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The structural, thermal, and dielectric properties of the ferroelectric phase of HfO, ZrO, and HfZrO (HZO) are investigated with carefully validated density functional computations. We find that the free bulk energy of the ferroelectric orthorhombic Pca2 phase is unfavorable compared to the monoclinic P2/c and the orthorhombic Pbca phase for all investigated stoichiometries in the HfZrO system. To explain the existence of the ferroelectric phase in nanoscale thin films, we explore the Gibbs/Helmholtz free energies as a function of stress and film strain and find them unlikely to become minimal in HZO films for technological relevant conditions. To assess the contribution of surface energy to the phase stability, we parameterize a model, interpolating between existing data, and find the Helmholtz free energy of ferroelectric grains minimal for a range of size and stoichiometry. From the model, we predict undoped HfO to be ferroelectric for a grain size of about 4 nm and epitaxial HZO below 5 nm. Furthermore, we calculate the strength of an applied electric field necessary to cause the antiferroelectric phase transformation in ZrO from the P4/nmc phase as 1 MV/cm in agreement with experimental data, explaining the mechanism of field induced phase transformation.


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