Domain walls in a perovskite oxide with two primary structural order parameters: First-principles study of BiFeO${}_3$

We present a first-principles study of ferroelectric domain walls (FE-DWs) in multiferroic BiFeO${}_3$ (BFO), a material in which the FE order parameter coexists with antiferrodistortive (AFD) modes involving rotations of the O${}_6$ octahedra. We find that the energetics of the DWs are dominated by the capability of the domains to match their O${}_6$ octahedra rotation patterns at the plane of the wall, so that the distortion of the oxygen groups is minimized. Our results thus indicate that, in essence, it is the discontinuity in the AFD order parameter, and not the change in the electric polarization, that decides which crystallographic planes are most likely to host BFO's FE-DWs. Such a result clearly suggests that the O${}_6$ rotational patterns play a primary role in the FE phase of this compound, in contrast with the usual (implicit) assumption that they are subordinated to the FE order parameter. Our calculations show that, for the most favorable cases in BFO, the DW energy amounts to several tens of mJ/m${}^2$, which is higher than what was computed for other ferroelectric perovskites with no O${}_6$ rotations. Interestingly, we find that the structure of BFO at the most stable DWs resembles the atomic arrangements that are characteristic of low-lying (meta)stable phases of the material. Further, we argue that our results for the DWs of bulk BFO are related with the nanoscale-twinned structures that Prosandeev et al. [Adv. Funct. Mater. (2012)] have recently predicted to occur in this compound, and suggest that BFO can be viewed as a polytypic material. Our work thus contributes to shape a coherent picture of the structural variants that BFO can present and the way in which they are related.

Figure 1

(a) Pseudocubic cell of BFO showing its three main axes (thin arrows) and the polarization along [111]${}_{\mathrm{pc}}$ (thick arrow). (b) Detail of the cell, where two octahedra have been removed to allow visualization of Fe cations and Fe-O bonds.

Figure 2

Atomic structures obtained upon relaxation of 120-atom unit cells: (a) [111](100)[1$\overline{1}$$\overline{1}$] 2/1 DW configuration; (b) [1$\overline{1}$1](110)[$\overline{1}$1$\overline{1}$] 3/0 configuration; (c) [111](110)[11$\overline{1}$] 1/1 configuration. The vertical lines indicate the approximate position of the center of the DWs.

Figure 3

Values of magnitudes that characterize the atomic displacements in the [111](100)[$1\overline{1}\overline{1}$] 2/1 DW configuration (polarizations at an angle of around ${109}^{\circ }$). The discontinuous lines show the values computed for bulk BFO. The continuous vertical lines mark the approximate position of the center of the DW. We describe in the text those magnitudes that are not obvious from the vertical axis labels.

Figure 4

Same measures as in Fig. 3, but for the [1$\overline{1}$1](110)[$\overline{1}$1$\overline{1}$] 3/0 DW configuration (polarizations at an angle of ${180}^{\circ }$).

Figure 5

Same measures as in Fig. 3, but for the [111](110)[11$\overline{1}$] 1/1 DW configuration (polarizations at an angle of around ${71}^{\circ }$).

Figure 6

Same measures as in Fig. 3, but for the [1$\overline{1}$1](110)[1$\overline{1}$$\overline{1}$] 1/1 DW configuration (polarizations at an angle of around ${71}^{\circ }$; mechanical matching condition not fulfilled).

Figure 7

Left panel: DW energy as a function of a measure of distortions in O-O edges of O${}_6$ octahedra (see text); the symbols refer to different types of octahedra mismatching at the DW. Right panel: Illustration of octahedra mismatches, as described in the text; each arrow indicates the axis around which each octahedron rotates.

Figure 8

LDoS of bulk BFO and of the three representative DW configurations of Table (simulation cells with 80 atoms). The density of states has been projected on the closest 4 Fe, 4 Bi, and 12 O ions to the DW (thick lines), and on the $d$ orbitals of those ions (shaded areas). The band gap is highlighted between two discontinuous vertical lines.