A generic dual d-band model for interlayer ferromagnetic coupling in a transition-metal doped MnBi ₂ Te ₄ family of materials

Realization of ferromagnetic (FM) interlayer coupling in magnetic topological insulators (TIs) of the MnBi₂Te₄ family of materials (MBTs) may pave the way for realizing the high-temperature quantum anomalous Hall effect (high-T QAHE). Here we propose a generic dual d-band (DDB) model to elucidate the energy difference (ΔE = E_AFM - E_FM) between the AFM and FM coupling in transition-metal (TM)-doped MBTs, where the valence of TMs splits into d-t_2g and d-e_g sub-bands. Remarkably, the DDB shows that ΔE is universally determined by the relative position of the dopant (X) and Mn d-e_g/t_2g bands, . If ΔE_d > 0, then ΔE > 0 and the desired FM coupling is favored. This surprisingly simple rule is confirmed by first-principles calculations of hole-type 3d and 4d TM dopants. Significantly, by applying the DDB model, we predict the high-T QAHE in the V-doped Mn₂Bi₂Te₅, where the Curie temperature is enhanced by doubling of the MnTe layer, while the topological order mitigated by doping can be restored by strain.