Moiré polar vortex, flat bands, and Lieb lattice in twisted bilayer BaTiO₃

Through first-principles calculations based on density functional theory, we investigate the crystal and electronic structures of twisted bilayer BaTiO₃. Our findings reveal that large stacking fault energy leads to a chiral in-plane vortex pattern that was recently observed in experiments. We also found nonzero out-of-plane local dipole moments, indicating that the strong interlayer interaction might offer a promising strategy to stabilize ferroelectric order in the two-dimensional limit. The vortex pattern in the twisted BaTiO₃ bilayer supports localized electronic states with quasi-flat bands, associated with the interlayer hybridization of oxygen p orbitals. We found that the associated bandwidth reaches a minimum at ∼19^∘ twisting, configuring the largest magic angle in moiré systems reported so far. Further, the moiré vortex pattern bears a notable resemblance to two interpenetrating Lieb lattices and the corresponding tight-binding model provides a comprehensive description of the evolution the moiré bands with twist angle and reveals the topological nature.