  We report on the realization of record low resistance Ohmic contacts to
MOVPE-grown heavily Si-doped $\beta$-Ga$_2$O$_3$ and
$\beta$-(Al$_x$Ga$_1-x$)$_2$ O$_3$ epitaxial films. Transfer length measurement
(TLM) patterns were fabricated on the heavily Si-doped homoepitaxial
$\beta$-Ga$_2$O$_3$ films with electron concentration (n) ranging from 1.77 to
3.23e20 cm^-3. Record low specific contact resistance and total contact
resistance (Rc) of 1.62e-7 Ohm.cm^2 and 0.023 Ohm.mm were realized for
$\beta$-Ga$_2$O$_3$: Si films with n > 3e20 cm^-3. TLM structures were also
fabricated on heavily Si doped coherently strained $\beta$-(Al$_x$Ga$_1-x$)$_2$
O$_3$ (x=12%, 17% and 22%) films. The film with 12% Al composition (n=1.23e20
cm^-3) showed \r{ho}c of 5.85e-6 Ohm.cm^2, but it increased to 2.19e-4 Ohm.cm^2
for a layer with a 22% Al composition. Annealing the samples post metal
deposition has generally led to a decrease in contact resistance, but for high
Al content $\beta$-(Al$_x$Ga$_1-x$)$_2$ O$_3$, the contact resistance did not
change significantly after the annealing process. The low contact resistance
values measured in this work are very promising for the fabrication of high
frequency power devices.
