Mechanical Properties of Monolayer MoS2 with Randomly Distributed Defects

The variation of elastic constants stiffness coefficients with respect to different percentage ratios of defects in monolayer molybdenum disulfide (MLMoS2) is reported for a particular set of atomistic nanostructural characteristics. The common method suggested is to use conventional defects such as single vacancy or di vacancy, and the recent studies use stone-walled multiple defects for highlighting the differences in the mechanical and electronic properties of 2D materials. Modeling the size influence of monolayer MoS2 by generating defects which are randomly distributed for a different percentage from 0% to 25% is considered in the paper. In this work, the geometry of the monolayer MoS2 defects modeled as randomized over the domain are taken into account. For simulation, the molecular static method is adopted and study the effect of elastic stiffness parameters of the 2D MoS2 material. Our findings reveals that the expansion of defects concentration leads to a decrease in the elastic properties, the sheer decrease in the elastic properties is found at 25%. We also study the diffusion of Molybdenum (Mo) in Sulphur (S) layers of atoms within MoS2 with Mo antisite defects. The elastic constants dwindle in the case of antisite defects too, but when compared to pure defects, the reduction was to a smaller extent in monolayer MoS2. Nevertheless, the Mo diffusion in sulfur gets to be more and more isotropic with the increase in the defect concentrations and elastic stiffness decreases with antisite defects concentration up to 25%. The distribution of antisite defects plays a vital role in modulating Mo diffusion in sulfur. These results will be helpful and give insights in the design of 2D materials.


Introduction
The early reports on graphene by Giem [1], specifcally considering two dimensional (2D) materials like transition metal dichalcogenide (MX 2 , M = Mo, W; X = S, Se, Te), or particularly Monolayer MoS 2 (MLMoS 2 ) have lure in the applications of electronics to structural and functional composites [2][3][4][5] owing to its exceptional electrical, optical, and mechanical properties. In structural applications, the most appealing feature of MoS 2 is the in-sheet elastic stiffness of perfect sp 3 covalently bonded structures [6,7]. Two dimensional MoS 2 is tri-layer as opposed to graphene, which is only single-layer, monolayer MoS 2 having a system of three atomic thickness layers with the transition metal (Mo) Wang et al. [16,18] proposed the low concentration defects on the topology, whereas we reported for defects varying from low to high concentration to mimic the mono-layer topology in MoS 2 . This method requires multiple defects with the varying defect concentration on the topology generated by using conventional molecular static modeling, namely topology-based atomistic defects caused by using random equilibrium distribution of the domain utilized in this study. Jinhua Hong et al. [26] pointed out that antisite defects with replacing molybdenum as sulfur, which are prevalent point defects in PVD-grown MoS 2 , while the sulfur vacancies are predominant in mechanical exfoliation and CVD specimens in MoS 2 .
The MoS 2 is also interesting due to the electrical, thermal, and optical properties. The paper [27] presents results of influence of vacancies on electrical properties of MoS 2 , this work focuses on studying the mechanical properties of monolayer MoS 2 with randomly distributed defects by systematically varying the vacancy concentration, The new results obtained for different levels of defects are presented. The results are shown for different random defect distributions. The results may be used in future works were the mechanical nanomachines and nanosystems based on MoS 2 sheets optimization are considered. The simulation methodology is detailed in Section 2.1, while the theoretical framework used for defectcted sheet analyzing is defined in Section 2.2. Results are presented in Section 3, and discussion and conclusions are presented in Section 4.

