Effect of Nonuniform Emission on Miram Curves

Analysis of temperature-limited flow, space-charge-limited flow, and the transition between them using a simple planar diode with a thermionic cathode, in which the cathode surface has spatially nonuniform emission properties, is presented. Our theoretical results, which are derived from a model based on solutions to the Vlasov and Poisson equations, compare well with the results of particle-in-cell simulations. We find that the location and the shape of the knee in the anode current versus temperature characteristic (Miram or "rollover" curve) are significantly affected by non-uniformities in the space-charge density in the A-K gap, but are relatively unaffected by the electron motion parallel to the electrode surfaces. In particular, emission from an actively emitting region is strongly affected by the forces (or lack thereof) exerted by the space-charge of the electrons emitted by their neighbors. Perhaps, most remarkably, we find that the limiting current reaching the anode is approximately given by the classical 1-D Child-Langmuir law, even if a significant fraction of the cathode surface is non-emitting.