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Theoretically and computationally describing the operation of nanodiodes requires characterizing the transitions between multiple electron emission mechanisms for nanodiodes with complicated geometries. This motivates our development of techniques to determine when simplified theories for individual mechanisms suffice compared to more complete, but more computationally expensive, models. Leveraging recent theories that define a canonical gap distance to translate planar theory to nonplanar diodes, we derive the conditions for the transitions among thermal emission, field emission, and space-charge-limited current density (SCLCD) in vacuum and with collisions for non-Cartesian coordinate systems, including spherical, cylindrical, and prolate spheroidal coordinate systems. Particle-in-cell (PIC) simulations of the current density as a function of applied voltage for a tip-to-plate geometry in vacuum agreed qualitatively with the asymptotes for thermal emission at low voltage and SCLCD at higher voltage using the canonical gap distance. This demonstrates the utility of this approach for guiding system design and suggests future extensions to save simulation time for more realistic geometries that are more computationally expensive.
Sinelli et al. (Tue,) studied this question.