## 3 Result and Discussion  
The displacements of interface atoms in various Mn\({}_{2}\)CoAl/Ag/Mn\({}_{2}\)CoAl structures are measured and exhibited in Figure 3. L1 and L2 indicate the interface and subinterface, respectively. It can be seen that the interfacial Mn\({}^{\rm A}\) atom has an obvious outward extension in MnCo\({}^{\rm B}\), MnCo\({}^{\rm T}\), MnMn\({}^{\rm T1}\), and MnMn\({}^{\rm B1}\) structures, revealing that the interface Mn atom prefers to bond with the interface atom from the spacer layer, and such behavior is also reported in previous theoretical and experimental studies [33; 34]. As for the interfacial Mn\({}^{\rm B}\) atom, it slightly stretches outward in MnMn\({}^{\rm T2}\) and MnMn\({}^{\rm B2}\)  
Figure 2: Various atomic terminations of the Mn\({}_{2}\)CoAl/Ag/Mn\({}_{2}\)CoAl (001) interface. and keeps its ideal position in MnAl\({}^{\text{T}}\), while it even shrinks inward in MnAl\({}^{\text{B}}\), showing that its bonding ability is weaker than interfacial Mn\({}^{\text{A}}\). The interface Co atom has a slight outward movement, and when the MnMn\({}^{\text{T1}}\) and MnMn\({}^{\text{B1}}\)-terminated interface are respectively formed by substituting interface Co with the Mn atom, such an outward action is enhanced. Besides, for various different structures, the outward movement of the subinterface atom is very small, and the position of the next subinterface atom is nearly unchanged, so their behaviors are extremely similar to the bulk (not shown in Figure 3). The interface free energy (\(\gamma\)) versus atomic chemical potentials is calculated in order to determine the interface stability. As was mentioned in Ref. [34], the \(\gamma\) can be expressed as  
\[\gamma=\frac{1}{2A}\left[G-\sum_{i}\mu_{i}N_{i}\right] \tag{1}\]  
where \(A\) is the interface area of a supercell, \(G\) stands for the Gibbs free energy of the slab; and \(\mu_{i}\) and \(N_{i}\) are the chemical potential and the number of the \(i\)th atom, respectively. The relationship among the interdependent chemical potentials can be expressed as \(2\mu_{Mn}+\mu_{Co}+\mu_{Al}=G_{bulk}\). Besides, the highest values of \(\mu_{Co}\) and \(\mu_{Mn}\) are \(\mu_{Co}=G_{Co}\) and \(\mu_{Mn}=G_{Mn}\), where \(G_{Co}\) and \(G_{Mn}\) are Gibbs free energies of the Co and Mn bulk, respectively. Besides, due to the reason that 2 Mn + CoAl = Mn\({}_{2}\)CoAl and Co + Mn\({}_{2}\)Al = Mn\({}_{2}\)CoAl, the lowest values of \(\mu_{Co}\) and \(\mu_{Mn}\) are  
\[\mu_{Mn}=\frac{1}{2}[G_{bulk}-G_{CoAl}],\ \mu_{Co}=\frac{1}{2}[G_{bulk}-G_{Mn _{2}Al}] \tag{2}\]  
In Figure 4, the MnMn\({}^{\text{T1}}\) and MnMn\({}^{\text{B1}}\) always stay in the thermodynamic accessible region (TAR) and occupy the maximum area. As \(\mu_{Mn}\) and \(\mu_{Co}\) increase, MnMn\({}^{\text{T2}}\), MnMn\({}^{\text{B2}}\), MnCo\({}^{\text{T}}\), and MnCo\({}^{\text{B}}\) are included in the TAR, revealing that these interfaces could be stable under the condition of Mn rich and Co rich. In addition, MnMn\({}^{\text{B2}}\) enters into the TAR under the condition of \(\mu_{Co}=-0.702\) Ryd and \(\mu_{Mn}=-0.523\) Ryd, while MnMn\({}^{\text{T2}}\) appears in the TAR when \(\mu_{Co}=-0.702\) Ryd and \(\mu_{Mn}=-0.694\) Ryd, indicating that the formation of a stable MnMn\({}^{\text{T2}}\) needs a richer Mn condition than MnMn\({}^{\text{B2}}\). In the case of MnCo\({}^{\text{B}}\), it enters into the TAR when \(\mu_{Co}\) and \(\mu_{Mn}\) respectively increase to \(-0.672\) Ryd and \(-0.477\) Ryd, while MnCo\({}^{\text{T}}\) appears in the TAR when \(\mu_{Co}=-0.68\) Ryd and \(\mu_{Mn}=-0.547\) Ryd. Therefore, it could be deduced that the MnCo terminated interface can be more easily obtained in a bridge-type  
Figure 3: The relaxed atomic positions in various terminations. L1 and L2 indicate the interface and subinterface, respectively. structure than in a top-type structure. However, as for MnAl terminated interfaces, they are excluded from the TAR in both top-type and bridge-type structures.