An Introduction to Condensed Matter Physics for the Nanosciences

355 Pages·2023·9.532 MB·English

by  Arthur McGurn

An Introduction to Condensed
Matter Physics for the
Nanosciences
The book provides an accessible introduction to the principles of condensed matter physics with a
focus on the nanosciences and device technologies. The basics of electronic, phononic, photonic,
superconducting, optics, quantum optics, and magnetic properties are explored, and nanoscience
and device materials are incorporated throughout the chapters. Many examples of the fundamental
principles of condensed matter physics are taken directly from nanoscience and device applications.
This book requires a background in electrodynamics, quantum mechanics, and statistical
mechanics at the undergraduate level. It will be a valuable reference for advanced undergraduates
and graduate students of physics, engineering, and applied mathematics.
Features
• Contains discussions of the basic principles of quantum optics and its importance to lasers,
quantum information, and quantum computation.
• Provides references and a further reading list to additional scientific literature so that read-
ers can use the book as a starting point to then follow up with a more advanced treatment
of the topics covered.
• Requires only a basic background in undergraduate electrodynamics, quantum mechanics,
and statistical mechanics.
Professor Emeritus Arthur McGurn, CPhys, FInstP, is a Fellow of the Institute of Physics, a
Fellow of the American Physical Society, a Fellow of Optica (formerly Optical Society of America),
a Fellow of the Electromagnetics Academy, and an Outstanding Referee for the journals of
the American Physical Society. He received his PhD in Physics in 1975 from the University of
California, Santa Barbara, followed by postdoctoral studies at Temple University, Michigan State
University, and George Washington University (NASA Langley Research Center). Prof. McGurn’s
research interests include the theory of: magnetism in disorder materials, electron conductivity, the
properties of phonons, ferroelectrics and their nonlinear dynamics, Anderson localization, amor-
phous materials, the scattering of light from disordered media and rough surfaces, the properties of
speckle correlations of light, quantum optics, nonlinear optics, the dynamical properties of nonlin-
ear systems, photonic crystals, and meta-materials. He has over 150 publications spread amongst
these various topics. Since 1981, he has taught physics for 38 years at Western Michigan University,
where he is currently a Professor Emeritus of Physics and a WMU Distinguished Faculty Scholar. A
number of PhD students have graduated from Western Michigan University under his supervision.

An Introduction to Condensed
Matter Physics for the
Nanosciences
Arthur McGurn
First edition published 2023
by CRC Press
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ISBN: 9780367466473 (hbk)
ISBN: 9780367517090 (pbk)
ISBN: 9781003031987 (ebk)
DOI: 10.1201/9781003031987
Typeset in Times
by codeMantra
Contents
Preface...............................................................................................................................................xi
Chapter 1 Introduction ..................................................................................................................1
1.1 Electrical Properties ..........................................................................................2
1.2 Optical Transport and the Interaction of Light with Matter ..............................4
1.3 Electrons in a Variety of Dimensions ................................................................4
1.4 Semiconductors .................................................................................................5
1.5 The Landauer Approach to Conductivity ..........................................................6
1.6 Photonic Crystals and Metamaterials ................................................................7
1.7 Quantum Optics.................................................................................................7
1.8 Anderson Localization and Mott Localization .................................................8
1.9 Quantum Hall Effect .........................................................................................9
1.10 Phenomena Related to the Hall Effect ............................................................10
1.11 Correlated Electron Systems ...........................................................................10
1.11.1 Superconductivity ...............................................................................10
1.12 Josephson Junctions .........................................................................................11
1.13 Fractional Quantum Hall Effect ......................................................................12
1.14 Coulomb Blockade ..........................................................................................12
1.15 Resonance ........................................................................................................13
1.16 Scaling and Renormalization Group ...............................................................13
References ..................................................................................................................15
Chapter 2 Conductivity ...............................................................................................................17
2.1 Basic Ideas of Conductivity .............................................................................17
2.2 Quantum Effects ..............................................................................................20
2.3 Magnetic Field Effects .....................................................................................25
2.3.1 Classical Treatment of a Two-Dimensional Gas in a Harmonic
Confining Potential ............................................................................28
2.3.2 Orbital and Landau Diamagnetism: Magnetism of a Two-
Dimensional Fermi Gas .....................................................................29
2.3.3 Magnetic Properties and Landau Diamagnetism of a Slab of
Fermi Gas ...........................................................................................33
2.3.4 Pauli Paramagnetism ..........................................................................34
2.4 Stoner Theory of Permanent Magnetism of Metals ........................................36
2.5 Localization Properties of the Fermi Gas Model ............................................40
2.5.1 Periodic Potential ...............................................................................45
2.5.2 Modes in the Anderson Localized Gas ..............................................46
2.5.3 Phase Coherence in Localization .......................................................48
2.5.4 Dependence of Localization on the System Dimension ....................50
2.5.5 Hopping and Variable-Range Hopping Conductivity ........................52
2.5.6 Ioffe–Regel Criterion and Minimum Metallic Conductivity .............56
2.6 Mott Transition ................................................................................................57
2.7 Wigner Crystal ................................................................................................58
2.8 Superconductivity ............................................................................................58
2.8.1 Planar Interface between a Superconductor and Normal Metal ........61
v
vi Contents
