Phase transitions in typical Pca2₁ ferroelectrics

While ferroelectric hafnia (HfO₂) has become a technically important material for microelectronics, the physical origin of its ferroelectricity remains poorly understood. The tetragonal P4₂/nmc phase is commonly assigned as its paraelectric mother phase, but it has no soft mode at the Brillouin zone center. In this work, we propose that the paraelectric-ferroelectric transition in hafnia-like Pca2₁ ferroelectric family can be described by a $Pcca$-Pca2₁ transition, where the $Pcca$ mother phase will evolve into either the Pca2₁ ferroelectric phase or the centrosymmetric P2₁/c monoclinic phase, depending on the strain conditions. The $Pcca$ phase is directly linked to both phases in the context of continuous phase transition. Hafnia is regarded as a special case of this family, in that it has accidental atomic degeneracy because all anions are oxygen. The theory is also correlated to the seven-coordination theory that explains the ferroelectricity in hafnia from a chemical perspective. In addition, the strain conditions to promote the ferroelectric phase in hafnia is discussed.