Emergence of magnetic topological states in topological insulators doped with magnetic impurities

Emergence of the topological invariant and the magnetic moment in topological insulators doped with magnetic impurities is studied based on a mutual cooperation between the spin-orbit coupling of electrons and the spin exchange of these electrons with magnetic impurity moments. The mutual cooperation is realized based on the Kane-Mele model in the presence of magnetic impurities. The topological invariants and the spontaneous magnetization are self-consistently determined within the dynamical mean-field theory. We find different magnetic topological phase transitions, depending on the electron filling. At half filling an antiferromagnetic topological insulator, which exhibits the quantum spin Hall effect, exists in the phase region between the paramagnetic topological insulator and the trivially topological antiferromagnetic insulator. At quarter and three-quarter fillings, a ferromagnetic topological insulator, which exhibits the quantum anomalous Hall effect, occurs in the strong spin-exchange regime.

Figure 1

The sign structure ${\nu }_{ij}$ of the SOC term in the honeycomb lattice.

Figure 2

The electron filling $n$ and the sublattice magnetization $m_A,m_B$ via the chemical potential $\mu$ for different values of the SE at fixed SOC $\lambda =0.5$. The horizontal dotted lines indicate fillings $n=0.5,1.0,1.5$.

Figure 3

The sublattice magnetization $m_A=-m_B$ and the spin Chern number $C_s$ at half filling and fixed SOC $\lambda =0.5$.

Figure 4

The sublattice DOS for up-spin (blue solid line) and down-spin (red dotted line) at half filling and fixed SOC $\lambda =0.5$.

Figure 5

Phase diagram of the half-filling case. The abbreviations AFMI, AFMTI, and PMTI denote the topologically trivial AFM insulator, AFM topological insulator, and PM topological insulator, respectively.

Figure 6

The sublattice magnetization $m_A=m_B$ and the band gap $\mathrm{\Delta }$ at quarter filling and fixed SOC $\lambda =0.5$.

Figure 7

The sublattice DOS for up-spin (blue solid line) and down-spin (red dotted line) at quarter filling and fixed SOC $\lambda =0.5$.

Figure 8

Phase diagram of the quarter-filling case. The abbreviations FMTI, FM M, and PM M denote the FM topological insulator, FM metal, and PM metal, respectively.