Methodology of Molecular Simulations
This study investigate the mechanical properties of pristine and defective monolayer MoS 2 structure including the relaxation strucutre through molecular static simulations and the calculations are performed in atomistic based LAMMPS (Large-Scale Atomic Molecular Massive Parallel Simulator, Sandia National Laboratories, Albuquerque, USA) [28,29], open source package developed by Sandia National Laboratories to model. The heart of the molecular simulations is the inter-atomic potential, which applies to describe the interaction among atoms. From the inter-atomic potential, we can obtain the new properties of any material like theoretical strength, elastic moduli, and Hooke's law. Stillinger-Weber (SW) potential employs an effective approach to describe the interactions in MoS 2 by considering all possible interactions between Mo and S [30][31][32].
To understand all its physical properties and know how to control these properties for specific usage, one needs to know the accurate interatomic potential. The potential used in our simulation representing Mo and S atoms developed by Stillinger-Weber (SW) potential of MoS 2, which includes all possible interactions Mo and S [30,31] as it is a many-body potential (the potential consists of one-,twoand three-body terms) which perfectly fitted to mono-layer MoS 2 . In this work, we performed molecular static (MS) simulations to generate the elastic constants of the monolayer MoS 2 . The parameters used in the simulation will influence the accuracy of the computed results. Consequently, we used the well parameterized molecular simulation that can describe the variety of bulk material properties. The bond interaction by two-body interaction acts towards the bond deformation while the three-body interaction conducts itself towards the angular rotation. The total potential of a system ϕ tot can be written as The two-body interaction potential Q 2 takes the following form. The three-body interaction potential Q 3 in Equation (1) is modeled as where exponential function gives a smooth decay of the potential to zero at the cut-off, which is essential to save the energy. Q 2 and Q 3 represents the two body(bond stretching) and three body (bond bending) interactions. The r ij , r ik and α ijk are the pair separations and angle between the separation on atom i respectively. The potential parameters are X, Y, Z, σ, along with r max cutoff radii and equilibrium angles and they rely upon on the atoms interacting with each other, for instance, Xij is the parameter for X for the pairwise interaction between atom i of category I and atom j of category J.

Crystal Structure
A cubic domain of monolayer MoS 2 with 1000 atoms was generted using LAMMPS is shown in Figure 1. The crystal orientation was aligned in order that all three principal directions of the crystal align with global coordiante system. The domain has 4.74 × 8.21 × 0.615 nm which is equal to 15a × 15a × 1a, as shown in Figure 1b. The atomistic model in the present work is developed by considering the hexagonal lattice structure of MoS 2 sheet with the lattice constant of a = 3.16 Å and c = 6.15 Å. The main objective of this work is to understand the topology of defects by varying size of domain. After the sheets had been relaxed over sufficiently long peiod of time by conjugate gradient optimization till the energy is conserved. The mechanical properties of MoS 2 are estimated in the next section. x (cos − cos 0, ) 2 where exponential function gives a smooth decay of the potential to zero at the cut-off, which is essential to save the energy. Q2 and Q3 represents the two body(bond stretching) and three body (bond bending) interactions. The rij, rik and αijk are the pair separations and angle between the separation on atom i respectively. The potential parameters are X, Y, Z, σ, along with r max cutoff radii and equilibrium angles and they rely upon on the atoms interacting with each other, for instance, Xij is the parameter for X for the pairwise interaction between atom i of category I and atom j of category J.

Crystal Structure
A cubic domain of monolayer MoS2 with 1000 atoms was generted using LAMMPS is shown in Figure 1. The crystal orientation was aligned in order that all three principal directions of the crystal align with global coordiante system. The domain has 4.74 × 8.21 × 0.615 nm which is equal to 15a × 15a × 1a, as shown in Figure 1b  The atomic model consists of 25,000 atoms in the nanosheet for the domain size of 12.95 nm 2 to 56.11 nm 2 . In this work for each simulation, the initial configuration of the MoS2 sheet was prepared, and the random defects were introduced.The periodic boundary conditions in all directions were enforced, and monolayer MoS2 sheets with 65 × 65 × 1 lattices are presented in the results section. Point defects (vacancies) are created by randomly removing atoms with probability from the lattice sites. The vacancies or defects are randomly placed in the crystal using a random number generator with uniform distribution. Defect concentrations range from Vc = 0.01% to 25% are considered. Hundreds of simulations were performed for each vacancy concentration with different defects locations .
In this work, to see the antisite defects, in particular, the molybdenum (Mo) atoms diffusing in sulfur layers with an assortment of random defect concentrations with different configurations is chosen. There were 5, 56, 120, 296, 571, 867, 1143 and 1427 atoms of molybdenum antisite defects in the MoS2 sheet that contains 25,000 atoms, which stand for a defect concentration of 0.1%, 1%, 2%, 5%, 10%, 15%, 20%, and 25%, respectively. Local structures and diffusion dynamics have a significant influence on the interaction of defects for significant defect fraction ratios. Thus, we explore the impact of antisite defects in MoS2. The atomic model consists of 25,000 atoms in the nanosheet for the domain size of 12.95 nm 2 to 56.11 nm 2 . In this work for each simulation, the initial configuration of the MoS 2 sheet was prepared, and the random defects were introduced.The periodic boundary conditions in all directions were enforced, and monolayer MoS 2 sheets with 65 × 65 × 1 lattices are presented in the results section. Point defects (vacancies) are created by randomly removing atoms with probability from the lattice sites. The vacancies or defects are randomly placed in the crystal using a random number generator with uniform distribution. Defect concentrations range from Vc = 0.01% to 25% are considered. Hundreds of simulations were performed for each vacancy concentration with different defects locations.

