The Spin State and Spectroscopic Modes of Multiferroic BiFeO3

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 \BF below $\TN \approx 640$ K. Three of the four modes measured by THz and Raman spectroscopies were recently identified using a simple microscopic model. While a Dzyaloshinskii-Moriya (DM) interaction $D$ along $[-1,2,-1]$ induces the cycloid with wavevector $(2\pi /a)(0.5+\delta, 0.5, 0.5-\delta)$ ($\delta \approx 0.0045$), easy-axis anisotropy $K$ along the $[1,1,1]$ direction of the electric polarization ${\bf P}$ 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 $D'$ 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, $D'$ produces cycloidal harmonics that split $\Psi_1 $ and activate $\Phi_2$. However, the observed mode frequencies and selection rules require that both $D'$ and $K$ are nonzero. This work also resolves an earlier disagreement between spectroscopic and inelastic neutron-scattering measurements.