  We report the use of suboxide molecular-beam epitaxy (S-MBE) to grow
$\beta$-Ga$_2$O$_3$ at a growth rate of ~1 ${\mu}$m/h with control of the
silicon doping concentration from 5x10$^{16}$ to 10$^{19}$ cm$^{-3}$. In S-MBE,
pre-oxidized gallium in the form of a molecular beam that is 99.98\% Ga$_2$O,
i.e., gallium suboxide, is supplied. Directly supplying Ga2O to the growth
surface bypasses the rate-limiting first step of the two-step reaction
mechanism involved in the growth of $\beta$-Ga$_2$O$_3$ by conventional MBE. As
a result, a growth rate of ~1 ${\mu}$m/h is readily achieved at a relatively
low growth temperature (T$_{sub}$ = 525 $^\circ$C), resulting in films with
high structural perfection and smooth surfaces (rms roughness of < 2 nm on ~1
${\mu}$m thick films). Silicon-containing oxide sources (SiO and SiO$_2$)
producing an SiO suboxide molecular beam are used to dope the
$\beta$-Ga$_2$O$_3$ layers. Temperature-dependent Hall effect measurements on a
1 ${\mu}$m thick film with a mobile carrier concentration of 2.7x10$^{17}$
cm$^{-3}$ reveal a room-temperature mobility of 124 cm$^2$ V$^{-1}$ s$^{-1}$
that increases to 627 cm$^2$ V$^{-1}$ s$^{-1}$ at 76 K; the silicon dopants are
found to exhibit an activation energy of 27 meV. We also demonstrate working
MESFETs made from these silicon-doped $\beta$-Ga$_2$O$_3$ films grown by S-MBE
at growth rates of ~1 ${\mu}$m/h.
