We present a first-principles study of the electronic, magnetic, and transport properties of the topological insulator Bi$_{2}$Te$_{3}$ doped with Mn atoms in substitutional (Mn$_{rm Bi}$) and interstitial van der Waals gap positions (Mn$_{rm i}$), which act as acceptors and donors, respectively. The effect of native Bi$_{rm Te}$- and Te$_{rm Bi}$-antisite defects and their influence on calculated electronic transport properties is also investigated. We have studied four models representing typical cases, namely (i) Bi$_{2}$Te$_{3}$ with and without native defects, (ii) Mn$_{rm Bi}$ defects with and without native defects, (iii) the same but for Mn$_{rm i}$ defects, and (iv) the combined presence of Mn$_{rm Bi}$ and Mn$_{rm i}$. It was found that lattice relaxations around Mn$_{rm Bi}$ defects play an important role for both magnetic and transport properties. The resistivity is strongly influenced by the amount of carriers, their type, and by the relative positions of the Mn-impurity energy levels and the Fermi energy. Our results indicate strategies to tune bulk resistivities, and also help to uncover the location of Mn atoms in measured samples. Calculations indicate that at least two of the considered defects have to be present simultaneously in order to explain the experimental observations, and the role of interstitials may be more important than expected.
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