Bloch-type domain walls in rhombohedral BaTiO$_3$

Ferroelectric domain walls (FDWs) are usually considered to be of Ising type, but there have been suggestions in recent years that Bloch-type FDWs, in which the polarization rotates in the plane of the FDW, are also possible. The mechanically compatible and electrically neutral FDWs in rhombohedral BaTiO$_3$ are of 71$^\circ$, 109$^\circ$, and 180$^\circ$ type. We have investigated these FDWs based both on first-principles calculations and on a Ginzburg-Landau-Devonshire (GLD) model.\footnote{ P. Marton, I. Rychetsky, and J. Hlinka, Phys. Rev. B \textbf{81}, 144125 (2010).} The results from both approaches confirm the Ising nature of the 71$^\circ$ FDW and the Bloch nature of the 180$^\circ$ FDW, and predict both Ising-type and Bloch-type FDWs are possible for the 109$^\circ$ case. Considering the relatively small rhombohedral strain in BaTiO$_3$, the competition between the energies of Bloch and Ising FDWs can be discussed in terms of a picture in which a Bloch wall is regarded as being composed of a pair of smaller-angle Ising ones. A reduction by 40\% in the parameters describing the gradient term in the GLD model brings it into better agreement with the first-principles results for detailed properties such as the energies and widths of the FDWs.