Space-Charge-Limited Current in Non-Planar Diodes with Nonzero Injection Velocity and Relativistic Effects

Space-charge limited current (SCLC) is an important threshold for multiple applications, including high power microwaves, thermionic converters, electric thrusters, time-resolved electron microscopy, and nano-vacuum transistors [1] . In this presentation, we first derive a method for identifying the bifurcation and SCLC thresholds with a nonzero injection velocity in any orthogonal coordinate system from first principles by applying variational calculus (VC), which was previously used to derive SCLC for zero injection velocity [2] . We further obtain the existing planar solutions for bifurcation [3] and SCLC [4] depending on the functional extremized in the Euler-Lagrange equation. After extending the nonzero injection velocity solutions to cylindrical and spherical geometries, we apply a similar technique for relativistic electrons. We recover the existing planar solution using VC, and compare our exact cylindrical and spherical solutions to previous theories and experiments [5] . We then report progress toward unifying these theories to account for incorporating both nonzero injection velocity and relativistic effects in SCLC calculations for any orthogonal coordinate system and conclude by discussing further extensions of this method for emission physics [1] .