Atomic Vacancy Defect, Frenkel Defect and Transition Metals (Sc, V, Zr) Doping in Ti4N3 MXene Nanosheet: A First-Principles Investigation

Using first-principles calculations based on the density functional theory, the effects of atomic vacancy defect, Frenkel-type defect and transition metal Z (Z = Sc, V and Zr) doping on magnetic and electric properties of the Ti4N3 MXene nanosheet were investigated comprehensively. The surface Ti and subsurface N atomic vacancies are both energetically stable based on the calculated binding energy and formation energy. In addition, the former appears easier than the latter. They can both enhance the magnetism of the Ti4N3 nanosheet. For atom-swapped disordering, the surface Ti-N swapped disordering is unstable, and then the Frenkel-type defect will happen. In the Frenkel-type defect system, the total magnetic moment decreases due to the enhancement of indirect magnetic exchange between surface Ti atoms bridged by the N atom. A relatively high spin polarizability of approximately 70% was detected. Furthermore, the doping effects of transition metal Z (Z = Sc, V and Zr) on Ti4N3 nanosheet are explored. All doped systems are structurally stable and have relatively large magnetism, which is mainly induced by the directed magnetic exchange between surface Z and Ti atoms. Especially in the doped Ti4N3-Sc system, the high spin polarizability is still reserved, suggesting that this doped system can be a potential candidate for application in spintronics.


Introduction
Graphene was discovered in 2004, breaking the classical theory that thermodynamic fluctuations do not happen in any two-dimensional (2D) crystal at a finite temperature [1,2]. Graphene had shown many excellent physical and chemical properties into wide material applications, such as big specific surface area is (2630 m 2 /g) [3], high electron mobility (1.5 × 10 5 cm 2 /v·s) [4], good thermal conductivity (5000 W/m·K) [5], high Young's modulus (1.0 TPa) [6] and visible light transmittance (approximately 97.7%) [7]. However, the zero-band gap of graphene limits its application in the field of electronic devices, leading to a series of studies on graphene-like 2D materials [8], such as hexagonal boron nitride, transition metal sulfur compounds, transition metal oxides, black phosphorus and various MXenes. Many graphene-like materials making up for the inadequacy of graphene materials showed better physical and chemical properties, and then were applied in optoelectronics, spintronics, catalysis, biological and chemical sensors, lithium/sodium ion batteries, supercapacitors, fuel cells, polymer composite materials and other fields [9]. moment, which is weak, and then may be eliminated by surface functional groups [15]. In this paper, to understand influences of the atomic defect, swap disordering and vacancy on the structural, magnetic and electronic properties of 2D MXene material, the Ti 4 N 3 MXene nanosheet was investigated. The structure, electronic properties, and magnetism of Ti 4 N 3 nanosheet without functional groups were analyzed firstly. Subsequently the Ti or N vacancy defect was investigated for the surface or subsurface layer of the Ti 4 N 3 nanosheet. Secondly, a swap defect (named as Frenkel-type defect) between the N atom on the subsurface layer and Ti atom on the surface layer was studied. The structure change, electronic properties and magnetism of atoms around the defect atom were calculated and analyzed. In the end, the surface Ti atom of Ti 4 N 3 nanosheet was replaced by Sc, V and Zr atoms, respectively. The effects of doping on the structure, electronic properties and magnetism of the atoms, and then the Ti 4 N 3 nanosheet were discussed. We hope our results are useful to elucidating the various electronic and magnetic behaviors of 2D materials and the application research of new 2D Ti-based MXene nanosheet in spintronics.

