  Although $\beta$-TeO$_2$ is a promising $p$-type transparent conducting oxide
(TCO) due to the large optical gap ($\sim$ 3.7 eV) and a light effective hole
mass, its hole dopability still remains unexplored. In this work, electronic
structure of $\beta$-TeO$_2$ and its point defects are investigated using the
HSEsol functional with the band-gap-tuned mixing parameter. Our calculations
reveal that $\beta$-TeO$_2$ exhibits a significant difference between the
fundamental and optical band gaps because lower energy optical transitions are
dipole forbidden. Additionally, it has a low hole effective mass, especially
in-plane. The point defect calculations show that $\beta$-TeO$_2$ is
intrinsically an insulator. From systematic calculations of the trivalent
dopants as well as hydrogen, Bi doping is suggested as the best candidate as an
acceptor dopant. This work paves the way for the material design of the
$p$-type $\beta$-TeO$_2$.
