Universal Gas Breakdown Theory from Microscale to the Classical Paschen Law

Summary form only given. Paschen's law for predicting breakdown voltage deviates at microscale dimensions and atmospheric pressure because field emission, rather than the Townsend avalanche, drives breakdown¹. One can more accurately predict this deviation by coupling field emission and the Townsend avalanche to derive “modified Paschen's curves¹.” Previous work extended one such modified Paschen's curve to derive a universal breakdown curve for any gas at atmospheric pressure with the secondary electron coefficient, γ_SE = 0, to demonstrate the limiting behaviors for large and small gap distances^2,3. Here, we further generalize this model for gap sizes larger than the mean free path of an electron (~100 nm), pressures below the transition to streamer discharge given by Meek's criterion (~1 to ~3000 Torr), and nonzero γ_SE. Analytic expressions agree well with numerical solutions and particle-in-cell simulations when γ_SE = 0. Moreover, we derive equations for a “modified Paschen minimum” that was previously masked by only considering atmospheric pressure and show that this minimum exists even when γ_SE = 0. Furthermore, including γ_SE in the analytic equations for breakdown voltage shows the transition from the field emission dominated regime to the coupled field-emission-Townsend regime to the classical Paschen law, where the conventional criterion for Townsend avalanche is satisfied. Thus, this study provides a universal breakdown model for any gas, pressure, and gap distance (between the electron mean free path and Meek's criterion) to aid system design and optimization.