Gas Breakdown for Nano- and Microscale Gaps: Linking Electron Emission and Avalanche Theories

Predicting discharge formation is critical for many applications. In some cases, such as combustion or medicine, the goal is to generate discharges to leverage reactive species formation. For other applications, such as pulsed power systems or microelectronics, plasma formation indicates device failure. Traditionally, Townsend avalanche drives discharge formation and is predicted mathematically by Paschen's law (PL), which shows that the breakdown voltage scales with the product of gas pressure and gap distance. Reducing device size increases the electric field at the cathode at breakdown, stripping electrons by field emission. These electrons ionize the gas near the cathode, which increases secondary emission and creates an additional electric field due to the positive space-charge. These phenomena reduce the voltage necessary to induce breakdown; therefore, the breakdown voltage decreases at smaller gaps rather than increasing as predicted by PL. This presentation summarizes theoretical work that derived a closed form solution that shows that the breakdown voltage in the field emission regime scales linearly with gap distance. We additionally apply particle-in-cell (PIC) simulations to determine the ionization coefficient for microscale gaps since the electric fields at breakdown fall outside the traditional semi-empirical range and show corrections for various pressures and gap distances for DC voltages. Further reducing the gap size ultimately makes the gap space-charge limited, meaning that current can no longer be increased. We describe the mathematical concept of "nexus theory" to characterize transitions between emission mechanisms and apply it to experiments of nanoscale gaps at atmospheric pressure and vacuum to demonstrate the proximity to these transitions and describe potential extensions to other mechanisms. We further outline the challenges of extending these approaches to AC fields, particularly in the microwave regime, and assess potential theories compared to particle-in-cell simulations, including the scalings of ionization frequency and breakdown voltage for various waveforms and gap conditions.