Empirically extending 1D Child-Langmuir theory to a warm electron beam

We use particle-in-cell (PIC) simulation to generalize the space charge limit for 1D diodes to the case of electron beams with finite temperature. As shown in our previous work, a certain class of nanoscale vacuum channel transistor (NVCT) devices can be practically considered a one dimensional diode within the gate-anode gap. 1D Child-Langmuir theory for a cold beam is a common estimate for the space charge limit, despite neglecting the finite temperature inherent in real world devices. Using a 1D approximation and collapsing the five dimensional input space (injected current density, voltage, gap distance, average beam velocity, and temperature) to just three reduced dimensions makes a systematic simulation sweep of the input space feasible. We find that the simulation data can be well approximated by a simple empirical formula with a handful of fitting parameters. The resultant function generally characterizes the current transmitted across the diode for a specific thermal electron emission profile. This empirical model can be easily applied to many diode-like devices, including the previously mentioned class of NVCT devices when coupled with an appropriate field emission model.