Calculation Method
All calculations were performed using the Vienna ab initio simulation package (VASP) based on the density functional theory (DFT). The Perdew-Burke-Ernzerhof (PBE) exchange correlation function under the generalized gradient approximation (GGA) is used in calculations [41][42][43]. To treat the electron-ion properly, the interaction between the core electron and the valence electron is described by plane wave super-soft pseudopotential. Ti: 3d 2 4s 2 , N: 2s 2 2p 3 , Sc: 3d 1 4s 2 , Zr: 4p 6 4d 2 5s 2 and V: 3d 3 4s 2 valence electron configurations were specifically considered. In this work, a 3 × 3 × 1 supercell was used and spin polarization was selected for structural optimization and single point energy calculation.
In the self-consistent field calculation, a global convergence standard of 1 × 10 −6 eV, cutoff energy of 500 eV, force per each atom of no more than 0.02 eV/Å, and 9 × 9 × 1 k-point grid precision and integral Brillouin zone were used. For structural optimization, the experimental lattice constant 2.991 Å of single Ti 4 N 3 [44] was set as the initial value of Ti 4 N 3 nanosheet. Then for defect and doping calculation, the optimized lattice constant of Ti 4 N 3 nanosheet was selected as the initial value.

Structure, Electronic Properties and Magnetism of Ti 4 N 3 Nanosheet
Ti 4 N 3 nanosheet is experimentally prepared as the first N-based series MXene nanosheet [15] by melting fluoride salt in argon gas environment and the temperature was 550 • C. The nanosheet was etched away from Ti 4 AlN 3 precursor powder layer. Figure 1 shows the construction processing of the Ti 4 N 3 MXene nanosheet by etching off the Al atomic layer from the Ti 4 AlN 3 bulk. The Ti 4 N 3 nanosheet had a hexagonal structure with sandwich layers. The N atomic layer was sandwiched between the upper and lower Ti atomic layers. Each N atom was bonded with six nearest Ti atoms. Three Ti atoms were located on the upper layer and the other three Ti atoms were on the lower layer. The whole Ti 4 N 3 nanosheet was composed of seven alternating layers of Ti and N layers. The structure was more complex than those of M 3 X 2 and M 2 X. Therefore, we deduced that M 4 X 3 had better physical and chemical properties than M 3 X 2 and M 2 X. In our work, the spin polarization was set for the structure optimization of Ti4N3 nanosheet. The optimized lattice constants a and c, bond length (dTix−Nx) and atomic layer thickness L are listed in Table 1 for comparing them with the structural parameters of the precursor Ti4AlN3 [44]. The structure parameters difference between Ti4N3 and Ti4AlN3 was relatively small. For the lattice constants, the error did not exceed 2%. This indicates that the Ti4N3 nanosheet was stable. There was no significant change between bonds due to etching, which also proves that the bond length of M-X is stronger than that of M-A [45]. Table 1. Calculated lattice constants a and c, atomic layer thickness L, bond lengths between the N2 atom at subsurface and its neighbor Ti1 atom (dTi1-N2) or Ti3 atom (dN2-Ti3) at the surface or the next subsurface, respectively.

