## 3 Results
### Strain Effect on the Electronic and Magnetic Properties  
In fact, the half-metallic character of HMs is very sensitive to external conditions (such as the temperature and train effect) in its application on spintronic devices. The change of the lattice constants can destroy the half-metallicity. Thus, the band structures and magnetism of FeRhCrSi and FePdCrSi under the strain effect (including hydrostatic strain and tetragonal deformation) have been calculated to test the thermal expansion and external strain effect on the half-metallicity of them. Here, the conduction band minimum (CBM) and the valence band maximum (VBM) as functions of the lattice constant and c/a ratio in the majority spin channel were calculated and plotted to describe the change of the half-metallic behavior of the FeRhCrSi and FePdCrSi compounds (as shown in Figures 6 and 7). Figure 6 shows the CBM and VBM values under different hydrostatic strain for FeRhCrSi and FePdCrSi. From Figure 6, it can be obviously found that the half-metallicity of FeRhCrSi and FePdCrSi compounds can be kept in wide lattice constants that range from 5.28 A-5.85 A for FeRhCrSi and 5.61 A-5.92 A for FePdCrSi, respectively. This indicates that these two EQHs can keep their half-metallic properties in the lattice distortion range of \(-\)9.3% to 0.5% and \(-\)4.4% to 1% (relative to their equilibrium lattice constants). For FeRhCrSi and FePdCrSi compounds, it can be clearly seen that its half-metallic property is very robust to the lattice constant compression. Their half-metallicity cannot be destroyed until the lattice constant is compressed to 5.28 A (by \(-\)9.3%) for FeRhCrSi and 5.61 A (by \(-\)4.4%) for FePdCrSi, their conduction bands have an overlapping with the Fermi level at 5.28 A and 5.61 A. However, the half-metallicity of FeRhCrSi is relatively sensitive to the lattice constant expansion and their half-metallic properties will be destroyed when their lattice constants are expanded by about 0.5% and 1% for FeRhCrSi and FePdCrSi compounds, respectively. Figure 7a,b show the VBM and CBM values under different c/a ratios in the range of 0.96-1.10 and 0.95-1.04 for FeRhCrSi and FePdCrSi, respectively. In this case, the cubic unit cells are compressed or stretched into a tetragonal one and the volumes of them are kept unchanged. As seen in Figure 7, it seems that their half-metallic properties are also robust to the tetragonal deformation of lattice constants for FeRhCrSi and FePdCrSi compounds. In detail, FeRhCrSi and FePdCrSi compounds can keep their half-metallic properties when the c/a ratios change from 0.99 to 1.08 and 0.97 to 1.02, respectively. As the c/a ratios of FeRhCrSi and FePdCrSi decreases or increases from 1.0, their CBM value decrease and the value of VBM increases. Finally, this leads to the reduction or disappearance of their half-metallic band gaps. In order to further observe the change of the half-metallic bang gap with the tetragonal distortion, several majority-spin band structures around Fermi level at different c/a ratios for FeRhCrSi compound have been shown in Figure 8. From Figure 8, it can be clearly seen that the half-metallic band gap of FeRhCrSi decreases with tetragonal distortion and disappears at c/a = 0.97 and c/a = 1.02. Figure 7: The CBM and VBM in the majority spin channel as functions of c/a ratios for (**a**) FeRhCrSi and (**b**) FePdCrSi compounds. In the next section, we will give a detailed discussion about the effect of the hydrostatic strain and tetragonal deformation on the magnetism of FeRhCrSi and FePdCrSi compounds. In Figures 9 and 10, the curves of the total and partial magnetic moments as functions of lattice constants and c/a ratios for FeRhCrSi and FePdCrSi are given. Clearly, one can observe that the total and partial magnetic moments of FeRhCrSi and FePdCrSi present quite a similar behavior with the change of the lattice constants and c/a ratios.