Space-charge-limited currents for cathodes with electric field enhanced geometry

Abstract

This paper presents the approximate analytic solutions of current density for annulus and circle cathodes. The current densities of annulus and circle cathodes are derived approximately from first principles, which are in agreement with simulation results. The large scaling laws can predict current densities of high current vacuum diodes including annulus and circle cathodes in practical applications. In order to discuss the relationship between current density and electric field on cathode surface, the existing analytical solutions of currents for concentric cylinder and sphere diodes are fitted from existing solutions relating with electric field enhancement factors. It is found that the space-charge-limited current density for the cathode with electric-field enhanced geometry can be written in a general form of J = g(βE)2J0, where J0 is the classical (1D) Child-Langmuir current density, βE is the electric field enhancement factor, and g is the geometrical correction factor depending on the cathode geometry.