  On-chip photonic devices based on SiO$_2$ are of interest for applications
such as microresonator gyroscopes and microwave sources. Although SiO$_2$
microdisk resonators have achieved quality factors exceeding one billion, this
value remains an order of magnitude less than the intrinsic limit due to
surface roughness scattering. Atomic layer etching (ALE) has potential to
mitigate this scattering because of its ability to smooth surfaces to
sub-nanometer length scales. While isotropic ALE processes for SiO$_2$ have
been reported, they are not generally compatible with commercial reactors, and
the effect on surface roughness has not been studied. Here, we report an ALE
process for SiO$_2$ using sequential exposures of Al(CH$_3$)$_3$
(trimethylaluminum, TMA) and Ar/H$_2$/SF$_6$ plasma. We find that each process
step is self-limiting, and that the overall process exhibits a synergy of 100%.
We observe etch rates up to 0.58 \r{A} per cycle for thermally-grown SiO$_2$
and higher rates for ALD, PECVD, and sputtered SiO$_2$ up to 2.38 \r{A} per
cycle. Furthermore, we observe a decrease in surface roughness by 62% on a
roughened film. The residual concentration of Al and F is around 1-2%, which
can be further decreased by O$_2$ plasma treatment. This process could find
applications in smoothing of SiO$_2$ optical devices and thereby enabling
device quality factors to approach limits set by intrinsic dissipation.
