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Topological insulators doped with magnetic impurities has become a promisingcandidate for Quantum Anomalous Hall Effect (QAHE) in the dilute doping limit.The crucial factor in realizing the QAHE in these systems is the spontaneousFerromagnetic (FM) ordering between the doped magnetic atoms. Hence,understanding the magnetic exchange interaction between the magnetic atomsbecomes essential. In this work, we use the Density functional theory (DFT) andMagnetic force theorem (MFT) to calculate the magnetic exchange interactionbetween magnetic impurities (V, Cr, Mn, Fe) in the host Bi2Se3. Through anorbital decomposition of the calculated exchange, we can identify the natureand origin of the exchange mechanism that depends on the type of magneticatoms, doping concentration, host material etc. Our results show that Cr dopingresults in an insulating state, a prerequisite for the QAHE, that remainsrobust against doping concentration and local correlation. In this case, theshort-ranged superexchange and long-ranged exchange via the p-orbitals of thehost results in an FM order. For other doped systems (V, Mn and Fe doped),their electronic configuration and local octahedral environment open thepossibility of finite carrier density at the Fermi energy. Depending on thetype of this carrier (electron/hole) and their localized/delocalized nature, ashort-ranged double exchange / long-ranged RKKY mechanism could occur betweenthe magnetic atoms.

Revisiting magnetic exchange interactions in transition metal doped Bi$_2$Se$_3$ using DFT+MFT