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.
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