Identifying the Magnetoelectric Modes of Multiferroic BiFeO${}_3$

We have identified three of the four magnetoelectric modes of multiferroic BiFeO${}_3$ measured using THz spectroscopy. Excellent agreement with the observed peaks is obtained by including the effects of easy-axis anisotropy along the direction of the electric polarization. By distorting the cycloidal spin state, anisotropy splits the ${\mathrm{\Psi }}_{\pm 1}$ mode into peaks at 20 and 21.5 cm${}^{-1}$ and activates the lower ${\mathrm{\Phi }}_{\pm 2}$ mode at 27 cm${}^{-1}$ (T=200 K). An electromagnon is identified with the upper ${\mathrm{\Psi }}_{\pm 1}$ mode at 21.5 cm${}^{-1}$. Our results also explain recent Raman and inelastic neutron-scattering measurements.