## 3 Results and Discussion
### Atomic Vacancy Effect of Ti\({}_{4}\)N\({}_{3}\) Nanosheet  
The atomic vacancy defect is an ordinary phenomenon in 2D materials. In general, high proportion vacancy may destruct properties of materials. We focused on the atomic vacancy defect of the Ti\({}_{4}\)N\({}_{3}\) nanosheet. In Figure 3, the structure of the surface Ti-vacancy (a) and subsurface N-vacancy (b), which are the two types of vacancy effects that possibly have a relatively great impact on Ti\({}_{4}\)N\({}_{3}\) nanosheet, were described dramatically. Moreover, in Ti-vacancy and N-vacancy system (labeled as Ti\({}_{4}\)N\({}_{3}\)-Ti and Ti\({}_{4}\)N\({}_{3}\)-N, respectively) in Table 2, the lattice constants \(a\), the atomic layer thickness \(L\), the bond length \(d\) between the vacancy and its neighbor atoms, the binding energy \(E_{\text{b}}\) and the forming energy \(E_{\text{form}}\), and the total magnetic moment \(M\)tot of the Ti\({}_{4}\)N\({}_{3}\)-Ti, Ti\({}_{4}\)N\({}_{3}\)-N and Ti\({}_{4}\)N\({}_{3}\) system are listed. As far as Ti-vacancy is concerned, a surface Ti atom had 3 neighbor N atoms and 4 neighbor Ti atoms in the Ti\({}_{4}\)N\({}_{3}\) system respectively, as shown in Figure 3a. When a surface Ti atom was lost, the bond length between Ti-vacancy and its neighbor surface Ti or subsurface N atom (labeled as \(d_{\text{v-Ti}}\) or \(d_{\text{v-N}}\)) shrank compared with the Ti\({}_{4}\)N\({}_{3}\) system. Therefore, the atomic layer thickness \(L\) decreased slightly. Atomic vacancy could cause the surrounding space to collapse in the same way as the atom vacancy in other materials. The magnetic moment of surface Ti atom near the vacancy increased, owing to the enhancement of localization of \(d\)-electrons of the Ti atom. Four coordination numbers between Ti-vacancy and its neighbor Ti atoms were lost. Thus, in the Ti\({}_{4}\)N\({}_{3}\)-Ti system, the total magnetic moment increased from 1.173 to 1.817 \(\upmu_{\text{B}}\). In the Ti\({}_{4}\)N\({}_{3}\) system, a subsurface N atom had four neighboring N atoms and six neighboring Ti atoms. In Table 2, we could see that in the N-vacancy (Ti\({}_{4}\)N\({}_{3}\)-N) system, the bond length \(d_{\text{v-N}}\) between the vacancy and its neighbor subsurface N atom in the subsurface layer shrank in comparison with the Ti\({}_{4}\)N\({}_{3}\) system. The atomic layer thickness \(L\) was stretched slightly due to the fact that the bridging interaction between two Ti-atomic layers supplied by N atoms was weakened. The system looks fluffy. Moreover, when a subsurface N atom was replaced by vacancy, six coordination numbers between N-vacancy and its neighbor Ti atoms at surface or the next subsurface might be lost. These atoms near the N-vacancy were far from each other. In the Ti\({}_{4}\)N\({}_{3}\) -N system, the total magnetic moment increased sharply to 2.373 \(\upmu_{\text{B}}\). In Table 2, the binding energies of Ti\({}_{4}\)N\({}_{3}\), Ti\({}_{4}\)N\({}_{3}\)-Ti and Ti\({}_{4}\)N\({}_{3}\)-N systems are listed. The binding energy can be defined as follows [49]:  
\[E_{\text{b}}=(E_{\text{tot}}-n_{\text{Ti}}u_{\text{Ti}}-n_{\text{N}}u_{\text{ N}})/n_{\text{tot}}, \tag{2}\]  
where \(E_{\text{tot}}\) is the total energy of the defective and ideal system, \(n_{\text{Ti}}\), \(n_{\text{N}}\) and \(n_{\text{tot}}\) indicate the number of Ti, N and total atoms, respectively. \(u_{\text{Ti}}\) and \(u_{\text{N}}\) are the chemical potential of Ti and N atoms, respectively. All the binding energies of Ti\({}_{4}\)N\({}_{3}\), Ti\({}_{4}\)N\({}_{3}\)-Ti and Ti\({}_{4}\)N\({}_{3}\)-N systems have a negative value. This finding  
Figure 3: Schematic of a vacancy defect: (**a**) surface Ti-vacancy and (**b**) subsurface N-vacancy on a Ti\({}_{4}\)N\({}_{3}\) nanosheet. can indicate that all systems were stable in structure. The binding energy of the Ti\({}_{4}\)N\({}_{3}\)-N system was the largest, which implies that the system had lost N atoms and might become unstable.