Introduction to the Physics of Electron Emission

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.

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.

Acknowledgements xiii

Part I: Foundations

1 Prelude 3

2 Units and evaluation 7

2.1 Numerical accuracy 7

2.2 Atomic-sized units 8

2.3 Units based on emission 11

3 Pre-quantum models 13

3.1 Discovery of electron emission 13

3.2 The Drude model and Maxwell–Boltzmann statistics 13

3.3 The challenge of photoemission 19

4 Statistics 25

4.1 Distinguishable particles 25

4.2 Probability and states 28

4.3 Probability and entropy 30

4.4 Combinatorics and products of probability 33

5 Maxwell–Boltzmann distribution 37

5.1 Classical phase space 37

5.2 Most probable distribution 39

5.3 Energy and entropy 41

5.4 The Gibbs paradox 42

5.5 Ideal Gas in a potential gradient 44

5.6 The grand partition function 45

5.7 A nascent model of electron emission 46

6 Quantum distributions 49

6.1 Bose–Einstein distribution 49

6.2 Fermi–Dirac distribution 50

6.3 The Riemann zeta function 50

6.4 Chemical potential 52

6.5 Classical to quantum statistics 56

6.6 Electrons and white dwarf stars 57

7 A box of electrons 61

7.1 Scattering 61

7.2 From classical to quantum mechanics 61

7.3 Moments and distributions 63

7.4 Boltzmann’s transport equation 64

8 Quantum mechanics methods 73

8.1 A simple model: the prisoner’s dilemma 73

8.2 Matrices and wave functions 78

9 Quintessential problems 91

9.1 The hydrogen atom 92

9.2 Transport past barriers 102

9.3 The harmonic oscillator 110

Part II: The canonical equations

10 A brief history 121

10.1 Thermal emission 121

10.2 Field emission 122

10.3 Photoemission 123

10.4 Secondary emission 124

10.5 Space-charge limited emission 124

10.6 Resources and further reading 124

11 Anatomy of current density 127

11.1 Supply function 128

11.2 Gamow factor 128

11.3 Image charge potential 131

12 Richardson–Laue–Dushman equation 135

12.1 Approximations 135

12.2 Analysis of thermal emission data 136

13 Fowler–Nordheim equation 139

13.1 Triangular barrier approximation 140

13.2 Image charge approximation 141

13.3 Analysis of field emission data 145

13.4 The Millikan–Lauritsen hypothesis 146

14 Fowler–Dubridge equation 149

14.1 Approximations 149

14.2 Analysis of photoemission data 153

15 Baroody equation 155

15.1 Approximations 155

15.2 Analysis of secondary emission data 160

15.3 Subsequent approximations 161

16 Child–Langmuir law 163

16.1 Constant density approximation 164

16.2 Constant current approximation 165

16.3 Transit time approximation 168

17 A General thermal–field–photoemission equation 173

17.1 Experimental thermal–field energy distributions 175

17.2 Theoretical thermal–field energy distributions 176

17.3 The N(n,s,u) function 181

17.4 Brute force evaluation 189

17.5 A computationally kind model 193

17.6 General thermal–field emission code 198

Part III: Exact tunneling and transmission evaluation

18 Simple barriers 209

18.1 Rectangular barrier 209

18.2 Triangular barrier: general method 213

18.3 Triangular barrier: numerical 222

19 Transfer matrix approach 227

19.1 Plane wave transfer matrix 227

19.2 Airy function transfer matrix 233

20 Ion enhanced emission and breakdown 245

20.1 Paschen’s curve 245

20.2 Modified Paschen’s curve 247

20.3 Ions and the emission barrier 250

Part IV: The complexity of materials

21 Metals 257

21.1 Density of states, again 257

21.2 Spheres in d dimensions 259

21.3 The Kronig Penny model 261

21.4 Atomic orbitals 264

21.5 Electronegativity 266

21.6 Sinusoidal potential and band gap 269

21.7 Ion potentials and screening 272

22 Semiconductors 277

22.1 Resistivity 277

22.2 Electrons and holes 279

22.3 Band gap and temperature 281

22.4 Doping of semiconductors 281

22.5 Semiconductor image charge potential 286

22.6 Dielectric constant and screening 287

23 Effective mass 291

23.1 Dispersion relations 291

23.2 The k ⋅ p method 293

23.3 Hyperbolic relations 296

23.4 The alpha semiconductor model 299

23.5 Current and effective mass 301

24 Interfaces 303

24.1 Metal–insulator–metal current density 303

24.2 Band bending 310

24.3 Accumulation layers 311

24.4 Depletion layers 319

