Immittance Spectroscopy

Mohammad A. Alim is a Professor in the Department of Electrical Engineering & Computer Science at Alabama A & M University (AAMU) where he joined as one of the founding faculty members in August 1998. He earned MS in Physics and PhD in Electrical Engineering & Computer Science from Marquette University in 1980 and 1986, respectively. Dr. Alim is a single-handed pioneering developer of the concurrent multiple complex plane analysis of the measured ac small-signal electrical data. His approach demonstrated lumped-parameter/complex-plane-analysis employing complex nonlinear least squares (CNLS) fitting using Levenberg-Marquardt algorithm. The achievement of the frequency-independent dielectric behavior for the polycrystalline varistors was a milestone. This outstanding work has been highly cited for a variety of complicated material systems. Thus, the immittance (impedance or admittance) spectroscopy turned to a powerful non-destructive tool in delineating underlying operative competing phenomena in a variety of material systems and devices. Most recently Dr. Alim had been instrumental in developing collaboratively MATLAB based CNLS curve fitting. His long time exposure in experiments with the state-of-the-art instruments and knowledge in supervision and maintenance is the asset for the semiconductor measurements and reverse engineering curricula. He possesses 100+ publications comprising of co-edited books, book chapters, NASA Technical Memorandum, peer reviewed journal papers, U.S. patents, and conference proceedings/abstracts, etc. beside international seminars.

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