Momentum-resolved spin splitting in Mn-doped trivial CdTe and topological HgTe semiconductors.

A series of experiments demonstrated that exchange-induced splitting of magnetooptical spectra of Cd1-xMnxTe at the L points of the Brillouin zone is, unexpectedly, more than one order of magnitude smaller compared to its magnitude at the zone center and show an unexpected sign of the effective Lande factor. We have determined spin-splitting of the valence and conduction bands in the whole Brillouin zone in Cd1-xMnxTe and in topologically-nontrivial Hg1-xMnxTe by means of relativistic first-principles density functional calculations. We find that spin splitting of bands is relatively large at the L points but, in contrast to the {\Gamma} point, effective exchange integrals have the same sign and similar magnitudes at the L points. This results in comparable energies of the optical transitions for two circular light polarizations leading to small splitting of optical spectra in Cd1-xMnxTe. Our results substantiate also previous suggestions that the antiferromagnetic sign and a relatively high magnitude of the effective exchange integral in the conduction band away from the {\Gamma} point results from an admixture of anion p-type wave functions and the proximity to upper Hubbard band of Mn d electrons. We use the obtained results to determine parameters of a minimal tight-binding model that describe rather accurately the band structure, including spin-orbit and exchange splittings of bands in the whole Brillouin zone of Cd1-xMnxTe and Hg1-xMnxTe.