This book emphasizes the use of four complex plane formalisms (impedance, admittance, complex capacitance, and modulus) in a simultaneous fashion. The purpose of employing these complex planes for handling semicircular relaxation using a single set of measured impedance data (ac small-signal electrical data) is highly underscored. The current literature demonstrates the importance of template version of impedance plot whereas this book reflects the advantage of using concurrent four complex plane plots for the same data. This approach allows extraction of a meaningful equivalent circuit model attributing to possible interpretations via potential polarizations and operative mechanisms for the investigated material system. Thus, this book supersedes the limitations of the impedance plot, and intends to serve a broader community of scientific and technical professionals better for their solid and liquid systems. This book addresses the following highlighted contents for the measured data but not limited to the:- (1) Lumped Parameter/Complex Plane Analysis (LP/CPA) in conjunction with the Bode plots; (2) Equivalent circuit model (ECM) derived from the LP/CPA; (3) Underlying Operative Mechanisms along with the possible interpretations; (4) Ideal (Debye) and non-ideal (non-Debye) relaxations; and (5) Data-Handling Criteria (DHC) using Complex Nonlinear Least Squares (CNLS) fitting procedures.

Table of Contents:
Background of this Book xiii Acknowledgments xxiii 1 Introduction to Immittance Spectroscopy 1 1.1 Basic Definition and Background 1 1.2 Scope and Limitation 5 1.3 Applications of the Immittance Studies to Various Material Systems 6 1.4 Concept of the Linear Circuit Elements: Resistance, Capacitance, and Inductance 9 1.5 Concept of Impedance, Admittance, Complex Capacitance, and Modulus 13 1.6 Immittance Functions 21 1.7 Series Resonant Circuit 22 1.8 Parallel Resonant Circuit 23 1.9 Capacitance and Inductance in Alternating Current 24 Problems 24 References 25 2 Basics of Solid State Devices and Materials 27 2.1 Overview of the Fundamentals of Physical Electronics 27 2.2 Basics of Semiconductors 33 2.3 Single-Crystal and Polycrystal Materials 35 2.4 SCSJ and MPCHPH Systems 37 2.5 Representation of the Competing Phenomena 42 2.6 Effect of Normalization of the Electrical Parameters 43 Problems 46 References 47 3 Dielectric Representation and Operative Mechanisms 49 3.1 Dielectric Constant of Materials: Single Crystals and Polycrystals 49 3.2 Dielectric Behavior of Materials: Single Crystals and Polycrystals 53 3.3 Origin of Frequency Dependence 58 3.4 Effect of Polarization 60 3.5 Equivalent Circuit Representation of the Mechanisms and Processes 67 3.6 Defects and Traps 69 3.7 Point Defects and Stoichiometric Defects 77 3.8 Leaky Systems 78 Problems 79 References 80 4 Ideal Equivalent Circuits and Models 85 4.1 Concept of Equivalent Circuit 85 4.2 Simple and Basic Circuits in Complex Planes: R, C, R-C Series, and R-C Parallel 86 4.3 Debye Circuits: Single Relaxation 89 4.4 Duality of the Equivalent Circuits: Multiple Circuits for a Single Plane 97 4.5 Duality of Equivalent Circuits between Z*- and M*-Planes for Relaxations without Intercept 98 4.6 Duality of Equivalent Circuits between Y*- and C*-Planes for Relaxations without Intercept 100 4.7 Duality of Equivalent Circuits for Simultaneous Z*-, Y*-, C*-, and M*-Planes’ Relaxations 102 4.8 Proposition of Equivalent Circuit: Polycrystalline Grains and Grain Boundaries 103 Problems 105 References 106 5 Debye and Non-Debye Relaxations 109 5.1 Ideal Systems 109 5.2 Non-Ideal Systems 116 5.3 Non-Ideal Systems Implying Distributed Time Constants 122 5.4 D-C Representation, Depression Parameter, and Equivalent Circuit: Conventional Domain 128 5.5 Depression Parameter Based on ωτpeak = 1: Complex Domain 134 5.6 Optimization of ZHF: Complex Domain 137 5.7 Depression Parameter β Based on ωτpeak = 1 139 5.8 Feature of the Depression Parameter β Based on ωτ π 1 145 5.9 Analysis of the Havriliak-Negami Representation 146 5.10 Geometrical Interpretation of H-N Relaxation at the Limiting Case 151 5.11 Extraction of the Relaxation Time τ and the H-N Depression Parameters α and β 154 5.12 Checking Generalized Depression Parameter β when α is Real 159 5.13 Checking Generalized Depression Parameter α when β is Real 160 5.14 Effect of α and β on the H-N Distribution Function 162 5.15 Meaning of the Depression Parameters α and β 166 5.16 Relaxation function with Respect to the Depression Parameters α and β 168 Problems 170 References 170 6 Modeling and Interpretation of the Data 175 6.1 Equivalent Circuit Model for the Single Complex Plane (SCP) Representation 175 6.2 Models and Circuits 177 6.3 Nonconventional Circuits 184 6.4 Multiple Equivalent Circuits for Multiple Relaxations in a Single Complex Plane 186 6.5 Single Equivalent Circuit for Multiple Complex Planes 187 6.6 Equivalent Circuit for Resonance 189 6.7 Single Equivalent Circuit