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Introduction to the Physics of Electron Emission

ISBN: 978-1-119-05189-3
712 pages
November 2017
Introduction to the Physics of Electron Emission (1119051894) cover image

Description

A practical, in-depth description of the physics behind electron emission physics and its usage in science and technology

Electron emission is both a fundamental phenomenon and an enabling component that lies at the very heart of modern science and technology. Written by a recognized authority in the field, with expertise in both electron emission physics and electron beam physics, An Introduction to Electron Emission provides an in-depth look at the physics behind thermal, field, photo, and secondary electron emission mechanisms, how that physics affects the beams that result through space charge and emittance growth, and explores the physics behind their utilization in an array of applications.

The book addresses mathematical and numerical methods underlying electron emission, describing where the equations originated, how they are related, and how they may be correctly used to model actual sources for devices using electron beams. Writing for the beam physics and solid state communities, the author explores applications of electron emission methodology to solid state, statistical, and quantum mechanical ideas and concepts related to simulations of electron beams to condensed matter, solid state and fabrication communities.

  • Provides an extensive description of the physics behind four electron emission mechanisms—field, photo, and secondary, and how that physics relates to factors such as space charge and emittance that affect electron beams.
  • Introduces readers to mathematical and numerical methods, their origins, and how they may be correctly used to model actual sources for devices using electron beams
  • Demonstrates applications of electron methodology as well as quantum mechanical concepts related to simulations of electron beams to solid state design and manufacture
  • Designed to function as both a graduate-level text and a reference for research professionals

Introduction to the Physics of Electron Emission is a valuable learning tool for postgraduates studying quantum mechanics, statistical mechanics, solid state physics, electron transport, and beam physics. It is also an indispensable resource for academic researchers and professionals who use electron sources, model electron emission, develop cathode technologies, or utilize electron beams.

