Origin of Reverse Size Effect in Ferroelectric Hafnia Thin Films

The persistence of ferroelectricity in ultrathin HfO$_2$ films challenges conventional theories, particularly given the paradoxical observation that the out-of-plane lattice spacing increases as the film thickness decreases, a reverse size effect absent in perovskite ferroelectrics. Here, we resolve this puzzle by revealing that this anomalous lattice expansion is counterintuitively coupled to a suppressed out-of-plane polarization. First-principles calculations combined with analytical modeling identify two mechanisms behind this expansion: a negative longitudinal piezoelectric response to the residual depolarization field and a positive surface stress that becomes significant at reduced thickness. Their interplay quantitatively reproduces the experimentally observed lattice expansion. Furthermore, (111)-oriented HfO$_2$ films can support out-of-plane polarization even under open-circuit conditions, in contrast to (001) films that stabilize a nonpolar ground state. This behavior points to the emergence of orientation-induced hyperferroelectricity, an unrecognized mechanism that enables polarization persistence through orientation engineering without electrode screening. We further demonstrate that this principle generalizes to conventional perovskites such as PbTiO$_3$, offering a strategy to eliminate the critical thickness limit by choosing the appropriate film orientation. As a practical pathway to device integration, we also identify the two-dimensional electride Ca$_2$N as a near-ideal electrode that fully restores the ferroelectric properties of HfO$_2$ in ultrathin capacitors.