Elastic Constants of Monolayer MoS2 Using Molecular Statics (MS)
In this work, to see the antisite defects, in particular, the molybdenum (Mo) atoms diffusing in sulfur layers with an assortment of random defect concentrations with different configurations is chosen. There were 5, 56, 120, 296, 571, 867, 1143 and 1427 atoms of molybdenum antisite defects in the MoS 2 sheet that contains 25,000 atoms, which stand for a defect concentration of 0.1%, 1%, 2%, 5%, 10%, 15%, 20%, and 25%, respectively. Local structures and diffusion dynamics have a significant influence on the interaction of defects for significant defect fraction ratios. Thus, we explore the impact of antisite defects in MoS 2 .

Elastic Constants of Monolayer MoS 2 Using Molecular Statics (MS)
Molecular statics calculations have been performed using LAMMPS code to generate the elastic constants of MoS 2 at 0 K. We know that the MoS 2 sheet consists of a tri-layer, and it experiences strong covalent bonding inward and weak van der Waal's interaction over the tri-layer due to the polarization effect [22]. The elastic properties of MoS 2 are determined as the derivative of the stress against the external strain according to Hooke's law. The generalized Hooke's law, for the number of independent elastic constants in MoS 2 , is three and can be written as There are three independent elastic constants for MoS 2 , i.e., C 11 is the coefficient of elastic constant relations due to σ 11 to ε 11 similarly for C 22 , and C 12 . The C ij values are correlated to the equal volume of the MoS 2 unit cell. Therefore, the vacuum space has been set up large enough in the z-axis to avoid the interlayer interactions in MoS 2 monolayer; the C ij constants then have to rescaled z = t 0 to the actual thickness of monolayer MoS 2 . So, we have set t 0 = 6.15 Å, i.e., one half of the out-of-plane lattice constant of bulk MoS 2 . The MoS 2 structure is fully optimized to its minimum energy by conjugate gradient minimization until the energy is converged. The specific finite lattice distortion of the simulation box leads to a change in energy during convergence, and the respective final elastic constants are obtained [33,34].
The second-order elastic constants for elastic matrix express as: where S 0 is the area of the sample, t 0 represents the thickness of MoS 2 monolayer, E is the elastic energy, and ε is the strain tensor. In polynomial form for 2D materials discussed in [34,35], the elastic energy E(ε) of MoS 2 is expressed as: The ε xx and ε yy are the longitudinal strain in x and y directions and can also be represented as ε 1 and ε 2 respectively in terms of Voight notations, and ε xy is the applied shear strain in xy plane. The MoS 2 sheet is arranged as zigzag and armchair in the x and y-axis. ε ij's and C ij's are the corresponding infinitesimal strain tensors and linear elastic constants [34,36]. Born set the benchmark mechanical stability for graphene-like 2D materials, which explains C 11 > 0, C 11 > C 12 , and C 12 > 0 and the condition to satisfy for the 2D materials to be isotropic is C 11 ≈ C 22 and C 12 . The elastic energy in general for 2D materials for finite distortion is expressed as After the MoS 2 sheet is perfectly relaxed or energy is fully converged, the independent elastic constants are extracted for respective strains.

