## 3 Results and Discussion
### Frenkel-Type Defects in Ti\({}_{4}\)N\({}_{3}\) Nanosheet
After geometrical optimization, the lattice constant \(a\), bond length \(d\), binding energy \(E_{\text{b}}\), forming energy \(E_{\text{form}}\), the neighbor Ti atomic magnetic moment of the doped atom \(M_{\text{Ti}}\), atomic magnetic moment of \(Z\) atom \(M_{Z}\) and total magnetic moment \(M_{\text{tot}}\) are listed in Table 3. The binding energy \(E_{\text{b}}\) is defined using the above formula. The formation energy \(E_{\text{form}}\) is defined as follows [51]:  
\[E_{\text{form}}=E_{Ti_{4}N_{3}-Z}-E_{Ti_{4}N_{3}}+n_{Ti}\mu_{Ti}-n_{Z}\mu_{Z} \tag{4}\]  
where \(E_{Ti_{4}N_{3}-Z}\) are \(E_{Ti_{4}N_{3}}\) are total energies of doped Ti\({}_{4}\)N\({}_{3}\)-\(Z\) system and \"ideal\" Ti\({}_{4}\)N\({}_{3}\) system, respectively. \(n_{Z}\) or \(n_{\text{Ti}}\) is the number of \(Z\) or Ti atom. \(\mu_{\text{Ti}}\) or \(\mu_{Z}\) is the chemical potential of \(Z\) or Ti atoms. Figure 6: TDOS and PDOS of the Frenkel-type defected Ti\({}_{4}\)N\({}_{3}\) nanosheet. \(\uparrow\) and \(\downarrow\) indicate the spin-up and spin-down orbitals, respectively. The Fermi level (dashed vertical line) is set to zero. The TDOS is transited to a primitive cell. Figure 7: Schematic of doping transition metal \(Z\) (\(Z\) = Sc, V, Zr) on the Ti\({}_{4}\)N\({}_{3}\) nanosheet. For the doped systems of Ti\({}_{4}\)N\({}_{3}\)-\(Z\) (\(Z\) = Sc, V, Zr), the calculated lattice barely changed, as shown by the comparison with the Ti\({}_{4}\)N\({}_{3}\) nanosheet. In the Ti\({}_{4}\)N\({}_{3}\)-V system, the thickness of layer \(L\) tended to extend, whereas the bond length of \(d_{\text{V-Ti}}\) or \(d_{\text{V-N}}\) tended to shrink. Moreover, to focus on structure stability, we show that all of the binding energies were negative in Table 3. This finding implies that this structure was stable. We predicted that Ti\({}_{4}\)N\({}_{3}\)-V was the most stable, due to the fact that the binding value of Ti\({}_{4}\)N\({}_{3}\)-V was the smallest among all structures. By comparing with the \"ideal\" Ti\({}_{4}\)N\({}_{3}\) system, the doping of V element on Ti\({}_{4}\)N\({}_{3}\) emitted energy because of the negative \(E_{\text{form}}\). For Zr-doping, a relatively large forming energy indicates the difficulty of fabrication. In Table 3, both Ti\({}_{4}\)N\({}_{3}\)-Sc and Ti\({}_{4}\)N\({}_{3}\)-V systems, which have one less electron and one more electron by comparing with the Ti\({}_{4}\)N\({}_{3}\) system separately, show high magnetism. Localization of the surface transition metal atoms was increased due to the relatively large bond length between surface impurity atoms and surface Ti atom or subsurface N atom. Although Ti\({}_{4}\)N\({}_{3}\)-Sc was a one-less-electron system, it can present high magnetism. For the same reason, the total magnetic moment was enhanced in the Ti\({}_{4}\)N\({}_{3}\)-Zr system. Finally, for the Ti\({}_{4}\)N\({}_{3}\)-V system, the added electron entered the spin-up channel and resulted in a magnetic moment increment of 1 \(\upmu_{\text{B}}\). In Table 3, the total magnetic moment derived from the surface V atom. However, the bond length of \(d_{\text{V-Ti}}\) or \(d_{\text{V-N}}\) in the Ti\({}_{4}\)N\({}_{3}\)-V system was smaller than that in the Ti\({}_{4}\)N\({}_{3}\) system. More direct magnetic hybridization partly weakened the total magnetism. On the whole, surface doping of transition metal on a Ti\({}_{4}\)N\({}_{3}\) nanosheet can mediate magnetism. This result provides a controlled way for us to design 2D MXene materials with high magnetic properties. In Figure 8, the TDOS of the doped system Ti\({}_{4}\)N\({}_{3}\)-\(Z\) (\(Z\) = Sc, V, Zr) and the PDOS of surface \(Z\) atom and its neighbor surface Ti and subsurface N atom are presented. In Figure 8a, because Sc is one electron less than the Ti atom, the TDOS near the Fermi level decreased and presented a large spin valley. The PDOS of surface Ti atom neighboring the Sc atom shifted toward low-energy orientation due to the fact that the whole system was one electron less.