Infrared phonon dynamics of multiferroic BiFeO$_3$ single crystal*

We discuss the first infrared reflectivity measurement on a BiFeO$_3$ single crystal between 5 K and room temperature. The 9 predicted \textit{ab}-plane $E$ phonon modes are fully and unambiguously determined. The frequencies of the 4 $A_1$ \textit{c}-axis phonons are found. These results settle issues between theory and data on ceramics. Our findings show that the softening of the lowest frequency $E$ mode is responsible for the temperature dependence of the dielectric constant, indicating that the ferroelectric transition in BiFeO$_3$ is soft-mode driven.