  In this article we review and discuss a mechanism for coupling between
electric polarization and magnetization that can ultimately lead to
electric-field switchable magnetization. The basic idea is that a ferroelectric
distortion in an antiferromagnetic material can "switch on" the
Dzyaloshinskii-Moriya interaction which leads to a canting of the
antiferromagnetic sublattice magnetizations, and thus to a net magnetization.
This magnetization M is coupled to the polarization P via a trilinear free
energy contribution of the form P(M x L), where L is the antiferromagnetic
order parameter. In particular, we discuss why such an invariant is present in
R3c FeTiO_3 but not in the isostructural multiferroic BiFeO_3. Finally, we
construct symmetry groups that in general allow for this kind of
ferroelectrically-induced weak ferromagnetism.