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Table of Contents

Acknowledgements 2

Part I Foundations 5

1 Prelude 7

2 Units and Evaluation 11

2.1 Numerical Accuracy 11

2.2 Atomic-sized Units 13

2.3 Units Based on Emission 18

3 Pre-Quantum Models 21

3.1 Discovery of Electron Emission 21

3.2 The Drude Model and Maxwell-Boltzmann Statistics 22

3.3 The Challenge of Photoemission 31

4 Statistics 37

4.1 Distinguishable Particles 37

4.2 Probability and States 43

4.3 Probability and Entropy 45

4.4 Combinatorics and Products of Probability 50

5 Maxwell-Boltzmann Distribution 55

5.1 Classical Phase Space 55

5.2 Most Probable Distribution 59

5.3 Energy and Entropy 62

5.4 The Gibbs Paradox 63

5.5 Ideal Gas in a Potential Gradient 65

5.6 The Grand Partition Function 67

5.7 A Nascent Model of Electron Emission 69

6 Quantum Distributions 73

6.1 Bose-Einstein Distribution 73

6.2 Fermi-Dirac Distribution 74

6.3 The Riemann Zeta Function 75

6.4 Chemical Potential 78

6.5 Classical to Quantum Statistics 84

6.6 Electrons and White Dwarf Stars 85

7 A Box of Electrons 91

7.1 Scattering 91

7.2 From Classical to Quantum Mechanics 92

7.3 Moments and Distributions 94

7.4 Boltzmann’s Transport Equation 96

8 Quantum Mechanics Methods 107

8.1 A Simple Model: The Prisoner’s Dilemma 107

8.2 Matrices and Wave Functions 115

9 Quintessential Problems 133

9.1 The Hydrogen Atom 134

9.2 Transport Past Barriers 151

9.3 The Harmonic Oscillator 162

Part II The Canonical Equations 175

10 A Brief History 177

10.1 Thermal Emission 178

10.2 Field Emission 180

10.3 Photoemission 181

10.4 Secondary Emission 181

10.5 Space Charge Limited Emission 182

10.6 Resources and Further Reading 183

11 Anatomy of Current Density 187

11.1 Supply Function 188

11.2 Gamow Factor 189

11.3 Image Charge Potential 194

12 Richardson-Laue-Dushman Equation 197

12.1 Approximations 197

12.2 Analysis of Thermal Emission Data 199

13 Fowler-Nordheim Equation 201

13.1 Triangular Barrier Approximation 203

13.2 Image Charge Approximation 205

13.3 Analysis of Field Emission Data 211

13.4 The Millikan-Lauritsen Hypothesis 213

14 Fowler-Dubridge Equation 217

14.1 Approximations 217

14.2 Analysis of Photoemission Data 221

15 Baroody Equation 225

15.1 Approximations 225

15.2 Analysis of Secondary Emission Data 232

15.3 Subsequent Approximations 234

16 Child-Langmuir Law 237

16.1 Constant Density Approximation 238

16.2 Constant Current Approximation 241

16.3 Transit Time Approximation 245

17 A General Thermal-Field-Photoemission Equation 251

17.1 Experimental Thermal-Field Energy Distributions 253

17.2 Theoretical Thermal-Field Energy Distributions 255

17.3 The N(n; s; u) Function 264

17.4 Brute Force Evaluation 275

17.5 A Computationally Kind Model 282

17.6 General Thermal Field Emission Code 288

Part III Exact Tunneling and Transmission Evaluation 297

18 Simple Barriers 299

18.1 Rectangular Barrier 300

18.2 Triangular Barrier - General Method 306

18.3 Triangular Barrier - Numerical 318

19 Transfer Matrix Approach 325

19.1 Plane Wave Transfer Matrix 326

19.2 Airy Function Transfer Matrix 335

20 Ion Enhanced Emission and Breakdown 353

20.1 Paschen’s Curve 353

20.2 Modified Paschen’s Curve 357

20.3 Ions and the Emission Barrier 360

Part IV The Complexity of Materials 367

21 Metals 369

21.1 Density of States, Again 370

21.2 Spheres in d��Dimensions 372

21.3 The Kronig Penny Model 375

21.4 Atomic Orbitals 381

21.5 Electronegativity 383

21.6 Sinusoidal Potential and Band Gap 388

21.7 Ion Potentials and Screening 391

22 Semiconductors 397

22.1 Resistivity 397

22.2 Electrons and Holes 400

22.3 Band Gap and Temperature 403

22.4 Doping of Semiconductors 406

22.5 Semiconductor Image Charge Potential 411

22.6 Dielectric Constant and Screening 414

23 Effective Mass 417

23.1 Dispersion Relations 417

23.2 The k _ p Method 420

23.3 Hyperbolic Relations 424

23.4 The Alpha Semiconductor Model 428

23.5 Current and Effective Mass 433

24 Interfaces 435

24.1 Metal-Insulator-Metal Current Density 435

24.2 Band Bending 445

24.3 Accumulation Layers 447

24.4 Depletion Layers 458

24.5 Modifications Due to Non-Linear Potential Barriers 466

25 Contacts, Conduction, and Current 471

25.1 Zener Breakdown 471

25.2 Poole-Frenkel Transport 472

25.3 Tunneling Conduction 477

25.4 Resonant Tunneling in Field Emission 483

26 Electron Density Near Barriers 489

26.1 An Infinite Barrier 489

26.2 Two Infinite Barriers 493

26.3 A Triangular Well 495

26.4 Density and Dipole Component 499

27 Many Body Effects and Image Charge 507

27.1 Kinetic Energy 508

27.2 Exchange Energy 509

27.3 Correlation Term 511

