## 3 Intermediate Band States  
The geometry optimizations of ZnTe and \(TM^{2+}{}_{0.25}Zn_{0.75}\)Te (\(TM\) = V, Cr, Mn) through DFT-GGA calculations were shown in Figures 2-4. We clearly see that the HOMO levels of ZnTe are localized on  
Figure 1: Atomistic representations of 8-atom ZnTe, V\({}^{2+}{}_{0.25}Zn_{0.75}\)Te, Cr\({}^{2+}{}_{0.25}Zn_{0.75}\)Te, Mn\({}^{2+}{}_{0.25}Zn_{0.75}\)Te, and 64-atom Cr\({}^{2+}{}_{0.03}Zn_{0.97}\)Te models. The atoms are represented by spheres: Zn (gray, dark), Te (yellow), V (gray, light), Cr (green), and Mn (purple). the Te atoms in the left panel of Figure 2. The LUMO isosurfaces are significantly different from the HOMO isosurfaces. The LUMO levels of ZnTe are not only around Te atoms but also around Zn atoms. In the right panel of Figure 2 it can be clearly seen that \(|1>\), \(|0>\), \(|-1>\), and \(|-2>\) levels of V\({}^{2+}\)\({}_{0.25}\)Zn\({}_{0.75}\)Te are in the bottom of the forbidden gap of the ZnTe or between the valence E\({}_{0}\) and conduction E\({}_{1}\) band of the ZnTe. The extremely narrow band gap cannot emit in the IR laser. Furthermore, the eight-atom zinc-blende ZnTe and Cr\({}^{2+}\)\({}_{0.25}\)Zn\({}_{0.75}\)Te models are shown in Figure 3 The results in the right panel of Figure 3 inform that the Cr\({}^{2+}\) ions in the ZnTe compounds lead to Jahn-Teller effects which further split the ground and excited states. It can be seen that Cr\({}^{2+}\)\({}_{0.25}\)Zn\({}_{0.75}\)Te is a good laser gain medium with the multi-level system, which the upper \(|1>\), \(|0>\), \(|-1>\), and \(|-2>\) levels may lead to a significant population inversion. Moreover, the eight-atom zinc-blende ZnTe and Mn\({}^{2+}\)\({}_{0.25}\)Zn\({}_{0.75}\)Te models are shown in Figure 4. It can be clearly seen that there is not any level of Mn\({}^{2+}\)\({}_{0.25}\)Zn\({}_{0.75}\)Te in the forbidden gap of the ZnTe. The Mn\({}^{2+}\)\({}_{0.25}\)Zn\({}_{0.75}\)Te is a two-level system, where it is hard to achieve population inversion in the two-level system. Figure 5a shows a two-level system with the ground state 0 and the excited state 3 indicating that the rate of absorption is the same as one of spontaneous emission or stimulated emission. It is hard to achieve population inversion in the two-level system. The idea laser gain medium with the multi-level system, such as four-level system as shown in Figure 5b, may lead to a significant population inversion. The stimulated emission occurs from level 2 to level 1 and is the basis of photon amplification and the basic mechanism of the laser action. Our eight-atom Cr\({}^{2+}\)\({}_{0.25}\)Zn\({}_{0.75}\)Te model is the only one of the four-level system where the population inversion could be achieved. Figure 2: Left (Right) panel: HOMO and LUMO slab isosurfaces of 8-atom ZnTe (V\({}^{2+}\)\({}_{0.25}\)Zn\({}_{0.75}\)Te) cells arranged by energy and denoted by the shade of purple. The atoms are represented by spheres: V (gray, light), Zn (gray, dark), and Te (yellow). Figure 4: Left (Right) panel: HOMO and LUMO slab isosurfaces of 8-atom ZnTe (Mn\({}^{2+}\)0.25 Zn0.75 Te) cells arranged by energy and denoted by the shade of purple. The atoms are represented by spheres: Mn (purple), Zn (gray, dark), and Te (yellow). Figure 3: Left (Right) panel: HOMO and LUMO slab isosurfaces of 8-atom ZnTe (Cr\({}^{2+}\)0.25 Zn0.75 Te) cells arranged by energy and denoted by the shade of purple. The atoms are represented by spheres: Cr (green), Zn (gray, dark), and Te (yellow). The calculated ground state structures for ZnTe and \(TM^{2+}{}_{0.25}Zn_{0.75}Te\) (\(TM\) = V, Cr, Mn) and their optical properties (band gaps) are summarized in Table 1. The lattice constants of strain relaxed lattices of V\({}^{2+}{}_{0.25}Zn_{0.75}Te\), Cr\({}^{2+}{}_{0.25}Zn_{0.75}Te\), Mn\({}^{2+}{}_{0.25}Zn_{0.75}Te\) are, respectively, 0.16%, 0.48%, and 0.16% larger than the one of ZnTe.