Finite-size effects on ferroelectricity in rhombohedral HfO2

In this work we analyze the finite-size effects on the structural properties and on the polarization of th e rhombohedral phase of ${\mathrm{HfO}}₂$ subjected to a biaxial compressive strain. We show how the presence of surface charges affects the polarization, leading to a strong reduction with respect to its bulk value. This reduction can be ascribed to two mechanisms: (i) the coupling between compressive strain and the phase-transition order parameter; (ii) the changes in the ferroelectric distortion. We give two alternative explanations of this phenomenon: From an atomistic point of view, analyzing the evolution of the bond lengths, and from a symmetryanalysis point of view, considering the changes in the amplitude of the symmetry-allowed distortions, when a slab configuration is considered. These results are in dependent of the slabthickness in the considered range, suggesting the absence of a critical thickness for ferroelectricity in ${\mathrm{HfO}}₂$, in agreement with the proposed improper nature of hafnia ferroelectricity.