Spin-current driven Dzyaloshinskii-Moriya interaction in multiferroic ${\mathrm{BiFeO}}_3$ from first principles

The electrical control of magnons opens up new ways to transport and process information for logic devices. In magnetoelectrical multiferroics, the Dzyaloshinskii-Moriya (DM) interaction directly allows for such control and hence is of major importance. We determine the origin and the strength of the (converse) spin-current DM interaction in the $R3c$ bulk phase of multiferroic ${\mathrm{BiFeO}}_3$ based on density functional theory. Our data support only the existence of one DM interaction contribution originating from the spin-current model. By exploring the magnon dispersion in the full Brillouin zone, we show that the exchange is isotropic, but the DM interaction and anisotropy prefer any propagation and any magnetization direction within the full (111) plane. Our work emphasizes the significance of the asymmetric potential induced by the spin current over the structural asymmetry induced by the anionic octahedron in multiferroics such as ${\mathrm{BiFeO}}_3$.

Figure 1

(a) Pseudocubic Brillouin zone (BZ) with high-symmetry points (red points and letters). Shown are the calculated propagation directions (colored lines) of spin spirals. Note that all calculations start from the same $R$ point. (b) Collinear magnetic states (FM, ferromagnetic; AFM, antiferromagnetic) connected to each high-symmetry point of the BZ.

Figure 2

Sketches of the two models for the Dzyaloshinskii-Moriya (DM) interaction used in this paper. (a) Levy-Fert model (LF model) [32, 49]: Two magnetic atoms ${\mathbf{S}}_i,{\mathbf{S}}_j$ are interacting via a heavy metal atom hosting large spin-orbit coupling (SOC). The triangle formed by the three atoms defines the DM vector ${\mathbf{D}}_{ij}$. (b) Converse spin-current (SC) model [22, 23]: In systems with a polarization $\mathbf{P}$, the spin-current vector $\mathbf{u}\times {\mathbf{e}}_{i,j}$ is perpendicular to the direction of polarization.

Figure 3

Energy dispersion $E\left(\mathbf{q}\right)$ without SOC of flat spin spirals along different directions within the pseudocubic Brillouin zone. Points show calculated energies from DFT, and curves represent a fit to the Heisenberg exchange interaction including seven neighbors. (a) Full energy dispersion. (b) Zoom around the ground state at $\mathbf{q}\to R$.

Figure 4

Dzyaloshinskii-Moriya interaction (DM interaction) in ${\mathrm{BiFeO}}_3$. (a) Energy contribution due to spin-orbit coupling ($\mathrm{\Delta }{E}_{\mathrm{SOC}}$) to the energy dispersion of spin spirals along different directions within the pseudocubic Brillouin zone [Fig. 1]. Points show calculated energies from density functional theory (DFT), the red curve corresponds to the LF model, and the blue curve corresponds to the SC model. Positive (negative) values prefer a clockwise (counterclockwise) sense of rotation of noncollinear states. (b) Zoom around the ground state at $\mathbf{q}\to R$ where only the DFT data and the SC model are presented.

Figure 5

Element-resolved energy contribution due to SOC ($\mathrm{\Delta }{E}_{\text{SOC}}$) to the energy dispersion. (a) Sketch of the ${\mathrm{BiFeO}}_3$ structure in the cubic approximation. (b), (c), and (d) $\mathrm{\Delta }{E}_{\text{SOC}}$ for two Bi, two Fe, and six O atoms in the unit cell, respectively. The total energy contribution is shown with gray points. Here, cw, clockwise; ccw, counterclockwise.

Figure 6

Calculated uniaxial anisotropy energy within the (111) plane of ${\mathrm{BiFeO}}_3$. (a) Sketch of the pseudocubic Brillouin zone (BZ) with different lattice directions. The shaded area shows the (111) plane, whereas the anisotropy calculations have been performed considering the G-type antiferromagnet with the application of spin-orbit coupling in all directions represented by the arrows. The colored arrows in the (111) plane represent $\left[1\overline{1}0\right]$ and $\left[11\overline{2}\right]$ and their equivalent directions. (b) Results of the calculations for part of the data. The results do not change for the complete rotation within the (111) plane. “SC-DFT” corresponds to self-consistent density functional theory calculations, and “FT-DFT” denotes calculations using the force theorem [44]. For each type of calculations, we considered different $k$-point sets, i.e., $20\times 20\times 20$ and $10\times 10\times 10$, in the whole BZ.

Figure 7

Energy contribution due to spin-orbit coupling ($\mathrm{\Delta }{E}_{\text{SOC}}$) to the energy dispersion of spin spirals along different directions within the pseudocubic Brillouin zone [BZ; see Fig. 3 in the main text]. Points show calculated energies from density functional theory (DFT), and the green curve corresponds to the spin-current model in the effective nearest-neighbor approximation. Positive (negative) values prefer a clockwise (counterclockwise) sense of rotation of noncollinear states. (a) Whole calculated paths. (b) Zoom around the ground state at $\mathbf{q}\to R$.