Techniques for Determining Space-Charge Limited Current for Nonplanar Crossed-Field Diodes

Calculating the space-charge-limited current (SCLC) is critical for numerous applications, including high-power microwaves (HPM), electric thrusters, and nano vacuum transistors
¹
. While the theory for one-dimensional (1–D), planar SCLC is over a century old, exact analytic solutions for more complicated geometries have been challenging
²
. Recent studies have used variational calculus, conformal mapping, and Lie-point symmetries to obtain exact solutions for 1–D SCLC for various geometries
^2,3
. Many HPM applications use crossed-field diodes, which incorporate a magnetic field perpendicular to the diode's electric field. While theories have been derived to determine the maximum current for planar 1–D crossed-field diodes, extensions to nonplanar geometries are more difficult. A recent study used variational calculus to derive an equation for SCLC for a crossed-field diode for a 1–D cylindrical diode that could be solved numerically;
⁴
however, this result is not readily extended to other geometries. In this paper, we apply Lie-point symmetries to transform equations derived from combinations of Poisson's equation, conservation of energy, and conservation of momentum. The resulting equations for unsolved 1–D diode geometries, specifically cylindrical, have similar mathematical forms as previously solved geometries (e.g., planar). This canonical solution can then be used to obtain solutions for previously intractable geometries. Implications of these results on assessing device operating limitations will be discussed.