from Z*- and M*-Planes 189 6.8 Temperature and Bias Dependence of the Equivalent Circuit Modeling 190 6.9 Equivalent Circuit: Zinc Oxide (ZnO) Based Varistors 191 6.10 Equivalent Circuit: Lithium Niobate LiNbO3 Single Crystal 196 6.11 Equivalent Circuit: Polycrystalline Yttria (Y2O3) 200 6.12 Equivalent Circuit: Polycrystalline Calcium Zirconate (CaZrO3) 201 6.13 Equivalent Circuit: Polycrystalline Calcium Stannate (CaSnO3) 202 6.14 Equivalent Circuit: Polycrystalline Titanium Dioxide (TiO2) 203 6.15 Equivalent Circuit: Multi-Layered Thermoelectric Device (Alternate SiO2/SiO2+Ge Thin-Film) 204 6.16 Equivalent Circuit: Polycrystalline Tungsten Oxide (WO3) 206 6.17 Equivalent Circuit: Biological Material – E. Coli Bacteria 207 Problems 208 References 209 7 Data-Handling and Analyzing Criteria 213 7.1 Acquisition of the Immittance Data 213 7.2 Lumped Parameter/Complex Plane Analysis (LP/CPA) 214 7.3 Spectroscopic Analysis (SA) 222 7.4 Bode Plane Analysis (BPA) 225 7.5 Misrepresentation of the Measured Data 227 7.6 Misinterpretation of the Bode Plot: Equivalent Circuit 230 Problems 232 References 233 8 Liquid Systems 241 8.1 Non-Crystalline Systems: Liquids 241 8.2 Warburg and Faradaic Impedances 245 8.3 Constant Phase Element (CPE) 249 8.4 Biological Liquid: E. Coli Bacteria 251 Problems 255 References 256 9 Case Study 259 9.1 Analysis of the Measured Data: Aspects of Data-Handling/Analyzing Criteria 259 9.2 Case 1: Proper Physical Geometrical Factors 260 9.3 Case 2: Improper Normalization 262 9.4 Case 3: Effect of Electrode and Lead Wire 264 9.5 Case 4: Identification of Contributions to the Terminal Immittance 265 9.6 Case 5: Use of Proper Unit 267 9.7 Case 6: Demonstration of the Invalid Plot 270 9.8 Case 7: Obscuring Frequency Dependence 271 9.9 Case 8: Misnomer Nomenclature for the Complex Plane Plot 273 9.10 Case 9: Extraction of Equivalent Circuit from the Straight Line or the Non-Relaxation Curve 274 Problems 277 References 278 10 Analysis of the Complicated Mott-Schottky Behavior 283 10.1 Capacitance – Voltage (C-V) Measurement 283 10.2 The Mott-Schottky Plot 287 10.3 Arbitrary Measurement Frequency and Construction of the Deceiving Mott-Schottky Plot 296 10.4 Frequency-Independent Representation 297 10.5 Extraction of the Device-Related Parameters 299 Problems 302 References 303 11 Analysis of the Measured Data 307 11.1 Introduction and Background of the Immittance Data Analysis 307 11.2 Measurement of the Immittance Data and Complex Plane Analysis 312 11.3 Nonlinear Least Squares Estimation 314 11.3.1 Gauss-Newton Method (Algorithm) of Least Squares Estimation 317 11.3.2 Levenberg-Marquardt Method (Algorithm) of Least Squares Estimation 320 11.3.3 Numerical Procedure to Calculate Jacobian Matrix 321 11.3.4 Error Analysis: Analysis of Errors in Regression 321 11.3.5 Selection of the Weights 322 11.4 Complex Nonlinear Least Squares (CNLS) Fitting of the Data 323 11.4.1 Procedure 1: Geometrical Fitting in the Complex Plane 323 11.4.2 Procedure 2: Simultaneous Fitting of Real and Imaginary Parts 328 11.5 Graphical User  Interface Implementation of the Nonlinear Least Square Procedures: Implementation of CNLS using MATLAB 330 11.5.1 Input Data Generation 330 11.5.2 Input Data Processing 331 11.5.2.1 Visualization of the Measured (Raw) Data 332 11.5.2.2 Selection of Data Points for Fitting 333 11.5.2.3 Fitting of the Semicircle: Geometric Fitting 334 11.5.2.4 Calculation of the Parameters from the Semicircle Fitting 335 11.5.2.5 Calculation of the Parameters from the Simultaneous Fitting of Real and Imaginary Parts 336 11.5.3 Output Generation: Output File 337 11.5.3.1 Parameters from the Semicircle Fitting 337 11.5.3.2 Nonlinear Regression: Semicircle Fitting Output 337 11.5.3.3 Linear Regression: Line Fitting Output 338 11.5.3.4 Parameters from Simultaneous Fitting of Real and Imaginary Data 338 11.5.3.5 Nonlinear Regression: Simultaneous Fitting of Real and Imaginary Data Output 338 11.5.3.6 Measured Data used in Analysis 339 11.6 Effect of Fitting Procedure, Measurement Noise, and Solution Algorithm on the Estimated Parameters 340 11.7 Case Studies: CNLS Fitting of the Measured Data in the Complex Planes 342 11.7.1 M*-Plane Fitting: R-C Parallel Circuit 343 11.7.2 C*- and M*-Plane Representations of the Lithium Niobate (LN) Crystal 344 11.7.3 Z*- and Y*-Plane Representations of Multi-Layered Junction Device 349 11.7.4 Y*-plane Representation of the E. Coli Bacteria in Brain Heart Infusion Medium 351 11.8 Summary 353 Problems 355 References 357 12 Items for Appendix 363 12.1 Appendix – A: Sample Input Data for the R-C Parallel Circuit 363 12.2 Appendix – B: R-C Parallel Circuit Data Analysis Output in Z*-Plane 364 12.3 Appendix – C: R-C Parallel Circuit Data Analysis Output in M*-Plane 368 12.4 Appendix – D: Lithium Niobate Crystal Data Analysis Output in C*-Plane 370 12.5 Appendix – E: Multilayer Junction Thermoelectric Device Data Analysis Output in Y*-Plane 372  Index