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The interplay between magnetic and topological order can give rise tophenomena such as the quantum anomalous Hall effect. The extension oftopological surface states into magnetic insulators (MIs) has been proposed asan alternative to using intrinsically magnetic topological insulators (TIs).Here, we theoretically study how this extension of surface states into amagnetic insulator are influenced both by the interface barrier potentialseparating a topological insulator and a magnetic insulator and by finite sizeeffects in such structures. We find that the the gap in the surface statesdepends non-monotonically on the barrier strength. A small, but finite, barrierpotential turns out to be advantageous as it permits the surface states topenetrate even further into the MI. Moreover, we find that due to finite sizeeffects in thin samples, increasing the spin-splitting in the MI can actuallydecrease the gap of the surface states, in contrast to the usual expectationthat the gap opens as the spin-splitting increases.

Barrier and finite size effects on the extension of topological surface-states into magnetic insulators