  $\beta$-Gallium Oxide (Ga$_{2}$O$_{3}$) is an extensively investigated
ultrawide-bandgap semiconductor for potential applications in power electronics
and RF switching. The room temperature bulk electron mobility ($\sim$200
cm$^{2}$V$^{-1}$s$^{-1}$) is comparatively low and is limited by the 30 phonon
modes originating from its 10-atom primitive cell. The theoretically calculated
saturation velocity is 1-2$\times$10$^{7}$ cms$^{-1}$ which is comparable to
GaN. The high field electron transport in the 2DEG is explored in this work
based on the first principles calculated parameters. A self-consistent
calculation on a given heterostructure design gives the confined eigenfunctions
and eigenenergies. The intrasubband and the intersubband scattering rates are
calculated based on the Fermi's golden rule considering LO phonon-plasmon
screening. The high field characteristics are extracted from the full-band
Monte Carlo simulation of heterostructures at 300 K. The motion of electrons in
the 2DEG and the bulk is treated through an integrated Monte Carlo program
which outputs the steady state zone population, transient dynamics and the
velocity-field curves for a few heterostructure designs. The critical field for
saturation does not change significantly from bulk values, however an improved
peak velocity is calculated at a higher 2DEG density. The velocity at low 2DEG
densities is impacted by the antiscreening of LO phonons which plays an
important role in shaping the zone population. A comparison with the
experimental measurements is also carried out and possible origins of the
discrepancies with experiments is discussed.
