Thermionic emission or tunneling? The universal transition electric field for ideal Schottky reverse leakage current in $beta$-Ga$_{2}$O$_{3}$

The reverse leakage current through a Schottky barrier transitions from a thermionic-emission dominated regime to a barrier-tunneling dominated regime as the surface electric field increases. In this study, we evaluate such transition electric field ($E_{rm T}$) in $beta$-Ga$_{2}$O$_{3}$ using a numerical reverse leakage model. $E_{rm T}$ is found to have very weak dependence on the doping concentration and barrier height, thus a near-universal temperature dependence suffices and is given by a simple empirical expression in Ga$_{2}$O$_{3}$. With the help of a field-plate design, we observed experimentally in Ga$_{2}$O$_{3}$ Schottky barrier diodes a near-ideal bulk reverse leakage characteristics, which matches well with our numerical model and confirms the presence of the transition region. Near the transition electric field, both thermionic emission and barrier tunneling should be considered. The study provides important guidance toward accurate design and modeling of ideal reverse leakage characteristics in $beta$-Ga$_{2}$O$_{3}$ Schottky barrier diodes.