## 3 Intermediate Band States
The data also show the band gap of \(TM^{2+}{}_{0.25}Zn_{0.75}Te\) decreases as the dopant atom is substituted by ZnTe in the alloy, yielding a long-wavelength emission or a narrow band gap from the \(TM^{2+}\) ion. The calculated band gap of ZnTe of 1.23 eV is in excellent agreement with the other theoretical value of 1.4 eV [22]. We note that the band gaps of V\({}^{2+}{}_{0.25}Zn_{0.75}Te\) and Cr\({}^{2+}{}_{0.25}Zn_{0.75}Te\) are somewhat lower than other theoretical gaps of V\({}_{0.25}Zn_{0.75}Te\) and Cr\({}_{0.25}Zn_{0.75}Te\) obtained by FP-LAPW calculations without using the two-valent chromium and vanadium [23; 24], respectively, showing that the band gaps are strongly influenced by the two-valent species. Previous studies reported that the infrared femtosecond lasers using transition metals doped II-VI materials with transition metal compositions up to 3% [25; 26; 27]. Therefore, our models with transition metals compositions of 25% cannot refer to reality. In order to compare with the experimental observations of transition metals Cr\({}^{2+}\) doped ZnTe materials with low concentration we modeled 64-atom zinc-blende ZnTe and Cr\({}^{2+}{}_{0.03}Zn_{0.97}Te\). In fact, as we will show below, the use of the 64-atom large cell allows us to prove that the ZnTe with Cr\({}^{2+}\) dopants as low as 3%, not included in the V\({}^{2+}\)- and Mn\({}^{2+}\)-doped ZnTe, is nearly identical within the four-level system as an 8-atom small cell. The reasons which we have not included doping with V\({}^{2+}\) and Mn\({}^{2+}\) for the larger cell are as following: (a) no level of Mn\({}^{2+}{}_{0.25}Zn_{0.75}Te\) in the forbidden gap of the ZnTe, showing a two-level system and limiting to the population inversion, (b) the defect levels of V\({}^{2+}{}_{0.25}Zn_{0.75}Te\) in the bottom of the forbidden gap of the ZnTe, showing that the extremely narrow band gap cannot emit in the IR laser, (c) the IR laser emitted in the small cell of Cr\({}^{2+}{}_{0.05}Zn_{0.75}Te\), suggesting that IR laser will be emitted in the large unit of Cr\({}^{2+}{}_{0.03}Zn_{0.97}Te\). According to experimental observations [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 15], it is perhaps not surprising that the II-VI compounds doped with transition metal Cr\({}^{2+}\) are mature laser materials. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Compounds** & **Lattice Constant (Å)** & **Energy Gap (eV)** \\ \hline ZnTe & 6.19 & 6.09  \\ V\({}^{2+}{}_{0.25}Zn_{0.75}Te\) & 6.20 & 6.22  \\ Cr\({}^{2+}{}_{0.25}Zn_{0.75}Te\) & 6.22 & 6.22  & 0.49 & 0.77  \\ Mn\({}^{2+}{}_{0.25}Zn_{0.75}Te\) & 6.20 & 0.68 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The lattice constants of optimized atomic structures of ZnTe and \(TM^{2+}{}_{0.25}Zn_{0.75}Te\) (\(TM\) = V, Cr, Mn) fit other theoretical values. The corresponding band gaps are compared with other calculated values. Figure 5: Schematic illustration of the normal population (**a**) and population inversion (**b**). Upward arrows show optical pumping which make electrons (solid stars) to rise from a lower energy level 0 to a higher one 3. Downward arrows indicate electrons emit photons when they relax from excited states 3 or 2 to lower states 1 or 0. The HOMO and LUMO of ZnTe and Cr\({}^{2+}\)\({}_{0.03}\)Zn\({}_{0.97}\)Te were calculated as shown in Figure 6. We can be clearly seen that \(|\)1\(>\), \(|\)0\(>\), \(|\)-1\(>\), and \(|\)-2\(>\) levels of Cr\({}^{2+}\)\({}_{0.03}\)Zn\({}_{0.97}\)Te are in the forbidden gap of the ZnTe. The fundamental gap between the conduction band and defect levels can be corrected with a \"scissor operator\" to match the photoluminescence (PL) peaks due to the well-known bandgap underestimation of DFT calculations. The calculated band gap of 0.94 eV is corrected by the scissor correction (the scissors operator is 1.63 eV) to approach the PL emission in the 2.57 eV spectral region. \(|\)0\(>\) and \(|\)-1\(>\) metastable excited states to E\({}_{0}\) ground state are scaled up to 0.49 eV and 0.46 eV, which the wavelengths are 2530 nm and 2695 nm, respectively.