Figure 1

Schematic sketch of the $\Sigma$5(201) grain boundaries with different polarization arrangements in adjacent grains (domains) for supercells with 20 unit cells. (a) 127${}^{°}$ DW; (b) 53${}^{°}$ DW; (c) 180${}^{°}$ DW; (d) 0${}^{°}$ polarization change (without a DW). Microscopic translations of grains with respect to each other are disregarded here.Reuse & Permissions

Figure 2

Supercell model of the $\Sigma$5(201) GB with a 53${}^{°}$ DW, corresponding to the spontaneous-polarization arrangement of Fig. 1b. The displayed supercell has 16 unit cells (two supercells are plotted in the $z$ direction). The position of the interface is marked by the thick vertical dashed line. Thin solid lines show the shapes of the PTO unit cells. Thin dashed lines indicate the TiO${}_6$ octahedra. The directions of spontaneous polarization are indicated by black arrows. The grains are shifted with respect to each other in this minimum-energy configuration.Reuse & Permissions

Figure 3

$\gamma$ surfaces as obtained from the atomistic SMP simulations. For $\Sigma$5 GB the (201)${}^{{53}^{°}}$ and (301)${}^{{37}^{°}}$ variants are displayed. Lighter gray indicates a higher energy, and darker gray indicates a lower energy (the grayscale is the same for all four plots). Nonequivalent local minima in the energy are referred to in the text by the displayed numbers. The broken lines in $\Sigma$3(111) indicate the $y$-$z$ cross-section area of the used supercell.Reuse & Permissions

Figure 4

Possible pinning mechanism of a 180${}^{°}$ DW at a GB formed by a combination of $\Sigma$5 facets known from STO. The GB forms a DW at the same time along its whole length. An applied external electric field prefers the shaded ferroelectric domains, which grow at the expense of the oppositely oriented domain. Thick solid lines represent the GB, thin dashed lines the original position of the 180${}^{°}$ DW, and thin solid lines the shifted DW under influence of the external electric field. Arrows indicate the direction of motion of the DW. If the DWs moved at their crossing point with the GB, a segment of GB without a DW would be created, which is energetically unfavorable. Therefore, the edges of this 180${}^{°}$ DW are likely well pinned.Reuse & Permissions

Figure 5

Polarization profiles across (a) the $\Sigma$3(111) and b) $\Sigma$(112) (variant 2) GBs. Filled squares, circles, and triangles represent the $P_{\mathrm{x}}$, $P_{\mathrm{y}}$, and $P_{\mathrm{z}}$ components of spontaneous ferroelectric polarization, respectively. Open circles stand for the magnitude of the spontaneous polarization $P_{\mathrm{s}}$. The irreducible part of the supercell with the interface in the center ($x=0$) is marked by shading. Bulk values of $P_{\mathrm{x}}$, $P_{\mathrm{y}}$, $P_{\mathrm{z}}$, and $P_{\mathrm{s}}$ are indicated by thin dotted lines.Reuse & Permissions

Figure 6

Polarization profiles across the four variants of a $\Sigma$5(201) GB in the supercell with 20 unit cells. The symbols used are the same as in Fig. 5.Reuse & Permissions

Figure 7

Polarization profiles across variant 2 of the $\Sigma$5(301) GB. The symbols used are the same as in Fig. 5.Reuse & Permissions

Figure 8

Comparison of the polarization profiles evaluated using DFT in the vicinity of the $\Sigma$5(201) GB for supercells with 16 formula units (dots, red) and 20 formula units (triangles, green), and using SMP with 20 formula units (squares, black). Results obtained with both SMP and DFT show good agreement.Reuse & Permissions