Field emission system with two emitters mathematical modeling

This paper presents a mathematical model of a two-dimensional emission diode system with two identical field emitters on a flat substrate; the anode is a plane parallel to the plane of the substrate. According to the proposed method, the real field cathode coincides with the virtual cathode, the shape of which is determined by the zero equipotential surface. The influence of each of the field emitters on the electrostatic potential distribution is replaced by the influence for a finite number of charged filaments. The solution of the boundary value problem for the Poisson equation is found in an analytical form. The potential distribution over the entire region of the emission system under study is presented in the form of expansions in terms of eigenfunctions. The coefficients of the series are calculated explicitly. The graphs presented demonstrate the dependence of the potential distribution on the distance between the emitters. All geometric dimensions of the system are the parameters of the problem.