Ferromagnetism in Defected TMD (MoX2, X = S, Se) Monolayer and Its Sustainability under O2, O3, and H2O Gas Exposure: DFT Study

Spin-polarized density-functional theory (DFT) has been employed to study the effects of atmospheric gases on the electronic and magnetic properties of a defective transition-metal dichalcogenide (TMD) monolayer, MoX2 with X = S or Se. This study focuses on three single vacancies: (i) molybdenum “VMo”; (ii) chalcogenide “VX”; and (iii) di-chalcogenide “VX2”. Five different samples of sizes ranging from 4 × 4 to 8 × 8 primitive cells (PCs) were considered in order to assess the effect of vacancy–vacancy interaction. The results showed that all defected samples were paramagnetic semiconductors, except in the case of VMo in MoSe2, which yielded a magnetic moment of 3.99 μB that was independent of the sample size. Moreover, the samples of MoSe2 with VMo and sizes of 4 × 4 and 5 × 5 PCs exhibited half-metallicity, where the spin-up state becomes conductive and is predominantly composed of dxy and dz2 orbital mixing attributed to Mo atoms located in the neighborhood of VMo. The requirement for the establishment of half-metallicity is confirmed to be the provision of ferromagnetic-coupling (FMC) interactions between localized magnetic moments (such as VMo). The critical distance for the existence of FMC is estimated to be dc≅ 16 Å, which allows small sample sizes in MoSe2 to exhibit half-metallicity while the FMC represents the ground state. The adsorption of atmospheric gases (H2O, O2, O3) can drastically change the electronic and magnetic properties, for instance, it can demolish the half-metallicity characteristics. Hence, the maintenance of half-metallicity requires keeping the samples isolated from the atmosphere. We benchmarked our theoretical results with the available data in the literature throughout our study. The conditions that govern the appearance/disappearance of half-metallicity are of great relevance for spintronic device applications.


Introduction
Point defects in semiconductors can significantly affect their electronic structures, transport, and optical properties. They can either enhance or hamper the performance of the host material in device applications depending on their impact on the material's properties. For instance, in gas-sensing applications, point defects can be useful in introducing dangling bonds on surfaces as intermediaries for capturing volatile organic compounds (VOCs) and toxic gas molecules [1][2][3][4]. On the other hand, the defects can also have a negative impact on the transport and the optical properties such as trapping centers or DX-centers in semiconductors [5]. These kinds of defects are well-characterized by deep-level transient spectroscopy (DLTS) [6].
In the characterization of defects in TMD ML, Conzalez and coworkers [7] used both scanning tunneling microscopy (STM) and Keldysh non-equilibrium Green-function formalism within density functional theory (DFT) to model the images in characterizing the defects on a free-standing MoS 2 ML. They showed that the most abundant defects were S and Mo vacancies, as well as S Mo and Mo S anti-sites. In a related work, Kc and coworkers [8] used STM and DFT to study point defects in MoS 2 MLs. They reported that V S is more energetically favorable than V Mo and thus more abundant. The research group of Liu [9] conducted a similar experimental study using STM and scanning tunneling spectroscopy (STS) on MoSe 2 bilayer and monolayer systems. They showed that the prominent point defects were Mo and Se vacancies and anti-sites.
In simulating the effects of defects on the electronic characteristics, Shafqat et al. [10] used the Amsterdam density functional (ADF) to study defects in MoSe 2 ML. They showed two kinds of defects (i.e., anti-site Mo Se2 and Mo vacancies, V Mo ) responsible for introducing magnetism into the system. Ma and coworkers [11] used DFT to study the effects of vacancy defects on the magnetic properties of TMD MLs. They used a supercell with a size of 4 × 4 primitive cells (PCs) to study H-adsorbed (MoSe 2 , MoTe 2 , WS 2 ) MLs and F-adsorbed (WS 2 , MoSe 2 ) MLs and showed the existence of long-range antiferromagnetic coupling between local moments up to a distance of about d c ∼ 12 Å. In a related computational work, Yang and coworkers [12] applied spin-polarized DFT to a 5 × 5 PC sample of WSe 2 ML to study the effect of simple and complex vacancies on the electronic and magnetic properties. These authors showed that only V W2 and V WSe6 can introduce magnetism into WSe 2 ML with magnetic moments of 2 µ B and 6 µ B , respectively. The magnetic moments are attributed to the atoms around the vacancies, and, in particular, WSe 2 with V W2 was reported to be half-metallic. It is worth mentioning that half-metallicity ought to be the primary distinguishing feature in the accommodating material for spintronic applications [13,14].
From the perspective of stability under ambient conditions, defect-free TMDs have shown long-range structural and optical stability in ambient conditions due to their inert surfaces and the absence of dangling bonds [15]. However, a thorough investigation of the literature reveals that TMDs can undergo environmental degradation and transformation, which is significant for managing development risks [16]. Pető, János, et al. [17] reported the spontaneous replacement of S atoms by oxygen atoms (one by one) in a mechanically exfoliated MoS 2 ML on long-range exposure to O 2, producing 2D MoS 2−x O x crystals. Nurdiwijayanto et al. [18] synthesized MoS 2 ML using Li-intercalation and stored it in ambient air. They detected degradation in the sample with oxidation after 2 months of exposure, which can be prevented if the material is stored in an inert atmosphere. In another study conducted by Gioele Mirabelli and coworkers, [19] the stabilities of five different TMDs (i.e., MoS 2 , MoSe 2 , MoTe 2 , HfS 2 , and HfSe 2 ) were studied. They reported that HfSe 2 was the least stable under ambient conditions, showing signs of degradation after 24 h due to the transformation into HfO 2 . On the other hand, MoS 2 was the most stable sample during the study period and, consequently, can be used for electronic applications. These experiments revealed a general tendency toward the deteriorating stability of TMD MLs under ambient conditions, particularly when the chalcogenide element shifts from S to Se and then Te (i.e., although MoS 2 is very stable, MoSe 2 started oxidizing on the ninth day and MoTe 2 was the most reactive among the Mo-based TMDs).

