Topological structures host rich dynamics which can be described by their equations of motion associated with their effective mass and spring constant. We work in collaboration with Argonne National Lab, Berkeley, SLAC, etc., to perform a combined experimental and theoretical study of the collective dynamics of polar vortices in PbTiO₃/SrTiO₃ superlattices. We predict the sub-THz collective phonon modes using dynamical phase-field simulations and show a condensation of a collective mode upon a second-order structural transition at a critical strain, analogous to the ferroelectric soft phonon mode at a ferroelectric transition. We also analytically establish the energetics and the equations of motion for the ferroelectric polar vortices and extract their effective mass, spring constant, and mode frequencies.

Figure. (Left) Polar vortex arrays in PbTiO₃/SrTiO₃ superlattices subjected to an applied electric-field pulse and (right) a pair of vortices undergoing core motions of a soft phonon mode with color arrows indicating the polarization field.