Micro-Raman spectroscopy has been used to study lattice dynamics associated with the ferroelectric domains of a BiFeO$_3$ single crystal at low temperature. The phonon assignment shows a large frequency splitting between the transverse and longitudinal components of the A$_1$ phonon mode related to the Bi-O bonds in contrast with thin films where the splitting is negligible. Applying an external electric field induces frequency shifts of the low energy modes related to the Bi-O bonds. These softenings are due to a tensile stress via the piezoelectric effect. We give estimates of the phonon deformation potentials.
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