Computational Method
Two computational methods were used to ensure the highest authenticity in our observations. The first one was based on the Spanish initiative for electronic simulations with thousand atoms (SIESTA), [20] whose efficiency stems from the use of a localized basis set to deal with large systems, and its accuracy can be competitive with plane-wave ab initio methods by selecting appropriate pseudo-potentials and exploring high K-mesh Nanomaterials 2023, 13,1642 3 of 20 grids. The second method is a widely popular and reliable method based on the Vienna Ab Initio Simulation Package (VASP), [21] which is used to benchmark our results and deal with more refined calculations such as in the case of the existence of half-metallicity. Furthermore, group theoretical analysis is involved in analyzing the splitting of states near the Fermi level to contrast the effect of the defect and trace the origins of half-metallicity in cases where it exists.
The density functional theory (DFT), which is common to SIESTA and VASP, employs the general gradient approximations (GGA) with the Perdew-Burke-Ernzerhof (PBE) [22] functional for the description of the exchange-correlation interaction between electrons. The basis sets of the Hamiltonians in the two methods are different, as mentioned above (i.e., SIESTA's basis set is based on localized orbitals, whereas VASP's basis set is based on plane waves). For the SIESTA method, we used a double zeta polarization (DZP) function basis set and a constant energy cutoff of E cut = 300 Ry. The Brillouin zone integration was carried out using the Monkhorst-Pack technique [23], with a K-mesh size of 10 × 10 × 1. The criteria for convergence for the total energy and force per atom were set at 10 −6 eV and 1 meV/Å, respectively. On the other hand, for the VASP package, the projector-augmented plane-wave (PAW) method was utilized. PAW pseudo-potential with non-local projectors for the molybdenum (Mo) 4s, 4p, 4d, and 5s atomic orbitals and the sulfur (S) 3s and 3p orbitals were included. The spin-orbit coupling (SOC) was self-consistently included [24]. An energy cutoff of E cut = 400 eV, with a K-mesh size of 5 × 5 × 1 in the Monkhorst-Pack scheme, was implemented in the geometry optimization with a tolerance of the total energy and force of about 0.01 eV/Å and 10 −5 eV, respectively. For both the SIESTA and VASP methods, a default value for the on-site U Hubbard parameter of U = 4.5 eV was used, which was almost the default value and in good agreement with the adjusted values calculated by Mann et al. [25]. The same value was also used in our previous work [26].
The binding energy (E b ), which is equivalent to the average cohesive energy per atom, of MoX 2 (where X = S, Se) with and without vacancy was calculated using the formula: where E tot is the total energy of the system, E Mo and E X are the energies of the isolated atoms of Mo and chalcogenide, and m and n are the numbers of Mo and X atoms in the system, respectively. The formation energy of vacancy (E f ), for instance, for V Mo in MoS 2 , is defined by: where E Bulk tot , E Mo tot , and E w/Vac tot represent the total energies of the bulk sample, isolated Mo atom, and sample with the Mo vacancy, respectively. If E f > 0, the formation of the vacancy is endothermic, and if E f < 0, the formation of the vacancy is exothermic.
The adsorption energy (E ads ) of the gas molecule on the adsorbent substrate is defined as: where E Sub/gas tot , E Sub tot , and E gas tot represent the total energies of the substrate with and without the gas molecule and the isolated gas molecule. It is worth mentioning that we had to carry out the spin-polarized DFT calculations, as the transition metal (Mo) atoms are included in our TMD systems.

Atomic Relaxations
Atomic relaxations were performed using the SIESTA code on supercells (SCell) of MoX 2 , where X represents a chalcogenide atom, either S or Se, in its monolayer form, comprising pristine and single-vacancy-defected cases. To simultaneously preserve the Nanomaterials 2023, 13, 1642 4 of 20 validity of translational symmetry (i.e., Bloch theorem) and assess the effect of the single vacancy defect on the electronic and magnetic properties, we selected the supercells' sizes: 4 × 4, 5 × 5, 6 × 6, 7 × 7, and 8 × 8 primitive cells. Three types of point defects were incorporated into each of these samples. Figure 1 displays the energy-optimized structures corresponding to the case of an SCell of MoS 2 with a size of 5 × 5 primitive cells after the achievement of full atomic relaxations, with the following starting configurations: (a) Pristine, (b) Mo vacancy "V Mo ", (c) S vacancy "V S ", and (d) S 2 di-vacancy "V S2 ". Some relevant geometry data of the relaxed structures of MoS 2 and MoSe 2 in both the pristine state and with the defects mentioned above are summarized in Tables S1 and S2, respectively. For instance, these tables show the Mo-X bond lengths and vacancy contents for all the studied samples, along with the corresponding average formation or binding energy for each configuration.
are included in our TMD systems.

