## 4 Conclusions  
The atomic vacancy defect, Frenkel-type defect and transition metal \(Z\) (\(Z\) = Sc, V, Zr) doping in Ti\({}_{4}\)N\({}_{3}\) MXene nanosheet were investigated comprehensively by the first-principles calculation  
Figure 8: The TDOS and PDOS of transition metal (**a**) Sc, (**b**) V and (**c**) Zr doped 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.  
based on density functional theory. The \"ideal\" Ti\({}_{4}\)N\({}_{3}\) had a magnetic moment of 1.173 \(\upmu_{\rm B}\) and a high spin polarization ration of more than 80%. Firstly, atomic vacancy in the Ti\({}_{4}\)N\({}_{3}\) MXene was studied. Although the system was stable even with the absence of a surface Ti or subsurface N atom, the Ti-vacancy appeared easier than the N-vacancy in the Ti\({}_{4}\)N\({}_{3}\) MXene nanosheet. Both atomic vacancies in Ti\({}_{4}\)N\({}_{3}\) could increase magnetism. Secondly, detection from swapped atom disordering indicates that the surface Ti-N swapped disordering was unstable and evolved into a Frenkel-type defect. In the Frenkel-type defect system, the total magnetic moment was decreased due to the enhancement of indirect magnetic exchange between surface Ti atoms bridged by the outside N atom and presented a relatively high spin polarization of approximately 70%. Finally, the doping effects of transition metal \(Z\) (\(Z\) = Sc, V, Zr) on the Ti\({}_{4}\)N\({}_{3}\) nanosheet were systematically investigated. The results revealed that all of the doped systems were structurally stable and had relatively large magnetism, which was derived from the directed magnetic exchange between surface \(Z\) and Ti atoms. Surface atomic relaxation also played an important role in magnetism and electronic properties. Especially in the doped system Ti\({}_{4}\)N\({}_{3}\)-Sc system, a high spin polarization ratio was reserved. The Ti\({}_{4}\)N\({}_{3}\) MXene nanosheet doped by Sc atom could be a potential candidate for application to spintronics. Studies on atomic defects and doping from transition metals (Sc, V, Zr) in MXene nanosheet Ti\({}_{4}\)N\({}_{3}\) can help elucidate the various electronic and magnetic behaviors of real 2D materials and contribute to the development of new 2D Ti-based MXene.  
Methodology, J.L. and B.W.; software, T.Z. and K.Y.; data curation, Y.L. and Q.Y.; writing-original draft preparation, T.Z. and W.Z.; writing-review and editing, J.L.; B.W. and T.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No. 11304410), Key Laboratory and Scientific Research Foundation of Guizhou Province (QJHKY[2019]055), Outstanding Youth Science and Technological Innovation Talents Training of Zunyi City (ZYOK[2018]7), Key Laboratory and Scientific Research Foundation of Zunyi City (SSKH[2015]55), Natural Science Foundation of Technology Department (QKHJZ-LKZS[2014]10, QJHJZ-LKZS[2012]03, and KHJZ[2014]2170), Youth Science Foundation of Education Ministry (QJHKZ[2012]084) and the Key Support Discipline ([2011]275) of Guizhou Province of China.  
The authors declare no conflict of interest. The authors claim that none of the material in our manuscript has been published or is under consideration for publication elsewhere. This manuscript is our original work.