Magnetism of $A$-site-ordered perovskites, CaCu${}_3$Ti${}_4$O${}_{12}$, CaCu${}_3$Ge${}_4$O${}_{12}$, CaCu${}_3$Sn${}_4$O${}_{12}$, and CaCu${}_3$Zr${}_4$O${}_{12}$, is comprehensively studied by means of ab initio electronic structure calculations. The magnetic exchange coupling constants between Cu-site spins, $J_1$, $J_2$, and $J_3$, are estimated within an effective Heisenberg model, revealing relative importance of $J_3$ despite its long interaction length. The ground-state magnetic order is reasonably explained by the combination of (i) relatively weak ferromagnetic superexchange interaction ($J_1$) that works for all the oxides and (ii) antiferromagnetic long-range superexchange interaction ($J_3$) that works selectively for CaCu${}_3$Ti${}_4$O${}_{12}$ and CaCu${}_3$Zr${}_4$O${}_{12}$.

• Received 1 April 2013

DOI:https://doi.org/10.1103/PhysRevB.87.224430