Mathematical Modeling of Emission in Small-Size Cathode

Preface

Chapter 1. Introduction

1.1. Brief history of the electron emission discovery

1.2. Types of electron emission

1.3. Statement of the problem

1.4. Mathematical statement of the problem. Heat transfer model

Chapter 2. Physical foundations of field emission

2.1. Band theory and Fermi levels

2.2. Specific conductance of semiconductors

2.2.2. Effective mass

2.2.3. Electron and hole mobility

2.2.4. Temperature dependence of specific conductance in silicon

2.3. Thermoelectricity

2.4. Heat conduction of solids

2.4.1. Electron heat conductivity

2.4.2. Heat conduction of crystal lattice

2.5. Emission current density and Nottingham effect

2.5.1. Support function in metals

2.5.2. Electron tunneling through potential barrier

2.5.3. Formula for the barrier transmission factor in the case of field emission cathode

2.5.4. Emission current density in metals

2.5.5. Specific characteristics of filed emission from semiconductor cathode

2.5.6. Approximation of the emission current density formula

2.5.7. Nottingham effect

2.5.8. Optimal values of approximation parameters

Chapter 3. Mathematical model

3.1. Phase field system and its use in heat transfer modeling

3.3. Asymptotic solution of the phase field system and modified Stefan problem

3.3.1. Construction of asymptotic solution

3.3.2. Examples

3.4. Weak solution of the phase field system and the melting zone model

3.4.1. Weak solutions and Hugoniot-type conditions

3.4.2. Wavetrain''-type solutions and the corresponding limit problem

3.5. Derivation of the limit Stefan-Gibbs-Thomson problem solution from numerical

solution of the phase field system

3.6. Generation and merging of dissipative waves

3.7. Cathode in the vacuum cube. Definition of a generalized solution to Poisson

equation for electric field potential

Chapter 4 Numerical modeling and its results

4.1. Nanocathode model

4.2. Computation of current density inside the cathode

4.3. Computation of emission current density and Nottingham effect modeling

4.4. Difference scheme

4.4.1. Difference scheme for the equation for the potential

4.4.2. Difference scheme for the equation for the function of order

4.4.3. Difference scheme for the heat conduction equation

4.4.4. Difference scheme stability

4.4.6. Choice of the difference scheme step

4.5. Algorithm for solving difference equations and possible versions of

its parallelization

4.6. Results of numerical experiments

4.6.1. Nonmonotone behavior of free boundaries

4.6.2. Results of modeling with physical parameters corresponding to

4.7. Formation of melting and crystallizing nuclei in the model