## 3 Results and Discussion
### Frenkel-Type Defects in Ti\({}_{4}\)N\({}_{3}\) Nanosheet  
Besides single atomic vacancy effects, atomic swap disordering effect, namely a potential atomic disordering phenomenon in 2D MXene materials, is also studied in this work. In Figure 5a, the surface Ti atom and its neighboring N atom at the subsurface layer were swapped. Unfortunately, such a Ti-N swapped structure was unstable, and then it would evolve into the configuration shown in Figure 5b after structure optimization. In Figure 5b, one of the N atoms at the subsurface moved to the hole-site of three surface Ti atoms in that the Ti-N bond moved from swapped atoms slides along the original crystalline direction to the surface of the 2D material. The disordering configuration shown in Figure 5b was called as a Frenkel-type defect. In the optimized Frenkel-type defect structure, the bond length of Ti-N was 1.896 A, the lattice constant was 2.995 A and the atomic layer thickness was 7.329 A. Compared with the \"ideal\" Ti\({}_{4}\)N\({}_{3}\) structure, the outer Ti-N bond was evidently decreased owing to surface effect. The whole system looked more relaxed. Compared with the \"ideal\" Ti\({}_{4}\)N\({}_{3}\) system, the outside N atom was shown to stick to the surface and exhibited a slightly inward contraction. However, for the whole system, the interlayer spacing increased, which was similar to the behavior of the N-vacancy system discussed above, which led to a decrease of the vacuum layer thickness. To focus on the structure stability of the disordering Ti\({}_{4}\)N\({}_{3}\) nanosheet, the binding energy \(E_{\text{b}}\) and the forming energy \(E_{\text{form}}\) were calculated by the above mentioned formula. Results show that the \(E_{\text{b}}\) and \(E_{\text{form}}\) were equal to \(-\)1.300 and \(-\)0.347 eV, respectively. The negative value indicates that these Frenkel-type defects were more stable than the \"ideal\" Ti\({}_{4}\)N\({}_{3}\) system. This result is not surprising, because most MXenes decorated by some functional groups are more stable [50]. The atomic magnetic moment in the Frenkel-type disordering structure was also calculated. The total magnetic moment decreased from 1.173 to 1.084 \(\mu_{\text{B}}\) due to the enhancement of indirect magnetic exchange between surface Ti atoms bridged by the outside N atom. More information on the electronic properties of Frenkel-type defective Ti\({}_{4}\)N\({}_{3}\) nanosheet is needed. In Figure 6, the TDOS and the PDOS of the outside N and its neighbor Ti atom in Frenkel-type defect structure are given and are compared with the Ti\({}_{4}\)N\({}_{3}\) configuration. The outer N was located at the hole-site of three surface Ti atoms, the surface Ti atom neighbor of the N atom would increase, and the spin split near the Fermi level in the defective structure would decrease. As a result, the high spin polarization in Ti\({}_{4}\)N\({}_{3}\) nanosheet was reduced sharply. The contribution of surface Ti atoms to TDOS was most significant from surface Ti atomic PDOS. The outside N and subsurface N provided little support. In this Frenkel-type defective Ti\({}_{4}\)N\({}_{3}\) nanosheet, we detected a relatively high spin polarization (approximately 70% can be retained). The spin-polarization ratio is described by the formula [48]. Figure 5: Schematic of the Ti-N swapped (**a**) before and (**b**) after geometric optimization on the Ti\({}_{4}\)N\({}_{3}\) nanosheet. Doping Effects of Transition Metal \(Z\) (\(Z\) = Sc, V, Zr) on the Ti\({}_{4}\)N\({}_{3}\) Nanosheet  
Impurities in materials are unavoidable in the preparation process and can easily affect physical and chemical properties. We implemented a theoretical calculation to reveal the doping properties of transition metal \(Z\) (\(Z\) = Sc, V, Zr) on the Ti\({}_{4}\)N\({}_{3}\) nanosheet. In Figure 7, a surface Ti atom was replaced by Sc, V and Zr, atoms in a Ti\({}_{4}\)N\({}_{3}\) supercell to simulate possible doping behavior. Impurity atom concentration can be close to 2% in this \(3\times 3\times 1\) supercell.