Spin-charge-orbital ordering in hollandite-type manganites studied by model Hartree-Fock calculation

We investigate spin-charge-orbital ordering in a Mn$^{3+}$/Mn$^{4+}$ mixed valence state on a hollandite-type lattice using unrestricted Hartree-Fock calculation on a multi-band Mn 3$d$-O 2$p$ lattice model. The calculations show that the Mn$^{3+}$-Mn$^{4+}$ double exchange interaction, the Mn$^{3+}$-Mn$^{3+}$ and Mn$^{4+}$-Mn$^{4+}$ superexchange interactions are ferromagnetic and play important roles to stabilize the charge and orbital ordering pattern. The most stable charge and orbital ordering pattern is consistent with the $1\times1\times1$ orthorhombic or monoclinic structure of K$_{1.6}$Mn$_8$O$_{16}$.