  Carbon contamination is a significant concern for proton-conducting oxides in
the cerate and zirconate family, particularly for BaCeO$_3$. Here, we use
first-principles calculations to evaluate carbon stability in SrCeO$_3$,
BaCeO$_3$, SrZrO$_3$, and BaZrO$_3$. The cerates require more carbon-poor
environments to prevent carbonate formation, though this requirement can be
loosened through the use of more oxygen-poor growth conditions. Carbonate
formation is not the only concern, however. We find that interstitial carbon
has lower formation energies in the cerates relative to the zirconates, leading
to higher carbon concentrations that compete with the desired oxygen vacancy
formation. We also examine the mobility of carbon interstitials, finding that
both migration barriers and binding energies to acceptors are lower in the
cerates. As a result, the cerates are likely to degrade when exposed to carbon
at operating temperatures. Our results show definitively why the cerates are
less stable than the zirconates with respect to carbon and elucidate the
mechanisms contributing to their instability, thereby helping to explain why
alloying with zirconium will enhance their operational efficiency.