MoS 2 Sheet with Pristine and Random Vacancy Defects
Our findings shows that the elastic constants for a MoS 2 sheet with an infinite system size are C 11 = C 22 = 149.42 N/m, C 12 = 52.29 N/m, which interpret the isotropic nature of the material. One Since the independent elastic constants for a 2D material MoS2 are five within the notation employed, the constant elastic tensor is a 3 × 3 symmetric matrix. Due to symmetry, some components vanishes and the elasticity matrix elements will get reduced to three independent elements. The calculated elastic constants of MoS2 are shown in Table 1.  [8] 180 ± 60 180 ± 60 -Li. M et al [19] 148.4 148.4 42.9 Nguyen T.H et al [28] 130.4 130.4 26.5 All elastic constants Cij calculated by conjugate gradient minimization using molecular statics simulation in comparison with literature results are given in Table 1. Due to symmetry C11 ≈ C22 the obtained elastic constants marginally diverse from the reference data. We can see that C11 = 149 N/m, which corresponds to a good Young's modulus of 242 GPa. Bertolazzi et al. [5] reported the elastic stiffness of MoS2 monolayer is 180 ± 60 N/m, which corresponds to a good Young's modulus of 270 ± 100 GPa by the atomic force microscope (AFM) experiment method. The experimental results are higher than our simulation results from this study because in AFM method, tip enforced on the sheet consists of monolayer or multilayer MoS2 suspended on the layer incorporate with an array of circular holes are under biaxial tensile stress whereas we have applied uniaxial stress to the monolayer. Li et al. [16] performed MD simulation under the uniaxial test presented that C11 is found to be 199 GPa for the 1H MoS2, our results are while Nguyen T.H. [25] obtained an average young's modulus of 201 GPa for monolayer MoS2. Note the deviation due to the performed DFT calculations, which are derived from the finite difference approach by Thermo-pw code, and we have used latest SW potential which can be used for higher temperatures as well. A comparison of elastic constants from this work are consistent with the experimental and simulation results. Regardless of vacancies, the average value of C12 and C21 is used to assess the physical properties of MoS2. It is apparent that C12 ≈ C21 due to the symmetric stiffness matrix.
We now describe the effects of modeling monolayer MoS2 sheet with randomly distributed defect fraction presented in Figure 3. The geometry optimized average elastic constants for MoS2 under different defect fractions are given in Table 2. The elastic constants of monolayer MoS2 vs. the defect percentage are illustrated in comparison to the perfect MoS2 sheet. The MoS2 monolayer sheet Since the independent elastic constants for a 2D material MoS 2 are five within the notation employed, the constant elastic tensor is a 3 × 3 symmetric matrix. Due to symmetry, some components vanishes and the elasticity matrix elements will get reduced to three independent elements. The calculated elastic constants of MoS 2 are shown in Table 1.  [8] 180 ± 60 180 ± 60 -Li. M. et al. [19] 148.4 148.4 42.9 Nguyen T.H. et al. [28] 130.4 130. 4 26.5 All elastic constants C ij calculated by conjugate gradient minimization using molecular statics simulation in comparison with literature results are given in Table 1. Due to symmetry C 11 ≈ C 22 the obtained elastic constants marginally diverse from the reference data. We can see that C 11 = 149 N/m, which corresponds to a good Young's modulus of 242 GPa. Bertolazzi et al. [5] reported the elastic stiffness of MoS 2 monolayer is 180 ± 60 N/m, which corresponds to a good Young's modulus of 270 ± 100 GPa by the atomic force microscope (AFM) experiment method. The experimental results are higher than our simulation results from this study because in AFM method, tip enforced on the sheet consists of monolayer or multilayer MoS 2 suspended on the layer incorporate with an array of circular holes are under biaxial tensile stress whereas we have applied uniaxial stress to the monolayer. Li et al. [16] performed MD simulation under the uniaxial test presented that C 11 is found to be 199 GPa for the 1H MoS 2 , our results are while Nguyen T.H. [25] obtained an average young's modulus of 201 GPa for monolayer MoS 2 . Note the deviation due to the performed DFT calculations, which are derived from the finite difference approach by Thermo-pw code, and we have used latest SW potential which can be used for higher temperatures as well. A comparison of elastic constants from this work are consistent with the experimental and simulation results. Regardless of vacancies, the average value of C 12 and C 21 is used to assess the physical properties of MoS 2 . It is apparent that C 12 ≈ C 21 due to the symmetric stiffness matrix.
We now describe the effects of modeling monolayer MoS 2 sheet with randomly distributed defect fraction presented in Figure 3. The geometry optimized average elastic constants for MoS 2 under different defect fractions are given in Table 2. The elastic constants of monolayer MoS 2 vs. the defect percentage are illustrated in comparison to the perfect MoS 2 sheet. The MoS 2 monolayer sheet is arranged as zig-zag and armchair in x & y directions, which denotes the C 11 , C 12 , and C 22 elastic moduli, respectively. It is clear that chirality slight effect on elastic constants irrespective of defect ratios. The elastic constants C ij nonetheless started dwindling as the defect fraction piled up from 0% to 25%. Its reduction becomes more expeditiously as the defects grow in the sheet. Piling up the defect fraction and maximum ratio up to 25% results in a considerable decline in the elastic constants, which implies the impact is significant. is arranged as zig-zag and armchair in x & y directions, which denotes the C11, C12, and C22 elastic moduli, respectively. It is clear that chirality slight effect on elastic constants irrespective of defect ratios. The elastic constants Cij nonetheless started dwindling as the defect fraction piled up from 0% to 25%. Its reduction becomes more expeditiously as the defects grow in the sheet. Piling up the defect fraction and maximum ratio up to 25% results in a considerable decline in the elastic constants, which implies the impact is significant.  The elastic constants of monolayer MoS 2 nanosheet vs. the defect percentage are presented in Figure 4. The dots denote average values of defects fraction with the fluctuating bar represents the standard deviation that shows maximum and minimum values for hundreds cases with random distributed defects. The result of chirality on the elastic properties of MoS 2 is negligible, despite the prevailing circumstances of the defect fraction. The elastic constants dwindle faster with the increase in the defect fraction, the maximum reduction of elastic constants is at 25%, more significant than 15%, 10%, and 5%, which implies the influence of defect fraction on the elastic constants is found to be substantial. The elastic constants of monolayer MoS2 nanosheet vs. the defect percentage are presented in Figure 4. The dots denote average values of defects fraction with the fluctuating bar represents the standard deviation that shows maximum and minimum values for hundreds cases with random distributed defects. The result of chirality on the elastic properties of MoS2 is negligible, despite the prevailing circumstances of the defect fraction. The elastic constants dwindle faster with the increase in the defect fraction, the maximum reduction of elastic constants is at 25%, more significant than 15%, 10%, and 5%, which implies the influence of defect fraction on the elastic constants is found to be substantial.  To give comprehensive and comparable studies of elastic properties of randomly distributed defects, the elastic constants of MoS2 with varying defect ratios have been studied. The elastic constants of this defective MoS2 versus defect fraction are shown in Figure 4. For comparison, the elastic constant of pristine MoS2 also included in the plots. Figure 4 shows the effect of defects on the elastic constants of MoS2 along with the individual independent elastic constants by comparing that of the pristine MoS2It will be hard to conclude the locations of the unperturbed vacancies as they are distributed randomly throughout the layer. We took an interest in determining that these vacancies To give comprehensive and comparable studies of elastic properties of randomly distributed defects, the elastic constants of MoS 2 with varying defect ratios have been studied. The elastic constants of this defective MoS 2 versus defect fraction are shown in Figure 4. For comparison, the elastic constant of pristine MoS 2 also included in the plots. Figure 4 shows the effect of defects on the elastic constants of MoS 2 along with the individual independent elastic constants by comparing that of the pristine MoS 2 . It will be hard to conclude the locations of the unperturbed vacancies as they are distributed randomly throughout the layer. We took an interest in determining that these vacancies need to form everywhere in the sheet, irrespective of defect fraction ratio. So, to achieve this, we repeated with different random seeds for a sufficient number of times and estimated the elastic constants for each seed. Table 2 displays the average values of the outcomes of the built-in elastic test after repeating the simulation. These results will also motivate us to study the tensile and other properties of MoS 2 with the defects. With the increase in defect ratio, we observe the difference in the elastic constants nonlinearly as expected. It was found that up to 1% of defects had little impact as it trims down to 2.1% rate of elastic constants when compared to defect-free MoS 2 and the impact of this vacancy defects on the elastic constants was not that obvious and can be neglected. When the defect fraction surpasses 2%, the elastic constants start trimming down at a rapid rate. Nevertheless, when defect ratios was 2%, 5%, 10%, and 25%, the decrease of the elastic constants was 4.02%, 13.42%, 28.8%, and 56.5%, respectively, compared with MoS 2 with no defects. This result showed that after exceeding some defect density, the vacancy had a substantial effect and damages the robustness and uniform symmetry of MoS 2 and has a full impact on the elastic tensile behavior of MoS 2 . It was also found that from Figure 4 as the elastic constants fluctuate within a certain range, and this fluctuation occurs due to the locations of the defects placed randomly and changes its location with each test. These results draw attention towards the foundation in randomly distributed vacancies in MoS 2 sheets.

