Fig. 3: Electric-field-induced motion of polar bubbles.

a Simulation of short-range displacement of the bubble induced by the PFM tip located 2 nm away from the bubble center (position of the tip indicated with a blue cross). The driving field profile with a maximum local value of 21 × 10⁷ V/m is assumed. As it can be seen, the bubble displaces from its initial position (left-most image) towards the point right beneath the PFM tip (right-most image) through a stretch-like motion (middle image). b Time evolution of the structure when PFM tip (blue cross) is applied 5.5 nm away from the bubble center. Here, the local maximum of the field profile is of 110 × 10⁷ V/m. In this case, the local electric field creates a bubble domain beneath the tip while simultaneously deleting the bubble domain at its initial position. In panels, a and b, the left to right sub-panels show the distribution of the z component of polarization at times t = 0, 0.6 ps, 0.75 ps, 0.8 ps, and 1 ps, respectively. Blue to red colors correspond to negative to positive polarization values. c, d Evolution of the Skyrmion number within the z = 2 and z = 4 planes and the normalized out-of-plane polarization. Panel (c) and (d) correspond to the continuous motion and teleportation, respectively. All presented simulation results were performed at 10 K and a background bias field E_z of 61 × 10⁷ V/m.