Fig. 2: Energy landscape connecting key HfO₂ polymorphs.

The black lines show the computed energy variation between the t and o-III phases (a), o-III and o-ref (b), and o-ref and m (c). The energies are computed for intermediate structures obtained by linear interpolation between the corresponding end-point polymorphs. The red line in (b) shows the energy variation of the o-ref state upon condensation of the \({\Gamma }_{2}^{-}\) distortions present in the o-III phase; the red line in (c) shows the analogous result when considering only the \({\Gamma }_{4}^{+}\) phonon distortions present in the m-phase. The blue line in (c) shows the result of condensing together the \({\Gamma }_{4}^{+}\) phonon and shear strain distortions present in the m-phase, while the green line shows the energy variation associated with the shear alone. In (b) and (c), the additional distortions leading to the black line are fully symmetric \({\Gamma }_{1}^{+}\) modes, including the normal cell strains.