Mathematical modeling of a diode system based on a field emitter

A mathematical model of a diode system based on a field emitter is considered. The field emitter is modeled in the following way: the tip is a conducting sphere; the “body” is a solid dielectric with a spindle surface of revolution. The anode is a part of the spherical surface and the substrate of the emitter is a spherical surface or a plane. The influence of a space charge is not taken into account. To find the electrostatic potential distribution, the methods of separation of variables and paired equations are used. The problem of finding unknown coefficients in the eigenfunction expansion of the potential is reduced to the solution of the system of equations, which includes linear algebraic equations and the Fredholm equation of the second kind. Thus, the computational problem of the distribution of electrostatic potential in the entire domain of the investigated system is solved.