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By using the first-principles electronic structure calculations, we havesystematically studied the magnetism in three recently synthesized iron-basedantiperovskite chalco-halides: Ba$_3$(FeS$_4$)Cl, Ba$_3$(FeS$_4$)Br, andBa$_3$(FeSe$_4$)Br. These compounds consist of edge-sharing Ba$Q_6$ ($Q$=Cl orBr) octahedra intercalated with isolated Fe$X_4$ ($X$=S or Se) tetrahedra. Wefind that even though the shortest distances between the nearest-neighboring Featoms in these three compounds already exceed 6 \AA, much larger than the bondlength of a chemical bonding, they all remarkably show antiferromagnetic (AFM)coupling along $b$ axis with very weak spin-spin correlation along $a$ axis.Our study shows that the mechanism underlying this novel AFM coupling is such anew type of exchange interaction between the nearest-neighboring Fe-basedsuper-moments mediated by Ba cations, which we call the super-super exchangeinteraction, in which each magnetic Fe atom partially polarizes its fournearest-neighboring $X$ atoms to form a super-moment through $p$-$d$ orbitalhybridization and the $X$ atoms in neighboring Fe$X_4$ tetrahedra along $b$axis antiferromagnetically couple with each others through the intermediate Bacations. Different from the conventional superexchange, here it is cationsrather than anions that mediate two neighboring super-moments. According to thecalculated strength of the AFM coupling, we predict that among these compoundsthe highest AFM phase transition temperature $T_N$ may reach 110 K inBa$_3$(FeSe$_4$)Br, in comparison with the observed $T_N$s of 84 K inBa$_3$(FeS$_4$)Br and 95 K in Ba$_3$(FeS$_4$)Cl.

The super-super exchange mechanism in iron-based antiperovskite chalco-halides