## 3 Results and Discussion
### Bulk Properties of VFeScZ (Z = Sb, As, P)
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Structure** & **Type** & \(\mathbf{E}\) & \(\mathbf{L}\) & \(\mathbf{M_{\text{V}}}\) & \(\mathbf{M_{\text{Fe}}}\) & \(\mathbf{M_{\text{Sc}}}\) & \(\mathbf{M_{\text{Sb}}}\) & \(\mathbf{M_{\text{Total}}}\) \\ \hline \multirow{3}{*}{VFeScSb} & I & \(-\)4270.94176 & 6.34 & 3.18 & \(-\)0.12 & 0.16 & \(-\)0.20 & 3.02 \\  & II & \(-\)4269.79299 & 6.38 & 2.08 & 0.48 & 0.54 & \(-\)0.12 & 2.98 \\  & III & \(-\)4270.39456 & 6.44 & 0.96 & 2.32 & \(-\)0.34 & 0.06 & 3.00 \\ \hline \multirow{3}{*}{VFeScAs} & I & \(-\)4292.08041 & 6.08 & 2.96 & 0.08 & 0.14 & \(-\)0.18 & 3.00 \\  & II & \(-\)4291.1535 & 6.14 & 2.32 & 0.02 & 0.60 & \(-\)0.14 & 2.80 \\  & III & \(-\)4292.05693 & 6.19 & \(-\)0.16 & 2.34 & \(-\)0.18 & 0.18 & 2.18 \\ \hline \multirow{3}{*}{VFeScP} & I & \(-\)4299.87526 & 5.93 & 2.82 & 0.22 & 0.10 & \(-\)0.16 & 2.98 \\  & II & \(-\)4298.93401 & 5.96 & 1.32 & 0.36 & 0.38 & \(-\)0.04 & 2.02 \\ \cline{1-1}  & III & \(-\)4299.84489 & 6.05 & \(-\)0.04 & 2.22 & \(-\)0.14 & 0.12 & 2.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ground state energy (\(E\), eV), lattice constant (\(L\), Å), and magnetic moment (\(M\), \(\mu_{\text{B}}\)) of type-I, -II, and -III structures of VFeScZ (Z = Sb, As, P). Figure 3: Total energy as a function of lattice constant in the Heusler alloy (**a**) VFeScSb, (**b**) VFeScAs, and (**c**) VFeScP with type-I, -II, and -III structures. Figure 2: Type-I structure of the quaternary alloy VFeScZ (Z = Sb, As, P). found that the total magnetic moment of the type-I alloy is 2.98 \(\mu_{\text{B}}\) and close to 3 \(\mu_{\text{B}}\), which abides by the Slater-Pauling rule. In type-II and -III structures, the total magnetic moments are 2.02 \(\mu_{\text{B}}\) and 2.16 \(\mu_{\text{B}}\), respectively, which is obviously a deviation from the Slater-Pauling rule and behave like the VFeScAs alloy. Here, in order to understand the relationship between the electronic structure and the geometric structure we carefully analyzed the DOS of VFeScZ (Z = Sb, As, P) type-I, -II, and -III structures after geometric optimization. As shown in Figure 4a for the VFeScSb alloy when it has type-I structure, two band gaps (spin-up and spin-down) can be clearly seen and the Fermi surface is located in the middle of the band gap, which is a typical semiconductor behavior. It can be basically judged as a typical ferromagnetic semiconductor combined with the previous magnetic analysis. For the type-II structure, although the spin-down band gap still exists, the spin-up state crosses the Fermi surface, which is a typical half-metallic property. And for the type-III structure, the spin-up and -down bands of the state density cross the Fermi surface, which is a general ferromagnetic alloy. In Figure 4b, when the alloy VFeScAs is in type-I structure, we found that it has a relatively wide spin-up gap. The Fermi level just locates at the edge of the spin-down gap, which can only be roughly judged as a semiconductor or half-metal. For the type-II and -III structures, the spin-up and -down bands cross the Fermi level, respectively, and are therefore ferromagnetic metals. In the VFeScP system, see Figure 4c, it is similar to the VFeScAs. The alloy with type-I structure is a narrow-band gap semiconductor, while with type-II or -III structure it is a common ferromagnetic alloy. The electronic properties of VFeScZ alloy need to be discussed in detail, which because of the type-I structure is the ground state configuration of the VFeScZ alloy. Figure 5 shows the total DOS and partial density of states (PDOS) of VFeScZ (Z = Sb, As, P). It can be found that the total DOS of VFeScZ in the low energy region from \(-\)6 eV to \(-\)3 eV is mainly contributed from the 2p electrons of Z atom and partially from the 3d electrons of V and Sc atom. In the high energy region from \(-\)3 eV to 0 eV, the total DOS mainly comes from the 3d electrons of Fe atom, V atom, and Sc atom. At and near the Fermi surface, the contribution of V and Fe atoms is prominent.