## 3 Results and Discussion
### Electronic Structures and Magnetic Properties
Taking Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)B\({}_{0.5}\) and Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)P\({}_{0.5}\), for example, we discussed the orbital-resolved spin polarization. The total density of the states (DOS) and partial DOS for Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)B\({}_{0.5}\) and Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)P\({}_{0.5}\) obtained using GGA, are presented in Figure 7. The DOS around the Fermi level is heavily dominated by the 3d states of the Ti and Co atoms. It was found that the 3d states of Co are mainly distributed from \(-1\) to \(-3\) eV, while the \(3d\) states of Ti are mainly distributed around and beyond the Fermi level. The electronic configurations of the Ti and Co atoms are 3d\({}^{2}\)4s\({}^{2}\) and 3d\({}^{7}\)4s\({}^{2}\), respectively. Thus, the Ti atom with less than half of the filling valence electron shell is mainly distributed at high energy levels, and the Co atom with more than half the filling valence electron shell is mainly distributed at low energy levels. It was also found that the impurity atoms play a key role in adjusting the valence electron concentration of the compound, and shift the Fermi level upwards or downwards. For Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)B\({}_{0.5}\), the acceptor doping moves the Fermi level downwards, and thus the Fermi level is submerged in the deeper energy region. For Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)P\({}_{0.5}\), the donor doping moves the Fermi level upwards, and thus the Fermi level is able to be pulled out of the filled bands and is located at the nearly zero band gap. Figure 5: Spin-resolved electronic band structures of (**a**) Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)B\({}_{0.5}\), (**b**) Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Al\({}_{0.5}\), and (**c**) Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Ga\({}_{0.5}\) at the equilibrium lattice parameters. The zero of the energy scale is set at the Fermi energy. The blue and red solid lines represent the spin-up and spin-down states, respectively. The results show that the spin-up and spin-down states around the Fermi level are predominantly derived from the Ti-d and Co-d states, and the d-d hybridization is the main origin for spin polarization. In order to understand this matter intuitively, we drew the spin density plots of Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)B\({}_{0.5}\) and Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)P\({}_{0.5}\) (see Figure 8), which are defined as the difference of spin-up and spin-down states. We can see that the spin density predominantly concentrates on the Ti and Co atoms, evincing that the spin polarization is mainly from the Ti and Co atoms. For these compounds, the magnetic moments are predominantly due to the Ti-d and Co-d electrons. The large exchange splitting of the Ti-d and Co-d states leads to a large magnetic moment. Figure 6: Spin-resolved electronic band structures of (**a**) Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)P\({}_{0.5}\), (**b**) Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)As\({}_{0.5}\), (**c**) and Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Sb\({}_{0.5}\) at the equilibrium lattice parameters. The zero of the energy scale is set at the Fermi energy. The blue and red solid lines represent the spin-up and spin-down states, respectively.