  Magnetically doped topological insulators enable the quantum anomalous Hall
effect (QAHE) which provides quantized edge states for lossless charge
transport applications. The edge states are hosted by a magnetic energy gap at
the Dirac point but all attempts to observe it directly have been unsuccessful.
The gap size is considered crucial to overcoming the present limitations of the
QAHE, which so far occurs only at temperatures one to two orders of magnitude
below its principle limit set by the ferromagnetic Curie temperature $T_C$.
Here, we use low temperature photoelectron spectroscopy to unambiguously reveal
the magnetic gap of Mn-doped Bi$_2$Te$_3$ films, which is present only below
$T_C$. Surprisingly, the gap turns out to be $\sim$90 meV wide, which not only
exceeds $k_BT$ at room temperature but is also 5 times larger than predicted by
density functional theory. By an exhaustive multiscale structure
characterization we show that this enhancement is due to a remarkable structure
modification induced by Mn doping. Instead of a disordered impurity system, it
forms an alternating sequence of septuple and quintuple layer blocks, where Mn
is predominantly incorporated in the septuple layers. This self-organized
heterostructure substantially enhances the wave-function overlap and the size
of the magnetic gap at the Dirac point, as recently predicted. Mn-doped
Bi$_2$Se$_3$ forms a similar heterostructure, however, only a large,
nonmagnetic gap is formed. We explain both differences based on the higher
spin-orbit interaction in Bi$_2$Te$_3$ with the most important consequence of a
magnetic anisotropy perpendicular to the films, whereas for Bi$_2$Se$_3$ the
spin-orbit interaction it is too weak to overcome the dipole-dipole
interaction. Our findings provide crucial insights for pushing the lossless
transport properties of topological insulators towards room-temperature
applications.
