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On the origin of the unusual strain morphologies and polar Moiré patterns in twisted ferroelectrics

Sergey Prosandeev, Charles Paillard, Laurent Bellaiche

Abstract

Density functional theory calculations are conducted to understand and reveal the origin of the complex shear strain morphology and of the polar Moiré 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 Moiré 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.

On the origin of the unusual strain morphologies and polar Moiré patterns in twisted ferroelectrics

Abstract

Density functional theory calculations are conducted to understand and reveal the origin of the complex shear strain morphology and of the polar Moiré topological pattern recently observed in twisted BaTiO 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 Moiré 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.

Paper Structure

This paper contains 13 equations, 2 figures.

Figures (2)

  • Figure 1: (Color online) Forces centered on the Ti ions in the bottom atomic layer of the BTO twisted bilayer in the 4$\times$4$\times$2 supercell, and related energies. (a) The "optical" forces and (b) The "acoustic" forces for the angle of twisting of 1$^\circ$. (c) The energy per 5-atom when varying the displacement of the optical mode. (d) The energy per 5-atom when varying the displacement of the acoustic mode. In panels (a) and (b), the locations of the two types of vortices are marked in red and blue while those of the antivortices are marked in green and brown. The amplitude of the acoustic force of Panel b is about 3.70 times larger than that of the optical one of Panel a.
  • Figure 2: (Color online) Map in a (x,y) plane of the shear strain morphology from Eq. (7) (panel a), of the polarization pattern from Eq. (\ref{['polarizationfinal']}) (panel b) and of the polarization self-organization from Eq. (\ref{['polarizationfinaloptical']}) (panel c) with selected values of $L$, $\mu_{xyxy}$, $k_{acoustic}$, $k_{optic}$, $Z^*$, $a_{lat}$ as well as for the $A$ and $\phi$ coefficients (see text). The locations of the two types of vortices are marked in red and blue while those of the antivortices are marked in green and brown in panels b and c.