Mathematical modeling of field emitter array

• The periodic system of the field emitters mathematical model for rectangular lattice is developed.
• The potential distribution for the whole region of the field emitters system is obtained in the analytical form.
• All geometrical dimensions and the anode potential are the parameters of the field emitter arrays mathematical model.
• The effect of electrostatic shielding on the emission characteristics is presented.

In this article a mathematical model of multi-tip field cathode is presented. The field emitters periodic system for rectangular lattice serves as a field cathode on a flat substrate. A flat anode is parallel to the substrate. A unit cell with the individual field emitter is considered to calculate the electrostatic potential distribution. The effect of the field emitter on the distribution of the electrostatic potential is simulated as the effect of a finite charged line that is located at the axis of the system. The variable separation method is used to solve the boundary-value problem. The three-dimensional analytical solution for the Laplace/Poisson equation is obtained in the Cartesian coordinate system. The potential distribution is represented as an Fourier expansion, and the expansion coefficients are found in an explicit form. Values of the emission characteristics are determined by FowlerNordheim equation. The effect of electrostatic shielding on the emission characteristics is presented.