## 3 Results
### Electronic, Magnetic, and Half-Metallic Properties  
In this part, the electronic and magnetic structure of FeRhCrSi and FePdCrSi at their equilibrium lattice constants have been discussed in detail. The calculated band structures of FeRhCrSi and FePdCrSi compounds have been displayed in Figure 4. From Figure 4a,b, it can be found that both FeRhCrSi and FePdCrSi compounds show quite similar band structure characters. In the majority-spin channel, the Fermi level is located in an indirect band gap. The conduction band minimum (CBM) appears at the X point and the valence band maximum (VBM) occurs at the G point of the Brillouin zone. The values of the indirect band gaps in the majority-spin channel are 0.336 eV and 0.177 eV for EQH compounds FeRhCrSi and FePdCrSi, respectively. However, their bands show a metallic cross over at the Fermi level in the minority-spin channel. This results in their 100% spin polarization. Thus, the EQH compounds FeRhCrSi and FePdCrSi are HMs. Figure 5a,b show the calculated total and partial density of states (DOS) of FeRhCrSi and FePdCrSi at their equilibrium lattice constants. From Figure 5, one can further understand their electronic structures. It can be seen that there is a metallic overlapping with the Fermi level in the minority-spin channel for both the EQH compounds FeRhCrSi and FePdCrSi. By contrast, in the majority-spin direction, the Fermi level locates in a relative wide energy gap (the half-metallic gap). Therefore, the EQH compounds FeRhCrSi and FePdCrSi are HMs that are consistent with the above discussion of their band structures. For FeRhCrSi and FePdCrSi compounds, because of the similar atomic configuration, they present two quite similar total and partial DOS. Here, we take the FeRhCrSi compound as an example and give a detailed discussion on the DOS. From Figure 5a, it can be found that the DOS is mainly dominated by the Si atom in the lower energy region (\(-\)7 eV-5 eV) for the two spin directions. In the energy region \(-\)5 eV to \(-\)3 eV, the DOS are mainly occupied by Rh-4d states. In the range of \(-\)3 eV to \(-\)1 eV, except a small contribution of Rh-4d and Cr-3d states, both spin channels are dominated by Fe-3d states. In the energy region around the Fermi level (\(-\)1 eV to 1 eV), the energy peak is the result of the common contribution of Fe, Rh, and Cr atoms with d orbitals. It can be observed that, in the minority spin, the main contribution to the DOS is the strong hybridization among the Fe-3d state, Rh-4d state, and Cr-3d state around the Fermi level. In the majority spin, the energy gap mainly comes from the covalent hybridization between the nearest neighbor atoms: Fe and Cr. [PERSON] et al. [34] have reported that the formation of the bonding and antibonding states can arise from the strong covalent hybridization between the lower-energy d states (higher-valent) transition metal atom (such as Fe) and the higher-energy d states (lower-valent) transition metal (such as Cr). And then, the half-metallic band gap generates between the bonding and antibonding bands. It has also been reported that, due to the strong intra-atomic exchange interaction, the low valence transition metal atom usually has a large spin splitting [35]. From Figure 5a, it can be found that, compared with Fe and Rh atoms, the Cr atom with a lower valent has a larger spin splitting. Thus, the Cr atom is the main contributor to the total magnetic moment of the FeRhCrSi compound. Figure 3: Energy optimization with different magnetic states of (**a**) FeRhCrSi and (**b**) FePdCrSi. Figure 4: The band structures of (**a**) FeRhCrSi and (**b**) FePdCrSi at their equilibrium lattice constants. The total and partial magnetic moments at the equilibrium lattice constant are listed in Table 2. The total magnetic moments (M\({}_{\rm{t}}\)) are integers for both FeRhCrSi and FePdCrSi compounds, which are 3 \(\upmu\)B and 4 \(\upmu\)B, respectively.