First-principles study of spin spirals in the multiferroic ${\mathrm{BiFeO}}_3$

We carry out density functional theory (DFT) calculations to explore the antiferromagnetic (AFM) spin cycloid in multiferroic ${\mathrm{BiFeO}}_3$ of the $R3c$ ground state structure. We calculate the energy dispersion $E\left(\mathbf{q}\right)$ of cycloidal spin spirals along the high symmetry directions of the pseudo-cubic unit cell and find a flat AFM spin spiral (or cycloid) ground state with a periodicity of $\sim 80$ nm, which is in good agreement with experiments. To investigate which structural distortion of the $R3c$ phase is the driving mechanism for the stabilization of this cycloid, we further study three artificial phases: cubic, $R\overline{3}c$, and $R3m$. In all cases, we find a large exchange frustration. The comparison between these phases provides detailed insight about how polarization and octahedral antiphase tilting affect the different magnetic interactions and the magnetic ground state in ${\mathrm{BiFeO}}_3$. In $R3c{\mathrm{BiFeO}}_3$, the magnetic ground state is driven by a competition between the frustrated exchange stemming from the hybridization between the elements Bi, Fe, O and the Dzyaloshinskii-Moriya (DM) interaction due to the Fe-Bi ferroelectric displacement. The cycloid appears to be stable because the anisotropy energy in $R3c{\mathrm{BiFeO}}_3$ is relatively small to enforce a collinear order.

Figure 1

Sketches of the studied ${\mathrm{BiFeO}}_3$ structures. Green spheres correspond to the Bi atoms, golden spheres in the center of each cube denote Fe, and small red spheres show O atoms. (a) Centrosymmetric cubic structure, (b) $R\overline{3}c$ structure where only the oxygen octahedral tilts are included, represented by curved arrows and $\pm \mathrm{\Omega }$. (c) $R3m$ structure, where only the ferroelectric displacement is included, denoted by arrows at the Bi sites and Fe sites. The displacement leads to a large spontaneous polarization $P$ in the [111] direction. (d) Structural $R3c$ ground state where both effects, oxygen octahedral tilts from (b) and ferroelectric displacement from (c), are included. Note that the structures of (a)–(c) are artificial to isolate the structural distortions of the $R3c$ phase.

Figure 2

Energy dispersion $E\left(\mathbf{q}\right)$ of homogeneous, flat spin spiral states for pseudocubic ${\mathrm{BiFeO}}_3$ [cf. Fig. 1] with respect to the G-type AFM structure energy $E\left(\mathrm{R}\right)$. (a) Energy dispersion without spin-orbit coupling (SOC) along the high-symmetry directions of the pseudocubic first Brillouin zone. The points are spin spiral energies computed from DFT and the lines are obtained by mapping the Heisenberg exchange Hamiltonian to the DFT data. The directions of high symmetry paths are shown in the inset. (b) Zoom around the $\mathrm{R}$ point for left (positive) and right-rotating (negative) spin spiral states with (red) and without (gray) SOC. Note that the energy scale is below 0.1 meV. The fit including SOC (red curve) contains exchange beyond first nearest neighbors, Dzyaloshinskii-Moriya interaction and uniaxial anisotropy energy. Note that the red circle at the $R$ point is only shifted by $K$/2 [$K$ being the uniaxial magnetocrystalline anisotropy defined in Eq. (4)] to keep the red points continuous. In that case, the ground state is at $E=0\mathrm{meV}$/Fe atom.

Figure 3

Energy dispersion $E\left(\mathbf{q}\right)$ of homogeneous, flat spin spiral states for $R\overline{3}c {\mathrm{BiFeO}}_3$ [cf. Fig. 1] with respect to the G-type AFM structure energy $E\left(\mathrm{R}\right)$. (a) Energy dispersion without spin-orbit coupling (SOC) along the high symmetry directions of the pseudocubic first Brillouin zone. The points and the line are spin spiral energies computed from DFT and obtained by mapping the Heisenberg exchange Hamiltonian to the DFT data, respectively. The directions of high symmetry paths are shown in the inset. (b) Zoom around the $\mathrm{R}$ point for left (positive) and right-rotating (negative) spin spiral states with (red) and without (gray) SOC. Note that the energy scale is below 0.4 meV. The fit including SOC (red curve) contains exchange beyond first nearest neighbors, Dzyaloshinskii-Moriya interaction and uniaxial anisotropy energy. Note that the red circle at the $R$ point is only shifted by $K/2$ to keep the red points continuous. In that case, the ground state is at $E=0\mathrm{meV}$/Fe atom.

