Magnetic interaction with the gapless surface states in topological insulator (TI) has been predicted to give rise to a few exotic quantum phenomena. However, the effective magnetic doping of TI is still challenging in experiment. Using first-princip
les calculations, the magnetic doping properties (V, Cr, Mn and Fe) in three strong TIs (Bi$_{2}$Se$_{3}$, Bi$_{2}$Te$_{3}$ and Sb$_{2}$Te$_{3}$) are investigated. We find that for all three TIs the cation-site substitutional doping is most energetically favorable with anion-rich environment as the optimal growth condition. Further our results show that under the nominal doping concentration of 4%, Cr and Fe doped Bi$_{2}$Se$_{3}$, Bi$_{2}$Te$_{3}$, and Cr doped Sb$_{2}$Te$_{3}$ remain as insulator, while all TIs doped with V, Mn and Fe doped Sb$_{2}$Te$_{3}$ become metal. We also show that the magnetic interaction of Cr doped Bi$_{2}$Se$_{3}$ tends to be ferromagnetic, while Fe doped Bi$_{2}$Se$_{3}$ is likely to be antiferromagnetic. Finally, we estimate the magnetic coupling and the Curie temperature for the promising ferromagnetic insulator (Cr doped Bi$_{2}$Se$_{3}$) by Monte Carlo simulation. These findings may provide important guidance for the magnetism incorporation in TIs experimentally.
