  Doping Bi$_2$Se$_3$ by magnetic ions represents an interesting problem since
it may break the time reversal symmetry needed to maintain the topological
insulator character. Mn dopants in Bi$_2$Se$_3$ represent one of the most
studied examples here. However, there is a lot of open questions regarding
their magnetic ordering. In the experimental literature different Curie
temperatures or no ferromagnetic order at all are reported for comparable Mn
concentrations. This suggests that magnetic ordering phenomena are complex and
highly susceptible to different growth parameters, which are known to affect
material defect concentrations. So far theory focused on Mn dopants in one
possible position, and neglected relaxation effects as well as native defects.
We have used ab initio methods to calculate the Bi$_2$Se$_3$ electronic
structure influenced by magnetic Mn dopants, and exchange interactions between
them. We have considered two possible Mn positions, the substitutional and
interstitial one, and also native defects. We have found a sizable relaxation
of atoms around Mn, which affects significantly magnetic interactions.
Surprisingly, very strong interactions correspond to a specific position of Mn
atoms separated by van der Waals gap. Based on the calculated data we performed
spin dynamics simulations to examine systematically the resulting magnetic
order for various defect contents. We have found under which conditions the
experimentally measured Curie temperatures ${T_{\rm{C}}}$ can be reproduced,
noticing that interstitial Mn atoms appear to be important here. Our theory
predicts the change of ${T_{\rm{C}}}$ with a shift of Fermi level, which opens
the way to tune the system magnetic properties by selective doping.