Atomic Relaxations
Atomic relaxations were performed using the SIESTA code on supercells (SCell) of MoX2, where X represents a chalcogenide atom, either S or Se, in its monolayer form, comprising pristine and single-vacancy-defected cases. To simultaneously preserve the validity of translational symmetry (i.e., Bloch theorem) and assess the effect of the single vacancy defect on the electronic and magnetic properties, we selected the supercells' sizes: 4 × 4, 5 × 5, 6 × 6, 7 × 7, and 8 × 8 primitive cells. Three types of point defects were incorporated into each of these samples. Figure 1 displays the energy-optimized structures corresponding to the case of an SCell of MoS2 with a size of 5 × 5 primitive cells after the achievement of full atomic relaxations, with the following starting configurations: (a) Pristine, (b) Mo vacancy "VMo", (c) S vacancy "VS", and (d) S2 di-vacancy "VS2". Some relevant geometry data of the relaxed structures of MoS2 and MoSe2 in both the pristine state and with the defects mentioned above are summarized in Tables S1 and S2, respectively. For instance, these tables show the Mo-X bond lengths and vacancy contents for all the studied samples, along with the corresponding average formation or binding energy for each configuration. For the sake of benchmarking our calculation of formation energy, as indicated in Tables S1 and S2, the results of the average binding energies are shown for the pristine and defected MoS2 and MoSe2 MLs, respectively. In the case of pristine MoS2 ML, we found that the average binding energy (i.e., cohesive energy per atom) was = −4.915 eV/atom, which is in good agreement with both the ab initio calculations of Ding and coworkers [27] (−5.00 eV/atom) using pseudo-atomic numerical orbitals and our previous For the sake of benchmarking our calculation of formation energy, as indicated in Tables S1 and S2, the results of the average binding energies are shown for the pristine and defected MoS 2 and MoSe 2 MLs, respectively. In the case of pristine MoS 2 ML, we found that the average binding energy (i.e., cohesive energy per atom) was E bind = −4.915 eV/atom, which is in good agreement with both the ab initio calculations of Ding and coworkers [27] (−5.00 eV/atom) using pseudo-atomic numerical orbitals and our previous work [28] (−4.890 eV/atom) using Troullier-Martins' norm-conserving relativistic pseudopotentials. In the case of pristine MoSe 2 ML, our result was E bind = −4.401 eV/atom, which is also in good agreement with the latter references (−4.530 eV/atom and −4.401 eV/atom, respectively). It should be emphasized that the greater the charge transfer from the metal layer to the chalcogenide layer, the higher the binding (or cohesive) energy. This is consistent with what is experimentally well-established, that is, MoS 2 is considered the most thermodynamically stable. From the pristine structure perspective, the metal atom "Mo" had a coordination of 6 and the chalcogenide atom "S" had a coordination of 3. Concerning the first type of vacancies, the removal of one Mo atom (creation of V Mo ) left the six sulfur atoms close to V Mo with just two neighbors each. The atomic reconstructions led to a decrease in Mo-S bond lengths but an increase in S-S separation, which increased from d S-S = 3.21 Å to 3.31 Å after the removal of the Mo atom. This increase in the S-S separation displayed an outward relaxation and can be attributed to the increase in electrostatic repulsion between two similarly charged atoms with high electronegativity [29]. It should be emphasized that the outward relaxation of S-S's first nearest neighbors (FNN) was larger than that of S-S's second nearest neighbors (SNN) [29].
Regarding the second type of vacancy, the removal of the S atom left the three neighboring Mo atoms with a coordination of less than 5. The average nearest neighbor Mo-Mo distance in the pristine MoS 2 monolayer was d Mo-Mo = 3.21 Å but it decreased to about 3.12 Å after the removal of the S atom. This decrease in the Mo-Mo distance in the neighborhood of the V S is indicative of an inward relaxation. In the case of the V S , the inward relaxation of the SNN atoms was larger than that of the FNN atoms, likely due to the decrease in electrostatic repulsion between the Mo atoms [30].
The third type of defect, V S2 , further decreased the coordination of its three neighboring Mo atoms to four, rather than six in the case of the pristine state. In Table S1, it can be seen that the bond length increased with respect to the V S , which can be attributed to the increase in the inward relaxations of Mo in the vicinity of the V S2 . The d Mo-Mo increased to 2.87 Å compared to 3.21 Å for the pristine monolayer.

MoSe 2 with Vacancy
In contrast to the previous study of MoS 2 material, the Mo vacancy in MoSe 2 resulted in the inward relaxation of the six Se atoms neighboring the V Mo . The distance d Se-Se = 3.34 Å in the pristine MoSe 2 decreased to 3.27 Å in the case of V Mo . Nevertheless, the single vacancy of Se persisted, causing an inward relaxation of the three Mo atoms neighboring the V Se because the d Mo-Mo decreased to 3.18 Å. This distance decreased even further to 2.85 Å in the case of the V S2, exhibiting a consistent trend and confirming more inward relaxations.
Tables S1 and S2 display the average binding energy, which should be an indicator of the stability of the structure. As a reference, the average binding energies for MoS 2 ML and MoSe 2 ML were found to be 4.915 eV/atom and 4.401 eV/atom, which are in good agreement with our previous studies [31]. The results of the absolute value of binding energy versus the sample size are displayed in Figure S1 for both (a) MoS 2 and (b) MoSe 2 . The common trend for the strength of the binding energies was as follows: | and was likely due to the size of the missing atom(s). Furthermore, the magnitude of the binding energy increased with the sample size as the system approached the bulk structure. In the case of MoSe 2 with V Se and V Se2 defects, the systems had E b converging fast to the bulk value (see, for instance, the samples of 7 × 7 and 8 × 8 PCs, which almost restored the binding energy of the bulk). Figure S1 shows that MoX 2 with V X and V X2 was more thermodynamically stable than MoX 2 with V Mo .
Our results of the formation energy of the V Mo single vacancy in MoS 2 E f = −1.341 eV are also in good agreement with the results of the ab initio calculations of Ding and coworkers [27] (E f = −1.420 eV). Moreover, in the case of the VMo single vacancy in MoSe 2, E f = −1.341 eV was also close to the value found by Ding et al. [27] (E f = −1.210 eV). As mentioned in Section 2, when E f < 0, the formation of such a vacancy requires an exothermic reaction. Our results are not only consistent with those in the literature but also reveal the thermodynamic stability of such defects.

