Magnetostructural Effect in the Multiferroic ${\mathrm{BiFeO}}_3\mathrm{\text{−}}{\mathrm{BiMnO}}_3$ Checkerboard from First Principles

Using first-principles calculations, we identify a magnetostructural effect in the ${\mathrm{BiFeO}}_3\mathrm{\text{−}}{\mathrm{BiMnO}}_3$ nanocheckerboard that is not to be found in either the bulk parent compound or in ${\mathrm{BiFeO}}_3\mathrm{\text{−}}{\mathrm{BiMnO}}_3$ superlattices with (001)-oriented Fe and Mn layers. The key role of the cation arrangement is explained by a simple model of the exchange coupling between cation spins, leading to magnetic frustration in the checkerboard. We also demonstrate that the atomic-scale checkerboard has a multiferroic ground state with the desired properties of each constituent material: polar and ferrimagnetic due to ${\mathrm{BiFeO}}_3$ and ${\mathrm{BiMnO}}_3$, respectively.

Figure 1

(i) ${\mathrm{BiFeO}}_3\mathrm{\text{−}}{\mathrm{BiMnO}}_3$ (001)-oriented layered superlattice with alternation of $\mathrm{Fe}/\mathrm{Mn}$ planes. (ii) (left) ${\mathrm{BiFeO}}_3\mathrm{\text{−}}{\mathrm{BiMnO}}_3$ checkerboard. Checkerboard ordering of $\mathrm{Fe}/\mathrm{Mn}$ atoms in the ($xy$) plane, pillars of the same composition form along the $z$ direction. (right) Ideal perovskite unit cell. Perovskite cells with $\mathrm{Fe}/\mathrm{Mn}$ atoms on the $B$ site repeat according to the checkerboard pattern (ii), or layered geometry (i).

Figure 2

From top left to bottom right: (i) $G$-AFM: rocksalt type antiferromagnetic (AFM) order; (ii) $C$-FIM: AFM order in horizontal planes, ferromagnetic (FM) order along $\mathrm{Fe}/\mathrm{Mn}$ pillars; (iii) FM order; (iv) FeAFMMnFM: AFM order along Fe pillars, FM order along Mn pillars; (v) FeFMMnAFM: FM order along Fe pillars, AFM order along Mn pillars; (vi) FeAFMMnAFM: AFM order along $\mathrm{Fe}/\mathrm{Mn}$ pillars, but FM order in horizontal planes.

Figure 3

Structural energetics of bulk ${\mathrm{BiFeO}}_3$. Energy difference per perovskite cell (f.u.) for different magnetic orderings (see Fig. 2) and structural distortions: (1) $Pm\overline{3}m$: no distortion—ideal perovskite; (2) ${\Gamma }_4^{-}(z)$: polar distortion along $z$ axis; (3) $M_3^{+}(z)$: $+$ oxygen octahedra tilts about $z$ axis; (4) $R_4^{+}(y)$: $-$ oxygen octahedra tilts about $y$ axis; (5) $R_4^{+}(y)$ and ${\Gamma }_4^{-}(y)\mathbf{(}R\Gamma (y)\mathbf{)}$: linear combination of (4) and (2) along $y$ axis, relaxes back to polar ${\Gamma }_4^{-}(y)$ with zero tilting angle; (6) $R_4^{+}([111])$ $\mathbf{(}{R}_4^{+}(d)\mathbf{)}$: $-$ oxygen octahedra tilts about [111] axis; (7) $R_4^{+}([111])$ and ${\Gamma }_4^{-}([111])$ ($R\Gamma (d)$): linear combination of (6) and (2) along [111] ($d$), where $d$ refers to the cube diagonal direction.

Figure 4

Structural energetics of bulk ${\mathrm{BiMnO}}_3$. Energy difference per perovskite cell (f.u.) for various structural distortions (see Fig. 3) and magnetic orderings (see Fig. 2); $P4/mmm$ corresponds to a tetragonally distorted perovskite cell with ideally positioned atoms and $M\Gamma (z)$ is a linear combination of rotational $M_3^{+}(z)$ and polar ${\Gamma }_4^{-}(z)$ modes.

Figure 5

Structural energetics of ${\mathrm{BiFeO}}_3\mathrm{\text{−}}{\mathrm{BiMnO}}_3$ checkerboard. Energy difference per perovskite cell (f.u.) for different magnetic orderings (see Fig. 2) and structural distortions: (1) $P4/mmm$, (2) ${\Gamma }_4^{-}(z)$, (3) $R_4^{+}(y)$ and ${\Gamma }_4^{-}(y)$, (4) $R_4^{+}([111])$ and ${\Gamma }_4^{-}([111])$ $\mathbf{(}{R}_4^{+},{\Gamma }_4^{-}(d)\mathbf{)}$. Inset: zoomed view of the magnetic energies of $c\mathrm{\text{−}}R3c$ (4) distortion.