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.

中文翻译：

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菱面体 HfO2 中铁电性的有限尺寸效应

在这项工作中，我们分析了有限尺寸对结构特性和菱面体相极化的影响${\mathrm{HfO}}_2$受到双轴压缩应变。我们展示了表面电荷的存在如何影响极化，从而导致其体积值的强烈降低。这种减少可归因于两种机制：（i）压缩应变与相变序参数之间的耦合；(ii) 铁电畸变的变化。我们对这种现象给出了两种不同的解释：从原子的角度分析键长的演变，从对称分析的角度考虑，当平板构型为经过考虑的。这些结果取决于所考虑范围内的板厚，表明铁电体中不存在临界厚度${\mathrm{HfO}}_2$，与所提议的氧化铪铁电的不当性质一致。