## 3 Results and Discussion  
### Interface Structures  
The electronic structure of Ti\({}_{2}\)CoSi is firstly calculated. In Figure 2, there is a large energy gap of 0.671 eV exists in spin down band, and the Fermi level locates at the top of the gap. Such an energy gap is a result of exchange splitting between spin down unoccupied antibonding bands (which are localized at Co, Ti\({}^{A}\) and Ti\({}^{B}\) atoms) and spin down occupied bands (which are predominantly of Co character). As for spin up band, it exhibits an obvious zero-width gap around the Fermi level. It also can be seen from Figure 3 that the maximum of the valence band sits at \(\Gamma\) point, while the minimum of the conduction bands locates at X point; such a closed spin up gap, therefore, is an indirect gap. When Fe atom is introduced into Ti\({}_{2}\)CoSi, it could lead to four doping structures, thus, X(Fe) doping structure where X atom (X = Ti\({}^{A}\), Ti\({}^{B}\), Co, Si) is substituted by an Fe atom. In order to determine which doping structure is more favorable, the formation energy is calculated by the following equation \(E_{f}=E^{\prime}-E-\sum\limits_{i}n_{i}\mu_{i}\), where \(E_{f}\) is formation energy, and \(E^{\prime}\) and \(E\) are total energy of the doping structure and undoping structure, respectively. The integer \(n_{i}\) is the number of atoms that has been removed from (\(n_{i}\) is negative value) or added to (\(n_{i}\) is positive value) to form the disorders, and \(\mu_{i}\) is the corresponding chemical potential which represents the energy of the reservoirs. According to our calculation, the highest \(E_{f}\) of 1.028 eV belongs to Ti\({}^{B}\)(Fe) doping structure, and that of Co(Fe) and Si(Fe) are also as high as \(-\)0.023 eV and \(-\)0.018 eV, respectively, while the minimal \(E_{f}\) of \(-\)0.927 eV occurs in Ti\({}^{A}\)(Fe) doping structure. It reveals that Ti\({}^{B}\), Co, and Si atoms are hard to be replace by Fe atom, however, Ti\({}^{A}\) atom could be easily replaced with Fe atom. Therefore, we focus on the Ti\({}^{A}\)\({}_{1-x}\)Fe\({}_{x}\)Ti\({}^{B}\)CoSi doping structure, where the doping concentration x = 0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, and 1. Figure 2: The electronic structure of Ti\({}_{2}\)CoSi. In Figure 4, when the Fe atom intervenes in Ti\({}_{2}\)CoSi and the doping concentration increases, the zero-width gap in spin up states is destroyed, and the spin up band strides over the Fermi level and shows metallic behavior. On the other hand, doping Fe atom produces a negative impact on the spin down energy gap, causing the width of the gap to vanish, and the Fermi level drops into a spin down conduction band. As a result, the SGS character of Ti\({}_{2}\)CoSi is completely destroyed by the impurity Fe atom. When the doping concentration x = 0.875, the top of the spin down valence band and the bottom of spin down conduction band touch at Fermi level, and form an indirect closed spin down gap. The doping structure, Ti\({}^{\text{A}}\)\({}_{0.125}\)Fe\({}_{0.875}\)Ti\({}^{\text{B}}\)CoSi, is a gapless half-metal. As the doping concentration increases up to x = 1, the closed spin down gap is opened, while the spin up states still cross the Fermi level. Therefore, when x = 1, the structure converts to a quaternary Heusler alloy TiFeCoSi with half-metallic character. It can be seen from Figure 5 that TiFeCoSi possesses a wide half-metallic energy gap of about 0.56 eV, and the Fermi level is located slightly above the middle of the gap and, hence, it is predicted to have stable half-metallicity. Both spin down occupied bonding bands and spin down unoccupied antibonding bands are mainly localized by Co and Fe atoms. Figure 6 shows the half-metallicity of TiFeCoSi as a function of the lattice constant, and it holds the half-metallic energy gap when its lattice constant increases from 5.4 A to 6.1 A.