Hidden Antipolar Order Parameter and Entangled Néel-Type Charged Domain Walls in Hybrid Improper Ferroelectrics

Hybrid improper ferroelectricity (HIF) denotes a new class of polar instability by the mixture of two octahedral-distortion modes and can feature the coexistence of abundant head-to-head and tail-to-tail polar domains, of which the domain walls tend to be charged due to the respective screening charges with an opposite sign. However, no such coexisting carriers are available in the materials. Using group-theoretical, microscopic, and spectroscopic analyses, we establish the existence of a hidden antipolar order parameter in model HIF ${(\mathrm{Ca},\mathrm{Sr})}_3{\mathrm{Ti}}_2{\mathrm{O}}_7$ by the condensation of a weak, previously unnoticed antipolar lattice instability, turning the order-parameter spaces to be multicomponent with the distinct polar-antipolar intertwining and accompanied formation of Néel-type twinlike antipolar domain walls (few nanometers) between the head-to-head and tail-to-tail domains. The finite-width Néel walls and correlated domain topology inherently lift the polar divergences between the domains, casting an emergent exemplification of charged domain-wall screening by an antipolar ingredient.

Figure 1

(a),(b) The $\mathrm{FE}\text{−}A{2}_{1}am$ crystal structures along $b^{’}$ and $a^{\prime }$ projections, respectively. The crystallographic sites are indicated (gray, $\mathrm{Ca}/\mathrm{Sr}$; cyan, Ti; red, oxygen). White arrows in (a), the antiparallel $\mathrm{Ca}1/\mathrm{Sr}1$ and $\mathrm{Ca}2/\mathrm{Sr}2$ displacements (black arrow, $P$). Gray arrows in (b), the antipolar $\mathrm{Ca}2/\mathrm{Sr}2$ displacements. Dashed gray lines in (a) and (b), guides for the off-center $\mathrm{Ca}/\mathrm{Sr}$ displacements. (c) The $a^{\prime }$-oriented polarization (black) of an individual perovskite slab in (a) and the atomistic contribution of each crystallographic site. (d) Group-theoretical analysis of the symmetry tree, with the black label indicating the space group. The symbols in parentheses depict the primary order-parameter directions. Solid (dashed) lines, reported (otherwise) transition pathways.

Figure 2

(a) The HAADF imaging of HH domains revealing a different feature in the DW (yellow). $\mathrm{Ca}1/\mathrm{Sr}1$, red rectangle. (b) The STEM-EELS chemical mapping. Gray (white) circles, $\mathrm{Ca}/\mathrm{Sr}$ (Ti) omitting the off-center distortions. (c) The $a^{\prime }$-projected HAADF image. Lower-bottom inset, an uc enlargement showing the accentuated antipolar $\mathrm{Ca}1/\mathrm{Sr}1$ (white-margined) and $\mathrm{Ca}2/\mathrm{Sr}2$ (red-margined arrows) displacements. Dashed white lines, guides for the eyes. (d) The various DF images (red, blue, and green) with the contrast reversal in each set unveiling the domain polarity. (e) The HAADF imaging of TT domains, with the DW structure (yellow) mimicking (a) and (c). $P$ in the DWs and (c), pointing in or out. White rectangles, projected uc.

Figure 3

(a) The calculated phonon dispersion of CTO with identical antipolar distortion to the CSTO (inset, $a^{\prime }$-projected HAADF of CTO). (b)–(d) The CBED patterns along respective $b^{’}$, $a^{\prime }$, and $c$ projections with the BF (gray margined) embedded in the center of the WP. $m$, mirror. The symmetry characteristics in (b) and (c) refer to the point group $2mm$ of $A{2}_{1}am$. In (d), the $2mm$ BF and twofold WP (guiding white, red arrows) symmetries suggest point-group 222-type distortion at the $P_5$ irrep.

Figure 4

(a)–(c) ${\mathrm{\Gamma }}_5^{-}$, $X_2^{+}$, and $X_3^{-}$ irreps with respective primary order parameters of $P$ (black arrows), $R$ (red), and $T$ (blue) along ($a,a$), (0, $a$), and (0, $a$), forming four-domain topologies by the degeneracy labeled on edges. $u$ (dashed lines), secondary order parameter of ferroelastic strain at the DWs. Gray (black) uc, $c$-projected parent (FE) lattice. (d) $P_5$ with effective ($0,0,b,b$). Green arrow, primary order parameter of antipolar $\mathrm{Ca}/\mathrm{Sr}$ displacements ($A$, double-headed). The eight domains (labels on edges) form a four-domain topology upon $ab$ degeneracy. (e) Domain topology (upper panel) upon ${\mathrm{\Gamma }}_5^{-}$-, $X_2^{+}$-, $X_3^{-}$-, and $P_5$-irrep condensations. $P$ and $A$, intertwined macroscopic order parameters in the four domains (1–4; eight domains upon $P$ reversals). (f) The HH and TT domains with a generically sandwiched antipolar-$A$, Néel-type DW by the inherent domain topology such as 1-2-3 or 1-4-3 in (e). (g) A cross-sectional view of the coexisting HH and TT domains in (f). (h) The experimental HAADF evidence for (g), with the $n=3$ defect also showing $P_5$- type antipolar distortion within the DW (yellow).