Preface ....................................................... vii
1 Introduction ................................................. 1
Quantum Tunneling Theory ........................................ 5
2 Quantum Physics and Quantum Formalism ........................ 7
2.1 Quantum Phenomena ....................................... 7
2.2 Quantum Characteristics ................................. 7
2.3 Quantum Formalism ....................................... 8
2.4 Probability Current and Current Conservation ........... 14
2.5 Quantum Physics versus Classical Physics ............... 16
2.6 Mesoscopic Physics and Characteristic Length ........... 18
2.6.1 Characteristic Length ........................... 18
2.6.2 Characteristic Transports ....................... 20
2.7 Mathematics in Classical and Quantum Worlds ............ 21
3 Basic Physics of Quantum Scattering and Tunneling ........... 23
3.1 Definitions of Quantum Scattering and Tunneling ........ 23
3.2 Description of Quantum Scattering and Tunneling ........ 24
3.3 Basic Physical Quantities in Quantum Tunneling ......... 26
3.3.1 Transmission and Reflection Coefficients ........ 26
3.3.2 Conductance: Landauer-Bьttiker Formula .......... 26
3.3.3 Charge Current .................................. 27
3.4 Relationships between Transmission Coefficient and
Scattering Matrix ...................................... 27
3.5 Basic Properties of Scattering and Transfer Matrices ... 29
3.6 Constraints of Scattering and Transfer Matrices ........ 35
4 Wave Function Matching Method ............................... 37
4.1 Square Barrier Model ................................... 38
4.2 Asymmetric Square Barrier Model ........................ 40
4.3 Double Square Barrier Model ............................ 43
4.4 Multi-Mode Square Barrier Model ........................ 45
4.5 Triangle Barrier ....................................... 47
4.6 Lattice Models ......................................... 51
4.6.1 One-dimensional Model ........................... 51
4.6.2 Two-chain Model ................................. 54
4.6.3 2D Square Lattice ............................... 58
5 WKB Method .................................................. 61
5.1 Mathematics of WKB Method .............................. 61
5.2 Validity ............................................... 63
5.3 Solution of Schrцdinger Equation ....................... 63
5.4 Quantum Tunneling ...................................... 64
5.5 Triangle Barrier ....................................... 65
5.6 Triangle and Image Potential Barrier ................... 67
6 Lippmann-Schwinger Formalism ................................ 71
6.1 Lippmann-Schwinger Equation ............................ 71
6.2 Wave Function and S Matrix ............................. 73
6.3 Green's Function and T Matrix .......................... 74
6.4 S Matrix ............................................... 76
6.5 Adiabatic Transport Model .............................. 77
6.6 Quantum Tunneling in Time-Dependent Barrier ............ 79
6.6.1 Floquet Theory .................................. 79
6.6.2 Time-Dependent Barrier .......................... 80
7 Non-Equilibrium Green's Function Method ..................... 83
7.1 Basic Physics of Non-Equilibrium Transport Problems .... 83
7.2 Model of Nanodevices ................................... 84
7.3 Green's Functions and Self-Energy ...................... 86
7.4 Spectral Function, Density of States, and Correlation
Function ............................................... 88
7.5 Definitions and Relationships .......................... 90
7.6 Current ................................................ 91
7.7 Tunneling Model and Master Equation .................... 93
8 Spin Tunneling .............................................. 97
8.1 Tunneling Magnetoresistance Phenomena .................. 97
8.2 Julliere Model ......................................... 98
8.3 Giant Magnetoresistance ............................... 101
8.4 Spin Tunneling in Spin-Orbital Coupling
Semiconductors ........................................ 102
8.4.1 Model and Issue ................................ 102
8.4.2 Ferromagnetic Nanowires ........................ 104
8.4.3 Spin-Orbital Coupling Semiconductor ............ 106
8.5 Spin Polarization ..................................... 110
8.6 Remarks ............................................... 117
9 Applications ............................................... 119
9.1 Josephson Effect ...................................... 119
9.2 Theory of Scanning Tunneling Microscopy ............... 121
9.2.1 Quantum Electron Tunneling and Bardeen's
Formula ........................................ 122
9.2.2 Tersoff-Hamann Formula ......................... 123
9.2.3 Non-Equilibrium Green's Function Method ........ 125
9.3 Conductance of Graphene ............................... 125
9.3.1 Graphene Nanoribbons Model ..................... 127
