  Beta gallium oxide ($\beta$-Ga$_2$O$_3$) shows significant promise in the
high-temperature, high-power, and sensing electronics applications. However,
long-term stable metallization layers for Ohmic contacts at high temperature
present unique thermodynamic challenges. The current most common Ohmic contact
design based on 20 nm of Ti has been repeatedly demonstrated to fail at even
moderately elevated temperatures (300-400$^{\circ}$C) due to a combination of
non-stoichiometric Ti/Ga$_2$O$_3$ interfacial reactions and kinetically favored
Ti diffusion processes. Here we demonstrate stable Ohmic contacts for
Ga$_2$O$_3$ devices operating up to 500-600$^{\circ}$C using ultrathin Ti
layers with a self-limiting interfacial reaction. The ultrathin Ti layer in the
5nm Ti / 100nm Au contact stack is designed to fully oxidize while forming an
Ohmic contact, thereby limiting both thermodynamic and kinetic instability.
This novel contact design strategy results in an epitaxial conductive anatase
titanium oxide interface layer that enables low-resistance Ohmic contacts that
are stable both under long-term continuous operation (>500 hours) at
600$^{\circ}$C in vacuum ($\leq$ 10$^{-4}$ Torr), as well as after repeated
thermal cycling (15 times) between room temperature and 550$^{\circ}$C in
flowing N$_2$. This stable Ohmic contact design will accelerate the development
of high-temperature devices by enabling research focus to shift towards
rectifying contacts and other interfacial layers.
