BEIJING SHANGHAI Quantum Tunneling and Field Electron Emission Theories Shi-Dong Liang Sun Yat-Sen University, China World Scientific NEW JERSEY LONDON SINGAPORE HONG KONG TAIPEI CHENNAI
Contents Preface vii 1 Introduction 1 Quantum Tunneling Theory 5 2 Quantum Physics and Quantum Formalism 7 21 Quantum Phenomena 7 22 Quantum Characteristics 7 23 Quantum Formalism 8 24 Probability Current and Current Conservation 14 25 Quantum Physics versus Classical Physics 16 26 Mesoscopic Physics and Characteristic Length 18 261 Characteristic Length 18 262 Characteristic Transports 20 27 Mathematics in Classical and Quantum Worlds 21 3 Basic Physics of Quantum Scattering and Tunneling 23 31 Definitions of Quantum Scattering and Tunneling 23 32 Description of Quantum Scattering and Tunneling 24 33 Basic Physical Quantities in Quantum Tunneling 26 331 Transmission and Reflection Coefficients 26 332 Conductance: Landauer-Biittiker Formula 26 333 Charge Current 27 34 Relationships between Transmission Coefficient and Scattering Matrix 27 xi
xii Quantum Tunneling and Field Electron Emission Theories 35 Basic Properties of Scattering and Transfer Matrices 29 36 Constraints of Scattering and Transfer Matrices 35 4 Wave Function Matching Method 37 41 Square Barrier Model 38 42 Asymmetric Square Barrier Model 40 43 Double Square Barrier Model 43 44 Multi-Mode Square Barrier Model 45 45 Triangle Barrier 47 46 Lattice Models 51 461 One-dimensional Model 51 462 Two-chain Model 54 463 2D Square Lattice 58 5 WKB Method 61 51 Mathematics of WKB Method 61 52 Validity 63 53 Solution of Schrodinger Equation 63 54 Quantum Tunneling 64 55 Triangle Barrier 65 56 Triangle and Image Potential Barrier 67 6 Lippmann-Schwinger Formalism 71 61 Lippmann-Schwinger Equation 71 62 Wave Function and S Matrix 73 63 Green's Function and T Matrix 74 64 S Matrix 76 65 Adiabatic Transport Model 77 66 Quantum Tunneling in Time-Dependent Barrier 79 661 Floquet Theory 79 662 Time-Dependent Barrier 80 7 Non-Equilibrium Green's Function Method 83 71 Basic Physics of Non-Equilibrium Transport Problems 83 72 Model of Nanodevices 84 73 Green's Functions and Self-Energy 86 74 Spectral Function, Density of States, and Correlation Function 88
Contents xiii 75 Definitions and Relationships 90 76 Current 91 77 Tunneling Model and Master Equation 93 8 Spin Tunneling 97 81 Tunneling Magnetoresistance Phenomena 97 82 Julliere Model 98 83 Giant Magnetoresistance 101 84 Spin Tunneling in Spin-Orbital Coupling Semiconductors 102 841 Model and Issue 102 842 Ferromagnetic Nanowires 104 843 Spin-Orbital Coupling Semiconductor 106 85 Spin Polarization 110 86 Remarks 117 9 Applications 119 91 Josephson Effect 119 92 Theory of Scanning Tunneling Microscopy 121 921 Quantum Electron Tunneling and Bardeen's Formula 122 922 Tersoff-Hamann Formula 123 923 Non-Equilibrium Green's Function Method 125 93 Conductance of Graphene 125 931 Graphene Nanoribbons Model 127 932 Impurity Effects 128 933 Vacancy and Impurity 130 934 Conclusion 131 94 Charge Transfer in DNA 132 941 G4-DNA Model 133 942 TG4 and Their Classifications 135 943 Anomalous Conductance in NCM(H)TG4 136 944 Topological Structure Transition versus Telomerase Activation and Inhibition 138 945 Conclusion 139 95 Remarks 140
xiv Quantum Tunneling and Field Electron Emission Theories Field Electron Emission Theory 141 10 Introduction 143 101 Field Electron Emission Phenomenon 143 102 Brief Histroy of Field Electron Emission 143 103 Basic Concepts of Field Electron Emission 144 1031 Electron Emissions from Solids 144 1032 Work Function and Field Emission Condition 145 1033 Basic Experiment Components of Field Emission 145 1034 Applications of Field Emission 146 104 Basic Issues of Field Electron Emission 146 1041 Theoretical Issues 146 1042 Engineering Issues 147 105 Novel Phenomena and Challenges of Field Emission 148 1051 New Phenomena 148 1052 Challenging Problems 149 11 Theoretical Model and Methodology 151 111 Theoretical Model of Field Emission 151 112 Theoretical Methodology 152 1121 Model and Analytic Solution 153 1122 Computer Simulation 153 1123 Empirical Method 153 113 Remarks 153 12 Fowler-Nordheim Theory 157 121 Assumptions of Fowler-Nordheim Theory 157 122 Fowler-Nordheim Theory 158 1221 Field Emission Equation I: Fowler-Nordheim Method 160 1222 Field Emission Equation II: Young-Gadzuk's Method 163 1223 Field Emission Equation III: R Forbes' Method 1224 Field Emission Equation VI: A Haug's Method 164 166 123 Remarks 167 124 Beyond Triangular Vacuum Potential Barrier 168 1241 General Formalism 169 1242 Generalized Triangular Barrier 171
