  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.
