Space-charge-limited current density for nonplanar diodes with monoenergetic emission using Lie-point symmetries

TL;DR

This paper derives exact analytic equations for space-charge limited current density in nonplanar diodes with monoenergetic emission using Lie-point symmetries, revealing universal behaviors and geometric independence.

Contribution

It introduces a novel Lie-symmetry based method to linearize and solve the nonlinear differential equations governing SCLCD in nonplanar diodes, providing explicit formulas for various geometries.

Findings

Derived exact SCLCD equations for cylindrical, spherical, t-t, and t-p diodes.

Found the correction factor depends only on initial energies, not geometry.

Showed SCLCD for t-p diodes is four times larger than for t-t diodes.

Abstract

Understanding space-charge limited current density (SCLCD) is fundamentally and practically important for characterizing many high-power and high-current vacuum devices. Despite this, no analytic equations for SCLCD with nonzero monoenergetic initial velocity have been derived for nonplanar diodes from first principles. Obtaining analytic equations for SCLCD for nonplanar geometries is often complicated by the nonlinearity of the problem and over constrained boundary conditions. In this letter, we use the canonical coordinates obtained by identifying Lie-point symmetries to linearize the governing differential equations to derive SCLCD for any orthogonal diode. Using this method, we derive exact analytic equations for SCLCD with a monoenergetic injection velocity for one-dimensional cylindrical, spherical, tip-to-tip (t-t), and tip-to-plate (t-p) diodes. We specifically demonstrate that the correction factor from zero initial velocity to monoenergetic emission depends only on the initial kinetic and electric potential energies and not on the diode geometry and that SCLCD is universal when plotted as a function of the canonical gap size. We also show that SCLCD for a t-p diode is a factor of four larger than a t-t diode independent of injection velocity. The results reduce to previously derived results for zero initial velocity using variational calculus and conformal mapping.