## 3 Results and Discussion
### Electronic Structures and Magnetic Properties  
The electronic band structures of the Ti\({}_{2.25}\)Co\({}_{0.75}\)Si alloy at its equilibrium lattice constant were calculated and are shown in Figure 3a. The spin-up and spin-down electronic bands are indicated by blue and red lines, respectively. When we replaced 25% cobalt with titanium in a cubic cell, a Ti\({}_{2.25}\)Co\({}_{0.75}\)Si alloy was achieved. It can be seen that the spin-up direction shows a metallic behavior and the spin-down direction evinces a semiconducting behavior with an indirect band gap of 0.22 eV around the Fermi level. The valence band maximum (VBM) and conduction band minimum (CBM) are located at R and M points, respectively. Thus, Ti\({}_{2.25}\)Co\({}_{0.75}\)Si is a half-metallic ferromagnetic material with an integer magnetic moment of 7 \(\mu_{\rm B}\) (see Table 1). The calculated values of P were 98% for Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)B\({}_{0.5}\) and Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Ga\({}_{0.5}\), and 96% for Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)P\({}_{0.5}\), and 100% for the other studied compounds in this work, where P = \(\left[N_{\uparrow}(E_{\rm F})-N_{\downarrow}(E_{\rm F})\right]/\left[N_{ \uparrow}(E_{\rm F})+N_{\downarrow}(E_{\rm F})\right]\), and \(N_{\uparrow,\downarrow}(E_{\rm F})\) are the spin-dependent density of states at the Fermi level [39; 40; 41]. It is also important to study the effect of the possible lattice distortion during the process of film deposition on the electronic structures. The influences of compression and tetragonalization on the magnetic properties of the Ti\({}_{2.25}\)Co\({}_{0.75}\)Si compound were investigated. Figure 4a presents the position of the CBM and VBM of the spin-down channel as a function of the lattice constant. Upon compressing the lattice strain, the half-metallic properties can be kept between a range from 5.78 A to 6.35 A, indicating that the half-metallic characteristics of Ti\({}_{2.25}\)Co\({}_{0.75}\)Si can be preserved when the lattice constants are changed by \(-\)4.9%-4.4% in relation to the equilibrium lattice constant. Figure 4b presents the position of the CBM and VBM of the spin-down channel as a function of the _c/a_ ratio. Under the influence of tetragonalization, we found that the alloy experiences the transformation from a magnetic metal to a traditional half metal to a gapless half metal, and finally to a magnetic metal. For a Ti\({}_{2.25}\)Co\({}_{0.75}\)Si compound, a gapless half metal can be obtained when the lattice parameter, c, is expanded by 12%. The corresponding electronic band structures (_c/a_ = 1.12) are plotted in Figure 3b. It can be seen that majority-spin electrons pass through the Fermi level, while the CBM and VBM of the minority spin just touch the Fermi level at R and A points, respectively. Figure 2: Crystal cell of Ti\({}_{2}\)CoSi with (**a**) regular cubic Heusler structure and (**b**) an inverse cubic Heusler structure. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Compound** & \(a_{\mathbf{o}}\) & \(\Delta\)_E_ & \(\mu_{\text{tot}}\) \\ \hline Ti\({}_{2.25}\)Co\({}_{0.75}\)Si & 6.08 & 1.171 & 7.00 \\ Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)B\({}_{0.5}\) & 6.01 & 0.607 & 5.00 \\ Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Al\({}_{0.5}\) & 6.13 & 0.592 & 5.00 \\ Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Ga\({}_{0.5}\) & 6.13 & 0.673 & 5.00 \\ Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)P\({}_{0.5}\) & 6.04 & 0.705 & 8.91 \\ Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)As\({}_{0.5}\) & 6.12 & 0.830 & 9.00 \\ Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Sb\({}_{0.5}\) & 6.25 & 0.963 & 9.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The optimized equilibrium lattice parameter, \(a_{\mathbf{o}}\) (Å); the energy difference \(\Delta\)_E_ (eV) between anti-ferromagnetic (AFM) and ferromagnetic (FM) states; and the total magnetic moment \(\mu_{\text{tot}}\) (\(\mu_{\text{B}}\)) of the studied compounds.