  Using first-principles calculations, we predict that tunable ferroelectricity
can be realized in oxide perovskites with the Grenier structure and ordered
oxygen vacancies. Specifically, we show that
$R_{1/3}A_{2/3}\mathrm{FeO}_{2.67}$ solids (where $R$ is a rare-earth ion and
$A$ an alkaline-earth cation) exhibit stable polar phases, with a spontaneous
polarization tunable by an appropriate choice of $R$ and $A$. We find that
larger cations combined with small $R$ elements lead to a maximum in the
polarization and to a minimum in the energy barriers required to switch the
sign of the polarization. Ferroelectricity arises from cooperative distortions
of octahedral and tetrahedral units, where a combination of rotational and
sliding modes controls the emergence of polarization within three-dimensional
connected layers. Our results indicate that polar Grenier phases of oxide
perovskites are promising materials for microelectronic applications and, in
general, for the study of phenomena emerging from breaking inversion symmetry
in solids.
