Unexpected Density Functional Dependence of the Antipolar $Pbcn$ Phase in HfO$_2$

The antipolar $Pbcn$ phase of HfO$_2$ has been suggested to play an important role in the phase transition and polarization switching mechanisms in ferroelectric hafnia. In this study, we perform a comprehensive benchmark of density functional theory (DFT) calculations and deep potential molecular dynamics (DPMD) simulations to investigate the thermodynamic stability and phase transition behavior of hafnia, with a particular focus on the relationship between the $Pbcn$ and ferroelectric $Pca2_1$ phases. Our results reveal significant discrepancies in the predicted stability of the $Pbcn$ phase relative to the $Pca2_1$ phase across different exchange-correlation functionals. Notably, the PBE and hybrid HSE06 functionals exhibit consistent trends, which diverge from the predictions of the PBEsol and SCAN functionals. For a given density functional, temperature-driven phase transitions predicted by DFT-based quasi-harmonic free energy calculations aligns with finite-temperature MD simulations using a deep potential trained on the same density functional. Specifically, the PBE functional predicts a transition from $Pca2_1$ to $Pbcn$ with increasing temperature, while PBEsol predicts a transition from $Pca2_1$ to $P4_2/nmc$. A particularly striking and reassuring finding is that under fixed mechanical boundary conditions defined by the ground-state structure of $Pca2_1$, all functionals predict consistent relative phase stabilities and comparable switching barriers as well as domain wall energies. These findings underscore the unique characteristics of the $Pbcn$ phase in influencing phase transitions and switching mechanisms in ferroelectric hafnia.