9.3.2 Impurity Effects ............................... 128
9.3.3 Vacancy and Impurity ........................... 130
9.3.4 Conclusion ..................................... 131
9.4 Charge Transfer in DNA ................................ 132
9.4.1 G4-DNA Model ................................... 133
9.4.2 TG4 and Their Classifications .................. 135
9.4.3 Anomalous Conductance in NCM(H)TG4 ............. 136
9.4.4 Topological Structure Transition versus
Telomerase Activation and Inhibition ........... 138
9.4.5 Conclusion ..................................... 139
9.5 Remarks ............................................... 140
Field Electron Emission Theory ................................ 141
10 Introduction ............................................... 143
10.1 Field Electron Emission Phenomenon .................... 143
10.2 Brief Histroy of Field Electron Emission .............. 143
10.3 Basic Concepts of Field Electron Emission ............. 144
10.3.1 Electron Emissions from Solids ................. 144
10.3.2 Work Function and Field Emission Condition ..... 145
10.3.3 Basic Experiment Components of Field
Emission ....................................... 145
10.3.4 Applications of Field Emission ................. 146
10.4 Basic Issues of Field Electron Emission ............... 146
10.4.1 Theoretical Issues ............................. 146
10.4.2 Engineering Issues ............................. 147
10.5 Novel Phenomena and Challenges of Field Emission ...... 148
10.5.1 New Phenomena .................................. 148
10.5.2 Challenging Problems ........................... 149
11 Theoretical Model and Methodology .......................... 151
11.1 Theoretical Model of Field Emission ................... 151
11.2 Theoretical Methodology ............................... 152
11.2.1 Model and Analytic Solution .................... 153
11.2.2 Computer Simulation ............................ 153
11.2.3 Empirical Method ............................... 153
11.3 Remarks ............................................... 153
12 Fowler-Nordheim Theory ..................................... 157
12.1 Assumptions of Fowler-Nordheim Theory ................. 157
12.2 Fowler-Nordheim Theory ................................ 158
12.2.1 Field Emission Equation I: Fowler-Nordheim
Method ......................................... 160
12.2.2 Field Emission Equation II: Young-Gadzuk's
Method ......................................... 163
12.2.3 Field Emission Equation III: R. Forbes'
Method ......................................... 164
12.2.4 Field Emission Equation VI: A. Haug's Method ... 166
12.3 Remarks ............................................... 167
12.4 Beyond Triangular Vacuum Potential Barrier ............ 168
12.4.1 General Formalism .............................. 169
12.4.2 Generalized Triangular Barrier ................. 171
12.4.3 Schottky-Nordheim Barrier: Image Potential
Effect ......................................... 172
12.4.4 Beyond Gamow Exponent Form ..................... 175
12.4.5 Emitter Curvature and Field Enhancement
Factor ......................................... 175
12.4.6 Space Charge Effect ............................ 176
12.4.7 Small-Scale Effect of Emitter .................. 178
12.4.8 Emission Area and Total Emission Current ....... 178
12.5 Energy Band Effect .................................... 178
12.5.1 Supply Function Density ........................ 179
12.5.2 Transmission Coefficient and Total Energy
Distribution ................................... 179
12.5.3 Emission Current Density ....................... 181
12.6 Finite Temperature Effect ............................. 182
12.7 Basic Characteristic of Current-Field Relation ........ 184
12.7.1 Current-Field Characteristic ................... 184
12.7.2 Maximum Emission Current Density ............... 185
12.7.3 FN Plot ........................................ 186
12.8 Energy Distribution of Emission Electrons ............. 191
12.8.1 Total Energy Distribution (TED) ................ 191
12.8.2 Normal Energy Distribution (NED) ............... 193
12.8.3 Basic Characteristics of TED and NED ........... 194
12.8.4 Measurement of Energy Distributions ............ 202
12.9 Nottingham Effect ..................................... 204
13 Field Emission from Semiconductors ......................... 209
13.1 Basic Properties of Semiconductors .................... 210
13.1.1 Energy Band Structure .......................... 210
13.1.2 Temperature Dependence of Energy Band Gap ...... 210
13.1.3 Carrier Concentration .......................... 211
13.2 Model of Field Emission from Semiconductors ........... 212
13.3 Supply Function Density ............................... 213
13.4 Vacuum Potential Barrier and Transmission
Coefficient ........................................... 213
13.5 Total Energy Distribution ............................. 215
13.6 Basic Characteristics of Total Energy Distribution .... 217
