Ordering of Oxygen Vacancies and Related Ferroelectric Properties in HfO_{2-δ}

Using density functional theory combined with an evolutionary algorithm, we investigate ferroelectricity in substoichiometric HfO_{2-δ} with fixed composition δ=0.25. We find that oxygen vacancies tend to cluster in the form of two-dimensional extended defects, revealing several patterns of local relative arrangements within an energy range of 100 meV per Hf atom. Two lowest-energy patterns result in polar monoclinic structures with different transformation properties. The lowest one elastically transforms to the ferroelectric orthorhombic structure via a shear deformation, overcoming an energy barrier, which is more than twice lower than in the stoichiometric hafnia. The second-lowest structure transforms at smaller volumes to a nonpolar tetragonal one. We discuss the experimentally observed wake-up effect, fatigue, and imprint in HfO_{2}-based ferroelectrics in terms of different local ordering of oxygen-vacancy extended defects, which favor specific crystallographic phases.