MoS 2 Sheet with Randomly Diffusing Sulfur to Molybdenum (S→Mo)
The concentration of antisite defects, i.e., sulfur atoms to molybdenum atoms, are also randomly distributed in sulfur layers of the MoS 2 sheet. Figure 5 shows the diffusion of sulfur to molybdenum as antisite defect for different percentages of sulfur diffused in monolayer MoS 2 for 25000 atoms nanosheet size. Table 3 shows that the elastic stiffness strengths of 0% to 25% molybdenum doped in sulfur layers in monolayer MoS 2 for sheet size of 65 Å × 65 Å (25000 atoms) increase by about 0.1%, 2%, 1%, 5%, and an impressive peak of 25% when compared with the MoS 2 is observed. The independent elastic constants of 1%, 2%, 5%, 10%, as well as 15% sulphur doped molybdenum in MoS 2 drops by about 1%, 1.5%, 4%, 8%, 11%, 15%, and 19% in comparison to the pristine MoS 2 . Increasing the percentage of sulfur doping in the MoS 2 sheet, the elastic properties decrease. Further, we found that the elastic properties due to sulfur vacancy defects with different percentages drop in great detail when compared the elastic properties due to diffusion at the respective percentage as deduce from Tables 2 and 3. Materials 2020, 13, x FOR PEER REVIEW 10 of 14  The independent elastic constants of 1%, 2%, 5%, 10%, as well as 15% sulphur doped molybdenum in MoS2 drops by about 1%, 1.5% , 4%, 8%, 11.%, 15%, and 19% in comparison to the pristine MoS2. Increasing the percentage of sulfur doping in the MoS2 sheet, the elastic properties decrease. Further, we found that the elastic properties due to sulfur vacancy defects with different percentages drop in great detail when compared the elastic properties due to diffusion at the respective percentage as deduce from Tables 2 and 3. Figure 6 shows the effect of antisite defects molybdenum diffusion in sulfur, also dwindle the elastic constants of MoS2 for the different defect fractions. The change of fractions were in ranges from  Figure 6 shows the effect of antisite defects molybdenum diffusion in sulfur, also dwindle the elastic constants of MoS 2 for the different defect fractions. The change of fractions were in ranges from 0.1% to 25% antisite defects. We started to pile-up the antisite defects and observed the elastic constants becomes more efficient and started to hinder further with the increase in the diffusion, unlike in case of pure defects where we seen the elastic properties drop dramatically with each defect fraction. The elastic constants C ij of antisite defect also shows the trend of decrease in nature when compared to defect-free structure, it decreases from 139.69 N/m for 5% defects to 113.83 N/m for 25 % defects when compared to the of pure defects as seen in Table 3. In order to provide the results of impact of the antisite defect in MoS 2 sheet, again we prepared the random antisite defect models. Hundreds of replications were considered for MoS 2 with 0% to 25% antisite defects. unlike in case of pure defects where we seen the elastic properties drop dramatically with each defect fraction. The elastic constants Cij of antisite defect also shows the trend of decrease in nature when compared to defect-free structure, it decreases from 139.69 N/m for 5% defects to 113.83 N/m for 25 % defects when compared to the of pure defects as seen in Table 3. In order to provide the results of impact of the antisite defect in MoS2 sheet, again we prepared the random antisite defect models. Hundreds of replications were considered for MoS2 with 0% to 25% antisite defects.

