The first known mention of magnetoelectricity is in the 1959 Edition of Landau & Lifshitz' Electrodynamics of Continuous Media which has the following comment at the end of the section on piezoelectricity: “Let us point out two more phenomena, which, in principle, could exist.
One is piezomagnetism, which consists of linear coupling between a magnetic field in a solid and a deformation (analogous to piezoelectricity).
The following describes the mechanisms that are known to circumvent this contraindication between ferromagnetism and ferroelectricity.
In geometric ferroelectrics, the driving force for the structural phase transition leading to the polar ferroelectric state is a rotational distortion of the polyhedra rather than an electron-sharing covalent bond formation.
The latter system, appears to be the first reported core-shell type relaxor ferroelectric multiferroic, where the magnetic structure in so-called "multiferroic clusters" is proposed to be due to Fe-Co ferrimagnetism, which can be switched by an electric field.
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Khomskii suggested the term type-I multiferroic for materials in which the ferroelectricity and magnetism occur at different temperatures and arise from different mechanisms.
When the pattern of localized electrons is polar, the charge ordered state is ferroelectric.
Usually the ions in such a case are magnetic and so the ferroelectric state is also multiferroic..
While magnetoelectric materials are not necessarily multiferroic, all ferromagnetic ferroelectric multiferroics are linear magnetoelectrics, with an applied electric field inducing a change in magnetization linearly proportional to its magnitude.
Magnetoelectric materials and the corresponding magnetoelectric effect have a longer history than multiferroics, shown in blue in the graph to the right.