_{2}Monolayer Heterostructures: Crack Propagation and Existing Notch Study

_{2}Monolayer Heterostructures: Crack Propagation and Existing Notch Study

_{2}Monolayer Heterostructures: Crack Propagation and Existing Notch Study

The outstanding thermal, optical, electrical and mechanical properties of molybdenum disolphide (MoS_{2}) heterostructures make them exceptional candidates for an extensive area of applications. Nevertheless, despite considerable technological and academic interest, there is presently a few information regarding the mechanical properties of these novel two-dimensional (2D) materials in the presence of the defects. In this manuscript, we performed extensive molecular dynamics simulations on pre-cracked and pre-notched all-molybdenum disolphide (MoS_{2}) heterostructure systems using ReaxFF force field. Therefore, we study the influence of several central-crack lengths and notch diameters on the mechanical response of 2H phase, 1T phase and composite 2H /1T MoS_{2} monolayers with different concentrations of 1T phase in 2H phase, under uniaxial tensile loading at room temperature. Our ReaxFF models reveal that larger cracks and notches decrease the strength of all 2D MoS_{2} single-layer heterostructures. Additionally, for all studied crack and notch sizes, 2H phase of MoS_{2} films exhibits the largest strength. Maximum tensile stress of composite 2H/1T MoS_{2} nanosheet with different concentrations are higher than those for the equivalent 1T phase, which implies that the pre-cracked composite structure is remarkably stronger than the equivalent 1T phase. The comparison of the results for cracked and notched all-MoS_{2} nanosheet heterostructures reveal that the load bearing capacity of the notched samples of monolayer MoS_{2} are higher than the cracked ones.

_{2})

Molybdenum disulphide (MoS_{2}) is well known for its various applications in industry and recently, its two-dimensional (2D) forms which are parts of the large family of so-called transition metal dichalcogenide (TMD), have attracted growing attention in high strength nanocomposites and in the nano-electronic technology. Like graphite, molybdenum disulfide crystals are composed of atomic layers with hexagonal lattices held together by van der waals forces. Even though MoS_{2} crystals exist in nature, its purification is difficult and expensive. On the other hand, natural gas and crude oil are sources of large amount of sulfur because they contain hydrogen sulphide (H_{2}S). Chemical companies produce pure MoS_{2} crystals _{2} including mechanical and chemical exfoliation of bulk crystals by peeling off the layers of MoS_{2} into 2D layers and vapour-phase growth of large-scale 2D monolayer MoS_{2} sheets [_{2}S and form large-areas of MoS_{2} monolayers [_{2} structures show extraordinary prospects for applications in flexible electrical and optical nanodevices for which mechanical stability is crucial [_{2} over graphene includes its direct-bandgap, _{2} is its polymorphism characteristic. The electrical characteristics of single-layer MoS_{2} significantly depends on the S atoms locations. Experimental approaches show that extra tuning of the electrical properties of MoS_{2} monolayers by the fabrication of mono-layer heterostructures is possible. A monolayer MoS_{2} sheet has triple atomic planes with different atomic stacking sequences, in which a close-packed of molybdenum (Mo) is encompassed by two atomic layers of close-packed sulfur (S), as shown in _{2} sheets, depending on (1) the coordination of sulphur atoms with respect to the central molybdenum atom and (2) the stacking order of each layer. The semi-conducting (2H) phase which is the original structure of this material, 2D atomic layers of MoS_{2} sheets indicate a hexagonal lattice and an (S_{top}-Mo-S_{Bot}) ABA atomic stacking sequence (as depicted in _{top}-Mo-S_{Bot}) ABC, where the S atoms on the bottom are located in the hollow center of the hexagonal lattice. Both 2H and 1T structures have a 30° angle of symmetry. The loading angles of 0 and 30° are generally known as armchair and zigzag directions, respectively as illustrated in _{2} heterostructure composed of semiconducting and metallic phases in a mono-layer configuration as shown in _{2} sheets maintain their structural integrity throughout service life, it is required to obtain a basic knowledge of the mechanical behavior of monolayer MoS_{2} nanosheets under different loading conditions. Several authors have studied 2H phase of MoS_{2} sheets both experimentally and theoretically [_{2} membranes will always contain different types of defects and impurities in their atomic lattices. For instance, crystal growth arising throughout the chemical vapor deposition (CVD) fabrication of monolayer MoS_{2} causes the formation of grain boundaries with different types of defects [_{2} nanosheets may also contain several atomic impurities like oxygen [

