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1 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
2 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 Quantum Physics versus Classical Physics Mesoscopic Physics and Characteristic Length Characteristic Length Characteristic Transports Mathematics in Classical and Quantum Worlds 21 3 Basic Physics of Quantum Scattering and Tunneling Definitions of Quantum Scattering and Tunneling Description of Quantum Scattering and Tunneling Basic Physical Quantities in Quantum Tunneling Transmission and Reflection Coefficients Conductance: Landauer-Biittiker Formula Charge Current Relationships between Transmission Coefficient and Scattering Matrix 27 xi
3 xii Quantum Tunneling and Field Electron Emission Theories 35 Basic Properties of Scattering and Transfer Matrices Constraints of Scattering and Transfer Matrices 35 4 Wave Function Matching Method Square Barrier Model Asymmetric Square Barrier Model Double Square Barrier Model Multi-Mode Square Barrier Model Triangle Barrier Lattice Models One-dimensional Model Two-chain Model D Square Lattice 58 5 WKB Method Mathematics of WKB Method Validity Solution of Schrodinger Equation Quantum Tunneling Triangle Barrier Triangle and Image Potential Barrier 67 6 Lippmann-Schwinger Formalism Lippmann-Schwinger Equation Wave Function and S Matrix Green's Function and T Matrix S Matrix Adiabatic Transport Model Quantum Tunneling in Time-Dependent Barrier Floquet Theory Time-Dependent Barrier 80 7 Non-Equilibrium Green's Function Method Basic Physics of Non-Equilibrium Transport Problems Model of Nanodevices Green's Functions and Self-Energy Spectral Function, Density of States, and Correlation Function 88
4 Contents xiii 75 Definitions and Relationships Current Tunneling Model and Master Equation 93 8 Spin Tunneling Tunneling Magnetoresistance Phenomena Julliere Model Giant Magnetoresistance Spin Tunneling in Spin-Orbital Coupling Semiconductors Model and Issue Ferromagnetic Nanowires Spin-Orbital Coupling Semiconductor Spin Polarization Remarks Applications Josephson Effect Theory of Scanning Tunneling Microscopy Quantum Electron Tunneling and Bardeen's Formula Tersoff-Hamann Formula Non-Equilibrium Green's Function Method Conductance of Graphene Graphene Nanoribbons Model Impurity Effects Vacancy and Impurity Conclusion Charge Transfer in DNA G4-DNA Model TG4 and Their Classifications Anomalous Conductance in NCM(H)TG Topological Structure Transition versus Telomerase Activation and Inhibition Conclusion Remarks 140
5 xiv Quantum Tunneling and Field Electron Emission Theories Field Electron Emission Theory Introduction Field Electron Emission Phenomenon Brief Histroy of Field Electron Emission Basic Concepts of Field Electron Emission Electron Emissions from Solids Work Function and Field Emission Condition Basic Experiment Components of Field Emission Applications of Field Emission Basic Issues of Field Electron Emission Theoretical Issues Engineering Issues Novel Phenomena and Challenges of Field Emission New Phenomena Challenging Problems Theoretical Model and Methodology Theoretical Model of Field Emission Theoretical Methodology Model and Analytic Solution Computer Simulation Empirical Method Remarks Fowler-Nordheim Theory Assumptions of Fowler-Nordheim Theory Fowler-Nordheim Theory Field Emission Equation I: Fowler-Nordheim Method Field Emission Equation II: Young-Gadzuk's Method Field Emission Equation III: R Forbes' Method 1224 Field Emission Equation VI: A Haug's Method Remarks Beyond Triangular Vacuum Potential Barrier General Formalism Generalized Triangular Barrier 171
6 Contents xv 1243 Schottky-Nordheim Barrier: Image Potential Effect Beyond Gamow Exponent Form Emitter Curvature and Field Enhancement Factor Space Charge Effect Small-Scale Effect of Emitter Emission Area and Total Emission Current Energy Band Effect Supply Function Density Transmission Coefficient and Total Energy Distribution Emission Current Density Finite Temperature Effect Basic Characteristic of Current-Field Relation Current-Field Characteristic Maximum Emission Current Density FN Plot Energy Distribution of Emission Electrons Total Energy Distribution (TED) Normal Energy Distribution (NED) Basic Characteristics of TED and NED Measurement of Energy Distributions Nottingham Effect Field Emission from Semiconductors Basic Properties of Semiconductors Energy Band Structure Temperature Dependence of Energy Band Gap Carrier Concentration Model of Field Emission from Semiconductors Supply Function Density Vacuum Potential Barrier and Transmission Coefficient Total Energy Distribution Basic Characteristics of Total Energy Distribution Emission Current Density Surface Effects and Resonance Field Emission Model with Surface Effects 221
