  The quantum anomalous Hall effect (QAHE) has recently been reported to emerge
in magnetically-doped topological insulators. Although its general
phenomenology is well established, the microscopic origin is far from being
properly understood and controlled. Here we report on a detailed and systematic
investigation of transition-metal (TM)-doped Sb$_2$Te$_3$. By combining density
functional theory (DFT) calculations with complementary experimental
techniques, i.e., scanning tunneling microscopy (STM), resonant photoemission
(resPES), and x-ray magnetic circular dichroism (XMCD), we provide a complete
spectroscopic characterization of both electronic and magnetic properties. Our
results reveal that the TM dopants not only affect the magnetic state of the
host material, but also significantly alter the electronic structure by
generating impurity-derived energy bands. Our findings demonstrate the
existence of a delicate interplay between electronic and magnetic properties in
TM-doped TIs. In particular, we find that the fate of the topological surface
states critically depends on the specific character of the TM impurity: while
V- and Fe-doped Sb$_2$Te$_3$ display resonant impurity states in the vicinity
of the Dirac point, Cr and Mn impurities leave the energy gap unaffected. The
single-ion magnetic anisotropy energy and easy axis, which control the magnetic
gap opening and its stability, are also found to be strongly TM
impurity-dependent and can vary from in-plane to out-of-plane depending on the
impurity and its distance from the surface. Overall, our results provide
general guidelines for the realization of a robust QAHE in TM-doped
Sb$_2$Te$_3$ in the ferromagnetic state.
