  Ferroelectric hafnia is being explored for next generation electronics due to
its robust ferroelectricity in nanoscale samples and its compatibility with
silicon. However, its ferroelectricity is not understood. Other ferroelectrics
usually lose their ferroelectricity for nanoscopic samples and thin films, and
the hafnia ground state is non-polar baddeleyite. Here we study hafnia with
density functional theory (DFT) under epitaxial strain, and find that strain
not only stabilizes the ferroelectric phases, but also leads to unstable modes
and a downhill path in energy from the high temperature tetragonal structure.
We find that for tensile epitaxial strains corresponding to a square substrate
of lattice constant $a_{epi} \geqslant 5.38$ \AA~the ferroelectric \oI is most
stable, even more stable than baddeleyite. Furthermore, we find that under
tensile epitaxial strain $\eta$ the tetragonal phase will distort to one of the
two ferroelectric phases: for $\eta > 1.5$\%, the $\Gamma^{-}_{5}$ mode is
unstable and leads to \oII , and at $\eta > 3.75$\% coupling between this mode
and the zone boundary M1 mode leads to \oI.