Electronic Structures
Spin-polarized calculations were carried out on all the samples previously relaxed to probe both the band structures and the total density of states (TDOS). Figures 2 and S2 show these results against sample sizes of 4 × 4, 5 × 5, 6 × 6, and 8 × 8 primitive cells and with the following point defects: (a) V Mo , (b) V X , and (c) V X2 in TMD of the MoX 2 monolayer, with X = S and Se, respectively. The in-band structures and spin-up and spindown bands are represented by black and red curves, respectively. The Fermi level is used as an energy reference (i.e., E F = 0) and is represented by horizontal red dashes. By looking at Figure S2, one can deduce that the defected MoS 2 ML is always paramagnetic, regardless of the existing vacancy defect because there is no distinction between spin-up and spin-down in the bands and TDOS plots. Similarly, the same observation can be made for MoSe 2 with chalcogenide vacancies (i.e., V Se and V Se2 ). The only case of V Mo in MoSe 2 was found to produce magnetism and yield magnetization in the samples, irrespective of their sizes.

Effect of Vacancies on Bandgap Energy
By looking at the case of MoS 2 , as shown in Table S1, it can be seen that the pristine form exhibited a direct band gap at the K-point of the Brillouin zone with a value of E g = 1.626 eV. This value is in good agreement with the results of other DFT methods [31,32] that reported values of 1.72 eV, 1.80 eV, and 1.70 eV, respectively, and is slightly lower than the experimental value of 1.80 eV reported by Boker and coworkers [31] using angleresolved photoelectron spectroscopy (ARPES). It is expected that DFT will underestimate the bandgap energy. . As mentioned in Section 2, when < 0, the formation of such a vacancy requires an exothermic reaction. Our results are not only consistent with those in the literature but also reveal the thermodynamic stability of such defects.

Electronic Structures
Spin-polarized calculations were carried out on all the samples previously relaxed to probe both the band structures and the total density of states (TDOS). Figures 2 and S2 show these results against sample sizes of 4 × 4, 5 × 5, 6 × 6, and 8 × 8 primitive cells and with the following point defects: (a) VMo, (b) VX, and (c) VX2 in TMD of the MoX2 monolayer, with X = S and Se, respectively. The in-band structures and spin-up and spin-down bands are represented by black and red curves, respectively. The Fermi level is used as an energy reference (i.e., EF = 0) and is represented by horizontal red dashes. By looking at Figure S2, one can deduce that the defected MoS2 ML is always paramagnetic, regardless of the existing vacancy defect because there is no distinction between spin-up and spindown in the bands and TDOS plots. Similarly, the same observation can be made for MoSe2 with chalcogenide vacancies (i.e., VSe and VSe2). The only case of VMo in MoSe2 was found to produce magnetism and yield magnetization in the samples, irrespective of their sizes.

