Mathematical Modeling of Emission in Small-Size Cathode

This book deals with mathematical modeling, namely, it describes the mathematical model of heat transfer in a silicon cathode of small (nano) dimensions with the possibility of partial melting taken into account. This mathematical model is based on the phase field system, i.e., on a contemporary generalization of Stefan-type free boundary problems. The approach used is not purely mathematical but is based on the understanding of the solution structure (construction and study of asymptotic solutions) and computer calculations. The book presents an algorithm for numerical solution of the equations of the mathematical model including its parallel implementation. The results of numerical simulation concludes the book. The book is intended for specialists in the field of heat transfer and field emission processes and can be useful for senior students and postgraduates.

Auteur

Vladimir G. Danilov received the Ph.D. degree from the Moscow Institute of Electronics and Mathematics, Moscow, Russia, in 1976, and the D.Sci. degree from Moscow State University, Moscow, in 1990. He is currently a Professor with the National Research University Higher School of Economics, Moscow. His current research interests include linear and nonlinear problems of PDE, asymptotic methods, and mathematical simulation. Roman K. Gaydukov received the M.S. degree from the Moscow Institute of Electronics and Mathematics, Moscow, Russia, in 2012, and the Ph.D. degree from National Research University Higher School of Economics, Moscow, Russia, in 2016. He is currently an Associate Professor with the National Research University Higher School of Economics, Moscow. His current research interests include asymptotic methods, mathematical and numerical simulation, field emission, fluid mechanics and boundary layer theory. Vadim I. Kretov received the M.S. degree from the Moscow Institute of Electronics and Mathematics, Moscow, Russia, in 2008, and the Ph.D. degree from National Research University Higher School of Economics, Moscow, Russia, in 2019. His current research interests include mathematical simulation, field emission, and numerical solution of PDE.

Contenu

**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 2.2.1. Electron and hole concentration

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 2.5.9. Inversion temperature dependence on the external electric field voltage

Chapter 3. Mathematical model

3.1. Phase field system and its use in heat transfer modeling 3.2. Phase field system as regularization of limit problems with free boundary

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 StefanGibbsThomson 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 3.8. Mathematical model of electron emission in a vacuum cube

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 4.4.5. One more version of the difference scheme

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 experimental parameters

4.7. Formation of melting and crystallizing nuclei in the model 4.8. Conclusion