  By using the first-principles electronic structure calculations, we have
systematically studied the magnetism in three recently synthesized iron-based
antiperovskite chalco-halides: Ba$_3$(FeS$_4$)Cl, Ba$_3$(FeS$_4$)Br, and
Ba$_3$(FeSe$_4$)Br. These compounds consist of edge-sharing Ba$Q_6$ ($Q$=Cl or
Br) octahedra intercalated with isolated Fe$X_4$ ($X$=S or Se) tetrahedra. We
find that even though the shortest distances between the nearest-neighboring Fe
atoms in these three compounds already exceed 6 \AA, much larger than the bond
length 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 a
new type of exchange interaction between the nearest-neighboring Fe-based
super-moments mediated by Ba cations, which we call the super-super exchange
interaction, in which each magnetic Fe atom partially polarizes its four
nearest-neighboring $X$ atoms to form a super-moment through $p$-$d$ orbital
hybridization and the $X$ atoms in neighboring Fe$X_4$ tetrahedra along $b$
axis antiferromagnetically couple with each others through the intermediate Ba
cations. Different from the conventional superexchange, here it is cations
rather than anions that mediate two neighboring super-moments. According to the
calculated strength of the AFM coupling, we predict that among these compounds
the highest AFM phase transition temperature $T_N$ may reach 110 K in
Ba$_3$(FeSe$_4$)Br, in comparison with the observed $T_N$s of 84 K in
Ba$_3$(FeS$_4$)Br and 95 K in Ba$_3$(FeS$_4$)Cl.