Cracks, holes and notches are among the most popular defects appearing in structures [_{2} films, ReaxFF reactive molecular dynamic simulation seems to be valuable choice. It worth mentioning that, in nonreactive force fields, molecules are formed through atomic static bonds that do not allow bond formations. This limitation prevents applying classical molecular dynamics (MD) methods to simulate chemical reactions. For the purpose of overcoming this limitation in MD simulation, the reactive force field (denoted ReaxFF) is employed [_{2}. However, the failure behavior of MoS_{2} nanosheet is more complicated than those of graphene and other graphene-like materials which have planar surfaces with single-layer atomic structures. Bao et al. [_{2} monolayer by using the Stillinger-Webber (SW) potential. In this work, the cracks were predefined by deleting some atoms in the nanosheet.

This research focuses on fracture of all phases of single-layer MoS_{2}. We therefore extended the studies in [_{2}) nanosheet with initial center nano-cracks and notches. Section 6 discusses the conclusions.

The main objective of this research is to investigate the fracture properties (maximum tensile stress and fracture strain) and also crack and notch propagation of all aforementioned phases of MoS_{2} single-layer structures with various pre-existing crack and notch shape defects. Therefore, we conduct ReaxFF based molecular dynamics (MD) simulations and study the effect of different nano-crack sizes (lengths) and nano-notch sizes (diameters) on the single-layer MoS_{2} mechanical and failure response, predicting the macroscopic maximum tensile stress and fracture strain under uniaxial tension. We employ the approach discussed in our previous study [

The focus of this study is to computationally predict the mechanical behavior of MoS_{2} films containing nano-defects _{2} structure, we created center crack and notch in the MoS_{2} nanosheet as depicted in

The results of MD modeling for pristine MoS_{2} nanosheets reveal that the elastic response of MoS_{2} phases depends on the loading direction. Along the armchair direction MoS_{2} exhibits a higher rigidity than in zigzag direction [_{2} films [_{2} presented in [_{system}) in ReaxFF is additively decomposed into several partial energy contributions given by [

where, E_{bond}, E_{val}, E_{tor}, E_{over}, E_{under}, E_{lp}, E_{vdW} and E_{Coulombic}, represent the bond energy, valence-angle, torsion-angle, over-coordinate, under-coordinate, lone pair, non-bonded van der Waals and coulomb contribution, respectively. The parameters of the ReaxFF potential are obtained by a quantum mechanical (QM) dataset which introduces bond dissociation and valence angle bending in different molecules and also the energy of formation and the state of condensed-phases equations of crystalline MoS_{2} nanomaterials [_{2} films, we first constructed models with different nanosheet sizes and obtained similar results which showed we have a concurrent multiscale method. Therefore, we select final models consisting of around 22,000 atoms with planar dimensions of about 25 nm × 25 nm and apply periodic boundary conditions in both planar directions. The time increment in all simulations is fixed at 0.25 fs. First, energy minimization was performed with a 10^{-6} stopping tolerance for energy. Then, the uniaxial loading condition is applied by increasing the periodic box size along the loading direction by a constant engineering strain rate of 10^{9} s^{-1}. Before subjecting the samples to uniaxial tension, the structures were relaxed and equilibrated to zero stress at room temperature taking advantage of the Nose-Hoover barostat and thermostat (NPT). This was done with damping parameters for 100 fs and 50 fs for the pressure and temperature, respectively. To apply the uniaxial load, the stress on the other two directions should remain small throughout the deformation procedure. As the atoms are in contact with vacuum along the nanosheet thickness orientation, the normal stress is insignificant. Moreover, the periodic simulation box along the width of the MoS_{2} nanosheet was kept at zero stress in the mentioned direction

The models were loaded in tension and the extracted uniaxial stress-strain results for defective MoS_{2} nanosheets are depicted on several graphs in order to calculate mechanical properties of MoS_{2} films. In the calculation of stresses, we choose the structure volume using a thickness of 6.1_{2} nanosheet.