2.8.2 Flux Quantization ...............................................................................63
2.9 Ahronov–Bohm Effect in Normal Metals .......................................................65
References ..................................................................................................................66
Chapter 3 Conductivity: Another View ......................................................................................69
3.1 The Landauer Formulation ..............................................................................69
3.2 Scattering within the Waveguide: Ohmic Limit ..............................................73
3.3 Landauer–Buttiker Formula ............................................................................77
3.4 Universal Conductance Fluctuations ...............................................................82
3.5 Nonzero Temperature ......................................................................................85
References ..................................................................................................................87
Chapter 4 Properties of Periodic Media......................................................................................89
4.1 Tight-Binding Model .......................................................................................91
4.1.1 A One-Dimensional Model of a Chain of Atoms ..............................93
4.1.2 A Two-Dimensional Tight-Binding Model of Graphene ...................97
4.1.2.1 The Tight-Binding Hamiltonian of Graphene ..................103
4.1.2.2 Dispersive Properties of the Excitations in Graphene ......106
4.1.3 Graphene Conductivity.....................................................................107
4.1.4 Graphene Nanotubes ........................................................................108
4.1.5 Some of the Interesting Properties of Graphene ..............................109
4.2 Quantum Dots, Quantum Wells, and Quantum Wires..................................110
4.2.1 Properties of GaAs and Ga Al As .................................................111
1−x x
4.2.2 Effective Mass Approximation ........................................................112
4.2.3 Envelope Function Approximation ..................................................113
4.2.4 Quantum Wells and Heterostructures ..............................................115
4.2.4.1 Quantum Wells .................................................................115
4.2.4.2 Quantum Heterostructures ...............................................118
4.2.5 Quantum Wires and Dots .................................................................121
4.3 Excitons .........................................................................................................123
4.4 Photonic and Phononic Crystals ....................................................................125
4.4.1 Properties of Waves in Periodic Media ............................................126
4.4.2 Examples of Periodic Media in One and Two Dimensions ..............128
4.4.2.1 Examples of Two-Dimensional Photonic Crystals ...........129
4.4.2.2 Two-Dimensional Semiconducting Photonic Crystal .......129
4.4.3 Applications of Photonic and Phononic Crystals .............................134
References ................................................................................................................135
Chapter 5 Basic Properties of Light and Its Interactions with Matter ......................................137
5.1 Quantized Electromagnetic Waves ................................................................137
5.1.1 General Form of 3D Quantized Electromagnetic Waves .................144
5.1.2 Cavity Modes ...................................................................................145
5.1.3 Coherent States .................................................................................147
5.2 Field Interactions with Atoms and Electrons ................................................151
5.2.1 Jaynes–Cumming Model .................................................................151
5.2.2 Jaynes–Cumming Model: Example of Fock States .........................154
5.2.3 Jaynes–Cumming Model: Example of Coherent States ...................156
Contents vii
5.2.4 Jaynes–Cumming Model: Temperature Effects ...............................156
5.3 Optical Correlations and Coherence .............................................................157
References ................................................................................................................164
Chapter 6 Basic Properties of Lasers, Masers, and Spasers .....................................................165
6.1 Stimulated Emission ......................................................................................166
6.1.1 Deviations of the System from Thermal Equilibrium .....................168
6.2 Rate Equation Model of Laser Operations ....................................................169
6.3 Resonator Cavity ...........................................................................................171
6.4 Maser .............................................................................................................173
6.4.1 The Model ........................................................................................173
6.4.2 The Hamiltonian: Absence of Maser Fields .....................................174
6.4.3 Density Matrix .................................................................................176
6.4.4 Hamiltonian: Phenomenological Dissipative Terms ........................177
6.4.5 Development of the Solutions with Dissipative Effects ...................179
6.4.6 Density Matrix of the Maser ............................................................180
6.4.7 The General Atomic Passage Maser Processes ...............................183
6.4.8 The Loss Terms in the Maser Processes ..........................................184
6.4.9 Statistical Properties of Maser Radiation .........................................187
6.5 Spasers and Atom Lasers ..............................................................................188
References ................................................................................................................189
Chapter 7 Semiconductor Junctions ..........................................................................................191
7.1 Semiconductor Model ....................................................................................192
7.1.1 Thermal Occupancy .........................................................................193
7.1.2 Extrinsic Semiconductors: n- and p- Type Materials .......................194
7.1.3 Positioning of the Chemical Potential ..............................................194
7.2 Semiconductor Junction Model .....................................................................195
7.2.1 Electrostatics at the Junction ............................................................197
7.2.2 Application of a Potential across the Junction .................................199
7.2.3 Resulting Current versus Voltage Relationship of the Junction .......201
References ................................................................................................................202
Chapter 8 Rectifiers and Transistors .........................................................................................203
8.1 Rectifiers and Transistors ..............................................................................204
8.1.1 The Transition Region in a p–n Junction .........................................205