Effect of Vacancies on Bandgap Energy
By looking at the case of MoS2, as shown in Table S1, it can be seen that the pristine form exhibited a direct band gap at the K-point of the Brillouin zone with a value of Eg = 1.626 eV. This value is in good agreement with the results of other DFT methods [31,32] that reported values of 1.72 eV, 1.80 eV, and 1.70 eV, respectively, and is slightly lower than the experimental value of 1.80 eV reported by Boker and coworkers [31] using angleresolved photoelectron spectroscopy (ARPES). It is expected that DFT will underestimate the bandgap energy.
Both pristine and vacancy-defected MoS2 were shown to be paramagnetic, as both spins were degenerate and spin-orbit coupling had a negligible effect on the band structures, as demonstrated in Figure 2. The bandgap energy versus the sample size is displayed in Figure S3a and shows that the bandgap energy decreased to 0.2 eV, 1.0 eV, and 1.1 eV after introducing the vacancies VMo, VS2, and VS, respectively. It seems that the dangling bonds formed in the vicinity of the vacancy and originating on the neighboring atoms introduced localized gap states. The charge states and characteristics of these gap states are examined in the next sub-section. In brief, the three types of vacancies in MoS2 maintained the semiconducting properties of the material but also introduced some deeplevel "trap" states in the bandgap.
The spin-polarized electronic structures of vacancy-defected MoSe2 MLs are presented in Figure 2. In pristine form, MoSe2 ML has direct bandgap energy at the K-point in the Brillouin zone with a value of Eg = 1.460 eV (see Table S1). This value is in good agreement with the theoretical results reported in the literature, for example, by Ma and coworkers [11] and Liu and coworkers, [33] who reported a similar value of 1.44 eV. Meanwhile, our bandgap value was very close to the experimental value of Ugeda and coworkers [34], who reported 1.61 ±0.11 eV using the scanning tunneling spectroscopy (STS) technique, and the value of 1.66 eV reported by Ross and coworkers [35] using photoluminescence (PL) spectroscopy.
The introduction of a chalcogenide atomic vacancy or di-vacancy (i.e., VSe and VSe2) did not introduce any magnetism into the system, irrespective of the sample size. The only vacancy that had the ability to introduce magnetism into the system was the molybdenum vacancy "VMo". Figure S3b displays the bandgap energy for each spin in the three cases of vacancies in the MoSe2 monolayers versus the sample sizes. Figure S3b shows that the effects of both VSe and VSe2 were very similar in introducing no magnetism but only deep Both pristine and vacancy-defected MoS 2 were shown to be paramagnetic, as both spins were degenerate and spin-orbit coupling had a negligible effect on the band structures, as demonstrated in Figure 2. The bandgap energy versus the sample size is displayed in Figure S3a and shows that the bandgap energy decreased to 0.2 eV, 1.0 eV, and 1.1 eV after introducing the vacancies V Mo , V S2, and V S , respectively. It seems that the dangling bonds formed in the vicinity of the vacancy and originating on the neighboring atoms introduced localized gap states. The charge states and characteristics of these gap states are examined in the next sub-section. In brief, the three types of vacancies in MoS 2 maintained the semiconducting properties of the material but also introduced some deep-level "trap" states in the bandgap.
The spin-polarized electronic structures of vacancy-defected MoSe 2 MLs are presented in Figure 2. In pristine form, MoSe 2 ML has direct bandgap energy at the K-point in the Brillouin zone with a value of E g = 1.460 eV (see Table S1). This value is in good agreement with the theoretical results reported in the literature, for example, by Ma and coworkers [11] and Liu and coworkers, [33] who reported a similar value of 1.44 eV. Meanwhile, our bandgap value was very close to the experimental value of Ugeda and coworkers [34], who reported 1.61 ± 0.11 eV using the scanning tunneling spectroscopy (STS) technique, and the value of 1.66 eV reported by Ross and coworkers [35] using photoluminescence (PL) spectroscopy.
The introduction of a chalcogenide atomic vacancy or di-vacancy (i.e., V Se and V Se2 ) did not introduce any magnetism into the system, irrespective of the sample size. The only vacancy that had the ability to introduce magnetism into the system was the molybdenum vacancy "V Mo ". Figure S3b displays the bandgap energy for each spin in the three cases of vacancies in the MoSe 2 monolayers versus the sample sizes. Figure S3b shows that the effects of both V Se and V Se2 were very similar in introducing no magnetism but only deep donor states in the gap at about 1.0 eV (i.e., at about E C − 0.46 eV). The lowest bandgap energy corresponded to a V Mo defect. Furthermore, it is remarkable that the V Mo introduced magnetization into MoSe 2 with a magnetic moment of about M ≈ 4 µ B , which can be attributed to the absence of four electrons in the d shell of the Mo vacancy and the localization of the wave function on the FNN chalcogenide atoms. In addition to the formation of the magnetic moment, the results of the band structures revealed the occurrence of half-metallicity in just two samples with respective sizes of 4 × 4 and 5 × 5 primitive cells, as shown in Figure S3b. Although spin-down exhibited semiconducting behavior with a bandgap energy of about 0.708-0.747 eV in the latter samples, the spinup exhibited a metallic transition by having few bands crossing the Fermi level. Halfmetallicity was also reported by Ma et al. [11] in 2011 on a sample of 4 × 4 PCs of MoSe 2 ML with V Mo . Because of its importance in spintronic device applications, the origins of the half-metallicity behavior should be further investigated (see the sub-sections below).

Type of Gap States Due to Vacancies
Defects, specifically vacancies, result in the formation of dangling bonds, which usually introduce gap states. Furthermore, it is known that in TMD, the metal atoms predominantly contribute to the electronic structures of the conduction-band minimum (CBM) and the valence-band maximum (VBM) [29]. As can be observed in Figure 3 and Figure S2, the existence of vacancies can introduce energy levels near the Fermi energy (within the bandgap) and can shift the VBM and CBM with respect to the vacuum level or the original band edges of the pristine state. The shifts of the VBM and CBM are more pronounced in the case of the V Mo than in any other case.

Magnetic Properties
Spin-polarized band structure calculations confirmed that all the studied systems were paramagnetic, except for one case that corresponded to the Mo vacancy in MoSe2 ML. The calculation of the magnetic moment, as shown in Table S2, showed that M = 3.99 µB, irrespective of the sample size. This value is comparable to the value of 3.27 µB reported by Ma and coworkers [11] using VASP on a small sample size of 4 × 4 PCs. Figure 3 dis-  No spin was included in the case of MoS 2, with and without defects, as the system was paramagnetic. Spin was included only in the case of V Mo -defected MoSe 2, as the system became ferromagnetic. Consistent with Figure S3, the energy levels corresponding to V Mo were always the lowest in energy and were located close to the Fermi level with respect to the VBM, as shown in Figure S4. By using the Fermi level as an energy reference in Figure S4, for MoS 2 (Figure S4a), the defect states corresponding to V S and V S2 were located above E F , whereas those of V Mo were located below E F . This indicates that the dangling bonds of V Mo were negatively charged, whereas those of V S and V S2 were empty of charges. The gap states due to the vacancies in MoS 2 ML were previously studied on a small sample of 4 × 4 PCs by Feng and coworkers [36].
Concerning the case of MoSe 2 presented in Figure S4b, the energy levels due to the V Se and V Se2 seemed to be located at or slightly above the CBM, making resonance states at the conduction band. The Mo vacancy was more interesting as it yielded not only magnetism but also half-metallicity in MoSe 2 . The energy levels corresponding to the spin-down states were always located above the Fermi level at an energy of about E ≈ E F + 0.2 eV, thus exhibiting semiconducting behavior. However, the energy levels corresponding to the spin-up states were located just near the Fermi level (i.e., in the two cases of the 4 × 4 and 5 × 5 PC samples, the gap states due to the defects met the Fermi level but in the other three 6 × 6, 7 × 7, and 8 × 8 PC samples, they were located slightly below the Fermi level). Hence, two samples were characterized to produce the half-metallicity characteristic. This is further investigated in the next sub-section, where the behaviors of the spin-polarized eigenfunctions of HUMO/LUMO and the Fermi-level states (of both spins) and the orbital density of states (ODOS) are studied.

