Multiferroics exhibit more than one primary ferroic ordering, including ferromagnetism, ferroelectricity, or ferroelasticity in the same phase^[1,2]. The terminology is usually extended to include non-primary orderings such as antiferromagnetism, antiferroelectricity, as well as composites of individual ferroics. The concept often used today is referred to magnetoelectric materiasl combing ferroelectric and magnetic behaviors (Fig. 1);. Such a combination of ferroic orders in multiferroics can lead to coupling between them and provide an efficient route to control magnetism by electric fields or vice versa.

Multiferroics can be classified into two categories: type-I mulitferroics refer to the ferroelectricity does not have magnetic origin, which usually leads to an indirect and weak magnetoelectric coupling; Type-II multiferroics are induced by some specific magnetic orders with a strong magnetoelectric coupling but a small electric polarization and thus low transition temperature (T_C). The two types are depicted in Fig. 2^[3]

In geometrically driven ferroelectrics, a small A-site cation permits rotations of its surrounding coordination polyhedral (Fig. 3b), which are either polar in their own right (for example, in BaNiF4^[6]) or couple to a secondary polar distortion (as in YMnO3^[7]). In this case, the polarization is both robust and small in magnitude. In h-RMnO3 (R=Sc, Y, In or Dy-Lu^[8,9]), unit-cell tripling drives the emergence of a ferroelectric order at Tc ≥ 1,200 K with a polarization of 5.6 μC/cm2, followed by a magnetic ordering at the Neel temperature (TN ≤ 120 K).

In magnetically induced ferroelectrics, a non-collinear magnetic ordering, such as a spin spiral, breaks the inversion symmetry and induces a spontaneous polarization. These magnetically induced, so-called improper ferroelectric materials are totally different from displacive ferroelectrics, in which magnetic ordering is inhibited. The most intensely discussed mechanism is the inverse Dzyaloshinskii–Moriya (DM) interaction. Spin-orbit coupling (SOC) is crucial for the inverse DM interaction. The polarization is essentially determined by the optimization of the spin configuration from the point of view of anitsymmetry exchange, expressed by the antisymmetric product S_i × S_j of neighboring spins Si,j (see Fig. 3d). The polarization vector is P ∝ e_ij × ( S_i × S_j), where eij is the unit vector connecting neighboring spins. Multiferroicity of this type was first found in Cr2BeO4^[11].