System
a (Å) c (Å) L (Å) dTi1−N2 (Å) dN2−Ti3 (Å) Ti4AlN3 [44] Table 2 shows some structural parameters of Ti4N3 nanosheet. For simplicity, the surface Ti atom, the subsurface N atom, the next subsurface Ti atom and the central layer N atom are marked as Ti1, N2, Ti3 and N4, respectively. In Table 1, the bond lengths dTi1-N2 and dN2-Ti3 were different, which shows that N atoms were located in a distorted octahedron. This structure was similar to the Ti2NTx calculated by Arkamita Bandyopadhyay et al. [40], but it is different from the single-layer Ti4C3 nanosheet, which is a regular octahedral structure [46].  In our work, the spin polarization was set for the structure optimization of Ti 4 N 3 nanosheet. The optimized lattice constants a and c, bond length (d Tix−Nx ) and atomic layer thickness L are listed in Table 1 for comparing them with the structural parameters of the precursor Ti 4 AlN 3 [44]. The structure parameters difference between Ti 4 N 3 and Ti 4 AlN 3 was relatively small. For the lattice constants, the error did not exceed 2%. This indicates that the Ti 4 N 3 nanosheet was stable. There was no significant change between bonds due to etching, which also proves that the bond length of M-X is stronger than that of M-A [45]. Table 1. Calculated lattice constants a and c, atomic layer thickness L, bond lengths between the N2 atom at subsurface and its neighbor Ti1 atom (d Ti1-N2 ) or Ti3 atom (d N2-Ti3 ) at the surface or the next subsurface, respectively.  Table 2 shows some structural parameters of Ti 4 N 3 nanosheet. For simplicity, the surface Ti atom, the subsurface N atom, the next subsurface Ti atom and the central layer N atom are marked as Ti1, N2, Ti3 and N4, respectively. In Table 1, the bond lengths d Ti1-N2 and d N2-Ti3 were different, which shows that N atoms were located in a distorted octahedron. This structure was similar to the Ti 2 NT x calculated by Arkamita Bandyopadhyay et al. [40], but it is different from the single-layer Ti 4 C 3 nanosheet, which is a regular octahedral structure [46].
In order to discuss electronic and magnetic properties of the Ti 4 N 3 nanosheet, the total magnetic moment of Ti 4 N 3 nanosheet is calculated as 1.173 µ B , which may be weakened, even eliminated by the action of the functional groups. This result accords with the conclusion that the magnetism may be removed by functionalization [47]. The atomic magnetic moment of Ti atoms on the surface and the next subsurface was 0.924 and 0.212 µ B , respectively, whereas the atomic magnetic moment of the N atom on the subsurface and central layers were 0.030 and 0.010 µ B , respectively. Therefore, the total magnetic moment was mainly contributed by the spin-polarized d-electrons of Ti atoms in the surface layer, which was similar to results in Reference [47], in which MXenes are magnetic and the magnetism is primarily due to surface Ti atoms. 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. 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 ↑ and N ↓ 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].
Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 14 In order to discuss electronic and magnetic properties of the Ti4N3 nanosheet, the total magnetic moment of Ti4N3 nanosheet is calculated as 1.173 µB, which may be weakened, even eliminated by the action of the functional groups. This result accords with the conclusion that the magnetism may be removed by functionalization [47]. The atomic magnetic moment of Ti atoms on the surface and the next subsurface was 0.924 and 0.212 µB, respectively, whereas the atomic magnetic moment of the N atom on the subsurface and central layers were 0.030 and 0.010 µB, respectively. Therefore, the total magnetic moment was mainly contributed by the spin-polarized d-electrons of Ti atoms in the surface 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 Ti4N3 nanosheet from the spin-polarizability Equation (1) , where N ↑ and N ↓ were spin-up and spin-down TDOS at the Fermi level, respectively. Therefore, the Ti4N3 nanosheet might be a kind of candidate material in 2D spintronics because of its high spin polarizability [48]. 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 Ti4N3 nanosheet.

Atomic Vacancy Effect of Ti4N3 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 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.

