Observations of vortex–antivortex structures. Panel (a) displays the cross-sectional HR-STEM image with an overlay of the polar displacement vectors (PPDs, indicated by yellow arrows) for a (SrTiO₃)₁₀/(PbTiO₃)₁₀ superlattice, showing that an array of vortex–antivortex pairs is present in each PbTiO₃ layer. Panel (b) shows a magnified image of a single vortex–antivortex pair, indicating the full density of data points (one for each atom) and the continuous rotation of the polarization state within such vortex–antivortex pairs. Panel (c) exhibits the curl of the polar displacement (∇ × PPD)[010] for the same vortex–antivortex pair, revealing the alternating rotation directions of the structures. The (∇ × PPD) [010] curl value is plotted with a red/blue colour scale where no-vorticity (curl = 0) is in white, clockwise (negative) is in blue and anticlockwise (positive) is in red. Panel (d) indicates polarization vectors from a phase-field simulation of the same (SrTiO₃)₁₀/(PbTiO₃)₁₀ superlattice, which predicts vortex–antivortex pairs that closely match the experimental observations. Reproduced from Ref. [22].