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We theoretically investigate the exchange interaction between magneticimpurities in ultrathin Bi$_2$Se$_3$ topological insulator films by taking intoaccount the low-energy states produced by the impurities. We find that thelocally induced impurity resonances strongly influence the exchange interactionbetween magnetic moments. In particular, we find a non-collinear alignmentbeing more favorable than the collinear ferromagnetic alignment preferred whenimpurity states are ignored and only the pristine topological insulator bandstructure is considered. Moreover, we show that by applying of an electricfield perpendicular to the ultrathin film, the exchange interaction can bedrastically enhanced. This opens for the possibility of highly tunablemagnetism by electric field.

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