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 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. Furthermore, under compressive epitaxial strain $eta < 0.55$% the ferroelectric oI is most stable, even more stable than baddeleyite.
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