Structure of $180^\circ$ ferroelectric domain walls in $\mathrm{HfO_2}$ and $\mathrm{ZrO_2}$

We report two series of structures representing two types of $180^\circ$ ferroelectric domain walls in $\mathrm{HfO_2}$ and $\mathrm{ZrO_2}$. We model the domain structures with different width by density functional theory calculations. The results reveal the dependency of domain wall energy and polarization on domain width. We show how the directions of polarization of ferroelectric domains can be determined from experimentally observed Hf atoms, despite the fact that the knowledge about the positions of O atoms is missing. The two series of structures thus provide more thorough interpretation and hence deeper understanding of experimental observations by transmission electron microscopy.