13.7 Emission Current Density .............................. 218
14 Surface Effects and Resonance .............................. 221
14.1 Field Emission Model with Surface Effects ............ 221
14.2 Double-Barrier Vacuum Potential and Transmission
Coefficient ........................................... 222
14.3 Total Energy Distribution ............................. 226
14.4 Emission Current Density .............................. 227
15 Thermionic Emission Theory ................................. 231
15.1 The Richardson Theory of Thermionic Emission .......... 231
15.2 Boundary of Field Emission and Thermionic Emission .... 233
16 Theory of Dynamical Field Emission ......................... 237
16.1 Adiabatic Process and Dynamic Field Emission Model .... 237
16.2 Supply Function and Time-Dependent Transmission
Coefficient ........................................... 238
16.3 Dynamic Total Energy Distribution ..................... 239
16.4 Dynamic Normal Energy Distribution .................... 240
16.5 Dynamic Emission Current .............................. 241
16.6 Quantum Tunneling Time ................................ 242
17 Theory of Spin Polarized Field Emission .................... 247
17.1 Basic Physics of Spin Polarized Field Emission ........ 247
17.2 Energy Band Spin-Split Model .......................... 249
17.2.1 Supply Function and Transmission Coefficient ... 249
17.2.2 Total Energy Distribution ...................... 250
17.2.3 Normal Energy Distribution ..................... 251
17.2.4 Emission Current Density and Spin
Polarization ................................... 252
17.3 Spin-Dependent Triangular Potential Barrier Model ..... 254
17.3.1 Spin-dependent Triangular Potential Barrier
and Transmission Coefficient ................... 254
17.3.2 Total Energy Distribution: ..................... 256
17.3.3 Normal Energy Distribution: .................... 256
17.3.4 Emission Current Density and Spin
Polarization ................................... 257
17.4 Spin-Dependent Image Potential Barrier Model .......... 259
17.4.1 Spin-dependent Image Potential Barrier and
Transmission Coefficient ....................... 259
17.4.2 Total and Normal Energy Distributions .......... 260
17.4.3 Emission Current Density and Spin
Polarization ................................... 261
17.5 Finite Temperature Effects ............................ 263
17.5.1 Energy-Band Spin-Split Model .................. 263
17.5.2 Spin-Dependent Triangular Potential Barrier
Model .......................................... 264
17.5.3 Spin-Dependent Image Potential Barrier Model ... 264
17.6 Comparison of Spin Polarizations ...................... 265
17.7 A Scheme of Pure Spin Polarized Electron Emission
Induced by Quantum Spin Hall Effect ................... 266
17.8 Difficulties and Possibilities of Spin Polarized
Field Emission ........................................ 268
18 Theory of Field Electron Emission from Nanomaterials ....... 271
18.1 Basic Physics of Field Emission from Nanoemitters ..... 271
18.2 Formulation of Field Emission Current Density ......... 273
18.2.1 Supply Function Density ........................ 274
18.2.2 Current Density ................................ 274
18.2.3 Density of States .............................. 274
18.2.4 Transmission Coefficient ....................... 274
18.2.5 Distribution Function .......................... 278
18.2.6 Total Energy Distribution ...................... 278
18.2.7 Emission Current Density ....................... 279
18.3 Computational Framework ............................... 279
18.4 Special Case I: Sommerfeld Model ...................... 280
18.5 Special Case II: Nanowires ............................ 280
18.6 Special Case III: Coupled Nanowires ................... 284
18.7 Thermionic Emission of Nanowires ...................... 290
18.8 Theory of Field Electron Emission from Carbon
Nanotubes ............................................. 292
18.8.1 Energy Dispersion and Density of States ........ 293
18.8.2 Density of States and Group Velocity ........... 293
18.8.3 Supply Function and Transmission Coefficient ... 294
18.8.4 Total Energy Distribution ...................... 295
18.8.5 Emission Current Density ....................... 295
18.8.6 Finite Temperature Effect ...................... 301
18.8.7 Thermionic Emission ............................ 301
18.9 Theory of Luttinger Liquid Field Emission ............. 303
19 Computer Simulations of Field Emission ..................... 305
19.1 Basic Idea on Computer Simulation .................... 305
19.2 Formulation of Field Emission Based on Non-
Equilibrium Green's Function Method ................... 306
19.2.1 Generalized Supply Function .................... 307
19.2.2 Transmission Coefficient ....................... 308
19.2.3 Total Energy Distribution and Emission
Current Density ................................ 308
