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 $1\times 1\times 1$ orthorhombic or monoclinic structure of K${}_{1.6}$Mn${}_8$O${}_{16}$.