## 3 Results and Discussion
### Electronic Structure: Magnetic Moments and DOS  
The quaternary Heusler compound TiZrCoIn with the configuration mentioned above showed half-metallic property and a broad indirect spin-down gap of 0.914 eV at the equilibrium lattice constant of 6.562 A. This result fits perfectly with the previous prediction in Reference [39]. Lattice parameters, spin polarizations, width of spin-down gap, and total magnetic moments in a unit cell are listed in Table 1. The spin polarizations at the Fermi levels of all the doped structures were 100%. Thus, all the derived compounds were half-metal. In addition, the magnetic moments increased from 2 u\({}_{\text{B}}\) to 3 u\({}_{\text{B}}\) at increased doping concentration. Before the detailed analysis of doping functional mechanism in quaternary Heusler TiZrCoIn, we initially discuss the ideal case of the compound TiZrCoIn. The spin-polarization total DOS and the atom PDOS are presented in the first line of Figures 1 and 2, respectively. In Figure 1, the spin-up band (shadowed area) of compound TiZrCoIn with ordered structure was metallic. Meanwhile, a wide energy gap was observed in the spin-down band at the Fermi level. The electrons at the Fermi level were totally polarized, and this condition resulted in 100% spin polarization. The width of the band gap was 0.914 eV, and the Fermi level was slightly near the bottom of the band gap. Thus, the half-metallicity of quaternary Heusler TiZrCoIn was susceptible to external influences. Meanwhile, the gap was primarily determined by the covalent hybridization between Ti and Co, as shown in Figure 2. The d states of Ti with low valence occupied the high energy area above the Fermi level, thereby forming the bonding bands. By contrast, the d states of Co with high valence were located at \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(x\) & \(E_{f}\) (eV) & \(a\) (Å) & \(P\)(\(\frac{1-z}{1+z}\)) (\%) & _Band Gap_ (eV) & \(M_{t}\) (\(\mu_{\text{B}}\)/\(f\).u.) \\ \hline x = 0 & - & 6.562 & 100 & 0.914 & 1.99 \\ x = 3.125\% & \(-\)9.256 & 6.633 & 100 & 0.681 & 2.03 \\ x = 6.25\% & \(-\)9.280 & 6.630 & 100 & 0.692 & 2.06 \\ x = 12.5\% & \(-\)9.333 & 6.609 & 100 & 0.721 & 2.13 \\ x = 25\% & \(-\)9.435 & 6.566 & 100 & 0.802 & 2.26 \\ x = 50\% & \(-\)9.604 & 6.496 & 100 & 0.904 & 2.50 \\ x = 75\% & \(-\)9.776 & 6.427 & 100 & 0.960 & 2.75 \\ x = 100\% & \(-\)10.109 & 6.356 & 100 & 1.130 & 3.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Formation energies (\(E_{f}\)), lattice parameters (unit in Å), spin polarizations presented as spin-up over spin-down rations, width of spin-down gap around the Fermi level (in units of eV), and total magnetic moments in a unit cell. Notice that x is the doping concentration. the low energy area below the Fermi level, thereby forming the antibonding bands. Ti and Co atoms have the same symmetry as in typical L2\({}_{1}\) full-Heuslers, and thus their d orbitals hybridize initially and form five bonding d hybrids and five nonbonding ones. Then, the five bonding d hybridized orbitals of Ti-Co hybridize with the d orbitals of the Zr atoms, thereby regenerating bonding and antibonding states. Given the large energy separation of d hybrids of the Ti and Co atoms, the five nonbonding d hybrids Ti-Co, namely, the t\({}_{\rm 1u}\) and e\({}_{\rm u}\) states, possess high energies and are unoccupied. A detailed d-d hybridization diagram is shown in Figure 3. Introduced by the sp element In, a single degenerate s band and a triple degenerated p band, lying deep in energy, are located below the d states and accommodate d charge from the transition metal atoms. Thus, 9 instead of 12 occupied states are usually observed in the spin-down band. In the quaternary Huesler alloy TiZrCoIn, the In atom carried three valence electrons, and Ti, Zr, and Co carried four, four, and nine, respectively. Therefore, 17 transition metal electrons were found, of which 5 were cached by the s and p bands of the In atom, 10 were filled in the bonding d bands, and 2 were uncompensated.