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A topological insulator doped with random magnetic impurities is studied. Thesystem is modelled by the Kane-Mele model with a random spin exchange betweenconduction electrons and magnetic dopants. The dynamical mean field theory fordisordered systems is used to investigate the electron dynamics. The magneticlong-range order and the topological invariant are calculated within the meanfield theory. They reveal a rich phase diagram, where different magneticlong-range orders such as antiferromagnetic or ferromagnetic one can exist inthe metallic or insulating phases, depending on electron and magnetic impurityfillings. It is found that insulator only occurs at electron half filling,quarter filling and when electron filling is equal to magnetic impurityfilling. However, non-trivial topology is observed only in half-fillingantiferromagnetic insulator and quarter-filling ferromagnetic insulator. Atelectron half filling, the spin Hall conductance is quantized and it is robustagainst magnetic doping, while at electron quarter filling, magnetic dopantsdrive the ferromagnetic topological insulator to ferromagnetic metal. Thequantum anomalous Hall effect is observed only at electron quarter filling anddense magnetic doping.

Impact of magnetic dopants on magnetic and topological phases in magnetic topological insulators