Spectroscopic modes provide the most sensitive probe of the very weak interactions responsible for the properties of the long-wavelength cycloid in the multiferroic phase of BiFeO${}_3$ below $T_N\approx 640$ K. Three of the four modes measured by terahertz (THz) and Raman spectroscopies were recently identified using a simple microscopic model. While a Dzyaloshinskii-Moriya (DM) interaction $D$ along $[-1,2,-1]$ induces a cycloid with wave vector $(2\pi /a)(0.5+\delta ,0.5,0.5-\delta )$ ($\delta \approx 0.0045$), easy-axis anisotropy $K$ along the $[1,1,1]$ direction of the electric polarization $\mathbf{P}$ induces higher harmonics of the cycloid, which split the ${\Psi }_1$ modes at 2.49 and 2.67 meV and activate the ${\Phi }_2$ mode at 3.38 meV. However, that model could not explain the observed low-frequency mode at about 2.17 meV. We now demonstrate that an additional DM interaction $D^{\prime }$ along $[1,1,1]$ not only produces the observed weak ferromagnetic moment of the high-field phase above 18 T but also activates the spectroscopic matrix elements of the nearly degenerate, low-frequency ${\Psi }_0$ and ${\Phi }_1$ modes, although their scattering intensities remain extremely weak. Even in the absence of easy-axis anisotropy, $D^{\prime }$ produces cycloidal harmonics that split ${\Psi }_1$ and activate ${\Phi }_2$. However, the observed mode frequencies and selection rules require that both $D^{\prime }$ and $K$ are nonzero. This work also resolves an earlier disagreement between spectroscopic and inelastic neutron-scattering measurements.

• Received 19 February 2013

DOI:https://doi.org/10.1103/PhysRevB.87.134416