Double-exchange via degenerate orbitals

We consider the double-exchange for systems in which doped electrons occupy degenerate orbitals, treating the realistic situation with double degenerate $e_g$ orbitals. We show that the orbital degeneracy leads in general to formation of anisotropic magnetic structures and that in particular, depending on the doping concentration, the layered magnetic structures of the A-type and chain-like structures of the C-type are stabilized. The phase-diagram that we obtain provides an explanation for the experimentally observed magnetic structures of some over-doped (electron-doped) manganites of the type Nd${}_{1-x}$Sr${}_x$MnO${}_3$, Pr${}_{1-x}$Sr${}_x$MnO${}_3$ and Sm${}_{1-x}$Ca${}_x$MnO${}_3$ with $x>0.5$.