Structure and spin dynamics of multiferroric BiFeO$_3$

Multiferroic materials have attracted much interest due to the unusual coexistence of ferroelectric and (anti-)ferromagnetic ground states in a single compound. They offer an exciting platform for new physics and potentially novel devices. BiFeO$_3$ is one of the most celebrated of multiferroic materials with highly desirable properties. It is the only known room-temperature multiferroic with $T_\mathrm{C}\approx 1100~ \mathrm{K}$ and $T_\mathrm{N}\approx 650~\mathrm{K}$, and exhibits one of the largest spontaneous electric polarisation, $P\approx 80~\mu\mathrm{C/cm}^2$. At the same time, it has a magnetic cycloid structure with an extremely long period of 630~\AA, which arises from a competition between the usual symmetric exchange interaction and antisymmetric Dzyaloshinskii-Moriya (DM) interaction. There is also an intriguing interplay between the DM interaction and the single ion anisotropy. In this review, we have tried to paint a complete picture of bulk BiFeO$_3$ by summarising the structural and dynamical properties of both spin and lattice parts, and their magneto-electric coupling.