Analytic theory for field emission driven microscale gas breakdown for a pin-to-plate geometry

Abstract

Decreasing electronics size necessitates better characterization of electron emission at the micro- and nanoscales for applications including microplasmas, micro- and nanoelectromechanical systems, and directed energy. While Paschen's law (PL) has historically predicted breakdown voltage based on the Townsend avalanche, field emission must be incorporated for gap sizes below ∼15 μm. Extensive studies have modified PL to explicitly include field emission for planar geometries; however, many practical experiments use pin-to-plate geometries. We modify a previous theory coupling PL and field emission to account for pin-to-plate geometries by replacing the field enhancement factor, which has been used primarily as a fitting parameter, with the appropriate vacuum electric field. This requires explicitly accounting for the spatial dependence of ionization and non-uniform space charge in Poisson's equation. We derive a breakdown equation of the form previously obtained for planar geometry [Venkattraman and Alexeenko, Phys. Plasmas 19, 123515 (2012)] that agrees well with experimental data with the work function as the fitting parameter. The work function was consistently lower (∼2 eV) than anticipated (∼4.5 eV) but was generally fairly consistent (∼ ± 7%). We then derived closed form solutions in the limit of low ionization, corresponding to the field emission regime, and recovered an analytic solution for a parallel plate geometry in the limit of small gap distance that differed from prior analytic results because of the explicit consideration of spatial dependence in charge density. This theory may ultimately be applied to other nonplanar geometries by applying the appropriate equation for the vacuum electric field.