## 3 Results and Discussion
### Structure, Electronic Properties and Magnetism of Ti\({}_{4}\)N\({}_{3}\) Nanosheet
Figure 1: Schematic of construction processing of the Ti\({}_{4}\)N\({}_{3}\) MXene nanosheet by etching off the Al atomic layer from the Ti\({}_{4}\)AlN\({}_{3}\) bulk. (**a**) Ti\({}_{4}\)AlN\({}_{3}\) bulk, (**b**) Ti\({}_{4}\)N\({}_{3}\) bulk and (**c**) front, (**d**) side and (**e**) top view of the Ti\({}_{4}\)N\({}_{3}\) MXene nanosheet. layer, which was similar to results in Reference [47], in which MXenes are magnetic and the magnetism is primarily due to surface Ti atoms. In Figure 2, the total density of states (TDOS) and partial density of states (PDOS) are shown. From Figure 2, near the Fermi surface the spin-up channel had a large valley and the Fermi level was located at the valley bottom. However, for the spin-down band a relatively large peak crossed over the Fermi level. This results in a great spin-polarization (more than 80%) in the Ti\({}_{4}\)N\({}_{3}\) nanosheet from the spin-polarizability Equation (1), where \(N_{\uparrow}\) and \(N_{\downarrow}\) were spin-up and spin-down TDOS at the Fermi level, respectively. Therefore, the Ti\({}_{4}\)N\({}_{3}\) nanosheet might be a kind of candidate material in 2D spintronics because of its high spin polarizability [48]. \[P=\frac{N_{\uparrow}-N_{\downarrow}}{N_{\uparrow}+N_{\downarrow}}\times 100\%, \tag{1}\]  
From Figure 2, TDOS near the Fermi level mainly originated from the surface Ti atoms, meaning that localization of \(d\)-electrons of surface Ti atoms played an important role in magnetism. The main reason was that the strong direct exchange magnetic coupling caused a remarkable spin split of energy bands near the Fermi level. This finding is in agreement with the above mentioned discussion on the total magnetic moment. Furthermore, the subsurface N2 atom was also spin polarized, which was induced by the neighbor magnetic atoms. This means that the indirect magnetic exchange between Ti1 atoms on the surface and the next subsurface Ti2 atoms might be bridged by the N2 atoms on the subsurface. Such a layer interaction could help stabilizing the whole structure of the Ti\({}_{4}\)N\({}_{3}\) nanosheet. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**System** & \(a\) (Å) & _L_(Å) & _d\({}_{v-\text{Ti}}\)_(Å) & _d\({}_{v-\text{N2}}\)_(Å) & _d\({}_{v-\text{Ti3}}\)_(Å) & _E\({}_{b}\)_(eV) & _E\({}_{form}\)_(eV) & _M\({}_{tot}\)_(\(\imath_{\text{IB}}\))** \\ \hline Ti\({}_{4}\)N\({}_{3}\) & 2.993 & 7.235 & 2.993(2.062) & 2.062(2.993) & 2.918(2.119) & \(-\)1.295 & - & 1.173 \\ Ti\({}_{4}\)N\({}_{3}\)-Ti & 2.995 & 7.176 & 2.986 & 2.045 & 2.841 & \(-\)1.281 & 2.167 & 1.817 \\ Ti\({}_{4}\)N\({}_{3}\)-N & 2.988 & 7.368 & 2.063 & 2.988 & 2.161 & \(-\)1.201 & 7.118 & 2.373 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculated lattice constants \(a\), atomic layer thickness \(L\), binding energy _E\({}_{b}\)_ and formation energy _E\({}_{form}\)_ and total magnetic moment _M\({}_{tot}\)_, where bond lengths between the vacancy atom at surface (or subsurface) and its neighbor Ti1, N2 or Ti3 atom at surface, subsurface or next subsurface, respectively. Figure 2: Total density of states (TDOS) and partial density of states (PDOS) 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.