Patch-field corrections to Child’s law and Schottky barrier lowering in cathodes

Patch-field effects from a nonuniform surface work function significantly alters net cathode emission, modeled here as Φo low work function patches of diameter D within a higher Φe average work function surface. Building on a 1985 technique by Eng to estimate nonlinear inhomogeneous Poisson’s equation solutions for planar diode thermionic emission, this analysis exhibits Child’s law for Space-Charge Limited (ScL) emission at low voltages, followed by Patch-Field Limited (PfL) emission, then transitioning to Temperature-Limited (TL) emission with Schottky barrier lowering at high voltages. Cathode contamination or aging that increases Φe can result in the cathode current density vs temperature (I–T) curves having rounder ScL-TL knee regions and observable I-T slopes far from the knee region, explaining a mystery often seen in data. Comparing this analytic model with 2022 data by Chen et al. on an S-type dispenser cathode shows a comparable or improved data fit to their numerical results. For these conditions, our model also shows PfL emission has nearly power-law behavior, with exponents increasing with temperature but staying below the 3/2 Child’s law limit (with 18 figures).