  In this article the mechanism of the linear magnetoelectric (ME) effect in
the rhombohedral multiferroic BiFeO$_3$ is considered. The study is based on
the symmetry approach of the GinzburgLandau type, in which polarization,
antiferrodistortion, and antiferromagnetic momentum vectors are viewed as
ordering parameters. We demonstrate that the linear ME effect in BFO is caused
by reorientation of the antiferrodistortion vector in either electric or
magnetic field. The numerical estimations, which show quantitative agreement
with the results of the recent measurements in film samples, have been
performed. A possibility of significant enhancement of the magnetoelectric
effect by applying an external static electric field has been investigated. The
considered approach is promising for explaining the high values of the ME
effect in composite films and heterostructures with BFO.
