figure 10.5 Generic zero-bias complex capacitance plot in the frequency range 10^–2 Hz ≤ f ≤ 10⁹ Hz showing the dc limit at f 0 Hz, resonance behavior at f → 100+ MHz, non-Debye trapping relaxations τ₄, τ₃, and τ₂ along with the heavily distorted τ₁ relaxation associating C₁.
Figure 10.6 Nonlinear current-voltage behavior of the ZnO-based varistor.
Figure 10.7 True Mott-Schottky behavior of the ZnO-based varistor reflecting frequency-independent straight line for the C_BL^–2 versus V_dc plot, where C_BL = C₅ + C₆ per Figure 10.5.

Chapter 11

Figure 11.1 Complex plane representation of the measured immittance data in the parallel mode comprising of C_P and G_P as the function of frequency f.
Figure 11.2 (a) Cole-Cole immittance response in complex plane where frequencies increase anti-clockwise along the arc of the semicircle for α = 0.8, I_LI = 0.1, I_RI = 1.1
Figure 11.2 (b) Determination of the peak frequency, f_peak, for α = 0.8 and τ = 10^-3 sec.
Figure 11.3 Flowchart of the program.
Figure 11.4 R-C parallel circuit with known values of the parameters used in data acquisition.
Figure 11.5 On-screen display of Matlab^® command window.
Figure 11.7 Measured data of the R-C circuit (Figure 11.4) displayed in the Z*-plane showing semicircular behavior.
Figure 11.8 Fitted semicircles to the R-C circuit (Figure 11.4) data in Z*-plane using both geometric and simultaneous fitting procedures.
Figure 11.9 Surface of Φ obtained from the semicircle fitting of the R-C circuit of Figure 11.4 data in Z*-plane.
Figure 11.10 Calculation of peak frequency for the R-C circuit of Figure 11.4 data in the Z*-plane.
Figure 11.11 Surface of Φ obtained from the simultaneous fitting of the R-C circuit of Figure 11.4 data in Z*-plane.
Figure 11.12 Comparison of results from CNLS fitting with parameters from the R-C parallel circuit.
Figure 11.13 Effect of the parameter α and fitting procedure on the estimated values of the parameters for synthetic Cole-Cole data.
Figure 11.14 Effect of measurement noise and fitting procedure on the estimated values of the parameters for synthetic Cole-Cole data.
Figure 11.15 Comparison between Gauss-Newton and Levenberg-Marquardt algorithms for the two fitting procedures.
Figure 11.16 Contour of Φ for Cole-Cole synthetic data with 2% Gaussian noise simulating Y*- or M*-plane with I_LI = 0.10, I_RI = 1.1, α = 0.8 and τ = 10^-3; (a) as function of the coordinates of center obtained from semicircle fitting resulting α = 0.812188 and τ = 1.002915 × 10^-3; (b) as function of α and τ obtained from simultaneous fitting resulting α = 0.807653 and τ = 1.006462 × 10^-3.
Figure 11.17 Estimated parameters in the complex planes from the geometric fitting.
Figure 11.18 Estimated parameters in the complex planes from the CNLS fitting.
Figure 11.19 Both geometric and simultaneous fitting procedures for the data of the R-C parallel circuit of Figure 11.4 in the M*-plane.
Figure 11.20 Complex plane representations of Lithium Niobate crystal data at 59 °C.
Figure 11.21 Fitted semicircles to Lithium Niobate data in C*-plane at 59 °C.
Figure 11.22 Determination of the peak frequency from the geometric fitting procedure in C*-plane for the Lithium Niobate crystal data at 59 °C.
Figure 11.23 Real part of complex capacitance versus frequency for Lithium Niobate crystal data at 59 °C and the comparison between the geometric (blue) and the CNLS (red) fitting procedure.
Figure 11.24 Imaginary part of complex capacitance versus frequency for Lithium Niobate crystal data at 59 °C and the comparison between the geometric (blue) and the CNLS (red) fitting procedure.
Figure 11.25 Magnitude of complex capacitance versus frequency for Lithium Niobate crystal data at 59 °C and the comparison between the geometric (blue) and the CNLS (red) fitting procedure.
Figure 11.26 Fitted semicircles to Lithium Niobate data in M*-plane at 59 °C.
Figure 11.27 Determination of the peak frequency from the geometric fitting procedure in complex modulus plane for Lithium Niobate crystal data at 59 °C.
Figure 11.28 Real part of complex modulus versus frequency for Lithium Niobate crystal data at 59 °C and the comparison between the geometric (blue) and the CNLS (red) fitting procedure.
Figure 11.29 Imaginary part of complex modulus versus frequency for Lithium Niobate crystal data at 59 °C and the comparison between the geometric (blue) and the CNLS (red) fitting procedure.
Figure 11.30 Four complex plane representations of the multi-layered junction device (SiO₂/SiO₂-Ge multi-layered un-bombarded thermoelectric device).
Figure 11.31 Fitted overlapping semicircles from geometric and CNLS fitting of the multi-layered junction device data in impedance plane. The data points used in fitting are denoted by their observation numbers.
Figure 11.32 Determination of the peak-frequency from the geometric fitting in the impedance plane for the multi-layered junction device data.
Figure 11.33 Fitted overlapping semicircles from geometric and CNLS fitting of the multi-layered junction device data in admittance plane. The data points used in fitting are denoted by their observation numbers.
Figure 11.34 Determination of the peak-frequency from the geometric fitting in the admittance plane for the multi-layered junction device data.
Figure 11.35 Measured data of the sterile BHI medium and E. Coli bacteria suspended in BHI medium in the Y*-plane. Legend: ∇- sterile BHI medium, E. Coli bacteria suspended in BHI medium after: 3 hours (•), 6.5 hours (+), 10 hours (×), 16.5 hours (Δ), and 24 hours (o) of growth.
Figure 11.36 Fitted semicircles (overlapped semicircles from both procedures of fitting) in the Y*-plane for the data of the E. Coli suspended BHI medium taken after 24 hours of growth.

Immittance Spectroscopy

Applications to Material Systems

Mohammad A. Alim

Department of Electrical Engineering and Computer Science Alabama A & M University Huntsville, Alabama, U.S.A.

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Library of Congress Cataloging-in-Publication Data

ISBN 978-1-119-18485-0

In the name of Allah, the Beneficent, the Merciful

Dedicated to

Father: Mohammad Alim Biswas, M.Sc., Ph.D., D.I.C., (1927 – 2009) son of Moulavi Mohammad Bhadu (Bahadur) Biswas (1885 – 1961) and Mosammat Saleha Khatun (1894 – 1970), who provided inspiration to comprehend general science beside navigating mathematics, chemistry, physics, history, geography, philosophy, and exposing world religions including exercising english and other global languages;
Mother: Mosammat Jinnatun Khatun (1934 –), daughter of Moulavi Mohammad Boytullah Mondol (1894 – 1976) and Mosammat Rowshan Ara Khatun (1904 – 1990), who guided to comprehend Bangla language making literature easy and walking through mathematics;
Wife: Sanjida Khanam, M.Sc., M.S., M.S.M.E., (1957 –), daughter of Muhammad Aminuddin (1919 – 2001) and Mosammat Anosha Khatun (1930 – 1989), who helped in achieivng academic and professional goals; and
Four Children: Nasiha Muna, Sabiha Runa, Maliha Lubna, and Mohammad E. Alim who helped in many ways to materialize this work.