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 b and the forming energy E form , and the total magnetic moment Mtot 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 v−Ti or d 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 µ B .
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 14 of the Ti4N3 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 Ti4N3 nanosheet, were described dramatically. Moreover, in Ti-vacancy and N-vacancy system (labeled as Ti4N3-Ti and Ti4N3-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 Eb and the forming energy Eform, and the total magnetic moment Mtot of the Ti4N3-Ti, Ti4N3-N and Ti4N3 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 Ti4N3 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 dv−Ti or dv−N) shrank compared with the Ti4N3 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 Ti4N3-Ti system, the total magnetic moment increased from 1.173 to 1.817 µB. In the Ti4N3 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 (Ti4N3-N) system, the bond length dv−N between the vacancy and its neighbor subsurface N atom in the subsurface layer shrank in comparison with the Ti4N3 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 Ti4N3 -N system, the total magnetic moment increased sharply to 2.373 µB.
In Table 2, the binding energies of Ti4N3, Ti4N3-Ti and Ti4N3-N systems are listed. The binding energy can be defined as follows [49]: where Etot is the total energy of the defective and ideal system, nTi, nN and ntot indicate the number of Ti, N and total atoms, respectively. uTi and uN are the chemical potential of Ti and N atoms, respectively. All the binding energies of Ti4N3, Ti4N3-Ti and Ti4N3-N systems have a negative value. This finding can indicate that all systems were stable in structure. The binding energy of the Ti4N3-N system was the largest, which implies that the system had lost N atoms and might become unstable. Moreover, in Table 2 the formation energy Eform [48] was calculated by the following formula: 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 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 µ 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]: where E tot is the total energy of the defective and ideal system, n Ti , n N and n tot indicate the number of Ti, N and total atoms, respectively. u Ti and u 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 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. Moreover, in Table 2 the formation energy E form [48] was calculated by the following formula: where E def and E id were the total energy of the defective and "ideal" system, n i is the number of removed or added atoms and u i is the chemical potential of the corresponding atoms. The positive formation energies imply that the "ideal" Ti 4 N 3 system is more stable than the defective systems. A relatively small value in Ti 4 N 3 -Ti system indicates that eliminating a surface Ti atom is easier than losing a subsurface N atom. In Figure 4, the total TDOS in a unit cell and PDOS of the neighbor Ti or N atom for Ti-vacancy and N-vacancy were plotted dramatically in Ti 4 N 3 -Ti and Ti 4 N 3 -N nanosheets. From Figure 4a, a relatively large spin-up peak appeared near the Fermi level in the Ti-vacancy system. The peak was mainly derived from spin-up d-orbitals of surface Ti neighboring vacancy. The high spin polarizability in Ti 4 N 3 was damaged. The subsurface N atom and the next subsurface Ti atom underwent a little change owing to the missing surface Ti atom. For the N-vacancy system, as shown in Figure 4b, the TDOS near Fermi level was contributed by the surface or the next subsurface Ti. A spin polarizability of more than 50% was still detected. In the two atomic vacancy systems, the middle layer N atom was not polarized visibly. where Edef and Eid were the total energy of the defective and "ideal" system, ni is the number of removed or added atoms and ui is the chemical potential of the corresponding atoms. The positive formation energies imply that the "ideal" Ti4N3 system is more stable than the defective systems. A relatively small value in Ti4N3-Ti system indicates that eliminating a surface Ti atom is easier than losing a subsurface N atom. In Figure 4, the total TDOS in a unit cell and PDOS of the neighbor Ti or N atom for Ti-vacancy and N-vacancy were plotted dramatically in Ti4N3-Ti and Ti4N3-N nanosheets. From Figure 4a, a relatively large spin-up peak appeared near the Fermi level in the Ti-vacancy system. The peak was mainly derived from spin-up d-orbitals of surface Ti neighboring vacancy. The high spin polarizability in Ti4N3 was damaged. The subsurface N atom and the next subsurface Ti atom underwent a little change owing to the missing surface Ti atom. For the N-vacancy system, as shown in Figure 4b, the TDOS near Fermi level was contributed by the surface or the next subsurface Ti. A spin polarizability of more than 50% was still detected. In the two atomic vacancy systems, the middle layer N atom was not polarized visibly.

Frenkel-Type Defects in Ti4N3 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

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 Å, the lattice constant was 2.995 Å and the atomic layer thickness was 7.329 Å. 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.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 14 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 Å, the lattice constant was 2.995 Å and the atomic layer thickness was 7.329 Å. Compared with the "ideal" Ti4N3 structure, the outer Ti-N bond was evidently decreased owing to surface effect. The whole system looked more relaxed. Compared with the "ideal" Ti4N3 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 Ti4N3 nanosheet, the binding energy Eb and the forming energy Eform were calculated by the above mentioned formula. Results show that the Eb and Eform 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" Ti4N3 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 µB 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 Ti4N3 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 Ti4N3 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 Ti4N3 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 Ti4N3 nanosheet, we detected a relatively high spin polarization (approximately 70% can be retained). The spin-polarization ratio is described by the formula [48]. 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 b and the forming energy E form were calculated by the above mentioned formula. Results show that the E b and E 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 µ 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].