Conclusions
The elastic constants for MoS2 monolayer using molecular statics simulation in great detail were investigated. MoS2 is flexible and isotropic for small deformations and the results obtained from this study are compared with the previous literature for defect-free MoS2 and progress towards higher

Conclusions
The elastic constants for MoS 2 monolayer using molecular statics simulation in great detail were investigated. MoS 2 is flexible and isotropic for small deformations and the results obtained from this study are compared with the previous literature for defect-free MoS 2 and progress towards higher defect fractions. The random distribution of defects in the MoS 2 sheet in addition to antisite defects were also discussed in great detail.
We have seen that the elastic constants of MoS 2 started dwindling at a rapid rate with the defects pile-up. It started to dwindle at a slow rate of up to 1% of defects. Just as the defects increase to 2% and beyond, its reduction began dramatically. Hence, when the defect percentage reaches 10%, the reduction in elastic constants was as huge as 28.8%. These vacancy defects greatly influenced the elastic behavior of the MoS 2 lattice. With the increase in defects fraction, the vector sum of displacement affected the geometrical symmetry of the MoS 2 sheet. Moreover, in this study, we reviewed the possibility of physical properties improvement, and strengthening the elastic stiffness properties due to defects in MoS 2 was confirmed.
We also study the elastic constants for Mo as antisite defects, molybdenum diffusion in sulfur layers in the MoS 2 nanosheet for different defect concentrations ratios ranging from (0.1% to 25%) by using molecular statics simulation. We confirmed that Mo diffusion as the antisite defects indeed decreases the elastic constants in the MoS 2 nanosheet. Nevertheless, with the increase in defect concentration, Mo diffusion has also shown the decrease in tendency of elastic properties. Mainly, when the antisite defects concentration at 5% to 25%, we see the elastic stiffness decreases less than in comparison to pure defect structures.