We show that a superstructure of antiferromagnetically interacting Fe$^{3+}$ ($S=5/2$) ions in double perovskites AFe$_{1/2}$M$_{1/2}$O$_{3}$ exhibits a ferrimagnetic ordering below $T_{fe} approx 5.6J_1$ ($J_1/k_B sim 50$~K), which is close to room
temperature. Small clusters of the same structure exhibit a superparamagnetic behavior at $T lesssim T_{fe}$. The possibility of formation of such clusters explains the room-temperature (superpara)magnetism in 3$d$-metal based oxides.
We show that the magnetism of double perovskite AFe_{1/2}M_{1/2}O_3 systems may be described by the Heisenberg model on the simple cubic lattice, where only half of sites are occupied by localized magnetic moments. The nearest-neighbor interaction J_
1 is more than 20 times the next-nearest neighbor interaction J_2, the third-nearest interaction along the space diagonal of the cube being negligible. We argue that the variety of magnetic properties observed in different systems is connected with the variety of chemical ordering in them. We analyze six possible types of the chemical ordering in 2x2x2 supercell, and argue that the probability to find them in a real compound does not correspond to a random occupation of lattice sites by magnetic ions. The exchange J_2 rather than J_1 define the magnetic energy scale of most double perovskite compounds that means the enhanced probability of 1:1 short range ordering. Two multiferroic compounds PbFe_{1/2}M_{1/2}O_3 (M=Nb, Ta) are exceptions. We show that the relatively high temperature of antiferromagnetic transition is compatible with a layered short-range chemical order, which was recently shown to be most stable for these two compounds [I. P. Raevski, {em et al.}, Phys. Rev. B textbf{85}, 224412 (2012)]. We show also that one of the types of ordering has ferrimagnetic ground state. The clusters with short-range order of this type may be responsible for a room-temperature superparamagnetism, and may form the cluster glass at low temperatures.
We analyze recent measurements [R. Blinc, V. V. Laguta, B. Zalar, M. Itoh and H. Krakauer, J. Phys. : Cond. Mat., v.20, 085204 (2008)] of the electric field gradient on the oxygen site in the perovskites SrTiO3 and BaTiO3, which revealed, in agreemen
t with calculations, a large difference in the EFG for these two compounds. In order to analyze the origin of this difference, we have performed density functional electronic structure calculations within the local-orbital scheme FPLO. Our analysis yields the counter-intuitive behavior that the EFG increases upon lattice expansion. Applying the standard model for perovskites, the effective two-level p-d Hamiltonian, can not explain the observed behavior. In order to describe the EFG dependence correctly, a model beyond this usually sufficient p-d Hamiltonian is needed. We demonstrate that the counter-intuitive increase of the EFG upon lattice expansion can be explained by a s-p-d model, containing the contribution of the oxygen 2s states to the crystal field on the Ti site. The proposed model extension is of general relevance for all related transition metal oxides with similar crystal structure.