27.4 Core Term 512

27.5 Exchange-Correlation and a Barrier Model 518

28 An Analytic Image Charge Potential 523

28.1 Work Function and Temperature 523

28.2 Work Function and Field 526

28.3 Changes to Current Density 528

Part V Application Physics 531

29 Dispenser Cathodes 533

29.1 Miram Curves and the Longo Equation 534

29.2 Diffusion of Coatings 539

29.3 Evaporation of Coatings 564

29.4 Knudsen Flow Through Pores 568

29.5 Lifetime of a Sintered Wire Controlled Porosity Dispenser Cathode 577

30 Field Emitters 581

30.1 Field Enhancement 582

30.2 Hemispheres and Notional Emission Area 586

30.3 Point Charge Model 590

30.4 Schottky’s Conjecture 594

30.5 Assessment of the Tip Current Models 601

30.6 Line Charge Models 603

30.7 Prolate Spheroidal Representation 608

30.8 A Hybrid Analytic-Numerical Model 616

30.9 Shielding 627

30.10 Statistical Variation 633

31 Photoemitters 643

31.1 Scattering Consequences 648

31.2 Basic Theory 650

31.3 Three Step Model 652

31.4 Moments Model 656

31.5 Reflectivity and Penetration Factors 664

31.6 Lorentz-Drude Model of the Dielectric Constant 667

31.7 Scattering Contributions 678

31.8 Low Work Function Coatings 695

31.9 Quantum Efficiency of a Cesiated Surface 706

32 Secondary Emission Cathodes 709

32.1 Diamond Amplifier Concept 710

32.2 Monte Carlo Methods 720

32.3 Relaxation Time 729

32.4 Monte Carlo and Diamond Amplifier Response Time 753

33 Electron Beam Physics 765

33.1 Electron Orbits and Cathode Area 767

33.2 Beam Envelope Equation 769

33.3 Emittance for Flat and Uniform Surfaces 776

33.4 Emittance for A Bump 796

33.5 Emittance and Realistic Surfaces 823

Part VI Appendicies 827

A Summation, Integration, and Differentiation 829

A.1 Series 829

A.2 Integration 830

A.3 Differentiation 840

A.4 Numerical Solution of an Ordinary Differential Equation 847

B Functions 851

B.1 Trigonometric Functions 851

B.2 Gamma Function 851

B.3 Riemann Zeta Function 852

B.4 Error Function 853

B.5 Legendre Polynomials 854

B.6 Airy Functions 855

B.7 Lorentzian Functions 858

C Algorithms 861

C.1 Permutation Algorithm 861

C.2 Birthday Algorithm 862

C.3 Least Squares Fitting of Data 863

C.4 Monty Hall Algorithm 867

C.5 Wave Function and Density Algorithm 868

C.6 Hydrogen Atom Algorithms 870

C.7 Root Finding Methods 873

C.8 Thermal-Field Algorithm 877

C.9 Gamow Factor Algorithm 879

C.10 Triangular Barrier D(E) 880

C.11 Evaluation of Hc(u) 881

C.12 Transfer Matrix Algorithm 883

C.13 Semiconductors and Doping Density 890

C.14 Band Bending: Accumulation Layer 892

C.15 Simple ODE Solvers 893

C.16 Current through a Metal-Insulator-Metal diode 897

C.17 Field Emission From Semiconductors 898

C.18 Roots of the Quadratic Image Charge Barrier 901

C.19 Zeros of the Airy Function 902

C.20 Atomic Sphere Radius rs 904

C.21 Sodium Exchange-Correlation Potential 906

C.22 Field Dependent Work Function 907

C.23 Digitizing an Image file 907

C.24 Lattice Gas Algorithm 909

C.25 Evaluation of the Point Charge Model Functions 912

C.26 Modeling of Field Emitter I(V ) Data 913

C.27 Modeling a Log-Normal Distribution of Field Emitters 915

C.28 Simple Shell and Sphere Algorithm 919

C.29 Gyftopoulos-Levine Work function Algorithm 921

C.30 Poisson Distributions 924

C.31 Electron-Electron Relaxation Time 927

C.32 Resistivity and the Debye Temperature 928

C.33 Orbits in a Magnetic Field 931

C.34 Trajectory of a Harmonic Oscillator 935

C.35 Trajectories for Emission from a Hemisphere 936

C.36 Monte Carlo and Integration 938

Index

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Author Information

Kevin Jensen, PhD is a research physicist in the Materials and Systems Branch, Materials Science and Technology Division, at the Naval Research Laboratory. Since 2001, he has been a visiting senior research scientist at the University of Maryland’s Institute for Research in Electronics and Applied Physics (IREAP). Dr. Jensen joined the theory section of the Vacuum Electronics Branch at NRL in 1990. He earned a doctorate in physics from New York University in 1987. He has been and is Principal Investigator for several research programs investigating the application of electron sources (particularly field and photoemission sources) to microwave devices and Free Electron Lasers. Over the years, he has authored or coauthored over 150 articles and conference proceedings. He became a Fellow of the American Physical Society in 2009 for his contributions to the theory and modeling of electron emission sources for particle accelerators and microwave tubes. He presently serves on the Editorial Board of Journal of Applied Physics.

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