  High quality Nb-based superconductor-insulator-superconductor (SIS) junctions
with Al oxide (AlO$_x$) tunnel barriers grown from Al overlayers are widely
reported in the literature. However, the thin barriers required for high
critical current density (J$_c$) junctions exhibit defects that result in
significant subgap leakage current that is detrimental for many applications.
High quality, high-J$_c$ junctions can be realized with AlN$_x$ barriers, but
control of J$_c$ is more difficult than with AlO$_x$. It is therefore of
interest to study the growth of thin AlO$_x$ barriers with the ultimate goal of
achieving high quality, high-J$_c$ AlO$_x$ junctions. In this work, 100\%\
O$_2$ and 2\%\ O$_2$ in Ar gas mixtures are used both statically and
dynamically to grow AlO$_x$ tunnel barriers over a large range of oxygen
exposures. In situ ellipsometry is used for the first time to extensively
measure AlO$_x$ tunnel barrier growth in real time, revealing a number of
unexpected patterns. Finally, a set of test junction wafers was fabricated that
exhibited the well-known dependence of J$_c$ on oxygen exposure (E) in order to
further validate the experimental setup.
