Competing Phases of HfO_{2} from Unstable Flat Phonon Bands of an Unconventional High-Symmetry Structure

We carry out first-principles calculations to demonstrate that the complex energy landscape and competing phases of HfO_{2} can be understood from the four unstable flat phonon bands of an unconventional high-symmetry structure of HfO_{2} with the space group Cmma. We consider structures generated from the Cmma reference structure by all possible combinations of the zone center and zone boundary modes belonging to the unstable flat phonon branches. We find 12 distinct locally stable structures, of which 5 correspond to well-known phases. We also show that 8 of these 12 structures can be described as period-2 superlattices of the ferroelectric Pca2_{1}(oIII), ferroelectric Pmn2_{1}(oIV), monoclinic P2_{1}/c(m), and distorted monoclinic P2_{1}/c(dm) structures. We demonstrate how the unstable flat phonon bands can explain the atomically thin grain boundaries in the various types of superlattices. Finally, we point out that arbitrary-period HfO_{2} superlattices derived from the 6 different types of period-2 superlattices are expected to form based on the flatness of the unstable phonon branches. The organizing principle provided by this work deepens our understanding of the underlying physics in the phase stability of HfO_{2} and provides guidance for functional phase stabilization.