  The performance of ultra-wide band gap materials like
$\beta$-Ga$_\mathrm{2}$O$_\mathrm{3}$ is critically dependent on achieving high
average electric fields within the active region of the device. In this report,
we show that high-k gate dielectrics like BaTiO$_\mathrm{3}$ can provide an
efficient field management strategy by improving the uniformity of electric
field profile in the gate-drain region of lateral field effect transistors.
Using this strategy, we were able to achieve high average breakdown fields of
1.5 MV/cm and 4 MV/cm at gate-drain spacing (L$_\mathrm{gd}$) of 6 um and 0.6
um respectively in $\beta$-Ga$_\mathrm{2}$O$_\mathrm{3}$, at a high channel
sheet charge density of 1.8x10$^\mathrm{13}$cm$^\mathrm{-2}$. The high sheet
charge density together with high breakdown field enabled a record power figure
of merit (V$^\mathrm{2}$$_\mathrm{br}$/R$_\mathrm{on}$) of 376
MW/cm$^\mathrm{2}$ at a gate-drain spacing of 3 um.
