Quantum Scale Gas Breakdown

Abstract

While commonly used to predict gas breakdown voltage, measured breakdown voltage deviates from the classical Paschen law for microscale gaps at atmospheric pressure due to the contribution of field emission 1. Recent studies have derived analytic expressions to predict breakdown voltage for gap distances down to the mean free path of an electron that agree well with numerical solutions, particle-in-cell simulations, and experiment 2. However, device sizes continue to shrink and some applications, such as electron emission in a graphene nanogap with sharp edges 3, consider gap sizes on the order of the electron mean free path. Furthermore, variations in critical field emission parameters, such as field enhancement factor and work function, due to surface roughness become more important for these smaller gaps. As a first step to elucidating this behavior, we couple the one-dimensional Schrodinger equation with Poisson’s equation, as done previously for the quantum extension of the Child-Langmuir law 4, and integrate this with previous analytic work for coupling field emission to Townsend avalanche to predict the deviation from the classical Paschen’s law 1. Applying a matched asymptotic analysis as we did at microscale 2 enables us to assess the transition from quantum breakdown back to the combined field emission and Townsend regime. Implications for developing a linked breakdown model from Townsend avalanche to field emission to quantum scale with decreasing gap distance and from field emission and quantum behavior to space-charge limited flow as a function of voltage and pressure will be discussed.