Realization of stable ferromagnetic order in a topological insulator: Codoping-enhanced magnetism in 4 f transition metal doped B i 2 S e 3

The quantum anomalous Hall effect (QAHE) originates from a combination of the spin-orbital coupling and the breaking of time-reversal symmetry due to intrinsic ferromagnetic ordering and was recently observed in Cr and V doped magnetic topological insulators (TIs). However, it was only observed at extremely low temperatures due to the low ferromagnetic Curie temperature and the tiny magnetically induced gap. To fully understand the mechanism of the ferromagnetic ordering, thereby improving the ferromagnetism, we investigated $4f$ transition metal doped $\mathrm{B}{\mathrm{i}}_{2}\mathrm{S}{\mathrm{e}}_{3}$, using density functional theory approaches. We predict that Eu and Sm can introduce stable long-range ferromagnetic states in $\mathrm{B}{\mathrm{i}}_{2}\mathrm{S}{\mathrm{e}}_{3}$, with large magnetic moments and low impurity disorders. Additionally, codoping is proposed to tune the Fermi level into the gap, which simultaneously improves the magnetic moment and the incorporation of magnetic ions. Our findings, thus, offer a step in facilitating the realization of QAHE in TI systems.