24.5 Modifications due to non-linear potential barriers 324

25 Contacts, conduction, and current 329

25.1 Zener breakdown 329

25.2 Poole–Frenkel transport 329

25.3 Tunneling conduction 333

25.4 Resonant tunneling in field emission 336

26 Electron density near barriers 341

26.1 An infinite barrier 341

26.2 Two infinite barriers 344

26.3 A triangular well 346

26.4 Density and dipole component 348

27 Many-body effects and image charge 353

27.1 Kinetic energy 353

27.2 Exchange energy 354

27.3 Correlation term 356

27.4 Core term 357

27.5 Exchange-correlation and a barrier model 360

28 An analytic image charge potential 363

28.1 Work function and temperature 363

28.2 Work function and field 363

28.3 Changes to current density 366

Part V: Application physics

29 Dispenser cathodes 371

29.1 Miram curves and the longo equation 371

29.2 Diffusion of coatings 375

29.3 Evaporation of coatings 391

29.4 Knudsen flow through pores 393

29.5 Lifetime of a sintered wire controlled porosity dispenser cathode 399

30 Field emitters 403

30.1 Field enhancement 403

30.2 Hemispheres and notional emission area 406

30.3 Point charge model 408

30.4 Schottky’s conjecture 412

30.5 Assessment of the tip current models 415

30.6 Line charge models 417

30.7 Prolate spheroidal representation 420

30.8 A hybrid analytic-numerical model 425

30.9 Shielding 433

30.10 Statistical variation 438

31 Photoemitters 443

31.1 Scattering consequences 446

31.2 Basic theory 448

31.3 Three-step model 449

31.4 Moments model 451

31.5 Reflectivity and penetration factors 457

31.6 Lorentz–Drude model of the dielectric constant 458

31.7 Scattering contributions 466

31.8 Low work function coatings 478

31.9 Quantum efficiency of a cesiated surface 485

32 Secondary emission cathodes 487

32.1 Diamond amplifier concept 487

32.2 Monte Carlo methods 494

32.3 Relaxation time 499

32.4 Monte Carlo and diamond amplifier response time 516

33 Electron beam physics 525

33.1 Electron orbits and cathode area 526

33.2 Beam envelope equation 528

33.3 Emittance for flat and uniform surfaces 533

33.4 Emittance for a bump 545

33.5 Emittance and realistic surfaces 563

Part VI: Appendices

Appendix 1 Summation, integration, and differentiation 569

A1.1 Series 569

A1.2 Integration 569

A1.3 Differentiation 577

A1.4 Numerical solution of an ordinary differential equation 582

Appendix 2 Functions 585

A2.1 Trigonometric functions 585

A2.2 Gamma function 585

A2.3 Riemann zeta function 585

A2.4 Error function 587

A2.5 Legendre polynomials 587

A2.6 Airy functions 588

A2.7 Lorentzian functions 590

Appendix 3 Algorithms 591

A3.1 Permutation algorithm 591

A3.2 Birthday algorithm 592

A3.3 Least squares fitting of data 593

A3.4 Monty Hall algorithm 595

A3.5 Wave function and density algorithm 596

A3.6 Hydrogen atom algorithms 598

A3.7 Root-finding Methods 601

A3.8 Thermal–field algorithm 604

A3.9 Gamow factor algorithm 606

A3.10 Triangular barrier D(E) 607

A3.11 Evaluation of Hc(u) 608

A3.12 Transfer matrix algorithm 610

A3.13 Semiconductors and doping density 616

A3.14 Band bending: accumulation layer 618

A3.15 Simple ODE solvers 619

A3.16 Current through a metal–insulator–metal diode 622

A3.17 Field emission from semiconductors 624

A3.18 Roots of the quadratic image charge barrier 626

A3.19 Zeros of the airy function 627

A3.20 Atomic sphere radius rs 629

A3.21 Sodium exchange-correlation potential 631

A3.22 Field-dependent work function 632

A3.23 Digitizing an image file 632

A3.24 Lattice gas algorithm 633

A3.25 Evaluation of the point charge model functions 636

A3.26 Modeling of field emitter I(V) data 638

A3.27 Modeling a log-normal distribution of field emitters 640

A3.28 Simple shell and sphere algorithm 643

A3.29 Gyftopoulos–Levine work function algorithm 645

A3.30 Poisson distributions 648

A3.31 Electron–electron relaxation time 650

A3.32 Resistivity and the Debye temperature 651

A3.33 Orbits in a magnetic field 655

A3.34 Trajectory of a harmonic oscillator 657

A3.35 Trajectories for emission from a hemisphere 658

A3.36 Monte Carlo and integration 660

References 663

Index 683

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