Ferromagnetic interactions between transition-metal impurities in topological and 3D Dirac semimetals

The magnitude of ferromagnetic coupling driven by inter-band (Bloembergen-Rowland - BR) and intra-band (Ruderman-Kittel-Kasuya-Yoshida - RKKY) spin polarization is evaluated within $kp$ theory for topological semimetals Hg$_{1-x}$Mn$_x$Te and Hg$_{1-x}$Mn$_x$Se as well as for 3D Dirac semimetal (Cd$_{1-x}$Mn$_x$)$_3$As$_2$. In these systems Mn$^{2+}$ ions do not introduce any carriers. Since, however, both conduction and valence bands are built from anion $p$-type wave functions, hybridization of Mn $d$ levels with neighboring anion $p$ states leads to spin-dependent $p-d$ coupling of both electrons and holes to localized Mn spins, resulting in sizable inter-band spin polarization and, thus in large BR interactions. We demonstrate that this ferromagnetic coupling, together with antiferromagnetic superexchange, elucidate a specific dependence of spin-glass freezing temperature on $x$, determined experimentally for these systems. Furthermore, by employing a multi-orbital tight-binding method, we find that superexchange becomes ferromagnetic when Mn is replaced by Cr or V. Since Cr should act as an isoelectronic impurity in HgTe, this opens a road for realization of ferromagnetic topological insulators based on (Hg,Cr)Te.