Origin of magnetic inhomogeneity in Cr- and V-doped topological insulators

The quantum anomalous Hall effect (QAHE) has been experimentally observed in magnetically doped topological insulators at ultralow temperatures. Inhomogeneous ferromagnetism is considered to be one of the main factors that lead to the unexpected low QAHE observation temperature. Dopant-induced disorder is usually the origin of the inhomogeneous ferromagnetism. Here, our systematic first-principles calculations demonstrate that inhomogeneous mixing of Bi and Sb in a ${(\mathrm{Bi},\mathrm{Sb})}_{2}{\mathrm{Te}}_{3}$ system is the intrinsic origin of inhomogeneous ferromagnetism. Different from diluted magnetic semiconductors, the mixing energy of Cr and V in ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, ${\mathrm{Sb}}_{2}{\mathrm{Te}}_{3}$, and ${(\mathrm{Bi},\mathrm{Sb})}_{2}{\mathrm{Te}}_{3}$ topological insulators clearly show that magnetic dopants are homogeneously distributed even in the presence of naturally formed crystalline defects. Surprisingly, our study shows that the mixing energies of Sb and Bi in a ${(\mathrm{Bi},\mathrm{Sb})}_{2}{\mathrm{Te}}_{3}$ system are all positive in the whole range of doping concentration, indicating that Bi elements in a ${(\mathrm{Bi},\mathrm{Sb})}_{2}{\mathrm{Te}}_{3}$ system are inhomogeneously distributed. Moreover, the formation energies of Cr and V suggest that they are relatively easy to substitute Bi sites in Bi inhomogeneously distributed ${(\mathrm{Bi},\mathrm{Sb})}_{2}{\mathrm{Te}}_{3}$ systems, which leads to inhomogeneous ferromagnetism of the experimental QAHE system. The influence of the inhomogeneous distribution of Bi on the electronic structures of bulk ${(\mathrm{Bi},\mathrm{Sb})}_{2}{\mathrm{Te}}_{3}$ systems is also analyzed. We believe that our finding of the intrinsic origin of magnetic inhomogeneity should be beneficial for the experimental enhancement of the QAHE observation temperature in magnetically doped topological insulators.