Quasi-2D and 3D phase-field simulation studies of polar vortex domain structures for high-density domain engineering

Phase-field simulations were performed on the polar vortex array structure appearing in the SrTiO₃/PbTiO₃/SrTiO₃ heterostructure to investigate the Kittel law relationship (w ∝ d^1/2) between the vortex array period (w) and the PbTiO₃ ferroelectric layer thickness (d). Quasi-two-dimensional simulation results show that the equilibrium period can be determined from the relationship between the size of the simulation box and the total free energy density and that the obtained period obeys the Kittel law. However, three-dimensional simulation results show that the polar vortex arrays are not homogeneous in the direction perpendicular to the vortex plane. In the pure vortex state, no obvious correlation exists between the simulation size and energy because of the dislocations caused by the vortex period mismatch. However, in the mixed state with ferroelectric a₁/a₂, the vortex domains exhibit equilibrium periods that satisfy the Kittel law relationship because of the confinement effect caused by phase separation. Further study of the confined vortex domains in these mixed states and the vortex domains of finite size due to mismatched dislocations are expected to enable domain engineering of higher domain densities.