Figure 2: Different microscopic mechanisms found in type-I multiferroics. (a) In “mixed” perovskites with ferroelectrically active $d^0$ ions (green circles) and magnetic $d^n$ ions (red), shifts of $d^0$ ions from the centers of ${\text{O}}_6$ octahedra (yellow plaquettes) lead to polarization (green arrows), coexisting with magnetic order (red arrows). (b) In materials like ${\text{BiFeO}}_3$ and ${\text{PbVO}}_3$, the ordering of lone pairs (yellow ”lobes”) of ${\text{Bi}}^{3+}$ and ${\text{Pb}}^{2+}$ ions (orange), contributes to the polarization (green arrow). (c) In charge ordered systems, the coexistence of inequivalent sites with different charges, and inequivalent (long and short) bonds, leads to ferroelectricity. (d) The “geometric” mechanism of generation of polarization in ${\text{YMnO}}_3$ [24] describes the tilting of a rigid ${\text{MnO}}_5$ block with a magnetic $\text{Mn}$ remaining at the center. Because of the tilting, the $\text{Y-O}$ bonds form dipoles (green arrows), and there appears two “down” dipoles per one “up” dipole so that the system becomes ferroelectric (and multiferroic when $\text{Mn}$ spins order at lower temperatures).