Doping Effects of Transition Metal Z (Z = Sc, V, Zr) on the Ti4N3 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 Ti4N3 nanosheet. In Figure 7, a surface Ti atom was replaced by Sc, V and Zr, atoms in a Ti4N3 supercell to simulate possible doping behavior. Impurity atom concentration can be close to 2% in this 3 × 3 × 1 supercell. After geometrical optimization, the lattice constant a, bond length d, binding energy Eb, forming energy Eform, the neighbor Ti atomic magnetic moment of the doped atom MTi, atomic magnetic moment of Z atom MZ and total magnetic moment Mtot are listed in Table 3. The binding energy Eb is defined using the above formula. The formation energy Eform is defined as follows [51]: are total energies of doped Ti4N3-Z system and "ideal" Ti4N3 system, respectively. nZ or nTi is the number of Z or Ti atom. µTi or µZ is the chemical potential of Z or Ti atoms.  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 × 3 × 1 supercell. After geometrical optimization, the lattice constant a, bond length d, binding energy E b , forming energy E form , the neighbor Ti atomic magnetic moment of the doped atom M Ti , atomic magnetic moment of Z atom M Z and total magnetic moment M tot are listed in Table 3. The binding energy E b is defined using the above formula. The formation energy E form is defined as follows [51]: 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 Ti is the number of Z or Ti atom. µ Ti or µ Z is the chemical potential of Z or Ti atoms.

Doping Effects of Transition Metal Z (Z = Sc, V, Zr) on the Ti4N3 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 Ti4N3 nanosheet. In Figure 7, a surface Ti atom was replaced by Sc, V and Zr, atoms in a Ti4N3 supercell to simulate possible doping behavior. Impurity atom concentration can be close to 2% in this 3 × 3 × 1 supercell. After geometrical optimization, the lattice constant a, bond length d, binding energy Eb, forming energy Eform, the neighbor Ti atomic magnetic moment of the doped atom MTi, atomic magnetic moment of Z atom MZ and total magnetic moment Mtot are listed in Table 3. The binding energy Eb is defined using the above formula. The formation energy Eform is defined as follows [51]: are total energies of doped Ti4N3-Z system and "ideal" Ti4N3 system, respectively. nZ or nTi is the number of Z or Ti atom. µTi or µZ is the chemical potential of Z or Ti atoms.   Table 3. Calculated lattice constants a, atomic layer thickness L, binding energy E b , forming energy E form , total magnetic moment M tot , magnetic moment of Z atom or its neighbor Ti atom at the surface, bond length between the doping Z atom at surface and its neighbor Ti 1 or N 2 atom at the surface or subsurface, respectively. 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 V-Ti or d 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 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 µ B . In Table 3, the total magnetic moment derived from the surface V atom. However, the bond length of d V-Ti or d 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. For subsurface N atom, the contribution to TDOS was extremely small. As a result, in the doped system Ti 4 N 3 -Sc system, a high spin polarization ratio was reserved, and the spin-polarization ratio was described by Formula [48]. For the Ti 4 N 3 -V system, as shown in Figure 8b, we can see that near the Fermi level, some spin-peaks appeared in the spin-up channel because of one more electron. Similar behaviors were detected in the PDOS of surface Ti atoms owing to rehybridization. However, for the doped Ti 4 N 3 -Zr system, as shown in Figure 8c, the strong Zr-Ti interaction could widen the spin split near the Fermi level owing to the short bond length of Zr-Ti. In the PDOS of surface Ti and V atoms, some peaks appeared near the Fermi level. This system presented a relatively low spin polarization but a large magnetism.

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 based on density functional theory. The "ideal" Ti 4 N 3 had a magnetic moment of 1.173 µ 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.