Fig. 1: Dipolar structure of electric bubbles.

a Schematic structure of an electric bubble. The arrows represent electric dipoles colored according to their out-of-plane Cartesian component. The downwards-pointing dipoles (red arrows) correspond to the core of the bubble. The light-gray surfaces correspond to the inner toroidal vortex (light gray torus) surrounding the core and the outer boundary of the bubble (light gray sphere). The electric bubble is confined between the top and bottom interfaces of the PZT film (not shown) so that the core is polarized along the out-of-plane z-axis. Electric bubbles most often form partially ordered arrays or hexagonal lattices. b, c The simulated planar cross-sections of a hexagonal bubble lattice one u.c. above (b) and below (c) the middle plane of the film. Within such cross-sections, the bubble gives rise to a Néel skyrmion pattern (red arrows). The skyrmions above and below the equatorial plane feature the same polarity and a mutually opposite sense of Néel rotations. In panels b and c, local dipoles are represented by arrows colored according to the dipole’s out-of-plane component.