The strain rate is an effective factor influencing the strength of materials. We know that because of the enormous amount of computational costs, MD simulations cannot capture strain rates as they often occur in engineering applications. However, some researches have been done to extract the mechanical behavior under quassi-static conditions [_{2} nanosheets. The tensile tests were performed for 2H phase and 1T phase of MoS_{2} material and also 2H/1T composite MoS_{2} structure with 5% concentration of 1T phase inside 2H phase at room temperature with strain rates ranging from 10^{8} to 10^{10} s^{-1}. Let us first consider a pristine 2H MoS_{2} nanosheet. The curves on ^{8} and 10^{9} s^{-1}, the results converge against a curve. For high strain rate of 10^{10} s^{-1}, first the results approximately converge against the other two curves until the strain of around 0.23 but it shows different pattern for high strains. Our results for 10^{8} s^{-1} strain rate predict an ultimate stress of 25.8 GPa and the elastic modulus of 265.6 GPa which are in fair agreement to the results in [_{2} nanosheet can be found in ^{8} and 10^{9} s^{-1}, the results approximately converge against a similar curve. This behavior is more pronounced for strains less than 0.15. The curve for high strain rate of 10^{10} s^{-1} is above the other two curves which shows higher results. The maximum tensile stress and corresponding failure strain for 10^{8} s^{-1} strain rate are 9.9 GPa and 0.18 respectively, which are very close to the results reported in [_{2} films in armchair loading direction are depicted in ^{8} s^{-1} strain rate are 19.36 GPa and 0.196, respectively which meet the results reported in [_{2} films. The failure properties of the aforementioned phases of MoS_{2} material which are obtained from _{2} nanosheets as a function of strain rate for different phases, as illustrated in _{2} nanosheet is more significant than 2H phase and hetero-structure.

Fracture is a phenomenon with size effects and the crack length influences the mechanical properties of the nanosheet. As mentioned before, we use ReaxFF reactive MD modeling and estimate the fracture properties of monolayer MoS_{2} nanosheets with initial center cracks under mode I loading condition in armchair direction. We created MD models with side length of 25.0 nm simulation box size consisting of around 22000 individual atoms. We consider several pre-crack lengths including L/6, L/3 and L/2 where L is the length of the square graphene-like MoS_{2} monolayer.

_{2} film with a center crack size of L/3 at different time steps and associated strain values. Additionally, strains at each stage are shown on each picture which is 0.104 at the end of the crack propagation process of the whole length. As can be seen, for 2H phase MoS_{2} nanosheet, first the crack opens by increasing uniaxial tensile strain. Then by increasing crack-driving force, the crack rapidly grows perpendicular to the loading direction throughout the nanosheet. Following the nanosheet failure, the bonds between atoms are ruined and larger deformation araise. Stress concentration areas can be distinguished near the tip of the crack. We compare the results of our MD modeling for the 2H-MoS_{2} nanosheets subjected to different crack lengths of L/6, L/3 and L/2 where L is 25 nm. The related stress-strain curves are depicted in _{2} nanosheet (0L). For L/6, L/3 and L/2 crack sizes, the max. tensile stresses are 13.42 GPa, 10.11 GPa and 7.79 GPa respectively and comparing them with crack-free MoS_{2} nanosheets, which exhibits a max. tensile stress of 25.8 GPa, shows the crack significantly decreases the max. tensile stress of 2D MoS_{2} material, _{2} nanosheet for the aforementioned crack sizes. Obviously, increasing the crack length, has a weakening effect on tensile strength of 2H MoS_{2} nanosheet. Additionally, strain-at-fracture decreases for larger crack sizes. According to _{2} models with crack sizes less than L/6, while larger crack sizes showed approximately a linear trend. Also, according to _{2} nanosheet.