19.3 Tight-Binding Approach ................................ 309
19.3.1 Computational Formulation ...................... 309
19.3.2 Carbon Nanotubes ............................... 310
19.3.3 Total Energy Distribution and Emission
Current ........................................ 312
19.3.4 Computational Framework ........................ 313
19.3.5 Basic Properties of Field Emission of SWCN ..... 314
19.4 Cap and Doping Effects ................................ 319
19.5 Field Penetration Effect and Field Enhancement
Factor ................................................ 320
19.6 First-Principle Method ................................ 321
19.6.1 The Multi-Scale Technique ...................... 321
19.6.2 The ab-initio Tight-Binding Method ............. 322
19.6.3 Lippman-Schwinger Scattering Formalism ......... 322
20 The Empirical Theory of Field Emission ..................... 323
20.1 The Empirical Theory of Field Emission ................ 323
20.2 The Generalized Empirical Theory of Field Emission .... 324
20.3 The Empirical Theory of Thermionic Emission ........... 325
20.4 Connection between Empirical Theory and Experimental
Data .................................................. 325
21 Fundamental Physics of Field Electron Emission ............. 327
21.1 Field Emission Behavior and Material Properties ....... 327
21.2 Equilibrium and Non-Equilibrium Currents .............. 328
21.3 Many-Body Effect ...................................... 329
21.4 Coherent and Non-Coherent Emission Currents ........... 330
21.5 Electron Emission Mechanism: Nano versus Bulk
Effects ............................................... 330
21.6 Universality versus Finger Effects .................... 331
21.7 Open Problems and Difficulties ........................ 332
21.8 Perspectives .......................................... 333
Appendix A Appendices ........................................ 335
A.1 Basic Properties of S and M Matrices .................. 335
A.1.1 Proof of Theorem 3.5 ........................... 335
A.1.2 Proof of Theorem 3.7 ........................... 336
A.1.3 Proof of Theorem 3.8 ........................... 336
A.1.4 Proof of Theorem 3.9 ........................... 338
A.2 Spin Tunneling ........................................ 340
A.2.1 Proof of Claim 8.1b and Claim 8.2b ............. 340
A.2.2 Proof of Claim 8.2 ............................. 341
A.2.3 Proof of Theorem 8.1 ........................... 341
A.2.4 Proof of Theorem 8.2 ........................... 342
A.2.5 Proof of Theorem 8.3 ........................... 343
A.3 Derivations in Non-Equilibrium Green's Function
Method ................................................ 343
A.3.1 Basic Relationships ............................ 343
A.3.2 Non-Equilibrium Current ........................ 344
A.4 Models of Solids ...................................... 346
A.4.1 Sommerfeld Model of Metals ..................... 346
A.4.2 Crystal Lattice Model and Bloch Theorem ........ 348
A.4.3 Tight-Binding Model ............................ 349
A.4.4 Remarks of Solid Model ......................... 351
A.5 Density of States ..................................... 351
A.5.1 Definition of Density of States ................ 351
A.5.2 Sommerfeld Model (Electron Gas) ................ 351
A.5.3 Beyond Sommerfeld Model ........................ 352
A.5.4 Non-Equilibrium Cases .......................... 353
A.6 Fermi Wave Vector and Fermi Wavelength ................ 354
A.6.1 Definitions of Fermi Wave Vector and Fermi
Wavelength ..................................... 354
A.6.2 Sommerfeld Model ............................... 355
A.7 The Widths of TED and NED ............................. 356
A.7.1 TED ............................................ 356
A.7.2 NED ............................................ 357
A.8 Spin Polarized Field Emission ......................... 358
A.9 Field Emission from Nanomaterials ..................... 360
A.9.1 Nanowire Integration ........................... 360
A.9.2 Coupled Nanowire ............................... 361
A.10 Carbon Nanotubes ...................................... 363
A.10.1 Graphene ....................................... 363
A.10.2 Lattice Structure of Single-Wall Carbon
Nanotubes (SWCN) ............................... 364
A.10.3 Unit Cell and Brillouin Zone of SWCN ........... 365
A.10.4 Energy Dispersion Relation of SWCN ............. 366
A.10.5 Energy Gap ..................................... 367
A.10.6 Density of States of SWCN ...................... 368
А.10.7 Multi-Wall Carbon Nanotubes (MWCN) ............. 368
A.ll Physical Constants .................................... 371
A.12 Field Emission Constants .............................. 372
A.13 Epilogue .............................................. 373
Bibliography .................................................. 375
Index ......................................................... 385
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