  Optimizing thermal anneals of Si-implanted $\beta$-Ga$_2$O$_3$ is critical
for low resistance contacts and selective area doping. We report the impact of
annealing ambient, temperature, and time on activation of room temperature
ion-implanted Si in $\beta$-Ga$_2$O$_3$ at concentrations from 5x10$^{18}$ to
1x10$^{20}$ cm$^{-3}$, demonstrating full activation (>80% activation,
mobilities >70 cm$^{2}$/Vs) with contact resistances below 0.29 $\Omega$-mm.
Homoepitaxial $\beta$-Ga$_2$O$_3$ films, grown by plasma assisted MBE on
Fe-doped (010) substrates, were implanted at multiple energies to yield 100 nm
box profiles of 5x10$^{18}$, 5x10$^{19}$, and 1x10$^{20}$ cm$^{-3}$. Anneals
were performed in a UHV-compatible quartz furnace at 1 bar with well-controlled
gas composition. To maintain $\beta$-Ga$_2$O$_3$ stability, $p_{O2}$ must be
greater than 10$^{-9}$ bar. Anneals up to $p_{O2}$ = 1 bar achieve full
activation at 5x10$^{18}$ cm$^{-3}$, while 5x10$^{19}$ cm$^{-3}$ must be
annealed with $p_{O2}$ <10$^{-4}$ bar and 1x10$^{20}$ cm$^{-3}$ requires
$p_{O2}$ <10$^{-6}$ bar. Water vapor prevents activation and must be maintained
below 10$^{-8}$ bar. Activation is achieved for anneal temperatures as low as
850 {\deg}C with mobility increasing with anneal temperature up to 1050
{\deg}C, though Si diffusion has been reported above 950 {\deg}C. At 950
{\deg}C, activation is maximized between 5 and 20 minutes with longer times
resulting in decreased carrier activation (over-annealing). This over-annealing
is significant for concentrations above 5x10$^{19}$ cm$^{-3}$ and occurs
rapidly at 1x10$^{20}$ cm$^{-3}$. RBS (channeling) suggests damage recovery is
seeded from remnant aligned $\beta$-Ga$_2$O$_3$ that remains after
implantation; this conclusion is also supported by STEM showing retention of
the $\beta$-phase with inclusions that resemble the $\gamma$-phase.
