field level. In Fig.4(a), (b) and (c), we plot the lowest four sub-band levels at Γ point as a function of exchange field. The level crossings (indicated by arrows) between the lowest conduction bands (blue lines) and valence bands (red lines) are found in all the three Bi$_{2}$Se$_{3}$ thin film systems with different layer thickness. The corresponding Hall conductance can be obtained using the following Kubo formula $^{3}$, with E$_{nk}$ being the dispersion of the quantum well sub-bands and ˆ v$_{x/y}$ being the velocity operators. In the insulator case, where the chemical potential is located inside the energy gap between conduction and valence sub-bands, the Hall conductance is determined by the first Chern number of the occupied bands and must be an exact integer in the unit of e $^{2}$/h . The calculated Hall conductance for Bi$_{2}$Se$_{3}$ thin film with three typical thickness are plotted in figure 4 (d) as function of exchange field, where a jump from ”0” to ”1” in the Hall conductance are observed at the corresponding critical exchange field for the level crossing. In Fig.5, we plot the sub-band dispersion and corresponding Hall conductance as function of chemical potential for 3 and 5 QL Bi$_{2}$Se$_{3}$ thin films with different exchange fields. We can find that the change induced by the exchange field is mainly in the area near Γ point. For the QAH phases ((d) and (h)), non-zero integer plateau is observed when the chemical potential falls into the energy gap. The difference between the 3 and 5 QL cases suggests that the QAH plateau is much narrower for thick film due to the narrower energy gap coming from the intersection of top and bottom surface states. In conclusion we have shown that magnetic dopants Cr and Fe form long range magnetic order in the insulating state of the Bi$_{2}$Se$_{3}$ family of topological insulators. The ferromagnetic order lead to a topologically non-trivial electronic structure with quantized Hall conductance without any external magnetic field. In real samples, a small amount of bulk carriers could always be present, however, if the concentration is low enough, they will be localized by disorder in two dimensions, and will not affect the precise quantization of the Hall plateau. The edge states of our QAH state conducts charge current without any dissipation in the absence of any external magnetic field. This effect could enable a new generation of low power electronic devices. 11 σ$_{xy}$ = e $^{2}$¯ h ∑ nmk Im [ 〈 nk | ˆ v$_{x}$ | mk 〉 〈 mk | ˆ v$_{y}$ | nk 〉 ] ( E$_{nk}$ − E$_{mk}$ ) 2 ( n$_{f}$ ( E$_{nk}$ ) − n$_{f}$ ( E$_{mk}$ )) (7)