Magnetic Properties
Spin-polarized band structure calculations confirmed that all the studied systems were paramagnetic, except for one case that corresponded to the Mo vacancy in MoSe 2 ML. The calculation of the magnetic moment, as shown in Table S2, showed that M = 3.99 µ B , irrespective of the sample size. This value is comparable to the value of 3.27 µ B reported by Ma and coworkers [11] using VASP on a small sample size of 4 × 4 PCs. Figure 3 displays the spin profiles (magnetic moment or spin-vector distribution) in the cases of samples 4 × 4, 5 × 5, 6 × 6, 7 × 7, and 8 × 8 PCs, which can be used to assess the origin of the orbitals/species contributing to the magnetization in MoSe 2 with a Mo vacancy. Figure 3 shows that the spin vectors of the atoms in the vicinity of the vacancy were aligned parallel to the same direction (i.e., indicating the formation of "ferromagnetism"). The spin distribution was localized on sites neighboring the Mo vacancy and was independent of the sample size. Different species of atoms (i.e., Se and Mo atoms) made different contributions to the magnetic moments. Se atoms, which were closer to the vacancy, made a greater contribution to the magnetization through the dangling bonds. In addition, there were six Se atoms consisting of the FNNs of V Mo , each contributing 0.414 µ B , and six Mo atoms consisting of the SNNs of V Mo , each contributing 0.212 µ B . Hence, the total magnetic moment due to the FNN and SNN contributions was 3.838 µ B (i.e., M = 6 × 0.414 + 6 × 0.212 = 3.838 µ B ). The rest of the atoms around the vacancy contributed very little to the magnetization.

Origin of Half-Metallicity
Half-metallicity is a collective physical phenomenon that cannot take place unless the system is ferromagnetic. Usually, the starting point is that both spins behave as semiconducting and then under certain conditions, one of them becomes conducting. Or vice versa, initially, the two spins are metallic, and if some physical conditions are provided, one spin changes character to become semiconducting. In our present case, both spins had a semiconducting character, but with the incorporation of a Mo vacancy into just two samples of MoSe 2 (sizes 4 × 4 and 5 × 5 PCs), the spin-up became conducting and the system became half-metallic. The coexistence of two characteristics (i.e., metallic and semiconducting) for the two independent spins can be attributed to the molar content of the Mo vacancy. The main requirement for half-metallicity is that the compromised content of V Mo cannot be too low or too high. This may be due to an interaction between the magnetic moments of the V Mo and its mirror vacancies, as the Hamiltonian is constructed to preserve the periodic boundary conditions. To investigate the half-metallic character in the MoSe 2 samples with V Mo of sizes 4 × 4 and 5 × 5 PCs, it is necessary to examine the eigenwave functions of the spin-up states at the Fermi level and both the HOMO and LUMO states of the spin-down states, as well as the orbital density of states (ODOS) for three sample sizes (i.e., 4 × 4, 5 × 5, and 6 × 6 PCs) to better understand the metallization of the spin-up state when the density of V Mo becomes moderately high, particularly in the small samples of sizes 4 × 4 and 5 × 5 PCs.
According to the point-group theoretical analysis, [29] in trigonal prismatic symmetry, the five d states of the metal atom in TMD (e.g., Mo atom) should split into three subgroups: (i) a singlet A 1 consisting of the d z 2 orbital, (ii) a doublet E' consisting of the d xy and d x 2 −y 2 orbitals, and (iii) a doublet E" consisting of the d yz and d zx orbitals. With this in mind, Figures 4 and 5 should be discussed simultaneously. Figure 4 displays the results of the PDOS and ODOS for two samples of sizes 4 × 4 and 5 × 5 PCs. The PDOS plots (in Figure 4a) show that the V Mo resulted in dangling bonds on the neighboring six Se atoms (i.e., two gap states of the spin-up and one gap state of the spin-down). Both the PDOS and ODOS (Figure 4a,b) show that the spin-up at the Fermi level was mainly due to the contributions of the d z 2 and d xy orbitals of the Mo atoms and the s and p orbitals of the Se atoms close to the V Mo vacancy. This is consistent with the group theory, which predicted that the preceding d states would be close to the Fermi level (i.e., comprising the pristine HOMO/LUMO states, respectively); therefore, the magnetic interaction between the vacancy and its mirror may be the reason for mixing the two d states and producing the metallization of the spin-up states. One can further see that the second gap state of the spin-up located below the Fermi level was mainly due to the contributions of the Se atoms and thus can be solely attributed to the dangling bonds attached to the Se atoms in the vicinity of the vacancy. The third gap state, located slightly higher than the Fermi level in energy, is considered the LUMO state of the spin-down states and was clearly localized on the (Mo and Se) atoms in the vicinity of the vacancy. Figure 5 displays the eigenfunction plots of the HOMO/LUMO states of the spindown and the Fermi states of the spin-up. The figure corroborates our discussions of the PDOS and ODOS. For both samples of sizes 4 × 4 and 5 × 5 PCS, regarding the spin-down states, the HOMO was mainly due to the contributions of both the d z 2 and d xy orbitals of the Mo atoms, whereas the LUMO was localized on the Se atoms in the vicinity of the vacancy. On the other hand, for the spin-up states at the Fermi level, the eigenstate can be attributed to the contributions of the d z 2 and d xy orbitals of the Mo atoms, located in the vicinity of the V Mo . In Figure 5, for the samples of sizes 6 × 6 and 8 × 8 PCs, the disappearance of half-metallicity can be attributed to the decoupling of the vacancy-vacancy interaction as the distance between them increased beyond the critical coherence length of ferromagnetic coupling (ferromagnetic coupling was estimated to be approximately d c ≈ 12 Å by Ma et al. [11]). Consequently, relaxations of both the spin-up and spin-down states were carried out for the SNN and FNN atoms near the V Mo in the 6 × 6 and 8 × 8 samples. This is consistent with Figure 3, which shows the robustness of spin localization near the defect to maintain a constant magnetization. Additionally, it reveals the ferromagnetic decoupling between the Mo vacancies and the loss of half-metallic character in the larger samples.   Figure 5, for the samples of sizes 6 × 6 and 8 × 8 PCs, the disappearance of half-metallicity can be attributed to the decoupling of the vacancy-vacancy interaction as the distance between them increased beyond the critical coherence length of ferromagnetic coupling (ferromagnetic coupling was estimated to be approximately dc ≈ 12 Å by Ma et al. [11]). Consequently, relaxations of both the spin-up and spin-down states were carried out for the SNN and FNN atoms near the VMo in the 6 × 6 and 8 × 8 samples. This is consistent with Figure 3, which shows the robustness of spin localization near the defect to maintain a constant magnetization. Additionally, it reveals the ferromagnetic decoupling between the Mo vacancies and the loss of half-metallic character in the larger samples.

