Polarity of domain boundaries in nonpolar materials derived from order parameter and layer group symmetry

Domain boundaries and other twin boundaries in crystalline materials are receiving increasing interest. They can carry unique functional properties, which in many cases are absent in the surrounding bulk material. One such property of domain boundaries can be their electric polarity. Phenomenological insight in the polarity of domain boundaries was so far based either on the knowledge of the order parameter and the form of Landau-Ginzburg free energy functional, or on the knowledge of the symmetry of the domain boundaries. In the present work we show on the concrete examples of potassium thiocyanate (KSCN) and lacunar spinel crystals that the concept of the primary order parameter can help to find the layer group describing the maximal possible symmetry of a given domain boundary. A combination of layer group and order parameter symmetries is then employed to clarify the nature of the polarity of domain boundaries.

Figure 1

(Left) Three-dimensional arrangement of atoms in the orthorhombic $Pbcm$ structure of KSCN. The vectors $\mathbf{a},\mathbf{b},\mathbf{c}$ of the primitive unit cell are marked by red dashed arrows. Purple = K atoms, yellow = S, gray = N. C atoms are omitted for clarity in the figures. Graphics made with vesta. (Right) $c$ projection of the structure including symmetry elements. K atoms at $z=\frac{1}{4}c$ levels and S, N atoms at $z=0,\frac{1}{2}c$ levels, respectively.

Figure 2

Four domain states of the $Pbcm$ structure of KSCN. The conventional unit cell with vectors ${\mathbf{a}}_t=(a_t,0,0)$, ${\mathbf{b}}_t=(0,a_t,0)$, and ${\mathbf{c}}_t=(0,0,c_t)$ of the high temperature tetragonal structure $I4/mcm$ is depicted in the center. The centering translation $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ is indicated by a dashed yellow arrow. It is lost at the phase transition to the $Pbcm$ phase $\left(1_1\right)$, leading to a doubling of the unit cell below $T_c$.

Figure 3

Sketch of the Landau free energy landscape in the OP space $({\eta }_1,{\eta }_2)$ of KSCN, representation of homogeneous domain states (points) and transition pathways (dotted lines) between them. 1 = straight path, also called linear antiphase boundary (LAPB), 2 and $2^{\prime }$ are side paths, also called rotational antiphase boundaries (RAPBs), 3 = ferroelastic domain boundary. Path 4 describes another ferroelastic boundary. The epikernel symmetries at the points or segments of the transition pathways [51] are also depicted. Generally they do not correspond to the correct local structures, since the rate of structural changes is usually not small, especially close and at the domain wall centers.

Figure 4

Top: Orientational DP $\{1_1,2_1\}=\{S_1,S_3\}$. The ferroelastic domain walls at positions $\mathbf{p}=p\mathbf{a}$ for $p=0,\frac{1}{2}$, and $-\frac{1}{2}$ are indicated by yellow and orange lines. Bottom: Symmetry elements of $J_{13}^{\prime }$ [that is $(1/000)$, $(2_z/\frac{1}{2}\frac{1}{2}\frac{1}{2})$, $(m_z/000)$, and $(\overline{1}/\frac{1}{2}\frac{1}{2}\frac{1}{2})$, drawn in black] and of $J_{13}^{\prime \prime }$ [that is $(2_{xy}/00\frac{1}{2})$, $(2_{\overline{x}y}/\frac{1}{2}\frac{1}{2}0)$, $(m_{\overline{x}y}/00\frac{1}{2})$, $(m_{\overline{x}y}/00\frac{1}{2})$ and $(m_{xy}/\frac{1}{2}\frac{1}{2}0)$, drawn in green] forming altogether $J_{13}^{\prime }\cup J_{13}^{\prime \prime }=C{\widehat{m}}_{xy}{\widehat{c}}_{\overline{x}y}{m}_z=D_{2h}^{17}$ attached to the structure of the DP. The unit cell (blue lines) of $Cmcm$ which is related to $Pbcm$ (deep red) according to ${\mathbf{a}}_o=\mathbf{a}-\mathbf{b}$, ${\mathbf{b}}_o=\mathbf{a}+\mathbf{b}$, and ${\mathbf{c}}_o=\mathbf{c}$ is also shown. To keep consistency with earlier work, the $y$ axis is drawn as horizontal. Graphics made with vesta.

Figure 5

Translational DP $\{1_1,1_2\}=\{S_1,S_2\}$ of the KSCN structure. The unit cell of the group $J_{12}=Ibam=D_{2h}^{26}$ is shown in magenta. Note, that the centring translation (dotted magenta vector) is conserved only if we omit the colors, i.e., if we treat it as an unordered DP. Graphics made with vesta.

Figure 6

Translational antiphase boundaries with $\mathbf{n}=(0,1,0)$ orientation at positions $p=0$ and $p=\frac{1}{4}$ together with corresponding layer groups $T_{12}$. Graphics made with vesta.

Figure 7

Top: OP profile of a ferroelastic domain boundary with wall with $n=(1,1,0)$ orientation. Bottom: Polarization profile $\mathbf{P}\left(\xi \right)=(P_{\left[1\overline{1}0\right]}\left(\xi \right),P_{\left[110\right]}\left(\xi \right))$, which is compatible with the symmetry $T_{13}=p{\underset{¯}{\overset{̂}{\mathrm{m}}}}_{xy}{\underset{¯}{\overset{̂}{2}}}_{x\overline{y}}{m}_z$ of the domain boundary. Note that $P_{\left[1\overline{1}0\right]}$ is symmetric with respect to $\xi$, whereas $P_{\left[110\right]}$ is antisymmetric to fulfill the symmetry requirements of $T_{13}$.

Figure 8

Mid panel: Variation of the order parameter ${\eta }_1$ in a linear antiphase boundary corresponding to path 1 (${\eta }_2=0$, LAPB). Upper and lower panels show RAPB corresponding to path 2 (${\eta }_2>0$) and path $2^{\prime }$ (${\eta }_2<0$). Blue lines show the corresponding antiphase polarizations as calculated from Eqs. (28), (29), (30), and (31).