Terahertz-Range Polar Modes in Domain-Engineered ${\mathrm{BiFeO}}_3$

The dielectric permittivity and properties of electrically active lattice resonances in nanotwinned ${\mathrm{BiFeO}}_3$ crystals have been studied theoretically using an earlier established interatomic potential. The results suggest that an array of 71° domain walls with about 2–5 nm spacing enhances the static permittivity of ${\mathrm{BiFeO}}_3$ by more than an order of magnitude. This enhancement is associated with an electrically active excitation, corresponding to a collective vibration of pinned domain walls at a remarkably high frequency of about 0.3 THz.

Figure 1

Shell-model-optimized 180-atom supercell with two 71° domain walls. The top panel shows profiles of order parameter components, the bottom panels show the real atomic structure. Solid symbols: Octahedral chains $C_1$ and $C_1^{\prime }$ depicted in ($e$), open symbols: $C_2$ and $C_2^{\prime }$. The description includes the adopted axis notation, where $r$ is the direction in which the polarization changes sign across the domain wall, $s$ is normal to the domain wall plane, and $t$ is normal to $r$ and $s$ [36].

Figure 2

Real (a) and imaginary (b) part of the principal components of the frequency-dependent permittivity of ${\mathrm{BiFeO}}_3$ with 71° domain walls (calculated for the supercell shown in Fig. 1). A damping factor of $2{\mathrm{cm}}^{-1}$ has been used. For comparison, the diagonal element of the monodomain permittivity tensor corresponding to the direction $r$ of the supercell with domain walls is also shown. Two different vertical scales were applied in order to better visualize the higher-frequency signal. The letters $A$, $B$, $C$, and $D$ mark resonances of ${ϵ}_{\mathrm{rr}}$ that are further discussed in the text.

Figure 3

Dispersion curves calculated for the monodomain [(a), left] and 71° domain wall [(b), right] structure of Fig. 1. The color scale indicates the projection onto the $r$ component of the Last-type mode with the wave vector $\mathbf{q}$ along the $s$ direction, with hotter colors marking a bigger participation ratio.

Figure 4

The dynamical polarization profiles associated with the modes marked in Figs. 2, 3. Each point symbol marks a polarization component (squares, $s$; triangles, $t$; and circles, $r$ direction) calculated from mode eigenvectors [47]. The static polarization profile $P_r$ is the same as in Fig. 1. Arrows mark wall movements caused by a particular mode. The panels (a)–(f) refer to modes marked A–F in Fig. 2.