Contents xv 1243 Schottky-Nordheim Barrier: Image Potential Effect 172 1244 Beyond Gamow Exponent Form 175 1245 Emitter Curvature and Field Enhancement Factor 175 1246 Space Charge Effect 176 1247 Small-Scale Effect of Emitter 178 1248 Emission Area and Total Emission Current 178 125 Energy Band Effect 178 1251 Supply Function Density 179 1252 Transmission Coefficient and Total Energy Distribution 179 1253 Emission Current Density 181 126 Finite Temperature Effect 182 127 Basic Characteristic of Current-Field Relation 184 1271 Current-Field Characteristic 184 1272 Maximum Emission Current Density 185 1273 FN Plot 186 128 Energy Distribution of Emission Electrons 191 1281 Total Energy Distribution (TED) 191 1282 Normal Energy Distribution (NED) 193 1283 Basic Characteristics of TED and NED 194 1284 Measurement of Energy Distributions 202 129 Nottingham Effect 204 13 Field Emission from Semiconductors 209 131 Basic Properties of Semiconductors 210 1311 Energy Band Structure 210 1312 Temperature Dependence of Energy Band Gap 210 1313 Carrier Concentration 211 132 Model of Field Emission from Semiconductors 212 133 Supply Function Density 213 134 Vacuum Potential Barrier and Transmission Coefficient 213 135 Total Energy Distribution 215 136 Basic Characteristics of Total Energy Distribution 217 137 Emission Current Density 218 14 Surface Effects and Resonance 221 141 Field Emission Model with Surface Effects 221
xvi Quantum Tunneling and Field Electron Emission Theories 142 Double-Barrier Vacuum Potential and Transmission Coefficient 222 143 Total Energy Distribution 226 144 Emission Current Density 227 15 Thermionic Emission Theory 231 151 The Richardson Theory of Thermionic Emission 231 152 Boundary of Field Emission and Thermionic Emission 233 16 Theory of Dynamical Field Emission 237 161 Adiabatic Process and Dynamic Field Emission Model 162 Supply Function and Time-Dependent Transmission 237 Coefficient 238 163 Dynamic Total Energy Distribution 239 164 Dynamic Normal Energy Distribution 240 165 Dynamic Emission Current 241 166 Quantum Tunneling Time 242 17 Theory of Spin Polarized Field Emission 247 171 Basic Physics of Spin Polarized Field Emission 247 172 Energy Band Spin-Split Model 249 1721 Supply Function and Transmission Coefficient 249 1722 Total Energy Distribution 250 1723 Normal Energy Distribution 251 1724 Emission Current Density and Spin Polarization 252 173 Spin-Dependent Triangular Potential Barrier Model 254 1731 Spin-dependent Triangular Potential Barrier and Transmission Coefficient 254 1732 Total Energy Distribution: 256 1733 Normal Energy Distribution: 256 1734 Emission Current Density and Spin Polarization 174 Spin-Dependent Image 257 Potential Barrier Model 259 1741 Spin-dependent Image Potential Barrier and Transmission Coefficient 259 1742 Total and Normal Energy Distributions 260 1743 Emission Current Density and Spin Polarization 261 175 Finite Temperature Effects 263 1751 Energy-Band Spin-Split Model 263
