Fig. 1.  (Color online) Schematic pictures of magnetic semiconductors with (a) wide band gaps and (b) narrow band gaps. The band gap is $ E_{\rm g} $. The top of valence band (VB) is dominated by p orbitals, and the bottom of conduction band (CB) is dominated by s orbitals. For the impurity with d orbitals, $ \epsilon_{d} $ is impurity level of d orbitals, and $ U $ is the on-site Coulomb interaction. Impurity bound state (IBS) is also developed due to the doping of impurity into the host. The density of state (DOS) as a function of energy, and the magnetic correlation $ \langle M_1^zM_2^z\rangle $ between two impurities as a function of the chemical potential $ \mu $ are depicted. (a) Due to strong mixing between the impurity and the VB, the position of the IBS $ \omega_{\rm{IBS}} $ (arrow) is close to the top of the VB. Due to weak mixing between the impurity and the CB, usually no IBS appears below the bottom of the CB^[19–21]. Thus, we have 0 $ \lesssim $ $ \omega_{\rm{IBS}} \ll E_{\rm g} $ for the wide band gap case. By the condition $ \mu \sim \omega_{\rm{IBS}} $, positive (FM coupling) $ \langle M_1^zM_2^z\rangle $ can develop^[22–24]. For p-type carriers ($ \mu\sim 0 $), ferromagnetic coupling can be obtained as the condition $ \mu $ $ \sim $ $ \omega_{\rm{IBS}} $ can be satisfied. For n-type carriers ($ \mu \sim E_{\rm g} $), no magnetic coupling is obtained between impurities because the condition $\mu \sim \omega_{\rm{IBS}}$ cannot be satisfied^[19–21]. (b) Case for narrow band gap $ E_{\rm g} $. By choosing suitable host semiconductors and impurities, the condition 0 $ \lesssim $ $ \omega_{\rm{IBS}} $ $ \lesssim $ $ E_{\rm g} $ can be obtained. For both p-type and n-type carriers, ferromagnetic coupling can be obtained because the condition $ \mu $ $ \sim $ $ \omega_{\rm{IBS}} $ is satisfied.

Fig. 2.  (Color online) Band structure of the ZnO host with wurtzite, zincblende, and rocksalt crystal structures. Adapted from Ref. [19].

Fig. 3.  (Color online) For Mn impurity in ZnO, hybridization parameter $ V_{\xi\alpha }({k}) $ of a Mn $ \xi $ orbital with the valence bands and the conduction bands. Adapted from Ref. [19].

Fig. 4.  (Color online) For Mn impurity in ZnO, square of the magnetic moment at the impurity site $ \langle(M^z)^2\rangle $ as a function of the chemical potential $ \mu $. The top of valence is energy zero, and the bottom of the conduction band is noted as vertical dashed lines. Adapted from Ref. [19].

Fig. 5.  (Color online) For Mn impurity in ZnO, impurity-impurity magnetic correlation function $ \langle M^z_{1}M^z_{2}\rangle $ as a functino of distance $ R $ between two impurities for the wurtzite, zincblende, and rocksalt structures. $ a $ is lattice constant. Adapted from Ref. [19].

Fig. 6.  (Color online) For N impurity in MgO, host band and hybridization. (a) MgO bands structure, where an direct band gap of 7.5 eV was obtained. Hybridization between 2$ {p} $ orbitals of N and (b) valence bands and (c) conduction bands of MgO. Adapted from Ref. [21].

Fig. 7.  (Color online) For N impurity in MgO, square of magnetic moment $ \langle(M_{\xi}^z)^2\rangle $ as a function of chemical potential $ \mu $. Adapted from Ref. [21].

Fig. 8.  (Color online) For N impurity in MgO, impurity-impurity magnetic correlation $ \langle M_{1\xi}^zM_{2\xi}^z\rangle $ as a function of distance $ R $, for the impurity level (a) $ \epsilon_{p} $ = –$ 1.5 $ eV and (b) $ \epsilon_{p} $ = –$ 0.5 $ eV. Adapted from Ref. [21].

Fig. 9.  (Color online) For Mn impurity in BaZn₂As₂, host band and impurity-host hybridization. (a) Energy bands off host BaZn₂As₂. Band gap of 0.2 eV was obtained by DFT calculations, consistent with experiment^[11]. The hybridization parameter between the 3d orbitals of Mn and (b) valence bands and (c) conduction bands of BaZn₂As₂. Adapted from Ref. [17].

Fig. 10.  (Color online) For Mn impurity in BaZn$ _2 $As$ _2 $, chemical potential $ \mu $ dependence of (a) impurity occupation number $ \langle n_{\xi}\rangle $ of $ \xi $, and (b) magnetic correlation $ \langle M_{1\xi}^zM_{2\xi}^z\rangle $ between impurities of the first-nearest neighbor. The band gap of 0.2 eV is noted by dash lines. Adapted from Ref. [17].

Fig. 11.  (Color online) For Mn impurities in BaZn$ _2 $As$ _2 $, magnetic correlation $ \langle M_{1\xi}^zM_{2\xi}^z\rangle $ as a function of distance $ R $. (a) Chemical potential is set as $ \mu $ = –0.3 eV to model p-type case. (b) It is set as $ \mu $ = 0.15 eV for n-type case. The first, second, and third nearest neighbors of $ R $ are noted. Adapted from Ref. [17].

Fig. 12.  (Color online) Cr impurity versus Mn impurity in host BaZn$ _2 $As$ _2 $. Chemical potential $ \mu $ dependence of (a) impurity occupation number $ \langle n_{\xi}\rangle $, and (b) magnetic correlation $ \langle M_{1\xi}^zM_{2\xi}^z\rangle $ between impurities of the 1st nearest neighbor. Adapted from Ref. [18].

Fig. 13.  (Color online) For Cr impurity in BaZn$ _2 $As$ _2 $, magnetic correlation $ \langle M_{1\xi}^zM_{2\xi}^z\rangle $ as a function of the distance $ R $. (a) chemical potential is set as $ \mu $ = –0.1 eV to model p-type case. (b) It is set as $ \mu $ = 0.15 eV for n-type case. The first, second, and third nearest neighbors of $ R $ are noted. Adapted from Ref. [18].

Fig. 14.  (Color online) Crystal structure of two-dimensional Cr$ _2 $Ge$ _2 $Se$ _6 $.

Fig. 15.  (Color online) Electron band structure of two-dimensional Cr$ _2 $Ge$ _2 $Se$ _6 $, obtained by the density functional theory calculations. Adapted from Ref. [39].

Fig. 16.  (Color online) For two-dimensional Cr$ _2 $Ge$ _2 $Se$ _6 $ with different tensile strains, the normalized magnetization as a function temperature. Adapted from Ref. [39].

Fig. 17.  (Color online) Crystal structure of two-dimensional PtBr$ _3 $.

Fig. 18.  (Color online) The band structure of two-dimensional PdBr$ _3 $, where Chern number C of the nontrivial band near Fermi energy $ E_{\rm F} $ is indicated, and the band gap is $ E_{\rm g} $ = 28.1 meV. The result is obtained by the density functional theory calculation. Adapted from Ref. [40].

Fig. 19.  (Color online) For two-dimensional PtBr$ _3 $,temperature dependence of the normalized magnetic moment obtained by the Monte Carlo simulation and the density functional theory calculation. Adapted from Ref. [40].