7 xvi Quantum Tunneling and Field Electron Emission Theories 142 Double-Barrier Vacuum Potential and Transmission Coefficient Total Energy Distribution Emission Current Density Thermionic Emission Theory The Richardson Theory of Thermionic Emission Boundary of Field Emission and Thermionic Emission Theory of Dynamical Field Emission Adiabatic Process and Dynamic Field Emission Model 162 Supply Function and Time-Dependent Transmission 237 Coefficient Dynamic Total Energy Distribution Dynamic Normal Energy Distribution Dynamic Emission Current Quantum Tunneling Time Theory of Spin Polarized Field Emission Basic Physics of Spin Polarized Field Emission Energy Band Spin-Split Model Supply Function and Transmission Coefficient Total Energy Distribution Normal Energy Distribution Emission Current Density and Spin Polarization Spin-Dependent Triangular Potential Barrier Model Spin-dependent Triangular Potential Barrier and Transmission Coefficient Total Energy Distribution: Normal Energy Distribution: Emission Current Density and Spin Polarization 174 Spin-Dependent Image 257 Potential Barrier Model Spin-dependent Image Potential Barrier and Transmission Coefficient Total and Normal Energy Distributions Emission Current Density and Spin Polarization Finite Temperature Effects Energy-Band Spin-Split Model 263
8 Contents xvii 1752 Spin-Dependent Triangular Potential Barrier Model Spin-Dependent Image Potential Barrier Model Comparison of Spin Polarizations A Scheme of Pure Spin Polarized Electron Emission Induced by Quantum Spin Hall Effect Difficulties and Possibilities of Spin Polarized Field Emission Theory of Field Electron Emission from Nanomaterials Basic Physics of Field Emission from Nanoemitters Formulation of Field Emission Current Density Supply Function Density Current Density Density of States Transmission Coefficient Distribution Function Total Energy Distribution Emission Current Density Computational Framework Special Case I: Sommerfeld Model Special Case II: Nanowires Special Case III: Coupled Nanowires Thermionic Emission of Nanowires Theory of Field Electron Emission from Carbon Nanotubes Energy Dispersion and Density of States Density of States and Group Velocity Supply Function and Transmission Coefficient Total Energy Distribution Emission Current Density Finite Temperature Effect Thermionic Emission Theory of Luttinger Liquid Field Emission Computer Simulations of Field Emission Basic Idea on Computer Simulation 305
9 xviii Quantum Tunneling and Field Electron Emission Theories 192 Formulation of Field Emission Based on Non-Equilibrium Green's Function Method Generalized Supply Function Transmission Coefficient Total Energy Distribution and Emission Current Density Tight-Binding Approach Computational Formulation Carbon Nanotubes Total Energy Distribution and Emission Current Computational Framework Basic Properties of Field Emission of SWCN Cap and Doping Effects Field Penetration Effect and Field Enhancement Factor First-Principle Method The Multi-Scale Technique The ab-initio Tight-Binding Method Lippman-Schwinger Scattering Formalism The Empirical Theory of Field Emission The Empirical Theory of Field Emission The Generalized Empirical Theory of Field Emission The Empirical Theory of Thermionic Emission Connection between Empirical Theory and Experimental Data Fundamental Physics of Field Electron Emission Field Emission Behavior and Material Properties Equilibrium and Non-Equilibrium Currents Many-Body Effect Coherent and Non-Coherent Emission Currents Electron Emission Mechanism: Nano versus Bulk Effects Universality versus Finger Effects Open Problems and Difficulties Perspectives 333 Appendix A Appendices 335 Al Basic Properties of S and M Matrices 335
10 Contents xix A 11 Proof of Theorem A12 Proof of Theorem A13 Proof of Theorem A14 Proof of Theorem A2 Spin Tunneling 340 A21 Proof of Claim 81b and Claim 82b 340 A22 Proof of Claim A23 Proof of Theorem A24 Proof of Theorem A25 Proof of Theorem 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
11 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
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