In this section, we derive our effort to compare the results of MD modeling for 1T-MoS_{2} structure subjected to different crack lengths including L/6, L/3 and L/2 where L is 25 nm. _{2} monolayer. Additionally, max. tensile stress and strain-at-fracture of 1T phase structure for the aforementioned crack lengths are depicted on _{2} material decreases under the effect of crack-shape defects. Obviously, crack length increase, has a weakening effect on the max. tensile stress of 1T-MoS_{2} structure, as the max. stress values decrease significantly by increasing the crack size. Max. tensile stresses for L/6, L/3 and L/2 crack sizes are 9.49 GPa, 7.42 GPa and 6.13 GPa respectively which are around 17%, 35% and 46% below the pristine 1T-MoS2 sample, respectively. Obviously, strain-at-fracture decreases for larger crack sizes of 1T-MoS_{2} nanosheet. As an example, based on

In this section, we use numerical simulations to evaluate the mechanical properties and to predict fracture geometry of 2H/1T singlelayer MoS_{2} heterostructures and investigate fracture initiation and crack propagation path in different samples of the nanosheet. It is likely the fracture properties (max. tensile stress) of 2H/1T heterostructures to be higher than that of the defective 1T phase and lower than defective 2H phase. This prediction is according to the fact that the results for the max. tensile stress of the defective 1T phase are almost half of that of the 2H phase. _{2} monolayer with 5%, 10% and 15% concentrations respectively. Our results for different 1T concentrations of 2H/1T MoS_{2} heterostructure and three crack sizes are summarized in _{2} heterostructure for the previously mentioned crack lengths are compared on _{2} nanosheet (0L) is also presented. As expected, the max. tensile stress of two dimensional composite 2H/1T MoS_{2} heterostructure decreases under the effect of crack-shape defects. Obviously, crack length increase, has a weakening effect on max. tensile stress of composite 2H/1T MoS_{2} heterostructure, as the max. stress values drop significantly by increasing the crack size. With reference to _{2} heterostructure with crack sizes less than L/6, while larger crack sizes showed approximately a linear trend. Additionally, according to

Finally we investigate the effect of notch shaped defects on the mechanical behavior of the MoS_{2} nanosheet. The impact of the notch on mechanical properties highly depends on the location of the notch in the samples. Furthermore, fracture is a phenomenon with size effects and the notch diameter influences mechanical properties of the 2D material. Therefore, we investigate notches located in the center of the nanosheet where their diameter range are L/6, L/3 and L/2. These sizes are selected similar to the length of previously studied cracks, enabling a comparison between these two kinds of defects. The existence of notch leads to a decrease in the max. tensile stress and strain-at-fracture compared to the samples without pre-existing notch. The obtained stress-strain curves for the samples with pre-existing notches in the 2H phase MoS_{2} nanosheet have been shown in _{2} models with notch sizes less than L/6, while larger notch sizes showed approximately a linear trend. As it can be seen in this figure, the max. tensile stress drops by 42%, 47% and 61% for samples with L/6, L/3 and L/2 of notch sizes, respectively. _{2} nanosheet. According to

In the next step we explore 2H/1T MoS_{2} heterostructures. _{2} heterostructure for the previously mentioned notch diameters are compared on _{2} nanosheet (0L) is also presented. As expected, the max. tensile stress of the 2D composite 2H/1T MoS_{2} heterostructure decreases under the effect of notch defects. Obviously, notch diameter increase, has a weakening effect on max. stress of composite 2H/1T MoS_{2} heterostructure, as the max. stress values drop significantly by increasing the notch size. Additionally, among the three mentioned studied composite samples, for 10% concentration model a drop in the max. tensile stress and fracture strain is more pronounced compared to the defect-free specimen.

