## 3 Results and Discussion
### Interface Magnetic Behaviors  
Table 2 summarizes the atom-resolved magnetic moments (AMMs) at the interface and subinterface layers in Ti\({}_{2}\)CoAl and at the interface layer in MgO for various terminations of the Ti\({}_{2}\)CoAl/MgO(100) interface. To facilitate comparisons with the bulk and surface values, the calculated atom-resolved magnetic moments per cell in Ti\({}_{2}\)CoAl bulk and (100) surfaces (TiAl termination and TiCo termination) are also listed in this table. The AMMs from the middle layers were close to the corresponding bulk values, thereby indicating that the implemented GGA + PBE scheme could reliably deal with the Heusler hertrojunction system. The middle layer is also called the bulk-like layer. Following the reduction of atomic coordination numbers at the surfaces, the crystal field was weakened, and the localization of d-electron atoms were enhanced, thereby resulting in the rehybridization of Ti and Co atoms. As a result, the surface Ti AMM was obviously enhanced when compared with the bulk value. For the surface and subsurface Co atoms, the AMM slightly decreased owing to the enhanced d-electronic hybridization caused by the surface Co atomic shrink. This same result had also been obtained in our previous work [34]. Similar surface behaviors were also observed in Ti\({}_{2}\)FeGe(001) [38] and Co\({}_{2}\)MnGe(111) [39].  
Interface AMMs are various and complex. The Ti may be located at the (0, 0, 0) or (0.25, 0.25, 0.25) sites and presents different AMMs in the bulk [29]. Table 2 shows that in the alloy-Mg terminations, the interface and subinterface Ti AMMs were slightly larger than those at the middle layers, thereby suggesting that the d-electron localization originating from relatively large metal-Mg bond lengths was enhanced. However, for the interface Co atom, the fierce inward shrink promoted the d-electron hybridization to resist d-electronic localization. Therefore, the interface Co AMMs, except for the CoCo-Mg termination, slightly decreased. In the alloy-O terminations, the relatively short metal-O bond lengths reduced the part of d-electron localization, thereby leading to a remarkable direct magnetic exchange. Therefore, the interface or subinterface Ti and Co AMMs in the alloy-O terminations, except for the CoCo-O termination, had low values. The modified CoCo-O or CoCo-Mg terminations were created by the surface atoms substituted for the Ti atoms in the \"ideal\" TiCo-Mg or TiCo-O terminations. Therefore, the two interface Co atoms had different magnetic properties. In the CoCo-Mg and CoCo-O terminations, given that the periodic structure of the Heusler crystal field was cut off and considering the competition between the localization and hybridization of d-electrons, the interface Co AMM was a large value, especially in the CoCo-Mg termination. Meanwhile, for the interface Al atom, the absolute value of AMMs decreased as a result of the reduced magnetic atoms. In MgO films, the interface or subinterface Co and Mg atoms suffered from an extremely small spin polarization and had zero spin magnetism in most cases.  
\begin{table}
\begin{tabular}{c c c} \hline
**Interface Termination** & **Bond Type** & **Bond Length (Å)** \\ \hline CoCo–Mg & Co–Mg & 4.28 \\ TiTi–Mg & Ti–Mg & 4.05 \\ AlAl–Mg & Al–Mg & 3.99 \\ TiAl–Mg & Al–Mg & 3.99 \\  & Ti–Mg & 4.06 \\ TiCo–Mg & Al–Mg & 3.55 \\  & Co–Mg & 4.02 \\ CoCo–O & Co–O & 2.07 \\ TiTi–O & Ti–O & 2.14 \\ AlAl–O & Al–O & 2.06 \\ TiAl–O & Al–O & 2.16 \\  & Ti–O & 2.15 \\ TiCo–O & Ti–O & 2.08 \\  & Co–O & 2.83 \\ \hline \end{tabular}
\end{table}
Table 1: Bond lengths at the interfaces.