The nature of polar distortions in ferroelectrics

Abstract

Polar distortion, the collective off-center displacements of atoms, is a fingerprint of a ferroelectric that governs its properties and functionalities. Since the 1970s, the concepts of proper, improper and triggered ferroelectrics have been established to shed light on a diversity of polar distortion mechanisms. Such concepts assign a single nature to polar distortion and are helpful to interpret how polar distortions occur in conventional ferroelectrics such as barium titanate. However, applying these concepts to complex ferroelectrics (e.g., polar orthorhombic hafnia) is notoriously challenging and can yield highly controversial arguments. Here we resolve this issue by developing a tailor-made graph theory for clarifying the nature of polar distortions in complex ferroelectrics, which emphasizes that polar distortions in such ferroelectrics usually exhibit multiple natures among proper, improper and triggered characteristics. We demonstrate the robustness of our theory by working with perovsktie superlattices and polar orthorhombic hafnia (i.e., two representative cases). We successfully identify the mixed proper-improper nature in perovsktite superlattices and reconcile the controversy on polar orthorhombic hafnia by confirming its mixed trigger-improper nature. Our work will definitely lead to a revisitation of concepts in ferroelectric physics and provide opportunities for discovering novel ferroelectrics and related phenomena.

Figures (6)

• Figure 1: Sketches of nonpolar hierarchy graph (a) and polar hierarchy graph (b). The set associated with each vertex (being omitted) is indicated by arrows pointing to that vertex. For instance, the vertexes marked by $G_{13}$ and $G_0$ have sets of $\{ q_1, q_3 \}$ and $\emptyset$, respectively. In panels (a) and (b), the vertexes with yellow (gray) color share identical crystal structures as that for $G_0$ ($G_{123}$); the vertexes with white color have crystal structures that are not equivalent to each other. We further assume that the $\tilde{G}_1$ vertex has a polar space group, vertexes with gray color have polar space groups, and other vertexes have nonpolar space groups.
• Figure 2: Ferroelectricity in the $AB$O$_3/A^\prime B$O$_3$ perovskite superlattice. Panels (a) and (b) sketch the nonpolar $M^{2+}$ and $M^{5-}$ distortions, where gray arrows represent ionic motions. The $A$, $A^\prime$, $B$ and O ions are represented by purple, pink, cyan and orange spheres, respectively. In panel (b), the motions of $A$, $A^\prime$ and $B$ ions (being very tiny) are not shown. Panel (c) is the nonpolar hierarchy graph for LaGaO$_3$/YGaO$_3$ and SrTiO$_3$/CaTiO$_3$ superlattices. Panel (d) shows the strain dependent polarizations. The strain for LaGaO$_3$/YGaO$_3$ and SrTiO$_3$/CaTiO$_3$ is defined with respect to $a_0=5.44$ Å and $a_0=5.46$ Å, respectively.
• Figure 3: Ferroelectricity in orthorhombic HfO$_2$. Panels (a)---(d) sketch four nonpolar distortion modes in HfO$_2$, where gray arrows represent ionic motions. The Hf and O ions are represented by green and yellow spheres, respectively. Panel (e) is the phase transition graph for HfO$_2$. Note that the $4_2$ screw axis in $P4_2/nmc$ is along the $x$ direction.
• Figure 4: The ferroelectricity in orthorhombic HfO$_2$ driven by the $X^{5+}$ distortion. Panels (a), (b), and (c) sketch the $X^{5+}_x$, $X^{5+}_y$, and $X^{5+}_z$ modes, respectively. The $X^{5+}$ mode, the compass and the axis labels can be found in Fig. \ref{['fig:hafnia']}. Panel (d) shows the polarization of HfO$_2$ as a function of the $X^{5+}_y$ distortion (i.e., $\mathcal{Q}_{X^{5+}_y}$). See Methods for the computational details.
• Figure 5: The strain-dependent membership coefficients for LaGaO$_3$/YGaO$_3$ and SrTiO$_3$/CaTiO$_3$ superlattices.