Mesoscopic Helices of Polar Domains in a Quadruple Perovskite

Abstract

A significant effort in condensed matter physics is dedicated to the search for exotic arrangements of electric dipoles in crystals. Non-collinear dipolar arrangements mimicking magnetic spin textures, such as polar vortices and skyrmions, have been realized, but others, like helices of dipoles, have remained elusive. While earlier work claimed the presence of a helical dipole modulation in ferroelectric BiCu$_{x}$Mn$_{7-x}$O$_{12}$, our results rule out such a structure, motivating a renewed search for helical textures. Using atomic-resolution imaging, we report a novel form of helical order in which polar domains self-organize into a mesoscopic helical pattern: from domain to domain, polarization rotation follows a consistent handedness. Both right- and left-handed chirality emerging from this helical ordering are observed. This discovery establishes mesoscopic ordering as novel mechanism for inducing helical textures, and hence chirality, in ferroelectric crystals, paralleling the efforts in liquid crystals and supramolecular assemblies.

Figures (3)

• Figure 1: Emergence of structural domains with copper doping. (A) Atomic structure of BiMn$_{7}$O$_{12}$ in the $Im\bar{3}$ (left panel) and $Cm$ (right panel) phase. The gray arrow in $Cm$ phase shows the polar displacement of Bi along $[1\bar{1}0]$$_{pc}$. All crystallography indexing are based on the pseudocubic lattice. (B) Previously proposed incommensurate dipole helical wave in BiCu$_{0.1}$Mn$_{6.9}$O$_{12}$. Left and right-handed chiralities are shown. (C) Simulated ADF-STEM image of structures with (left panel) and without (right panel) dipole helical wave. The projection is along [$11\bar{2}$]$_{pc}$ (D) Experimental ADF-STEM of BiCu$_{0.1}$Mn$_{6.9}$O$_{12}$ along [$11\bar{2}$]$_{pc}$ shows no helical Bi modulation. (E, F) $250 \times 250~\mathrm{nm}^2$ field-of-view LAADF-STEM image of (E) BiMn$_{7}$O$_{12}$ and (F) BiCu$_{0.1}$Mn$_{6.9}$O$_{12}$, revealing the presence of high density of domain boundaries in the latter. Inserts are the (114)$_{pc}$ Bragg peak. An incommensurate peak along the $[\bar{1}\bar{1}1]$$_{pc}$ direction is evident only in the doped system. The right panel of (F) is the Fourier transform from regions highlighted by yellow and blue rectangles. The incommensurate peak is absent without a domain boundary (lower panel). (G) Lattice spacing along [110]$_{pc}$ in BiCu$_{0.1}$Mn$_{6.9}$O$_{12}$ measured by 4D-STEM (see SI for more details). The field-of view is the same as in (F).
• Figure 2: A mesoscopic helix of polar domains. (A) Schematic graph of three distinct $<110>_{\text{pc}}$ polar displacements lying on the ($\bar{1}\bar{1}1$)$_{pc}$ plane, corresponding to the three ferroelastic domains (I, II and III). (B) HAADF-STEM image of BiCu$_{0.1}$Mn$_{6.9}$O$_{12}$ viewed along [$1\bar{1}0$]$_{pc}$. Right panel is the definition of projected lattice spacing along [110] (d$_{[110]pc}$) and Bi polar displacement ($\mathbf{\Delta}_{\mathrm{Bi}}$) measured in experiment. (C) Atomic structures viewed along [$1\bar{1}0$]$_{pc}$ directions in three ferroelastic domains (I, II and III). The gray arrow shows $\mathbf{\Delta}_{\mathrm{Bi}}$. The measured d$_{[110]pc}$ is at the bottom. (D) HAADF-STEM image overlaid with measured $\mathbf{\Delta}_{\mathrm{Bi}}$ in three ferroelastic domains. (E) Polar histogram of $\mathbf{\Delta}_{\mathrm{Bi}}$ in three ferroelastic domains (I, II and III). The radius of the histogram 40 pm. Each polar displacement vector has unique planar projection. (F) d$_{[110]pc}$ from $50 \times 35~\mathrm{nm}^2$ field-of view HAADF-STEM image. (G) Projected $\mathbf{\Delta}_{\mathrm{Bi}}$ measured from same field-of view HAADF-STEM image as (F). The color and transparency represent the polar direction and amplitude, respectively. (H) Schematic representation of ordering of polar domains.
• Figure 3: Left- and right-handed chirality of mesoscale polar domains. (A) Schematic graph shows left-handed and right-handed chirality, coupled to helical order of polar domains. (B) Difference of Gaussian applied to large scale 4D-STEM map of polarization reveals long-range repeat of right- (left panel) and left-handed (right panel) helical ordering of polar domains. Domains I (green), II (yellow) and III (red) are labeled. (C) Atomic-scale mapping of polar displacement of sub-regions regions highlighted by white rectangles in (A). (D) Polar histograms of displacements collected from each domain. The radius of the histogram is 40 pm.