Rotation and Metastability of BiFeO$_{3}$ Domains in a Magnetic Field*

Earlier models for the room-temperature multiferroic BiFeO$_{3}$ assumed that the domain wavevectors \textbf{q} are restricted to the three hexagonal axis normal to the static polarization parallel to a cubic diagonal like [1,1,1]. However, increasing evidence suggests that the domain wavevectors rotate so that \textbf{q} lies perpendicular to the field orientation \textbf{m} for fields above about 9 T. We show that previously neglected hexagonal anisotropy restricts the wavevectors to lie along the hexagonal axis in zero field. For fields below B$_{c1\, }=$ 7 T, the domain with wavevector \textbf{q} along \textbf{m} remains metastable while the two wavevectors of the other domains rotate perpendicular to \textbf{m}. For B$_{c1}$ \textless B \textless B$_{c2}$, the domain with wavevector along \textbf{m} disappears as it is no longer even metastable. For fields above B$_{c2} \quad =$ 9 T, the stable domain has wavector \textbf{q} normal to \textbf{m}. These new results explain recent measurements of the critical field as a function of field orientation, small-angle neutron scattering measurements of the wavevectors, as well as spectroscopic measurements for field along [0,1,-1].