Empirically extending Child-Langmuir theory to a thermal electron distribution

While the self-consistent potential and current flow in a parallel plate diode can be analytically calculated in the 1D limit for a cold beam of electrons, this work explores the non-trivial extension to finite temperature. The input parameter space (injected current density, potential boundary conditions, gap distance, beam velocity, and temperature) is reduced to just three dimensionless parameters. A systematic sweep over these three parameters is performed to explore space charge effects on the average anode current, as well as to characterize the oscillatory behavior expected near the space charge limit. The results from 1D simulation allow the construction of a simple, empirical model for the diode characteristics for the case of a thermal electron distribution. This model can be applied to nanoscale vacuum channel transistors (NVCT). Specifically, we have shown using full-device 3D PIC simulations that NVCT arrays comprising a large number of uniform tips field-emitting into a large gap are practically one dimensional as far as space charge effects are concerned, meaning the gap environment is well described by the aforementioned empirical model for diodes.