Magnetic structure of Fe$_5$O$_6$: group-theoretical analysis and DFT calculations

The magnetic structure of Fe$_5$O$_6$ is studied using a combination of the group-theoretical analysis and $\mathrm{DFT}+U$ calculations of the electronic spectrum. The calculations are performed for the magnetic $\mathbf{k} = (0, 0, 0)$ vector. The magnetic ground state corresponds to the orthogonal ordering of two magnetic subsystems:  the magnetic moments of Fe$^{2+}$/Fe$^{3+}$ ions located at the octahedral sites (slabs of octahedra) are directed along the $c$ axis and are antiferromagnetically ordered, whereas the magnetic moments of Fe$^{2+}$ ions in trigonal prisms forming one-dimensional chains are directed along the $b$ axis and are antiferromagnetically coupled along the $c$ axis. The formation of a nonzero antiferromagnetic component of magnetic moments in the slabs of octahedra directed along the $b$ axis is caused by the effect of magnetic chains on the three-dimensional magnetic structure.