On the origin of the unusual strain morphologies and polar Moir\'e patterns in twisted ferroelectrics

Density functional theory calculations are conducted to understand and reveal the origin of the complex shear strain morphology and of the polar Moir\'e topological pattern recently observed in twisted BaTiO$_3$ bilayers. Our first-principles calculations, along with an original analysis of them allowing the decomposition of forces into the acoustic and optical contributions, point out to the occurrence of forces mostly acting on the {\it acoustic-related} motions to produce the standing waves of the shear strain. Such acoustic waves naturally generate a striking self-organization of the shear strains, and hence create a peculiar gradient of these shear strains. A Moir\'e dipole pattern, consisting of the interpenetrated arrays of vortices and antivortices made of the electric dipoles, then mostly arises due to the coupling of this gradient of the shear strain with the electric dipoles. Furthermore, other forces, namely acting on the motions associated with the {\it optical phonons}, could also play a role in the formation of these polar vortices and antivortices, but at a smaller extent.