Domain walls of ferroelectric ${\text{BaTiO}}_3$ within the Ginzburg-Landau-Devonshire phenomenological model

Mechanically compatible and electrically neutral domain walls in tetragonal, orthorhombic, and rhombohedral ferroelectric phases of ${\text{BaTiO}}_3$ are systematically investigated in the framework of the phenomenological Ginzburg-Landau-Devonshire model with parameters of J. Hlinka and P. Marton, Phys. Rev. B 74, 104104 (2006). Polarization and strain profiles within domain walls are calculated numerically and within an approximation leading to the quasi-one-dimensional analytic solutions applied previously to the ferroelectric walls of the tetragonal phase [W. Cao and L. E. Cross, Phys. Rev. B 44, 5 (1991)]. Domain-wall thicknesses and energy densities are estimated for all mechanically compatible and electrically neutral domain-wall species in the entire temperature range of ferroelectric phases. The model suggests that the lowest-energy walls in the orthorhombic phase of ${\text{BaTiO}}_3$ are the $90°$ and $60°$ walls. In the rhombohedral phase, the lowest-energy walls are the $71°$ and $109°$ walls. All these ferroelastic walls have thickness below 1 nm except for the $90°$ wall in the tetragonal phase and the $60°$ $S$ wall in the orthorhombic phase, for which the larger thickness on the order of 5 nm was found. The antiparallel walls of the rhombohedral phase have largest energy and thus they are unlikely to occur. The calculation indicates that the lowest-energy structure of the $109°$ wall and few other domain walls in the orthorhombic and rhombohedral phases resemble Bloch walls known from magnetism.

Figure 1

Directions of spontaneous polarization in (a) the tetragonal, (b) orthorhombic, and (c) rhombohedral phase. Angles between polarization direction “0” and its symmetry equivalent ones are indicated.

Figure 2

Set of mechanically compatible and electrically neutral domain walls in the three ferroelectric phases of ${\text{BaTiO}}_3$. In the case of $180°$ domain walls, where the orientation is not determined by symmetry, walls with the most important crystallographic orientations are displayed.

Figure 3

Profile of the “reversed” polarization component $P_{\text{r}}$ [Eq. (20)] for a domain wall with the shape factor $A=1$ (solid) and $A=3$ (dashed). Both profiles have the same derivative for $s/2\xi =0$ (dotted) and therefore also the same thickness (indicated by vertical lines) according to the definition in Eq. (23).

Figure 4

Dependence of correction factor in the expression for domain-wall energy density [Eq. (25)] as a function of the shape coefficient $A$. Full points indicate values of $A$ for particular domain walls considered in Table .

Figure 5

Course of strain components along the domain-wall normal coordinate $s$ for the T90 domain wall. Indices refer to cubic axes of the parent phase.

Figure 6

Temperature dependence of the thickness (a) and energy density (b) for various mechanically compatible and neutral domain walls of ${\text{BaTiO}}_3$ estimated within SPP approximation. The SPP values for domain walls for which relaxing of the $P_{\text{t}}$ component results in a lower-energy CPP solution are shown by dashed lines. Thickness of O180{001}, O90 has a very close temperature dependence. The results for T180 as well as for R180 walls are not distinguished by specific orientations of the wall normal since for both cases the angular dependence of thickness and energy of SPP solutions on the domain-wall normal is negligible (see Table ). Vertical lines mark phase-transition temperatures.

Figure 7

Equipotential contours of the Euler-Lagrange potentials showing ELPS associated with the $P_{\text{s}}=P_{\text{s}}\left(\pm \infty \right)$ plane for a set of ${\text{BaTiO}}_3$ domain walls considered in Table . Polarization scale given for the bottom left inset (in ${\text{Cm}}^{-2}$) is equally valid for all shown ELPSs. Energetically most favorable domain-wall solutions with trajectories restricted to the $P_{\text{s}}=P_{\text{s}}\left(\pm \infty \right)$ plane are indicated with bold lines. The solutions corresponding to a linear segment are denoted in the text as SPP walls, as opposed to the CPP solutions which have a curved order-parameter trajectory.

Figure 8

Predicted profiles of polarization components for O60 domain wall in ${\text{BaTiO}}_3$ at $T=208\text{K}$. Full line stands for the $P_{\text{r}}$ component of polarization vector and broken line for the $P_{\text{t}}$ one. Analytically obtained (a) SPP solution practically does not differ from the numerically obtained (b) CPP solution with an unconstrained $P_{\text{t}}$ component in this case.

Figure 9

Profiles of polarization components in $\text{R}180\left\{1\overline{1}0\right\}$ domain wall showing (a) the SPP stationary trajectory as well as the (b) lower-energy Bloch-type solution, both for the model parameters corresponding to $T=118\text{K}$. Full line stands for the $P_{\text{r}}$ component of polarization vector and broken line for the $P_{\text{t}}$ one. The bottom panel demonstrates that Bloch solution may considerable modify the domain-wall profile.