Magnetic, orbital and charge ordering in the electron-doped manganites

The three dimensional perovskite manganites in the range of hole-doping $x > 0.5$ are studied in detail using a double exchange model with degenerate $e_g$ orbitals including intra- and inter-orbital correlations and near-neighbour Coulomb repulsion. We show that such a model captures the observed phase diagram and orbital-ordering in the intermediate to large band-width regime. It is argued that the Jahn-Teller effect, considered to be crucial for the region $x<0.5$, does not play a major role in this region, particularly for systems with moderate to large band-width. The anisotropic hopping across the degenerate $e_g$ orbitals are crucial in understanding the ground state phases of this region, an observation emphasized earlier by Brink and Khomskii. Based on calculations using a realistic limit of finite Hund's coupling, we show that the inclusion of interactions stabilizes th e C-phase, the antiferromagnetic metallic A-phase moves closer to $x=0.5$ while th e ferromagnetic phase shrinks in agreement with recent observations. The charge ordering close to $x=0.5$ and the effect of reduction of band-width are also outlined. The effect of disorder and the possibility of inhomogeneous mixture of competing states have been discussed.