  In this article, we perform the symmetry analysis of perovskite-based
multiferroics: bismuth ferrite (BiFeO3)-like, orthochromites (RCrO3), and
Ruddlesden-Popper perovskites (Ca3Mn2O7-like), being the typical
representatives of multiferroics of the trigonal, rhombic, and tetragonal
crystal families and explore the effect of crystallographic distortions on
magnetoelectric properties. We determine the principal order parameters for
each of the considered structures and obtain their invariant combinations
consistent with the particular symmetry. This approach allowed us to analyze
the features of the magnetoelectric effect observed during structural phase
transitions in BixR1-xFeO3 compounds and show that the rare-earth sublattice
gives an impact into the linear magnetoelectric effect allowed by the symmetry
of the new structure. It is shown that the magnetoelectric properties of
ortho-chromites are attributed to the couplings between the magnetic and
electric dipole moments arising near Cr3+ ions due to distortions linked with
rotations and deformations of the CrO6 octahedra. For the first time, such
symmetry consideration was implemented in the analysis of the Ruddlesden-Popper
structures, which demonstrates the possibility of realizing the magnetoelectric
effect in the Ruddlesden-Popper phases containing magnetically active cations
and allows to estimate conditions required for its optimization.