Anti-Ferromagnetism versus Ferromagnetism
Usually, the magnetic interaction between two localized magnetic moments is antiferromagnetic in order to form the ground state. However, this is only the case when the localized moments are in close proximity. For instance, in our recent work [37], we carried out a comparative assessment of a double-atom catalyst consisting of manganese Mn 2 embedded in a large pore of C 2 N by comparing the anti-ferromagnetic (AFM) and ferromagnetic (FM) states. The results confirmed that the AFM state was the ground state, as it had a total energy lower than the FM state (i.e., ∆E = E AFM − E FM = −0.595 eV). Furthermore, the bond length of the Mn 2 dimer in the C 2 N pore was shorter in the AFM state (i.e., d (AFM) Mn−Mn = 2.207 Å [37]. So, the impact of our previous work on the current investigation is that one should question what would happen when the vacancies physically exist in one sample and possibly interact via AFM coupling, particularly if the AFM state is the ground state and the FM state is the excited state. In this case, what would the effect be on half-metallicity? So, here we used a large sample of MoSe 2 consisting of 8 × 8 PCs, and four vacancies of Mo (i.e., V Mo ) were included in a uniform distribution by repeating the sample of 4 × 4 PCs, with a single vacancy, V Mo , in both xy directions, as shown in Figure 6a. The new sample was four times the size of the original 4 × 4 PC sample. We assessed both the AFM and FM states on this large new sample. The results showed that the atomic relaxation always led to an AFM state, which represented the ground state. This trend might be due to the fact that the distance between the magnetic impurities was so large (i.e., d Mn−Mn = 13.20 Å), which favored the FM state as the ground state over the FM state. Figure 6b shows the spin-polarized band structure and TDOS, which confirm the existence of the half-metallic character. Figure 6b shows that the bands of the spin-up states (in orange) were metallic, dispersive, and crossed the Fermi level, whereas the bands of the spin-down states (in black) remained semiconducting. Moreover, the spin vectors, as shown in the side-view of Figure 6a, were localized within the vacancies. This confirms that the ground state was ferromagnetic and the material maintained its half-metallic behavior. In other words, half-metallicity requires the provision of a magnetic-coupling interaction of intermediate strength at the level of ferromagnetic coupling.

Anti-Ferromagnetism versus Ferromagnetism
Usually, the magnetic interaction between two localized magnetic moments is antiferromagnetic in order to form the ground state. However, this is only the case when the localized moments are in close proximity. For instance, in our recent work [37], we carried out a comparative assessment of a double-atom catalyst consisting of manganese Mn2 embedded in a large pore of C2N by comparing the anti-ferromagnetic (AFM) and ferromag- of the spin-up states (in orange) were metallic, dispersive, and crossed the Fermi level, whereas the bands of the spin-down states (in black) remained semiconducting. Moreover, the spin vectors, as shown in the side-view of Figure 6a, were localized within the vacancies. This confirms that the ground state was ferromagnetic and the material maintained its half-metallic behavior. In other words, half-metallicity requires the provision of a magnetic-coupling interaction of intermediate strength at the level of ferromagnetic coupling.

Environmental Effects
It is necessary to assess the environmental effects on the properties of TMD (MoX2, X = S, Se) MLs, especially the half-metallic character. We mainly focused on atmospheric gases that are oxidizing (e.g., O2, H2O, and O3) to study their adsorption properties on our

