Strain-induced moiré polar vortex in twisted paraelectric freestanding bilayers

The recent discovery of two-dimensional (2D) moiré vortex patterns reveals new possibilities for nanoscale polar topology engineering and unexplored physical phenomena. However, the physical origin and detailed topological characteristics of these moiré vortex patterns have still not been understood. In this study, based on the lattice polarization coupling of ferroelectrics, we analytically determined the discovered strain state in twisted bilayer systems by elastic theory. Furthermore, the resulting moiré vortex patterns are investigated via phase-field simulations. Our findings demonstrate that the in-plane moiré vortex patterns arise from periodic displacement vorticity induced by moiré stacking. The complex interplay among elastic, flexoelectric, and gradient energy is identified as the energetic driving force behind the formation of these vortex patterns. Through three-dimensional simulation, we reveal that each polar vortex exhibits significant in-plane divergence and out-of-plane chirality, with the latter being tunable via external electric fields. These findings offer new avenues for manipulating nanoscale ferroelectric topologies.