Figure 4

Energy dispersion $E\left(\mathbf{q}\right)$ of homogeneous, flat spin spiral states for $R3m {\mathrm{BiFeO}}_3$ [cf. Fig. 1] with respect to the G-type AFM structure energy $E\left(\mathrm{R}\right)$. (a) Energy dispersion without spin-orbit coupling (SOC) along the high symmetry directions of the pseudocubic first Brillouin zone. The points and the line are spin spiral energies computed from DFT and obtained by mapping the Heisenberg exchange Hamiltonian to the DFT data, respectively. The directions of high symmetry paths are shown in the inset. (b) Zoom around the $\mathrm{R}$ point for left (positive) and right-rotating (negative) spin spiral states with (red) and without (gray) SOC. Note that the energy scale is below 0.2 meV. The fit including SOC (red curve) contains exchange beyond first nearest neighbors, Dzyaloshinskii-Moriya interaction and uniaxial anisotropy energy. Due to slight deviations stemming from the exchange interaction, the value for the DM interaction has been adapted to better describe the energy dispersion with SOC. Note that the red circle at the$\mathrm{R}$ point is only shifted by $K/2$. (c) Element-resolved energy contribution due to spin-orbit coupling $\mathrm{\Delta }{E}_{\text{SOC}}$ to the energy dispersion of spin spirals calculated in the $\left[11\overline{2}\right]$ direction. Shown are the total (black), Fe (red), Bi (green), and O (blue) contributions. Due to an antiferromagnetic unit cell, each atom appears twice with the same contribution. Here, the lines serve as a guide to the eye.

Figure 5

Energy dispersion $E\left(\mathbf{q}\right)$ of homogeneous, flat spin spiral states for $R3c {\mathrm{BiFeO}}_3$ [cf. Fig. 1] with respect to the G-type AFM structure energy $E\left(\mathrm{R}\right)$. (a) Energy dispersion without spin-orbit coupling (SOC) along the high symmetry directions of the pseudocubic first Brillouin zone. The points and the line are spin spiral energies computed from DFT and obtained by mapping the Heisenberg exchange Hamiltonian to the DFT data, respectively. The directions of high symmetry paths are shown in the inset. (b) Zoom around the $\mathrm{R}$ point for left (positive) and right-rotating (negative) spin spiral states with (red) and without (grey) SOC. Note that the energy scale is below 0.1 meV. The fit including SOC (red curve) contains exchange beyond first nearest neighbors, Dzyaloshinskii-Moriya interaction and uniaxial anisotropy energy. Due to slight deviations stemming from the exchange interaction, the value for the DM interaction has been adapted to better describe the energy dispersion with SOC. Note that the red circle at the $\mathrm{R}$ point is only shifted by $K/2$. (c) Element resolved contribution due to spin-orbit coupling $\mathrm{\Delta }{E}_{\text{SOC}}$ to the energy dispersion of spin spirals calculated in the $\left[11\overline{2}\right]$ direction. Shown are the total (black), Fe (red), Bi (green) and O (blue) contributions. Due to an antiferromagnetic unit cell, each atom appears twice with the same contribution. Here, the lines serve as a guide to the eye.

Figure 6

Energy dispersions of homogeneous flat spin spiral states without (gray) and with (red) spin-orbit coupling (SOC) for ${\mathrm{BiFeO}}_3$ (BFO) at $\left|\mathbf{q}\right|\to \mathrm{R}$ of the pseudocubic Brillouin zone. The points show the DFT calculated energies (dark gray without SOC, red with SOC) whereas lines represent the fit to the extended Heisenberg model including the exchange interaction beyond first nearest neighbors (without SOC, gray lines), the Dzyaloshinskii-Moriya interaction and magnetocrystalline anisotropy energy (with SOC, red lines). For positive (negative) values of $\mathbf{q}$, the AFM spin spirals prefer a counterclockwise (clockwise) rotation. (a) $R3m$ phase and (b) $R3c$ phase. Note that here, all lines are based on the complete fitting of the DFT data.