Intrinsic polar vortex crystals in A-site layer-ordered perovskites

a, Projected image obtained by summing up the slices in the ptychography reconstruction. The intensity profile on the right is extracted along the dashed arrow indicated in a, revealing a “dead” surface layer of 1 ~ 2 unit-cell thick. b, Structural model elucidating the lattice distortions in the [100] p projection, which include the antipolar A-site displacement (indicated by thick arrows) as well as the complex octahedral rotation (indicated by dashed ovals). The bottom image cropped from the region in a reveals the elongated B-O columns and dumbbell-like equatorial O columns corresponding to octahedral rotation. The a−b+c+ tilting boundary can be identified by the elongation direction of the B-O column shape, as also depicted in the top structural model. c, Polar displacement field mapped from a by calculating the relative displacement of Na/Nd against the four nearest B neighbors. d, Map of the out-of-phase octahedral rotation around [100] p axis derived from the dumbbell-like equatorial O columns in a. e, A-site displacement magnitude (left) and octahedral rotation angle (right) horizontally averaged from the maps in c and d. Only the A-site displacement neighboring the tilting boundaries (cyan lines) is selected for plotting. The sign of the displacement in the plot is to indicate their antiparallel nature. f, Significant polar displacement within the (001) p plane in c. g, Negligible out-of-plane component of the polar displacement in c. The polar displacement components in f and g are vertically averaged from c while the displacement directions are indicated in each panel by color arrows. The locations of the a−b+c+ tilting boundaries are depicted by the vertical dashed lines.