A Tutorial on Theoretical and Computational Techniques for Gas Breakdown in Microscale Gaps

Paschen's law (PL), derived based on the Townsend avalanche (TA) condition, is commonly used to predict gas breakdown. For microscale gaps near atmospheric pressure, TA is insufficient to drive breakdown and ion-enhanced field emission (FE) dominates. Accurately predicting breakdown voltages for these gaps is critical for numerous applications, including environmental remediation, medicine, combustion, and propulsion. This tutorial summarizes theoretical and computational approaches for predicting this behavior and demonstrating the transition between the TA and FE mechanisms. It focuses on the derivation of closed-form solutions from a theory that accounts for the generation of additional positive space charge at the cathode due to electrons generated by the strong FE-induced electric fields. Appropriate simplifications using a matched asymptotic analysis in terms of the total ionization in the gap agree well with simulations and experiments and show the transition from FE for small gaps to PL at larger gaps. Specifically, this theory shows that the breakdown voltage varies linearly with gap distance when FE dominates, agreeing with both the experimental and simulation results. The particle-in-cell Monte Carlo collision (PIC/MCC) simulations used to predict the ionization coefficient provide additional insight into the mechanisms involved. Future benefits of extended theoretical and computational research for examining microscale and nanoscale breakdown and electron emission, particularly assessing the impact of electrode surface structure and device design and coupling with additional emission mechanisms, will be discussed.