Highly tunable magnetic coupling in ultrathin topological insulator films due to impurity resonances

We theoretically investigate the exchange interaction between magnetic impurities in ultrathin Bi$_2$Se$_3$ topological insulator films by taking into account the low-energy states produced by the impurities. We find that the locally induced impurity resonances strongly influence the exchange interaction between magnetic moments. In particular, we find a non-collinear alignment being more favorable than the collinear ferromagnetic alignment preferred when impurity states are ignored and only the pristine topological insulator band structure is considered. Moreover, we show that by applying of an electric field perpendicular to the ultrathin film, the exchange interaction can be drastically enhanced. This opens for the possibility of highly tunable magnetism by electric field.