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Earlier theoretical results on $p$-$d$ and $d$-$d$ exchange interactions forzinc-blende semiconductors with Cr$^{2{+}}$ and Mn$^{3{+}}$ ions are revisitedand extended by including contributions beyond the dominating ferromagnetic(FM) superexchange term [i.e., the interband Bloembergen-Rowland-Van Vleckcontribution and antiferromagnetic (AFM) two-electron term], and applied totopological Cr-doped HgTe and non-topological (Zn,Cr)Te and (Ga,Mn)N inzinc-blende and wurtzite crystallographic structures. From the obtained valuesof the $d$-$d$ exchange integrals $J_{ij}$, and by combining the Monte-Carlosimulations 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 availableexperimental 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 arelatively large magnitude of the AF $p$-$d$ exchange energy $N_0\beta$ canstabilize the quantum spin Hall effect, but may require the application of atilted magnetic field to observe the quantum anomalous Hall effect in HgTequantum wells doped with Cr.

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