  The idea that surface effects may play an important role in suppressing
$e_g'$ Fermi surface pockets on Na$_x$CoO$_2$ $(0.333 \le x \le 0.75)$ has been
frequently proposed to explain the discrepancy between LDA calculations
(performed on the bulk compound) which find $e_g$' hole pockets present and
ARPES experiments, which do not observe the hole pockets. Since ARPES is a
surface sensitive technique it is important to investigate the effects that
surface formation will have on the electronic structure of Na$_{1/3}$CoO$_2$ in
order to more accurately compare theory and experiment. We have calculated the
band structure and Fermi surface of cleaved Na$_{1/3}$CoO$_2$ and determined
that the surface non-trivially affects the fermiology in comparison to the
bulk. Additionally, we examine the likelihood of possible hydroxyl cotamination
and surface termination. Our results show that a combination of surface
formation and contamination effects could resolve the ongoing controversy
between ARPES experiments and theory.
