  Layered transition metal dichalcogenide (TMD) semiconductors oxidize readily
in a variety of conditions, and a thorough understanding of this oxide
formation is required for the advancement of TMD-based microelectronics. Here,
we combine scanning transmission electron microscopy (STEM) with spectroscopic
ellipsometry (SE) to investigate oxide formation at the atomic scale of the
most widely-studied TMD, MoS$_2$. We find that aggressive thermal oxidation
results in $\alpha$-phase plate-like crystalline MoO$_3$ with sharp interfaces,
voids, and a textured alignment with the underlying MoS$_2$. Experiments with
remote substrates and patterned MoS$_2$ prove that thermal oxidation proceeds
via vapor-phase mass transport and redeposition - a challenge to forming thin,
conformal planar oxide films. We accelerate the kinetics of oxidation relative
to the kinetics of mass transport using a non-thermal oxygen plasma process, to
form a smooth and conformal amorphous oxide. The resulting amorphous MoO$_3$
films can be grown several nanometers thick, and we calibrate the oxidation
rate for varying plasma processing conditions. Our results illustrate how TMD
semiconductor oxidation differs significantly from oxidation of legacy
semiconductors, most notably silicon, and provide quantitative guidance for
managing both the atomic scale structure and thin film morphology of oxides in
the design and processing of MoS$_2$ semiconductor devices.
