We choose the concentration of the magnetic ion to be 5%, the total angular momentum of a single magnetic ion to be J = 3 / 2 and the effective exchange coupling J$_{eff}$ = 2 . 0 eV , which are in the same order with the quantitative first-principles calculations (as described later). The above mean field equations have been solved self-consistently, and the ordered moment as well as Curie temperature can be calculated. In Fig.1(b), the calculated magnetization versus temperature are plotted for both bulk material and thin film system with 3, 4, 5 quintuple layers (QL) thicknesses. The Curie temperature for the bulk system is around 70K, which can be well compared with previous experimental studies $^{28–30}$. The ferromagnetic ordering is anisotropic, and the Curie transition is of the Ising type, with a finite transition temperature in 2D. Our mean field calculation also indicates that for the thin film system, the Curie temperature is only slightly reduced due to the finite size effect, which give rise to the possible QAHE as discussed below. In this section, we shall show, by parameter-free first principles calculations, that the insulating magnetic ground state discussed above can be indeed obtained by proper choice of magnetic dopants. It has been suggested by experiments 28–30 that the magnetic dopants, such as Ti, V, Cr and Fe, will mostly substitute the Bi ions (or the Sb ions in the case of Sb$_{2}$Te$_{3}$), we therefore concentrate on this situation in the following discussions. Since the nominal valence of Bi (or Sb) ions are 3+, a general rule is to find a transition metal element which may have a stable 3+ chemical state, so that no free carriers are introduced by this iso-electronic substitution. In addition, due to the coexistence of orbital and spin degrees of freedom of magnetic dopants (due to the partially filled d -shells), we need a mechanism to quench these degrees of freedom and stabilize the insulating state. We shall show that such conditions can be satisfied by the combination of crystal field splitting and large Hund’s rule coupling for Cr and Fe dopants. We performed self-consistent first-principle calculations 33 for Bi$_{2}$Se$_{3}$ doped with various transition metal elements, Ti, V, Cr and Fe. We use a large supercell (of size 3 × 3 × 1) with one of the Bi atoms substituted by a magnetic dopant. The crystal structure and the magnetic state are fully optimized to obtain the ground state. The results shown in Fig.2 8 IV. ELECTRONIC STRUCTURES OF BI$_{2}$SE$_{3}$ DOPED WITH TRANSITION METAL ELEMENTS