In order to compare fracture properties of all monolayer MoS_{2} nanosheet heterostructures in the presence of defects, we show all the obtained ReaxFF MD results for pre-cracks and pre-notches in _{2} films has larger max. stress. Also, for this phase a drop in max. stress and fracture strain is more pronounced compared to the defect-free specimen. The lowest max. stress belongs to 1T phase. The results for 2H/1T MoS_{2} heterostructure with different concentrations are below the equivalent 2H phase but higher than 1T phase. However, max. tensile stress of cracked and notched nanosheets of 1T phase is well below both single-layer 2H and all studied 2H/1T composite structures of MoS_{2}. According to _{2} samples of 2H phase, hetero (5%), hetero (10%), hetero (15%) and 1T phase, shifting the samples from crack-free to having L/6 initial center crack, leads to a large drop in the max. tensile stress by 48%, 39%, 35%, 24% and 17%, respectively. For the aforementioned phases, when the initial center crack length increases from L/6 to L/3 a decrease in max. tensile stress of 25%, 29%, 29%, 15% and 22% is observed, respectively. Also, according to _{2} samples of 2H phase, hetero (5%), hetero (10%), hetero (15%) and 1T phase at room temperature, shifting the samples from pristine to having L/6 pre-existing center notch, lead to a large drop in the max. tensile stress by 43%, 17%, 34%, 21% and 18%, respectively. A drop in max. tensile stress of 9%, 40%, 24%, 8% and 24% is observed when the pre-existing center notch diameter increases from L/6 to L/3, respectively.

_{2} material. The comparison of the results for cracked and notched 2H-MoS_{2} nanosheet shows that the max. stress and fracture strain of the notched samples are higher than the samples with crack. Therefore, the load bearing capacity of the notched samples of 2H-MoS_{2} nanosheets are higher than the cracked ones. Additionally, for 1T-MoS_{2} models, the comparison of results for cracked and notched samples with identical crack length and notch diameter indicates that the max. stress and strain-at-fracture of the notch and crack samples are close and therefore load bearing capacity of both cases are almost identical (see

In the present contribution we used tensile loading simulations by performing MD calculations to predict the mechanical response of all phases of MoS_{2} single-layer heterostructures and studied the response of defective MoS_{2} mono-layers, where defects were assumed to be center-cracks and notches. To this goal realistic atomistic models were simulated with specified concentration and domain size for the metallic phase inside the semiconducting. The molecular dynamic simulations were performed for different crack lengths and notch diameters including, L/6, L/3 and L/2 all located at the center of the nonosheet. Our predictions based on a parameterized ReaxFF potential for the mechanical properties of the defective MoS_{2} monolayes, showed that MoS_{2} films subjected to crack and notch shape defects have significantly lower ultimate tensile strength compared to their pristine material. Our molecular dynamic (MD) modeling results confirm that the fracture is a phenomenon with size effect and the crack size and notch diameter non-linearly influence fracture properties of all-MoS2 heterostructures. For all studied crack and notch sizes, an increasing crack length and notch diameter, decreases the ultimate tensile strength of the monolayer MoS_{2} material as well as the Young’s modulus. Furthermore, according to our classical molecular dynamics simulations, semi-conducting (2H) phase of MoS_{2} films has the largest strength. Fracture properties of all studied concentrations of composite 2H/1T MoS_{2} nanosheet are higher than those for the equivalent metallic (1T) phase. We can imply that the pre-cracked and pre-notched composite MoS_{2} structure is remarkably stronger than equivalent metallic (1T) phase. Also, they are remarkably strong and flexible materials, even in the presence of the defects. This study provides valuable result for employing all strong, flexible MoS_{2} films for several industrial applications

The first author would like to give a special thanks to Dr. Bohayra Mortazavi for providing a wealth of helpful advice and guidance related to the area of molecular dynamics simulation.

_{2}transistors

_{2}single-layer heterostructures: A reax investigation

_{2}

_{2}

_{2}nanosheets

_{2}nanostructures from a theoretical perspective

_{2}

_{2}under tension via atomistic simulations

_{2}and 2H-NbSe

_{2}extracted from measured dispersion curves and linear compressibilities

_{2}- an atomistic simulation

_{2}monolayer

_{2}nanosheets

_{2})

_{3}; a molecular dynamics study

_{2}