First-principles study of spin spirals in the multiferroic 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 $E(\mathbf{q})$ 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 $\ensuremath{\sim}80$ 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 $R3c\phantom{\rule{4pt}{0ex}}{\mathrm{BiFeO}}_{3}$, 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 $R3c\phantom{\rule{4pt}{0ex}}{\mathrm{BiFeO}}_{3}$ is relatively small to enforce a collinear order.