LAUSR

The layered magnetic topological insulator $\mathrm{MnB}{\mathrm{i}}_{2}\mathrm{T}{\mathrm{e}}_{4}$ is a promising platform to realize the quantum anomalous Hall effect because its layers possess intrinsic ferromagnetism. However, it is not well understood why the high-spin ${d}^{5}$ magnetic ions $\mathrm{M}{\mathrm{n}}^{2+}$ forming the Mn-Te-Mn spin exchange paths prefer ferromagnetic (FM) coupling, contrary to the prediction of the Goodenough-Kanamori rule that a TM-$L$-TM spin exchange where TM and $L$ are a transition-metal magnetic cation and a main group ligand, respectively, is antiferromagnetic (AFM) even when the bond angle of the exchange path is 90\ifmmode^\circ\else\textdegree\fi{}. Using density functional theory calculations, we show that the presence of $\mathrm{B}{\mathrm{i}}^{3+}$ ions is essential for the FM coupling in $\mathrm{MnB}{\mathrm{i}}_{2}\mathrm{T}{\mathrm{e}}_{4}$. Then, using a tight-binding model Hamiltonian, we find that high-spin ${d}^{5}$ ions ($S=5/2$) in TM-$L$-TM spin exchange paths prefer FM coupling if the empty $p$ orbitals of a nonmagnetic cation $M$ (e.g., $\mathrm{B}{\mathrm{i}}^{3+}$ ion) hybridize strongly with those of the bridging ligand $L$ but AFM coupling otherwise.