Fig. 3: Phonon bands of the o-ref phase.
From: Theoretical approach to ferroelectricity in hafnia and related materials
a shows the computed bands, presenting imaginary frequencies as negative values. The most important unstable modes are marked in (a). We also show the corresponding eigenmodes and the polymorphs they lead to: the \({\Gamma }_{2}^{-}\) soft mode (b) and the corresponding o-III phase (c); the \({X}_{2}^{-}\) soft mode (d) and the corresponding o-I phase (e); the \({\Gamma }_{4}^{+}\) soft mode (f) and the associated m-phase (g); and the \({X}_{4}^{+}\) soft mode (h) and the corresponding o-I* phase (i). We mark in red the active oxygens whose displacements characterize these phonons. For the polymorphs, we indicate the energy with respect to o-ref.
