The quantum anomalous Hall effect (QAHE), originates form a combination of the spin-orbital coupling and the breaking of time-reversal symmetry due to intrinsic ferromagnetic ordering, was recently observed in Cr-and V-doped magnetic topological insulators (TI). However, it was only observed at extremely low temperatures due to the low ferromagnetic Curie temperature and tiny magnetically-induced gap. To fully understand the mechanism of the ferromagnetic ordering whereby improving the ferromagnetism, we investigated 4f transition-metal-doped Bi 2 Se 3 , using density-functional-theory approaches. We predict that Eu and Sm can introduce stable long-range ferromagnetic states in Bi 2 Se 3 , with large magnetic moments and low impurity disorders .Additionally, codoping is proposed to tune the Fermi level into the gap, which simultaneously improve the magnetic moment and the incorporation of magnetic ions. Our findings thus offer a critical step in facilitating the realization of QAHE in TI systems.PACS numbers: 71.20.Nr, 61.72.U-, 75.50.Pp
I. INTRODUCTIONThe Quantum Hall effect (QHE), characterized by the quantized Hall conductance, with the formation of Laudau levels under an external magnetic field, was observed in two-dimensional electron systems more than 30 years ago[1,2]. As first proposed by Haldane[3], by circulating currents loop on a honeycomb lattice, the QHE can exist even without the external magnetic field and the formation of Laudau levels, namely the Quantum anomalous Hall effect (QAHE). However, the Haldane model is difficult to realize experimentally. Later, numerous models were proposed to realize QAHE in realistic materials, including magnetically doped quantum wells [4][5][6][7], transition metal oxide heterostructures [8][9][10][11][12][13], grapheme /silicene [14][15][16][17][18][19], magnetic topological insulator (TI) thin films [20-26], etc. [27-29]. However, the QAHE was only reported to be successful in Cr-and V-doped (Bi,Sb) 2 Te 3 magnetic TI thin films [30][31][32][33][34][35][36].The key idea for the realization of QAHE is to introduce the time-reversal symmetry (TRS) breaking perturbations, e.g. a ferromagnetic order in topological insulators, usually by doping the TI with magnetic impurities [30][31][32][33][34][35][36]. The introduced exchange splitting can destroy the band inversion in one of the spin channels, whilst keeping the nontrivial band topology in the other spin channel due to the spin-orbital coupling, which opens up a finite gap at the Dirac point and lead to the QAHE [37]. The reason for chose the TI system, especially the Bi 2 Se 3 family which has been proved successful in QAHE, is that besides its strong SOC and a single-surface Dirac cone, the ferromagnetic orders can be derived directly from the large Van Vleck spin susceptibility in the host without the mediation of itinerant carriers [24], which eventually makes the QAHE being possible. However, the QAHE observed from recent experimental observations typically occurs at extremely low temperatures because of t...