8.1.2 The Transition Region Characteristics .............................................206
8.2 Field Effect Transistor ...................................................................................208
8.2.1 p–n–p Transistor ..............................................................................209
8.2.2 Basic Transistor Circuit ....................................................................210
8.2.3 Geometry of the Subregion of Net Positive Charge within the
n-Material Layer ...............................................................................211
8.2.4 Connected Region of Charge-Neutral n-Material between the
Source and Drain ..............................................................................212
8.2.5 Disconnected Region of Charge-Neutral n-Material between
the Source and Drain ........................................................................215
8.2.6 Conditions of Zero Drain Current ....................................................217
viii Contents
8.2.7 Switches and Amplifier Circuits ......................................................218
8.2.8 n–p–n Transistors .............................................................................220
8.3 Bipolar Transistor ..........................................................................................220
References ................................................................................................................224
Chapter 9 Toward Single-Electron Transistors: Coulomb Blockade ........................................225
9.1 A Single Island Device ..................................................................................227
9.2 Single-Electron Transistor .............................................................................230
9.2.1 Electron Transitions between the Source and the Island .................233
9.2.2 Electron Transitions between the Drain and the Island ...................235
9.2.3 Stability of N Net Uncompensated Electrons on the Island ............236
9.3 Applications of Single-Electron Transistors ..................................................238
References ................................................................................................................238
Chapter 10 Quantum Hall Effect ................................................................................................241
10.1 Two Quantum Hall Effects ............................................................................242
10.1.1 Integer Quantum Hall Effect ............................................................242
10.1.2 Fractional Quantum Hall Effects .....................................................244
10.2 Classical Model of the Hall Effect ................................................................245
10.3 Theory of the Integer Quantum Hall Effect ..................................................247
10.3.1 Transverse Conductivity ...................................................................249
10.3.1.1 Another Approach to the Conductivity ............................250
10.3.2 Shubnikov-de Hass Effect ................................................................252
10.4 Fractional Quantum Hall Effect ....................................................................254
10.4.1 Free Electrons in High Fields ...........................................................254
10.4.2 Wave Function for the Fractional Quantum Hall Effect ..................258
10.4.3 The Electron–Electron Interactions .................................................260
1
10.4.4 Conductivity and Resistivity of the Fractional Hall State ............261
3
10.4.5 Conclusions ......................................................................................262
10.5 Spin ................................................................................................................263
10.6 Spin Hall Effect and the Spin Hall Effect of Light .......................................264
References ................................................................................................................265
Chapter 11 Resonance Properties ...............................................................................................267
11.1 Metamaterial Responses and Simple Spatial Resonances ............................268
11.2 Standard Resonance Involving Quantum Wells ............................................270
11.3 Fano Resonance Involving Quantum Wells ..................................................275
11.4 Topological Excitations .................................................................................279
References ................................................................................................................280
Chapter 12 Josephson Junction Properties and Basic Applications ............................................281
12.1 Time-Dependent Ginzburg–Landau Free Energy .........................................281
12.1.1 Schrodinger Gauge Symmetry .........................................................282
12.1.2 Ginzburg–Landau Form ...................................................................283
12.1.2.1 An Example ......................................................................284
12.2 Josephson Junction ........................................................................................285
12.3 Spatial Dependent Effects: Magnetic Fields .................................................288
Contents ix
12.4 Josephson Junction in a Static Magnetic Field ..............................................289
12.5 Real-World versus Ideal Josephson Junction .................................................290
12.6 Josephson Junctions of Finite Cross-Sectional Area .....................................292
12.6.1 Two Junctions in Parallel..................................................................293
12.7 Type I and Type II Superconductors and Interfaces with Normal Metals ....295
12.8 High-Temperature Superconductors ..............................................................301
References ................................................................................................................304
Chapter 13 Scaling and Renormalization ...................................................................................305
13.1 One-Dimensional Ferromagnet .....................................................................307
13.2 Second-Order Phase Transitions ...................................................................310
13.3 First-Order Phase Transitions ........................................................................313
13.4 Scaling Theory ..............................................................................................315
13.4.1 Examples of the Two-Dimensional Ising Model and Landau
Theory ..............................................................................................319
13.4.2 Natural Length Scales ......................................................................321
13.5 Renormalization Group Approach ................................................................322
13.6 Application of the Renormalization Group to the Landau Formulation .......332
Appendix ..................................................................................................................336
References ................................................................................................................337
Index ..............................................................................................................................................339