Nanoscale local structures of rhombohedral symmetry in the orthorhombic and tetragonal phases of BaTiO${}_3$ studied by convergent-beam electron diffraction

The symmetries of the rhombohedral, orthorhombic, and tetragonal phases of barium titanate (BaTiO${}_3$) are investigated using convergent-beam electron diffraction. Nanometer-sized local structures with rhombohedral symmetry are observed in both the orthorhombic and tetragonal phases. This indicates that an order-disorder character exists in phase transformations of BaTiO${}_3$. The nanostructures in these phases are discussed in terms of an order-disorder model with off-centered Ti in the $\langle 111\rangle$ directions.

Figure 1

CBED pattern of the rhombohedral phase of BaTiO${}_3$ taken at 90 K with the ${\left[100\right]}_{\mathrm{pc}}$ ($={\left[100\right]}_{\mathrm{rhom}}$) incidence. Difference map between original and mirror-operated patterns $\Delta I/I$ is attached with a color presentation, where the mirror is indicated by a yellow line. Corresponding projected structure is shown as an inset, the projected direction of ferroelectric polarization being indicated by a red arrow.

Figure 2

CBED patterns of the orthorhombic phase of BaTiO${}_3$ taken at 200 K with (a) the ${\left[001\right]}_{\mathrm{pc}}$ ($={\left[0\overline{1}1\right]}_{\mathrm{orth}}$) incidence and (b) the ${\left[100\right]}_{\mathrm{pc}}$ ($={\left[100\right]}_{\mathrm{orth}}$) incidence, respectively. Difference maps with respect to the mirror symmetries indicated by the dotted and solid lines are attached, where the projected structures are shown as insets.

Figure 3

CBED patterns of the tetragonal phase of BaTiO${}_3$ taken at room temperature with (a) the ${\left[001\right]}_{\mathrm{pc}}$ ($={\left[001\right]}_{\mathrm{tet}}$) incidence and (b) & (c) the ${\left[100\right]}_{\mathrm{pc}}$ ($={\left[100\right]}_{\mathrm{tet}}$) incidence, respectively. Pattern (c) shows a ${\left[100\right]}_{\mathrm{pc}}$ pattern averaged over 25 patterns taken each from an area of 5 nm in diameter. Difference maps with respect to the mirror symmetries indicated are attached, where the projected structures are shown as insets.

Figure 4

Schematic diagrams of the model of Ti off-centering in $\langle 111\rangle$ directions (Ref. 13). Models of (a) the rhombohedral phase, (b) the orthorhombic phase, (c) the tetragonal phase, and (d) the cubic phase.

Figure 5

(a) Experimental and (b) simulated CBED patterns of the tetragonal phase of BaTiO${}_3$. Difference maps with respect to mirror symmetries indicated are attached.