Contents xvii 1752 Spin-Dependent Triangular Potential Barrier Model 264 1753 Spin-Dependent Image Potential Barrier Model 264 176 Comparison of Spin Polarizations 265 177 A Scheme of Pure Spin Polarized Electron Emission Induced by Quantum Spin Hall Effect 266 178 Difficulties and Possibilities of Spin Polarized Field Emission 268 18 Theory of Field Electron Emission from Nanomaterials 271 181 Basic Physics of Field Emission from Nanoemitters 271 182 Formulation of Field Emission Current Density 273 1821 Supply Function Density 274 1822 Current Density 274 1823 Density of States 274 1824 Transmission Coefficient 274 1825 Distribution Function 278 1826 Total Energy Distribution 278 1827 Emission Current Density 279 183 Computational Framework 279 184 Special Case I: Sommerfeld Model 280 185 Special Case II: Nanowires 280 186 Special Case III: Coupled Nanowires 284 187 Thermionic Emission of Nanowires 290 188 Theory of Field Electron Emission from Carbon Nanotubes 292 1881 Energy Dispersion and Density of States 293 1882 Density of States and Group Velocity 293 1883 Supply Function and Transmission Coefficient 294 1884 Total Energy Distribution 295 1885 Emission Current Density 295 1886 Finite Temperature Effect 301 1887 Thermionic Emission 301 189 Theory of Luttinger Liquid Field Emission 303 19 Computer Simulations of Field Emission 305 191 Basic Idea on Computer Simulation 305
xviii Quantum Tunneling and Field Electron Emission Theories 192 Formulation of Field Emission Based on Non-Equilibrium Green's Function Method 306 1921 Generalized Supply Function 307 1922 Transmission Coefficient 308 1923 Total Energy Distribution and Emission Current Density 308 193 Tight-Binding Approach 309 1931 Computational Formulation 309 1932 Carbon Nanotubes 310 1933 Total Energy Distribution and Emission Current 312 1934 Computational Framework 313 1935 Basic Properties of Field Emission of SWCN 314 194 Cap and Doping Effects 319 195 Field Penetration Effect and Field Enhancement Factor 320 196 First-Principle Method 321 1961 The Multi-Scale Technique 321 1962 The ab-initio Tight-Binding Method 322 1963 Lippman-Schwinger Scattering Formalism 322 20 The Empirical Theory of Field Emission 323 201 The Empirical Theory of Field Emission 323 202 The Generalized Empirical Theory of Field Emission 324 203 The Empirical Theory of Thermionic Emission 325 204 Connection between Empirical Theory and Experimental Data 325 21 Fundamental Physics of Field Electron Emission 327 211 Field Emission Behavior and Material Properties 327 212 Equilibrium and Non-Equilibrium Currents 328 213 Many-Body Effect 329 214 Coherent and Non-Coherent Emission Currents 330 215 Electron Emission Mechanism: Nano versus Bulk Effects 330 216 Universality versus Finger Effects 331 217 Open Problems and Difficulties 332 218 Perspectives 333 Appendix A Appendices 335 Al Basic Properties of S and M Matrices 335
Contents xix A 11 Proof of Theorem 35 335 A12 Proof of Theorem 37 336 A13 Proof of Theorem 38 336 A14 Proof of Theorem 39 338 A2 Spin Tunneling 340 A21 Proof of Claim 81b and Claim 82b 340 A22 Proof of Claim 82 341 A23 Proof of Theorem 81 341 A24 Proof of Theorem 82 342 A25 Proof of Theorem 83 343 A3 Derivations in Non-Equilibrium Green's Function Method 343 A31 Basic Relationships 343 A32 Non-Equilibrium Current 344 A4 Models of Solids 346 A41 Sommerfeld Model of Metals 346 A42 Crystal Lattice Model and Bloch Theorem 348 A43 Tight-Binding Model 349 A44 Remarks of Solid Model 351 A5 Density of States 351 A51 Definition of Density of States 351 A52 Sommerfeld Model (Electron Gas) 351 A53 Beyond Sommerfeld Model 352 A54 Non-Equilibrium Cases 353 A6 Fermi Wave Vector and Fermi Wavelength 354 A61 Definitions of Fermi Wave Vector and Fermi Wavelength 354 A62 Sommerfeld Model 355 A7 The Widths of TED and NED 356 A71 TED 356 A72 NED 357 A8 Spin Polarized Field Emission 358 A9 Field Emission from Nanomaterials 360 A91 Nanowire Integration 360 A92 Coupled Nanowire 361 A 10 Carbon Nanotubes 363 A 101 Graphene 363 A 102 Lattice Structure of Single-Wall Carbon Nanotubes (SWCN) 364 A 103 Unit Cell and Brillouin Zone of SWCN 365
xx Quantum; Tunneling and Field Electron Emission Theories A104: Energy Dispersion!Relation of SWCN 366 A 105 Energy'Gap 367 A106 Density, of States, of-swcn 368 A107 Multi-Wall Carbon Nanotubes (MWCN) 368 A 11 Physical Constants 371 A 12 Field Emission'Constants 372 A 13 Epilogue 373 Bibliography 375 Index 385