Features of the electronic structure of the Bi$_2$Se$_3$ topological insulator digitally doped with $3d$ transition metals

The electronic and optical spectra of the Bi$_2$Se$_3$ topological insulator digitally doped with V, Cr, Mn, Fe, and Co atoms are calculated using density functional theory. It is shown that the introduction of magnetic atoms into separate delta layers (one per 2–9 Bi$_2$Se$_3$ quintuple layers) multiply enhances magnetic effects. A special emphasis is put on Mn doping, which gives rise to the ferromagnetic spin ordering. The sensitivity of the spin order to the concentration and location of magnetic atoms is revealed. The study of an analytical model describing the resonant scattering of electrons in Bi$_2$Se$_3$ by atomic layers of transition metals also suggests the existence of spin-polarized states within the Bi$_2$Se$_3$ band gap. Our ab initio calculations show that transitions between the nested branches of the electronic spectrum, which exist near the Fermi level, are responsible for features of the optical conductivity at $\hbar\omega\approx0.15{-}0.3\,$eV, the infrared plasmon, and the Kerr angle $\theta _K>12^\circ$ in the infrared spectral range.