Symmetry of ferroelectric switching and domain walls in hafnium dioxide

Hafnium dioxide $(\mathrm{Hf}{\mathrm{O}}₂)$ is a promising ferroelectric (FE) material for achieving high-density nonvolatile memory and neuromorphic computing due to its compatibility with the mainstream integrated circuit technology and the surprisingly enhanced ferroelectricity by reduced thickness. The FE switching dynamics is essential to the device performance, but the complexity of $\mathrm{Hf}{\mathrm{O}}₂$ atomic structure causes unknown of various FE switching paths and domain-wall configurations. Here, we demonstrate that its low-barrier paths and domain walls can be comprehensively found and understood from a perspective of topological symmetry. By discussing pseudochirality and equivalent transformation relations in a crystal with first principles and lattice modes, we classify and analyze 4 low-barrier FE switching paths and 93 irreducible topology domain-wall configurations in $\mathrm{Hf}{\mathrm{O}}₂$. The anisotropic switching mechanism is inferred based on the investigation for 12 types of ${180}^{\ensuremath{\circ}}\text{-side}$ domain walls. This methodology is expected to be generally applicable to displacive ferroelectrics with low unit-cell point-group symmetries and lay a foundation for mechanism study of the switching dynamics.