Tight-binding theory of spin-spin interactions, Curie temperatures, and quantum Hall effects in topological (Hg,Cr)Te in comparison to non-topological (Zn,Cr)Te, and (Ga,Mn)N

Earlier theoretical results on $p$-$d$ and $d$-$d$ exchange interactions for zinc-blende semiconductors with Cr$^{2{+}}$ and Mn$^{3{+}}$ ions are revisited and extended by including contributions beyond the dominating ferromagnetic (FM) superexchange term [i.e., the interband Bloembergen-Rowland-Van Vleck contribution and antiferromagnetic (AFM) two-electron term], and applied to topological Cr-doped HgTe and non-topological (Zn,Cr)Te and (Ga,Mn)N in zinc-blende and wurtzite crystallographic structures. From the obtained values of the $d$-$d$ exchange integrals $J_{ij}$, and by combining the Monte-Carlo simulations with the percolation theory for randomly distributed magnetic ions, we determine magnitudes of Curie temperatures $T_{\text{C}}(x)$ for $\mathrm{Zn}_{1-x}\mathrm{Cr}_x\mathrm{Te}$ and $\mathrm{Ga}_{1-x}\mathrm{Mn}_x\mathrm{N}$ and compare to available experimental data. Furthermore, we find that competition between FM and AFM $d$-$d$ interactions can lead to a spin-glass phase in the case of $\mathrm{Hg}_{1-x}\mathrm{Cr}_x\mathrm{Te}$. This competition, along with a relatively large magnitude of the AF $p$-$d$ exchange energy $N_0\beta$ can stabilize the quantum spin Hall effect, but may require the application of a tilted magnetic field to observe the quantum anomalous Hall effect in HgTe quantum wells doped with Cr.