## 3 Results and Discussion
### Interface Structures
When the lattice constant is 5.4 A, the Fermi level lies at the top of spin down valence band, and when TiFeCoSi is further compressed, the Fermi level would drop into the valence band and lose its half-metallicity. With the lattice constant increases, the Fermi level gradually moves from a low energy zone to a high energy zone. As the lattice constant increases up to 6.1 A, the Fermi level locates at the bottom of spin down conduction band, and when TiFeCoSi is further stretched, the Fermi level would move into the conduction band, and half-metallicity is also destroyed. Figure 3: The band structure of Ti\({}_{2}\)CoSi. (**a**) Spin-up band structure. (**b**) Spin-down band structure. Figure 5: The electronic structure of TiFeCoSi. The atom-resolved spin magnetic moment (AMM) and total magnetic moment (TMM) of Fe doped Ti\({}_{2}\)CoSi as a function of Fe concentration is shown in Figure 7. As for pure Ti\({}_{2}\)CoSi, Ti\({}^{A}\) and Ti\({}^{B}\) contribute the main part of TMM, while Co offers a small part, due to the reason that low valence transition metal Ti atom has larger spin splitting than the high valence transition metal Co. When Fe atom doping in Ti\({}_{2}\)CoSi forms Ti\({}^{A}\)\({}_{1-x}\)Fe\({}_{x}\)Ti\({}^{B}\)CoSi, the contribution of Fe atom to the TMM is lower than Ti\({}^{A}\) atom. As the Fe doping concentration increases, the AMM of Ti\({}^{A}\), Ti\({}^{B}\), and Fe atoms lineally decline, while AMM of the Co atom is enhanced, and it is almost as high as Fe atom when the doping concentration increases to x = 1. It also can be seen that when x ranges from 0 to 0.625, the AMMs of Ti\({}^{A}\), Ti\({}^{B}\), Fe, and Co are positive values, indicating the ferromagnetic arrangement among these transition metal elements. While, when x > 0.625, the AMM of Ti\({}^{B}\) atom reverses to a negative value, leading to the fact that the doped structure changes into a ferrimagnet. Overall, the TMM lineally decreases with Fe concentration increases, and it is in good agreement with the Slater-Pauling rule, which can be written as TMM = \(Z-24\), where Z is the total number of valence electrons. Figure 6: Dependence of the half-metallic state on the lattice constant. The dotted line indicates the half-metallic state at equilibrium lattice constant. Figure 7: The calculated total magnetic moment and atom-resolved spin magnetic moments (AMMs) of the Co, Fe, Ti\({}^{A}\), and Ti\({}^{B}\) atoms of Ti\({}^{A}\)\({}_{1-x}\)Fe\({}_{x}\)Ti\({}^{B}\)CoSi as a function of impurity Fe concentration x. Based on the half-metallic TiFeCoSi, we continue to introduce the impurity, Fe atom, and impurity Fe may occupy Ti, Co, or Si atoms, forming a Ti(Fe), Co(Fe), or Si(Fe) doping structure, respectively. The formation energy of these three possible doping structures are also calculated by Equation (1). Si(Fe) and Ti(Fe) doping structures respectively exhibit high formation energies of 1.895 eV and 1.437 eV, revealing that Si and Ti are hard to be replaced by Fe atom. While Co(Fe) doping structure possesses the minimal formation energy of \(-\)0.153 eV, indicating that the Co atom is easily occupied by Fe atom and forms the TiFeCo\({}_{1-y}\)Fe\({}_{y}\)Si doping structure, where y is the concentration of the impurity Fe atom, and y = 0.125, 0.25,..., 0.875, 1. Figure 8 shows the band structures of TiFeCo\({}_{1-y}\)Fe\({}_{y}\)Si. When impurity Fe concentration y ranges from 0.125 to 0.625, spin up bands stride over the Fermi level, while there is an obvious energy gap that exists in the spin down bands. Therefore, TiFeCo\({}_{1-y}\)Fe\({}_{y}\)Si maintains its half-metallicity when the impurity Fe concentration increases from 0.125 to 0.625. As y increases to 0.75, the Fermi level drops into the spin down conduction band, and the doping structure loses its half-metallicity, and the spin polarization decreases to about 74%. In addition, with an increase of the impurity Fe concentration, the overlap degree between spin up valence band and spin down valence band increases.