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
