## 3 Results and Discussion
### Electronic Structures and Magnetic Properties
Figure 4: The energy of conduction band minimum (CBM) and valence band maximum (VBM) of the spin-down channel as a function of (**a**) the lattice constant and (**b**) the \(c\)/\(a\) ratio for Ti\({}_{2.25}\)Co\({}_{0.75}\)Si. Figure 3: Spin-resolved electronic band structures of Ti\({}_{2.25}\)Co\({}_{0.75}\)Si (**a**) at an equilibrium lattice constant and (**b**) with \(c\)/\(a\) = 1.12 lattice distortion. The zero of the energy scale is set at the Fermi energy. The blue and red solid lines represent spin-up and spin-down states, respectively. In the following, we substituted half of the silicon with B, Al, Ga, P, As, and Sb atoms in the cubic cell of Ti\({}_{2.25}\)Co\({}_{0.75}\)Si compound. The spin-resolved electronic band structures of the investigated compounds in the equilibrium lattice parameters were computed and are presented in Figures 5 and 6. For Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)B\({}_{0.5}\), Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Al\({}_{0.5}\), and Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Ga\({}_{0.5}\), although the Fermi level shifts into a low region of energy with the decrease of the valence electrons, the majority-spin electrons still pass through the Fermi level and the minority-spin bands evince semiconducting characteristics with a band gap, indicating that they are also half metallic. In particular, we found that for Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)B\({}_{0.5}\) and Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Ga\({}_{0.5}\), the VBM of the spin-down channel touched the Fermi level at T point, demonstrating two quasi-gapless half metals. We also found that the minority spin band gap of Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)B\({}_{0.5}\) was smaller than that of Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Al\({}_{0.5}\) and Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Ga\({}_{0.5}\). The reason for this may be that the atomic radius of the boron atom is relatively small, which leads to a certain chemical pressure, and then reduces the band gap. For the cases of Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)P\({}_{0.5}\), Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)As\({}_{0.5}\), and Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Sb\({}_{0.5}\), the spin-up bands open a seamless gap, and the Fermi level is located at this zero-gap as a result of the increase of valence electrons. Generally speaking, spin-gapless semiconductors can be divided into four categories, as follows: (i) one spin channel is semiconducting and the other spin channel is gapless; (ii) one spin channel is semiconducting and the VBM touches the Fermi level, while the other spin channel is gapless; (iii) there is a gap for both spin-up and spin-down channels, nonetheless, both the VBM of the spin-up channel and the CBM of the spin-down channel are in contact with the Fermi level; and (iv) one spin channel is gapless, and for the other spin channel, the Fermi level touches the edge of the conduction bands. For Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)P\({}_{0.5}\), the spin-up channel is gapless and the CBM touches the Fermi level in the spin-down channel. Thus, it belongs to the fourth type. For Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)As\({}_{0.5}\) and Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Sb\({}_{0.5}\), the spin-up channel is gapless, while the spin-down channel is semiconducting. As a consequence, they should belong to the first type. Nevertheless, for Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)As\({}_{0.5}\), the CBM of the down spins stand too close to the Fermi level. Thus, the singular electromagnetic transport effect can only be observed at very low temperatures. For Ti\({}_{2.25}\)Co\({}_{0.75}\)Si\({}_{0.5}\)Sb\({}_{0.5}\), there is also a very small gap in the spin-up channel. Thereby, strictly speaking, it would be a spin-polarized narrow band-gap semiconductor. However, at a finite temperature, the compound would present the typical electromagnetic transport phenomenon of the spin-gapless semiconductor.