Ferroelectric memories experienced a revival in the last decade due to the discovery of ferroelectricity in ${\mathrm{HfO}}_2$-based nanometer-thick thin films. These films exhibit exceptional silicon compatibility, overcoming the scaling and integration obstacles that impeded perovskite ferroelectrics' use in high-density integrated circuits. The exact phase responsible for ferroelectricity in hafnia films remains debated with no single factor identified that could stabilize the ferroelectric phase thermodynamically. Here, supported by density functional theory (DFT) high-throughput (HT) calculations that screen a broad range of epitaxial conditions, we demonstrate conclusively that specific epitaxial conditions achievable with common substrates such as yttria-stabilized zirconia (YSZ) and ${\mathrm{SrTiO}}_3$ can favor the polar $Pca{2}_1$ phase thermodynamically over other polar phases such as $R3m$ and $Pmn{2}_1$ and nonpolar $P{2}_1/c$ phase. The substrate's symmetry constraint-induced shear strain is crucial for the preference of $Pca{2}_1$. The strain-stability phase diagrams resolve experiment-theory discrepancies and can guide the improvement of ferroelectric properties of epitaxial hafnia thin films.

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• Epitaxial strain
• Ferroelectrics
• Density functional theory