Neural-network-backed effective harmonic potential study of the ambient pressure phases of hafnia

Machine learning methods, and in particular neural-network force fields (NNFF), are bridging the gap between macroscopically relevant parameters and quantum mechanical simulations. This work describes an NNFF training strategy and uses it as the back end for an effective harmonic potential study of phase transitions in hafnia. While good agreement with experiment is found regarding the monoclinic-tetragonal transition, the commonly assumed transition to a Fm$\overline{3}$m cubic phase is found to be unlikely for the stoichiometric material