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The insulating iron compounds Eu$_2$SrFe$_2$O$_6$ and Sr$_3$Fe$_2$O$_4$Cl$_2$have high-temperature antiferromagnetic (AF) order despite their differentlayered structures. Here we carry out density functional calculations and MonteCarlo simulations to study their electronic structures and magnetic propertiesaided with analyses of the crystal field, magnetic anisotropy, andsuperexchange. We find that both compounds are Mott insulators and in thehigh-spin (HS) Fe$^{2+}$ state ($S$ = 2) accompanied by the weakened crystalfield. Although they have different local coordination and crystal fields, theFe$^{2+}$ ions have the same level sequence and ground-state configuration$(3z^2-r^2)^2(xz,yz)^2(xy)^1(x^2-y^2)^1$. Then, the multiorbital superexchangeproduces strong AF couplings, and the $(3z^2-r^2)/(xz,yz)$ mixing via thespin-orbit coupling (SOC) yields a small in-plane orbital moment andanisotropy. Indeed, by tracing a set of different spin-orbital states, ourdensity functional calculations confirm the strong AF couplings and the easyplanar magnetization for both compounds. Moreover, using the derived magneticparameters, our Monte Carlo simulations give the N\'eel temperature $T_{\rm N}$= 420 K (372 K) for the former (the latter), which well reproduce theexperimental results. Therefore, the present study provides a unified picturefor Eu$_2$SrFe$_2$O$_6$ and Sr$_3$Fe$_2$O$_4$Cl$_2$ concerning their electronicand magnetic properties.

Spin-Orbital States and Strong Antiferromagnetism of Layered Eu$_2$SrFe$_2$O$_6$ and Sr$_3$Fe$_2$O$_4$Cl$_2$