Environmental Effects
It is necessary to assess the environmental effects on the properties of TMD (MoX 2 , X = S, Se) MLs, especially the half-metallic character. We mainly focused on atmospheric gases that are oxidizing (e.g., O 2 , H 2 O, and O 3 ) to study their adsorption properties on our current samples of MoX 2 (X = S, Se) monolayers. Figure 7 shows the DFT results of the atomic relaxations of these three gases on the four samples: (a) MoS 2 :V Mo (i.e., on a Mo vacancy); (b) MoS 2 :V S (i.e., on a S vacancy); (c) MoSe 2 :V Mo (i.e., on a Mo vacancy); and (d) MoSe 2 :V Se (i.e., on a Se vacancy). The computational supercell contained 4 × 4 PCs, with the inclusion of one vacancy. The gas molecule was initially placed on the top of the vacancy within a distance of about 3.0 Å to start the process of atomic relaxation. The results of the atomic relaxations, which are summarized in Table 1, comprise the adsorption energy, charge transferred with the molecule, and the closest distance between the molecule and substrate (adsorbent bed). The results of the atomic relaxations can be summarized as follows: (i) The common trend among all gas adsorption processes was the weak physisorption exhibited by O 2 molecules on all four samples. The adsorption energy reported in Table 1 was small and within the range of about [−0.18, −0.10] eV. (ii) Ozone molecule O 3 seemed to be the strongest and interacted with all samples through either a chemisorption associated with a dissociation (e.g., see Figure 7a,c,d) or a strong physisorption (e.g., see Figure 7b). (iii) The vapor molecule H 2 O seemed to exhibit weak physisorption on most of the samples (e.g., see Figure 7a,c,d) but it exhibited chemisorption associated with dissociation on one sample, MoS 2 :V S (e.g., Figure 7b).   Table 1, comprise the adsorption energy, charge transferred with the molecule, and the closest distance between the molecule and substrate (adsorbent bed). The results of the atomic relaxations can be summarized as follows: (i) The common trend among all gas adsorption processes was the weak physisorption exhibited by O2 molecules on all four samples. The adsorption energy reported in Table 1 was small and within the range of about [−0.18,−0.10] eV. (ii) Ozone molecule O3 seemed to be the strongest and interacted with all samples through either a chemisorption associated with a dissociation (e.g., see Figure 7a,c,d) or a strong physisorption (e.g., see Figure 7b). (iii) The vapor molecule H2O seemed to exhibit weak physisorption on most of the samples (e.g., see Figure 7a,c,d) but it exhibited chemisorption associated with dissociation on one sample, MoS2:VS (e.g., Figure 7b).

Conclusions
The effects of intrinsic point defects on the electronic structures and magnetic and spintronic properties of TMD MoX2 monolayers (X = S and Se) are presented based on both localized and plane-wave base sets using SIESTA and VASP packages, respectively. Among the studied defects, we selected (i) a molybdenum vacancy "VMo", (ii) a chalcogenide vacancy "VX", and (iii) a di-chalcogenide vacancy "VX2". The vacancies were studied as single-point defects in periodic samples of sizes ranging from 4 × 4 to 8 × 8 PCs. The scaling in the sample size was explored to stimulate the quantum and magnetic effects of the vacancy-vacancy interactions between a real vacancy and its six mirror images. Fur-

Conclusions
The effects of intrinsic point defects on the electronic structures and magnetic and spintronic properties of TMD MoX 2 monolayers (X = S and Se) are presented based on both localized and plane-wave base sets using SIESTA and VASP packages, respectively. Among the studied defects, we selected (i) a molybdenum vacancy "V Mo ", (ii) a chalcogenide vacancy "V X ", and (iii) a di-chalcogenide vacancy "V X2 ". The vacancies were studied as single-point defects in periodic samples of sizes ranging from 4 × 4 to 8 × 8 PCs. The scaling in the sample size was explored to stimulate the quantum and magnetic effects of the vacancy-vacancy interactions between a real vacancy and its six mirror images. Furthermore, the effects of the adsorption of atmospheric gases (e.g., H 2 O, O 2 , O 3 ) on the electronic and magnetic properties were analyzed to access their robustness under ambient conditions. The results can be summarized as follows: (1) In pristine monolayer form, both MoS 2 and MoSe 2 were paramagnetic semiconductors with direct bandgaps at the K-point of energies 1.626 eV and 1.460 eV, respectively. (2) The samples of MoS 2 MLs with any kind of vacancy were found to be paramagnetic.
(3) The samples of MoSe 2 MLs with V S and V S2 were also found to be paramagnetic.
Only in the case of the "V Mo " Mo vacancy could magnetism be introduced into the MoSe 2 samples to become ferromagnetic. The magnetic moment due to the V Mo was estimated to be 3.99 µ B , which remained robust independent of the sample size. However, the samples of sizes 4 × 4 and 5 × 5 PCs were found to exhibit half-metallicity, where the spin-up state changed and became metallic. (4) We investigated the origins of half-metallicity, in particular for the small-sized samples (i.e., 4 × 4 and 5 × 5 PCS) of MoSe 2 with a single vacancy of V Mo . So, we calculated the PDOS, ODOS, and HOMO/LUMO, as well as the eigenstates at the Fermi energy for both the spin-down and spin-up states. Our results showed that half-metallicity can be attributed to the mixing of the d z 2 and d xy orbitals of the Mo atoms triggered by the FMC interactions. Half-metallicity is a fundamental requirement for the development of spintronic devices. It requires the presence of FMC interactions, which are magnetic-coupling interactions of intermediate strength. However, the effects of the environment (atmospheric gases) must be minimized in order to preserve the half-metallic properties. Understanding the conditions that lead to the appearance/disappearance of half-metallicity is crucial for spintronic device applications.
Supplementary Materials: The following supporting information can be downloaded at https: //www.mdpi.com/article/10.3390/nano13101642/s1, Table S1: Geometrical and electronic structure parameters of MoS 2 ML, with and without vacancies. The data include the supercell size (N x × N y primitive cells), number of atoms (N), bond length in the vicinity of the vacancy (b in Å), average formation energy (Eb eV/atom), and bandgap energy (Eg in eV). D and I stand for the direct and indirect bandgap, respectively. Table S2: Geometric and electronic structure parameters of MoSe 2 ML, with and without vacancies. The data include the supercell size (N x × N y primitive cells), number of atoms (N), bond length in the vicinity of the vacancy (b in Å), average formation energy (E b eV/atom), and bandgap energy (E g in eV). D and I stand for the direct and indirect bandgap, respectively. Figure  Funding: This project was sponsored by the National Water and Energy Center at the UAE University (Grants # 12R125 and 12R162).

Data Availability Statement:
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.