  Rare-earth doped barium zirconate (BaZrO$_3$) ceramics are of interest as
proton-conducting and luminescent materials. Here, we report a study of
dysprosium (Dy) and other relevant point defects in BaZrO$_3$ using hybrid
density-functional defect calculations. The tetravalent Dy$^{4+}$ is found to
be structurally and electronically stable at the Zr lattice site (i.e., as
Dy$_{\rm Zr}^0$), but most often energetically less favorable than the
trivalent Dy$^{3+}$ (i.e., Dy$_{\rm Zr}^-$) in as-synthesized BaZrO$_3$, due to
the formation of low-energy, positively charged oxygen vacancies and the
mixed-site occupancy of Dy in the host lattice. The Dy$^{4+}$/Dy$^{3+}$ ratio
can, in principle, be increased by preparing the material under highly
oxidizing and Ba-rich conditions and co-doping with acceptor-like impurities;
however, post-synthesis treatment may still be needed to realize a
non-negligible Dy$^{4+}$ concentration. We also find that certain unoccupied Dy
$4f$ states and the O $2p$ states are {\it strongly hybridized}, a feature not
often seen in rare-earth-containing materials, and that the isolated Dy$_{\rm
Zr}$ defect might be the source of a broad blue emission in band-to-defect
("charge-transfer") luminescence.
